diff --git "a/Low Fermi Level/paper/all_papers.json" "b/Low Fermi Level/paper/all_papers.json"
new file mode 100644--- /dev/null
+++ "b/Low Fermi Level/paper/all_papers.json"
@@ -0,0 +1 @@
+[    {        "title": "0901.1299v1.Fermi_surface_topology_and_low_lying_quasiparticle_structure_of_magnetically_ordered_Fe1_xTe.pdf",        "content": "arXiv:0901.1299v1  [cond-mat.supr-con]  9 Jan 2009Fermi surface topology and low-lying quasiparticle struct ure of magnetically ordered\nFe1+xTe\nY. Xia,1D. Qian,1,2L. Wray,1,3D. Hsieh,1G.F. Chen,4J.L. Luo,4N.L. Wang,4and M.Z. Hasan1,5,6\n1Joseph Henry Laboratories of Physics, Department of Physic s, Princeton University, Princeton, NJ 08544\n2Department of Physics, Shanghai Jiao Tong University, Shan ghai 200030, China\n3Lawrence Berkeley National Laboratory, University of Cali fornia, Berkeley, CA 94305\n4Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing , China\n5Princeton Center for Complex Materials, Princeton Univers ity, Princeton, NJ 08544\n6Princeton Institute for the Science and Technology of Mater ials, Princeton University, Princeton, NJ 08544∗\n(Dated: November 21, 2018)\nWe report the first photoemission study of Fe 1+xTe - the host compound of the newly discovered\niron-chalcogenidesuperconductors(maximumT c∼27K).Ourresultsrevealapairofnearlyelectron-\nhole compensated Fermi pockets, strong Fermi velocity reno rmalization and an absence of a spin-\ndensity-wave gap. A shadow hole pocket is observed at the ”X” -point of the Brillouin zone which is\nconsistent with a long-range ordered magneto-structural g roundstate. No signature of Fermi surface\nnesting instability associated with Q=( π/2,π/2) is observed. Our results collectively reveal that\nthe Fe 1+xTe series is dramatically different from the undoped phases o f the high T cpnictides and\nlikely harbor unusual mechanism for superconductivity and magnetic order.\nPACS numbers:\nThe discovery of superconductivity in the pnictides\n(FeAs-based compounds) has generated interest in un-\nderstandingthegeneralinterplayofquantummagnetism,\nelectronic structure and superconductivity in iron-based\nlayered compounds [1, 2]. The recent observation of un-\nusual superconductivity and magnetic order in the struc-\nturally simpler compounds such as FeSe and Fe 1+xTe are\nthe highlights of current research [3, 4, 5, 6, 7]. The ex-\npectation is that these compounds may provide a way\nto isolate the key ingredients for superconductivity and\nthe nature of the parent magnetically order state which\nmaypotentiallydifferentiatebetweenvastlydifferentthe-\noretical models [8, 9, 10]. The crystal structure of these\nsuperconductors comprises of a direct stacking of tetra-\nhedral FeTe layers along the c-axis and the layers are\nbonded by weak van der Waals coupling. Superconduc-\ntivity with transition temperature up to 15K is achieved\nin the Fe 1+x(Se,Te) series [3, 4] and T cincreases up to\n27K under a modest application of pressure [5]. Den-\nsity functional theories (DFT) predict that the electronic\nstructure is very similar to the iron-pnictides and mag-\nnetic order in FeTe originates from very strong Fermi\nsurface (FS) nesting leading to the largest SDW gap in\nthe series. Consequently, the doped FeTe compounds\nare expected to exhibit T cmuch higher than that ob-\nserved in the iron-pnictides if superconductivity would\nindeed be originating from the so called ( π,0) spin fluc-\ntuations [10]. These predictions critically base their ori-\ngin on the Fermi surfacetopologyand the band-structure\ndetails, however, no experimental results on the Fermi-\nology and band-structure exist on this sample class to\n∗Electronic address: mzhasan@Princeton.edu0 100 200 3000.0030.0060.009\nH//c\nH//a-bχ (emu/mol)\nT (K)SrFe As2 2 FeTe\n(a) (b) (c)Fe Te1+x\nab\nc\nFIG. 1: Magneto-structural transition and long-range orde r\nin Fe1+xTe: (a)Temperature dependence of the magnetic sus-\nceptibility [6]. (b) The spins in SrFe 2As2are collinear and\nQAFpoints along the ( π,0) direction. (c) The ordering vec-\ntor,QAFin FeTe, is rotated by 45◦and points along the\n(π\n2,π\n2) direction [15].\nthis date. Here we report the first angle-resolved photoe-\nmission (ARPES) study of the Fe 1+xTe - the host com-\npound of the superconductor series. Our results reveal a\npair of nearly electron-hole compensated Fermi pockets,\nstrong band renormalization and a remarkable absence\nof the predicted large SDW gap. Although the observed\nFermi surface topology is broadly consistent with the\nDFT calculations, no Fermi surface nesting instability\nassociated with the magnetic order vector was observed.\nAn additional hole pocket is observed in our data which\ncan be interpreted to be associated with a local-moment\nmagneto-structuralgroundstateinclearcontrasttointer-\nband nesting. Our measurements reported here collec-\ntively suggest that the FeTe compound series is dramati-\ncally different from the parent compound of the pnictide\nsuperconductorsand mayharbornovelformsofmagnetic\nand superconducting instabilities not present in the high\nTcpnictides.2\n0□meV -20□meV -40□meV -60□meV\n(a) (b) (c) (d)\nΓM\nXLDA Schematic\n(f)Γ\nMX( ,□0)π\n(□□□, )π\n2π\n2Γ M\n(e)ΓM\nEF\n(g)Band\nSchematic\nFIG. 2: Fermi surface topology : (a) Fermi surface topol-\nogy of FeTe. (a-d) Connection to the underlying band-\nstructure is revealed by considering the evolution of the de n-\nsity of states, n(k), as the chemical potential is rigidly lo w-\nered by 20, 40 and 60 meV. The Fermi surface consists of hole\npockets centered at Γ, and electron pockets centered at M. An\nadditional hole-like feature is observed at M in (b-d), whic h\nis attributed to a band lying below E F(see schematic in (e)).\n(e) presents a schematic of the band dispersions in the Γ-M\ndirection forming the observed electron and hole pockets. A\nqualitative agreement is observed between the (f) the exper -\nimentally measured FS topology and (g) the LDA calculated\n[10] FS although measured bands are strongly renormalized.\nSingle crystals of Fe 1+xTe were grown from a mixture\nofgroundedFe and Te powderusing the Bridgemantech-\nnique. The powder was heated to 920◦C in an evacuated\ntube thenslowlycooled, formingsinglecrystals. Theiron\nconcentrationwasmeasured by ICP (inductively-coupled\nplasma) technique and a value of x was determined to be\nless than 0.05. High-resolution ARPES measurements\nwere then performed using linearly-polarized 40eV pho-\ntons on beamline 10.0.1 of the Advanced Light Source\nat the Lawrence Berkeley National Laboratory. The en-\nergy and momentum resolution was 15meV and 2% of\nthe surface Brillouin Zone (BZ) using a Scienta analyzer.\nThe in-plane crystal orientation was determined by Laue\nx-raydiffractionpriortoinsertingintotheultra-highvac-\nuum measurement chamber. The magnetic order below\n65K was confirmed by DC susceptibility measurements\n(Fig-1). The samples were cleaved in situat 10K under\npressures of less than 5 ×10−11torr, resulting in shiny\nflat surfaces. Cleavage properties were characterized by\nSTM topography and by examining the optical reflection\nproperties.\nFigure 2 presents the momentum dependence of the\nphotoemission intensity n(k) from the sample at 10K at\nseveral different binding energies (0, 20, 40 and 60 meV)\nintegratedovera finite energywindow ( ±5 meV) at each\nbinding energy. The non-zero spectral intensity at the\nFermi level ( µ) confirmed that the low temperature state\nof Fe1+xTe is metallic-like. At the Fermi level, electrons\nare mostly distributed in one broad hole-like pocket at Γ\nand another similar size electron-like pocket around M.\nTo reveal the band shapes we present a gradual binding-energy evolution of the band features and present the\ndata in a way to simulate the effect of rigidly lowering\nof the chemical potential down to 60meV. The FS (at\n0 meV) is seen to consist of circular hole pockets cen-\nteredatΓ, andellipticalelectronpocketsattheMpoints.\nAn elliptical-like feature at the M point expands as the\nbinding energy is increased, suggesting that the associ-\nated band is hole-like. We will subsequently show that\nthis band lies below µand the M-point Fermi pockets\nare only electron-like ( 2(e)). By considering the domi-\nnant intensity patterns in panel-(a-d), apart from a weak\nfeature at ”X”-point, the FS topology is similar to that\nexpected from the DFT calculations ( 2(g)) [10]. In ad-\ndition, the area of the hole-like FS pocket at Γ and the\nelectron-like FS pocket at M are approximately equal in\nsize suggesting nearly equal number of electron and hole\ncarrier densities in this material. Therefore, if the ex-\ncess Fe atoms are contributing to the carrier density it\nis likely to be small and beyond our k-resolution of the\nexperiment.\nIn order to systematically study the low lying energy\nband structure, ARPES spectra are taken along different\nk-spacecut directionsin the 2DBrillouinzone(BZ). Two\ndifferent electron-photon scattering geometries are used\nto ensure that all bands are imaged. In a σscattering\ngeometry where the polarization vector is parallel to the\nkyaxis, photoelectron signal is predominantly from the\ndxy,dxz, anddz2energybands due to the dipole emission\nmatrix elements [11, 12]. Similarly, when the polariza-\ntion vector is parallel to kx, theπ-geometry, the dyzand\ndx2−y2states are predominantly excited. Figure 3 (a)\nand (b) present scans along the Γ −Mdirection in the\nσandπgeometries. In both sets of spectra, one finds\na broad hole-like band centered at Γ. However, near M\nthe scanstakenat twodifferent geometriesaredrastically\ndifferent. In cut 1, two hole-like bands ( α2,α3as marked\nin panel (a)) are observed to be approaching µ. Under\ntheπgeometry the band emission pattern near M is dra-\nmatically different (see the β1band in cut 2), while the\nα2andα3band signals are significantly weaker. The po-\nlarization dependence suggests that the β1band should\nhavedyzordx2−y2orbital character.\nIn order to fully resolve the broad band feature cen-\ntered at Γ, high resolution scans are performed along\ndifferent k-cut directions through the zone center. Fig-\nure 3(c) presents one cut in the Γ −Mdirection inside\nthe first zone, together with the corresponding energy\ndistribution curve (EDC). Two hole-like bands are re-\nsolved, labeled α2andα3. Since there are traces of mul-\ntiple bands near M, one might wonder whether the α3\nband crosses µnear the Mpoint, thus forming a hole-like\nFermi pocket. To systematically investigate this, a series\nof spectra are taken along the Γ −Mdirection with π-\ngeometry ( 3(d)-(f)) through multiple k-cuts intersecting\ntheM-point Fermi pocket. At M, cut-4 shows a strong\nelectron-like band forming the FS pocket. This band can\nbe attributed to the β1band (also observed in cut-2). As\none moves away from M, the band intensity becomes in-3\nFIG. 3: Low-lying band topology : ARPES spectra along the Γ −Mdirection in the (a) σand (b)π-scattering geometries.\nvFof the quasiparticle band forming the Γ hole pocket is calcul ated from the dashed line. (c) The EDC of a high momentum\nresolution scan through Γ shows two hole-like bands crossin g the E F. (d)-(f) Cuts along the M-Γ direction through the electron\nFS pocket show that the hole band near M remains at least 10meV below E F. For each scan direction the corresponding\nEDC is also presented, within the momentum range marked by gr een dashed lines. Comparison with (g) a band dispersion\nschematic of the LDA result [10] shows a good agreement betwe en experiment and theory. (h) The directions of the six scans\nare summarized and identified in the first BZ. The direction of the electric field is shown.\ncreasinglymore hole-like, indicating the emergenceofthe\nα2band. Nevertheless, the hole band lies completely and\nconsistently below the Fermi level, with some weak elec-\ntron quasiparticle intensity above the band maximum.\nTheβ1intensity becomes the weakest near the edge of\nthe M pocket (cut-6). But even at that location, the hole\npocket lies at least 10meV below the chemical potential.\nThe result shows that there are no hole pocket features\nin the FS near Mwhich supports the interpretation that\nthis material is nearly electron-hole compensated. Ad-\nditionally, one can map the observed bands to the band\nstructure estimated by DFT calculations (schematic in\n3(g) [10]). The calculation finds three bands ( α1-α3)\ncrossing Fermi level near Γ and two near ( β1,β2) M,\nforming the electron and hole pockets. While the DFT\ncalculated bands agree fairly well with our data, within\nour resolution or the sample surface roughness, we have\nnot succeeded in fully resolving the α1band near Γ. The\nβ2band, which forms the second M electron pocket, is\nobserved in the Fermi surface topology (Fig. 2). The\noverall band narrowing is about factor of 2 compared to\nthe DFT calculations highlighting the importance of cor-\nrelationeffects. This isalsoconsistentwith asmallFermi\nvelocity (v Fermi∼0.7 eV·˚A) observed (Fig.3(a)).\nWe now revisit the details of the FS map (Fig.-2) and\ndiscuss a weak feature observed at X=(π\n2,π\n2) between\ntwo Γ points (Fig. 4(e)). Such a feature is not expected\nfrom the DFT calculations. A similar feature, whose ori-\ngin is debated, has been observed in AFe 2As2(A=Ba,\nSr, Ca), which is attributed to a 2 ×1 surface reconstruc-\ntion of the Aatom layer [13, 14], in addition to a weak\nbulk structural distortion. However, in FeTe there are\nno additional atoms (such as Sr or Ba) between the Felayers and the crystal cleaves at a weak van der Waals\nbond bewteen two adjacent layers no strong 2 ×1 long-\nrange ordered A-type reconstruction is expected except\nfor a weak bulk-like structural orthorhombicity tied to\nthe magnetic order (magneto-structural effect). Recent\nneutron[15, 16] andx-raydiffraction[17, 18]studieshave\nshown that FeTe undergoes a bulk structural distortion\nfrom the tetragonal to weakly-monoclinic or orthorhom-\nbic phasenear65K,accompaniedbylong-rangemagnetic\norder Q AF=(π\n2,π\n2).\nIn the parent compound of the pnictide superconduc-\ntors such as the SrFe 2As2or BaFe 2As2, the SDW vector\nQSDW=(π, 0) coincides with a Fermi surface nesting of\nvector connecting the hole-pocket at Γ and the electron\npocket at M. Currently, it is believed that this nesting\nis responsible for opening a gap in the low temperature\nphysical properties [6, 11, 19]. In the case of FeTe, while\na nesting vector can indeed be drawn along QSDW=(π,\n0) between a pair of electron and hole pockets, all avail-\nableneutronscatteringmeasurementsreportthat thean-\ntiferromagnetic ordering vector is 45◦away from that in\nSrFe2As2, namely, in Fe 1+xTe,QAF=(π\n2,π\n2). The order-\ning shows a commensurate to incommensurate cross-over\nif the concentration of excess iron, x, is increased. An-\nother remarkable difference is that the magnetic suscep-\ntibility is Curie-Weiss like in FeTe suggesting that the\nmagnetism is of local-moment origin. Within a local\nmoment-like AFM long-range ordered state which also\ncouples to a weak structural distortion one should ex-\npect relatively intense shadow Fermi surfaces along the\nNeel vector QAF. The X-point Fermi surface we ob-\nserve thus can be related to the vector observed in neu-\ntron scattering X= Γ+QAF. The weak shadow-like X4\n-0.05 0.00\n0.0 0.5 1.0 1.5-0.3-0.2-0.10.0\nE (eV)BΓ1 X\nk(Å )-1SrFe As2 2 FeTe\nΓ1\nXMSrFe As2 2 FeTeFe Sr(a) (b)\n(c)\n2 1□Sr□Order/c180 No□Sr□□OrderM\nXΓ\nQSDW\nMXΓ\nQSDW?\n(e)QAFQAFE (eV)BGold\nM\nΓ\n(d)\n(f)Norm.□Intensity\nFIG. 4:Electronic structure and Magnetism : While (a)\nthe electron and hole pockets in SrFe 2As2can be kinemat-\nically nested by QAF=(π,0), there exists (b) no FS pocket\nwhich can be nested by QAF= (π\n2,π\n2) in the FeTe BZ. A\nFS pocket observed at X in SrFe 2As2has been attributed to\nthe (c) 2 ×1 surface order of the Sr atomic layer (red), which\noccurs in addition to a weak bulk distortion (blue) across th e\nSDW transition. However, there are no additional Sr atoms\nbetween the Fe layers in FeTe, so no Sr order is possible, al-\nthough a weak bulk distortion is not excluded. (d) The EDCs\nmeasured at near the Γ and M pockets exhibit no evidence of\nenergygaps. Nevertheless, (e)aweakFSpocketisobserveda t\nX,(f) correspondingtotwohole-like bandsdispersingtowa rds\nthe chemical potential which is consistent with a long-rang e\nlocal moment magneto-structural order.\npocket FS might therefore arise from a band folding due\nto long-range magnetic order. However, unlike SrFe 2Se2,\nwhere the Γ electron pocket nests with the M hole pock-\nets viaQSDW= (π,0) [20], an analogousnesting channel\nis unavailable at (π\n2,π\n2) in the FeTe ( 4(b)) clearly ruling\nout FS nesting as the origin of magnetic order. In the\nabsence of nesting FS gapping is not expected in FeTe\nwhich is consistent with our results in Fig-2 and 3. A\nlarge low temperature specific heat value [6] is thus con-sistent with our observation of a Fermi surface in the\ncorrelated magneto-structurally ordered state. To exam-\nine the band dispersion character of the X pocket, figure\n4 (f) presents a spectra along the Γ −Xdirection. Re-\nsults show two bands dispersing towards µ, forming the\nhole pocket in the ”X”-FS whose shapes are indeed very\nsimilar to the bands that form the central Γ-pocket FS.\nFurther evidence against nesting comes from the absence\nof a large gap at the overall low temperature electron\ndistributions presented in Fig-2 and 3 ( 4(d)). Our re-\nsults seem to suggest that the groundstate is a nearly\nelectron-hole compensated semimetal. Absence of a gap\nis consistent with recent bulk optical conductivity, spe-\ncific heat and Hall measurements in FeTe [6] (while most\nmeasurements do report a gap in SrFe 2As2[19]).\nA recent density functional calculation [21] suggests\nthat the excess Fe is in a valence state near Fe+and\ntherefore donates electron to the system. Due to the in-\nteraction of the magnetic moment of excess Fe with the\nitinerant electrons of FeTe layer a complex magnetic or-\ndering pattern is realized. In this scenario, excess Fe\nwould lead to an enlargement of the electron pocket FS,\nhowever, within our resolution electron and hole Fermi\nsurface pockets are measured to be of very similar in size\nsuggestingalackofsubstantialelectrondopingduetoex-\ncess Fe. Finally, we note that a gaplessyet long-rangeor-\nderedlocal-momentmagneto-structuralgroundstatecon-\nsistent with ARPES and neutron data taken together is\ncaptured in both first-principles electronic structure [8]\nandmany-bodyspinmodel[9]calculations. However,the\nbroad agreement of DFT calculations with experimental\nband-structure data except for about a factor 2 renor-\nmalization is also remarkable. A complete understand-\ning of electron correlation, moment localization and the\ntrue nature of the gapless antiferromagnetic state would\nrequire further systematic ARPES study [22].\nIn conclusion, we have presented the first ARPES\nstudy of Fermi surface topology and band structure of\nFe1+xTe. Our results reveal a pair of nearlyelectron-\nhole compensated Fermi pockets, strong renormalization\nand the remarkable absence of a spin-density-wave gap.\nThe observed shadow hole pocket is consistent with a\nlong-rangeorderedlocal moment-likemagneto-structural\ngroundstate whereas, most remarkably, no Fermi surface\nnesting instability associated with the antiferromagnetic\norderwasobserved. Contraryto band theorysuggestions\n[23], results collectively suggest that the Fe 1+xTe series\nis dramatically different from the undoped phases of the\nhigh T cpnictides and likely harbor a novel mechanism\nfor superconductivity and quantum magnetism.\n[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,\nJ. Am. Chem. Soc. 130, 3296 (2008); Z.A. Ren et al.,\nChin. Phys. Lett. 25, 2215 (2008).\n[2] G.F. Chen et al., Phys. Rev. Lett. 100, 247002 (2008);X. H. Chen et al., Nature 453, 761 (2008).\n[3] F.C. Hsu et al., Proc. Natl. Acad. Sci. 105, 14262 (2008).\n[4] M.H. Fang et al., Phys. Rev. B 78, 224503 (2008).\n[5] Y. Mizuguchi et al., Appl. Phys. Lett. 93, 152505 (2008).5\n[6] G.F. Chen et al., arXiv:0811.1489 (2008).\n[7] C. Xu and S. Sachdev, Nature Phys. 4, 898 (2008).\n[8] F.J. Ma et al., arXiv:0809.4732 (2008).\n[9] C. Fang et al., arXiv:0811.1294 (2008).\n[10] A. Subedi et al., Phys. Rev. B 78, 134514 (2008).\n[11] D. Hsieh et al., arXiv:0812.2289 (2008).\n[12] D. Qian et al., Phys. Rev. Lett. 97, 186405 (2006).\n[13] M.C. Boyer et al., arXiv:0806.4400 (2008).\n[14] Y. Yin et al., arXiv:0810.1048 (2008).\n[15] S. Li et al., arXiv:0811.0195 (2008).[16] W. Bao et al., arXiv:0809.2058 (2008).\n[17] K. Yeh et al., arXiv:0808.0474 (2008).\n[18] Y. Mizuguchi et al., arXiv:0810.5191 (2008).\n[19] W.Z. Hu et al., Phys. Rev. Lett. 101, 257005 (2008).\n[20] J. Zhao et al., Phys. Rev. B 78, 140504 (2008).\n[21] L. Zhang et al., arXiv:0810.3274 (2008).\n[22] Y. Xia, M.Z. Hasan et al.,.\n[23] M. Johannes, Physics 1, 28 (2008)"    },    {        "title": "1903.04107v1.Spontaneous_charge_current_in_a_doped_Weyl_semimetal.pdf",        "content": "arXiv:1903.04107v1  [cond-mat.mes-hall]  11 Mar 2019Journal of the Physical Society of Japan\nSpontaneous Charge Current in a Doped Weyl Semimetal\nYositake Takane\nDepartment of Quantum Matter, Graduate School of Advanced S ciences of Matter,\nHiroshima University, Higashihiroshima, Hiroshima 739-8 530, Japan\n(Received )\nA Weyl semimetal hosts low-energy chiral surface states, wh ich appear to connect a pair of Weyl nodes\nin reciprocal space. As these chiral surface states propaga te in a given direction, a spontaneous circulating\ncurrent is expected to appear near the surface of a singly con nected Weyl semimetal. This possibility is\nexamined by using a simple model with particle-hole symmetr y. It is shown that no spontaneous charge\ncurrent appears when the Fermi level is located at the band ce nter. However, once the Fermi level deviates\nfrom the band center, a spontaneous charge current appears t o circulate around the surface of the system\nand its direction of flow is opposite for the cases of electron doping and hole doping. These features are\nqualitatively unchanged even in the absence of particle-ho le symmetry. The circulating charge current is\nshown to be robust against weak disorder.\n1. Introduction\nA Weyl semimetal possesses a pair of, or pairs of, non-\ndegenerate Dirac cones with opposite chirality.1–10)The\npair of Dirac cones can be nondegenerate if time-reversal\nsymmetry or inversion symmetry is broken. The elec-\ntronic property of a Weyl semimetal is significantly influ-\nenced by the position of a pair of Weyl nodes in recipro-\ncal and energy spaces, where a Weyl node represents the\nband-touchingpointofeachDiraccone.Inthe absenceof\ntime-reversalsymmetry,apairofWeylnodesisseparated\nin reciprocal space. In this case, low-energy states with\nchirality appear on the surface of a Weyl semimetal3)if\nthe Weyl nodes are projected onto two different points in\nthe corresponding surface Brillouin zone. A notable fea-\nture of such chiral surface states is that they propagate\nonly in a given direction, which depends on the position\nof the Weyl nodes. This gives rise to an anomalous Hall\neffect.5)If inversion symmetry is also broken in addition\nto time-reversal symmetry, a pair of Weyl nodes is also\nseparated in energy space. In this case, the propagating\ndirection of chiral surface states is tilted accordingto the\ndeviation of the Weyl nodes. If only inversion symmetry\nis broken,no chiralsurface state appearssince the pairof\nWeyl nodes coincides in reciprocal space. To date, some\nmaterials have been experimentally identified as Weyl\nsemimetals.11–18)\nLet us focus on the case in which a Weyl semimetal\nwith a pair of Weyl nodes at k±= (0,0,±k0) is in the\nshape of a long prism parallel to the zaxis. In this case,\nchiral surface states appear on the side of the system. In\nthe presence of inversion symmetry, they typically prop-\nagate in a direction perpendicular to the zaxis; thus,\nwe expect that a spontaneous charge current appears to\ncirculate in the system near the side surface. If inver-\nsion symmetry is additionally broken, the propagating\ndirection is tilted to the zdirection; thus, an electron in\nthe chiral surface state shows spiral motion around the\nsystem.19)Thus, we expect that a spontaneous charge\ncurrent has a nonzero component in the zdirection. This\nlongitudinalcomponentmust becanceledoutbythe con-tribution from bulk states if they are integrated over a\ncross section parallel to the xyplane.\nIn this paper, we theoretically examine whether a\nspontaneous charge current appears in the ground state\nof a Weyl semimetal. Our attention is focused on the\ncase where time-reversal symmetry is broken. We calcu-\nlatethespontaneouschargecurrentinducedneartheside\nof the system by using a simple model with particle-hole\nsymmetry. We find that no spontaneous charge current\nappears when the Fermi level, EF, is located at the band\ncenter, which is set equal to 0 hereafter, implying that\nthe contribution from chiral surface states is completely\ncanceled out by that from bulk states. However, once EF\ndeviates from the band center, the spontaneous charge\ncurrent appears to circulate around the side surface of\nthe system and its direction of flow is opposite for the\ncases of electron doping (i.e., EF>0) and hole doping\n(i.e.,EF<0). The circulating chargecurrent is shown to\nbe robust against weak disorder. In the absence of inver-\nsion symmetry, we show that chiral surface states induce\nthe longitudinal component of a spontaneous charge cur-\nrent near the side surface, which is compensated by the\ncontribution from bulk states appearingbeneath the side\nsurface.Thislongitudinalcomponentisshowntobe frag-\nile against disorder.\nIn the next section, we present a tight-binding model\nfor Weyl semimetals and show the absence of a sponta-\nneous charge current when the Fermi level is located at\nthe band center. In Sect. 3, we derive a tractable con-\ntinuum model from the tight-binding model and analyt-\nically determine the magnitude of a spontaneous charge\ncurrent induced by the deviation of the Fermi level from\nthe band center. In Sect. 4, we numerically study the\nbehaviors of a spontaneous charge current by using the\ntight-binding model. We also examine the effect of disor-\nder on the spontaneous charge current. The last section\nis devoted to a summary and discussion. We set /planckover2pi1= 1\nthroughout this paper.\n1J. Phys. Soc. Jpn.\n2. Model\nLet us introduce a tight-binding model for Weyl\nsemimetals on a cubic lattice with lattice constant a. Its\nHamiltonian is given by H=H0+Hx+Hy+Hzwith4,5)\nH0=/summationdisplay\nl,m,n|l,m,n/an}bracketri}hth0/an}bracketle{tl,m,n|, (1)\nHx=/summationdisplay\nl,m,n{|l+1,m,n/an}bracketri}hthx/an}bracketle{tl,m,n|+h.c.},(2)\nHy=/summationdisplay\nl,m,n{|l,m+1,n/an}bracketri}hthy/an}bracketle{tl,m,n|+h.c.},(3)\nHz=/summationdisplay\nl,m,n{|l,m,n+1/an}bracketri}hthz/an}bracketle{tl,m,n|+h.c.},(4)\nwhere the indices l,m, andnare respectively used to\nspecify lattice sites in the x,y, andzdirections and\n|l,m,n/an}bracketri}ht ≡[|l,m,n/an}bracketri}ht↑,|l,m,n/an}bracketri}ht↓] (5)\nrepresents the two-component state vector with ↑,↓cor-\nresponding to the spin degree of freedom. The 2 ×2 ma-\ntrices are\nh0=/bracketleftbigg2tcos(k0a)+4B 0\n0 −2tcos(k0a)−4B/bracketrightbigg\n,(6)\nhx=/bracketleftbigg−Bi\n2A\ni\n2A B/bracketrightbigg\n, (7)\nhy=/bracketleftbigg−B1\n2A\n−1\n2A B/bracketrightbigg\n, (8)\nhz=/bracketleftbigg−t+iγ0\n0t+iγ/bracketrightbigg\n, (9)\nwhere 0<k0<π/a, and the other parameters, A,B,t,\nandγ, are assumed to be real and positive. The Fourier\ntransform of His expressed as\nH(k) =/bracketleftbiggΛ(k)+2γsin(kza) Θ −(kx,ky)\nΘ+(kx,ky)−Λ(k)+2γsin(kza)/bracketrightbigg\n,\n(10)\nwhere Λ( k) = ∆(kz) + 2B/summationtext\nα=x,y[1−cos(kαa)] and\nΘ±(kx,ky) =A[sin(kxa)±isin(kya)] with\n∆(kz) =−2t[cos(kza)−cos(k0a)].(11)\nIt canbe seenthat inversionsymmetryisbrokenif γ/ne}ationslash= 0.\nThe energy dispersion of this model is given as\nE=2γsin(kza)±/braceleftBig\nA2/parenleftbig\nsin2(kxa)+sin2(kya)/parenrightbig\n+/bracketleftbig\n∆(kz)+2B/parenleftbig\n2−cos(kxa)−cos(kya)/parenrightbig/bracketrightbig2/bracerightBig1\n2,(12)\nindicating that a pair of Weyl nodes appears at k±=\n(0,0,±k0). Note that E= 0 at the band center. The\nenergy of each Weyl node is located at the band center\nofE= 0 atγ= 0, whereas it deviates from there if\nγ/ne}ationslash= 0. In Sect. 4, we focus on the lattice system of a\nrectangular parallelepiped with Lsites in both the xand\nydirections and Nsites in the zdirection (see Fig. 1).\nIn this setup, chiral surface states appear on its side.\nNow, we show that no spontaneous charge current ap-\npears ifEFis located at the band center on the basis ofFig. 1. Lattice system considered in the text: a rectangular par-\nallelepiped with Lsites in both the xandydirections and Nsites\nin thezdirection.\nthe particle-hole symmetry inherent in the model intro-\nduced above. Let us define the operator Γ phas\nΓph=σxK, (13)\nwhereσxandKare respectively the xcomponent of the\nPauli matrices and the complex conjugate operator. The\ntight-binding Hamiltonian Hsatisfies\nΓ−1\nphHΓph=−H. (14)\nLet us denote eigenstates of Hin the valence band\nas|q/an}bracketri}htvand those in the conduction band as |q/an}bracketri}htcwith\nq= 1,2,3,..., whereH|q/an}bracketri}htv=ǫv\nq|q/an}bracketri}htvwithǫv\nq<0 and\nH|q/an}bracketri}htc=ǫc\nq|q/an}bracketri}htcwithǫc\nq>0. Here,qlabels the eigenstates\nin descending order (i.e., 0 ≥ǫv\n1≥ǫv\n2≥ǫv\n3≥...) in the\nvalence band and in ascending order (i.e., 0 ≤ǫc\n1≤ǫc\n2≤\nǫc\n3≤...) in the conduction band. Equation (14) allows\nus to setǫv\nq=−ǫc\nqwith\n|q/an}bracketri}htc= Γph|q/an}bracketri}htv. (15)\nWe introduce the charge current operator jαdefined\non an arbitrary site, where α=x,y,zspecifies the direc-\ntion of flow. For example, the current operator jxon the\n(l,m,n)th site is given by\njx=−e(−i)/bracketleftbig\n|l+1,m,n/an}bracketri}hthx/an}bracketle{tl,m,n|−h.c./bracketrightbig\n.(16)\nIt may be more appropriate to state that this is defined\non the link connecting the ( l,m,n)th and (l+1,m,n)th\nsites. We can show that any jαis invariant under the\ntransformation of Γ phas\nΓ−1\nphjαΓph=jα. (17)\nIn the ground state with EF= 0, the expectation value\nof anyjαis expressed as\n/an}bracketle{tjα/an}bracketri}ht=/summationdisplay\nq≥1v/an}bracketle{tq|jα|q/an}bracketri}htv. (18)\nBy using the relations given above and the completeness\nof the set of eigenstates consisting of {|q/an}bracketri}htv}and{|q/an}bracketri}htc},\nwe can show that\n/an}bracketle{tjα/an}bracketri}ht=1\n2/summationdisplay\nq≥1[v/an}bracketle{tq|jα|q/an}bracketri}htv+c/an}bracketle{tq|jα|q/an}bracketri}htc]\n=1\n2tr{jα}= 0, (19)\nindicating that a spontaneous charge current completely\nvanishes everywhere in the system at EF= 0. That is,\nalthough chiral surface states carry a circulating charge\n2J. Phys. Soc. Jpn.\ncurrent, their contribution is completely canceled out by\nthat from bulk states.\nNote that the above argument based on particle-hole\nsymmetry is not restricted to the two-orbital model used\nin this study and is also applicable to the four-orbital\nmodel introduced in Ref. 20.\n3. Analytical Approach\nBefore performing numerical simulations in the case\nofEF/ne}ationslash= 0, we analytically study the behaviors of chi-\nral surface states in a cylindrical Weyl semimetal. To do\nso, we apply the analytical approach given in Ref. 21,\nwhich was developed to describe unusual electron states\nin a Weyl semimetal: chiral surface states22)and chiral\nmodes along a screw dislocation.23,24)It is convenient to\nmodify the tight-binding Hamiltonian by taking the con-\ntinuum limit in the xandydirections, leaving the lattice\nstructure in the zdirection so that the resulting model\nhas a layered structure. After the partial Fourier trans-\nformation in the zdirection, the Hamiltonian is reduced\nto\nH=/bracketleftbigg˜Λ+2γsin(kza)˜A(ˆkx−iˆky)\n˜A(ˆkx+iˆky)−˜Λ+2γsin(kza)/bracketrightbigg\n,(20)\nwhere˜Λ = ∆(kz) +˜B(ˆk2\nx+ˆk2\ny) withˆkx=−i∂x,ˆky=\n−i∂y,˜A=Aa, and˜B=Ba2. We adapt this model\nto a cylindrical Weyl semimetal of radius Rby using\nthe cylindrical coordinates ( r,φ) withr=/radicalbig\nx2+y2and\nφ= arctan(y/x). Let Ψ(r,φ) =t(F,G) be an eigenfunc-\ntion ofHfor a given kz. It is convenient to rewrite F\nandGasF=eiλφf(r) andG=ei(λ+1)φg(r), where\nλis the azimuthal quantum number. Then, in terms of\nψ(r,φ) =t(f,g) for givenkzandλ, the eigenvalue equa-\ntion is written as\n/bracketleftbigg\n∆(kz)−˜BDλ˜A/parenleftbig\n−i∂r−iλ+1\nr/parenrightbig\n˜A/parenleftbig\n−i∂r+iλ\nr/parenrightbig\n−∆(kz)+˜BDλ+1/bracketrightbigg\nψ=˜Eψ,(21)\nwhere˜E=E−2γsin(kza) and\nDλ=∂2\nr+1\nr∂r−λ2\nr2. (22)\nAs demonstrated in Ref. 21, if ˜Bis finite but very\nsmall, the eigenvalue equation, Eq. (21), can be decom-\nposed into two separate equations: the Weyl and sup-\nplementary equations. The Weyl equation for fandgis\ngiven by\n(Dλ−Λ−)f= 0,(Dλ+1−Λ−)g= 0,(23)\nwhile the supplementary equation is\n(Dλ−Λ+)f= 0,(Dλ+1−Λ+)g= 0,(24)\nwhere\nΛ−=−˜E2−∆2\n˜A2,Λ+=˜A2\n˜B2.(25)\nHere,fandgarerelated by the originaleigenvalue equa-\ntion with a finite but very small ˜B. Note that the restric-\ntion on ˜B(i.e.,˜Bis very small) does not significantly\naffect the behaviors of chiral surface states. Indeed, the\nenergy of chiral surface states does not depend on ˜Bas\nseen in Eq. (28).We hereafter focus on chiral surface states, which ap-\npear only in the case of |∆(kz)|>|˜E|. The solutions\nof both the Weyl and supplementary equations are ex-\npressed by modified Bessel functions in this case. By su-\nperposing two solutions that asymptotically increase in\nan exponential manner, we can describe spatially local-\nized states near the side. With η≡/radicalbig\n∆2−˜E2/˜Aand\nκ≡˜A/˜B, the general solution for given λandkzis writ-\nten as\nψ=a/bracketleftBiggI|λ|(ηr)\n−i∆−˜E√\n∆2−˜E2I|λ+1|(ηr)/bracketrightBigg\n+b/bracketleftbiggI|λ|(κr)\niI|λ+1|(κr)/bracketrightbigg\n,\n(26)\nwhere the first and second terms respectively arise from\nthe Weyl and supplementary equations. The boundary\ncondition of ψ(R) =t(0,0) requires\n∆−˜E/radicalbig\n∆2−˜E2=−I|λ+1|(κR)\nI|λ|(κR)I|λ|(ηR)\nI|λ+1|(ηR),(27)\nindicating that a relevant solution is obtained only in the\ncase of ∆(kz)<0, which holds when kz∈(−k0,k0). The\neigenvalue of energy is approximately determined as\nE=˜A\nR/parenleftbigg\nλ+1\n2/parenrightbigg\n+2γsin(kza). (28)\nIn the case of γ= 0, the dispersion is flat (i.e., inde-\npendent of kz), representing a characteristic feature of\nthe chiral surface state. An electron in the chiral surface\nstate propagates in the anticlockwise direction viewed\nfrom above. The dispersion becomes dependent on kzif\nγ/ne}ationslash= 0, indicating that the groupvelocity is tilted to the z\ndirection. Consequently, an electron in the chiral surface\nstate circulates around the side surface in a spiral man-\nner.19)This implies that a spontaneous charge current\nin thezdirection can be induced near the side surface if\nγ/ne}ationslash= 0.\nNow, we roughly determine the magnitude of a spon-\ntaneous circulating current in the system consisting of\nNlayers. The circulating charge current carried by each\nchiral surface state is\nJ0\nφ=−e˜A\n2πR, (29)\nwhich flows in the clockwise direction. Note that the\nchiral surface state with E(λ) appears only when kz∈\n(−k0,k0). Ifkzdeviates from this interval, the state is\ncontinuously transformed to a bulk state, which is spa-\ntially extended over the entire system. Let us determine\nthe total chargecurrent Jφ. SinceJφvanishes at EF= 0,\nweneedtocollectthecontributionsto Jφarisingfromthe\nchiral surface states with E(λ) satisfying 0 <E <E Fif\nEF>0. Here, it is assumed that the contribution from\nbulk states in the same interval of energy is not impor-\ntant since their wavelength is relatively long and hence\nthey cannot induce a short-wavelength response local-\nized near the surface. If EF<0, the total charge current\nis obtained by collecting the contributions to Jφarising\nfrom the states with E(λ) satisfying EF< E <0 and\nthen reversingits sign. In addition to the condition for λ,\nit is important to note that the chiral surface states are\n3J. Phys. Soc. Jpn.\n-1 0 1\n 0(a)-πE/A\nkza\n-1 0 1\n 0(b)-πE/A\nkza\nFig. 2. Energy dispersion as a function of kzin the cases of (a)\nγ/A= 0 and (b) 0 .1. For the purely bulk states, only a quarter of\nthe corresponding branches are shown for clarity.\nstabilized only when |kz|<k0. Assuming that kzis given\nbykj\nz=jπ/[(N+1)a] withj= 1,2,3,...in the system\nconsisting of Nlayers, we require kj\nz< k0. From the\nobservation given above, we find that the spontaneous\ncharge current per layer is\nJφ\nN=−ek0a\n2π2EF, (30)\nwhich depends on EFas well ask0. This indicates that\nthe direction of flow reverses depending on the sign of\nEF. That is, the spontaneous current flows in opposite\ndirections for the cases of electron doping and hole dop-\ning.Notethat,since Jφ/Nisindependentof R,weexpect\nthat the circulating charge current will be insensitive to\nthe geometry of the Weyl semimetal. Thus, we expect\nthat Eq. (30) can be applied to the system of a rectan-\ngular parallelepiped, which is treated in the next section.\n4. Numerical Results\nWe focus on the lattice system of the rectangular par-\nallelepiped shown in Fig. 1, which occupies the region of\n1≤l,m≤Land 1≤n≤N, under the open boundary\ncondition in the three spatial directions. We set L= 50\nandN= 30 with the following parameters: B/A= 0.5,\nt/A= 0.5, andk0a= 3π/4. We consider the cases of\nγ/A= 0 and 0.1 forEF/A=±0.1 and±0.2. The energy\ndispersion for the infinitely long system with a cross-\nsectional area of L2is shown in Fig. 2 in the cases of\n(a)γ/A= 0 and (b) 0 .1 as a function of kz. In the case\nofγ/A= 0, branches with flat dispersion are uniformly 0 10 20 30 40 50 0 10 20 30-0.020.000.02\n(a)\nlnjy-0.020.000.02\n 0 10 20 30 40 50 0 10 20 30-0.020.000.02\n(b)\nlnjy\n 0 10 20 30 40 50 0 10 20 30-0.020.000.02\n(c)\nlnjy\n 0 10 20 30 40 50 0 10 20 30-0.020.000.02\n(d)\nlnjy\nFig. 3. (Color online) Spatial distribution of jynormalized by\neAin the cross section parallel to the xzplane; (a) EF/A= 0.2,\n(b) 0.1, (c)−0.1, and (d) −0.2.\ndistributed near E= 0 in the region of −k0< kz< k0.\nThey represent chiral surface states localized near the\nside surface of the system. These states should induce\na circulating charge current near the side surface when\nEF/ne}ationslash= 0. The dispersion of these states becomes slightly\nupward to the right in the case of γ/A= 0.1, implying\nthe appearance of a spontaneous charge current in the z\ndirection. Hereafter, we numerically study how the spon-\ntaneous charge current appears in this system depending\nonEFandγ.\nFirstly, let us examine the behaviors of a spontaneous\nchargecurrentthatcirculatesaroundthesystemnearthe\nside surface.Todoso,wecalculatethe distributionofthe\n4J. Phys. Soc. Jpn.\nspontaneouschargecurrentinthe ydirectionthroughthe\ncross section parallel to the xzplane at the center of the\nsystem (dotted line in Fig. 1). Precisely speaking, jyon\neach link connecting the ( l,25,n)th and (l,26,n)th sites\nis calculated for 1 ≤l≤50 and 1 ≤n≤30. Figure 3\nshows the results for jynormalized by eAforγ/A= 0\nandEF/A= 0.2, 0.1,−0.1, and−0.2. The results for\nγ/A= 0.1 are not shown since they are almost identi-\ncal to those for γ/A= 0. We observe that jy>0 near\nl= 1 andjy<0 nearl= 50 in the case of EF>0,\nwhereasjy<0 nearl= 1 andjy>0 nearl= 50 in\nthe case of EF<0. This indicates that the spontaneous\ncharge current circulates around the system in the clock-\nwise direction viewed from above when EF>0, whereas\nit circulates in the anticlockwise direction when EF<0\n(see Fig. 1). That is, its direction of flow is opposite for\nthe cases of EF>0 andEF<0. We also observe that jy\nincreaseswithincreasing EFinaccordancewithEq.(30).\nEquation (30) predicts |Jy|/N≈0.024×eAin the case\nofEF/A=±0.2, whereJyrepresents the total charge\ncurrent induced near each side surface of height N. This\nresult is consistent with those shown in Figs. 3(a) and\n3(d).\nSecondly, let us examine the behaviors of a sponta-\nneous charge current in the longitudinal direction. We\ncalculate the distribution of the charge current in the\nzdirection through the cross section parallel to the\nxyplane at the center of the system (broken line in\nFig. 1). Precisely speaking, jzon each link connecting\nthe (l,m,15)th and ( l,m,16)th sites is calculated for\n1≤l≤50 and 1 ≤m≤50. Figure 4 shows the results\nforjznormalized by eAforγ/A= 0.1 andEF/A= 0.2,\n0.1,−0.1, and−0.2. The results for γ/A= 0 are not\nshown since jzvanishes everywhere in this case. Again,\nFig. 4 indicates that jzincreases with increasing EFand\nthat its sign is opposite for the cases of EF>0 and\nEF<0. Note that a relatively large current appears\nnear the side surface, particularly near the corners, while\na small current flowing in the opposite direction is dis-\ntributed beneath the side surface. The former is induced\nby chiral surface states, while the latter originates from\nbulk states. These two contributions cancel each other\nout if they are integrated over the cross section; thus,\nthe total charge current in the zdirection completely\nvanishes.\nFinally, we examine the effect of disorder25–30)on the\nspontaneous charge current by adding the impurity po-\ntential term\nHimp=/summationdisplay\nl,m,n|l,m,n/an}bracketri}ht/bracketleftBigg\nV(l,m,n)\n1 0\n0V(l,m,n)\n2/bracketrightBigg\n/an}bracketle{tl,m,n|\n(31)\nto the Hamiltonian H, whereV1andV2are as-\nsumed to be uniformly distributed within the interval\nof [−W/2,+W/2]. Previous studies have shown that a\nWeyl semimetal phase is robust against weak disorder\nup to a critical disorder strength, Wc,25,26)and that chi-\nral surface states also persist as long as W < W c.29)In\nthe case of B/A= 0.5,t/A= 0.5, andk0a= 3π/4,\nthe critical disorder strength is Wc/A∼4. The ensemble 0 10 20 30 40 50 0 10 20 30 40 50-0.0040.0000.004\n(a)\nlmjz-0.0040.0000.004\n 0 10 20 30 40 50 0 10 20 30 40 50-0.0040.0000.004\n(b)\nlmjz\n 0 10 20 30 40 50 0 10 20 30 40 50-0.0040.0000.004\n(c)\nlmjz\n 0 10 20 30 40 50 0 10 20 30 40 50-0.0040.0000.004\n(d)\nlmjz\nFig. 4. (Color online) Spatial distribution of jznormalized by\neAin the cross section parallel to the xyplane; (a) EF/A= 0.2,\n(b) 0.1, (c)−0.1, and (d) −0.2.\naverages, /an}bracketle{tjy/an}bracketri}htand/an}bracketle{tjz/an}bracketri}ht, are calculated over 500 samples\nwith different impurity configurations at EF/A= 0.1\nfor a given value of W/A. In calculating jyandjzfor a\ngiven impurity configuration,we take accountofonly the\ncontribution from electron states with an energy Esat-\nisfying 0<E <E F, assuming that electron states below\nthe band center have no contribution owing to cancella-\ntion between them. This assumption is not strictly jus-\ntified here since particle-hole symmetry is broken by the\nimpurity potential. Nonetheless, this should be a good\napproximation after taking the ensemble average.\nFigure 5 shows the results for /an}bracketle{tjy/an}bracketri}htnormalized by eA\nforγ/A= 0 andEF/A= 0.1 withW/A= 3, 4, and\n4.5. We observe that the circulating charge current is ro-\n5J. Phys. Soc. Jpn.\n 0 10 20 30 40 50 0 10 20 30-0.010.000.01\n(a)\nln< jy > -0.010.000.01\n 0 10 20 30 40 50 0 10 20 30-0.010.000.01\n(b)\nln< jy >\n 0 10 20 30 40 50 0 10 20 30-0.010.000.01\n(c)\nln< jy >\nFig. 5. (Color online) Spatial distribution of /angbracketleftjy/angbracketrightnormalized by\neAin the cross section parallel to the xzplane at EF/A= 0.1; (a)\nW/A = 3, (b) 4, and (c) 4 .5.\n 0 10 20 30 40 50 0 10 20 30 40 50-0.0020.0000.002\n(a)\nlm< jz >-0.0020.0000.002\n 0 10 20 30 40 50 0 10 20 30 40 50-0.0020.0000.002\n(b)\nlm< jz >\nFig. 6. (Color online) Spatial distribution of /angbracketleftjz/angbracketrightnormalized by\neAin the cross section parallel to the xyplane at EF/A= 0.1; (a)\nW/A = 2, and (b) 3.bust against disorder up to W/A∼4 but is suppressed\nwhenW/Aexceeds 4. This behavior is consistent with\nan observation reported previously.29)Figure 6 shows\nthe results for /an}bracketle{tjz/an}bracketri}htnormalized by eAforγ/A= 0.1 and\nEF/A= 0.1withW/A= 2and3.We observethat /an}bracketle{tjz/an}bracketri}htis\nsignificantlysuppressedinthecaseof W/A= 3,although\nthe circulatingchargecurrentis almost unaffected in this\ncase. This indicates that the charge current in the zdi-\nrectionis morefragilethan the circulatingchargecurrent\nagainstthe mixingofchiralsurfacestatesandbulk states\ndue to disorder.\n5. Summary and Discussion\nWe theoretically studied a spontaneous charge current\ndue to chiral surface states in the ground state of a Weyl\nsemimetal. We analytically and numerically determined\nthemagnitudeofthechargecurrentinducedneartheside\nsurface of the system. It is shown that no spontaneous\ncharge current appears when the Fermi level, EF, is lo-\ncated at the band center. It is also shown that, once EF\ndeviates from the band center, the spontaneous charge\ncurrent appears to circulate around the side surface of\nthe system and its direction of flow is opposite for the\ncases of electron doping and hole doping. The circulating\ncurrent is shown to be robust against weak disorder.\nLet us focus on the two features revealed in this paper:\nthe appearance of a spontaneous charge current except\nat the band center and the reversal of its direction of\nflow as a function of EF. As they are derived by using a\nmodel possessingparticle-hole symmetry, a natural ques-\ntion arises: do these features manifest themselves even in\nthe absence of particle-hole symmetry? The answer is\nyes. The disappearance of the spontaneous charge cur-\nrent reflects the fact that the contribution from chiral\nsurface states is completely canceled out by that from\nbulk states. As the spontaneous charge current due to\nchiral surface states is localized near the side surface,\nthis cancellation should be mainly caused by bulk states\nwith a short wavelength, occupying the bottom region\nof the energy band far from the band center. Hence, if\nEFis varied near the band center, the contribution from\nbulk states is almost unaffected but that from chiral sur-\nface states is significantly changed, depending on EFin\na roughly linear manner. This behaviorshould take place\nregardlessofthepresenceorabsenceofparticle-holesym-\nmetry. Thus, we expect that the features of the sponta-\nneouschargecurrentstillmanifest themselveseveninthe\nabsence of particle-hole symmetry, although the point of\nthe disappearance shifts away from the band center.\nAcknowledgment\nThis work was supported by JSPS KAKENHI Grant\nNumbers JP15K05130 and JP18K03460.\n1) R. Shindou and N. Nagaosa, Phys. Rev. Lett. 87, 116801\n(2001).\n2) S. Murakami, New J. Phys. 9, 356 (2007).\n3) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov,\nPhys. Rev. B 83, 205101 (2011).\n4) K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 84, 075129\n6J. Phys. Soc. Jpn.\n(2011).\n5) A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205\n(2011).\n6) A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84,\n235126 (2011).\n7) W. Witczak-Krempa and Y. B. Kim, Phys. Rev. B 85, 045124\n(2012).\n8) P.Delplace, J.Li, and D.Carpentier, Europhys.Lett. 97, 67004\n(2012).\n9) G. Hal´ asz and L. Balents, Phys. Rev. B 85, 035103 (2012).\n10) A. Sekine and K. Nomura, J. Phys. Soc. Jpn. 82, 033702 (2013).\n11) H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Phys.\nRev. X5, 011029 (2015).\n12) S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang , B.\nWang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia,\nA. Bansil, H. Lin, and M. Z. Hasan, Nat. Commun. 6, 7373\n(2015).\n13) S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian , C.\nZhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang,\nH. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou,\nP. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, Science 349,\n613 (2015).\n14) B.-Q. Lv, H.-M. Weng, B.-B. Fu, X.-P. Wang, H. Miao, J. Ma,\nP. Richard, X.-C. Huang, L.-X. Zhao, G.-F. Chen, Z. Fang, X.\nDai, T. Qian, and H. Ding, Phys. Rev. X 5, 031013 (2015).\n15) B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C.\nHuang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N.\nStrocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H.\nDing, Nat. Phys. 11, 724 (2015).\n16) S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T. -R.\nChang, H. Zheng, V. N. Strocov, D. S. Sanchez, G. Chang, C.Zhang, D. Mou, Y. Wu, L. Huang, C.-C. Lee, S.-M. Huang,\nB. Wang, A. Bansil, H.-T. Jeng, T. Neupert, A. Kaminski, H.\nLin, S. Jia, and M. Z. Hasan, Nat. Phys. 11, 748 (2015).\n17) S. Souma, Z. Wang, H. Kotaka, T. Sato, K. Nakayama, Y.\nTanaka, H. Kimizuka, T. Takahashi, K. Yamauchi, T. Oguchi,\nK. Segawa, and Y. Ando, Phys. Rev. B 93, 161112 (2016).\n18) K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille, A. A. Nu-\ngroho, P. Goswami, M. Ochi, M. Ikhlas, M. Nakayama, S.\nAkebi, R. Noguchi, R. Ishii, N. Inami, K. Ono, H. Kumigashira ,\nA. Varykhalov, T. Muro, T. Koretsune, R. Arita, S. Shin, T.\nKondo, and S. Nakatsuji, Nat. Mater. 16, 1090 (2017).\n19) P. Baireuther, J. A. Hutasoit, J. Tworzyd/suppress lo, and C. W. J.\nBeenakker, New J. Phys. 18, 045009 (2016).\n20) M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 111, 027201\n(2013).\n21) Y. Takane, J. Phys. Soc. Jpn. 86, 123708 (2017).\n22) R.Okugawa and S.Murakami, Phys.Rev.B 89, 235315 (2014).\n23) K.-I. Imura and Y. Takane, Phys. Rev. B 84, 245415 (2011).\n24) H. Sumiyoshi and S. Fujimoto, Phys. Rev. Lett. 116, 166601\n(2016).\n25) C.-Z. Chen, J. Song, H. Jiang, Q.-F. Sun, Z. Wang, and X. C.\nXie, Phys. Rev. Lett. 115, 246603 (2015).\n26) H. Shapourian and T. L. Hughes, Phys. Rev. B 93, 075108\n(2016).\n27) S. Liu, T. Ohtsuki, and R. Shindou, Phys. Rev. Lett. 116,\n066401 (2016).\n28) E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and P. O. Sukh a-\nchov, Phys. Rev. B 93, 235127 (2016).\n29) Y. Takane, J. Phys. Soc. Jpn. 85, 124711 (2016).\n30) Y. Yoshimura, W. Onishi, K. Kobayashi, T. Ohtsuki, and K. -I.\nImura, Phys. Rev. B 94, 235414 (2016).\n7"    },    {        "title": "1006.5179v1.Competition_between_paramagnetism_and_diamagnetism_in_charged_Fermi_gases.pdf",        "content": "arXiv:1006.5179v1  [cond-mat.stat-mech]  27 Jun 2010Competition between paramagnetism and diamagnetism in cha rged Fermi gases\nXiaoling Jian, Jihong Qin, and Qiang Gu∗\nDepartment of Physics, University of Science and Technolog y Beijing, Beijing 100083, China\n(Dated: November 10, 2018)\nThe charged Fermi gas with a small Lande-factor gis expected to be diamagnetic, while that with\na larger gcould be paramagnetic. We calculate the critical value of th eg-factor which separates\nthe dia- and para-magnetic regions. In the weak-field limit, gchas the same value both at high and\nlow temperatures, gc= 1/√\n12. Nevertheless, gcincreases with the temperature reducing in finite\nmagnetic fields. We also compare the gcvalue of Fermi gases with those of Boltzmann and Bose\ngases, supposing the particle has three Zeeman levels σ=±1,0, and find that gcof Bose and Fermi\ngases is larger and smaller than that of Boltzmann gases, res pectively.\nPACS numbers: 05.30.Fk, 51.60.+a, 75.10.Lp, 75.20.-g\nI. INTRODUCTION\nMagnetism of electron gases has been considerably\nstudied in condensed matter physics. In magnetic field,\nthe magnetization of a free electron gas consists of two\nindependent parts. The spin magnetic moment of elec-\ntrons results in the paramagnetic part (the Pauli para-\nmagnetism), while the orbital motion due to charge de-\ngreeoffreedom in magneticfield induces the diamagnetic\npart (the Landau diamagnetism)[1]. The Pauli paramag-\nnetism and the Landau diamagnetism compete with each\nother. For electrons whose Lande-factor g= 2, the zero-\nfield paramagnetic susceptibility is two times stronger\nthan the diamagnetic susceptibility, so the free electron\ngas exhibits paramagnetism altogether.\nMagnetic properties of relativistic Fermi gases have\nalso been under extensive investigation. Daicic et al.\ndeveloped statistical mechanics for the magnetized pair-\nfermion gases and found that the intrinsic spin causes\nimportant effects upon the relativistic para- and dia-\nmagnetism[2].\nThe study of ultracold atoms has stimulated renewed\nresearch interest in the magnetism of quantum gases[3–\n6]. When atomic gases are confined in optical traps[3, 4],\ntheir spin degree of freedom becomes active, leading\nto the manifestation of magnetism. Theoretically, the\nparamagnetism[7] and ferromagnetism[8–10] in a neutral\nspin-1 Bose gas have ever been studied. In experiments,\nmagnetic domains have been directly observed in87Rb\ncondensate, a typical ferromagneticspinor condensate[5].\nVery recently, the exploration of magnetism in quantum\ngases has been extended to the Fermi gas. It is already\nobserved that an ultracold two-component Fermi gas of\nneutral6Li atoms exhibits ferromagnetism caused by re-\npulsive interactions between atoms[6].\nFurthermore, it is possible to realize charged quantum\ngases from neutral ultracold atoms. So far, cold plasma\nhas been created by photoionization of cold atoms[11].\nThe temperatures of electrons and ions are as low as 100\n∗Corresponding author: qgu@ustb.edu.cnmK and 10 µK, respectively. The ions can be regarded\nas charged Bose or Fermi gases. Once the quantum gas\nhas both the spin and charge degrees of freedom, there\narises the competition between paramagnetism and dia-\nmagnetism, as in electrons. Different from electrons, the\ng-factor for different magnetic ions is diverse, ranging\nfrom 0 to 2.\nTheg-factor is a characteristic parameter to measure\nthe intensity of paramagnetic effect. It is expected that\nthequantumgasdisplaysdiamagnetisminsmall gregion,\nbut paramagnetism in large gregion. The main purpose\nof this paper is to calculate the critical value of g. We\nalso present a comparison with results of charged spin-1\nBose gases which have been obtained previously[12].\nII. THE MODEL\nWe consider a charged Fermi gas of Nparticles, with\nthe effective Hamiltonian written as\n¯H−µN=DL/summationdisplay\nj,kz,σ(ǫjkz+ǫσ−µ)njkzσ,(1)\nwhereµis the chemical potential and ǫjkzis the quan-\ntized Landau energy in the magnetic field B,\nǫjkz= (1\n2+j)/planckover2pi1ω+/planckover2pi12k2\nz\n2m∗(2)\nwithj= 0,1,2,...labeling different Landau levels and\nω=qB/(m∗c) being the gyromagnetic frequency of a\nfermionwithcharge qandmassm∗.DL=qBSxy/(2π/planckover2pi1c)\nmarks the degeneracy of each Landau level with Sxythe\ntotal section area in x, y directions of the system. ǫσ\ndenotes the Zeeman energy,\nǫσ=−g/planckover2pi1q\nm∗cσB=−gσ/planckover2pi1ω, (3)\nwheregis the Lande-factor and σrefers to the spin- z\nindex of Zeeman state |F,σ/angbracketright.\nThe grand thermodynamic potential of the Fermi gas\nis expressed as\nΩT/negationslash=0=−1\nβDL/summationdisplay\nj,kz,σln[1+e−β(ǫjkz+ǫσ−µ)],(4)2\nwhereβ= (kBT)−1. Performing Taylor expansions and\nthen integrating out kz, Eq. (4) becomes\nΩT/negationslash=0=−ωV\n/planckover2pi12/parenleftbiggm∗\n2πβ/parenrightbigg3/2\n×��/summationdisplay\nl=1/summationdisplay\nσ(−1)l+1l−3\n2e−lβ(/planckover2pi1ω\n2−gσ/planckover2pi1ω−µ)\n1−e−lβ/planckover2pi1ω,(5)\nwhereVis the volume of the system. For simplicity, the\nfollowing notation is introduced,\nFσ\nτ[α,δ] =∞/summationdisplay\nl=1(−1)l+1lα/2e−lx(ησ+δ)\n(1−e−lx)τ,(6)\nwherex=β/planckover2pi1ωandησ= (/planckover2pi1ω/2−µ+ǫσ)/(/planckover2pi1ω). Then\nEq. (5) is rewritten as\nΩT/negationslash=0=−ωV\n/planckover2pi12/parenleftbiggm∗\n2πβ/parenrightbigg3/2/summationdisplay\nσFσ\n1[−3,0].(7)\nFor a system with the given particle density n=N/V,\nthe chemical potential is obtained according to the fol-\nlowing equation\nn=−1\nV/parenleftbigg∂Ω\n∂µ/parenrightbigg\nT,V=x/parenleftbiggm∗\n2πβ/planckover2pi12/parenrightbigg3/2/summationdisplay\nσFσ\n1[−1,0].(8)\nA similar treatment has been employed to study diamag-\nnetism of the charged spinless Bose gas[13] and extended\nto the study of competition of diamagnetism and param-\nagnetism in charged spin-1 Bose gases[12]. This method\nis more applicable at relatively high temperatures.\nTo determine gc, we need calculate the magnetization\ndensityMas a function of the magnetic field Band tem-\nperatureT. The system is paramagnetic when M >0\nwhile diamagnetic when M <0.Mcan be derived from\nthe thermodynamic potential by the standard procedure\nMT/negationslash=0=−1\nV/parenleftbig∂Ω\n∂B/parenrightbig\nT,V, which yields\nMT/negationslash=0=/planckover2pi1q\nm∗c/parenleftbiggm∗\n2πβ/planckover2pi12/parenrightbigg3/2/summationdisplay\nσ/braceleftbigg\nFσ\n1[−3,0]\n+x(gσ−1\n2)Fσ\n1[−1,0]−xFσ\n2[−1,1]/bracerightbigg\n.(9)\nFor carrying out numerical calculations, it is useful\nto make the parameters dimensionless, such as M=\nm∗cM/(n/planckover2pi1q),ω=/planckover2pi1ω/(kBT∗), andt=T/T∗. HereT∗\nis the characteristic temperature of the system, which is\ngiven bykBT∗= 2π/planckover2pi12n2\n3/m∗. Then we have x=ω/t,\nησ= 1/2−µ′/ω−gσandµ′=µ/(kBT∗) in the notation\nFσ\nτ[α,δ]. Equations (8) and (9) can be re-expressed as\n1 =ωt1/2/summationdisplay\nσFσ\n1[−1,0] (10)/s45/s48/s46/s53/s48/s48/s46/s48/s48/s48/s46/s53/s48\n/s48/s46/s48/s48/s48/s46/s52/s48/s48/s46/s56/s48\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56/s45/s48/s46/s52/s56/s45/s48/s46/s52/s52/s40/s99/s41\n/s32/s32/s77/s40/s97/s41\n/s40/s98/s41\n/s32\n/s32/s32/s77\n/s112\n/s32/s32/s77\n/s100\n/s103\nFIG. 1: (a) The total magnetization density ( M), (b) the\nparamagnetization density ( Mp), and (c) the diamagnetiza-\ntion density ( Md) as a function of gatt= 0.2. The dotted\nand dashed lines correspond to ω= 5 and 10, respectively.\nand\nMT/negationslash=0=t3/2/summationdisplay\nσ/braceleftbigg\nFσ\n1[−3,0]+x(gσ−1\n2)\n×Fσ\n1[−1,0]−xFσ\n2[−1,1]/bracerightbigg\n.(11)\nIn following discussions, ωis used to indicate the magni-\ntude of magnetic field since it is proportional to B.\nIII. RESULTS AND DISCUSSIONS\nFirst, we look at a spin-1\n2Fermi gas, setting σ=±1\nto present the two Zeeman levels. The dimensionless\nmagnetization density Mas a function of gis shown in\nFig. 1(a). As expected, Mis negative in the small g\nregion, which means that the diamagnetism dominates.\nFor each given value of ω,Mgrows with gand changes\nits sign from negative to positive in the larger gregion,\nindicating that the paramagnetic effect is enhanced due\nto increase of g. This phenomenon is also observed in the\nBose system[12]. gcis just the value of gwhereM= 0.3\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s46/s50/s56/s48/s46/s51/s53/s48/s46/s52/s50/s48/s46/s52/s57\n/s32 /s32/s61/s32/s50/s48\n/s32 /s32/s61/s32/s49/s48\n/s32 /s32/s61/s32/s53\n/s32/s32/s103\n/s99\n/s49/s47/s116\nFIG. 2: Plots of the critical value of g-factor as a function of\n1/t. The magnetic field is chosen as ω= 20 (solid line), 10\n(dashed line) and 5 (dotted line), respectively.\nThe total magnetization Mshown in Fig. 1(a) con-\nsists of both the paramagnetic and diamagnetic contri-\nbutions. Figure 1(b) depicts the pure paramagnetic con-\ntribution to M, which is calculated by Mp=gmwhere\nm=n1−n−1. Figure 1(c) plots the diamagnetic con-\ntribution to M,Md=M−Mp. For each fixed value\nofω, the diamagnetization Mdis slightly weakened with\nincreasingg. Interestingly, for a charged spin-1 Bose gas\nin a constant magnetic field, the diamagnetism is not\nsuppressed but enhanced as gbecomes larger[12]. Com-\nparing Figs 1(a), 1(b) and 1(c), it can be seen clearly\nthat the increase of Mwithgis mainly owing to the\nparamagnetization Mp.\ngcis an important parameter to describe the competi-\ntionbetweenthediamagnetismandparamagnetism. Fig-\nure 2 plotsgcfor the chargedspin-1\n2Fermi gas. The tem-\nperature is described by 1 /tand the magnetic fields are\nchosen to be relatively larger, since the method adopted\nhere is more applicable at higher temperatures or in\nstronger fields. In the high temperature limit, gctends\nto a universal value, gc|t→∞= 0.28868, regardless of the\nstrength of the magnetic field. In a fixed magnetic field,\ngcincreasesmonotonicallyasthe temperaturefalls down.\nButthetrendofincreasingissloweddownifthemagnetic\nfield is weakened.\nPresent method can not produce valid results in the\nlowtemperatureregionifasmallmagneticfieldischosen.\nIn this case that gω≪t≪µ′, the grand thermodynamic\npotential in Eq. (4) can be calculated using the Euler-\nMaclaurin formula\n∞/summationdisplay\nj=0ψ(j+1\n2)≈/integraldisplay∞\n0ψ(x)dx+1\n24ψ′(0),(12)\nand then Eq. (4) is transformed into\nΩT/negationslash=0=−V\nβλ3/summationdisplay\nσf5\n2(z)+Vβ\n24λ3(/planckover2pi1ω)2/summationdisplay\nσf1\n2(z),(13)/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s46/s51/s53/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48\n/s32/s61/s32/s53\n/s32/s32/s103\n/s99\n/s49/s47/s116/s32/s61/s32/s49/s48\nFIG. 3: The gc−1/tcurves for charged spin-1 gases obey-\ning the Bose-Einstein (BE, dotted line), Maxwell-Boltzman n\n(MB, solid line) and Fermi-Dirac (FD, dashed line) statisti cs,\nrespectively. The magnetic field is chosen as ω= 10 and 5.\nwhereλ=hβ1\n2/(2πm∗)1\n2,z=eβ(µ−ǫσ)and the Fermi-\nDirac integral fn(z) is normally defined as\nfn(z)≡1\nΓ(n)/integraldisplay∞\n0xn−1\nz−1ex+1dx, (14)\nwhere Γ(n) is a usual gamma function and x=βǫ. Then\nthe magnetization density, M, is obtained from grand\nthermodynamic potential in Eq. (13),\nMT/negationslash=0=gt3/2/summationdisplay\nσσf3\n2(z′)−ωt1/2\n12/summationdisplay\nσf1\n2(z′)\n−gω2t−1/2\n24/summationdisplay\nσσf−1\n2(z′), (15)\nwherez′=e(µ′+gσω)/t. After some algebra, we get from\nEq. (15) that gc|ω→0= 1/√\n12≈0.28868. So in the weak\nfield limit, gchas the same value both at the high and\nlow temperature limit. This implies that the gc−1/t\ncurve is likely to flatten out in the weak-field limit.\nTo proceed, we make a comparison between Fermi\nand Bose gases. Considering that gcfor charged spin-\n1 Bose gas has already been studied[12], we discuss a\nFermi gas with three sublevels, σ=±1,0 (a pseudo-\nspin-1 Fermi gas) to ensure the comparability. Results\nfor the spin-1 Boltzmann gas are also obtained. Fig-\nure 3 shows the gc−1/tcurves for the three kinds of\ncharged spin-1 gases. In a given value of ω,gcof all the\nthree gases displays similar temperature-dependence: gc\nincreases monotonously as treduces. In the high tem-\nperature limit, gcgoes to the same value in all magnetic\nfields,gc|t→∞= 1/√\n8, reflecting that the Bose-Einstein\n(BE) and Fermi-Dirac (FD) statistics coincide with the\nMaxwell-Boltzmann (MB) statistics in this case.\nFigure 3 also demonstrates the difference among the\nthree kinds of statistics. For each given fixed value of ω,4\nthegc−1/tcurves of Bose and Fermi gases always locate\nat the two sides of that of the Boltzmann gas. Given the\nsame temperature and magnetic field, gcof Fermi gas\nis the smallest. According to our previous research[12],\ngc|t→0= 1/2 for the Boltzmann gas regardless of the\nmagnetic field. This means that gcof the Fermi gas does\nnever exceed 1 /2 in the low temperature, no matter how\nstrong the field is.\nIV. SUMMARY\nIn summary, we study the interplay between paramag-\nnetism and diamagnetismofchargedFermi gasessuppos-\ning that the Lande-factor gis a variable. The gas under-\ngoes a shift from diamagnetism to paramagnetism at thecritical value of gandgcincreases monotonically as the\ntemperature tdecreases in a fixed magnetic field ω, and\nthe rise ingcis lowered as ωis reduced. We conjecture\nthatgcholds a constant at all temperatures in the weak\nfield limit. For a spin-1\n2Fermi gas,gc|ω→0= 1/√\n12. We\nalso briefly compare gcof charged spin-1 gases obeying\nthe Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann\nstatistics. The gc−1/tcurves of Boltzmann gases are al-\nways between those of Bose and Fermi gases in the same\nmagnetic field. In the high temperature limit, gcof all\nthe three gases tends to the same value.\nThisworkwassupportedbytheFokYingTungEduca-\ntion Foundation of China (No. 101008), the Key Project\nof the Chinese Ministry of Education (No. 109011), and\nthe Fundamental Research Funds for the Central Univer-\nsities of China.\n[1] L.D. Landau, E.M. Lifshitz, Statistical Physics. Part 1 ,\nButterworth-Heinemann, Oxford, 1980.\n[2] J. Daicic , N.E. Frankel, R.M. Gailis, V. Kowalenko,\nPhys. Rep. 63 (1994) 237 and references therein.\n[3] D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S.\nInouye, H.-J. Miesner, J. Stenger, W. Ketterle, Phys.\nRev. Lett. 80 (1998) 2027.\n[4] J. Stenger, S. Inouye, D.M. Stamper-Kurn, H.-J. Mies-\nner, A.P. Chikkatur, W. Ketterle, Nature 396 (1998) 345.\n[5] L.E. Sadler, J.M. Higbie, S.R. Leslie, M. Vengalattore,\nD.M. Stamper-Kurn, Nature 443 (2006) 312.\n[6] G.B. Jo, Y.R. Lee, J.H. Choi, C.A. Christensen, T.H.\nKim, J.H. Thywissen, D.E. Pritchard, W. Ketterle, Sci-\nence 325 (2009) 1521.\n[7] K. Yamada, Prog. Theor. Phys. 76 (1982) 443; M.V.Simkin, E.G.D. Cohen, Phys. Rev. A 59 (1999) 1528.\n[8] T.-L. Ho, Phys. Rev. Lett. 81 (1998) 742; T. Ohmi, K.\nMachida, J. Phys. Soc. Jpn. 67 (1998) 1822.\n[9] Q. Gu, R.A. Klemm, Phys. Rev. A 68 (2003) 031604(R);\nC. Tao, P. Wang, J. Qin, Q. Gu, Phys. Rev. B 78 (2008)\n134403.\n[10] K. Kis-Szabo, P. Szepfalusy, G. Szirmai, Phys. Rev. A\n72 (2005) 023617; S. Ashhab, J. Low Tempt. Phys. 140\n(2005) 51.\n[11] T.C. Killian, S. Kulin, S.D. Bergeson, L.A. Orozco, C.\nOrzel, S.L. Rolston, Phys. Rev. Lett. 83 (1999) 4776.\n[12] X. Jian, J. Qin, Q. Gu, unpublished.\n[13] G.B. Standen, D.J. Toms, Phys. Rev. E 60 (1999) 5275."    },    {        "title": "2303.14534v1.Spin_momentum_locking_and_ultrafast_spin_charge_conversion_in_ultrathin_epitaxial_Bi___1_x__Sb__x__topological_insulator.pdf",        "content": "Spin-momentum locking and ultrafast spin-charge conversion in\nultrathin epitaxial Bi 1\u0000xSbxtopological insulator.\nE. Rongione,1, 2L. Baringthon,1, 3, 4D. She,1, 3, 4G. Patriarche,4R. Lebrun,1\nA. Lemaître,4M. Morassi,4N. Reyren,1M. Mičica,2J. Mangeney,2J. Tignon,2\nF. Bertran,3S. Dhillon,2, 5P. Le Fèvre,3H. Jaffrès,1, 5and J.-M. George1, 5\n1Unité Mixte de Physique, CNRS, Thales,\nUniversité Paris-Saclay, F-91767 Palaiseau, France\n2Laboratoire de Physique de l’Ecole Normale Supérieure,\nENS, Université PSL, CNRS, Sorbonne Université,\nUniversité Paris Cité, F-75005 Paris, France\n3Synchrotron SOLEIL, L’Orme des Merisiers,\nDépartementale 128, F-91190 Saint-Aubin, France\n4Université Paris-Saclay, CNRS, Centre de Nanosciences\net de Nanotechnologies, F-91120 Palaiseau, France\n5Corresponding authors: sukhdeep.dhillon@phys.ens.fr,\nhenri.jaffres@cnrs-thales.fr, jeanmarie.george@cnrs-thales.fr\n(Dated: March 28, 2023)\n1arXiv:2303.14534v1  [cond-mat.mtrl-sci]  25 Mar 2023Abstract\nThe helicity of 3D topological insulator surface states has drawn significant attention\ninspintronicsowingtospin-momentumlockingwherethecarriers’spinisorientedper-\npendicular to their momentum. This property can provide an efficient method to con-\nvert charge currents into spin currents, and vice-versa, through the Rashba-Edelstein\neffect. However, experimental signatures of these surface states to the spin-charge\nconversion are extremely difficult to disentangle from bulk state contributions. Here,\nwe combine spin- and angle-resolved photo-emission spectroscopy, and time-resolved\nTHz emission spectroscopy to categorically demonstrate that spin-charge conversion\narises mainly from the surface state in Bi 1\u0000xSbxultrathin films, down to few nanome-\nters where confinement effects emerge. We correlate this large conversion efficiency,\ntypically at the level of the bulk spin Hall effect from heavy metals, to the complex\nFermi surface obtained from theoretical calculations of the inverse Rashba-Edelstein\nresponse. Both surface state robustness and sizeable conversion efficiency in epitaxial\nBi1\u0000xSbxthinfilmsbringnewperspectivesforultra-lowpowermagneticrandom-access\nmemories and broadband THz generation.\nKeywords: topological insulator, ARPES, spin-resolved ARPES, THz-TDS, spin-charge conversion, surface\nstates\n2INTRODUCTION\nThe discovery of metallic quantum states at the surface of 3D topological insulators\n(TIs) [1–3] has opened exciting new functionalities in spintronics owing to their topological\nprotection and spin-momentum locking (SML) properties [4, 5]. Indeed, the combination\nof band inversion and time reversal symmetry (TRS) results in a peculiar spin texture in\nmomentumspace. Injectingacurrentinthesestatesresultsthereforeinanoutofequilibrium\nspin density (also called spin-accumulation) along the transverse direction [4–6]. This is the\nRashba-Edelstein effect (REE) [7] which can be used to exert a spin-orbit torque (SOT)\nonto the magnetization of an adjacent ferromagnet (FM) [8]. The reciprocal phenomenon,\nby which a spin density produces an in-plane transverse charge current, is called the Inverse\nRashba-Edelstein effect (IREE) [6, 9].\nImportantly, the resulting spin-charge conversion (SCC) efficiencies in topological surface\nstates (TSS) combining strong spin-orbit coupling (SOC) and SML is expected to be at least\noneorderofmagnitudelargercomparedtothespinHalleffect(SHE)of5 dheavymetals[10–\n12]. SCC has been demonstrated in a range of Bi-based TI coumpounds, including bismuth\nselenide Bi 2Se3, bismuth telluride Bi 2Te3, Bi 2(Se,Te) 3[13] or Bi 1\u0000xSbx(BiSb) [14]. To\nbenefit fully from IREE, the charge currents should be confined in the surface states and\nany current flowing through the bulk states should be avoided. The prerequisites are hence\ni) a sizeable bandgap typically larger than 0.2 to 0.3 eV, and ii) a perfect control of the\nFermi level position, usually achieved by stoichiometry and/or strain engineering. In this\nrespect, Bi 1\u0000xSbxalloys, although displaying clear topological surface states [15–17], have\nbeen mostly neglected for spintronic applications as a result of their modest bulk bandgap\n(about 40 meV for x= 0:07) and relatively complex band structure. However, quantization\neffects in ultrathin films have shown to lead to much larger gap [18, 19] while retaining their\nband inversion near the M point in the x= 0:07\u00000:3composition range [20], unlike pure\nBi[21–23]. BiSbthereforehasconsiderablepotentialascandidateforspintronicsapplications\nas well as for recently engineered efficient spintronic THz emitters [24–31].\nIn this letter, we report on our detailed investigation of the surface state SML properties\nof ultrathin (111)-oriented Bi 1\u0000xSbxepitaxial films. They exhibit a topological phase as\nrecently confirmed by our angular-resolved photo-emission spectroscopy (ARPES) measure-\nments [18]. Here, we focus on spin-resolved ARPES (SARPES) performed on ultrathin BiSb\n3films and extract the in-plane spin texture for the different electron and hole pockets charac-\nterizing the complex BiSb Fermi surface. Moreover, the SCC mediated by the BiSb surface\nstates is probed at the sub-picosecond timescale using an adjacent metallic Co layer acting\nas a spin injector. Unprecedentedly large SCC is measured with efficiencies beyond the level\nof carefully optimized Co/Pt systems. Our results also indicate that surface state related\nIREE is the mechanism responsible for SCC mechanism. Tight-binding (TB) calculation\nand linear response theory account for our findings.\nI. SPIN-RESOLVED ARPES\nSARPES is the method of choice to probe the spin-textured Fermi contour of TI surfaces.\nUltrathin epitaxial Bi 1\u0000xSbxfilms withx= 0:07, 0.1, 0.15, 0.21, 0.3, 0.4 and thicknesses\ndown to 2.5 nm were grown by molecular beam epitaxy (MBE) on clean 7\u00027reconstructed\nSi(111) surfaces. They all exhibit a non-trivial topological phase. Details of the growth [32]\nare quickly recalled in the Methods section where scanning transmission electron microscopy\n(STEM) and energy-dispersive X-ray spectroscopy (EDX) characterizations are also dis-\ncussed. We focus first on the 5 nm thick Bi 0:85Sb0:15sample. Fig. 1a shows its experimental\nFermi surface within the 2D surface Brillouin zone, as measured by ARPES. It is composed\nof three pockets labelled P1(the hexagonal electron pocket surrounding \u0000), and two elon-\ngatedP2hole andP3electron pockets along each of the six equivalent \u0000M directions (one\nhas been chosen as the kxaxis). Fig. 1b displays the energy dispersion along the \u0000M di-\nrection. The signature of two surface S1andS2states [17, 18] are visible in the bandgap.\nThe valence band state energy dispersion is also visible, from a series of confined states with\nenergy splitting increasing with reducing thickness [18]. The corresponding experimental\nSARPES polarization map for the \u001byspin-polarization component is given in Fig. 1c. For\nthis experiment, the incident photon energy is 20eV (5 meV resolution).\nOwing to the electron analyzer movable entrance optics, the \u001bxand\u001byspin polarizations\nwere measured on a large part of the surface first Brillouin zone (see Methods). On Fig. 1e,\nthe spin polarization is represented as a vector field. For clarity, the measured polarization is\nonly displayed for positions where the DOS is the largest. The norm is roughly proportional\ntothespin-resolvedDOS(s-DOS).Thevectorfieldissuperimposedoveracolormapyielding\nthe sum of all the signals measured by the spin detector, which is proportional to the\n4DOS at the Fermi level. For a clearer representation, the vector fields were averaged over\n0:02\u00020:02Å\u00002areas, corresponding to twenty data points. The experimental data reveals\nthe helical spin texture of the inner P1Fermi contour, the opposite spin polarization of the\nP2hole pocket, and finally, the same spin chirality for weaker P3pocket. We recover from\nexperiments the symmetry property imposing the orthogonality between the spin direction\nandtheverticalsymmetryplanes \u001b\u0000M\nVcontainingthe \u0000Mlines. Suchpropertyremainspartly\ntrue concerning the in-plane spin components for the \u0000K directions even if the corresponding\nvertical planes \u001b\u0000K\nVdo not represent perfect symmetry operators. The lack of symmetry for\n\u001b\u0000K\nVleads to the warping term responsible for the appearance of a \u001bzcomponent [2], not\npresently discussed.\nWe now compare the S1andS2surface state spin-texture with a TB model. Calculations\nare implemented by considering relaxed bulk lattice parameters following the Vegard’s law\n(refer to Ref. [18] and Methods). The electronic band structure and energy band dispersion\nalongM\u0000M are plotted in the Suppl. Mat. ??showing a good agreement with ARPES\ndata. The \u001bys-DOS at the Fermi surface, originating from S1andS2, and projected on\nthe top surface (first BL, n= 1) is displayed in Fig. 1d. The color code represents the \u001by-\nDOSN\u001by\nDOSprojected onto the first BL. Around \u0000, positive (negative) values are observed\nfor positive (negative) kx. The sign of this Rashba field is opposite around the M point\n(close to 0.8 Å\u00001). Two electron ( P1,P3) and one hole ( P2) pockets emerge, in very good\nagreementagainwith(S)ARPESdata. ByintroducingtheRashbasurfacepotentialimposed\nby the surface symmetry breaking, we are able to reproduce the spin-resolved map over the\nprojected 2D-Brillouin onto the first BiSb BL (see Methods).\nII. ULTRAFAST SPIN-CHARGE CONVERSION IN Bi 1\u0000xSbx/Co PROBED BY\nTHz-TDS EMISSION SPECTROSCOPY.\nWith the clear demonstration of the spin resolved surface states, we now discuss the\ndynamical spin-charge conversion in Bi 1\u0000xSbxcapped by a thin Co layer as a spin-injector.\nTHz emission spectroscopy in the time domain (THz-TDS) has recently emerged as a pow-\nerful spectroscopic technique to investigate ultrafast SCC in materials with strong SOC [24]\nand, in particular, in TI/FM structures [25–27, 31]. The thin FM layer is excited by a\nfemtosecond laser pump leading to ultrafast demagnetization. This generates both spin\n5-0.10.00.10.2\n-\n0.20.00.20.4\n-\n0.2-0.10.00.10.20.30.4-0.10.00.1\n-0.20.00.20.4-0.10.00.1\n0\n.00.20.40.60.8-0.4-0.3-0.2-0.10.0ky (A-1)(a)M\n(d)P\n1P2P3\nky (A-1)k\nx (A-1)(e)0°\n°P\n1P2P3ky (A-1)(c)°\nk\nx (A-1)Binding energy (eV)(b)0 °\n° S1S2ΓFigure 1.Spin-resolved surface states of a 5-nm thick Bi 0:85Sb0:15film grown on Si(111).\n(a) High-resolution ARPES map at the Fermi energy (integrated over 20 meV). (b) ARPES energy\ndispersion along the \u0000M direction ( kxdirection). As a guide to the eye, red and blue lines underline\ntheS1andS2surface states. All ARPES measurements were performed at 20 K. (c) \u001bypolarization\nDOS measured at the Fermi level, and (d) corresponding TB modelling of the s-DOS projected on\nthe first BL. The experiment is performed at room temperature, while the calculations are at T= 0.\nThe color bar indicates the spin polarization \u001bybetween -1 and +1 in (d). The color scale in (c) is\nproportional to the polarization with scale extrema of P\u0001S=\u00060:1. (e) Color map representing the\nmeasured ARPES intensity (arb. units, proportional to the DOS at the Fermi level) close to Fermi\nenergy integrated on 25 meV with arrows representing the measured spin polarization direction and\namplitude at room temperature. The different electron and hole pockets labeled P1,P2andP3in\n(d) are easily identified in the experimental measurements.\ndensity ^\u0016sand spin-currentJsdiffusing toward the BiSb/Co interface owing to the two spin\npopulations and mobilities introduced by the sp-band spin-splitting in Co [33]. The spin is\nafterwards converted into a transverse charge flow Jcon sub-picosecond timescales leading\nto a THz transient emission that is directly probed in the time-domain (see Refs. [24, 33]\nand Methods). Two SCC mechanisms can contribute: the IREE from the surface states and\nthe ISHE from the bulk. The THz electric field can be expressed as:\n6Figure 2. SCC and THz emission from Bi 1\u0000xSbx/Co bilayers. (a) THz time-\ntrace from Bi 0:85Sb0:15(5)/Co(2)/AlO x(3) for\u0006Bcompared to Co(2)/Pt(4) (grown on high\nresistivity Si substrates). THz phase reversal is a signature of SCC-mediated THz emis-\nsion. (b) Spectral components of the THz emission from Bi 0:85Sb0:15(5)/Co(2)/AlO x(3) and\nCo(2)/Pt(4) for +B. Inset) Normalized THz efficiency \u0011THzdependence on the azimuthal\nangle\u001efor Bi 0:85Sb0:15(5)/Co(2)/AlO x(3). (c) THz efficiency \u0011THzas a function of the\nBi0:79Sb0:21layer thickness (2.5, 5 and 15 nm) compared to Co(2)/Pt(4) (high resistivity Si sub-\nstrates). (d) THz signals from Bi 0:79Sb0:21(15)/Al(5)/Co(4)/AlO x(3), Bi 0:79Sb0:21(15)/AlO x(3),\nBi0:79Sb0:21(15)/Co(4)/AlO x(3)onBaF 2(Suppl.Mat. ??)andCo(2)/Pt(4). Timetracesareshifted\nin time for clarity. Inset) Fluorescence map obtained from a TEM cross-section for Bi 0:79Sb0:21\n(15)/Co(4)/AlOx(3)(brownframe)andBi 0:79Sb0:21(15)/Al(5)/Co(4)/AlOx(3)(greenframe)grown\non BaF 2for the elements and the color code given below the maps.\n78\n>>><\n>>>:EISHE\nTHz(\u0012)/@JISHE\nc\n@t/i!\u0012SHE\u0012\nJs\u0002M(\u0012)\njMj\u0013\nEIREE\nTHz(\u0012)/@JIREE\nc\n@t/i!\u0003IREE\u0012\n^\u0016s\u0002M(\u0012)\njMj\u0013 (1)\nwhere M(\u0012)is the Co layer magnetization vector, ^\u0016sis the spin-accumulation vector relaxing\nonS1andS2and!isthefrequency. M(\u0012)canbecontrolledbyanexternalin-planemagnetic\nfieldBat an angle \u0012from theyaxis. In the above equation, \u0012SHE=Jc=Jsis the spin Hall\nangle scaling the bulk ISHE-mediated SCC, whereas \u0003IREEis the Rashba-Edelstein length\nscaling the IREE from the surface states (refer to Methods and Suppl. Mat. ??).\nOn Fig. 2a, we report the THz signal acquired in reflection geometry from\nBi0:85Sb0:15(5)/Co(2)sample(numbersinparenthesisarethicknessesinnm)atroomtemper-\nature with a saturating in-plane magnetic field B'\u0006100mT insuring that the magnetiza-\ntion follows the external field within much less than a degree. It is compared to the emission\nfrom our optimized Co(2)/Pt(4) metallic ISHE-type sample. In both cases, we observe a\nshort picosecond THz pulse with some minor oscillations within a 3 ps wide envelope. The\nTHz signal phase changes sign when reversing the magnetic field, in full agreement with\nSCC-mediated THz emission (Suppl. Mat. ??). We note several differences: first, the am-\nplitude from the BiSb/Co layer is 1.5 times larger than the Co/Pt revealing the large SCC\nefficiency in BiSb. For the same magnetic field orientation, the BiSb/Co THz signal phase\nis opposite to the one found in the Co/Pt sample (Fig. 2a): this sign inversion is related to\nthe inverted layer stacking of the FM and non magnetic layer, giving an opposite phase and\nindicating that Pt and BiSb share the same conversion sign.\nWe also report in Fig. 2b the THz spectra (Fourier transform) obtained from Co(2)/Pt(4)\nand Bi 1\u0000xSbx(5)/Co(2), demonstrating a relative power enhancement by a factor '2:3\nfor the Bi 1\u0000xSbx/Co sample (field enhancement by a factor \u00021.5). The THz amplitude\nof Bi 1\u0000xSbx/Co is also shown to scale linearly with the pump fluence (Suppl. Mat. ??),\nmeasured up to a few tens of µJ.cm\u00002. The azimuthal angular dependence of the THz\nemission obtained by rotating the sample in the plane by an angle \u001e, while keeping fixed the\nmagnetic field, is shown in the inset of Fig. 2b. The emission is almost isotropic revealing a\npure SCC phenomenon with no evidence of non-linear optical effects such as shift or surge\ncurrent contributions ( e.g.photon drag, photogalvanic effects, etc.). This is in contrast\nwith the recent report given by Park et al.[30] where additional but small non-magnetic\n8contributions were observed. At normal incidence of the optical pump pulse, a pure isotropic\nTHz response vs.\u001eis expected from the linear response theory for both ISHE and IREE\nscenario (Suppl. Mat. ??) as discussed here. The role of the capping layer (metallic Au or\noxidizedAl,AlO x)hasbeencarefullyexcludedbycontrolexperiments(Suppl.Mat. ??). Two\nadditional control samples were grown: i) a Bi 0:79Sb0:21(15nm) sample free of ferromagnetic\nCo and only capped with naturally oxidized AlO x(3nm) almost emitting no THz radiation\n(Fig. 2d) and ii) a sample with a 5 nm thick Al metallic spacer inserted between BiSb and\nCo. The Al insertion strongly reduces the THz emission. It may be explained either by a\nlarger near-infrared (NIR) and THz absorption in Al, or by interfacial spin-loss or weakening\nof the surface SCC induced by the strong degradation of the interface quality. Indeed, TEM\npictures displayed in the inset of Fig. 2d reveal a strong intermixing of the Al interlayer that\nmay induce a loss of an efficient spin-injection and/or the alteration of the surface states\n(more details in Suppl. Mat. ??).\nTo get a better insight into the SCC mechanism, the thickness dependence of the THz\nefficiency,\u0011THz, is displayed in Fig. 2c for the Bi 0:79Sb0:21series (2.5, 5 and 15 nm thick\nlayers).\u0011THzrepresents the spin-injection and conversion efficiencies and is obtained by\nwithdrawing the NIR and THz absorptions in the heterostructure following the procedure\nproposed in Refs. [34–36] (Suppl. Mat. ??). Strikingly, \u0011THzfor BiSb remains constant for\nthe whole thickness series where the BiSb bandgap widens dramatically by several hundreds\nof meV as the thickness is reduced, as calculated previously [18]. Moreover, in agreement\nwith our ARPES results [18], this demonstrates the absence of coupling between the top and\nbottom surface states [37] down to 2.5 nm, unlike previously argued in the case of sputtered\nBiSb materials [28]. The characteristic evanescence length is indeed ultrashort, typically 2\nBL (0.8 nm) near \u0000as confirmed by density functional theory [38] for pure Bi and confirmed\nby our TB calculation (Suppl. Mat. ??). Such behavior is therefore more in favor of an\ninterfacial origin of the SCC and thus strongly hints towards IREE from the surface states\nrather than ISHE from bulk.\nFurthermore, when considering the possible contribution of bulkISHE, BiSb thickness\nhas to be compared with its spin diffusion length. It has been evaluated in bulk BiSb at\n\u0015sf'8 nm by Sharma et al.[28], which is much larger than 2.5 nm. If bulk states were\nto contribute viaISHE, the spin current would therefore flow across the whole layer depth,\nand upper and lower interfaces would contribute similarly to the SCC but with an opposite\n9Figure 3. Sb content dependence of the THz efficiency \u0011THzand calculated IREE re-\nsponse for \u001cs= 10fs.(a)\u0011THzfrom Bi 1\u0000xSbx(5)/Co(2-3)/AlO x(3) withx=0.07, 0.15 and 0.3.\n(b)\u0011THzfrom Bi 1\u0000xSbx(15)/Co(4)/Au(4-6) with x=0.1, 0.21, 0.3 and 0.4. (c-d) Values of the IREE\nlength \u0003IREE\nxyillustrating the conversion efficiency vs.energy\"for (c) Bi 0:79Sb0:21layers with thick-\nnesses2.5, 5and15nmand(d)Bi 1\u0000xSbx(5nm)forx=0.07, 0.15and0.3. Theinsetpresentsthefull\nIREE response outside the bandgap in the bulk valence and conduction bands for Bi 0:79Sb0:21(5nm).\nThe calculations have been performed on for a spin-relaxation time \u001cs=\u001c= 10fs after integration\nof the bands from to the top BL to the middle of the layer.\nsign. The net charge current would drop to zero, as tanh\u0010\ntTI\n\u0015sf\u0011\ntanh\u0010\ntTI\n2\u0015sf\u0011\n/t2\nTI\n2\u00152\nsf[39] for\nsmall TI thicknesses tTIwhen considering the multiple spin current reflections at the TI\ninterfaces [36]. This is in contrast with the thickness-independent SCC observed here, in the\nultrathin limit. We thus anticipate that the bulk states are hardly involved, and that the\n10surface states at the FM/TI interface are mainly responsible for a net charge current through\nIREE. This conclusion is also supported by i) the large increase of the surface state DOS\nof BiSb at the interface compared to the DOS bulk states as the BiSb thickness increases\n(see Suppl. Mat. ??) and by ii) the assumption of the preservation of BiSb surface states in\nexchange contact with Co as it seems to be the case.\nWe now focus on the THz efficiency \u0011THz vs.the Sb content x, plotted for\nBi1\u0000xSbx(5)/Co(2)/AlO x(3) in Fig. 3a and for Bi 1\u0000xSbx(15)/Co(4)/Au in Fig. 3b\n(Suppl. Mat. ??). In each series, a large \u0011THzis measured, comparable to that of Co/Pt\nwith a maximum for x= 0:3for the 15 nm series, i.e.at the limit of the topological phase\ndiagram [20]. Importantly, \u0011THzremains similar for all the samples for the tTI= 5nm series\n(Fig. 3a), highlighting again the prominent role of the surface states in the SCC. It also\nemphasizes the robustness of the IREE over a wide range of Sb content. The persistence of\na large signal outside the bulk topological window ( x= 0:4for 15 nm thickness, in Fig. 3b)\nindicates that bulk states may start to contribute by shunting a part of the spin current.\nIII. SCC AND IREE TENSOR FROM LINEAR RESPONSE THEORY\nWe now compare the enhanced THz emission observed on ultrathin BiSb films to that\nobtained from the linear response theory described by the IREE response, namely \u0003IREE\nxy\nmatching with the so-called inverse Edelstein length (see Suppl. Mat. ??and Methods).\nWe consider an extended formalism to the one recently developed to address the direct\nREE response as given in Refs. [40–42]. We evaluate here the intraband response to an\nongoing spin-current relaxing onto the Fermi surface generating, viaan out-of-equilibrium\nspin-density, a charge current according to: Jx\nc= \u0003IREE\nxyJy\ns;zwith \u0003IREE\nxy =P\nn;k\u0010\n\u001by\nnkvx\nnk\u001cs@fnk\n@\"\u0011\nP\nn;k\u0010@fnk\n@\"\u0011\nwhereyis the direction of the spin injected with a flow along zwhereasxis the direction of\nthein-planechargecurrent. Intheaboveexpression, nisthebandindex, vx\nnkthecorrespond-\ning band velocity along xand\u001csis the typical (spin) relaxation time assumed to be constant\nonto the Fermi surface. fnkis the occupation number for the band nand wavevector k\nwhereas\u0000@fnk\n@\"=NDOS(\";k;n)represents the local DOS in the kspace.\nCalculations are performed on bare BiSb bilayers (BLs) free of any Co overlayer as in-\nvestigated by SARPES. We show, on Fig. 3c, the energy dependence of \u0003IREE\nxyobtained\nfor Bi 0:79Sb0:21of different film thicknesses: 2.5, 5 and 15 nm (see Methods). The Fermi\n11level position corresponds to \"= 0. One observes that \u0003IREE\nxyis largely enhanced in the\nbandgap region in the (-0.2 - 0.2) eV window where the S2surface state and S1TSS are lo-\ncated, however without being able to differentiate their individual contributions (see Fig. ??\nfor the corresponding DOS in the Suppl. Mat. ??).\u0003IREE\nxyremains constant vs.BiSb film\nthickness in this window range, as experimentally observed in THz data of Fig. 2c. Last,\n\u0003IREE\nxyis maximum at the Fermi level, where it reaches values in the range of the equiva-\nlent SCC efficiency of Pt, product of the spin-Hall angle by the spin-diffusion length \u0015sfas\n\u0012SHE\u0002\u0015sf\u00190:2\u00000:3nm for the same spin or momentum relaxation time ( 10fs). Increas-\ning the spin-relaxation onto the surface states to 30 fs would yield \u0003IREE\nxy\u00191nm. \u0003IREE\nxy\nis displayed in Fig. 3c as a function of the Sb content xfor the 5 nm series. Although one\nobserves a slight increase of the SCC response from x=0.1 to 0.3 in the gap window (still in\nline with the presence of the surface states, and possibly indicating a volume contribution),\none may conclude that the SCC remains roughly constant at the vicinity of the Fermi en-\nergy (\"=\"F= 0). This is observed experimentally from THz-TDS measurements, and thus\nsuggests an interfacial IREE nature of the SCC, at least for the thinner films. One cannot\ntotally rule out a certain ISHE contribution arising from the bulk propagating bands asso-\nciated to a very short spin-diffusion length. Nonetheless, the strong impedance mismatch\nand subsequent spin-backflow at Co/BiSb interface would be strongly in disfavor of such\nscenario.\nCONCLUSIONS\nAlthough less investigated compared to other Bi-based families because of its modest\nbandgap, BiSb ultrathin films still exhibit very robust surface states, as revealed by our\nspin-resolved ARPES measurements. This is in part related to the confinement effects with-\nout being detrimental to the surface states. In particular, the presence of a topological\nsurface state S1in a wide Sb-composition and thickness ranges is clearly observed, dis-\nplaying helical spin texture at the Fermi surface with a specific opposite chirality between\nthe electron pocket near the \u0016\u0000point and the six hole pockets away from \u0016\u0000.Viaultrafast\nTHz emission spectroscopy, we demonstrate that the complex spin-texture gives rise to a\nvery efficient spin-charge conversion, resulting from the spin-injection from a Co overlayer\nexcited by an ultrashort laser pulse, mainly occurring viainverse Rashba-Edelstein effect\n12(IREE) owing to the strong localization of the surface states. Our results address the role\nof spin-textured hybridized Rashba-like surface states offering unprecedented SCC efficiency\ndespite the breaking of the TRS symmetry due to the local exchange interactions imposed\nby the magnetic contact. These results hold promise for efficient and integrated structures\nbased on BiSb. Future investigations will concern the dynamics of the spin relaxation onto\nthe BiSb surface states excited by ultrashort pulses.\nACKNOWLEDGMENTS\nWe thank T. Kampfrath and G. Bierhance (Freie Universität Berlin) for very fruitful\ndiscussions. This work was supported by a grant overseen by the French National Research\nAgency (ANR) as part of the “Generic Project Call - 2021” Programme (ANR-21-CE24-\n0011 TRAPIST), public grant overseen by the French National Research Agency (ANR)\nas part of the “Investissements d’Avenir” program (Labex NanoSaclay, reference: ANR-10-\nLABX-0035 SPICY) and the program ESR/EquipEx+ (Grant No. ANR-21-ESRE-0025).\nWe acknowledge financial support from the Horizon 2020 Framework Programme of the\nEuropean Commission under FET-Open Grant No. 863155 (s-Nebula) and Grant No. 64735\n(Extreme-IR).\nAUTHOR CONTRIBUTIONS\nJ.-M.G., H.J., P.LF., S.D. and A.L. conceived and designed the experiment. J.-M.G.\nsupervised the project. L.B., D.S., Ma.Mo., A.L., N.R., F.B., P.LF. and J.-M.G. grew\nthe samples and performed ARPES and SARPES experiments at Synchrotron Soleil. G.P,\nL.B. and A.L. performed TEM cross-section and SR-TEM microscopy. E.R, S.D., Ma.Mi.,\nJu.M., J.T. performed THz-TDS experiments at room temperature. H.J. performed TB\ncalculations. E.R., P.LF., S.D., A.L., R.L., N. R., H.J, J.-M.G. analysed the data. E.R.,\nP.LF., S.D., A.L., N. R., R.L., H.J., J.-M.G. wrote the paper.\n13METHODS\nMBE growth of Bi 1\u0000xSbx.The Bi 1\u0000xSbxsamples were grown by Molecular Beam\nEpitaxy (MBE). A Si(111) substrate was annealed in ultra-high vacuum (UHV) at 1370 K\nin order to obtain a 7 \u00027 surface reconstruction, observed by reflection high energy electron\ndiffraction (RHEED). The Bi 1\u0000xSbxalloy is grown by co-deposition from two Knudsen cells.\nEach cell flux is calibrated prior to the deposition using a quartz microbalance and the\nrelative deposition rates are directly used to estimate the Sb-concentration x. RHEED mea-\nsurements are performed throughout the deposition. Up to 5 nm, we observe a 2D-growth;\nfor thicker films, we stop the evaporation at 7 nm for a 10 min intermediate annealing at\n500 K before completing the film growth up to the targeted thickness. This procedure is\nfurther described in Ref. [18]. BiSb films with thicknesses from 2.5 to 15 nm and with var-\nious compositions (0.03< x<0.3) were grown using this method and their crystallographic\nquality was demonstrated by RHEED, X-ray-diffraction or Scanning Transmission Electron\nMicroscopy (STEM) [18].\nAll our samples were grown in the MBE chamber of the CASSIOPEE beamline installed\non the SOLEIL synchrotron. After their elaboration, they can be transferred in UHV to\nan ARPES (Angle-Resolved PhotoEmission Spectroscopy) or a SARPES (Spin-Resolved\nARPES) experiments where we can characterize their electronic structure. After photoe-\nmission measurements, some of the sample were transferred again into the MBE chamber\nfor Co electron beam deposition. These bilayer systems were used for SCC measurements\nusing THz emission spectroscopy.\nSEM - FIB milling and EDX. Lamellae for STEM observation were prepared from\nthe sample using Focused Ion Beam (FIB) ion milling and thinning. Prior to FIB ion\nmilling, the sample surface was coated with 50 nm of carbon to protect the surface from the\nplatinum mask deposited used for the ion milling process. Ion milling and thinning were\ncarried out in a FEI SCIOS dual-beam FIB-SEM. Initial etching was performed at 30 keV,\nand final polishing was performed at 5 keV. The lamellae were prepared following the two\ndifferent zone axis ( h110iandh112i) of the BaF 2substrate. All samples were observed in an\naberration-corrected FEI TITAN 200 TEM-STEM operating at 200 keV. The convergence\nhalf-angle of the probe was 17.6 mrad and the detection inner and outer half-angles for\nHAADF-STEM were 69 mrad and 200 mrad, respectively. All micrographs where 2048 by\n142048 pixels. The dwell time was 8 µs and the total acquisition time 41 s. EDX measurements\nwere performed in the Titan microscope featuring the Chemistem system, that uses a Bruker\nwindowless Super-X four-quadrant detector and has a collection angle of 0.8 sr.\n(Spin-)Angular-resolved photoemission spectroscopy. The photoemission experi-\nments were performed on the CASSIOPEE beamline installed on the SOLEIL storage ring\n(Saint-Aubin, France). The beamline hosts two endstations. A high-resolution ARPES end-\nstation, which was used in this work for the measurement of the Fermi surface and of the\nband dispersion, using 20 eV incident photons with a linear horizontal polarization. It is\nequipped with a Scienta R4000 electron analyzer. The photon spot size on the sample is\nof the order of 50 \u000250µm2and the overall kinetic energy resolution (taking into account\nboth the photon energy and the electron kinetic energy resolutions) was of the order of 10\nmeV. The second endstation is a spin-resolved ARPES experiment, where the beam size\nis around 300\u0002300µm2. It is equipped with a MBS A1-analyzer with a 2D detector for\nARPES measurements. Close to this 2D detector, a 1 \u00021 mm2hole collects photoelectron\nwith well-defined kinetic energy and momentum. They are sent into a spin manipulator\nable to orient any spin component along the magnetization axis of a FERRUM VLEED\nspin-detector, made of a Fe(100)-p(1 \u00021)O surface [43, 44] deposited on a W-substrate.\nThe spin polarization along the selected direction is proportional to the difference of\nthe two signals collected for opposite magnetizations of the Fe-oxide target. To reduce as\nmuch as possible the measurement asymmetries stemming from the instrument ( i.e., not\ndue to the spin polarization), four measurements per polarization direction are acquired,\nreversing both the ferrum magnetization direction, and the electron spin direction. This four\nmeasurements are combined into a geometrical average. The polarization is then determined\nbyP=S\u00001(I\u001b\n+\u0000I\u001b\n\u0000)=(I\u001b\n++I\u001b\n\u0000).\nThe 1\u00021 mm2hole introduces an integration on both the kinetic energy and the wave\nvector. For the kinetic energy, it corresponds to 0.23% of the used pass energy (10 eV in our\ncase), so around 23 meV. Convoluted with the energy resolution of the analyzer (10 meV for\nthis pass energy and an entrance slit of 400 µm), it gives an overall kinetic energy resolution\nof 25 meV. For the wave vector, the 1 mm aperture corresponds to an integration on 4%\nof the total (30\u000e) angular range, which gives 1.2\u000e. At 20 eV photon energy, for electrons at\nthe Fermi level, this gives a k-resolution of around 0.048 Å\u00001. This explains the relatively\n15broad features of the SARPES in Fig. 1c,e compared to Fig. 1a.\nThe analyzer optics is movable and can collect electrons in a large 2D (30\u000e\u000230\u000e) angular\nrange. To map the spin texture at the Fermi level, the analyzer is set to the appropriate\nkinetic energy while the optics is moved by 0.2\u000estep along two X and Y perpendicular\ndirections. The two in-plane spin components are measured at each step.\nTHz emission spectroscopy. Ultrafast near-infrared (NIR) pulses ( '100 fs) centered\nat\u0015NIR=810 nm are derived from a Ti:Sapphire oscillator to photo-excite the spin carriers\ndirectly from the front surface (Co side). Average powers of up to 600 mW were used with a\nrepetition rate of 80 MHz (the energy per pulse is around 3 nJ). The typical laser spot size\non the sample was about 200 µm\u0002200µm. The optical pump is initially linearly polarized\nand irradiate the TI/FM heterostructure under normal incidence. The generated THz pulses\nwere also collected from the front surface of the samples ( i.e., reflection geometry) using a\nset of parabolic mirrors of 150 and 75 mm focal length to focus on the detection crystal. The\nsamples were placed on a mount with a small magnetic field (around 100 mT) in the plane of\nthe thin films. Both the sample orientation (angle \u001e) and the in-plane magnetic field (angle\n\u0012) can be independently rotated in the sample plane. Standard electro-optic sampling was\nused to detect the electric field of the THz pulses, using a 500 µm-thickh1 1 0iZnTe crystal.\nA chopper was placed at the focal point between the second and third parabolic mirror to\nmodulate the THz beam at 6 kHz for heterodyne lock-in detection. A mechanical delay line\nwas used to sample the THz ultrafast pulse as a function of time. The THz propagation\npath was enclosed in a dry-atmosphere purged chamber (typically <2% humidity) to reduce\nwater absorption of the THz radiation.\nTight-binding calculations of Bi 1\u0000xSbxmultilayers. We have developed a tight\nbinding (TB) model in order to describe the Bi 1\u0000xSbxelectronic band structure as well as\ntheir surface topological properties [45]. This approach is indeed well suited for TI and gives\na fair description of the surface state spin texture in close agreement with the one derived\nfrom Density Functional Theory (DFT) developed for pure Bi surfaces [21, 22, 46]. The\nrhombohedral A7 structure is described by two atoms per unit cell, forming then a bilayer\n(BL) of thickness of about 0.4 nm. Bi 1\u0000xSbxslabs are obtained by stacking the BL along\nthe (111) direction ( zaxis) with two different plane-to-plane distances. We constructed our\nHamiltonian on the basis of the work of Ref. [45] using the generalization of the sp3TB-\n16model Hamiltonian proposed for bulk Bi and Sb crystals [47], adapted to Bi 1\u0000xSbxalloys [15]\nand complemented by the introduction of additional surface potential terms when dealing\nwith thin layers (treatment in slabs) [45, 48, 49]. In particular, the hopping parameters for\nthe BiSb alloys are obtained by using the virtual crystal approximation (VCA) according\nto [15]:\nVBiSb\nC=x VSb\nC+ (1\u0000x2)VBi\nC (2)\nwherexis the antimony content and VSb\nCandVBi\nCare the respective hopping parameters of\nSb and Bi taken from Ref. [47]. One notes ^\u001b\u000bthe spin index on each atom where \u000bstands\nfor the directional index. The hopping terms among the atomic orbitals are decomposed\ninto inter- and intra-BL hopping terms. The inter-BL off-diagonal hopping term between\natoms (plane) 1 and atoms (plane) 2 consists of the nearest-neighbor coupling in the bulk\nBiSb Hamiltonian, whereas the intra-BL hopping term consists of two parts which represents\nrespectively the third and second nearest neighbor contributions. We considered the overall\nTB Hamiltonian according to:\n^H=^H0+^HSO+^H\r (3)\nwhere ^H0= \u0006j\u0017\ni\u0016ji\u0016iVj\u0017\ni\u0016hj\u0017jrepresents the hopping Hamiltonian ( i,jare the atomic posi-\ntions,\u0016,\u0017are the orbitals), ^HSO=~\n4m2c2\u0010\u0000 !rV(r)\u0002^p\u0011\n\u0001^\u001bthe SOC term and ^H\rthe Rashba\nsurface potential induced by the deformation of the surface orbitals due to the local electric\nfield. Indeed, due to the symmetry breaking at the surface, a Rashba SOC term must be\ntaken into account in the Hamiltonian at the two surface planes. We model such effect for\nthesp3basis by using the approach of Ast and Gierz [49] for ^H\rconsidering two additional\nsurface hopping terms \rspand\rppacting respectively between the s\u0000pzandpx\u0000pz(or\npy\u0000pz) surface orbitals. We thus add the ^H\rHamiltonian term of the form:\n^H\r=8\n>>>>><\n>>>>>:\rsp (i;i)\u0011(s;pz)\n\u0006\rsp1 (i;j)\u0011(s;pz)\n\u0006\rppcos(\u0012) (i;j)\u0011(px;pz)\n\u0006\rppsin(\u0012) (i;j)\u0011(py;pz)(4)\n17where the + (-) sign corresponds to the uppermost (lowermost) atomic plane and \u0012is the\nangle between the direction joining the two atoms considered and the x-direction. We then\nrestrained ourselves to the in-plane surface hopping as for a pure 2D system. The best\nagreement with ARPES results is found by adding, as proposed in Ref. [49], additional\non-sites\u0000pzcoupling\rsp=\u00000:2eV, and surface hopping terms \rsp1=0.3 eV and \rpp=-\n0.6 eV forx=0.15 slightly departing from the values given for pure Bi, i.e.\rsp=0.45 eV and\n\rpp=-0.27 eV [45], with opposite sign for the top and bottom surfaces due to the opposite\ndirection of the potential gradient. We emphasize that this surface terms are required to\ncorrectly reproduce the surface state dispersion as observed by ARPES experiments. The\nsize of the Hamiltonian ^H(kx;ky)to diagonalize is 16N\u000216NwhereNis the number of\nbilayers (BLs). Once the Green function of the multilayer system is defined as:\n^G(\";kx;ky) =h\n\"+i\u000e\u0000^H(kx;ky)i\u00001\n(5)\nthe partial density of state (DOS) NDOS(\")vs.the energy \"equalsN(n;\") =\n\u0000(1=\u0019)ImTr[^G(\";n;kx;ky)]whereas the spin density of states (spin-DOS) with spin along\nthe\u000bdirection is s\u000b(\") = (1=\u0019)ImTr[^\u001b\u000b^G(\";kx;ky)].\u000eis the typical energy broadening ( '\n10 meV) and the trace ( Tr) is applied over the considered sp3orbitals on a given BL index\n(n2[1;N]). The energy zero ( \"=0) refers to the Fermi level position.\nThe modelling of IREE were performed by TB method. We have summed the contribu-\ntions from the different Fermi surface pockets within the 2D-BZ after i) having introduced\nthe Rashba potentials at the BiSb surfaces, required to match both the TSS electronic dis-\npersion and SML measured in (S)ARPES experiments [18]; and after ii) having considered\na same (spin) relaxation time of \u001cs=\u001c0= 10fs involved in the intraband transitions. The\ninverse Edelstein length has been then evaluated on the whole Fermi surface according to the\nfollowing expression \u0003xy(\") =P\nnR\nd2kh nkj^vx\u001c0j nkih nkj^\u001byj nkiNDOS(\";k;n)\u001cP\nnR\nd2kNDOS(\";k;n)(see Suppl. Mat. ??).\nThe typical value of \u001cs= 10fs corresponds to an energy broadening \u0000 =~=(2\u001cs)'\n50meV) and a Fermi velocity of about 5\u0002105m.s\u00001like extracted from ARPES data. The\ntrace on\u0014xyis performed by summing the BL contributions from the top surface down to\nthe middle of the BL, typically matching the typical finite TSS evanescence or extension\n18length.\n[1] M.Z.HasanandC.L.Kane, “ Colloquium : topologicalinsulators,” Reviews ofModernPhysics,\nvol. 82, pp. 3045–3067, Nov. 2010.\n[2] Y. Ando, “Topological Insulator Materials,” JournalofthePhysical SocietyofJapan, vol. 82,\np. 102001, Oct. 2013.\n[3] X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Reviews ofModern\nPhysics, vol. 83, pp. 1057–1110, Oct. 2011.\n[4] Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak, K. Segawa, Y. Ando, and E. Saitoh,\n“Spin-Electricity Conversion Induced by Spin Injection into Topological Insulators,” Physical\nReviewLetters, vol. 113, p. 196601, Nov. 2014.\n[5] Y. Ando and M. Shiraishi, “Spin to Charge Interconversion Phenomena in the Interface and\nSurface States,” JournalofthePhysical SocietyofJapan, vol. 86, p. 011001, Jan. 2017.\n[6] W. Han, Y. Otani, and S. Maekawa, “Quantum materials for spin and charge conversion,” npj\nQuantum Materials, vol. 3, p. 27, Dec. 2018.\n[7] V. Edelstein, “Spin polarization of conduction electrons induced by electric current in two-\ndimensional asymmetric electron systems,” SolidStateCommunications, vol. 73, pp. 233–235,\nJan. 1990.\n[8] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi,\nA. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, “Spin-transfer torque generated by a\ntopological insulator,” Nature, vol. 511, pp. 449–451, Jul 2014.\n[9] S. Zhang and A. Fert, “Conversion between spin and charge currents with topological insula-\ntors,”Physical ReviewB, vol. 94, p. 184423, Nov. 2016.\n[10] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera,\nB. Rodmacq, A. Schuhl, and P. Gambardella, “Perpendicular switching of a single ferromag-\nnetic layer induced by in-plane current injection,” Nature, vol. 476, pp. 189–193, Aug. 2011.\n[11] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, “Current-Induced\nSwitching of Perpendicularly Magnetized Magnetic Layers Using Spin Torque from the Spin\nHall Effect,” Physical ReviewLetters, vol. 109, p. 096602, Aug. 2012.\n[12] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, “Spin-Torque\n19Switching with the Giant Spin Hall Effect of Tantalum,” Science, vol. 336, pp. 555–558, May\n2012.\n[13] K. Kondou, R. Yoshimi, A. Tsukazaki, Y. Fukuma, J. Matsuno, K. S. Takahashi, M. Kawasaki,\nY. Tokura, and Y. Otani, “Fermi-level-dependent charge-to-spin current conversion by Dirac\nsurface states of topological insulators,” NaturePhysics, vol. 12, pp. 1027–1031, Nov 2016.\n[14] N. H. D. Khang, Y. Ueda, and P. N. Hai, “A conductive topological insulator with large\nspin Hall effect for ultralow power spin–orbit torque switching,” NatureMaterials, vol. 17,\npp. 808–813, Sept. 2018.\n[15] J. C. Y. Teo, L. Fu, and C. L. Kane, “Surface states and topological invariants in three-\ndimensional topological insulators: Application to Bi 1\u0000xSbx,”Phys.Rev.B, vol. 78, p. 045426,\nJul 2008.\n[16] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “A topological\nDirac insulator in a quantum spin Hall phase,” Nature, vol. 452, pp. 970–974, Apr. 2008.\n[17] H. M. Benia, C. Straßer, K. Kern, and C. R. Ast, “Surface band structure of Bi 1\u0000xSbx(111),”\nPhysical ReviewB, vol. 91, p. 161406, Apr. 2015.\n[18] L. Baringthon, T. H. Dang, H. Jaffrès, N. Reyren, J.-M. George, M. Morassi, G. Patriarche,\nA. Lemaitre, F. Bertran, and P. Le Fèvre, “Topological surface states in ultrathin Bi 1\u0000xSbx\nlayers,”Physical ReviewMaterials, vol. 6, p. 074204, July 2022.\n[19] S. Ito, M. Arita, J. Haruyama, B. Feng, W.-C. Chen, H. Namatame, M. Taniguchi, C.-M.\nCheng, G. Bian, S.-J. Tang, T.-C. Chiang, O. Sugino, F. Komori, and I. Matsuda, “Surface-\nstate Coulomb repulsion accelerates a metal-insulator transition in topological semimetal\nnanofilms,” ScienceAdvances, vol. 6, no. 12, p. eaaz5015, 2020.\n[20] B. Lenoir, H. Scherrer, and T. Caillat, “An overview of recent developments for BiSb Alloys,”\ninSemiconductors andSemimetals, vol. 69, pp. 101–137, Elsevier, 2001.\n[21] T. Hirahara, T. Nagao, I. Matsuda, G. Bihlmayer, E. V. Chulkov, Y. M. Koroteev, P. M.\nEchenique, M. Saito, and S. Hasegawa, “Role of Spin-Orbit Coupling and Hybridization Effects\nin the Electronic Structure of Ultrathin Bi Films,” Phys.Rev.Lett., vol. 97, p. 146803, Oct\n2006.\n[22] Y. M. Koroteev, G. Bihlmayer, E. V. Chulkov, and S. Blügel, “First-principles investigation\nof structural and electronic properties of ultrathin Bi films,” Phys.Rev.B, vol. 77, p. 045428,\nJan 2008.\n20[23] I. Aguilera, H.-J. Kim, C. Friedrich, G. Bihlmayer, and S. Blügel, “ Z2topology of bismuth,”\nPhysical ReviewMaterials, vol. 5, p. L091201, Sept. 2021.\n[24] T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. Freimuth,\nA. Kronenberg, J. Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov, P. M. Oppeneer, M. Jour-\ndan, G. Jakob, D. Turchinovich, L. M. Hayden, M. Wolf, M. Münzenberg, M. Kläui, and\nT. Kampfrath, “Efficient metallic spintronic emitters of ultrabroadband terahertz radiation,”\nNaturePhotonics, vol. 10, pp. 483–488, Jul 2016.\n[25] X. Wang, L. Cheng, D. Zhu, Y. Wu, M. Chen, Y. Wang, D. Zhao, C. B. Boothroyd, Y. M. Lam,\nJ.-X. Zhu, M. Battiato, J. C. W. Song, H. Yang, and E. E. M. Chia, “Ultrafast Spin-to-Charge\nConversion at the Surface of Topological Insulator Thin Films,” Advanced Materials, vol. 30,\nno. 52, p. 1802356, 2018.\n[26] M. Tong, Y. Hu, Z. Wang, T. Zhou, X. Xie, X. Cheng, and T. Jiang, “Enhanced Terahertz\nRadiation by Efficient Spin-to-Charge Conversion in Rashba-Mediated Dirac Surface States,”\nNanoLetters, vol. 21, no. 1, pp. 60–67, 2021. PMID: 33331788.\n[27] X.Chen,H.Wang,C.Wang,C.Ouyang,G.Wei,T.Nie,W.Zhao,J.Miao,Y.Li,L.Wang,and\nX. Wu, “Efficient Generation and Arbitrary Manipulation of Chiral Terahertz Waves Emitted\nfrom Bi 2Te3–Fe Heterostructures,” Advanced Photonics Research, vol. 2, no. 4, p. 2000099,\n2021.\n[28] V. Sharma, W. Wu, P. Bajracharya, D. Q. To, A. Johnson, A. Janotti, G. W. Bryant, L. Gund-\nlach, M. B. Jungfleisch, and R. C. Budhani, “Light and microwave driven spin pumping across\nFeGaB–BiSb interface,” Phys.Rev.Materials, vol. 5, p. 124410, Dec 2021.\n[29] H. Park, K. Jeong, I. Maeng, K. I. Sim, S. Pathak, J. Kim, S.-B. Hong, T. S. Jung, C. Kang,\nJ. H. Kim, J. Hong, and M.-H. Cho, “Enhanced Spin-to-Charge Conversion Efficiency in\nUltrathin Bi 2Se3Observed by Spintronic Terahertz Spectroscopy,” ACSApplied Materials &\nInterfaces, vol. 13, no. 19, pp. 23153–23160, 2021. PMID: 33945256.\n[30] H. Park, S. Rho, J. Kim, H. Kim, D. Kim, C. Kang, and M. Cho, “Topological Surface-\nDominated Spintronic THz Emission in Topologically Nontrivial Bi 1\u0000xSbxFilms,”Advanced\nScience, p. 2200948, May 2022.\n[31] E. Rongione, S. Fragkos, L. Baringthon, J. Hawecker, E. Xenogiannopoulou, P. Tsipas,\nC. Song, M. Mičica, J. Mangeney, J. Tignon, T. Boulier, N. Reyren, R. Lebrun, J.-M. George,\nP. Le Fèvre, S. Dhillon, A. Dimoulas, and H. Jaffrès, “Ultrafast Spin-Charge Conversion at\n21SnBi 2Te4/Co Topological Insulator Interfaces Probed by Terahertz Emission Spectroscopy,”\nAdvanced OpticalMaterials, vol. 10, no. 7, p. 2102061, 2022.\n[32] L.Baringthon, Élaboration et caractérisation de phases topologiques de l’alliage Bi 1\u0000xSbxpour la conversion spin charge.\nPhD thesis, Université Paris-Saclay, 2022.\n[33] T.H.Dang, J.Hawecker, E.Rongione, G.BaezFlores, D.Q.To, J.C.Rojas-Sanchez, H.Nong,\nJ. Mangeney, J. Tignon, F. Godel, S. Collin, P. Seneor, M. Bibes, A. Fert, M. Anane, J.-M.\nGeorge, L. Vila, M. Cosset-Cheneau, D. Dolfi, R. Lebrun, P. Bortolotti, K. Belashchenko,\nS. Dhillon, and H. Jaffrès, “Ultrafast spin-currents and charge conversion at 3 d-5dinter-\nfaces probed by time-domain terahertz spectroscopy,” Applied PhysicsReviews, vol. 7, no. 4,\np. 041409, 2020.\n[34] O.Gueckstock, L.Nádvorník, M.Gradhand, T.S.Seifert, G.Bierhance, R.Rouzegar, M.Wolf,\nM. Vafaee, J. Cramer, M. A. Syskaki, G. Woltersdorf, I. Mertig, G. Jakob, M. Kläui, and\nT.Kampfrath,“TerahertzSpin-to-ChargeConversionbyInterfacialSkewScatteringinMetallic\nBilayers,” Advanced Materials, vol. 33, no. 9, p. 2006281, 2021.\n[35] M. Meinert, B. Gliniors, O. Gueckstock, T. S. Seifert, L. Liensberger, M. Weiler, S. Wim-\nmer, H. Ebert, and T. Kampfrath, “High-Throughput Techniques for Measuring the Spin Hall\nEffect,”Phys.Rev.Applied, vol. 14, p. 064011, Dec 2020.\n[36] J. Hawecker, E. Rongione, A. Markou, S. Krishnia, F. Godel, S. Collin, R. Lebrun, J. Tignon,\nJ. Mangeney, T. Boulier, J.-M. George, C. Felser, H. Jaffrès, and S. Dhillon, “Spintronic THz\nemitters based on transition metals and semi-metals/Pt multilayers,” AppliedPhysicsLetters,\nvol. 120, no. 12, p. 122406, 2022.\n[37] Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai,\nW.-Y. Shan, S.-Q. Shen, Q. Niu, X.-L. Qi, S.-C. Zhang, X.-C. Ma, and Q.-K. Xue, “Crossover\nof the three-dimensional topological insulator Bi 2Se3to the two-dimensional limit,” Nature\nPhysics, vol. 6, pp. 584–588, Aug. 2010.\n[38] H. Ishida, “Decay length of surface-state wave functions on Bi(111),” Journal ofPhysics:\nCondensed Matter, vol. 29, p. 015002, nov 2016.\n[39] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, “Spin Hall effects,”\nRev.Mod.Phys., vol. 87, pp. 1213–1260, Oct 2015.\n[40] J.-C. Rojas-Sánchez and A. Fert, “Compared Efficiencies of Conversions between Charge and\nSpin Current by Spin-Orbit Interactions in Two- and Three-Dimensional Systems,” Phys.Rev.\n22Appl., vol. 11, p. 054049, May 2019.\n[41] A. Johansson, B. Göbel, J. Henk, M. Bibes, and I. Mertig, “Spin and orbital Edelstein effects\nin a two-dimensional electron gas: Theory and application to SrTiO 3interfaces,” Phys.Rev.\nRes., vol. 3, p. 013275, Mar 2021.\n[42] A. Roy, F. T. Cerasoli, A. Jayaraj, K. Tenzin, M. B. Nardelli, and J. Sławińska, “Long-range\ncurrent-induced spin accumulation in chiral crystals,” npjComputational Materials, vol. 8,\np. 243, Nov 2022.\n[43] R. Bertacco and F. Ciccacci, “Oxygen-induced enhancement of the spin-dependent effects in\nelectron spectroscopies of Fe(001),” Phys.Rev.B, vol. 59, pp. 4207–4210, Feb 1999.\n[44] M. Escher, N. B. Weber, M. Merkel, L. Plucinski, and C. M. Schneider, “FERRUM: A New\nHighly Efficient Spin Detector for Electron Spectroscopy,” e-Journal ofSurfaceScienceand\nNanotechnology, vol. 9, pp. 340–343, 2011.\n[45] K. Saito, H. Sawahata, T. Komine, and T. Aono, “Tight-binding theory of surface spin states\non bismuth thin films,” Phys.Rev.B, vol. 93, p. 041301, Jan 2016.\n[46] Y. M. Koroteev, G. Bihlmayer, J. E. Gayone, E. V. Chulkov, S. Blügel, P. M. Echenique, and\nP. Hofmann, “Strong Spin-Orbit Splitting on Bi Surfaces,” Phys.Rev.Lett., vol. 93, p. 046403,\nJul 2004.\n[47] Y. Liu and R. E. Allen, “Electronic structure of the semimetals Bi and Sb,” Phys.Rev.B,\nvol. 52, pp. 1566–1577, Jul 1995.\n[48] L.PetersenandP.Hedegård,“Asimpletight-bindingmodelofspin–orbitsplittingof sp-derived\nsurface states,” SurfaceScience, vol. 459, no. 1, pp. 49–56, 2000.\n[49] C. R. Ast and I. Gierz, “ sp-band tight-binding model for the Bychkov-Rashba effect in a two-\ndimensional electron system including nearest-neighbor contributions from an electric field,”\nPhys.Rev.B, vol. 86, p. 085105, Aug 2012.\n23"    },    {        "title": "0903.4406v2.p_type_Bi2Se3_for_topological_insulator_and_low_temperature_thermoelectric_applications.pdf",        "content": " \n1 \n p-type Bi 2Se3 for topological insula tor and low temperature \nthermoelectric applications  \n \nY.S. Hor1, A. Richardella2, P. Roushan2, Y. Xia2, J.G. Checkelsky2,  \nA. Yazdani2, M.Z. Hasan2, N.P. Ong2, and R.J. Cava1 \n \n1Department of Chemistry, Princeton University \n2Department of Physics, Princeton University \n \nAbstract \n \nThe growth and elementary properties of p-type Bi 2Se3 single crystals are reported. Based \non a hypothesis about the defect chemistry of Bi 2Se3, the p-type behavior has been induced \nthrough low level substitutions (1% or less) of  Ca for Bi. Scanning tunneling microscopy is \nemployed to image the defects and establish their charge. Tunneling and angle resolved \nphotoemission spectra show that the Fermi level has been lowered into the valence band by \nabout 400 meV in Bi 1.98Ca0.02Se3 relative to the n-type material. p-type single crystals with ab \nplane Seebeck coefficients of + 180 µV/K at room temperature ar e reported. These crystals show \na giant anomalous peak in the Seebeck coeffi cient at low temperatures, reaching +120 µVK-1 at 7 \nK, giving them a high thermoelectric power fact or at low temperatures. In addition to its \ninteresting thermoelectric properties, p-type Bi 2Se3 is of substantial interest for studies of \ntechnologies and phenomena proposed for topological insulators.   \n  \n2 \n Introduction \nBi2Se3 is one of the binary e nd-members of the (Bi,Sb) 2(Te,Se) 3 family of thermoelectric \nmaterials. Decades of work in the chemistry, physics, and processing of these materials has led to complex formulations of compounds and microstr uctures optimized for use as thermoelectrics \nunder various conditions (see e.g. refs. 1 and 2). Due to the superiority of Bi\n2Te3-based materials \nfor these applications, Bi 2Se3, while well studied, has not been subject to the same degree of \nintensive research as has its heavier mass analog. One of the major issues for Bi 2Se3-based \nthermoelectrics has been the di fficulty in making the material p-type. Unlike Bi 2Te3, which can \nsimply be made n- or p-type through variation of the Bi:T e ratio, the defect chemistry in Bi 2Se3 \nis dominated by charged selenium vacancies, wh ich act as electron donors [3], resulting in n-type \nbehavior for virtually all of the reported transport st udies (see e.g. refs. 3-11). p-type behavior for \npure Bi 2Se3 was reported in an early study [9], but neve r since. Modern studies have shown that \np-type behavior is possible when beginning with an n-type host material in the (Bi 2-xSbx)Se 3 \nsolid solution with x = 0.4, and then doping that composition with small amounts of Pb to create \na quaternary p-type material [10]. \nThe present study is motivated by the desire to find a chemically less complex p-type \nBi2Se3-based material to address the recent emergence of Bi 2Se3 as one of the prime candidates \nfor the study of topological su rface states (see e.g. refs. 12-16), and for thermoelectric \napplications. The character and stab ility of the surface states in Bi 2Se3 at room temperature has \nmotivated the suggestion that they may be usef ul for quantum computi ng applications if the \nFermi level can be lowered to the Dirac point through p-type doping of the normally n-type \ncompound [15,17,18]. Therefore high quality p-type crystals are important for both fundamental \nand applied research on Bi 2Se3.   \n3 \n Unlike the case for Bi 2Te3, where the chemical similarity of Bi and Te leads to antisite \ndefects as the primary source of carrier doping in binary compounds , there is little tendency for \nBi/Se mixing in Bi 2Se3, and the primary structural defect giving rise to electron doping is doubly \ncharged selenium vacancies V Se••  [3]: \nSeSe → VSe•• + Se(g) + 2e ′   \nOrdinarily, one compensates for the presence of  donors in this chemical family through doping \nwith Pb on the Bi site, as Pb has one fewer elect ron than Bi. This substitution does not, however, \nlead to the formation of p-type material for Bi 2Se3. We hypothesize that this  is due to the fact \nthat Pb in Bi 2Se3 is ambipolar, much as is the case for Cu  [11]. This indicates that a more ionic \nsubstitution on the Bi site may be required for hole-doping of Bi 2Se3, suggesting the use of Ca \nsubstitution, with the defect reaction:  2Ca     →     2Ca\nBi′ + 2h• \n        Bi 2Se3 \nwith Ca substitution for Bi creating a negatively charged defect (Ca Bi′) that in turn generates \nholes (h•) to compensate the electrons created by the Se vacancies. Our results, described below, \nsupport this scenario as a good representation of the defect chemistry in this compound. \nExperimental   \nThe single crystals of Bi 2-xCaxSe3 were grown via a process of two-step melting, starting \nwith mixtures of high purity elements (B i, 99.999 %, Se, 99.999 %, Ca, 99.8 %). First, \nstoichiometric mixtures of Bi and Se were me lted in evacuated quartz ampoules at 800 ºC for 16 \nhours. The melts were then stirred before bein g allowed to solidify by air-quenching to room \ntemperature. Second, the stoichiometric amount of  Ca was added in the form of pieces, using \ncare to avoid direct contact of th e added Ca with the quartz. The materials were then heated in \nevacuated quartz ampoules at 400 ºC for 16 hours followed by 800 ºC for a day. The crystal  \n4 \n growth process involved cooling from 800 to  550 ºC over a period  of 24 hours and then \nannealing at 550 ºC for 3 days. The crystals were then furnace cooled to room temperature. They \nwere cleaved very easily along the basal plane,  and were cut into approximately 1.0 × 1.0 × 6.0 \nmm3 rectangular bar samples for the thermal and el ectronic transport measurements. Resistivity, \nSeebeck coefficient, and ther mal conductivity measurements were performed in a Quantum \nDesign PPMS, using the standard four-probe t echnique, with silver paste cured at room \ntemperature used for the contacts. Hall Effect m easurements to determine carrier concentrations \nwere performed at 1.5 K in a home-built apparatus.  In all cases, the elect ric- and thermal-currents \nwere applied in the basal plane ( ab plane in the hexagonal setting of the rhombohedral cell) of \nthe crystals. In order to probe  the electronic states of nati ve and Ca related defects, Bi 2Se3 and \nBi1.98Ca0.02Se3 samples were studied in a cryogenic scan ning tunneling microscope (STM) at 4.2 \nK. The samples were cleaved in situ in ultrahigh vacuum to expose a pristine surface. The \nsurfaces of the p-type crystals are fully stable in th e STM experiments. High-resolution angle \nresolved photoemission spectrosc opy (ARPES) measurements we re performed using 22-40 eV \nphotons at Beamline-12 of the A dvanced Light Source (Berkeley Laboratory) and on beamline 5 \nat the Stanford Synchrotron Radiation Laborat ory. The energy and momentum resolutions were \n15 meV and 2 % of the surface Brillouin Zone, re spectively, obtained using a Scienta analyzer. \nThe samples were cleaved at 10 K under pressures of less than 5 × 10-11 torr, resulting in shiny \nflat surfaces. The spectra employed are those clos est to the time of cleavage (about 10 minutes), \nwhich are the most repr esentative of the bulk.  \nResults  \n The rhombohedral crystal structure of Bi 2Se3 consists of hexagonal planes of Bi and Se \nstacked on top of each other along the [001] crysta llographic direction (hexagonal setting), with  \n5 \n the atomic order: Se(1)-Bi-Se(2)-Bi-Se(1), wh ere (1) and (2) are refer to different lattice \npositions [19]. The unit cell consists of three of  these units stacked on top of each other with \nweak van-der-Waals bonds between Se(1)-Se(1) layers, making the (001) plane the natural \ncleavage plane. Scanning tunneli ng spectroscopy (dI/dV) was perf ormed to measure the density \nof states. The spectra of Bi 2Se3 were n-type, as expected, due to th e presence of Se vacancies \nwith the Fermi energy near the conduction band, c onsistent with previous  STM results [20]. In \ncontrast, a clear shift in the position of the Ferm i level towards the valance band occurs in the \nBi1.98Ca0.02Se3 sample, as shown in Fig. 1(a) . The finite density of states  inside the gap is due to \nsurface states, for which novel topological properties have been predicted [12-18]. \nWe were able to identify vari ous defects and the si gn of their charge state from the STM \ntopographies of the filled states and unoccupied  states. The STM topogr aphies of the native \nBi2Se3 (001) surface are dominated by one type of defect, which a ppears as a bright triangular \nprotrusion in the topographies of the unoccupied stat es (Fig. 1(b)). On aver age, these defects are \nabout 40 Å across, but vary in size between def ects indicating that they are located in various \nlayers beneath the surface. Given that no othe r defects were observed, we attribute these \ntriangular defects to Se vacancies. Figs. 1(c) a nd (d) show the topography of the unoccupied and \noccupied states of the Ca-doped Bi 2Se3, respectively. Comparison between 1(b) and (c) makes it \nclear that the density of tria ngular defects has been reduced significantly in the Ca-doped \nsamples. In addition, the STM topography of the Bi 1.98Ca0.02Se3 (001) surface shows distinct \ndefects that were not present in Bi 2Se3 samples. In topographic imag es of the empty states, the \nshape of these three-fold symmetric defects resemb les a cloverleaf (Fig. 1(c)). Based on their two \ndistinct spatial extents we conclude that they  are located at two di fferent crystallographic \npositions; most likely the smaller one is located in a layer nearer the surface and other one in a  \n6 \n layer deeper beneath the surface. Because thes e defects occur only in Ca-doped samples, we \nidentify them as Ca-related defects.    The charge state of the defects can be inferr ed by observing the bending of the host bands \ncaused by the coulomb field surrounding a charge d defect. A positively charged defect lowers \nthe electronic energy level in its neighboring re gion, leading to a depression area in the STM \ntopography of the filled states, and an enhancement in the topogr aphy of the unoccupied states. \nWe expect this to be observed in imaging the ionized donors, and the opposite effect for imaging \nnegatively charged defects, such as acceptors. The cloverleaf shape defects are surrounded by a \nregion of enhancement in the topography of the fille d states (Fig. 1(d)), implying that they are \nnegatively charged, consistent wi th a Ca acceptor. A comparison between the triangular defect \nclose to the center of the image in Fig. 1(c) and (d) shows a region of depression in the \ntopography of the filled states (c), and hence imp lies the presence of a  positively charged defect. \nThis observation is expected for Se vacancies, which are known to be electron donors and in \ntheir ionized state be positively charged. Thus th e STM data support the defect chemistry model \ndescribed above for Ca-doped Bi\n2Se3, with the added observation that the concentration of \ndefects in general is lower in the Ca-doped samples than for native Bi 2Se3. \nThe ARPES data showing valence band energy dispersion curves in the vicinity of the Γ \npoint of the Brillouin Zone for n- and  p- type Bi 2Se3 are presented in Figs. 2a and 2b. For the n-\ntype crystal, the valence band is clearly obse rved below E F, as is a small pocket of the \nconduction band near the Γ point. For the  p-type crystal, only the vale nce band is observed. This \nchemical potential shift indicates that the Ca  doped crystal is hole doped relative to the n-type \nBi2Se3 crystal. In the p-type doped samples, the surface state bands are not observed, indicating \nthat the Fermi level is below the energy of th e Dirac point. Judging from the position of the  \n7 \n strongest valence bands, the ener gy dispersion curves through the Γ point show that the chemical \npotential in p-type Bi 1.98Ca0.02Se3 is shifted by approximately 400 meV relative to the n-type \ncrystal (Fig. 2c). Lower hol e doping levels, which would be obtained by tuning the Ca \nconcentration, are expected to lead to materials whose chemical potential is very close to the \nDirac point, and therefore of interest in fundamental and a pplied studies of  the topological \nsurface states [15].   \nThe 2–300 K resistivities in the ab plane for undoped Bi 2Se3 and Bi 2-xCaxSe3 crystals for \nx = 0.005, 0.02 and 0.05 are shown in Fig. 3. All show the weakly meta llic resistivities \ncommonly seen in high carrier concentration sma ll band gap semiconductors, with resistivities in \nthe 0.3 to 1.5 m Ω-cm range at temperatures near 10 K.  The lightly doped x = 0.005 material is p-\ntype with a carrier concentrati on at room temperature determin ed by Hall effect measurements \n(lower inset, Fig. 3) to  be approximately 1 × 1019 cm-3. This crystal has a resistivity ratio, \nρ300/ρ4.2 of about 3, as does the undoped n-type crystal. The upper inset shows the low \ntemperature resistivity region in more detail for n- and p-type crystals. The decreasing resistivity \nseen on cooling is arrested at these low te mperatures and rises slightly below 20-30 K, \nsuggesting that carriers have been fro zen out in this temperature regime. \n The corresponding ab plane Seebeck coefficients are shown in Fig. 4. The room \ntemperature Seebeck coefficient for the n-type undoped Bi 2Se3 crystal, -190 µVK-1, is large \ncompared to those usually reported for this mate rial, which are typically in the -50 to -100 µVK-1 \nrange (3-11). The carrier concentration for this n-type Bi 2Se3 is 8 × 1017 cm-3 (lower inset, Fig. \n2). The magnitude of the Seebeck  coefficient decreases smoothl y with temperature until around \n15 K (inset Fig. 4), where it beco mes distinctly more negative in a small peak with an onset \ntemperature roughly corresponding to that of the upturn in the resistivity. A similar peak in  \n8 \n Seebeck coefficient has been observed previously in this temperature range in n-type Bi 2Se3 [3]. \nThe p-type materials show similar but more unusua l behavior. The room temperature Seebeck \ncoefficient for the lightly doped p-type material is again very high, reaching approximately +180 \nµVK-1. This value is larger than what is observed in the quaternary p-type (Bi,Sb,Pb) 2Se3 \nmaterials [10] and remains large and p-type over the full temperature range of measurement. \nCrystals with Ca-doping levels as low as Bi 1.9975Ca0.0025Se3 are p-type. A dramatic low \ntemperature peak in the Seebeck coeffi cient (inset Fig. 3) is seen for all p-type compositions \nstudied. The low temperature Seebec k coefficient peaks of -60 µVK-1 and +120 µVK-1 for pure \nand doped Bi 2Se3 at 7 K represent very high values when compared to other small band gap \nsemiconductors at this low temperature. Inte restingly, similarly anomalous low temperature \npeaks have been observed in Bi 1-xSbx alloys, another material in th e vicinity of a Dirac point [12-\n14], in the critical “zero gap” comp osition region near  x = 0.07 [21].  \n The total thermal conductivities ( κ) in the ab plane are shown in Fig. 5.  They display the \nbehavior typical of high quality crystals, in creasing with decreasing temperature as the phonon \ncontribution to κ grows, until low temperatures, where the phonon mean free path becomes \ncomparable to intra-defect distan ces. The low temperature regime for n- and p-type crystals is \nshown in the inset. The thermal conductivity for the n-type crystal reaches the relatively high \nvalue of about 100 WK-1m-1 at 10 K, and all the Ca-doped materials show a somewhat lower \nmaximum thermal conductivity, in the 40-50 WK-1m-1 range. \n The efficiency of a material for thermo electric cooling or power generation at \ntemperature T is described by the thermoelect ric figure of merit ZT [22]. The materials \nparameter is Z = (S2/ρ)(1/κ), where the power factor, define d as the square of the Seebeck \ncoefficient S divided by the electrical resistiv ity, has been separated from the total thermal  \n9 \n conductivity κ. Due to the very high thermal conductiv ities of the single crystals, the Bi 2Se3-\nbased n- and p-type materials are of limited use as practical thermoelectric materials in single \ncrystal form, especially at low temperatures. The resulting low values of the thermoelectric \nfigure of merit are shown in the inset to Fig.  6. The Seebeck coefficients and electrical \nresistivities do, however, suggest that thes e materials may have potential for use as \nthermoelectrics in the very low temperature regime . To illustrate this fact we have plotted the \ntemperature dependences of the power factors in the main pane l of Fig. 6.  The anomalous low \ntemperature peaks in all the ma terials, especially for the p-type material, where the power factor \nat 7 K is higher than it is at 300 K, and comparable to that of the n-type material at 300 K, are \nparticularly intriguing for possible low temperatur e thermoelectric applications. The underlying \norigin of these peaks, and also the more subtle anomalies seen in the transport properties in the \n30-40 K range in all material s, is not presently known. \nConclusions \n Through consideration of th e defect chemistry of Bi 2Se3, we have identified Ca as a \ndopant that when present in sub-percent quantities results in the formation of a p-type material. \nThe use of p-type Bi 2Se3 for the fundamental study of topologi cal surface states  and for devices \nbased on these states for quantum computing will be of considerable future interest. The dramatic low temperature peaks found in the Seebeck coefficients in Bi\n1-xSbx and Bi 2Se3, \nmaterials that have strong spin orbit coupling and nearby Dirac points [12-16,19], suggests that \nthese characteristics may be related. In addi tion, processing of th e microstructure of n- and p-\ntype Bi 2Se3 to reduce the thermal conductivity to the < 1 WK-1m-1 range typically observed for \npolycrystalline materials in this family, if it can be performed while maintaining the power \nfactors observed in the current work, promises the possibility of low temperature thermoelectric  \n10 \n applications. Finally, the development of more s ophisticated crystal grow th methods for lightly-\ndoped p-type Bi 2-xCaxSe3 crystals will be of interest for th e study of topological surface states.  \n \nAcknowledgements \n This work was supported by the AFOSR,  grant FAA9550-06-1-0530, and also by the \nNSF MRSEC program, grant DMR-0819860.         \n11 \n References \n[1] H. Scherrer and S. Scherrer, in Handbook of Thermoelectrics , D.M. Rowe, ed. CRC press, \nNY (1994) pp 211-237. [2] M. Stordeur, Ibid, pp. 239-255. \n[3] J. Navrátil, J. Horák, T. Plechacek, S. Kamba, P. Losta´ k, J. S. Dyck, W. Chen, and C. Uher, \nJ. Sol. Stat. Chem.  177 1704 (2004). \n[4] A. Von Middendorf, H.F. Ko ehler, and G. Landwehr,  Phys. Status Solidi B  57 203 (1973).  \n[5] L.P. Caywood, Jr. and G.R. Miller, Phys. Rev. B  2 3209 (1970). \n[6] V.F. Boechko and V. I. Isarev,  Inorg. Mater. 11 1288 (1975). \n[7] M. Stordeur, K.K. Ketavong, A. Prei muth, H. Sobotta, and V. Reide,  Phys. Status Solidi B  \n169 505 (1992). \n[8] A. Al Bayaz, A. Giani, A. Fou caran, F. Pascal-Delannoy, A. Boyer, Thin Solid Films  441 1 \n(2003). \n[9] H. Köhler , and A. Fabricius, Phys. Status Solidi B 71 487 (1975). \n[10] J. Kašparová, È. Drašar,  A. Krejèová, L. Beneš P. Lošt’ák, Wei Chen, Zhenhua Zhou, and \nC. Uher, J. Appl. Phys.  97 103720 (2005). \n[11] A. Vaško , L. Tichy, J.  Horák and J. Weissenstein, Appl. Phys.  5 217 (1974).  \n[12] L. Fu and C.L.  Kane, Phys. Rev. B  76 045302 (2007). \n[13] S. Murakami, New J. Phys.  9 356 (2007).  \n[14] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nature  452 970 \n(2008). [15] Y. Xia, L. Wray, D. Qian, D. Hsieh, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. \nCava, and M.Z. Hasan, arXiv:0812.2078 (2008).  \n12 \n [16] Haijun Zhang, Chao-Xing Liu, Xiao-Lia ng Qi, Xi Dai, Zhong Fang, Shou-Cheng Zhang, \narXiv: 0812.1622 (2008). [17] J. Zaanen, Science  323 888 (2009). \n[18] L. Fu and C. L. Kane, Phys. Rev. Lett.  100 096407 (2008). \n[19] S. K. Mishray, S. Satpathy, and O. Jepsen, J. Phys.: Condens. Mattter  9 461 (1997). \n[20]  S. Urazhdin, D. Bilc, S. H. Tessmer, S. D.  Mahanti, T. Kyratsi, and M. G. Kanatzidis, Phys. \nRev. B  66 161306(R) (2002). \n[21] B. Lenoir, A. Dauscher, X. Devaux, R. Ma rtin-Lopez, Yu.I. Ravich, H. Scherrer and S. \nScherrer, Proceedings of the 15th International Conference on Thermoelectrics, IEEE  (1996) pp. \n1-13. [22] G.J. Snyder and E.S. Toberer, Nature Materials  7 105 (2008). \n   \n13 \n Figures \nFig. 1.  (Color on line)  (a) Spatially averaged density of st ates (dI/dV) measurements showing \nthe shift of E F between n-type Bi 2Se3 and p-type Bi 2Ca0.02Se3. (b) STM topography of the empty \nstates of Bi 2Se3 (V B = +1 V, and I = 10 pA) showing tria ngular shaped defects observed at \nvarious intensities, indicating they are located in  different layers beneath the surface. (c)  \nTopography of the empty states of Bi 1.98Ca0.02Se3 (VB = +2.0 V, and I = 10 pA) showing clover \nleaf looking defects and a substantial reduction in the density of the triangular defects, which \ndominated the undoped samples.  (d) Topography of the filled states over the same area as in (c) \n(VB = -1.0 V, and I = 10 pA).  The area around th e triangular defect near center shows a \ndepression around it in the filled states, demonstrat ing it is positively charged. In contrast, the \nCa-related defects exhibit an area of enhancem ent around them, indicati ng they are negatively \ncharged. All STM topographies are 500 Å by 500 Å. Fig. 2. (color on line)  (a) The valence band structure of n-type Bi\n2Se3 measured by ARPES near \nthe Fermi energy and the Γ point of the Brillouin Zone. The ch emical potential (the Fermi level) \nis about 0.4 eV above the top of the valence band, about 0.3 eV  above the surface Dirac point \n(15). (b) The band stru cture of a Ca-doped Bi 2Se3 crystal (Bi 1.98Ca0.02Se3) measured under \nsimilar conditions. The chemical potential is with in 30 meV of the valence band. For (a) and (b) \nthe Γ point is 0.0 on the horizontal axes and the band dispersions are shown in the ±M directions. \n(c) The momentum-selective densit y of states (DOS) near the Γ point, compared for the n- and p-\ntype crystals, covering a large bi nding energy range. A large shift of the chemical potential with \nCa doping can be traced by consid ering the shift of DOS  peaks in the valence band. The drop of \nthe chemical potential by about 380 meV conf irms the hole doped nature of the Ca doped \nsamples.  \n14 \n Fig. 3.  (color on line) Temperatur e dependent resistivity be tween 2 and 300 K in the ab plane of \nsingle crystals of Bi 2Se3 and Ca-doped variants. The stoichiometric material is n-type and the \nCa-doped materials are all p-type. The upper inset shows the resistivities in the low temperature \nregime. The lower inset shows the Hall effect  data employed to de termine the carrier \nconcentrations for undoped Bi 2Se3 and one Ca-doped crystal. \nFig. 4.  (color on line) Temperature dependent Seeb eck coefficients between 2 and 300 K in the \nab plane of single crystals of Bi 2Se3 and Ca-doped variants. The inset shows the low temperature \nregion for both pure and lightly Ca-doped Bi 2Se3. The undoped Bi 2Se3 is n-type, with a carrier \nconcentration of 8 × 1017 cm-3, and the Ca-doped materi al with x = 0.005 is p-type with a carrier \nconcentration of 1 × 1019 cm-3.   \nFig. 5.  (color on line) Temperature dependent to tal thermal conductivities between 2 and 300 K \nin the ab plane of single crystals of Bi 2Se3 and Ca-doped variants. The inset shows the low \ntemperature region for both pure and lightly Ca-doped Bi 2Se3.  \nFig. 6.  (color on line) Temperature dependent ther moelectric power factor between 2 and 300 K \nin the ab plane of single crystals of Bi 2Se3 and Ca-doped variants. All materials show a peak at \nlow temperatures but the peak in the p-type variant with 1 % Ca doping (Bi 2-xCaxSe3 with x = \n0.02) is particularly dramatic. The inset shows the thermoelectric figure of merit in the same \ntemperature range.        \n15 \n  \n \n \n \nFigure 1 \n \n  \n16 \n  \n \n \n \nFigure 2        \n17 \n \n \nFigure 3 \n  \n18 \n  \n \nFigure 4 \n  \n19 \n  \n \nFigure 5 \n  \n20 \n  \n \nFigure 6  "    },    {        "title": "2404.06532v1.Solvable_models_of_two_level_systems_coupled_to_itinerant_electrons__Robust_non_Fermi_liquid_and_quantum_critical_pairing.pdf",        "content": "Solvable models of two-level systems coupled to itinerant electrons:\nRobust non-Fermi liquid and quantum critical pairing\nEvyatar Tulipman,1,∗Noga Bashan,1,∗J¨ org Schmalian,2, 3and Erez Berg1\n1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel\n2Karlsruher Institut f¨ ur Technologie, Institut f¨ ur Theorie der Kondensierten Materie, 76049, Karlsruhe, Germany\n3Karlsruher Institut f¨ ur Technologie, Institut f¨ ur Quantenmaterialien und Technologien, 76021, Karlsruhe, Germany\n(Dated: April 11, 2024)\nStrange metal behavior is traditionally associated with an underlying putative quantum critical\npoint at zero temperature. However, in many correlated metals, e.g., high- Tccuprate supercon-\nductors, strange metallicity persists at low temperatures over an extended range of microscopic\nparameters, suggesting the existence of an underlying quantum critical phase , whose possible phys-\nical origins remain poorly understood. Systematic investigations of physical scenarios giving rise to\nsuch a critical, non-Fermi liquid (NFL) phase are therefore crucial to better understand this puz-\nzling behavior. In a previous work [1], we considered a solvable large- Nmodel consisting of itinerant\nelectrons coupled to local two-level systems (TLSs) via spatially random interactions, inspired by\nthe possibility of emergent metallic glassiness due to frustrated competing orders, and found that\nthe system hosts an NFL phase with tunable exponents at intermediate couplings. In this work,\nwe expand our investigation to the following: (i) We study the extent to which this NFL phase is\ngeneric by considering various deformations of our theory, including coupling of electrons to multiple\noperators of the TLSs and arbitrarily directed TLS-fields. We find that the physical picture obtained\nin [1] qualitatively persist in a wide region of parameter space, showcasing the robustness of the\nNFL phase; (ii) We analyze the superconducting instability due to coupling of TLSs to electrons,\nand find a rich structure, including quantum critical pairing associated with the NFL phase and\nconventional BCS pairing in the weak and strong coupling limits; (iii) We elaborate on the analysis\nof Ref. [1], including single-particle, transport and thermodynamic properties.\nI. INTRODUCTION\nOne of the central problems in condensed matter\nphysics concerns the low-temperature anomalous normal-\nstate transport properties of correlated metals such as\nhigh- Tccuprate superconductors and others [2–6]. A\nhallmark of the anomalous behavior is the linear-in-\ntemperature scaling of the dc resistivity, known as\n‘strange metal’ behavior. Such behavior stands at odds\nwith conventional Fermi-liquid theory, where a T2-scaling\nis predicted, and is believed to indicate that quantum\nfluctuations are so pronounced as to completely invali-\ndate the Landau Fermi-liquid quasiparticle picture [7–\n10].\nThe traditional theoretical approach to describe\nstrange metals and other non-Fermi liquids (NFLs) in-\nvolves coupling a Fermi surface to bosonic collective fluc-\ntuations of an order parameter, sometimes leading to\nNFL behavior when tuning the system to a quantum crit-\nical point (QCP) [11–13]. In this case, the NFL behavior\nmanifests in a critical fan, emanating from a single (crit-\nical) point at T→0. Interestingly, while consistent with\nsome materials, e.g. heavy fermion systems [14], there\nare numerous examples, e.g. cuprates [15–18] as well as\ntwisted bilayer graphene, organic superconductors, and\nother systems [6, 19–26], where NFL behavior persists\nover an extended region of non-thermal parameters at\n∗These authors contributed equally to this work.T→0 and thus cannot be ascribed to a single QCP.\nRather, the extended NFL behavior raises the possibility\nof a quantum critical phase . Since no general prescription\nexists for this scenario, capturing such behavior within a\ncontrolled, physically motivated theory is considerably\nchallenging.\nOne potential route to realize an extended critical NFL\nphase requires an efficient source of scattering for itin-\nerant electrons over an extended region of non-thermal\nparameters, e.g., by identifying a physical setting where\n“critical” low-energy excitations (i.e., with support at\nthe lowest energies) exist. To this end, in a recent work\n[1], we demonstrated that an NFL phase can arise when\nitinerant electrons are interacting with fluctuations of\na metallic glass (e.g., charge or stripe glass), described\nas a collection of two-level systems (TLSs) that corre-\nspond to quasi-local collective excitations, analogously\nto the excitations in structural glasses1. Physically, our\ntheory is motivated by the complex phase diagrams of\nsuch strongly correlated materials that often host multi-\nple frustrated, competing orders which can give rise to\nglassiness, even in the absence of impurity disorder [28].\nThe presence of disorder could further stabilize such ex-\ntended NFL behavior, as was observed in Ref. [29].\nIn fact, it has been long recognized that inelastic scat-\ntering of electrons off of local TLSs can result in a T-\nlinear resistivity in the weak coupling limit [30]. Nonethe-\n1For a recent detailed analysis of the density of tunneling defects\nin metallic glasses, see Ref. [27].arXiv:2404.06532v1  [cond-mat.str-el]  9 Apr 20242\nless, going beyond weak coupling, the interaction with\nelectrons may dramatically alter the properties of the\nTLSs, e.g., by renormalization of the bare TLS param-\neters or inducing inter-TLS correlations analogously to\nthe Ruderman–Kittel–Kasuya–Yosida (RKKY) mecha-\nnism [31–33]. Clearly these effects can further affect the\nelectrons themselves.\nTo study the evolution of the physical picture beyond\nweak coupling, in Ref. [1], we considered a large- Nthe-\nory consisting of itinerant electrons interacting with local\nTLSs via spatially random couplings. We found that the\nnon-Gaussian saddle point of the theory hosts a robust\nNFL phase with tunable exponents, which is not asso-\nciated with quantum criticality; see Sec. V for a sum-\nmary of our findings. Roughly speaking, the electrons\nconstitute an ohmic bath for the TLSs which results in\na renormalization of the TLS energy splitting towards\nlower energies. At sufficiently strong couplings, a signifi-\ncant portion of the TLSs are renormalized to low energies,\nwhich in turn provides an efficient source of scattering for\nthe electrons, resulting in an NFL behavior over a finite\nrange of coupling strengths.\nIn Ref. [1], we have considered the special case where\nelectrons are coupled to a single operator of TLSs (i.e.,\nσxof spin-1 /2 Pauli operators), which allowed us to fo-\ncus on the effects of inelastic electron-TLS scattering in a\nsimple setting. Specifically, we considered the case where\nelectrons are coupled to σxof the TLSs, which we shall\nrefer to as the ‘ x-model’ henceforth. Aiming for a broader\nunderstanding, it is natural to ask whether the behavior\nof the x-model persist to the generic case, where elec-\ntrons may couple to all three operators of the TLSs. An-\nother important aspect of the physical picture concerns\nTLS-induced pairing, which is expected to take over at\nsufficiently low temperatures.\nIn this work, we investigate a class of large- Nmod-\nels, generalizing our study of the x-model of Ref. [1] to\nmore generic settings. We begin by considering the ef-\nfect of coupling of spinless electrons to all operators of\nthe TLSs. Importantly, the low-energy behavior is qual-\nitatively identical to that of the x-model within a wide\nrange of parameters, showcasing the robustness of the\nNFL phase found in Ref. [1]. Towards a more realistic\nscenario, we generalize our analysis to spinful electrons\nand find a transition to a superconducting ground state\ndue to TLS-induced pairing. Remarkably, the rich phe-\nnomenology of the normal state in our model is also man-\nifested in the form of the critical temperature Tc, exhibit-\ning various crossovers, where in particular the transition\nfrom the NFL phase assumes a quantum critical form.\nIn addition to these key finding, we further study various\naspects of the model, including transport and thermody-\nnamic properties and 1 /Ncorrections.\nThe structure of the paper is as follows. In Sec. II\nwe present the model and discuss physically motivated\nchoices of parameters. In Sec. III we provide a brief sum-\nmary of our main results. In Sec. IV we provide a map-\nping between our model to a set of decoupled spin-boson(SB) models. In Sec. V we provide an extensive review\nof the x-model, previously studied in Ref. [1], which cor-\nresponds to the special case where the electrons interact\nonly with the ˆ xcomponent of the TLSs. In Sec. VI we\nanalyze the general case where the electrons interact with\nall operators of the TLSs (dubbed the xyz−model), and\nin Sec.VII, to the most general case where the field acting\non the TLS is allowed to point in arbitrary directions. In\nSec. VIII, Sec. IX and Sec. X we use our results to study\ntransport, thermodynamic and superconducting proper-\nties of the model, respectively. In Sec. XI we consider\nseveral finite- Ncorrections. Sec. XII contains a sum-\nmary of our work and discuss possible implications and\nfurther directions.\nII. MODEL\nWe consider the following Hamiltonian, defined on a\nd-dimensional hypercubic lattice:\nH=NX\nk;i=1(εk−µ)c†\nikcik−MX\nr;l=1hl,r·σl,r+Hint,(1)\nwhere\nHint=NX\nr;i,j=1\"\n1\nN1/2Vij,r+1\nNMX\nl=1gijl,r·σl,r#\nc†\nircjr.\n(2)\nEach site contains Nelectronic “orbitals” i= 1, ..., N ,\nandMspecies of TLSs l= 1, ..., M where σl,ris a vec-\ntor of spin-1 /2 Pauli operators at position r.εkand\nµdenote the electronic dispersion and chemical poten-\ntial, respectively, and are assumed to be diagonal and\ni−independent in orbital space. For simplicity, we con-\nsider spinless fermions. We will reintroduce the spin in-\ndex later when we discuss superconductivity. Thus, elec-\ntrons create a dynamic local field that acts on the TLSs\nwhile, at the same time, they scatter off those local de-\ngrees of freedom.\nEach TLS is subjected to a field hl,r= (hx\nl,0, hz\nl),\nwith hzbeing the asymmetry and hxthe tunneling rate\nbetween the two states associated with each TLS, both\ntaken to be independent random variables drawn from a\nprobability distribution Pβa(ha) with a=x, z. Below, we\nrefer to this as the physical basis for h. Note that hy≡0\nin order to respect time-reversal symmetry. We focus on\npower-law distributions, Pβa(ha)∝(ha)βa, supported on\nthe interval 0 < ha< ha\nc. It suffices to consider positive\nfields as the sign of hacan be absorbed into the definition\nofσz.ha\ncdenote the TLS bare bandwidth and βa>\n−1 is a tunable parameter. The generalization to other\ndistributions is straightforward. In some cases, it will\nbe convenient to rotate to the eigenbasis where h=hˆz,\nwith h=p\n(hx)2+ (hz)2being the energy splitting of\nthe TLS. We call this the diagonal basis of h.\nWhile we present throughout this work general results\nfor arbitrary βxandβz, there are two physically moti-3\nvated choices for the splitting distributions. Working in\nthe “physical” basis (i.e. the eigenbasis of σz), we as-\nsume that the distribution of the level asymmetry, hz,\nhas finite weight around hz= 0 [34], which corresponds\nto a uniform distribution with βz= 0. It is then natural\nto consider the cases where the width of the distribu-\ntion of the tunneling rate, hx, is either comparable or\nnegligible compared to the asymmetry, corresponding to\nβx= 0, hx\nc=hz\nc≡hc, or setting hx\nc≡0, respectively. In\nthe latter the diagonal basis coincides with the physical\nbasis, while in the former, changing to the diagonal basis\nresults in a linear distribution of the eigenvalues h, i.e.\nβz= 1.\nThe couplings gijl,r=\u0010\ngx\nijl,r, gy\nijl,r, gz\nijl,r\u0011\nare taken\nto be uncorrelated Gaussian random variables with zero\nmean and variance g2\na. For a=x, z, we consider real-\nvalued couplings with\nga\nijl,rga\ni′j′l′,r′=g2\naδr,r′δll′(δii′δjj′+δij′δji′), (3)\nand to ensure that His time reversal symmetric, gymust\nbe purely imaginary with\ngy\nijl,rgy\ni′j′l′,r′=−g2\nyδr,r′δll′(δii′δjj′−δij′δji′).(4)\nThe gaare all real valued. The different components\nare uncorrelated, i.e. gaga′= 0 if a̸=a′.(·) denotes\naveraging over realizations of the coupling constants.\nSimilarly, the onsite potential disorder, Vij,r, is nor-\nmally distributed with zero mean and variance V2. Note\nthat setting the couplings gxandgzto be uncorrelated in\nthe physical basis or in the eigenbasis is not equivalent in\nthe cases where both hx, hz>0. Here we first consider\nsimple variants of the model where gxandgzare uncor-\nrelated in the diagonal basis (i.e. setting hx≡0) and\nlater show that the qualitative physical picture persists\nwhen they are uncorrelated in the physical basis (with\nhx̸= 0).\nThe Fermi energy, EF, sets the largest energy scale in\ntheory, and also corresponds to the cutoff energy of the\nelectronic bath, traditionally denoted by ωc(=EF) in the\nspin-boson literature [35–37].\nThe TLS bandwidth satisfies hc≪EF, and we re-\nstrict the on-site disorder strength, V2, such that Γ =\n2πρFV2≪EF(Γ being the elastic scattering rate),\ntherefore considering ‘good metals.’ We do not restrict\nthe interaction strengths ga, namely, our study covers\nthe range from weak to strong coupling. We focus on the\nlow-energy limit of the model, defined by ω, T≪hc,R,\nwhere hc,Ris the renormalized cutoff of the TLSs, to be\ndefined below.\nThe dimensionless coupling parameters that will be\nused in the following are related to the interaction\nstrengths by ( a=x, y, z ):\nαa=ρ2\nFg2\na\nπ2, (5)\nλa=M\nNρFg2\na\nhc,R. (6)The parameters αa(defined in accordance to the spin-\nboson literature conventions) represent the strength of\nthe dissipation acting on the TLSs, while λaquantify\nthe strength of the scattering of electrons by TLSs at\nlow energies.\nThroughout this work we consider the limit N, M → ∞\nwith a fixed ratio M/N . We will see below that the\nlimit M→ ∞ enables us to (i) reduce the electron’s self-\nenergy to a summation over rainbow diagrams containing\nonly two-point correlation functions of the TLSs, which\nis not clear a priori as Wick’s theorem does not hold\nfor the TLSs; and (ii) invoke self-averaging of the TLSs,\nsuch that sums over the TLS flavors can be replaced by\naverages over the splitting distribution:PM\nl=1f(hl)→\nM´\nf(h)Pβ(h)dh. Importantly, since the splitting dis-\ntribution is independent of position, the self-averaging\nassumption translates to statistical translation invariance\nof the model. Note further that the N→ ∞ limit is es-\nsential for the mapping of our model to the SB model,\nwhere the bosonic bath coupled to the TLSs is composed\nof particle-hole pairs, see Sec. IV.\nIII. BRIEF SUMMARY OF RESULTS\nIn the following sections we expound on the properties\nof different variants of the model. However, for the ben-\nefit of the reader, we first briefly outline the key conclu-\nsions of our work. We first describe the physical picture\nof the x-model and then show that this picture qualita-\ntively persists to generic variants of the model.\nNormal state. Consider the normal state properties\nat low T, corresponding to regions (I) and (II) in Fig. 1.\nIn region (I), as the dimensionless coupling α=αxis\nincreased, the system crosses over from a FL, MFL, and\nNFL, up to a critical value α≈1 where the TLS freeze\natT= 0. These regimes are defined by the exponent of\nthe single-particle scattering rate: Σ′′(ω)−Σ′′(0)∝ |ω|γ,\nγ(α, β) = (1 + β)(1−α) (shown for β= 1 in Fig. 1).\nThis is also manifested in the dc resistivity: ρ−ρ0∝Tγ.\nIn region (II), the TLSs are frozen at T= 0, such that\nscattering off of TLS is mainly elastic. At finite Tfor\nα≳1, however, residual quantum fluctuations of the\nTLSs provides a source for inelastic scattering, leading to\nan additional sequence of NFL-MFL-FL crossovers with\nan inelastic scattering exponent 2( α−1). This residual\ncontribution corresponds to a weak T-variation of the dc\nresistivity as shown in Fig. 1. The behavior in the critical\nregion α≈1 and T→0, separating regions (I) and (II),\nis more involved and show logarithmic T-dependence of\nthe single-particle scattering rate (and the dc resistivity)\nthat smoothly interpolates between the two regions.\nSuperconductivity. Considering the model with\nspinful electrons, a superconducting transition occurs be-\nlow a critical temperature Tcdue to TLS-induced pairing;\nsee Fig. 1. Interestingly, Tcis a non-monotonic function\nof the coupling αx, with remarkably rich pairing phe-\nnomenology. Specifically, at intermediate couplings (cor-4\nFigure 1. (a,b) Illustration of the lattice model Eqs. (1,2). (a) a unit cell containing a large number of electronic states\nand local two-level systems interacting via random couplings; (b) the electrons hop between unit cells. (c)Phase diagram\nof the x−model in α−Tplane ( α=αxis the dimensionless coupling strength) for a linear splitting distribution ( β= 1).\nRegion (I), defined by T≪T⋆∼hc,R,hc,Rbeing the renormalized cutoff of the TLS-splitting, is characterized by the leading\ninelastic scattering exponent 2(1 −α) as manifested in the dc resistivity, see (d); the system crosses over from a Fermi-liquid\n(FL) for α <1/2, to a marginal Fermi-liquid (MFL) for α= 1/2, and a non Fermi-liquid (NFL) for 1 /2< α < 1. At α≈1,\nthe TLSs undergo a freezing transition at T= 0. In Region (II), for α≳1 at finite T, scattering off of the TLSs is mainly\nelastic with small inelastic corrections that scale ∼T2(α−1), namely, another set of crossovers from NFL (1 < α < 3/2) to MFL\n(α= 3/2) to FL ( α >2); see (e). Region (II) is defined up to T∼T⋆⋆, where T⋆⋆is the scale at which standard FL behavior\nbecomes dominant. The gray line denotes the transition temperature to the superconducting state, Tc, in the spinful version\nof the x-model. (d,e) Resistivity as a function of Tfor regions (I) and (II), respsectively, where ρ∞is the resistivity due to\nsaturated classical TLSs. Note that, in region (II), the α-dependent coefficient ηα∝(hc/EF)2≪1, corresponding to the weak\nT-variation of the dc resistivity.\nresponding, e.g., to the NFL phase of region (I) in Fig. 1)\nTcassumes an algebraic, quantum critical scaling form,\nand, as the coupling is further increased beyond a certain\nthreshold (but still at intermediate values), crosses over\nto an Allen-Dynes-like, strong coupling form [38]. In ad-\ndition, Tcassumes the standard BCS form at weak cou-\npling, but also at very strong coupling (e.g., for α >3/2\nin Fig. 1), which corresponds to pairing due to the resid-\nual quantum fluctuations of the nearly frozen spins.\nRobustness. To test the extent to which the physical\npicture of the x-model is generic, we allow for interactions\nwith other operators of the TLSs. In Fig. 2, we show how\nthe normal state T= 0 phase diagram of the x-model\nchanges upon introducing coupling to the y(top row) and\nz(bottom row) operators of the TLSs for constant and\nlinear TLS-splitting distributions (left and right columns,\nrespectively). We shall refer to these variants as the xy-\nandxz-models.\nIn all cases, the qualitative behavior of the x-model\npersists provided that the largest coupling is orthogonal\nto the direction of the TLS-field (i.e. to hzσz), namely,\nthe characterizing exponent γ(α) varies from 1 + βto\n0 as the dominant coupling is increased, up to a crit-ical value at which the TLSs freeze. This sequence of\ncrossovers corresponds to region (I) in Fig. 1, while the\nresidual crossovers of region (II) are expected to qualita-\ntively change for sufficiently strong perturbations due to\nnon-universal corrections. Note that when the dominant\ncoupling is parallel to the TLS-field (i.e., when αz> αx),\nthe TLSs are essentially static and the system shows\nFermi-liquid behavior with TLS-induced elastic scatter-\ning along with weak FL-like corrections.\nIV. MAPPING TO SPIN-BOSON MODEL\nIn this section, we use an effective action approach\nto map our theory to the SB model. We set V2= 0\nfor simplicity. An alternative diagrammatic derivation of\nthe mapping is given in App. A.\nWe begin by considering the spin coherent-state path\nintegral representation for the TLSs. The partition func-\ntion is given by\nZ[h,g] =ˆ\nD[σ, c,c]e−S, (7)5\nFigure 2. T= 0 normal-state phase diagram of the xy-model\n(top) and xz-model (bottom), for constant ( β= 0) and linear\n(β= 1) TLS splitting ditributions. The color represents the\nexponent in the low- Tbehavior of the resistivity: ρ−ρ0∝Tγ\n(analogous to region (I) in Fig. 1). The dashed line denotes\nthe BKT transition over which the TLSs freeze, and beyond\nwhich there is a non-universal version of the NFL phase (sim-\nilarly to region (II) of Fig. 1 for αx>1).\nwith the action, S=S0+Sint:\nS0=X\nrMX\nl=1SBerry[σl,r]−X\nrMX\nl=1ˆ\nτhl,r·σl,r+NX\ni=1X\nkˆ\nτcik(∂τ+εk−µ)cik, (8)\nSint=1\nNX\nrNX\ni,j=1MX\nl=1ˆ\nτgijl,r·σl,rcircjr. (9)\nHere we kept the same symbols σl,rfor the unit vectors\nthat result from the coherent state representation of the\nPauli operators. SBerry denotes the Berry’s phase of the\nTLSs, see e.g. [39].\nTo proceed, we average over the random couplings us-\ning the replica method and introduce the bilocal fields\nGr,r′(τ, τ′) =1\nNX\nicir(τ)cir′(τ′), (10)\nχa,r(τ, τ′) =1\nMX\nlσa\nl,r(τ)σa\nl,r(τ′). (11)\nThe constraints, (10) and (11), are enforced via conju-\ngated fields, Σ and Π, respectively. Notice that, for now,\nwe are considering spinless fermions. In this case, there is\nno pairing instability to leading order in 1 /N. Later on,\nwe shall consider a model of spinful fermions, where the\nanomalous part of the Green’s function must be consid-\nered, and an instability towards a superconducting state\nwith an intra-flavor, on-site order parameter occurs [40].\nTo proceed, we integrate over the fermions and\nsubstitute a replica-diagonal Ansatz , which allows\nus to express the partition function as Z[h] =´\nD[G,χ,Σ,Π,σ]e−Seff, where the effective action is\ngiven by\nSeff=−NTr ln\u0000\nG−1\n0−Σ\u0001\n−Nˆ\nτ,τ′X\nr,r′X\nσGr,r′(τ, τ′) Σr,r′(τ, τ′) +M\n2ˆ\nτ,τ′X\nr,aχa,r(τ, τ′) Πa\nr(τ′, τ)\n+M\n2ˆ\nτ,τ′X\nrX\nag2\naGr(τ, τ′)Gr(τ′, τ)χa,r(τ, τ′) +X\nrMX\nl=1SBerry [σl,r]−ˆ\nτX\nrMX\nl=1hl,r·σl,r\n−1\n2ˆ\nτ,τ′X\nr,aΠa\nr(τ′, τ)MX\nl=1σa\nl,r(τ)σa\nl,r(τ′). (12)\nIn the limit of large MandN, with fixed ratio M/N ,\nwe can analyze the problem in the saddle point limit.\nPerforming the variation with respect to Gand Σ gives\nΣr,r′(τ) =δr,r′M\nNX\nag2\naGr,r(τ)χa,r(τ) (13)\nas well as\nGr,r′(iω) =\u0000\nG−1\n0(iω)−Σ (iω)\u0001−1\f\f\f\nr,r′. (14)\nHere, we have used thermal equilibrium to write the\nsaddle-point equations with time-translation-invariantcorrelation functions and their Fourier transforms. In\naddition, the stationary point that follows from the vari-\nation with respect to χis\nΠa,r(τ) =−g2\naGr,r(τ)Gr,r(−τ). (15)\nThe Berry phase term SBerry, that reflects the fact that\nno Wick theorem exists for Pauli operators, implies that\nthe TLSs cannot be simply integrated over as a Gaussian\nintegral. However, it allows us to recast the TLS problem\nto that of Mdecoupled TLSs per site r,PM\nr,l=1Sr,l[σr,l],\ncoupled to a bosonic bath of particle-hole excitations.6\nEach TLS is governed by the spatially local effective ac- tion\nSr,l[σ] =SBerry [σ]−ˆ\nτhl,r·σ(τ)−ˆ\nτ,τ′Πr(τ′−τ)σa(τ)σa(τ′). (16)\nThis is indeed the action of the spin-boson model after\nthe bosonic bath degrees of freedom have been integrated\nout [37]. The latter give rise to the non-local in time\ncoupling Πa\nr(τ′, τ) that is, in general, different for each\nsite. Of course, in our problem the origin of the bath\nfunction are not bosons but the conduction electrons. For\nthe solution of this local problem this makes, however, no\ndifference. Sr,lstill depends on the random configuration\nhl,rof the fields.\nFor a given realization of the fields hl,rthe problem\nis not translation invariant and correlation functions likeD\nσa\nl,r(τ)σa\nl,r(0)E\nfluctuate in space. However, to deter-\nmine the self energy in Eq. (13) we only need to know\nthe average χa,r(��) of this correlation function over the\nMflavors. To proceed we assume that the model is self-\naveraging in the M→ ∞ limit, such that sums over\nthe TLS flavors can be replaced with averaging over the\nTLS splitting distribution (PM\nl=1→M´\nP(hr)dhr).\nSince the splitting distribution is independent of position,\nthe self-averaging assumption translates to a statistical\ntranslation invariance of the model, at least for the av-\nerage of interest. Hence, χa,r(τ) =χa(τ) is independent\nonr. The same must then hold for the bath function\nΠa,r(τ) = Π a(τ). From the saddle point equations (15)\nit follows that the local fermionic Green’s function and\nthrough Eq. (14) the self energy are both space indepen-\ndent. Hence we can go to momentum space and find that\nthe theory is goverened by a momentum-independent\nself-energy and the Dyson equation for the electrons read\nΣ (τ) =M\nNX\nag2\naχa(τ)G(τ), (17)\nGk(iω) =1\niω−εk−Σ (iω), (18)\nwhere G(τ) =´\nkGk(τ) is the local Green’s function.\nFor a momentum independent fermionic self energy we\nobtain in the limit of large electron bandwidth\nG(iω) =ˆ\nkGk(iω)≈ −iπρFsgn (ω). (19)\nThe particle-hole correlation function can now be evalu-\nated. We find\nΠa(ω) =ρ2\nFg2\na\n2π|ω|, (20)\nirrespective of the electronic self-energy. We thus con-\nclude that each TLS is coupled to an Ohmic bath ofparticle-hole excitations that is independent of the back\nreaction of the TLS on the electronic degrees of freedom.\nThis is a consequence of the fact that the Π aare inde-\npendent of Σ. Thus, we have shown that the (spatially\nlocal) TLS-correlator\nχa(τ−τ′) =1\nMX\nl⟨σa\nl(τ)σa\nl(τ′)⟩, (21)\nis determined by the behavior of Mdecoupled SB models.\nThe strategy of the solution of our model in the large-\nNlimit is therefore: (i) solve the spin-boson problem\nwith ohmic bath for a given realization of the field h, (ii)\naverage over the TLS distribution function of the fields,\nand (iii) use the resulting propagator χa(ω) of the TLSs\nto determine the fermionic self energy from Eq.(18). The\nnon-linear character of the problem is rooted in the rich\nphysics of the spin-boson problem, along with the averag-\ning over the distribution functions of the fields h. Given\nthe importance of the spin-boson model for our analysis\nwe will give a summary of this model in the next section.\nV. THE x−MODEL\nWe begin with a review of the solution of the model for\nthe simplest case where the electrons interact only with\nσxof the TLSs’ pseudospins, i.e., we are working in the\ndiagonal basis and setting gz, gy= 0, as in Ref. [1]. This\nspecial case allows for a more transparent discussion of\nthe key steps of our analysis. In addition, we will see\nthat the more general problem reduces in many cases to\nthis model in the limit of sufficiently low energies. In\nterms of the mapping provided in Sec. IV, the model is\nmapped into the SB model with one bath. Throughout\nthis section, since αy, αz= 0 we will use the notation\nα=αxfor simplicity.\nA. A simple view of the physical picture for α <1\nConsider first the weak coupling limit, α→0, where\nthe effect of interactions can be studied perturbatively.\nTo leading order in g2the decay rate of an electron with\nenergy ωis proportional to the amount of TLSs at acces-\nsible energies, namely,\nΣ′′(ω)∝ˆω\n0Pβ(h)dh∝ |ω|1+β, (22)7\nwhere Σ′′denotes the imaginary part of the electronic\nretarded self energy ( F′′will denote the imaginary part\nof the retarded function Fthroughout the paper). For\nβ= 0 this weak coupling analysis yields marginal Fermi\nliquid behavior. On the other hand, for any β >0, i.e.\nfor distribution functions that vanish for h→0 one only\nfinds Fermi liquid behavior. Increasing the strength of\nthe interaction modifies the behavior of the TLSs in two\nmain aspects: it renormalizes the energy splitting of each\nTLS, such that hℓ→hR(hℓ); and broadens the TLS spec-\ntral function. The broadening has negligible effect on the\nfrequency dependence of Σ′′(at sufficiently low energies).\nHowever, as we show in detail below, the renormalization\nof the energy splitting leads to a renormalization of the\nbare TLS-splitting distribution, decreasing its exponent\nfrom βtoβ−(1 + β)α. Thus, increasing the interac-\ntion strength transfers the spectral weight of the TLSs\ntowards lower energies (at the limit α→1, the spectral\nweight is pushed to zero energy, signaling a BKT tran-\nsition of the TLSs to a localized phase). Consequently,\nthe naive perturbative argument will hold for the renor-\nmalized splitting distribution, resulting in a tunable ex-\nponent as a function of α.\nB. Summary of the single-bath spin-boson model\nThe single-bath spin-boson model (1bSB), or the\nCaldeira-Leggett model [37, 41], is given by the Hamil-\ntonian (note that the commonly used convention in the\nSB literature swaps σx↔σzrelative to our convention)\nH1bSB =−hσz+gσxϕ+Hϕ. (23)\nThe bosonic field ϕ=P\ni(ai+a†\ni) is to be inter-\npreted in terms of a bath of oscillators whose spectral\nfunction, dictated by the Hamiltonian Hϕ, is assumed to\nbe of power law form below some high energy cutoff ωc:\nΠ(ω≪ωc)∝ |ω|s. The cases where s <1, s= 1, s > 1\nare respectively called the subohmic, ohmic, and super-\nohmic baths. Throughout our work we will exclusively\nbe interested in the ohmic case Π( ω) =π\n2αω, using this\nas the definition of the dimensionless coupling constant\nα. For extensive reviews, see, e.g., Refs. [35, 37].\nHistorically, this model was proposed as a toy model\nfor the study of quantum dissipation and decoherence\n[42]. For an ohmic bath, it was found that the spin grad-\nually loses its coherence as αis increased and becomes\noverdamped (in terms of the one-point function ⟨σx(t)⟩)\nbeyond α= 1/2 [35, 36]. At α= 1, the spin undergoes\na Berezinskii-Kosterlitz-Thouless (BKT) phase transition\nafter which it becomes localized in one of the two ˆ xstates\n[35, 37].\nFor our purposes, the most important corollary is that\nin the delocalized regime ( α <1), the TLS-splitting, h, is\nrenormalized due to the high-frequency modes of the bath\nwhich must adjust to different positions whenever the hσz\nterm attemps to flip the TLS between the two ˆ xstates(similarly to the Frank-Condon effect of electron-phonon\ncoupling). This renormalization process, along with the\nBKT transition, are governed by the beta functions of α\nand the rescaled splitting ˜h≡h/ωc, which, to order ˜h2,\nare given by\ndα\ndℓ=−α˜h2, (24)\nd˜h\ndℓ= (1−α)˜h, (25)\nwhere eℓis the renormalization group rescaling factor.\nThe flow dictated by Eqs. (24) and (25) on the α−˜h\nplane is shown in Fig. 3: For αnear 1 there exists a\nconstant of flow, x≡(1−α)2−˜h2, such that the BKT\nseperatrix corresponds to the rightmost of the two x= 0\nlines, with the localized (strong coupling) phase to the\nright of it. The effective energy scales, i.e. the renormal-\nized splittings, hR, in the different regions of the phase\ndiagram are given by\nhR=cαωc×\n\n\u0010\nh\nωc\u00111\n1−αIh/ωc≪(1−α)\ne−bωc\nh IIh/ωc≈(1−α) or\nh/ωc≫ |1−α|\ne−π√\n|x|III h/ωc→α−1\n0 IV else(26)\nwhere cαis a numerical prefactor which cannot be de-\ntermined solely from the RG equations (but can be ex-\ntracted using exact techniques such as bosonization or\nBethe Ansatz [43–46]), and b=b(α, x) is a slowly vary-\ning function whose value is of order 1 away from the BKT\ntransition (both are given explicitly in App. B). The lo-\ncalization of the TLSs in region IV is due to the fact that\nthe effective tunneling hRbetween the the two ˆ xstates\nflows to zero in that region.\nUsing this information on the effective low-energy the-\nory, we now turn to the study of correlation functions. As\nwe apply the RG process, the operator σxremains sup-\nported at low energies, such that correlation functions of\nσxdepend solely on parameters of the low-energy the-\nory:αandhR, and not explicitly on horωc. That is, at\nzero temperature, the x-susceptibility can be expressed\nin terms of a one-parameter scaling form [47],\nχ′′\nx(ω)≡1\nωfα\u0012ω\nhR\u0013\n, (27)\nwith hRgiven by (26), and fα(x) being an α-dependent\nscaling function. This relation between the frequency and\nthehRdependence of the susceptibility will be crucial for\nobtaining exact results in the rest of this section. Unlike\nσx, the correlation functions of the operators σz, σycon-\ntain substantial spectral weight at high energies (of order\nEF), and hence cannot be reduced to a single-parameter\nscaling form as in Eq. (27), and will depend on ωcexplic-\nitly2. This fact will be important when discussing the\n2The difference between σxandσy, σzis due to the fact that after8\nFigure 3. Schematic flow diagram of the 1bSB model.\nWe present the RG flow of Eqs. (24) and (25). The solid\nred line is the BKT speratrix between the dynamic phase\n(I−III) and the localized phase (IV), and the dashed black\nline represents the set of points which map into the isotropic\nKondo model. For our purposes, we separate the dynamic\nphase into subregions according to the functional dependence\nof the renormalized scale hRon the bare ˜h. In region I, the\nflow is nearly vertical (the beta function of αis small), and the\nrenormalization of hRis power law like. In region II, the flow\nbehaves similarly to that of the isotropic Kondo model, and so\nthe renormalized scale is exponential, with a weakly varying\nprefactor in the exponent. Finally, in region III the flow slows\ndown significantly due to the vicinity to the BKT transition,\nand the renormalized scale is exponential and depends on the\ndistance from the transition, vanishing exactly on the critical\nline.\nsubleading contributions to the electronic self-energy in\nSec. VI.\nAnother approach to the solution of the 1bSB model is\nvia a mapping to the anisotropic Kondo model (AKM) or\nthe resonant level model (RLM) [35, 49–51]. Remarkably,\natα= 1/2, known as the “Toulouse Point,” the model\nmaps to the non-interacting point of the RLM, and is\nthus exactly solvable. At this point the effective energy\nscale is hR=πh2/4ωc, and the temperature dependent\nsusceptibility is given by\nχ′′\nx(ω, T) =4\nπ2hR\nω2+ 4h2\nR\u0014\nImΦ ( ω, T)−2hR\nωReΦ ( ω, T)\u0015\n,\n(28)\nwhere\nΦ (ω, T) =ψ\u00121\n2+hR\n2πT\u0013\n−ψ\u00121\n2+hR\n2πT−iω\n2πT\u0013\n(29)\nintegrating out the fast modes of the bath, the operators σy, σz\nin the emergent low energy theory are highly dressed polaronic\nversions of the bare ones which move the high-energy bath modes\nalong with the TLS. However, since the bath couples to the TLS\nin the x−basis, integrating it out does not dress σx, and thus\nthe “high energy” and “low energy” σxoperators coincide. See\n[48] for more details.(ψ(z) being the Digamma function). At zero temperature\nthe scaling form takes the exact form\nf1/2(x) =4\nπ21\nx2+ 4\u0002\nxtan−1(x) + ln\u0000\n1 +x2\u0001\u0003\n.(30)\nHaving summarized the essential properties of the SB\nmodel, we shall now turn to evaluate the electronic prop-\nerties of the x−model using the fact that each TLS is\nequivalent to a spin with a randomly distributed split-\ntingh. Note also that the scaling behavior at T= 0 can\nbe extended to sufficiently small T(to be defined below),\nas can be verified explicitly in the Toulouse point and at\nweak coupling, which enables us to estimate the finite- T\nproperties of the model in the following.\nC. Averaged TLS-susceptibility and electronic self\nenergy\nOnce we have determined the TLS susceptibility\nχi(ω), we can determine the fermionic self energy from\nEq.(18). We compute the imaginary part of the retarded\nself energy using\nΣ′′(ω) =M\nNX\na=x,y,zg2\naˆ∞\n−∞dν\n2πχ′′\na(ν)G′′\nR(ω+ν)\n×\u0012\ncoth\u0010ν\n2T\u0011\n−tanh\u0012ν+ω\n2T\u0013\u0013\n. (31)\nIf we use Eq. (19) for the fermion propagator we obtain\nthat at T= 0 (here gy=gz= 0 and gx=g)\nΣ′′(ω) =−M\nNρFg2ˆω\n0dν \n1\nMMX\ni=1χ′′\nx,i(ν)!\n.(32)\nSince each of the MTLSs contributing to this sum has a\nrandomly distributed splitting hi, the self energy of each\nelectron will be a randomly distributed variable. How-\never, since M≫1 the central limit theorem guarantees\nthat this random variable will be normally distributed,\nwith width of order M−1/2. We can thus replace this\nrandom variable by its expectation value (we will revisit\nthis assumption, and consider when it breaks down for\nlarge but finite M, at the end of this section):\n1\nMMX\ni=1χ′′\nx,i(ω)→χ′′x(ω) =ˆ\nPβ(h)χ′′\nx(ω, h)dh. (33)\nOur task thus reduces to the calculation of the TLS\nsusceptibility averaged over the distribution of splittings\nPβ(h). We note however that the distribution which is\nof interest to us is not that of the bare splittings, but\nof the renormalized splittings hR. We can instead define\nthe renormalized distribution\nPr(hR) =\u0012dhR\ndh\u0013−1\nPβ(h), (34)9\nwhich is nonzero up to the renormalized bandwidth\nhc,R≡hR(hc). Combined with Eq. (27), we may write\nthe averaged susceptibility as\nχ′′x(ω) = sgn( ω)ˆ∞\n|ω|/hc,RPr\u0012|ω|\nx\u0013fα(x)\nx2dx. (35)\nWe will now evaluate this integral for the different func-\ntional forms of hR(h) corresponding to the regions I-IV\nin Fig. 3 at zero temperature.\n1.α <1−hc/EF\nIn region I of Fig. 3, where the flow of αis weak, hR\nis a power of h. Thus the effect of the renormalization\nwould be to alter the exponent in the distribution:\nPr(hR) =γhγ−1\nR\nhγ\nc,R, γ≡(1 +β)(1−α).(36)\nFor energy scales far below the renormalized cutoff ω≪\nhc,R, the result would not be sensitive to the exact form of\nthe scaling function fα, and we retrieve results similar to\nthose presented in the perturbative argument in Sec. V A,\nalbeit with a modified exponent:\nχ′′x(ω) = sgn( ω)Aαγ|ω|γ−1\nhγ\nc,R, (37)\nwith Aα≡´∞\nω/hc,Rfα(x)\nxγ+1dx. Since the scaling func-\ntion fα(x→0)∝x2(corresponding to the universal\n1/t2decay of the real-time correlation function at late\ntimes [35, 37, 48]), we may continue the lower limit of\nthe integral to 0 (provided γ < 2), such that Aα=´∞\n0fα(x)\nxγ+1dx+O\u0010\u0010\n|ω|\nhc,R\u0011γ\u0011\n3.\nRemarkably, observe that at low energies, the leading\nfrequency dependence of the response of the whole col-\nlection of TLSs (which is the effective degree of freedom\ncoupled to the electrons) is independent of broadening ef-\nfects of the individual TLSs. Rather, it is governed solely\nby the renormalized distribution Pr, while the functional\nform of the susceptibility of each individual TLS fα(x)\nis absorbed into the prefactor Aα4. Hence, for ω≪hc,R,\nwe find that\nΣ′′(ω) =−AαM\nNρFg2\f\f\f\fω\nhc,R\f\f\f\fγ\n(38)\n3This result follows from dimensional analysis: If the upper in-\ntegration limit can be continued to ∞, the only place where an\nenergy scale appears is in the normalization of the distribution,\n1/hγ\nc,R, which sets the frequency dependence since the suscepti-\nbility is dimensionless.\n4Note that the low-frequency dependence of χ′′\nxserves as a “cut-\noff” for χ′′x: The exponent in the averaged susceptibility does not\nexceed that of the individual susceptibilities, namely, χ′′x∝ |ω|\nfor all γ >2. This is equivalent to the cases where the integral\ninAαdiverges at the lower limit, resulting in a modification of\nthe frequency dependence.=−λAαhc,R\f\f\f\fω\nhc,R\f\f\f\fγ\n. (39)\nWe see that the self-energy depends on the parameters\nαandβonly via γ. In particular, for any initial β≥\n0, the self energy realizes a MFL form upon tuning the\ncoupling to α=β\n1+β, and realizes any NFL exponent γ <\n1 by increasing αtowards 1. Note, however, that as we\nincrease the coupling α, the effective TLS bandwidth hc,R\ndecreases such that smaller NFL exponents are restricted\nto narrower low-energy intervals; see Fig. 1.\nThe temperature dependence of Σ′′at low Tand zero\nfrequency follows from similar considerations. The con-\ntribution of an individual TLS to the self energy can be\nwritten as a scaling function:\nΣ1(ω= 0, T, h) =−ρFg2M\nNfΣ(T/h R). (40)\nWe can thus perform the averaging over hRat this stage,\nand analogously to Eq. (35) we find that\nΣ′′(ω= 0, T) =A′\nαM\nNρFg2\u0012T\nhc,R\u0013γ\n, (41)\nwith A′\nα=γ´∞\n0fΣ(x)\nx1+γdx. Since fΣ(x≪1)∝x2and\nfΣ(x→ ∞ )→1, this is well defined for 0 < γ < 2. This\nsuggests an ω/T scaling of the form Σ′′∝max (|ω|, T)γ.\n2.α >1 +hc/EF\nForα > 1 +hc/EFall the TLSs have undergone the\nBKT transition and are in the localized phase, where the\ndominant TLS contribution at T= 0 is an elastic scat-\ntering term. However, residual quantum fluctuations of\nthe TLSs at finite frequencies provide a weak inelastic\nscattering mechanism which becomes the leading contri-\nbution to the temperature dependence of the dc resistiv-\nity. The finite frequency behavior follows from scaling\nconsiderations (App. D 4, [48]) and is given by\nχ′′x(ω) = (1 −ηα)δ(ω) + 2ηα(α−1)E2−2α\nF\n|ω|3−2α(42)\nηα=2α(1 +β)\n(α−1)(3 + β)\u0012hc\nEF\u00132\n≪1 (43)\nwith ηα∝(hc/EF)2≪1. We thus find in this regime\nthat the leading inelastic contribution to the self energy\nis of the form\nΣ′′(ω) =−ρFg2M\nN \n1−ηα+ηα\u0012|ω|\nEF\u00132α−2!\n.(44)\nThe low energy excitations of the system are thus those\nof a NFL for 1 < α < 3/2, a MFL at α= 3/2 and FL for\nα >3/2. However, note that unlike in the regime α <1,\nhere the elastic contribution is much larger than the in-\nelastic. Note that while this behavior persists up to a10\nlarge energy scale (a fraction of EF), it is expected to be\nthe dominant contribution to the self energy only below\nan energy scale of order 1 /α4−2α(hc/EF)α−1\n2−α, defined as\nthe scale at which conventional FL-like corrections to re-\nsistivity become comparable (i.e. assuming ρcontains an\nadditional T2/EFcontribution).\n3.α≈1\nIn the “critical” region |1−α|< h c/EFthe above\ndescriptions are not valid, since the flow of hslows down\nand becomes comparable to the flow of α. This slowdown\nleads to a logarithmic behavior of the self energy, which\ninterpolates between the regimes described by Eq. (39)\nand Eq. (42). As an example of the behavior in region\nII of Fig. 3, we consider specifically the case α= 1. The\nrenormalized scale is given by hR=c1ωcexp\u0000\n−πωc\n2h\u0001\n,\nand as a result the renormalized distribution becomes\nlogarithmic:\nPr(hR) =(1 +β) log1+β\u0010\nωc\nhc,R\u0011\nhRlog2+β\u0010\nωc\nhR\u0011, (45)\nwhere we ignore subleading corrections (in log( ωc/hc,R)),\nrelated to the prefactor c1; see App. D 2. Inserting\nEq. (45) into Eq. (35), we obtain that\nχ′′x(ω) =1\nω(1+β) log1+β\u0012ωc\nhc,R\u0013ˆ∞\n|ω|\nhc,Rf1(x)\nxlog2+β\u0010\nωc\n|ω|x\u0011dx.\n(46)\nSince the function f1(x) must decay faster than 1 /xfor\nx≫1 (due to the sum rule Eq. (C4)), and also since\n|ω|/ωc≪ |ω|/hc,R, we may ignore the xinside the log\nin the denominator and then, as in the previous case,\ncontinue the lower limit of integration to 0. The resulting\nself energy is given by\nΣ′′(ω) =−M\nNρFg2\n2\nlog\u0010\nωc\nhc,R\u0011\nlog\u0010\nωc\n|ω|\u0011\n1+β\n. (47)\nNote that when αis not exactly 1, the only difference\nwould be that the factor of π/2 in the exponent of hR\nwill vary slightly. Repeating the above calculations, this\nwill only alter the value of hc,R, but not the functional\nform of the self energy.\nFor 1 < α < 1 +hc/EF, we must distinguish the TLSs\ninto those which are dynamical ( h/E F> α−1), and those\nwhich are localized/frozen ( h/E F≤α−1). The contri-\nbution of the dynamical ones will be similar to that of the\nα= 1 case (shown explicitly in App. D 3, the exact result\nis somewhat more involved, but maintains the logarith-\nmic form of (47)), while the frozen ones will contribute an\nelastic scattering term to leading order (i.e. a delta func-\ntion peak around ω= 0), plus higher order Fermi-liquidterms which we ignore. Defining mα≡α−1\nhc/ωcas the frac-\ntion of frozen TLSs, the self energy will include both a\nconstant elastic contribution along with the NFL contri-\nbution attained earlier. For simplicity, setting β= 1 we\nfind that the self energy will be:\nΣ′′(ω) =−M\nNρFg2\nmα+ (1−mα)Bα\nlog\u0010\nωc\nhc,R\u0011\nlog\u0010\nωc\n|ω|\u0011\n2\n.\n(48)\nAsαapproaches 1 + hc/ωcfrom below, both the relative\nweight, 1 −mα, of the inelastic contribution, as well as\nthe energy scale, hc,R, vanishes.\nVI. THE xyz−MODEL\nWe consider a generalized variant of the model, where\nwe allow electron-TLS coupling in arbitrary directions,\ni.e.,g2\na>0 for a=x, y, z (keeping the field hparallel to\nthezdirection), which we dub ‘ xyz−model’. Remark-\nably, we will show that throughout much of the parame-\nter space (of gx−gy−gz), the behavior is qualitatively\nsimilar to that of the x−model, namely, increasing the\ncouplings will generically drive the model towards a BKT\ntransition, leading to a tunable exponent in the electronic\nself-energy which depends on the distance from the tran-\nsition.\nTo proceed we recall that the electronic self energy in\nthe multi-channel model is given by\nΣ′′(ω) =−M\nNX\na=x,y,zρFg2\naˆω\n0χ′′a(ν)dν\n2π.(49)\nAs before, the TLSs are decoupled such that the dynam-\nics of each TLS are determined by solving an independent\nSB model coupled to three ohmic baths. The correspond-\ning multibath SB model (mbSB) for a single TLS is\nHmbSB =−hσz+X\na=x,y,zgaσa\nxϕa+X\na=x,y,zHϕa,(50)\nwhere the bosonic field of (23) is generalized to three\nfields, ϕx, ϕy, ϕz, corresponding to three independent\nbaths which couple to the three spin directions. In our\ncase, all three baths are ohmic and have the same cutoff\n(because ωc=EF), such that the interaction strengths\nare measured via the three dimensionless couplings, αa≡\nρ2\nFg2\na/π2,a=x, y, z . Let us point out that the model\n(50) has two high-symmetry points: the U(1) symmetric\npoint, corresponding to αx=αy; and the SU(2) sym-\nmetric point corresponding to h= 0, αx=αy=αz(for\nrelevant works see, e.g., Refs. [52–54]). These points will\nnot be of particular interest to us, as they are both un-\nstable fixed points (see below) and require fine tuning.\nSimilarly to the 1bSB model, the low-energy proper-\nties of the model can be obtained from an analysis of the11\nRG flow. Here, the RG equations can be derived pertur-\nbatively in two out the of three couplings; while one of\nthe couplings, αa, is allowed to be arbitrarily large, the\nRG equations are valid to linear order in αb̸=a[52–54].\nLoosely speaking, the RG analysis is valid near the axes\nin the ( αx, αy, αz)-coordinate system. The beta functions\nfor the mbSB model are given by\nd˜h\ndℓ= (1−αx−αy)˜h, (51)\ndαa\ndℓ=−\n2X\nb̸=aαb+ (1−δaz)˜h2\nαa.(52)\nIn order to simplify the analysis, we first discuss the cases\nwhere only two of the baths are active (by setting αz= 0\norαy= 0), treating one bath as dominant and the other\nas a perturbation. We then generalize the discussion to\nthe case where all three baths are active, where, as we will\nshow, the physical pictures can be essentially reduced to\nthe simplified cases (of the two active baths). Further, as\nthe model does not contain any other stable fixed points,\nwe expect that the RG analysis will qualitatively capture\nthe physics for all couplings.\nIt is useful to think about the RG of the multi-bath\ncases as a two-step process: the first step describes the\n‘fast’ flow of the couplings, where the dominant bath as-\nsumes a weakly renormalized coupling while the other ir-\nrelevant baths flow to weak coupling; in the second step,\nthe baths renormalize the TLS-fields according to the\nrenormalized couplings. We now proceed to analyze the\ndifferent cases, where the usefulness 2-step perspective\nbecomes apparent.\nA. The xy−model\nWe start by setting αz= 0, i.e., the case where there\nare two active baths acting in the direction perpendicular\nto the field. The beta functions are given by\nd˜h\ndℓ= (1−αx−αy)˜h, (53)\ndαx\ndℓ=−2αyαx−˜h2αx, (54)\ndαy\ndℓ=−2αxαy−˜h2αy. (55)\nIt is insightful to define the bath anisotropy parameter,\nθ≡\u0012αx−αy\nαx+αy\u00132\n, (56)\nwhose beta function is\ndθ\ndℓ= (αx+αy)θ(1−θ). (57)\nThe anisotropy is thus relevant whenever the couplings\nare not finely tuned to the U(1) symmetric point θ= 0,\nFigure 4. Initial flow of the couplings in the xy-model.\nθ=\u0010\nαx−αy\nαx+αy\u00112\nis the bath-anisotropy parameter.\nand flows towards the maximally anistropic case θ= 1,\ni.e. where the larger of the two couplings dominates and\nthe other one becomes irrelevant, as depicted in Fig. 4.\nAs we will now see, since the subdominant bath is ir-\nrelevant it can be integrated out easily, leading to a low\nenergy description similar to that of the x−model with\nrenormalized coupling.\nConsider the case αx≫αy. We focus on the regime\n1−αx≫˜h2(i.e., far enough from the BKT transi-\ntion), where simple analytical estimations can be made\nas the effect of ˜hon the flow of the couplings is neg-\nligible. Indeed, the equations can be solved by utiliz-\ning the fact that δα=αx−αyis an approximate con-\nstant along the flow. The resulting low-energy theory\nis described by the renormalized splitting, hR, and cou-\nplings, αx,R=δα≫αy,R= (hR/ωc)δααy. The BKT\ntransition is determined by the renormalized value of\nthe dominant coupling, αx,R, such that the system be-\ncomes localized when αx,R>1 +O(h/ωc), and below\nthis value the effective energy scale assumes the familiar\nform hR=ωc(h/ωc)1/(1−αx,R)(note that the exponent\ndepends on αx,Rand not on the bare value). More details\non the RG flow are shown in App. E.\nTo proceed, we note that the low energy theory we have\narrived at is nearly identical to that in the 1bSB, the one\ndifference being a remaining weak coupling to the ybath\n(αy,R≪1) which we may now treat perturbatively. The\noperator σxis only weakly dressed (it is renormalized\nonly in the short first section of the flow when αyis of\norder 1), and thus we can once again conclude that its\ncorrelation functions will assume a one parameter scaling\nform, as in Eq. (27):\nχ′′\nx(ω) =1\nωfαx,R\u0012ω\nhR\u0013\n+δχ′′\nx(ω), (58)\nwhere fαx,Ris a scaling function and the perturbative\ncorrection due to the coupling to the losing bath is of the\nform δχ′′\nx(ω) =αy,R1\nω˜fαx,R\u0010\nω\nhR\u0011\n, with a different scaling\nfunction ˜fαx,R. When averaging over the second term.,\nthe strong renormalization of the losing bath, αy,R=\n(hR/ωc)δααy, effectively enhances the exponent in the\nrenormalized distribution which results in a subleading\nfrequency dependence of the averaged δχ′′\nx; see App. F.\nIn contrast, the y−susceptibility does not assume a\none parameter scaling form. Rather, it is suppressed by\nadditional factors of hR/ωc, such that in the IR limit all\nspectral weight is shifted to frequencies of order ωc; see\nApp. F.12\nThe electronic self-energy can be evaluated following\nthe analysis of Sec. V C. Remarkably, the leading term in\n(58) assumes the same form as in the x−model:\nΣ′′(ω) =−Aαx,RρFg2\nx\f\f\f\fω\nhc,R\f\f\f\fγ\n, (59)\nwhere γ= (1+ β)(1−αx,R), and Aδαx,R=´∞\n0fαx,R(x)\nx1+γdx.\nThe subleading correction due to δχ′′\nxis of the form\nδΣ′′(ω) = −Bαyγ\nγ+δαρFg2\nx\f\f\fω\nhc,R\f\f\fγ\f\f\fω\nωc\f\f\fδα\nwith B=\n´∞\n0˜fδα(x)\nx1+γ+δαdx. An additional subleading contribution\nto Σ′′is related to the coupling to the y−susceptibility,\nwhich we denote by δΣ′′\ny. While an explicit evaluation of\nδΣ′′\nyis more challenging, the fact that δΣ′′\nyis also sublead-\ning follows from the additional “non-universal” factors of\nhR/ωcwhich it contains, similarly to the case of δχ′′\nx.\nAs mentioned before, this RG-based analysis is pertur-\nbative in the strength of the weaker coupling, αy. How-\never, at strong coupling (when αx≳αy) the value of αx,R\nis no longer equal to δα, and the BKT line αx,R= 1\nchanges accordingly. The problem has been solved nu-\nmerically for varying coupling strengths by [55], who have\nfound that at strong coupling the BKT line approaches\nthe line αx=αyasymptotically. This behavior is de-\npicted schematically in Fig. 2.\nNote that for U(1)-symmetric points, αx=αy=α,\nthe coupling to the two baths is frustrated and the system\nflows to weak coupling [53, 54, 56]. This case will be\ndiscussed in an upcoming work [57].\nB. The xz−model\nConsider now the case where αx, αz>0 and αy= 0,\ndubbed xz−model. The major difference in this case\ncompared to the xy−model is the fact that the z-bath\nis aligned with the “field” hz, making the two baths in-\nequivalent. The flow equations in this case are:\nd˜h\ndℓ= (1−αx)˜h, (60)\ndαx\ndℓ=−2αzαx−˜h2αx, (61)\ndαz\ndℓ=−2αxαz. (62)\nAs before, we neglect the effect of ˜hon the initial flow of\nthe couplings, assuming that it is sufficiently small.\nLet us start with the case αx≫αz. Then, as the\nx-bath dominates, the flow is essentially identical to the\nxy−model with zreplacing y. The main difference is\nthat σz, albeit being strongly dressed in the low energy\ntheory, has a non-zero equilibrium expectation value, i.e.,\n⟨σz⟩ ̸= 0. The z−susceptibility therefore contains a term\nproportional to ⟨σz⟩2δ(ω). Fortunately, ⟨σz⟩may beevaluated using the sum rule Eq. (C5), yielding\n⟨σz⟩∞=\u0012hR\nωc\u0013δα\na+b1−\u0010\nhR\nωc\u00111−2δα\n1−2δα\n,(63)\nwith aandbbeing numerical constants which depend\nonαx, αz. Interestingly, this static piece contributes an\nelastic scattering term to the electronic self energy:\nΣ′′\nel(ω)∝ −ρFg2\nz\n\n\u0010\nhc,R\nωc\u00112δα\nδα < 1/2,\nhc,R\nωclog2\u0010\nωc\nhc,R\u0011\nδα= 1/2,\n\u0010\nhc,R\nωc\u00112−2δα\n1/2< δα < 1,\n(64)\nwhere O(1) numerical coefficients have been suppressed\nfor clarity. The leading “inelastic” part of Σ′′due to\nthex−bath is identical to Eq. (59), with appropriate\nrenormalized value αx,R. It is worth recalling that the\ninteraction-induced elastic term of Eq. (64) adds to the\nelastic scattering term due to onsite potential in the gen-\neral model. We comment on this matter further in the\ndiscussion on the dc resistivity in Sec. VIII.\nWe move on to the second scenario, where αz≫αx. In\nthis case, the z−bath dominates such that the coupling\nflows to a finite value, αz,R=δα=αz−αx. Unlike the\nprevious case, hzis only marginally renormalized, such\nthat the low-energy parameters are given by\nhR=δα\nαzhz, (65)\nαx,R=αx\u0012hR\nωc\u0013δα\n. (66)\nNote that when δα→0,hRassumes the form shown\nin the xy−isotropic case, although the system is not\nU(1)−symmetric due to hz. For αx= 0 the TLS is static,\nand the scattering is solely elastic. The effect of αxis an\naddition of a weak inelastic term to χ′′\nz, as well as a cou-\npling of the electrons to χ′′\nx. This will result in an inelastic\ncontribution in the self energy such that Σ′′= Σ′′\nel+Σ′′\ninel\nwith\nΣ′′\nel≈ −ρFg2\nz\n2π(67)\nΣ′′\ninel(ω)∝ −|ω|γ+2δα\nhγ\nc,Rω2δαc(68)\nwhere γ= (1+ β). While this is the leading inelastic con-\ntribution, it does not give rise to any MFL/NFL behavior\nfor the considered splitting distributions with β≥0. For\nmore details, see App. F.\nC. The xyz−model\nUnderstanding the physical picture in the more general\ncase where the electrons are coupled to all opeartors of13\nthe TLSs (i.e. αa>0 for a=x, y, z andhx≡0), dubbed\nxyz−model, rests upon the fact that the anisotropy re-\nmains relevant such that one bath dominates over the\nothers at low energies. The qualitative behavior is thus\nreduced to one of the two previous cases. In the case\nwhere the coupling to the x−ory−baths is the largest,\nthe behavior is qualitatively similar to the x−model, al-\nbeit with a renormalized coupling, αR(α) (of the domi-\nnating bath), which depends on the initial values of the\ncouplings. Specifically, the leading inelastic contribution\nto the self-energy satisfies\nΣ′′(ω)−Σ′′(0)∼ −|ω|γ, γ = (1 + βz)(1−αR),(69)\nas we have demonstrated above. In addition, above a\ncritical value, αR(α)≥1, the TLSs undergo a BKT tran-\nsition to the localized phase, where most of the scattering\nis elastic. If, on the other hand, the z−bath dominates,\nthe TLSs act essentially as static impurities, with an ad-\nditional weak, FL-like inelastic contribution to the self\nenergy (as in the xz−model with dominant z−bath).\nVII. “BIASED” MODEL\nAll of our above analyses relied on the assumption that\nthe field, h, is parallel to the ˆ zdirection. Note that this\ncannot always be made the case by rotating a generic field\nh= (hx,0, hz) to point in this direction, since this would\ninduce correlations between the the couplings gx, gz, and\nin turn will lead to a mbSB model with correlated baths.\nWe thus keep the couplings uncorrelated and treat the\ncase were hx, hz̸= 0.\nA. Biased x−model\nWe start by considering the x−model with finite par-\nallel fields hx>0. We thus allow hto be randomly\ndistributed in the x−zplane with a joint distribution\nPβx,βz(hx, hz)∝hβxxhβzzforhx, hz< h c. This variant\nmaps into the “biased” 1bSB, where a field parallel to the\nbath coupling, hxσx, is added to the Hamiltonian [35, 37].\nThe main difference in this case is that hxis unaffected\nby the bath (it commutes with the interaction), while hz\nis renormalized as before. In addition, the presence of a\nnon-zero hximplies that ⟨σx(t→ ∞ )⟩ ̸= 0, which leads\nto an “elastic” delta function term in χ′′\nx(ω), namely, the\nx−susceptibility can be written as χ′′\nx≡χ′′\nel+χ′′\ninel, with\nχ′′\nx,el(ω) =\u00122\nπtan−1\u0012hx\nhR\u0013\u00132\nδ(ω), (70)\nχ′′\nx,inel (ω) =1\nωf\u0012ω\nhR,ω\nhx\u0013\n. (71)To leading order in hc,R/hc, we find that the low-energy\nself-energy is given by Σ′′≡Σ′′\nel+ Σ′′\ninel, with\nΣ′′\nel(ω) =−M\nNρFg2, (72)\nΣ′′\ninel(ω)∝ −M\nNρFg2\f\f\f\fω\nhc\f\f\f\f1+βx\f\f\f\fω\nhc,R\f\f\f\f(1+βz)(1−α)\n.(73)\nNote that since hc/hc,R∝(ωc/hc)α/(1−α)≫1, the\nelastic term contributes most of the spectral weight to\nχx, and correspondingly, Σ′′\nel≫Σ′′\ninelat low energies.\nWe provide an explicit calculation using the 2-parameter\nscaling-form along with exact evaluation at the TP in\nApp. D 5. Notice that for the reasonable case of βx≥0\nthis results in a FL for α < 1 and approaches a MFL\nbehaviour near α= 1.\nB. Biased xyz−model\nWe now treat the most general variant of our model,\nwhere the couplings and fields are all allowed to point\nin generic directions. The behavior of the biased\nxyz−model can be similarly understood from the RG\nanalysis, where Eqs. (52) are modified as [58]\nd˜hz\ndℓ= (1−αx−αy)˜hz, (74)\nd˜hx\ndℓ= (1−αz−αy)˜hx, (75)\ndαa\ndℓ=−\nX\nb̸=aαb+ (1−δaz)˜h2\nz+ (1−δax)˜h2\nx\nαa,\n(76)\nwhere ˜hx≡hx/ωc. Importantly, there are no cross terms\nbetween the fields hx, hzsuch that the qualitative be-\nhavior can be understood in terms of the approximately\nindependent flow of the individual fields. Furthermore,\nsince a bias along ˆ yis forbidden in order to respect time-\nreversal symmetry, the physical picture in the biased\nxyz−model is determined by whether or not the y−bath\ndominates. In the case where the y−bath dominates,\nboth the ‘field’, hz, and the ‘bias’, hx, will be renormal-\nized according to Eq. (26) with α→αy,R. Consequently,\nthe leading inelastic contribution to the self-energy will\ntake the form\nΣ′′(ω)−Σ′′(0)∼ −|ω|γ, (77)\nγ= (2 + βx+βz)(1−αy,R), (78)\nalong with elastic scattering terms as in Eq. (64). In\ncontrast, if the x−orz−baths dominate, the behavior\nwill be analogous the biased x−model with the appro-\npriate renormalized couplings of the dominant bath; see\nEq. (73).14\nVIII. TRANSPORT\nLet us begin by considering the electronic contribution\nto the electrical conductivity, and later incorporate the\neffect of the TLSs on the optical conductivity. Using the\nKubo formula, the real part of the conductivity (associ-\nated with the electrons) is given by\nσel(Ω) =ImΠR\nJx(Ω)\nΩ, (79)\nwhere ΠR\nJxis the retarded current correlator (along the\nxdirection). The current operator is given by J=P\na´\nkvkc†\nakcakandvk=∇kεk. The evaluation of ΠR\nJxis greatly simplified since all vertex corrections vanish due\nto the spatial randomness of the couplings to the local\nTLSs (similarly to Refs. [59, 60]). The electronic optical\nconductivity is thus given by\nσel(Ω) =1\nΩˆ\nωˆ\nkv2\nkAk(ω)Ak(Ω + ω) [f(ω)−f(Ω + ω)].\n(80)\nHereAk(ω)≡ −1\nπImGR\nk(ω) is the electronic spectral\nfunction and f(ω) denotes the Fermi distribution func-\ntion.\nIn the dc limit, the conductivity is given by\nσel(Ω→0) =v2\nFρF\n16Tˆdω\n2π1\n|Σ′′(ω)|sech2\u0010ω\n2T\u0011\n.(81)\nHence, the T-scaling of the dc resistivity follows the\nsingle-particle lifetime. It is instructive to first consider\nρ(T) in the x−model. For α < 1, the low- Tbehavior,\nT≪hc,R, is of the form\nρ(T) =ρ0+ATγ, (82)\nwhere here we have restored the on-site disorder by set-\nting V2>0, corresponding to the residual resistiv-\nity term, ρ0, and γ= (1 + β)(1−α). Similarly, for\nα= 1, we have that ρ(T)−ρ0∝1/|log (T)|1+β; and\nforα > 1 +hc/ωc, the TLSs are frozen at T= 0, and\nthus contribute to the residual resistivity, with FL-like\nfinite- Tcorrections ( ρ(T)−ρ0∝T2). In the intermedi-\nate regime, 1 < α < 1+hc/ωc, the resistivity interpolates\nsmoothly between these two behaviors. In the more gen-\neralxyz−model, the resistivity follows the behavior of\nthex−model whenever the transverse couplings αxor\nαydominate (as discussed extensively in the Sec. VI).\nWhereas, if the parallel coupling αzdominates, the scat-\ntering is mainly elastic with weak FL-like temperature\nscaling. Similarly, the biased xyz−model follows analo-\ngous behavior to that of the x−model provided that the\ny−bath dominates, and to the biased x−model if the x−\norz−bath dominates.\nWe proceed to consider the optical conductivity. In ad-\ndition to the contribution due to the itinerant electrons,\nwe also assume that each TLS carries a randomly dis-\ntributed dipole moment (recall the TLS are phenomeno-\nlogically related to charged collective degrees of freedom)which depends on the state of the TLS:\nHdipole =X\nr,lEr·\u0000\ndz\nr,lσz\nr,l+dx\nr,lσx\nr,l\u0001\n. (83)\nHereEris the local electric field and dx,z\nr,ldenote un-\ncorrelated Gaussian random dipole moments of the TLS\nflavors, with variances d2\nx,z. In total, the (longitudinal)\noptical conductivity takes the two-component form\nσ(Ω) = σel(Ω) + σTLS(Ω). (84)\nThe electronic contribution is standard and follows\nstraightforwardly from the form of the self energy. In\nparticular, at low energies, where −Σ′′(ω) =Γ\n2+c|ω|γ\n(i.e. we restore the elastic scattering term that does not\naffect any of the previous results), if the scattering is\nmainly inelastic (Γ ≪c|Ω|γ):\nσel(Ω)∼\n\n1\nΩγ γ <1,\n1\nΩ log2(1/Ω)γ= 1,\n1\nΩ2−γ γ >1.(85)\nwhile if the scattering is mainly elastic (Γ ≫c|Ω|γ):\nσel(Ω)∼1\nΓ−2γ+1c\n(γ+ 1)Γ2|Ω|γ. (86)\nAt higher energies, Ω ≳hc,R, the TLS contribution to Σ′′\nsaturates to a constant such that σel(Ω≳hc,R)∼1/Ω.\nThe TLS contribution is given by\nσTLS(Ω) = Ω\u0000\nd2\nxχ′′x(Ω) + d2\nzχ′′z(Ω)\u0001\n. (87)\nInterestingly, the TLS contribution follows the frequency\ndependence of the inelastic part of Σ′′(provided that the\ny−bath is not dominant). In particular, if the dipole mo-\nments are not negligibly small, σTLSmight constitute the\nleading frequency dependence, leading to a positive slope\nand non-monotonic behavior of the optical conductivity.\nDefining the energy scale Ω ∗=√ρFvF/da, we find that\nif the scattering is dominantly elastic and Γ ≫Ω∗then\nthere will be an increasing optical conductivity around\nzero frequency. If inelastic scattering dominates, Γ ≪Ω∗,\nthe optical conductivity will always be decreasing around\nzero frequency, but will begin increasing for frequencies\nof order Ω mIR∼ZΩ∗if the system is a FL (with Zthe\nquasiparticle weight), or Ω mIR∼(Ω∗/c)1/γif the system\nis a NFL (i.e. if γ≤1), leading to a so-called mid-\nIR peak around energies of order hc,R(assuming that\nΩmIR< hc,R) [1].\nThe assumption that led to Eq. (87) was that there\nare sufficiently many TLSs that carry a dipole moment\nand can therefore be optically excited. At the same\ntime one expects that there are TLSs that locally come\nwith a quadrupole moment. For example, they could lo-\ncally distort a state of four-fold rotation symmetry to a\nlower symmetry. In this case one can excite the TLS\nvia inelastic light scattering and the Raman response15\nfunction[61, 62] will measure directly the TLS suscep-\ntibilities\nRα,β(Ω) =\u0000\nqx\nα,β\u00012χ′′x(Ω) +\u0000\nqx\nα,β\u00012χ′′z(Ω). (88)\nHere qκ\nα,βis the quadrupole moment due to the κ-\ncomponent of the TLS pseuodspin. The individual tensor\nelements can be detected by an appropriate combination\nof the polarization of the incoming and scattered light.\nHence, the presence of TLS can, at least partially account\nfor the broad Raman continuum that has been observed\nin many correlated electron materials [61].\nIt is intriguing to examine the MFL/NFL transport\nproperties of our model through the viewpoint of Planck-\nian dissipation and the putative bound on transport\ntimes [8]. Since there is no unique definition for the\ntransport time, we consider two different approaches.\nFollowing Ref. [8], we can associate the transport time\nto the single-particle lifetime as the two are propor-\ntional in our model. In that case, the inverse transport\ntime is Planckian in the sense that 1 /τtr∼Twith an\nO(1) coefficient for the NFL phase while at the MFL\npoint the coefficient is O(1/ln(1/T)). In particular, our\nmodel trivially satisfies a ‘Planckian bound’ due to the\nKramers-Kroning relations between the real and imagi-\nnary parts of Σ5. Alternatively the inverse transport time\ncan be defined in terms of the energy scale for which\nthe dc and ac conductivities become comparable [59]:\nσ\u0000\nτ−1\ntr(T), T= 0\u0001\n∼σ(Ω = 0 , T). This procedure agrees\nwith the single-particle lifetime result for NFLs while for\nthe MFL case the transport time contains an additional\nlog correction: τ−1\ntr∼T/log2(T).\nLastly, relying on the analysis of the weakly disor-\ndered MFL (or NFL) model in Ref. [63], we note that\nthe Wiedemann-Franz law is obeyed as T→0, regard-\nless of the existence of well-defined Landau quasiparticles\n(in the absence of vertex correction, as we have here, the\nanalysis is essentially identical).\nIX. THERMODYNAMICS\nIn this section, we study thermodynamic properties of\nthe model. We mainly consider the x−model and further\ndiscuss the expected behavior in the xyz−model. It is\nworth noting that a direct evaluation of the free energy\nfrom the saddle point of the large- Neffective action is\nchallenging due to its non-Gaussian nature. Instead, we\nobtain the specific heat from the internal energy, and\n5Note that for MFLs or NFLs, the Kramers-Kroning relations\nrelate the low energy regimes of the real and imaginary parts of\nthe self energy, such that one is completely determined by the\nother (independent of the UV cutoff). This is in contrast to FLs\nwhere, since Σ′′(ω)∼ |ω|γ+O(|ω|γ) (with γ >1), the low and\nhigh energy sectors both contribute to the leading linear in ω\nbehavior of ReΣ ret(ω) (making Kramers Kroning relations not\nparticularly useful if only low-energy information is accessible).corroborate our results with an alternative derivation of\nthe specific heat from the entropy, where in particular\nwe confirm the absence of T= 0 residual entropy in our\nmodel.\nConsider the internal energy density\nU≡1\nNV⟨Hel+HTLS+Hint⟩, (89)\nwhere HelandHTLScorrespond to the first two terms in\n(1), respectively, and Vis the volume of the system. Let\nus henceforth suppress the factor 1 /(NV) and assume r=\nN/M = 1 for simplicity. By employing the equation of\nmotion for the retarded and advanced electronic Green’s\nfunctions (see App. G), we may write\n⟨Hel+Hint⟩=ˆ\nkˆ\nωωnF(ω)Ak(ω)≡Uel,0,(90)\nwhere nF(ω) is the Fermi function. Note that due to\nthe locality of the self energy, Uel,0corresponds to the\ninternal energy of non-interacting electrons. To see this,\nwe use the fact that´\nkA(ω,k) = ρF, hence Uel,0=\nρF´\nωωnF(ω). The specific heat related to Uel,0is given\nby\ncel,0=ρFˆ\nωω∂nF(ω)\n∂T=π2\n3ρFT, (91)\ni.e., the specific heat of non-interacting electrons. Inter-\nestingly, all interaction effects are encoded in the renor-\nmalized TLS part of the internal energy, UTLS=⟨HTLS⟩,\nwhich we will now evaluate. Using the sum rule Eq. (C5),\nwe may express the TLS specific heat as\ncTLS=1\n2ˆ\nωω∂χ′′x(ω, T)\n∂T. (92)\nAshc,Ris the only energy scale, for ω, T≪hc,Rone can\nwrite the TLS susceptibility as a 2-component scaling\nform, i.e. χ′′(ω, T) =1\nωF\u0010\nω\nhc,R,ω\nT\u0011\n. For concreteness,\nwe assume the following scaling form:\nχ′′x(ω, T) =χ′′x(ω, T= 0)× \n|ω|p\nω2+ (aT)2!φ\n(93)\nwith a∼ O(1) some numerical coefficient and scaling\nexponent φ >0,6which both affect the result only by a\nnumerical prefactor and not the Tdependence.\nUsing Eq. (92), we obtain that\ncTLS∼\n\nT\nhc,Rγ >1,\nT\nhc,Rloghc,R\nTγ= 1,\u0010\nT\nhc,R\u0011γ\nγ <1.(94)\n6Exact results for α≪1 and α= 1/2 indicate that φ= 1.16\nIn addition, Eq. (92) can be evaluated numerically at the\nToulouse point where the exact temperature dependence\nofχ′′(ω, T) is known analytically. The results, confirming\ntheT-dependence found in Eq. (94), are shown in Fig. 5\nforhdistributions corresponding at the Toulouse point\nto a FL, MFL and NFL.\nWe corroborate the above discussion with an alterna-\ntive derivation of the specific heat from the entropy. To\ndo so, we consider the addition of a single TLS per site\nto the theory with M= 0. In the language of the SB\nmodel, the excess entropy added to the system, defined\nbyδS≡S(M= 1)−S(M= 0), is known as the ‘impurity\ncontribution’ [36]. Importantly, δSis determined by the\nspectral function of the bath, the renormalized splitting\nand the temperature. Hence, since the TLSs are decou-\npled for any Mprovided that N≫1 (as gijlgijl′= 0\nforl′̸=l), and the particle-hole bath is ohmic, we may\nwrite the entropy of MTLSs by adding the impurity\ncontributions of the individual TLSs.\nConsidering the x−model, the impurity contribution\nof a single TLS for α <1 is given by [36, 48, 64, 65]\nδS(x) =(\nαπ\n3x+O\u0000\nx3\u0001\nx≪1,\nlog 2 x≫1,(95)\nwhere x=T/h R. The entropy of the full system (i.e. in\nthe large- M, N limit) can therefore be written as\nS(T) =S0(T) + ∆ S(T), (96)\nwhere S0=S(M= 0) denotes the contribution of the\nnon-interacting electrons and\n∆S(T) =MVˆhc,R\n0Pr(hR)δS\u0012T\nhR\u0013\ndhR.(97)\nTo evaluate ∆ S, we divide the integral over hRtohR< T\nandhR> T, denoted by S<andS>, respectively, and,\nsubstituting (36) in (97), we obtain\nS<(T)≈παγ\n3hγ\nc,RT\nhc,Rhγ−1\nc,R−Tγ−1\nγ−1, (98)\nS>(T)≈πα\n3aγ\u0012T\nhc,R\u0013γ\n, (99)\nwhere aγ≡´1\n0δS\u00001\nx\u0001\nxγ−1dx≈1/γ. Hence, for T≪\nhc,R,\n∆S(T)∼\n\nT\nhc,Rγ >1,\nT\nhc,Rloghc,R\nTγ= 1,\u0010\nT\nhc,R\u0011γ\nγ <1.(100)\nSince S0∼T, the total entropy, S, obeys the same scaling\nas ∆S, which also holds for the specific heat, in agree-\nment with Eq. (94).\nIt is also worth noting that there is no residual ex-\ntensive entropy at T= 0, in contrast to theories of\nFigure 5. Specific heat, as extracted from the internal energy\nat the Toulouse point, for values of γfor FL, MFL and NFL\nbehavior (corresponding to β= 2,1,0). As expected from the\nabove scaling arguments (99), at low temperatures the ratio\nc/Tapproaches a constant for the FL, has logarithmic diver-\ngence for the MFL, and polynomial divergence with exponent\n1−γfor the NFL.\nMFLs constructed from variants of the Sachdev-Ye-\nKitaev model [7].\nPhysically, the T−scaling of the impurity contribution\nin Eq. (96) stems from scattering of the low-energy modes\nof the bath by the TLS. Hence in the cases where αxor\nαydominate, the impurity contribution is expected to\nfollow Eq. (96) since the low energy theory is identical\nto that of the x−model (up to weak perturbations). It\ntherefore follows that S(T) satisfies Eq. (100). Moreover,\nwhen αzdominates, the system realizes a FL (with addi-\ntional static impurities) and weak renormalization of the\nsplittings, such that S(T)∝T.\nX. SUPERCONDUCTIVITY\nIn order to study the superconducting instability, we\nintroduce spinful electrons to the Hamiltonian Eq. (1,2),\nnamely, we let c†\ni,r→c†\ni,s,rwith s={↑,↓}. Note that\nsince the TLSs have no spin structure (assuming that\nthe underlying glass is non-magnetic), the couplings g\ndo not depend on spin index s, and the interaction term\nis diagonal in spin space. The mapping of the spinful\nvariant of the Hamiltonian onto a spin-boson problem is\nsimilar to the spinless case, but one must also consider\nthe anomalous bilocal field, analogously to Eqs. (10,11),\ndefined as\nFr,r′(τ, τ′) =1\nNX\nici,↓,r(τ)ci,↑,r′(τ′), (101)\nand enforced via the anomalous self-energy Φr,r′(τ, τ′)\n[40, 66]; see App. H.17\nWe study the critical temperature Tcas a function of\nthe couplings α. Approaching the SC state from the\nnormal state, where Φ = 0, we obtain Tcas the solu-\ntion for the linearized Eliashberg equation (in imaginary\nfrequency) for the local anomalous self-energy Φ( iω):\nΦ(iω) =TX\nω′DΦ(iω−iω′)\n|ω′+iΣ(iω′)|Φ(iω′), (102)\nwhere the bosonic propagator is\nDΦ(iω) =X\na=x,y,zλahc,Rχa(iω)×(−1)δa,y. (103)\nNotice that while the interactions via gxandgzmedi-\nate pairing, gyis pair breaking, as it couples to a cur-\nrent in fermion-flavor space (i.e., to an anti-symmetric\nfermionic bilinear). By rewriting Eq. (102) as Φ( iω) =P\nω′K(iω, iω′)Φ(iω′), we see that Tcis determined as the\nminimal temperature for which the largest eigenvalue of\nthe kernel Kis equal to 1.\nIn the following we first obtain Tcin the spinful variant\nof the x-model as a representative example and later com-\nment on the behavior of Tcin other variants. Specifically,\nwe estimate Tcanalytically by focusing on the “weak”\n(λ≡λx≪1) and “strong” ( λ≫1) coupling regimes,\nwhere Tcis much smaller or larger than the characteris-\ntic energy scale hc,R, respectively. Note that “weak” and\n“strong” coupling regimes do not necessarily correspond\nto small or large values of α. For simplicity we further\nassume that hc/EF≪ |1−α|(to avoid the complications\nresulting from the slowdown of the RG flow of haround\nα= 1) and mainly focus on the parametric dependence\nofTc, ignoring various O(1) coefficients.\n1. Weak coupling (λ≪1)\nIn the weak coupling regime, we rely on the α <1 and\nT= 0 form of the susceptibility given in Eq. (37), and\nassume an ω/T scaling with exponent φ >0 (similar to\nEq. (93))\nχ′′x(ω, T) = sgn( ω)γAα|ω|γ−1\nhγ\nc,R×min\u0012\n1,|ω|\nT\u0013φ\n.(104)\nWe perform analytical continuation to imaginary fre-\nquencies (see App. H) and obtain\nDΦ(iωn)∝γλ\nγ−1 \n1−\f\f\f\fωn\nhc,R\f\f\f\fγ−1!\n.(105)\nIn the FL phase γ >1, the leading piece of DΦat low T\nis constant, resulting in the conventional BCS form of Tc.\nHowever, in the MFL point or NFL phase, where γ≤1,\nDΦ(iω) diverges logarithmically or with exponent γ−1,\nrespectively. Consequently, Tccrosses over from a BCS\nform to an algebraic, quantum critical form. Explicitly,by solving the Eliashberg equation in this regime, we\nobtain\nTc∝hc,R\n\nexp\u0010\n−γ−1\nγλ\u0011\n, γ > 1 +O(p\nλ)\nexp\u0010\n−1√\nλ\u0011\n, γ= 1\n\u0010\nγλ\n1−γ\u00111\n1−γ, γ < 1− O(p\nλ))(106)\nThe results for γ≤1 are similar to those found by [67–69]\nfor other cases of quantum critical pairing. For consis-\ntency, we must require that Tc≪hc,R, which translates\nintoλ≪1. Due to the vanishing of hc,Rasα→1,\nthe problem will eventually cross over to the strong cou-\npling regime, where Tc≫hc,R, beyond some intermedi-\nate value α <1.\n2. Strong coupling (λ≫1)\nWe now consider the transition to superconductivity\nat temperatures T≫hc,R. In this regime, we obtain the\nfinite- TTLS-susceptibility via a combination of scaling\narguments with known results. For details, see App. H.\nWe find that\nχ′′x(ω, T)∝h2\nc\nE2α\nF(max ( |ω|, aαT))2−2α\nω(107)\nwhere aα=O(1). Similarly to the weak coupling limit,\nwe obtain the parametric form of Tcby performing the\nanalytical continuation and solving the linearized Eliash-\nberg equation; see App. H. Remarkably, in an analogous\nfashion to the “weak coupling” regime, we find that Tc\nexhibits a series of crossovers from a quantum critical,\nto a ‘marginal BCS’ [67], to a conventional BCS form as\nthe coupling is increased (rather than decreased, as in\nthe “weak coupling” case). Explicitly, we have that\nTc∝EF\n\n\u0010\nα2\n3−2αϵ\u00111\n3−2α, α < 3/2− O(√ϵ)\nexp\u0010\n−1\nα√ϵ\u0011\n, α= 3/2\nexp\u0000\n−2α−3\nα2ϵ\u0001\n, α > 3/2 +O(√ϵ)(108)\nwhere we defined the small parameter ϵ≡M\nN\u0010\nhc\nEF\u00112\n≪\n1. Interestingly, Tcdecreases up to α= 3, where it has\na local minimum. For larger values of α, it increases and\napproaches the limiting form Tc∝EFexp(−1/αϵ). We\nexpect our results to hold as long as α≪EF/hc(such\nthat αϵ≪1). For consistency of the strong coupling\nanalysis we must require that Tc≫hc,R. While this is\ntrivially fulfilled when α >1, for α <1 this results in the\nrequirement λ≫1, which is, as expected, complimentary\nto the weak coupling condition.\nIntuitively, the reduction of Tcfor larger values of the\ncoupling corresponds to the fact that at finite T, while\nthe TLSs are nearly frozen (i.e. resemble classical impu-\nrities), they preserve their quantum mechanical nature18\nand can thus mediate pairing. The accessible low-energy\nspectral weight for pairing diminishes with the coupling\nstrength and therefore suppresses superconductivity (this\ntrend is reversed beyond α= 3, where the increase in the\ncoupling strength is more significant than the shift of the\nremaining spectral weight to high frequencies).\n3. Superconductivity in other model variants\nFollowing from the discussion of the x-model, we com-\nment on the expected behavior of Tcin generic variants\nof the model. Let us first consider cases with gy= 0\n(i.e., without pair-breaking interactions). In the “weak\ncoupling” regime, in the sense defined above, Tcis deter-\nmined by the behavior of the dominant bath, such that\nit qualitatively follows that of the x-model if αxis domi-\nnant, or otherwise assumes a conventional BCS form (see\nFig. 2). In the “strong coupling” limit, however, the be-\nhavior of Tcis non-universal, namely, it is determined by\nthe susceptibility of the least irrelevant operator. Map-\nping out the quantitative form of Tc(α) necessitate the\nexact renormalized exponents of the TLS-susceptibilities\nand is beyond the scope of our work (given the leading ex-\nponents, the analysis is identical to that of the x-model).\nHowever, recalling that in both cases where αxorαz\ndominates, the strong coupling behavior approaches a\nBCS form, we expect that at intermediate couplings, Tc\nwill smoothly interpolate from a quantum critical to a\nBCS form as αzis increased for fixed αx. Lastly, in-\ntroducing pair-breaking interactions, i.e. a non-zero αy,\nsuppresses Tc[66].\nXI. 1/NCORRECTIONS\nIn this section, we discuss two perturbative corrections\nthat arise at leading order in 1 /N: the validity of the self-\naveraging assumption and the effect of electron-mediated\nTLS-TLS interactions, i.e. RKKY-like interactions.\nA. Validity of self averaging\nAn important assumption of our above analysis lies in\nthe self averaging of the model, which allows us to replace\nthe sum over many TLS-susceptibilities by its mean value\n[with respect to Pr(h)], due to the fact that its variance\nis suppressed by a factor of 1 /M. While this assumption\nis clearly valid in the limit M→ ∞ , for any finite (but\nstill large) Mthe standard deviation may dominate over\nthe mean at sufficiently low energies due to its different\nfrequency dependence. Indeed, consider the variance of\nthe average TLS auto-correlation function in imaginary\ntime:\nVar \n1\nMMX\ni=1\nσi\nx(τ)σi\nx(0)\u000b!\n=1\nMˆ\nPr(hR)S(τ)2dhR.(109)\nFor simplicity let us focus on the regime of interest α <1\nin the x−model. By dimensional considerations, at long\ntimes the dimensionless integral must be proportional to\n(hc,Rτ)−γ(assuming that there is no obstruction to tak-\ning the upper integration limit to ∞, which is the case for\nγ <4). As a result, by taking the square root and trans-\nforming to the frequency domain, we obtain the root-\nmean-square the TLS-susceptibility:\nq\n(χ′′x)2(ω)∼1√\nM|ω|γ/2−1\nhγ/2\nc,R. (110)\nComparing Eq. (110) to the mean in Eq. (37), we con-\nclude that statistical fluctuations can be neglected above\na parametrically small energy scale: ω∼M−1/γhc,R. For\nenergies below this scale, the self-averaging assumption\nis no longer valid and a more systematic treatment of the\n1/M(and 1 /N) fluctuations is needed to determine the\nbehavior of the model.\nB. RKKY interactions\nAnother effect arising when Nis taken to be large but\nfinite, is the emergence of RKKY-like interactions be-\ntween the different TLSs, mediated by the itinerant elec-\ntrons. We analyze this perturbative effect in the spirit of\nRef. [70]. We shall consider the x-model for simplicity,\nthe generalization to other variants is straightforward.\nIncluding the RKKY-like term,\nHRKKY =X\njkgijk,rgi′j′k′,r′\nN2g2\n×Πjk(r−r′, τ−τ′)σx\ni,r(τ)σx\ni′,r′(τ′),(111)\neach TLS will now feel the effect of a sub-ohmic bath aris-\ning from the RKKY coupling to other TLSs, in addition\nto the ohmic particle-hole bath. Following the analysis\nof Ref. [70], this contribution to the bath will be propor-\ntional to χx(iω), and thus the full bath will be of the\nform\nΠ(iω) =α|ω|+λ2\nNh2−γ\nc,R|ω|γ−1(112)\nwith sub-ohmic exponent 2 −γ.\nIn the limit of large yet finite M, N , the sub-ohmic con-\ntribution to the bath may be neglected above the small\nenergy scale ω∼\u0010\nλ2\nαN\u00111/(2−γ)\nhc,R. However, even for\nvery large Nthis energy scale will eventually approach\nhc,Rnearα→1 since λdiverges as hc,R→0. Below this\nscale, the self-consistent approximation of a TLS-induced\nsub-ohmic bath acting on itself breaks down, and a more\nsystematic analysis is needed to determine the behavior\nat very low energies. The low-energy behavior in similar19\ncases [71] suggests that this state remains non-trivial in\nthe sense that γis expected to remain less than 2.\nLastly, note that the subohmic nature of the TLS-\ninduced bath considered above is a result of perturbing\naround the N, M → ∞ saddle point. In a more realistic\nfinite-but-large- Msetting, we expect the subohmic be-\nhaviour to crossover to ohmic below a small energy scale,\ncorresponding to the lowest renormalized splitting of the\nnearby TLSs. In this case, a qualitative change in the\nbehaviour of the TLSs is less obvious, and the system\nmight remain stable to the weak RKKY-like interactions\neven at low energies.\nXII. DISCUSSION AND OUTLOOK\nIn this work, we have studied a class of large Nmod-\nels of itinerant electrons interacting with local two-level\nsystems via spatially random couplings. These mod-\nels, inspired by the possibility of metallic glassiness in\nstrongly correlated materials, exhibit a remarkably rich\nphenomenology at low energies. Most strikingly our the-\nory hosts a robust extended NFL phase in a considerable\npart of parameter space. At the crossover from FL to\nNFL our theory realizes a MFL that shows strange metal-\nlic behavior with T-linear resistivity and Tlog(1/T) spe-\ncific heat. Note that the MFL/NFL behavior does not ne-\ncessitate the existence of a quantum critical point. Physi-\ncally, the departure from FL behavior is rooted in the fact\nthat the characteristic energy of each TLS is algebraically\nsuppressed by the interaction, thus providing significant\nspectral weight of low-energy excitations which consti-\ntute an efficient scattering mechanism for the electronic\ndegrees of freedom. These abundant low-energy excita-\ntions further manifest in a rich phenomenology of the\ncritical transition temperature to the superconducting\nground state of the system.\nThe physical picture of the simplest variant of our the-\nory (the x-model), studied in Ref. [1], qualitatively per-\nsists upon relaxing several simplifying assumptions, such\nas allowing for interactions with different operators of\nthe TLSs and introducing arbitrary TLS-fields. Aiming\nat more realistic models, we further considered the ef-\nfects of relaxing additional simplifications, such as 1 /N\ncorrections, spatial correlations in gijl,rand the self av-\neraging assumption. While these tend to suppress the\nNFL behaviour found in this work below some energy\nscale suppressed by powers of N, there are physical rea-\nsons to think that this scale remains small in a realistic\nsetting. Specifically, recalling that TLSs in physical sys-\ntems are extended objects, the interaction would retain a\nhigh degree of connectivity (i.e. each TLS would interact\nwith many electrons and vice versa), which in turn could\npreserve the self averaging property, and frustrate effects\nof RKKY-like interactions.\nIt is interesting to ask what is the relation between the\ninteraction strengths ( αx,y,z) to actual physical knobs in\nrealistic systems. This is a complicated question as themicroscopic origin of such TLSs is not well understood.\nHowever, there have been many studies attempting to\nprovide a microscopic theoretical framework for under-\nstanding these objects [72–77]. It is possible that as the\nsystem approaches a glassy charge or spin ordering tran-\nsition, the shape, size and other properties of these TLSs\nchange, affecting the magnitude of their coupling to elec-\ntrons, or the relative sizes of the couplings to the x, y, z\noperators. Thus, tuning a physical knob of the system\ncould be parameterized as a nontrivial path in the space\nof couplings, leading to a nontrivial variation of the ex-\nponent in the electronic self energy.\nTo this end, another issue concerns the density of states\nof TLSs, which is parametrically larger than that of the\nelectrons (i.e. h−1\nc,R≫ρF). A direct consequence is the\nseemingly enhanced coupling λ≡M\nNρF\nhc,Rαthat appears\nin the electronic self energy. It appears, however, natural\nto expect that α∼λat least up to some intermediate\ncoupling strength. This is the case if M/N ∼h−1\nc,R/ρF≪\n1, i.e., if the TLSs are sparse compared to the electrons.\nPhysically, this seems plausible based on the mesoscopic\nconsiderations mentioned above.\nNon-Fermi liquid behavior is ultimately tied to an\nanomalous spectrum of gapless excitations. Such a spec-\ntrum is usually believed to emerge from collective modes\nwith soft long wavelength fluctuations. As we showed in\nthis paper, it can also be the result of quantum fluctu-\nations of modes that are localized in a region of size l,\nwhere each mode has an excitation gap Emin∼hRbut\nis governed by a singular distribution function P(hR)∝\nhγ−1\nRwith γ >0. Even if the correlation function for a\ngiven hRdecays rapidly in time, χhR(τ)∼exp (−hRτ),\nthe average χav(τ) =´\ndhRP(hR)χhR(τ) then decays\nlike a powerlaw ∝τ−γand the system becomes critical.\nFor the static susceptibility, χav(T) =´1/Tdτχav(τ), it\nfollows that χav(T)∝Tγ−1andC∝Tγfor the heat\ncapacity. Non-Fermi liquid behavior occurs for γ <1.\nSuch a singular distribution function was also obtained\nfrom quantum Griffiths behavior [78]. Let us therefore\ncompare and contrast our results with the ones that fol-\nlow from quantum Griffiths physics, where rare, large\ndroplets of size loccur with probability pl∝e−cldand\npossess an exponentially small gap hl∝e−bld[78]. This\nyields a power-law for\nP(hR) =ˆ\ndldplδ(hR−hl)∝hγ−1\nR, (113)\nwith non-universal exponent γ=b/c. Exponentially\nsmall gaps occur for the random transverse field Ising\nmodel [78]. However, as soon as one includes the coupling\nto conduction electrons, large droplets will freeze by the\nCaldeira-Leggett mechanism, and one rather finds super-\nparamagnetic behavior of classical droplets [79, 80]. On\nthe other hand, for systems with a continuous order pa-\nrameter symmetry power law quantum Griffiths behav-\nior becomes possible even in the presence of particle-hole\nexcitations [81]. This behavior was also seen in recent20\nnumerical simulations [29]. In contrast to this quantum\nGriffiths behavior, in our approach we consider the cou-\npling of TLSs of characteristic size of several lattice spac-\nings to conduction electrons. While isolated TLSs are\ngoverned by P(h)∝hβthat is, on its own, not suffi-\nciently singular ( β > 0), strong local quantum fluctua-\ntions due to the coupling to conduction-electrons renor-\nmalize the excitation gap h→hR∼h1/(1−α), which\nreduces the exponent β+ 1→βR+ 1 = ( β+ 1)(1 −α).\nWhile our theory does not aim to realistically describe\nany specific material, the existence of a tunable non-\nFermi liquid phase in a controlled microscopic theory\ncould shed new light on some aspects of strange metallic-\nity. It provides a novel viewpoint on the widely observed\nextended strange metal regime [15–17, 82, 83] that does\nnot rely on a putative quantum critical point. Further,\nwhile conventional wisdom typically interprets the resis-\ntivity in terms of a T−linear and a T2components, i.e.\nρ−ρ0=AT+BT2[5, 15, 16, 82, 84–87], our theory\noffers an alternative interpretation7where the exponent\nis a continuous parameter, ρ−ρ0=CTγ. Interestingly,\nthis interpretation (also known as power-law liquid) has\nbeen shown to be consistent with experimental data of\nstrange metals [26, 88–91].\nSeveral natural questions remain open. Aiming to\nbetter understand more realistic scenarios, a systematicstudy of our model for finite- Nis called for, either by\nanalytical or numerical methods. In addition, the behav-\nior deep inside the superconducting state might exhibit\ninteresting new physics, as the electrons constituting the\nOhmic bath in the normal state are becoming gapped,\nwhich has non-trivial effects on the TLSs and vice versa.\nMore broadly, one may consider various other physical\nsystems containing a coexistence of electrons and two-\nlevel systems, where the framework developed in this\nwork can be applied.\nAcknowledgements.— We thank G. Grissonnanche, S.\nA. Kivelson, C. Murthy, A. Pandey, B. Ramshaw, and B.\nSpivak for numerous discussions and for a collaboration\non prior unpublished work. We are grateful to Natan An-\ndrei, Girsh Blumberg, Andrey Chubukov, Rafael Fernan-\ndes, Tobias Holder, Yuval Oreg, and Alexander Shnirman\nfor helpful discussions. J.S. was supported by the Ger-\nman Research Foundation (DFG) through CRC TRR 288\n“ElastoQMat,” project B01 and a Weston Visiting Pro-\nfessorship at the Weizmann Institute of Science. E.B. was\nsupported by the European Research Council (ERC) un-\nder grant HQMAT (Grant Agreement No. 817799) and\nby the Israel-US Binational Science Foundation (BSF).\nSome of this work was performed at KITP, supported\nin part by the National Science Foundation under PHY-\n1748958.\n[1] N. Bashan, E. Tulipman, J. Schmalian, and E. Berg, Tun-\nable non-fermi liquid phase from coupling to two-level\nsystems (2023), arXiv:2310.07768.\n[2] C. M. Varma, Colloquium: Linear in temperature resis-\ntivity and associated mysteries including high tempera-\nture superconductivity, Reviews of Modern Physics 92,\n031001 (2020).\n[3] C. Proust and L. Taillefer, The Remarkable Underlying\nGround States of Cuprate Superconductors, Annual Re-\nview of Condensed Matter Physics 10, 409 (2019).\n[4] J. a. N. Bruin, H. Sakai, R. S. Perry, and A. P. Macken-\nzie, Similarity of Scattering Rates in Metals Showing T-\nLinear Resistivity, Science 339, 804 (2013).\n[5] A. Legros, S. Benhabib, W. Tabis, F. Lalibert´ e, M. Dion,\nM. Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Z.\nLi, P. Auban-Senzier, N. Doiron-Leyraud, P. Fournier,\nD. Colson, L. Taillefer, and C. Proust, Universal T -\nlinear resistivity and Planckian dissipation in overdoped\ncuprates, Nature Physics 15, 142 (2019).\n[6] Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-\nBigorda, K. Watanabe, T. Taniguchi, T. Senthil,\nand P. Jarillo-Herrero, Strange Metal in Magic-Angle\nGraphene with near Planckian Dissipation, Physical Re-\nview Letters 124, 076801 (2020).\n[7] D. Chowdhury, A. Georges, O. Parcollet, and S. Sachdev,\nSachdev-Ye-Kitaev models and beyond: Window into\n7Following the procedure in Ref. [82], the derivative dρ/dT can\ndistinguish the two behaviors provided that the T→0 resistivity\nis measured with sufficient accuracy.non-Fermi liquids, Reviews of Modern Physics 94, 035004\n(2022).\n[8] S. A. Hartnoll and A. P. Mackenzie, Colloquium: Planck-\nian dissipation in metals, Rev. Mod. Phys. 94, 041002\n(2022).\n[9] P. W. Phillips, N. E. Hussey, and P. Abbamonte, Stranger\nthan metals, Science 377, eabh4273 (2022).\n[10] S. Sachdev, Quantum statistical mechanics of the\nSachdev-Ye-Kitaev model and strange metals (2023).\n[11] H. v. L¨ ohneysen, A. Rosch, M. Vojta, and P. W¨ olfle,\nFermi-liquid instabilities at magnetic quantum phase\ntransitions, Reviews of Modern Physics 79, 1015 (2007).\n[12] C. M. Varma, Z. Nussinov, and W. van Saarloos, Singular\nor non-Fermi liquids, Physics Reports 361, 267 (2002).\n[13] S.-S. Lee, Recent Developments in Non-Fermi Liquid\nTheory, Annual Review of Condensed Matter Physics 9,\n227 (2018).\n[14] P. Gegenwart, Q. Si, and F. Steglich, Quantum criticality\nin heavy-fermion metals, Nature Physics 4, 186 (2008).\n[15] R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lipscombe,\nS. M. Hayden, Y. Tanabe, T. Adachi, Y. Koike, M. No-\nhara, H. Takagi, C. Proust, and N. E. Hussey, Anomalous\nCriticality in the Electrical Resistivity of La 2–xSrxCuO 4,\nScience 323, 603 (2009).\n[16] N. E. Hussey, J. Buhot, and S. Licciardello, A tale of\ntwo metals: contrasting criticalities in the pnictides and\nhole-doped cuprates*, Reports on Progress in Physics 81,\n052501 (2018).\n[17] N. E. Hussey, H. Gordon-Moys, J. Kokalj, and R. H.\nMcKenzie, Generic strange-metal behaviour of overdoped\ncuprates, Journal of Physics: Conference Series 449,21\n012004 (2013).\n[18] R. L. Greene, P. R. Mandal, N. R. Poniatowski, and\nT. Sarkar, The Strange Metal State of the Electron-\nDoped Cuprates, Annual Review of Condensed Matter\nPhysics 11, 213 (2020).\n[19] L. Taillefer, Scattering and pairing in cuprate supercon-\nductors, Annu. Rev. Condens. Matter Phys. 1, 51 (2010).\n[20] A. Jaoui, I. Das, G. Di Battista, J. D´ ıez-M´ erida, X. Lu,\nK. Watanabe, T. Taniguchi, H. Ishizuka, L. Levitov, and\nD. K. Efetov, Quantum critical behaviour in magic-angle\ntwisted bilayer graphene, Nature Physics 18, 633 (2022).\n[21] A. Ghiotto, E.-M. Shih, G. S. S. G. Pereira, D. A.\nRhodes, B. Kim, J. Zang, A. J. Millis, K. Watanabe,\nT. Taniguchi, J. C. Hone, L. Wang, C. R. Dean, and\nA. N. Pasupathy, Quantum criticality in twisted transi-\ntion metal dichalcogenides, Nature 597, 345 (2021).\n[22] C. Pfleiderer, Non-Fermi liquid puzzle of MnSi at high\npressure, Physica B: Condensed Matter Proceedings of\nthe Second Hiroshima Workshop on Transport and The\nrmal Properties of Advanced Materials, 328, 100 (2003).\n[23] C. Pfleiderer, D. Reznik, L. Pintschovius, H. v. L¨ ohney-\nsen, M. Garst, and A. Rosch, Partial order in the non-\nFermi-liquid phase of MnSi, Nature 427, 227 (2004).\n[24] N. Doiron-Leyraud, I. R. Walker, L. Taillefer, M. J.\nSteiner, S. R. Julian, and G. G. Lonzarich, Fermi-liquid\nbreakdown in the paramagnetic phase of a pure metal,\nNature 425, 595 (2003).\n[25] S. Paschen and Q. Si, Quantum phases driven by strong\ncorrelations, Nature Reviews Physics 3, 9 (2021).\n[26] H. Zhao, J. Zhang, M. Lyu, S. Bachus, Y. Tokiwa,\nP. Gegenwart, S. Zhang, J. Cheng, Y.-f. Yang, G. Chen,\nY. Isikawa, Q. Si, F. Steglich, and P. Sun, Quantum-\ncritical phase from frustrated magnetism in a strongly\ncorrelated metal, Nature Physics 15, 1261 (2019).\n[27] F. C. Mocanu, L. Berthier, S. Ciarella, D. Khomenko,\nD. R. Reichman, C. Scalliet, and F. Zamponi, Micro-\nscopic observation of two-level systems in a metallic glass\nmodel, The Journal of Chemical Physics 158, 014501\n(2023).\n[28] J. Schmalian and P. G. Wolynes, Stripe Glasses: Self-\nGenerated Randomness in a Uniformly Frustrated Sys-\ntem, Physical Review Letters 85, 836 (2000).\n[29] A. A. Patel, P. Lunts, and S. Sachdev, Localization of\noverdamped bosonic modes and transport in strange met-\nals (2023), arXiv:2312.06751 [cond-mat.str-el].\n[30] J. L. Black, B. L. Gyorffy, and J. J¨ ackle, On the resistivity\ndue to two-level systems in metallic glasses, Philosophical\nMagazine B 40, 331 (1979).\n[31] M. A. Ruderman and C. Kittel, Indirect Exchange Cou-\npling of Nuclear Magnetic Moments by Conduction Elec-\ntrons, Physical Review 96, 99 (1954).\n[32] T. Kasuya, A Theory of Metallic Ferro- and Antifer-\nromagnetism on Zener’s Model, Progress of Theoretical\nPhysics 16, 45 (1956).\n[33] K. Yosida, Magnetic Properties of Cu-Mn Alloys, Physi-\ncal Review 106, 893 (1957).\n[34] W. A. Phillips, Tunneling states in amorphous solids,\nJournal of Low Temperature Physics 7, 351 (1972).\n[35] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.\nFisher, A. Garg, and W. Zwerger, Dynamics of the dissi-\npative two-state system, Reviews of Modern Physics 59,\n1 (1987).\n[36] T. A. Costi and G. Zar´ and, Thermodynamics of the dissi-\npative two-state system: A Bethe-ansatz study, PhysicalReview B 59, 12398 (1999).\n[37] U. Weiss, Quantum Dissipative Systems, 4th ed.\n(WORLD SCIENTIFIC, 2012).\n[38] P. B. Allen and R. C. Dynes, Transition temperature of\nstrong-coupled superconductors reanalyzed, Phys. Rev.\nB12, 905 (1975).\n[39] A. Auerbach, Interacting Electrons and Quantum Magnetism\n(1998).\n[40] I. Esterlis and J. Schmalian, Cooper pairing of incoherent\nelectrons: An electron-phonon version of the Sachdev-Ye-\nKitaev model, Physical Review B 100, 115132 (2019).\n[41] A. O. Caldeira and A. J. Leggett, Influence of Dissipation\non Quantum Tunneling in Macroscopic Systems, Physical\nReview Letters 46, 211 (1981).\n[42] A. O. Caldeira and A. J. Leggett, Quantum tunnelling in\na dissipative system, Annals of Physics 149, 374 (1983).\n[43] K. L. Hur, Entanglement entropy, decoherence, and\nquantum phase transitions of a dissipative two-level sys-\ntem, Annals of Physics 323, 2208 (2008).\n[44] G. Camacho, P. Schmitteckert, and S. T. Carr, Ex-\nact equilibrium results in the interacting resonant level\nmodel, Physical Review B 99, 085122 (2019).\n[45] V. M. Filyov and P. B. Wiegmann, A method for solving\nthe Kondo problem, Physics Letters A 76, 283 (1980).\n[46] V. V. Ponomarenko, Resonant tunneling and low-energy\nimpurity behavior in a resonant-level model, Physical Re-\nview B 48, 5265 (1993).\n[47] T. A. Costi and C. Kieffer, Equilibrium Dynamics of the\nDissipative Two-State System, Physical Review Letters\n76, 1683 (1996).\n[48] F. Guinea, Dynamics of simple dissipative systems, Phys-\nical Review B 32, 4486 (1985).\n[49] P. W. Anderson, G. Yuval, and D. R. Hamann, Exact\nResults in the Kondo Problem. II. Scaling Theory, Qual-\nitatively Correct Solution, and Some New Results on\nOne-Dimensional Classical Statistical Models, Physical\nReview B 1, 4464 (1970).\n[50] F. Guinea, V. Hakim, and A. Muramatsu, Bosonization\nof a two-level system with dissipation, Physical Review\nB32, 4410 (1985).\n[51] M. Sassetti and U. Weiss, Correlation functions for dis-\nsipative two-state systems: Effects of the initial prepara-\ntion, Physical Review A 41, 5383 (1990).\n[52] G. Zar´ and and E. Demler, Quantum phase transitions\nin the Bose-Fermi Kondo model, Physical Review B 66,\n024427 (2002).\n[53] A. H. Castro Neto, E. Novais, L. Borda, G. Zar´ and, and\nI. Affleck, Quantum Magnetic Impurities in Magnetically\nOrdered Systems, Physical Review Letters 91, 096401\n(2003).\n[54] E. Novais, A. H. Castro Neto, L. Borda, I. Affleck, and\nG. Zarand, Frustration of decoherence in open quantum\nsystems, Physical Review B 72, 014417 (2005).\n[55] H. Kohler, A. Hackl, and S. Kehrein, Nonequilibrium dy-\nnamics of a system with quantum frustration, Physical\nReview B 88, 205122 (2013).\n[56] R. Belyansky, S. Whitsitt, R. Lundgren, Y. Wang,\nA. Vrajitoarea, A. A. Houck, and A. V. Gorshkov,\nFrustration-induced anomalous transport and strong\nphoton decay in waveguide QED, Physical Review Re-\nsearch 3, L032058 (2021).\n[57] N. Bashan, E. Tulipman, J. Schmalian, and E. Berg, To\nappear.\n[58] D. V. Khveshchenko, Quantum impurity models of noisy22\nqubits, Physical Review B 69, 153311 (2004).\n[59] D. Chowdhury, Y. Werman, E. Berg, and T. Senthil,\nTranslationally Invariant Non-Fermi-Liquid Metals with\nCritical Fermi Surfaces: Solvable Models, Physical Re-\nview X 8, 031024 (2018).\n[60] A. A. Patel, J. McGreevy, D. P. Arovas, and S. Sachdev,\nMagnetotransport in a Model of a Disordered Strange\nMetal, Physical Review X 8, 021049 (2018).\n[61] T. P. Devereaux and R. Hackl, Inelastic light scatter-\ning from correlated electrons, Rev. Mod. Phys. 79, 175\n(2007).\n[62] U. Karahasanovic, F. Kretzschmar, T. B¨ ohm, R. Hackl,\nI. Paul, Y. Gallais, and J. Schmalian, Manifestation of\nnematic degrees of freedom in the raman response func-\ntion of iron pnictides, Phys. Rev. B 92, 075134 (2015).\n[63] E. Tulipman and E. Berg, A criterion for strange metal-\nlicity in the Lorenz ratio, npj Quantum Materials 8, 1\n(2023).\n[64] R. G¨ orlich and U. Weiss, Specific heat of the dissipative\ntwo-state system, Physical Review B 38, 5245 (1988).\n[65] T. A. Costi, Scaling and Universality in the Anisotropic\nKondo Model and the Dissipative Two-State System,\nPhysical Review Letters 80, 1038 (1998).\n[66] D. Hauck, M. J. Klug, I. Esterlis, and J. Schmalian,\nEliashberg equations for an electron–phonon version of\nthe sachdev–ye–kitaev model: Pair breaking in non-fermi\nliquid superconductors, Annals of Physics 417, 168120\n(2020), eliashberg theory at 60: Strong-coupling super-\nconductivity and beyond.\n[67] D. T. Son, Superconductivity by long-range color mag-\nnetic interaction in high-density quark matter, Phys.\nRev. D 59, 094019 (1999).\n[68] A. Abanov and A. V. Chubukov, Interplay between su-\nperconductivity and non-fermi liquid at a quantum criti-\ncal point in a metal. i. the γmodel and its phase diagram\natT= 0: The case 0 < γ < 1, Phys. Rev. B 102, 024524\n(2020).\n[69] Y.-M. Wu, A. Abanov, Y. Wang, and A. V. Chubukov,\nInterplay between superconductivity and non-fermi liq-\nuid at a quantum critical point in a metal. ii. the γmodel\nat a finite Tfor 0 < γ < 1, Phys. Rev. B 102, 024525\n(2020).\n[70] V. Dobrosavljevi´ c and E. Miranda, Absence of conven-\ntional quantum phase transitions in itinerant systems\nwith disorder, Phys. Rev. Lett. 94, 187203 (2005).\n[71] A. M. Sengupta and A. Georges, Non-fermi-liquid behav-\nior near a T= 0 spin-glass transition, Phys. Rev. B 52,\n10295 (1995).\n[72] V. Lubchenko and P. G. Wolynes, Intrinsic quantum exci-\ntations of low temperature glasses, Physical Review Let-\nters87, 195901 (2001).\n[73] J. Burnett, L. Faoro, I. Wisby, V. L. Gurtovoi,\nA. V. Chernykh, G. M. Mikhailov, V. A. Tulin,\nR. Shaikhaidarov, V. Antonov, P. J. Meeson, etal., Evi-\ndence for interacting two-level systems from the 1/f noise\nof a superconducting resonator, Nature Communications\n5, 4119 (2014).\n[74] D. Khomenko, C. Scalliet, L. Berthier, D. R. Reichman,\nand F. Zamponi, Depletion of two-level systems in ultra-\nstable computer-generated glasses, Physical review let-\nters124, 225901 (2020).\n[75] D. Khomenko, D. R. Reichman, and F. Zamponi, Re-\nlationship between two-level systems and quasilocalized\nnormal modes in glasses, Physical Review Materials 5,055602 (2021).\n[76] W. Ji, T. W. de Geus, E. Agoritsas, and M. Wyart, Mean-\nfield description for the architecture of low-energy excita-\ntions in glasses, Physical Review E 105, 044601 (2022).\n[77] F. C. Mocanu, L. Berthier, S. Ciarella, D. Khomenko,\nD. R. Reichman, C. Scalliet, and F. Zamponi, Micro-\nscopic observation of two-level systems in a metallic glass\nmodel, The Journal of Chemical Physics 158(2023).\n[78] M. Thill and D. Huse, Equilibrium behaviour of quantum\nising spin glass, Physica A: Statistical Mechanics and its\nApplications 214, 321 (1995).\n[79] A. Millis, D. Morr, and J. Schmalian, Local defect in\nmetallic quantum critical systems, Physical Review Let-\nters87, 167202 (2001).\n[80] A. J. Millis, D. K. Morr, and J. Schmalian, Quantum grif-\nfiths effects in metallic systems, Phys. Rev. B 66, 174433\n(2002).\n[81] T. Vojta and J. Schmalian, Quantum griffiths effects in\nitinerant heisenberg magnets, Phys. Rev. B 72, 045438\n(2005).\n[82] N. E. Hussey, R. A. Cooper, X. Xu, Y. Wang,\nI. Mouzopoulou, B. Vignolle, and C. Proust, Dichotomy\nin the T-linear resistivity in hole-doped cuprates, Philo-\nsophical Transactions of the Royal Society A: Mathemat-\nical, Physical and Engineering Sciences 369, 1626 (2011).\n[83] P. M. Lozano, G. D. Gu, Q. Li, and J. M. Tranquada,\nStrange-metal behavior in La 2−xSrxCuO 4beyond the\nstripe-percolation transition (2023), arXiv:2307.13740\n[cond-mat].\n[84] T. Park, V. A. Sidorov, F. Ronning, J.-X. Zhu,\nY. Tokiwa, H. Lee, E. D. Bauer, R. Movshovich, J. L.\nSarrao, and J. D. Thompson, Isotropic quantum scatter-\ning and unconventional superconductivity, Nature 456,\n366 (2008).\n[85] N. Doiron-Leyraud, P. Auban-Senzier, S. Ren´ e de Cotret,\nC. Bourbonnais, D. J´ erome, K. Bechgaard, and L. Taille-\nfer, Correlation between linear resistivity and Tcin\nthe bechgaard salts and the pnictide superconduc-\ntor Ba(Fe 1−xCox)2As2, Physical Review B 80, 214531\n(2009).\n[86] K. Jin, N. P. Butch, K. Kirshenbaum, J. Paglione, and\nR. L. Greene, Link between spin fluctuations and electron\npairing in copper oxide superconductors, Nature 476, 73\n(2011).\n[87] M. Berben, J. Ayres, C. Duffy, R. D. H. Hinlopen, Y.-T.\nHsu, M. Leroux, I. Gilmutdinov, M. Massoudzadegan,\nD. Vignolles, Y. Huang, T. Kondo, T. Takeuchi, J. R.\nCooper, S. Friedemann, A. Carrington, C. Proust, and\nN. E. Hussey, Compartmentalizing the cuprate strange\nmetal (2022).\n[88] Y. Dagan, M. M. Qazilbash, C. P. Hill, V. N. Kulkarni,\nand R. L. Greene, Evidence for a Quantum Phase Transi-\ntion in Pr 2−xCexCuO 4−δfrom Transport Measurements,\nPhysical Review Letters 92, 167001 (2004).\n[89] T. J. Reber, X. Zhou, N. C. Plumb, S. Parham,\nJ. A. Waugh, Y. Cao, Z. Sun, H. Li, Q. Wang, J. S.\nWen, Z. J. Xu, G. Gu, Y. Yoshida, H. Eisaki, G. B.\nArnold, and D. S. Dessau, Power Law Liquid - A\nUnified Form of Low-Energy Nodal Electronic Interac-\ntions in Hole Doped Cuprate Superconductors (2015),\narXiv:1509.01611 [cond-mat].\n[90] S. Smit, E. Mauri, L. Bawden, F. Heringa, F. Ger-\nritsen, E. van Heumen, Y. K. Huang, T. Kondo,\nT. Takeuchi, N. E. Hussey, T. K. Kim, C. Cacho,23\nA. Krikun, K. Schalm, H. T. C. Stoof, and M. S. Golden,\nMomentum-dependent scaling exponents of nodal self-\nenergies measured in strange metal cuprates and mod-\nelled using semi-holography (2021), arXiv:2112.06576\n[cond-mat].\n[91] K. Harada, Y. Teramoto, T. Usui, K. Itaka, T. Fu-\njii, T. Noji, H. Taniguchi, M. Matsukawa, H. Ishikawa,\nK. Kindo, D. S. Dessau, and T. Watanabe, Revised\nphase diagram of the high- Tccuprate superconductor Pb-\ndoped Bi 2Sr2CaCu 2O8+δrevealed by anisotropic trans-port measurements, Physical Review B 105, 085131\n(2022).\n[92] W. Gasser, E. Heiner, and K. Elk,\nGreensche Funktionen in Festk¨ orper- und Vielteilchenphysik\n(Wiley, 2002).\n[93] A. V. Chubukov and J. Schmalian, Superconductivity\ndue to massless boson exchange in the strong-coupling\nlimit, Phys. Rev. B 72, 174520 (2005).\n[94] A. J. Bray and M. A. Moore, Influence of dissipation on\nquantum coherence, Phys. Rev. Lett. 49, 1545 (1982).\nAppendix A: Diagrammatic approach to mapping\nWe now present an alternative approach for the mapping to the spin-boson model, where we demonstrate, by a\nperturbative expansion of the interaction, that the electrons constitute an Ohmic bath to the TLSs. Importantly, due\nto the spatial randomness of the couplings, the bath is that of non-interacting particle-hole pairs, i.e., it is independent\nof the electronic self energy. Consider the correlation function for a single TLS of flavor s:Sx\ns(τ)≡ ⟨Tτ{σx\ns(τ)σx\ns(0)}⟩.\nThe expansion in interaction vertices reads (we suppress the spatial index rsince all operators act on the same site):\nSx\ns(τ) =*\nTτ(\nσx\ns(τ)σx\ns(0)∞X\nn=0(−1)n\nn!\"Y\ni=1...n ˆ\ndτiX\nabcgabc\nNσx\na(τi)c†\nb(τi)cc(τi)!#)+\n, (A1)\nwhich decouples into a sum of terms of the form\nIn=ˆ\nτ1,τ2...τnX\na1b1c1a2b2c2...\u0010ga1b1c1\nNga2b2c2\nN...\u0011\nTτ\b\nσx\ns(τ)σx\ns(0)\u0000\nσx\na1(τi)...σx\nan(τn)\u0001\t\u000b\n×D\nTτn\nc†\nb1(τi1)cc1(τi1)c†\nb2(τi2)cc2(τi2)...oE\n(A2)\nBy integrating over the realizations of gabc, we note that (i) terms where all interaction TLS indices a1, ..., a n̸=sare\n“disconnected” and cancel with the vacuum diagrams; (ii) if only some (but not all) ai=s, the contribution is either\nsubleading in 1 /Nor corresponds to a self-energy insertion for the electrons (see Fig. 6). Thus if we treat the electrons\nself consistently as being fully dressed, the only relevant insertions of the interaction are those in which ai=s.\nFigure 6. Examples of diagrams considered in the mapping. a) If all insertions are of other TLSs, the bubble disconnects from\nthe “external vertices” and cancels with the vacuum diagrams. b) Other insertions of different TLSs become subleading in N.\nc) The contribution of insertions of a different TLS a̸=scan be absorbed into the full electron green function. d) Example of\na contributing diagram, with the wavy lines representing particle hole pairs (as defined in e)). Here electrons are denoted by\nsolid line, contractions over realizations of gabcby dashed lines and TLS operators, who do not admit a direct diagrammatic\nexpansion, by full circles.\nHence Incan be written as (all TLS indices = s)\nIn=ˆ\nτ1,τ2...X\nb1c1b2c2...\u0010gsb1c1\nNgsb2c2\nN...\u0011\n⟨Tτ{σx\ns(τ)σx\ns(0) (σx\ns(τi)...σx\ns(τn))}⟩24\n×D\nTτn\nc†\nb1(τi1)cc1(τi1)c†\nb2(τi2)cc2(τi2)...oE\n(A3)\nConsidering the electronic part (including the couplings gsbici), we see that the leading order contribution in 1 /N\ncorresponds to terms with b1̸=c1̸=b2...(all indices are distinct), namely,\nCn≡X\nb1c1b2c2...1\nNn\u0000\ng2\nsb1c1g2\nsb2c2...\u0001D\nTτn\nc†\nb1(τi1)cc1(τi1)c†\nc1(τi2)cb1(τi2)oE\n(A4)\n×D\nTτn\nc†\nb2(τi1)cc2(τi3)c†\nc2(τi4)cb2(τi4)oE\n...+ permutations + O\u00121\nN\u0013\n. (A5)\nThe particle-hole pairs obey the bosonic Wick’s theorem since the couplings gabcobey it. Therefore,\nCn=˜J(τi1−τi2)˜J(τi3−τi4)...+ permutations (A6)\nwhere we have denoted\n˜J(τi1−τi2)≡g2G(τi1−τi2)G(τi2−τi1) (A7)\nAs mentioned earlier, the fact that the bath is Ohmic follows from spatial randomness of the couplings, which translates\nto the TLS being coupled to the local particle-hole correlators, with G(τ) =´\nkG(τ,k). Upon analytical continuation,\nthe spectral function of the bath is given by\n˜J(ω) =g2ρ2\nF\n2πω≡ηω, (A8)\nsuch that the dimensionless coupling strength (using the conventions of Ref. [37]) is given by\nα≡2\nπη=g2ρ2\nF\nπ2. (A9)\nGeneralizing the derivation to the xyz−model is straightforward.\nAppendix B: Details of RG flow of 1bSB\nWe present here the calculation of the renormalized scale hRin the different regimes I −III in the 1bSB. As a\nreminder, the flow equations are, to order ˜h2:\ndα\ndℓ=−˜h2α (B1)\nd˜h\ndℓ= (1−α)˜h (B2)\nWe allow the couplings to flow until there is only one energy scale in the problem, i.e. until the cutoff Λ = ωc∗exp(−ℓ)\nand TLS energy hR=˜hΛ′become equal, which is given by ˜h(ℓ∗) = 1.\nWe start with the regime where, (1 −α)≫˜h2, so that the flow of αis much slower than the flow of ˜h. We can thus\nsolve the flow of ˜hwhile treating αas a constant, and the weak change in αat the end of the flow (when ˜happroaches\n1) will only change the result by a multiplicative factor, which is absorbed into the definition of the prefactor cαin\nEq. (B4). Therefore, allowing ˜hto flow until it reaches the value 1 we find that\n(1−α)ℓ∗= logωc\nh(B3)\nInserting Eq. (B3) into the definition of hR, we find that\nhR=cαωc\u0012h\nωc\u00131\n1−α\n(B4)\nAs mentioned in the main text, this prefactor cαcannot be determined merely from the RG flow. However, it can be\nextracted using exact techniques such as bosonization or Bethe ansatz [43–46], and is given by\ncα= (Γ(1 −α) exp ( αlogα+ (1−α) log(1 −α)))1\n1−α (B5)25\nwhich satisfies the two known limits c0= 1, c1/2=π/4. We now study the regime near the BKT transition, α≈1.\nIn this regime, we define J= 1−αsuch that |J| ≪1. The RG equations then approximately become:\ndJ\ndℓ=˜h2(B6)\nd˜h\ndℓ=˜hJ (B7)\nNote that the combination x0≡˜h2−J2obeys:\n1\n2dx0\ndℓ=˜hd˜h\ndℓ−JdJ\ndℓ= 0 (B8)\nsox0is constant along flow lines. Using this relation the equations can thus be solved easily\nd˜h\ndℓ=˜hq\n˜h2−x0 (B9)\nℓ∗=ˆ1\n˜h0d˜h\n˜hp˜h2−x0. (B10)\nwhere ˜h0=h/ωc. We now separate to the cases where x0>0 and x0<0. If x0<0, we have that\nℓ∗=asinh\u0010√−x0\n˜h0\u0011\n−asinh (√−x0)\n√−x0(B11)\nhR=ωc\n√−x0\n˜h0+q\n1−x0\n˜h2\n0√−x0+√1−x0\n−1√\n−x0\n(B12)\nIf˜h2\n0≪Jthis expression simplifies to the power law given earlier. On the other hand, for x0→0 this expression\nbecomes the familiar Kondo scale of the isotropic Kondo model hR∝ωcexp(−1/˜h0). For x0>0, we obtain that\nℓ∗=atan\u0010q\n1−x0\nx0\u0011\n−sgn(J)atan\u0012q\n˜h2\n0−x0\nx0\u0013\n√x0. (B13)\nTaking x0→0 also gives the isotropic Kondo result ℓ= 1/˜h0. Setting J= 0 we find that ℓ=π/2˜h0. Therefore,\nin this regime we can approximately think of the renormalized scale as taking the form hR∝ωcexp(−b(J,˜h0)/˜h0)\nwith bbeing a slowly varying function of order 1. However, when the flow approaches the BKT line J=−˜hthis\napproximation does not hold, and instead the renormalized scale is set by the distance from the transition:\nhR∝ωcexp\u0012\n−π√x0\u0013\n. (B14)\nAppendix C: Sum rules for the 1bSB\nThis is based on a short analysis first derived in [48]. We define the correlation function\n⟨σx(t)σx(0)⟩=ˆ\nωeiωtAx(ω) (C1)\nThis is related to the dynamical susceptibility by the fluctuation dissipation theorem [39]:\nAx(ω) =\u0012\n1 + coth\u0012βω\n2\u0013\u0013\nχ′′\nx(ω) (C2)26\nAdditionally, the following equation of motion derives from the Hamiltonian Eq. (23):\nidσx\ndt=−2hσy (C3)\nThus, by Fourier transforming ⟨σx(t)σx(0)⟩,\ndσx\ndt(t)σx(0)\u000b\nand\ndσx\ndt(t)dσx\ndt(0)\u000b\n, setting t= 0 and using the antisym-\nmetry of χ′′\nx(ω) we obtain the three sum rules:\n1 =ˆ\nωχ′′\nx(ω)coth\u0012βω\n2\u0013\n(C4)\n2h⟨σz⟩=ˆ\nωωχ′′\nx(ω) (C5)\n4h2=ˆ\nωω2χ′′\nx(ω)coth\u0012βω\n2\u0013\n(C6)\nAppendix D: Explicit calculation of χ′′xfor the x−model\nThe average (imaginary part of the) susceptibility is given by\nχ′′x=ˆhc\n0χ′′\nx(ω, h)Pβ(h)dh (D1)\n=ˆhc,R\n01\nωfα\u0012ω\nhR\u0013\nPr(hR)dhR (D2)\n= sgn( ω)ˆ∞\n|ω|/hc,Rfα(x)\nx2Pr(|ω|/x)dx (D3)\nThe result of this integral thus depends on the renormalized distribution Pr.\n1.α <1\nStarting with hR=cαωc(h/ωc)1/(1−α)⇒h= (hR/cα)1−αω−α\nc:\nP(hR) =P(h)\u0012dh\ndhR\u0013\n(D4)\n=Nhβ(1−α)−α\nR (D5)\nSince the distribution is cut off at hc,R=hR(hc), the normalization constant must be N=γ/hγ\nc,R, with γ=\nβ(1−α)−α+ 1 = (1 + β)(1−α). Inserting this into the averaged susceptibility gives:\nχ′′x= sgn( ω)ˆ∞\n|ω|/hc,Rfα(x)\nx2γ|ω|γ−1\nhγ\nc,Rxγ−1dx (D6)\n=1\nω\f\f\f\fω\nhc,R\f\f\f\fγ\n×γˆ∞\n|ω|/hc,Rfα(x)\nxγ−1dx (D7)\nUsing that fact that at long times χ(t)∝1/t2, we see that χ′′\nx(ω/hR)∝ω⇒fα(x≪1)∝x2. Near the lower\nintegration limit the integrand is ∝1/xγ−3. Ifγ < 2 then the integral converges when taking ω/hc,R→0, and\ncan thus be considered as a constant. (If γ >2 then the integral diverges and the resulting frequency dependence\nisχ′′x∝ω. This is because the averaged susceptibility cannot decay faster than the susceptibility of the TLSs with\nhighest h.)27\n2.α≈1\nIn this case we use hR=cαωcexp(−bωc/h)⇒h=bωc\n2 log\u0010\ncαωc\nhR\u0011, which gives the renormalized distribution:\nP(hR) =N\nhRlog2+β\u0010\ncαωc\nhR\u0011 (D8)\nand the normalization can be found to be N= (1 + β) log1+β(ωc/hc,R). We neglect for simplicity the factor of\ncα∼ O(1) inside the logarithm. The averaged susceptibility is then given by\nχ′′x= sgn( ω)(1 + β) log1+β(ωc/hc,R)ˆ∞\n|ω|/hc,Rfα(x)\nxlog2+β(xωc/|ω|)dx (D9)\nIn order to simplify the integral, we rely on the fact that fα(x≫1)∝1/x4−2α[48] and f(x≪1)∝x2, such that\nmost of the weight of the integral is around x∼ O(1), for which |log(x)| ≪log(ωc/|ω|). Thus we may neglect the\nxdependence inside the log, giving the form of the susceptibility presented in the main text (using the sum rule\nEq. (C4) for´∞\n0f(x)/xdx = 1/2).\n3. Through the BKT transition ( 1< α < 1 +hc/EF)\nThe behavior around the BKT transition is slightly more convoluted, since when x0→0−the dependence of\nhRon˜his slightly different. However, if we work close enough to the transition, we can just change variables to\ny0(˜h) =√−x0, and use the form (B14) which explicitly depends only on y0. Changing variables we thus find that:\nP(y0) =1 +β\nh1+β\ncy0\u0000\ny2\n0+J2\u0001−1+β\n2(D10)\nwith the cutoff yc=p\n(hc/wc)2−J2, and note that the range y∈(0, yc) covers only the range h∈(|J|, hc), since the\nTLSs with h <|J|are in the localized phase. Thus, if yc≪Jwe can approximate P(y0)∝y0for any β, and we will\ntherefore find that the distribution of P(hR) will be identical to (D8) with β= 1. Thus, while for α= 1 the exponent\nof the log will be 1 + β, it will change smoothly to 2 near the end of the transition. We evaluate this numerically\nfor any value of 1 < α < 1 +hc/ωcusing the form given in (B13), and for the sake of the computation using the\nsimplification χ′′(ω) =δ(|ω| −2hR) (since the results should not depend on the actual function f(x) but rather on\nthe form of the distribution Pr). The results, confirming the analysis presented in this subsection and the previous\none, are shown in Fig. 7.\nFigure 7. Averaged susceptibility of TLSs around the BKT transition, for β= 0,1,2 and varying values of 1 < α < 1 +hc/ωc.\nAs expected, for α= 1 ( J= 0) the susceptibility is ∝1/ωlogβ+2(ωc/ω), while as α→1 +hc/ωc(J→hc/ωc) this changes\nsmoothly into ∝1/ωlog3(ωc/ω). Note that the change in the cutoff of the values in the xaxis with increasing Jis due to the\nlowering of hc,R.28\n4. Localized phase ( α >1 +hc/EF)\nIn the localized phase, where hR= 0, most of the weight of χ′′\nx(ω) lies in a delta function at zero frequency. However,\nthere are still weak residual quantum fluctuations. The form of these fluctuations can be found using a simple scaling\nanalysis: we write the susceptibility at finite frequency as some function χ′′\nx(ω) =1\nωF(h/E F,|ω|/EF). Reducing the\ncutoff to EF→EF/b, the field rescales to h/E F→h/E F/b(1−a). Since the result must be independent of b, we can\nsetb=|ω|/EFand find that\nχ′′\nx=1\nωF \nh\nEF\u0012EF\n|ω|\u00131−α\n,1!\n=1\nωF\u0012h(ω)\n|ω|,1\u0013\n(D11)\nwith h(ω) =h(|ω|/EF)αthe frequency dependant energy scale. Since h(ω)≪ω, we may expand to second order\nusing Fermi’s golden rule, and find\nχ′′\nx(ω)∝1\nω\u0012h(ω)\nω\u00132\n∝sign(ω)h2\nE2α\nF|ω|3−2α(D12)\nThe constant of proportionality may be set using the sum rule Eq. (C6), and then averaging over hwe obtain Eq. (42)\nof the main text.\n5. Biased case\nFor the biased case, the ⟨σx⟩has an equilibrium value, so that ⟨σx(t→ ∞ )σx(0)⟩ → ⟨ σx⟩2. We therefore decompose\nχ′′\nx(ω) =⟨σx⟩2δ(ω) +χ′′\ninel(ω) (D13)\nThe equilibrium value, which contributes to the elastic scattering rate, is given by [51]\n⟨σx⟩=2\nπatan\u0012hx\nhR\u0013\n(D14)\nSince hc/hc,R∝(ωc/hc)α/(1−α)≫1, when averaging over hx, hRwill not have much effect, and thus\n⟨σx⟩2= 1− O\u0012hc,R\nhc\u0013\n(D15)\nThe inelastic contribution χ′′\ninelwill now have a two-parameter scaling form:\nχ′′\ninel(ω) =1\nωfα\u0012ω\nhR,hx\nhR\u0013\n(D16)\nForα <1, the distributions are P(hR) =γ\nhγ\nc,Rhγ−1\nR,P(ϵ) =1+βx\nh1+βxchβxx.Thus:\nχ′′inel(ω) =ˆ\nP(hR, hx)1\nωfα\u0012ω\nhR,hx\nhR\u0013\ndhRdhx\n= (1 + βx)γ1\nhγ\nc,Rhβx+1\ncˆ\nhγ+βx\nRyβxfα\u0012ω\nhR, y\u0013dhR\nωdy\n= (1 + βx)γωγ+βx\nhγ\nc,Rhβx+1\ncˆ∞\nω/hc,Rdx\nxγ+βx+2ˆhcx/ω\n0dyf(x, y)\nSince hc/hc,R≫1, the upper limit of the yintegral is large for any value of x > ω/h c,R. Therefore defining\n˜fα(x) =´∞\n0fα(x, y)dy, we can rewrite the susceptibility in a form similar to earlier:\nχ′′\ninel(ω)≈(1 +βx)γωγ+βx\nhγ\nc,Rh1+βxcAα29\nAα=ˆ∞\n0˜fα(x)\nxγ+βx+2dx\nHere we have assumed that the upper limit of the yintegral and the lower limit of the xintegral can be continued\nto∞safely. In this case, the validity of this assumption is not as clear as it was in the unbiased case. We verify this\nby an explicit calculation at the TP. There, the scaling function is given exactly by [Ulrich Maura]:\nf(x, y) =4\nπ1\nx2+ 4\nxatan ( x+y) +xatan ( x−y) + ln\n\u0010\n1 + (x+y)2\u0011\u0010\n1 + (x−y)2\u0011\n(1 +y2)2\n\n (D17)\nFor large y,f(x, y)∝1\ny2, so the integral over yindeed converges (this should generically be the case since the\nsingle-TLS susceptibility is an analytic symmetric function of ywhich vanishes for y→ ∞ ). In this case the integral\ncan be evaluated exactly, and we find that:\n˜f(x) =4x2\nx2+ 4(D18)\nWe confirm that ˜f(x≪1)∝x2, just as in the 1bSB, and our approximation is justified as long as γ+βx≤1.\nNear the critical point α→1, the splitting distribution takes the form P(hR)∝1\nhR(logωc/hR)2+βz. When integrating\novery, the effective distribution will change to P(hR)∝hβx\nR\n(logωc/hR)2+βz. Therefore the self energy will be of the form:\nΣ′′\ninel(ω)∝ω1+βx\n(logωc/hR)2+βz(D19)\nNote that for the physical case βx=βz= 0 this will result in MFL-like behavior around α≈1.\nAppendix E: RG flow of 2bSB\nWe now discuss the details of the RG flow of the 2-bath SB model. We will mainly consider the region of interest\nwhich is analogous to the α <1 region in the 1bSB, where the effect of hon the flow of the couplings is negligible\nand the renormalization of hRis a power law. Therefore we begin by examining the effect of the two couplings on\neach other. For example, for the xy−model the RG equations will be (as in Eq. (55))\ndαx\ndℓ=dαy\ndℓ=−2αxαy+O(˜h2). (E1)\nWe can simplify these equations by using the constant of flow δα=αx−αy, which is approximately conserved along\nflow. We assume δα > 0 without loss of generality. We can then simply integrate the equations:\n2ℓ=ˆα0\nx\nαx(ℓ)dαx\nαx(αx−δα)=log\u0010\nr\n1−δα/α x\u0011\nδα(E2)\nwhere r=α0\ny/α0\nx, and α0\naare the bare couplings. We thus find that:\nαx(ℓ) =δα\n1−re−2δαℓ(E3)\nαy(ℓ) =rδα\ne2δαℓ−r(E4)\nAssuming that the initial h/ωc≪1 is small enough, the flow will reach δαℓ≪1, at which point the dominant coupling,\nwhich in this case is αx, saturates at the value αx,R=δα, while the subleading coupling continues to decrease:\nαy,R=rδα(Λ′/ωc)δα. Since the flow stops when hR= Λ′then in the low energy theory αy,R=rδα(hR/ωc)δα. Once\nthis point has been reached, we can examine the beta function of ˜h:\nd˜h\ndℓ= (1−αx−αy)˜h (E5)30\nAs mentioned above, after some “time” ℓ∼δα−1(which importantly does not depend on the initial value of h/ωc),\nαxwill saturate, while αybecomes negligible. Thus at this point the flow is identical to the flow of the 1bSB, with\nα=δα. We therefore find that if δα > 1 the tunneling flows to zero and the TLS becomes localized, while for\nδα < 1 the renormalized tunneling assumes the familiar form hR∝ωc(h/ωc)1/(1−δα). Note that in this case the\nproportionality constant will depend on the time it took αxto saturate, which is a quantity which depends on α0\nx, α0\ny\nand not on h, ωc. This can be found exactly by insetring αx,y(ℓ) into (E5) and integrating:\nlogωc\nh=ˆℓ∗\n0\u0012\n1 +δα1 +re−2δαℓ\n1−re−2δαℓ\u0013\ndℓ= (1−δα)ℓ∗−log\u00121−re−2δαℓ∗\n1−r\u0013\n(E6)\n⇒hR∝(1−r)2\n1−δαωc\u0012h\nωc\u0013 1\n1−δα\n(E7)\nThe flow of the couplings in the xz−model is identical. However, if the dominant coupling is αzthen after ℓ≳δα−1\nthe flow of hwill slow down, and thus hwill only by renormalized by a multiplicative factor. We find in this case\n(analogous to only inserting αy(ℓ) into (E5)):\nlogωc\nh=ˆℓ∗\n0\u0012\n1 +δαre−2δαℓ\n1−re−2δαℓ\u0013\ndℓ=ℓ∗−log\u00121−re−2δαℓ∗\n1−r\u0013\n(E8)\n⇒hR=δα\nαzh (E9)\nAppendix F: Subleading corrections in xyz−model\nFollowing the methods of Ref. [48], we characterize the magnitude of the different subleading corrections in the\nmulti-bath case. There are two types of subleading corrections: the susceptibilities of the subdominant baths which\nappear in the self energy, and perturbative corrections to the susceptiblity of the dominant bath due to the weak\ncoupling to the losing baths. We will study these in the xy−model and in the xz−model, and the generalization\nto the xyz−model is straightforward since the couplings to the subleading baths are perturbative in the low energy\ntheory.\n1.xy−model\nAs usual we will assume without loss of generality that αx> α y. As mentioned above, there are two types of\ncorrections to the self energy. We start with that due to perturbative corrections to χx. As presented in [48], if only\nthexbath was present after integrating out the high energy modes, we could expand the ground and excited states\nas (performing perturbation theory in the low-energy modes):\n|g⟩0=\f\f\f˜↓E\n+ϕx\nhR\f\f\f˜↑E\n+1\n2\u0012ϕx\nhR\u00132\f\f\f˜↓E\n+··· (F1)\n|ωi, x⟩0=b†\nx,i|g⟩+··· (F2)\nwhere ϕα=P\ni√\nΠa(ωi)\nhR(b†\na,i+ba,i), Πa(ω)∝αr\naandb†\na,i, ba,iare respectively the bath operator, bath spectral function,\nand boson creation and annihilation operators of the abath. Importantly, the states\f\f\f˜↑,˜↓E\n= 1/√\n2\u0000\f\f˜+\u000b\n±\f\f˜−\u000b\u0001\nare\nsuperpositions of the high-frequency-moded dressed xstates\f\f˜±\u000b\n. This gives the expected χx(ω≪hR)∝αx,Rω/h2\nRat\nlow frequencies, but for general frequencies should be treated in a non-perturbative manner in αx,R. However, since\nαy,Ris small, we can add it perturbatively only to first order:\n|g⟩ ≈ | g⟩0−iϕy\nhR \n1 +\u0012ϕx\nhR\u00132\n+···!\f\f\f˜↑E\n(F3)\nand the relevant excited states will involve insertions of one yboson with multiple xbosons. Using the spectral\ndecomposition for χx:\nχx(ω) =X\nn|⟨n|σx|g⟩|2δ(En−ω) (F4)31\nwe find that the leading correction will come from matrix elements of the form ⟨ω1,···ω2k, x;ωj, y|σx|g⟩. While\nthe summation over the many orders of ϕxis nontrivial, we know that it must produce a scaling function that only\ndepends on ω/hR, and we may thus write:\nχ′′\nx(ω, αy,R) = χ′′\nx(ω,0) +αy,R1\nω˜fαx,R\u0012ω\nhR\u0013\n(F5)\n=1\nωfαx,R\u0012ω\nhR\u0013\n+αy\u0012hR\nωc\u0013αx,R1\nω˜fαx,R\u0012ω\nhR\u0013\n(F6)\nwhere in the second line we inserted the expression for αy. While averaging over the first term will give the usual\ncontribution, in the second term we can treat the distribution as effectively having an increased exponent ˜Pr∼\nhγ−1+αx,R\nR , which will in turn produce a term with a subleading frequency dependence in the averaged susceptibility\n∝ωγ−1+αx,R.\nWe now consider the susceptibility χy, whose spectral decomposition is\nχy(ω) =X\nn|⟨n|σy|g⟩|2δ(En−ω). (F7)\nUsing the fact that the bare σyflips the TLS without properly adjusting the high-energy bosons, we have that\nD\n˜↑|σy|˜↓E\n∝hR\nh∝\u0012hR\nωc\u0013αx,R\n. (F8)\nFor small frequencies we may use the perturbative form (F2) and find that χ′′\ny∝\u0010\nhR\nωc\u00112αx,R\nχ′′\nx, which will naively\ntranslate into a frequency dependence ωγ−1+2αx,Rin the averaged susceptibility. However, the averaged susceptibility\ndepends on the full χ′′\ny, and since for intermediate and high frequencies this perturbation theory is not applicable, we\ncannot fully determine the nonuniversal prefactor, and can only argue that χ′′y∝ωγ−1+ϵwith ϵ >0.\n2.xz−model\nWe begin by studying the similar case where αx> αz. The susceptibility χ′′\nxwill now acquire similar corrections\ndue to αz. However, the matrix elements with single insertions of σzϕzvanish, and we must instead go to second\norder in σzϕz. This means that the corresponding correction to the averaged susceptibility will be ∝ωγ−1+2αx,R.\nThe susceptibility χ′′\nzwill be nonuniversal due to considerations identical to (F8), and will thus be suppressed by\na prefactor\u0010\nhR\nωc\u00112αx,R\nat low frequencies, although we do not know the generalization of it to higher frequencies.\nHowever, in addition this susceptibility includes a static delta function peak due to the equilibrium value of ⟨σz⟩∞.\nFor small αz≪1 this can be calculated using the sum rule Eq. (C5):\n⟨σz⟩=1\nhˆωc\n0f(ω/hR)dω=\u0012hR\nωc\u0013αx\n× ˆωc/hR\n0f(x)dx!\n(F9)\nWe must therefore find if this integral converges or diverges when the upper limit is taken to ωc/hR→ ∞ . Ref. [48]\nshows that f(x≫1)∝1/x2−2α, so that for α <1/2 this integral converges to some constant = Aαwhile for α >1/2\nthis integral diverges as a power law = Aα(ωc/hR)2α−1(forα= 1/2 it diverges logarithmicaly ∝log(ωc/hR)). Thus\nwe can write\nχ′′\nz≈δ(ω)×Aαx\u0012hc,R\nωc\u0013min(αx,1−αx)\n+··· (F10)\nwith the dots referring to the subleading frequency-dependent terms.\nWe now turn to the case where αz> αx. Here, with no αxthe ground state is exactly a coherent state of all the\nbosons centered around the location corresponding to |↓⟩. Acting with the high-frequency mode dressed ˜ σxwill only\nagitate the low energy modes, and thus we can writeD\n˜↑|˜σx|˜↓E\n=sx∼ O(1). Therefore incorporating the effects of\nαx,Rperturbatively will modify the ground state as:\n|g⟩= \n1 +1\n2s2\nx\u0012ϕx\nhR\u00132!\f\f\f˜↑E\n+sxϕx\nhR\f\f\f˜↓E\n+··· (F11)32\nIn terms of the resulting modification to χ′′\nz, we can easily see that the elastic peak will decrease by a small amount\nproportional to αx,R, and the inelastic part will be modified by a term proportional to ( αx,R)2ω/h2\nR(assuming that\nω/hR≪1/αx,R), which in turn will give a correction to χ′′zproportional to ωγ−1+2αz,R. The susceptilibty χ′′\nxis\nsimply a delta function time the factor corresponding to (F8):\nχ′′\nx(ω) =\u0012hR\nωc\u00132αz,R\nδ(ω±2hR) (F12)\nχ′′x(ω)∝ωγ−1+2αz\nx (F13)\nAnd we can thus conclude that the frequency dependence of the self energy in this case will be\nΣ′′(ω)−Σ′′(0)∝|ω|γ+2αz,R\nhγ\nc,Rω2αz,Rc(F14)\nNote that when averaging over hRwe neglect the contribution from TLS whose splitting obeys ω/hR≫(ωc/hR)αz,R→\nhR≪ω(ω/ω c)αz,R/(1−αz,R), for which the perturbation theory breaks down.\nAppendix G: Derivation of specific heat from internal energy\nHere we derive Eq. (90) that enables us to obtain the specific heat of the model following Ref. [92]. Considering\nthe Hamiltonian (1):\nH=X\nαεαc†\nαcα+X\nαβγ,lgl\nαβγc†\nαcβσl\nγ+X\nγhγσz\nγ. (G1)\nThe single-particle quantum numbers stand for combinations of momenta and flavor indices and we also supress factors\nofMandNfor brevity. It is useful to introduce an arbitrary retarded fermionic Green’s function\n⟨⟨A, B⟩⟩r\nt≡ −iθ(t)\n[A(t), B]+\u000b\n(G2)\nand its Fourier transform ⟨⟨A, B⟩⟩r\nω. Here [ A, B]+is the anti-commutator. The advanced Green’s function is given\nby⟨⟨A, B⟩⟩a\nω= (⟨⟨A, B⟩⟩r\nω)∗where ( ·)∗denotes complex conjugation. We use the fact that the retarded and advanced\nGreen’s functions both obey the equation of motion:\nω⟨⟨A, B⟩⟩ω=\n[A, B]+\u000b\nω+\n\n[A, H]−, B\u000b\u000b\nω. (G3)\nWe use Eq. (G3) to obtain the equation for motion of the retarded/advanced fermionic Green’s function:\n(ω−εα)\n\ncα, c†\nα\u000b\u000b\nω= 1 +X\nαβγ,lgl\nαβγ\n\ncβσl\nγ, c†\nα\u000b\u000b\nω. (G4)\nWe proceed to consider the internal energy U≡ ⟨H⟩:\nU=X\nαεα\nc†\nαcα\u000b\n+X\nαβγ,lgl\nαβγ\nc†\nαcβσl\nγ\u000b\n+X\nγhγ\nσz\nγ\u000b\n(G5)\nUsing the identity (that follows from the spectral representation)\n⟨AB⟩=iˆ∞\n−∞dω\n2π⟨⟨A, B⟩⟩r\nω− ⟨⟨A, B⟩⟩a\nω\neβω+ 1, (G6)\nand Eq. (G4) we may express the first two terms in Uas\nX\nαεα\nc†\nαcα\u000b\n+X\nαβγ,lgl\nαβγ\nc†\nαcβσl\nγ\u000b\n=−ˆdω\nπX\nαIm\n\ncα, c†\nα\u000b\u000br\nω. (G7)\nThe right-hand-side can be written in terms of the electronic spectral function:´\nkAk(ω) =−1\nπP\nαIm\n\ncα, c†\nα\u000b\u000br\nω.\nInserting this form to the expression for the internal energy, we obtain the form given in Eq. (90).33\nAppendix H: Superconductivity\n1. Effective action\nWe generalize our model to spin-1 /2 fermions by altering the interaction term in Eq. (1) to\nHint=1\nNX\nr,s,ijlgijl,r·σl,rc†\nirscjrs. (H1)\nHere, c†\nikαis the fermionic creation operator for momentum k,spins=↑,↓, and flavor index i= 1,···, N/2 (so that\nthe total number of electron flavors remains N), and the rest of the definitions are identical to the case in the main\ntext.\nAfter averaging over the coupling constants via replica trick, introducing the bilocal fields (including the new pairing\nfieldF)\nGr,r′s(τ, τ′) =1\nNX\ni¯cirα(τ)cir′s(τ′),\nFr,r′(τ, τ′) =1\nNX\nicir↓(τ)cir′↑(τ′),\nχa,r(τ, τ′) =1\nMX\nlσa\nl,r(τ)σa\nl,r(τ′), (H2)\nand integrating over the fermions, we obtain the effective action:\nS=−Ntr log\u0010\nˆG−1\n0−ˆΣ\u0011\n−NˆX\nr,sGr,r′s(τ, τ′) Σr′,rs(τ′, τ)\n−NˆX\nr,σ\u0010\nF†\nr,r′(τ, τ′) Φr′,r(τ′, τ) +Fr,r′(τ, τ′) Φ†\nr′,r(τ′, τ)\u0011\n−MX\naX\nrg2\naˆ\nχa,r(τ, τ′)\n×\"X\nsGr,rs(τ, τ′)Gr,rs(τ′, τ)−(−1)δa,y2F†\nr,r(τ, τ′)Fr,r(τ′, τ)#\n+MˆX\nrsχa,r(τ, τ′) Πa,r(τ′, τ) +X\nrMX\nl=1STLS[σl,r], (H3)\nwhere\nSTLS[σ] = SBerry [σ]−ˆ\ndτhl,r·σ(τ)\n−ˆ\ndτdτ′X\naΠa,r(τ′−τ)σa(τ)σa(τ′) (H4)\nis the action of a spin-boson problem with multiple baths.\nIn the first term we use a 2 ×2 Nambu-Gor’kov formulation, sufficient for singlet pairing\nˆΣrr′(τ, τ′) =\u0012Σrr′↑(τ, τ′) Φrr′(τ, τ′)\nΦ†\nrr′(τ, τ′)−Σr′r↓(τ′, τ)\u0013\n. (H5)\nGeneralizations to triplet pairing are straightforward but can only play a role for odd-frequency pairing. We use a\nsimilar expression for the propagator\nˆGrr′(τ, τ′) =\u0012Grr′↑(τ, τ′)Frr′(τ, τ′)\nF†\nrr′(τ, τ′)−Gr′r↓(τ′, τ)\u0013\n. (H6)34\nThe bare propagator in frequency and momentum space is\nˆG0k(iω)−1=\u0012\niω−εk 0\n0 iω+εk\u0013\n. (H7)\nIn the limit of large MandN, with fixed ratio M/N , we can analyze the saddle point limit. We consider a saddle\npoint that does not break time-reversal symmetry Grr′↑(τ, τ′) =Grr′↓(τ, τ′) and drop the spin index. Performing\nthe variation with respect to ˆΣ gives\nˆGr,r′(iω) =\u0010\nˆG−1\n0(iω)−ˆΣ (iω)\u0011−1\f\f\f\f\nr,r′. (H8)\nThe variation w.r.t. GandFyield\nΣr,r′(τ) = δr,r′M\nNX\nag2\naGr,r(τ)χa,r(τ) (H9)\nΦr,r′(τ) =−δr,r′M\nNX\na(−1)δa,yg2\naFr,r(τ)χa,r(τ) (H10)\nThese two equations resemble the ones that occur for electrons that couple to bosonic modes with propagator χa,r(τ)\nvia a Yukawa coupling. The stationary point that follows from the variation with respect to χis\nΠa,r(τ) =−2g2\na\u0002\nGr,r(τ)Gr,r(−τ)−(−1)δa,yF†\nr,r(τ)Fr,r(−τ)\u0003\n, (H11)\nan expression that is also analogous to the self energy of a bosonic problem.\nThe TLS-correlation functionD\nσa\nl,r(τ)σa\nl,r(τ′)E\nis determined from the solution of the spin boson problem.\n2. Linearized Eliashberg equations\nAs long as we are only interested in the onset of pairing and the superconducting phase transition is of second order\nwe can focus on the linearized gap equation. In this case we can neglect the feedback of superconductivity on the\nohmic bath. The solution of the spin-boson problem then yields the local propagator χa(ω). The equation for the\nmomentum-independent normal self energy is:\nΣ (iω) =M\nNTX\nω′,ag2\naG(iω′)χa(iω−iω′)\n=M\nNTX\nω′,ag2\naρFˆ\ndεk1\niω′−εk−Σ (ω′)χa(iω−iω′). (H12)\nThe linearized equation for the s-wave anomalous self energy is (assuming particle-hole symmetry for simplicity, we\nuse the fact that iΣ (iω′) is real):\nΦ (iω) =−M\nNTX\nω′,ag2\na(−1)δa,yF(iω′)χa(iω−iω′)\n=M\nNTX\nω′,aˆ\ndεkρFg2\na(−1)δa,yΦ (iω′)χa(iω−iω′)\n(iω′−εk−Σ (iω′)) (−iω′−εk+ Σ ( iω′))\n=M\nNTX\nω′,aˆ\ndεkρFg2\na(−1)δa,yΦ (iω′)χa(iω−iω′)\n(iω′+iΣ (iω′))2+ε2\nk. (H13)\nIf we perform the integration over εk, we get for both self energies the Eliashberg equations.\nΣ (iω) =−iTX\nω′sign ( iω′)DΣ(iω−iω′),35\nΦ (iω) = TX\nω′Φ (iω′)\n|ω′+iΣ (iω′)|DΦ(iω−iω′), (H14)\nwith\nDΣ(iω) =M\nNX\naρFg2\naχa(iω),\nDΦ(iω) =M\nNX\na(−1)δa,yρFg2\naχa(iω). (H15)\nThe contribution due to the coupling gyis pair breaking and sufficiently large gycan partially or fully destroy\nsuperconductivity.\nFor the solution of the linearized gap equation we introduce\n∆ (iωn) =ωnΦ (iωn)\nωn+iΣ(iωn), (H16)\nwhich yields a closed equation\n∆ (iωn) =TX\nn′sign(iωn′)\u0012∆ (iωn′)\nωn′DΣ(iωn−iωn′)−∆ (iωn)\nωnDΦ(iωn−iωn′))\u0013\n. (H17)\nOne nicely finds that if DΣ(iω) =DΦ(iω)≡D(iω) ( i.e. gy= 0) the zeroth bosonic Matsubara frequency does\nnot contribute to the solution of the coupled equation. Static fluctuations are irrelevant for the pairing problem, in\nagreement with Anderson’s theorem. From now on we will assume that gy= 0.\nHence, in what follows, we can just skip n′=nin the sum. Then we do not have any problem with a potentially\ndivergent D(0) for γ≤1.\n∆ (iωn) =TX\nn′̸=n\u0012\n∆ (iωn′)−ωn′\nωn∆ (iωn)\u0013D(iωn−iωn′)\n|ωn′|. (H18)\nFor even frequency pairing we have ∆ ( iωn) = ∆ ( −iωn). Hence we can write (we only consider ωn>0)\n∆ (iωn) = TX\nn′≥0∆ (iωn′)(1−δn,n′)D(iωn−ωn′) +D(iωn+iωn′)\nωn′\n−∆ (iωn)TX\nn′≥0(1−δn,n′)D(iωn−iωn′)−D(iωn+iωn′)\nωn(H19)\nNext we introduce\nΨn=∆ (iωn)\n|ωn|1/2(H20)\nand obtain\nΨn=TX\nn′≥0Ψn′(1−δn,n′)D(iωn−iωn′) +D(iωn+iωn′)√ωnωn′\n−ΨnTX\nn′≥0(1−δn,n′)D(iωn−iωn′)−D(iωn+iωn′)\nωn(H21)\nWe can write this as a matrix equation, where Nmaxis the maximum number of Matsubara frequencies included:\nΨn=∞X\nn′=0Kn,n′Ψ′\nn (H22)\nKn,n′=δn,n′Kdiag\nn+ (1−δn,n′)D(iωn−ωn′) +D(iωn+ωn′)\nπp\n(2n+ 1)(2 n′+ 1)(H23)36\nKdiag\nn =∞X\nm=0(1−δn,m)D(iωn−ωm) + (1 + δn,m)D(iωn+ωm)\nπ(2n+ 1)(H24)\nGiven this matrix equation, we first consider the behavior of Tcfor a power law form of the pairing propagator (with\nsome high energy cutoff Λ, so that the Matsubara sum runs up to Nmax= Λ/T)\nD(iωn) =A\f\f\f\fΛ\nωn\f\f\f\fa\n≡A\u0012Λ\nT\u0013a\ndn (H25)\nwhere we introduced the rescaled propagator dn=1\n(2πn)a. Since Kn,n′is linear in D, we may also define its rescaled\nversion Kn,n′=A\u0000Λ\nT\u0001akn,n′where kn,n′is defined by replacing D(iωn) with dnin the definition of Kn,n′. Defining\nκ(a, N max) as the largest eigenvalue of kn,n′, the Eliashberg equation simplifies to:\n1 =A\u0012Λ\nTc\u0013a\nκ(a, N max) (H26)\nThe qualitative behavior of the eigenvalue can be estimated by inspecting the diagonal elements of k, namely, a\nsimple power counting indicates whether the sum converges or not, which determines the depedence on the cutoff\nNmax. In particular, for a= 0, κ≈κ0log(Nmax) and we find the BCS solution Tc∝Λ exp( −1/Aκ 0). For a >0\nthe series converges for n, n′→ ∞ andκ=κadepends only on the exponent a, giving the critical temperature Tc=\nΛ(Aκa)1/a. A numerical calculation of κais shown in Fig. 8. Finally, consider the case where D(iωn) =Alog(Λ /|ωn|)\n(corresponding to a MFL behavior of the electrons). Using dn= log( Nmax/|n|) one can show numerically that\nκ≈κ′\n0log2(Nmax) such that Tc∝Λ exp( −1/√Aκ0). An analytic derivation of this result using the Eliashberg\nequation is given in [93].\n3. Analytic continuation of the bosonic propagator\nWe start with the imaginary part of the bosonic propagator, assuming a power-law form:\nD′′(ω) = sign( ω)A\f\f\f\fΛ\nω\f\f\f\fa\n(H27)\nAnalytically continuing to Matsubara frequencies (for a <2), we obtain\nD(iΩ) = −1\nπˆ∞\n−∞dω\niΩ−ωD′′(ω) (H28)\n=2\nπˆ∞\n0ωdω\nΩ2+ω2D′′(ω) (H29)\n=2AΛa\nπˆΛ\n0ω1−adω\nΩ2+ω2(H30)\n=2A\n(2−a)πΛ2\n|Ω|22F1\u0012\n1,−a\n2,1−a\n2,−Λ2\nΩ2\u0013\n(H31)\nwhere 2F1is the Gaussian hypergeometric function. In order to obtain anayltical results, we approximate:\nD(iΩ)≈2AΛa\nπ ˆΩ\n0ω1−a\nΩ2dω+ˆΛ\nΩω−1−adω!\n(H32)\n=2A\nπ\u00121\n2−a\f\f\f\fΛ\nΩ\f\f\f\fa\n−1\na\u0012\n1−\f\f\f\fΛ\nΩ\f\f\f\fa\u0013\u0013\n(H33)\nwhich coincides with the appropriate limits of the function 2F1. Note that the leading behavior for |Ω| ≪Λ is of the\nform\nD(iΩ)≈2A\nπ×\n\n1\n|a|a <0\nlog\u0010\nΛ\n|Ω|\u0011\na= 0\n2\na(2−a)\f\fΛ\nΩ\f\fa0< a < 2(H34)\nWe now use the above to obtain Tcin the x-model.37\nFigure 8. Numerical calculation of the largest eigenvalue κ(a, N max) as a function of a(top) and for the special cases a= 0\n(bottom left) and a= 0′(i.e. logarithmic DΦ) (bottom right). For any finite value of athis approaches a constant as Nmax→ ∞ ,\nwhile for a= 0,0′it scales as log( Nmax),log2(Nmax) respectively at large Nmax.\n4. Detailed analysis of the x−model\nWe analyze the scaling of Tcfor two parameter regimes: “weak coupling”, for which Tc≪hc,R, and “strong\ncoupling” for which Tc≫hc,R. We will find approximate solutions by analytically continuing the TLS susceptibility\nχxand identifying the most singular contribution to DΦ(iΩ), from which we obtain Tcusing Eq. (H26).\na. Weak coupling\nHere we study the susceptibility for T, ω≪hc,R. From dimensional considerations, we write the averaged suscep-\ntibility as a scaling function:\nχ′′\nx(ω, T) =Aαγ\nω\f\f\f\fω\nhc,R\f\f\f\fγ\nmin\u0012\n1,\u0010ω\nbT\u0011δ\u0013\n(H35)\nwith some δ >0 and b∼ O(1). Analyzing the susceptibility at weak coupling and at the Toulouse point suggests that\nδ= 1, although as we will see the exact value of δdoes not qualitatively change Tc. The corresponding Matsubara38\nfrequency correlator is given by\nχx(iΩn)≈2γAα\nπhc,R\u0012\n1\nγ−1−2\nγ2−1\f\f\fΩ\nhc,R\f\f\fγ−1\n−δ\n(γ+1)(γ+1+δ)Tγ\nΩ2hγ−1\nc,R\u0013\n(H36)\nThe most singular contribution to DΦis therefore (the temperature dependent term is not significant and can be\nignored, since Ω ≳T)\nDΦ(iΩ)≈2πAαλ×\n\nγ\nγ−1γ >1\nlog\u0010\nhc,R\n|Ω|\u0011\nγ= 1\n2γ\n1−γ2\f\f\fhc,R\nΩ\f\f\f1−γ\nγ <1(H37)\nUsing Eq.(H26) we find that\nTc/hc,R∝\n\nexp\u0010\n−γ−1\n2πAαγκ0λ\u0011\nγ >1 +O(√\nλ)\nexp\u0012\n−1√\n2πAακ′\n0λ\u0013\nγ= 1\n\u0010\n4πAαγκγ−1\n1−γ2λ\u00111\n1−γγ <1− O(√\nλ)(H38)\nwhere the requirement |γ−1|>O(√\nλ) in the BCS and quantum critical regimes is necessary for self consistency.\nAdditionally, demanding that Tc≪hc,R, which is assumed in taking the low- Tform of the TLS-susceptibility, requires\nλ≪1, i.e.\n1≪αM\nNEF\nhc,R∝αM\nN\u0012EF\nhc\u00131\n1−α\n. (H39)\nThis condition will always break down at some α <1. For M/N ∼ O(1) this will happen at very small values of α:\nα∝hc/EF, while in the limit where TLSs are extremely sparse:M\nN≪hc\nEF, this occurs at α≈1−log\u0010EF\nhc\u0011\nlog(N\nM).\nb. Strong coupling\nWe now turn to the regime T≫hc,R. Note that for α >1 this is always the case since hc,R= 0. For frequencies\nω≪Tthe TLS correlation function decays exponentially with rate [94],[35] (Eq 5.29),\nΓ =ch2\nT\u0012T\nEF\u00132α\n(H40)\nwith csome αdependent prefactor. Note that T≫Γ for T≫hR. In this regime, the TLS-susceptibility can be\napproximated as\nχ′′\nx(ω) =1\n2πΓ\nTω\nΓ2+ω2(H41)\n(The prefactor Γ /Tis due to the sum rule Eq. (C4)). For ω≫T, hRthe analysis of App. D 4 can be extended for\nα̸= 1 (or more precisely |α−1|> hc/EF). Overall for ω≫hc,Rone finds that\nχ′′\nx(ω, T) = 4 αh2\nE2α\nF(max ( ω, bT ))2−2α\nω(H42)\nwith b∼ O(1). Analytically continuing and seperating the different frequency regimes, we define\nχx(iΩ) =2\nπ ˆΓ\n0+ˆbT\nΓ+ˆΩ\nbT+ˆEF\nΩ!\ndωωχ′′\nx(ω)\nω2+ Ω2≡χ1+χ2+χ3+χ4. (H43)39\nThus, for Ω > aT≫Γ\nχ1≈1\n3π2Γ2\nTΩ2(H44)\nχ2≈1\nπ2Γ\nΩ2\u0012\n1−Γ\nbT\u0013\n(H45)\nχ3≈2α\nπh2\nE2α\nF|Ω|2α−3\n2α−1 \n1−\u0012bT\n|Ω|\u00132α−1!\n(H46)\nχ4≈2α\nπh2\nE3\nF1\n2α−3 \n1−\u0012|Ω|\nEF\u00132α−3!\n(H47)\nThe most singular contribution to DΦis given by\nDΦ(iΩ) = 2 πϵα2bβ×\n\n2α\n1−2αE3−2α\nF\nT1−2αΩ2 α <1/2\nE2\nF\nΩ2log\u0010\n|Ω|\nT\u0011\nα= 1/2\n2\n(2α−1)(3−2α)\f\fEF\nΩ\f\f3−2α1/2< α < 3/2\nlog\u0010\nEF\n|Ω|\u0011\nα= 3/2\n1\n2α−3α >3/2(H48)\nwhere bβ=\u0010\n1+β\n3+β\u0011\ncomes from averaging over h2. Inserting these into Eq. (H26), we obtain\nTc/EF∝\n\n\u0010\n4πα2κ3−2αbβ\n(2α−1)(3−2α)ϵ\u00111\n3−2αα <3/2− O(√ϵ)\nexp\u0012\n−1\nα√\n2πbβκ′\n0ϵ\u0013\nα= 3/2\nexp\u0010\n−2α−3\n2πα2βκ0ϵ\u0011\nα >3/2 +O(√ϵ)(H49)\nNote that for α≤1/2 the prefactor in the parentheses changes, according to the corresponding expression in Eq. (H48).\nHowever, the dependence on the small parameter ϵremains ϵ1/(3−2α)for all α <3/2.\nOnce again, for α <1 consistency requires that Tc≫hc,R, which translates into\n1≪αM\nN\u0012EF\nhc\u00131\n1−α\n⇐⇒ λ≫1 (H50)\nwhich is complementary to the requirement for weak coupling."    },    {        "title": "1908.04099v2.Low_energy_type_II_Dirac_fermions_and_spin_polarized_topological_surface_states_in_transition_metal_dichalcogenide_NiTe__2_.pdf",        "content": "Low-energy type-II Dirac fermions and spin-polarized topological surface states in\ntransition-metal dichalcogenide NiTe 2\nBarun Ghosh,1Debashis Mondal,2Chia-Nung Kuo,3Chin Shan Lue,3Jayita\nNayak,1Jun Fujii,2Ivana Vobornik,2,\u0003Antonio Politano,4,yand Amit Agarwal1,z\n1Department of Physics, Indian Institute of Technology - Kanpur, Kanpur 208016, India\n2Istituto O\u000ecina dei Materiali (IOM)-CNR, Laboratorio TASC,\nin Area Science Park, S.S.14, Km 163.5, I-34149 Trieste, Italy.\n3Department of Physics, National Cheng Kung University, 1 Ta-Hsueh Road 70101 Tainan, Taiwan\n4Dipartimento di Scienze Fisiche e Chimiche (DSFC),\nUniversit\u0012 a dell'Aquila, Via Vetoio 10, I-67100 L'Aquila, Italy\nUsing spin- and angle- resolved photoemission spectroscopy (spin-ARPES) together with ab initio\ncalculations, we demonstrate the existence of a type-II Dirac semimetal state in NiTe 2. We show\nthat, unlike PtTe 2, PtSe 2, and PdTe 2, the Dirac node in NiTe 2is located in close vicinity of the\nFermi energy. Additionally, NiTe 2also hosts a pair of band inversions below the Fermi level along\nthe \u0000\u0000Ahigh-symmetry direction, with one of them leading to a Dirac cone in the surface states.\nThe bulk Dirac nodes and the ladder of band inversions in NiTe 2support unique topological surface\nstates with chiral spin texture over a wide range of energies. Our work paves the way for the\nexploitation of the low-energy type-II Dirac fermions in NiTe 2in the \felds of spintronics, THz\nplasmonics and ultrafast optoelectronics.\nThe discovery of topological semimetals has ushered\nin a new era of exploration of massless relativistic quasi-\nparticles in crystalline solids[1{4]. These arise as emer-\ngent quasi-particles in crystals with linearly dispersing\nbands in vicinity of a degenerate band crossing point (ei-\nther accidental or symmetry-enforced) and are protected\nby crystalline symmetries[5]. Double, triple and quadru-\nple degeneracy of the band crossing leads to topologi-\ncally protected Weyl[6{14], triple point[15{21] and Dirac\nfermions[22{30], respectively. In contrast to their high\nenergy counter-parts, these emergent quasi-particles are\nnot protected by Lorentz symmetry, and can also occur\nin a tilted form, giving rise to type-I and type-II Dirac\nfermions. Speci\fcally, Na 3Bi[26, 29] and Cd 3As2[25, 30]\nare type-I Dirac semimetal (DSM), while the transition-\nmetal dichalcogenides (TMDs) PtTe 2[31{33], PtSe 2[34],\nand PdTe 2[32, 35, 36] are type-II DSM.\nIn group X Pd- and Pt- based dichalcogenides, the\nbulk Dirac node lies deep below the Fermi level ( \u00180.6,\n\u00180.8 and\u00181.2 eV in PdTe 2, PtTe 2, and PtSe 2, respec-\ntively) [31{36], hindering their successful exploitation in\ntechnology. In contrast, NiTe 2has been predicted to host\ntype-II Dirac fermions in vicinity of the Fermi energy [37].\nThe so far performed experimental studies on NiTe 2have\nprimarily focused on its crystal structure, and transport\nproperties while its topological band structure remains\nunexplored [37{44]. Motivated by this, we explored the\nelectronic band structure of NiTe 2by means of spin- and\nangle-resolved photoemission spectroscopy (ARPES) in\ncombination with density functional theory (DFT).\nOur spin-resolved ARPES measurements explicitly\ndemonstrate the existence of a pair of type-II Dirac nodes\nin NiTe 2along the C3rotation axis, lying just above\n(within 20 meV) the Fermi energy. Additionally, we\nshow that NiTe 2also hosts a series of inverted band-gaps\nFIG. 1. (a) The side view and (b) hexagonal crystal structure\nof NiTe 2with theC3rotation axis. Layers of Ni are stu\u000bed\nbetween two Te layers. (c) The LEED pattern of (0001)-\noriented NiTe 2single crystals, acquired at a primary electron\nbeam energy of 84 eV, clearly indicates its purity and the six-\nfold symmetry along along the (001) direction. (d) The bulk\nand the (001) surface Briluoin zone (BZ) of NiTe 2.\n(IBG). Especially, one of the IBG below the Fermi level\nsupports a Dirac cone in the surface states. Together, the\nbulk Dirac node and the pair of IBG in NiTe 2give rise\nto topological spin-polarized surface states over a wide\nrange of energies. This non-trivial band morphology in\nNiTe 2originates primarily from the 5 p-orbital manifold\nof the Te atoms modi\fed by the intra-layer hybridization,\ntrigonal crystal \feld splitting and spin-orbit coupling.arXiv:1908.04099v2  [cond-mat.mes-hall]  13 Aug 20192\nFIG. 2. (a) Band structure of NiTe 2(including SOC) clearly showing the tilted type-II Dirac node along the \u0000 \u0000Adirection\n(at D). The irreducible representation of the bands (close to Fermi energy) at the \u0000 and Apoints are also marked. (b) The\nFermi surface of NiTe 2originating from the crossing Dirac bands. The type-II Dirac points appear at the touching points of the\nelectron and hole pockets. (c) kzdispersion along \u0000 \u0000Adeduced from the h\u0017-dependent data measured along \u0016\u0000\u0000\u0016Kdirection\n(shown in part in (d-h)); red dashed lines represent the DFT calculations; (d)-(h) The measured band dispersion along the\n\u0016K\u0000\u0016\u0000\u0000\u0016Kdirection for di\u000berent kzvalues, with the red dashed lines indicating the bulk DFT band structure. The blue arrows\nin panel (f) mark the surface states. Data were taken at photon energies of 17, 19, 21, 23 and 25 eV, respectively. Panel (f)\ncorresponding to kz= 0:34c\u0003, is closest to the location of the Dirac point ( kz= 0:35c\u0003in our DFT calculations). Note that,\nfor matching with the experimental data, we have shifted the DFT band structure downward by 100 meV.\nBulk NiTe 2crystallizes in the CdI 2type trigonal struc-\nture (space group P\u00163m1, number 164). It has a layered\nstructure with individual monolayer stacked together via\nweak van der Walls force. As shown in Fig. 1(a)-(b),\neach monolayer has three sub-layers, with the central Ni\nlayer being sandwiched between two adjacent Te layers\n(Ni-Te bond length 2.60 \u0017A). The observation of sharp\nspots in the low-energy di\u000braction pattern (LEED) in\nFig. 1(c) con\frms the high quality of the NiTe 2crystals\ncleaved along the (001) direction, along with the pres-\nence of six-fold symmetry. Surface cleanliness of the as-\ncleaved samples was checked by high-resolution electron\nenergy loss spectroscopy and X-ray photoelectron spec-\ntroscopy. The details of crystal preparation and charac-\nterization, ARPES measurements and DFT calculations\nare presented in Sec. S1, S2 and S3, of the Supplemen-\ntary material (SM) [45].\nThe electronic band structure of NiTe 2including spin-\norbit coupling is shown in Fig. 2(a). It clearly depicts\nthe presence of a pair of tilted band crossings along the\u0000\u0000Adirection. The presence of inversion and time-\nreversal symmetry mandates these bands to be doubly\ndegenerate. Furthermore, the \u0000 \u0000Ahigh-symmetry di-\nrection is the invariant subspace of the three fold rotation\n(C3) symmetry, and a symmetry analysis reveals that the\ncrossing bands have opposite rotation character. This\nprevents their hybridization, resulting in a pair of gapless\nquadruply degenerate type-II Dirac points. The type-II\nnature of the DSM phase is also con\frmed by the fact\nthat the Dirac point appears at the touching point of the\nelectron and hole pocket, as highlighted in Fig. 2(b).\nOur photon-energy dependent ARPES data, in part\npresented in Fig. 2(d-h), results in the kzdispersion pre-\nsented in Fig. 2(c). Our experimental results are con-\nsistent with the DFT-based bulk band structure calcu-\nlations. Extrapolating the \ftted DFT band structure,\nwe \fnd that the Dirac cone is located just above ( \u001820\nmeV) the Fermi energy. in contrast to other TMD-\nbased type-II DSMs like PtTe 2[31{33], PtSe 2[34], and\nPdTe 2[32, 35, 36]. Our attempt to electronically dope3\nFIG. 3. (a) The evolution of the Te 5p orbitals in the formation of Dirac-cone states in NiTe 2. Step (I) shows the the creation of\nbonding and anti-bonding orbitals. Step (II) shows the e\u000bect of the strong trigonal crystal \feld which separates the pzorbitals\nfrom thepx;yorbitals. In step (III), we show the splitting of these states into the jJ;jmJjistates in the presence of SOC. In step\n(IV), we demonstrate the e\u000bect of out of plane dispersion and the formation of the Dirac point. (b) The orbital-resolved band\nstructure and various band inversions along the \u0000 Ahigh-symmetry direction is shown along with the irreducible representations\nof the bands. In panels (c) and (d) we show the ARPES data (for ~\u0017= 24 eV) along \u0016K\u0000\u0016\u0000\u0000\u0016Kand \u0016M\u0000\u0016\u0000\u0000\u0016M(for~\u0017= 30\neV) directions of the (001) surface BZ, respectively. ARPES results are consistent with the DFT predictions in panels (e) and\n(f). To match the ARPES results with DFT, we have used a surface potential of \u00000:14 eV.\nthe sample via alkali metal (potassium) deposition (see\nFig. S6 in the SM [45]), and shift the Fermi energy above\nthe Dirac point, revealed that only the surface states in\nNiTe 2are impacted by surface deposition. Bulk doing is\nneeded to shift the bulk bands.\nExtended energy range ARPES spectra for the two\nhigh-symmetry directions are shown in Fig. 3(c) and\n(d). These spectra are mainly dominated by the surface\nstates, as seen from the comparison with the calculated\nband structure in Fig. 3 (e) and (f). In order to under-\nstand their origin, we note that there are two symmetry\ninequivalent Te atoms (Te1and Te2) and a single Ni atom\nin an unit cell of NiTe 2. The electronic con\fguration of\nNi is 3 s23d8and that of Te is 4 d105p4. We \fnd that sim-\nilar to other group-X TMDs [32, 46], the Te 5 porbital\nmanifold in NiTe 2, aided by the interplay between intra-\nlayer hopping, crystal \feld splitting, and SOC strength,\ngives rise to most of the bulk Dirac nodes and multiple\ninverted band-gaps. To highlight this, we show the evolu-\ntion of the p-orbital manifold of the Te atoms in Fig. 3(a).\nTo start with (step I in Fig. 3(a)), strong intra-layer hy-bridization between the Te1and Te2porbitals results\nin bonding and anti-bonding states. These orbitals are\nfurther split (in step II), due to a strong trigonal crystal\n\feld generated by the layered crystal structure of NiTe 2,\nseparating pzfrom the px; pyorbitals. Inclusion of SOC\n(step III) further splits the orbitals into jJ;jmJjistates.\nStep IV of Fig. 3(a) highlights the e\u000bect of the dispersion\nalong the \u0000\u0000Adirection, and the formation of the bulk\ntype-II Dirac point along with multiple band-inversions\nin the valance band.\nThe irreducible representation of some of these states\nat the at \u0000 and Apoints and along the \u0000 \u0000Ahigh-\nsymmetry line is shown in Fig. 3(b). The bulk Dirac\npoint originates from the crossing of the \u0001 4and \u0001 5;6\nstates along the \u0000 \u0000Adirection. As discussed earlier,\nthe doubly degenerate \u0001 4and \u0001 5;6bands have opposite\nrotation characters (+1 and \u00001, respectively) and, there-\nfore, the Dirac point is protected from gap opening by\nany perturbation which respects the C3symmetry. Ad-\nditionally, Fig. 3(b) also highlights the existence of a pair\nof IBGs in the valance band at the Apoint. However,4\nFIG. 4. Measured and calculated spin texture for the bands along \u0016\u0000\u0000\u0016K(a, b) and \u0016\u0000\u0000\u0016M(f, g) in the BZ. Experiments\nwere performed in the same conditions as in Figure 3; (c-e) spin-polarized spectra and spin polarization for the points along\n\u0016\u0000\u0000\u0016Kmarked by black arrows in (a); in all \fgures red/blue indicate positive/negative polarization perpendicular to the\nhigh-symmetry direction. Ebindenotes the binding energy.\nin comparison to PdTe 2, the parity of the crossing bands\nat the Apoint for NiTe 2is di\u000berent and only the lower\nIBG supports a Dirac node in the surface states. See\nSec. S5 in the SM [45] for a more detailed discussion and\ncomparison of the topological band structure and surface\nstates with PdTe 2.\nThe Dirac-like conical crossing in the surface states of\nNiTe 2(at -1.4eV) is evident in the ARPES data taken\nalong the two high-symmetry directions \u0016K\u0000\u0016\u0000\u0000\u0016Kand\n\u0016M\u0000\u0016\u0000\u0000\u0016Mas shown in Figs. 3(c) and (d). The domi-\nnant surface bands are indicated by the red arrows. In\naddition to the Dirac cone, several other surface states\nare present in NiTe 2, owing to several band inversions\nbelow and above the Fermi level. Along the \u0016M\u0000\u0016\u0000\u0000\u0016M\ndirection, the surface states near the Fermi energy has its\norigin from a band inversion above the Fermi energy (see\nFig. 3(b), and Fig. S3 in SM [45]) and the correspond-\ning surface Dirac cone lies above the Fermi level. While\nits Dirac-like nature is signi\fcantly altered far away from\nthe\u0016\u0000 point, we demonstrate its topological origin by dis-\nplaying its chiral spin texture in Fig. 4. Similar surface\nstates have also been observed in other Te-based TMDs\nlike PtTe 2and PdTe 2, while they are absent in the Se\nbased compounds like PtSe 2. This is a consequence of\nthe avoided band inversion in PtSe 2resulting from the\nreduced interlayer hopping [31, 47].\nSince these surface states have a topological origin,\nwe now focus on the spin polarization of the bands us-\ning spin-polarized ARPES. In Fig. 4(a) we display the\nspin-resolved data superimposed directly onto the spin-\nintegrated band structure shown earlier in Fig. 3(c) and\n(d). The measured spin polarization matches reason-\nably well with the calculated spin textures reported in\nFig. 4(b) and (g). In the present dataset, the spin com-\nponent is always perpendicular to the dispersion direc-tion. The most prominent feature in Fig. 4(a)-(b) is\nthe crossover of two opposite spin polarizations of almost\nequal magnitude for the surface state bands crossing at\nthe \u0000 point at a binding energy of \u0018\u00001:4 eV. This con-\n\frms the helical nature of the spin-momentum locking\nin vicinity of the surface Dirac point, resulting from the\nIBG with the Z2= 1 topological order. In Fig. 4(c)-\n(e), we display the spin-polarized spectra and the spin\npolarization for the points marked by black arrows in\nFig. 4(a). The measured polarization perpendicular to\n\u0016\u0000\u0000\u0016Kreaches almost 50%.\nIn the case of the bands along \u0016M\u0000\u0016\u0000\u0000\u0016Mdirection of\nthe surface BZ [Fig. 4 (f) and (g)], the polarization was\nmeasured for the electron-pocket-like surface states close\nto the Fermi energy. As discussed previously, although\ntheir shape is considerably di\u000berent than a usual topo-\nlogical surface states, the clear spin polarization demon-\nstrates their topological origin (see Fig. S3 of SM [45]).\nThese indeed appear to be the most prominent spin-\npolarized features also in the calculated spin polarization\nperpendicular to the \u0016\u0000\u0000\u0016Mdirection. The high values of\nthe measured and calculated spin polarization indicates\nthat NiTe 2belongs to the recently identi\fed topological-\nladder family of Pt/PdTe2 [32, 46].\nTo summarize, we have established the existence\nof type-II DSM phase in NiTe 2single crystals using\nspin-resolved ARPES measurements in conjunction with\nDFT-based ab-initio calculations. We show that, in con-\ntrast to similar class of materials like PtTe 2, PdTe 2, and\nPtSe 2, where the Dirac point is buried deep in the valence\nband, the Dirac point in NiTe 2is located in vicinity of\nthe Fermi energy. In addition to the bulk Dirac node, the\nTep-orbital manifold in NiTe 2also gives rise to a series\nof IBGs with non-trivial Z2topological orders. Together,\nthese give rise to topological Dirac nodes in the surface5\nstates characterized by the particular spin texture over\na wide range of energies. Our \fndings establish NiTe 2\nas a prime candidate for exploration of Dirac fermiology\nand applications in TMD-based spintronic devices and\nultrafast optoelectronics.\nACKNOWLEDGEMENTS\nA.A. acknowledges funding support by Dept. of Sci-\nence and Technology, Government of India, via DST\nGrant No. DST/NM/NS/2018/103(G), and from SERB\nGrant No. CRG/2018/002440. B.G. acknowledges CSIR\nfor senior research fellowship. A. A. and B. G. acknowl-\nedges HPC- IIT Kanpur for its computational facilities.\nThis work has been partly performed in the framework of\nthe nanoscience foundry and \fne analysis (NFFA-MIUR\nItaly, Progetti Internazionali) facility.\nNote: B. G. and D. M. contributed equally to this work\nas \frst authors.\n\u0003ivana.vobornik@elettra.trieste.it\nyantonio.politano@iit.it\nzamitag@iitk.ac.in\n[1] A. Bansil, Hsin Lin, and Tanmoy Das, \\ Colloquium :\nTopological band theory,\" Rev. Mod. Phys. 88, 021004\n(2016).\n[2] M. Z. Hasan and C. L. Kane, \\Colloquium: Topological\ninsulators,\" Rev. Mod. Phys. 82, 3045{3067 (2010).\n[3] N. P. Armitage, E. J. Mele, and Ashvin Vishwanath,\n\\Weyl and dirac semimetals in three-dimensional solids,\"\nRev. Mod. Phys. 90, 015001 (2018).\n[4] Desheng Kong and Yi Cui, \\Opportunities in chemistry\nand materials science for topological insulators and their\nnanostructures,\" Nat. Chem. 3, 845 (2011).\n[5] Barry Bradlyn, Jennifer Cano, Zhijun Wang, M. G.\nVergniory, C. Felser, R. J. Cava, and B. Andrei Bernevig,\n\\Beyond dirac and weyl fermions: Unconventional quasi-\nparticles in conventional crystals,\" Science 353, 6299\n(2016).\n[6] Su-Yang Xu, Ilya Belopolski, Nasser Alidoust, Mad-\nhab Neupane, Guang Bian, Chenglong Zhang, Raman\nSankar, Guoqing Chang, Zhujun Yuan, Chi-Cheng Lee,\nShin-Ming Huang, Hao Zheng, Jie Ma, Daniel S. Sanchez,\nBaoKai Wang, Arun Bansil, Fangcheng Chou, Pavel P.\nShibayev, Hsin Lin, Shuang Jia, and M. Zahid Hasan,\n\\Discovery of a weyl fermion semimetal and topological\nfermi arcs,\" Science 349, 613{617 (2015).\n[7] Shin-Ming Huang, Su-Yang Xu, Ilya Belopolski, Chi-\nCheng Lee, Guoqing Chang, BaoKai Wang, Nasser Ali-\ndoust, Guang Bian, Madhab Neupane, Chenglong Zhang,\nShuang Jia, Arun Bansil, Hsin Lin, and M. Zahid Hasan,\n\\A weyl fermion semimetal with surface fermi arcs in the\ntransition metal monopnictide TaAs class,\" Nature Com-\nmunications 6, 7373 (2015).\n[8] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,\nJ. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,\nZ. Fang, X. Dai, T. Qian, and H. Ding, \\Experimen-tal discovery of weyl semimetal TaAs,\" Phys. Rev. X 5,\n031013 (2015).\n[9] Su-Yang Xu, Nasser Alidoust, Ilya Belopolski, Zhu-\njun Yuan, Guang Bian, Tay-Rong Chang, Hao Zheng,\nVladimir N. Strocov, Daniel S. Sanchez, Guoqing Chang,\nChenglong Zhang, Daixiang Mou, Yun Wu, Lunan\nHuang, Chi-Cheng Lee, Shin-Ming Huang, BaoKai\nWang, Arun Bansil, Horng-Tay Jeng, Titus Neupert,\nAdam Kaminski, Hsin Lin, Shuang Jia, and M. Za-\nhid Hasan, \\Discovery of a weyl fermion state with fermi\narcs in niobium arsenide,\" Nature Physics 11, 748 (2015).\n[10] Chen Fang, Matthew J. Gilbert, Xi Dai, and B. Andrei\nBernevig, \\Multi-weyl topological semimetals stabilized\nby point group symmetry,\" Phys. Rev. Lett. 108, 266802\n(2012).\n[11] Hongming Weng, Chen Fang, Zhong Fang, B. Andrei\nBernevig, and Xi Dai, \\Weyl semimetal phase in noncen-\ntrosymmetric transition-metal monophosphides,\" Phys.\nRev. X 5, 011029 (2015).\n[12] Bahadur Singh, Ashutosh Sharma, H. Lin, M. Z. Hasan,\nR. Prasad, and A. Bansil, \\Topological electronic struc-\nture and weyl semimetal in the TlBiSe 2class of semicon-\nductors,\" Phys. Rev. B 86, 115208 (2012).\n[13] Tay-Rong Chang, Su-Yang Xu, Daniel S. Sanchez, Wei-\nFeng Tsai, Shin-Ming Huang, Guoqing Chang, Chuang-\nHan Hsu, Guang Bian, Ilya Belopolski, Zhi-Ming Yu,\nShengyuan A. Yang, Titus Neupert, Horng-Tay Jeng,\nHsin Lin, and M. Zahid Hasan, \\Type-ii symmetry-\nprotected topological dirac semimetals,\" Phys. Rev. Lett.\n119, 026404 (2017).\n[14] Su-Yang Xu, Nasser Alidoust, Guoqing Chang, Hong Lu,\nBahadur Singh, Ilya Belopolski, Daniel S. Sanchez, Xiao\nZhang, Guang Bian, Hao Zheng, Marious-Adrian Hu-\nsanu, Yi Bian, Shin-Ming Huang, Chuang-Han Hsu, Tay-\nRong Chang, Horng-Tay Jeng, Arun Bansil, Titus Neu-\npert, Vladimir N. Strocov, Hsin Lin, Shuang Jia, and\nM. Zahid Hasan, \\Discovery of lorentz-violating type II\nweyl fermions in LaAlGe,\" Science Advances 3(2017),\n10.1126/sciadv.1603266.\n[15] B. Q. Lv, Z. L. Feng, Q. N. Xu, X. Gao, J. Z. Ma, L. Y.\nKong, P. Richard, Y. B. Huang, V. N. Strocov, C. Fang,\nH. M. Weng, Y. G. Shi, T. Qian, and H. Ding, \\Ob-\nservation of three-component fermions in the topologi-\ncal semimetal molybdenum phosphide,\" Nature 546, 627\n(2017).\n[16] Wenshuai Gao, Xiangde Zhu, Fawei Zheng, Min Wu, Jin-\nglei Zhang, Chuanying Xi, Ping Zhang, Yuheng Zhang,\nNing Hao, Wei Ning, et al. , \\A possible candidate\nfor triply degenerate point fermions in trigonal layered\nPtBi 2,\" Nature communications 9(2018).\n[17] Ziming Zhu, Georg W. Winkler, QuanSheng Wu, Ju Li,\nand Alexey A. Soluyanov, \\Triple point topological met-\nals,\" Phys. Rev. X 6, 031003 (2016).\n[18] Jiabin Yu, Binghai Yan, and Chao-Xing Liu, \\Model\nhamiltonian and time reversal breaking topological\nphases of antiferromagnetic half-heusler materials,\"\nPhys. Rev. B 95, 235158 (2017).\n[19] I. C. Fulga and Ady Stern, \\Triple point fermions in a\nminimal symmorphic model,\" Phys. Rev. B 95, 241116\n(2017).\n[20] Hao Yang, Jiabin Yu, Stuart S. P. Parkin, Claudia Felser,\nChao-Xing Liu, and Binghai Yan, \\Prediction of triple\npoint fermions in simple half-heusler topological insula-\ntors,\" Phys. Rev. Lett. 119, 136401 (2017).6\n[21] Jianfeng Wang, Xuelei Sui, Wujun Shi, Jinbo Pan,\nShengbai Zhang, Feng Liu, Su-Huai Wei, Qimin\nYan, and Bing Huang, \\Prediction of ideal topolog-\nical semimetals with triply degenerate points in the\nNaCu 3Te2family,\" Phys. Rev. Lett. 119, 256402 (2017).\n[22] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J.\nMele, and A. M. Rappe, \\Dirac semimetal in three di-\nmensions,\" Phys. Rev. Lett. 108, 140405 (2012).\n[23] Steve M. Young and Charles L. Kane, \\Dirac semimetals\nin two dimensions,\" Phys. Rev. Lett. 115, 126803 (2015).\n[24] Benjamin J. Wieder, Youngkuk Kim, A. M. Rappe, and\nC. L. Kane, \\Double dirac semimetals in three dimen-\nsions,\" Phys. Rev. Lett. 116, 186402 (2016).\n[25] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,\nD. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai,\nZ. Hussain, and Y. L. Chen, \\Discovery of a three-\ndimensional topological dirac semimetal, Na 3Bi,\" Science\n343, 864{867 (2014).\n[26] Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M.\nWeng, D. Prabhakaran, S-K. Mo, H. Peng, P. Dudin,\nT. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L.\nFeng, Z. Hussain, and Y. L. Chen, \\A stable three-\ndimensional topological dirac semimetal Cd 3As2,\" Na-\nture Materials 13, 677 (2014).\n[27] Madhab Neupane, Su-Yang Xu, Raman Sankar, Nasser\nAlidoust, Guang Bian, Chang Liu, Ilya Belopolski, Tay-\nRong Chang, Horng-Tay Jeng, Hsin Lin, Arun Bansil,\nFangcheng Chou, and M. Zahid Hasan, \\Observation of\na three-dimensional topological dirac semimetal phase in\nhigh-mobility Cd 3As2,\" Nature Communications 5, 3786\n(2014).\n[28] Sergey Borisenko, Quinn Gibson, Danil Evtushinsky,\nVolodymyr Zabolotnyy, Bernd B uchner, and Robert J.\nCava, \\Experimental realization of a three-dimensional\ndirac semimetal,\" Phys. Rev. Lett. 113, 027603 (2014).\n[29] Zhijun Wang, Yan Sun, Xing-Qiu Chen, Cesare Fran-\nchini, Gang Xu, Hongming Weng, Xi Dai, and Zhong\nFang, \\Dirac semimetal and topological phase transitions\nin A 3Bi (a = Na, K, Rb),\" Phys. Rev. B 85, 195320\n(2012).\n[30] Zhijun Wang, Hongming Weng, Quansheng Wu, Xi Dai,\nand Zhong Fang, \\Three-dimensional dirac semimetal\nand quantum transport in Cd 3As2,\" Phys. Rev. B 88,\n125427 (2013).\n[31] Mingzhe Yan, Huaqing Huang, Kenan Zhang, Eryin\nWang, Wei Yao, Ke Deng, Guoliang Wan, Hongyun\nZhang, Masashi Arita, Haitao Yang, Zhe Sun, Hong Yao,\nYang Wu, Shoushan Fan, Wenhui Duan, and Shuyun\nZhou, \\Lorentz-violating type-ii dirac fermions in tran-\nsition metal dichalcogenide PtTe 2,\" Nature Communica-\ntions8, 257 (2017).\n[32] M. S. Bahramy, O. J. Clark, B. J. Yang, J. Feng, L. Baw-\nden, J. M. Riley, I. Markovi\u0013 c, F. Mazzola, V. Sunko,\nD. Biswas, S. P. Cooil, M. Jorge, J. W. Wells, M. Lean-\ndersson, T. Balasubramanian, J. Fujii, I. Vobornik, J. E.\nRault, T. K. Kim, M. Hoesch, K. Okawa, M. Asakawa,\nT. Sasagawa, T. Eknapakul, W. Meevasana, and P. D. C.\nKing, \\Ubiquitous formation of bulk dirac cones and\ntopological surface states from a single orbital manifold in\ntransition-metal dichalcogenides,\" Nature Materials 17,\n21 EP { (2017).\n[33] Antonio Politano, Gennaro Chiarello, Barun Ghosh, Kr-\nishanu Sadhukhan, Chia-Nung Kuo, Chin Shan Lue, Vit-\ntorio Pellegrini, and Amit Agarwal, \\3d dirac plasmonsin the type-ii dirac semimetal PtTe 2,\" Phys. Rev. Lett.\n121, 086804 (2018).\n[34] Kenan Zhang, Mingzhe Yan, Haoxiong Zhang, Huaqing\nHuang, Masashi Arita, Zhe Sun, Wenhui Duan, Yang\nWu, and Shuyun Zhou, \\Experimental evidence for type-\nii dirac semimetal in PtSe 2,\" Phys. Rev. B 96, 125102\n(2017).\n[35] Han-Jin Noh, Jinwon Jeong, En-Jin Cho, Kyoo Kim, B. I.\nMin, and Byeong-Gyu Park, \\Experimental realization\nof type-ii dirac fermions in a PdTe 2superconductor,\"\nPhys. Rev. Lett. 119, 016401 (2017).\n[36] Fucong Fei, Xiangyan Bo, Rui Wang, Bin Wu, Juan\nJiang, Dongzhi Fu, Ming Gao, Hao Zheng, Yulin Chen,\nXuefeng Wang, Haijun Bu, Fengqi Song, Xiangang Wan,\nBaigeng Wang, and Guanghou Wang, \\Nontrivial berry\nphase and type-ii dirac transport in the layered material\nPdTe 2,\" Phys. Rev. B 96, 041201 (2017).\n[37] Chunqiang Xu, Bin Li, Wenhe Jiao, Wei Zhou, Bin Qian,\nRaman Sankar, Nikolai D. Zhigadlo, Yanpeng Qi, Dong\nQian, Fang-Cheng Chou, and Xiaofeng Xu, \\Topologi-\ncal type-ii dirac fermions approaching the fermi level in\na transition metal dichalcogenide NiTe 2,\" Chemistry of\nMaterials 30, 4823{4830 (2018).\n[38] Enji Uchida and Hisamoto Kondoh, \\Magnetic properties\nof nickel telluride,\" Journal of the Physical Society of\nJapan 11, 21{27 (1956).\n[39] M. Ettenberg, K.L. Komarek, and E. Miller, \\Thermo-\ndynamic properties of nickel-tellurium alloys,\" Journal of\nSolid State Chemistry 1, 583 { 592 (1970).\n[40] G Y Guo and W Y Liang, \\Study of the electronic\nstructures of ni-group metal ditellurides: NiTe 2, PdTe 2\nand PtTe 2by the self-consistent LMTO-ASA method,\"\nJournal of Physics C: Solid State Physics 19, 5365{5380\n(1986).\n[41] P J Orders, J Liesegang, R C G Leckey, J G Jenkin,\nand J D Riley, \\Angle-resolved photoemission from the\nvalence bands of NiTe 2, PdTe 2and PtTe 2,\" Journal of\nPhysics F: Metal Physics 12, 2737{2753 (1982).\n[42] Joo Frederico Haas Leandro Monteiro, Mariana Braga\nMarciniak, Alcione Roberto Jurelo, Ezequiel Costa\nSiqueira, Fbio Teixeira Dias, and Jorge Luiz Pimentel\nJnior, \\Synthesis and microstructure of NiTe 2,\" Journal\nof Crystal Growth 478, 129 { 133 (2017).\n[43] Bei Zhao, Weiqi Dang, Yuan Liu, Bo Li, Jia Li, Jun Luo,\nZhengwei Zhang, Ruixia Wu, Huifang Ma, Guangzhuang\nSun, Yu Huang, Xidong Duan, and Xiangfeng Duan,\n\\Synthetic control of two-dimensional NiTe 2single crys-\ntals with highly uniform thickness distributions,\" Jour-\nnal of the American Chemical Society 140, 14217{14223\n(2018).\n[44] Qianqian Liu, Fucong Fei, Bo Chen, Xiangyan Bo,\nBoyuan Wei, Shuai Zhang, Minhao Zhang, Faji Xie,\nMuhammad Naveed, Xiangang Wan, Fengqi Song, and\nBaigeng Wang, \\Nontopological origin of the planar hall\ne\u000bect in the type-ii dirac semimetal NiTe 2,\" Phys. Rev.\nB99, 155119 (2019).\n[45] Supplementary material detailing 1) Sample prepara-\ntion and characterization, 2) computational details, 3)\nARPES measurements, 4) Topological band inversions\nand 5) ARPES measurements with potassium doping, is\navailable at https://bit.ly/2YWZf1o .\n[46] O J Clark, F Mazzola, I Markovi\u0013 c, J M Riley, J Feng,\nB-J Yang, K Sumida, T Okuda, J Fujii, I Vobornik, T K\nKim, K Okawa, T Sasagawa, M S Bahramy, and P D C7\nKing, \\A general route to form topologically-protected\nsurface and bulk dirac fermions along high-symmetry\nlines,\" Electronic Structure 1, 014002 (2019).\n[47] O. J. Clark, M. J. Neat, K. Okawa, L. Bawden,\nI. Markovi\u0013 c, F. Mazzola, J. Feng, V. Sunko, J. M. Ri-\nley, W. Meevasana, J. Fujii, I. Vobornik, T. K. Kim,M. Hoesch, T. Sasagawa, P. Wahl, M. S. Bahramy, and\nP. D. C. King, \\Fermiology and superconductivity of\ntopological surface states in PdTe 2,\" Phys. Rev. Lett.\n120, 156401 (2018)."    },    {        "title": "1304.3661v1.Electric_Field_Tuning_of_the_Surface_Band_Structure_of_Topological_Insulator_Sb2Te3_Thin_Films.pdf",        "content": " \n1 \n Electric Field Tuning of the Surface Band Structure of Topological \nInsulator Sb 2Te3 Thin Films \n \nTong Zhang1,2, Jeonghoon Ha1,2,3, Niv Levy1,2, Young Kuk3, Joseph Stroscio1* \n \n1Center for  Nanoscale Science and Technology, NIST, Gaithersburg, MD 20899, USA  \n2Maryland NanoCenter, University of Maryland, College Park, MD 20742, USA  \n3Department of Physics and Astronomy, Seoul National University, Seoul 151 -747, Korea  \n \nWe measured  the response of the surface state spectrum of epitaxial Sb 2Te3 thin films to \napplied gate electric fields by low temperature scanning tunneling microscopy.  The gate \ndependen t shift of the Fermi level and the screening effect from bulk carrie rs vary as a function \nof film thickness.  We observe d a gap opening at the Dira c point for films thinner than four  \nquintuple layers, due to the coupling  of the top and bottom surfaces .  Moreover,  the top surface \nstate band gap of the three  quintuple layer  films was found to be tunable by back gate , indicating \nthe possibility of observing a topological phase transition in this system .  Our result s are well \nexplained by an effective model of 3D topological insulator thin films with structure inversion \nasymmetry , indicati ng that three quintuple layer Sb2Te3 films are topologically nontrivial and \nbelong  to the quantum spin Hall insulator class . \n   \n2 \n Topological insulators  (TI) represent  a novel state of quantum matter that has \ntopologically protected edge or surface states   [1,2] .  Theory and experimental verification of TI \nmatter rapidly increased  since the first predictions of these states of matter in 2 -dimensions (2D) .  \n2D TIs  are equivalent to quantum spin Hall (QSH) insulators which host 1-dimensional (1D) \nspin polarized edge states   [3,4] .  The concept of QSH insulators has been generalized to three -\ndimensional (3D) TIs, which are 3D ban d insulators surrounded by 2D metallic surface states \nwith helical  spin texture   [1,2] .  These surface stat es are protected by time reversal symmetry and \nare not  eliminated by scattering from weak non-magnetic disorder.  Recently, t he family of \nbismuth chalcogenide materials (Bi 2Se3, Bi 2Te3 and Sb 2Te3) have been confirmed to be 3D TIs \nwith a single Dirac cone  at the Γ point  [5–11].  In sufficiently thin 3D TI  films  the top and \nbottom surface states hybridize , disrupt ing the Dirac cone and creat ing an energy gap at k=0, \nallowing a  range of topologically interesting phases  [12–21].  In the ultrathin  film geometry, the \nlow-lying physics is described by two degenerate Dirac hyperbolas.  Each massive band has a k-\ndependent spin configuration determined by the energy gap and spin -orbit coupling.  As a result, \nultrathin 3D TI film s can be considered a 2D QSH system with 1D spin helical edge states if the \nsystem is  topologically nontrivial.  Calculations show that the surface band  gap of the ultrathin \n3D TIs displays an oscillating dependence on thickness   [13–21], and the system may alterna te \nbetween being a topologically trivial or no ntrivial insulator .  More interestingly, the band \ntopology is predicted to be tunable by external electric fields , which cause  structure inversion \nasymmetry (SIA), leading to the possibility of controlling a ph ase transition between \ntopologically trivial and nontrivial phases   [15–21].  For practical applications it is important to \nstudy the  surface state properties of  different phases.  This may enable control  of the QSH  edge \nstates with potential applications  in future spintronic devices.    \n3 \n In this letter , we stud y the gate tunable electron ic structure of the 3D TI Sb2Te3 using low \ntemperature scanning tunneling microscopy and spectroscopy (STM/STS)  on in-situ epitaxiall y \ngrown thin films on back gated SrTiO 3 substrates .  We find that t he field effect gating of the thin \nfilms varies strongly  with film thickness . This was determined by measuring the shift of the thin \nfilm quantum well states with back gate potential.  For films below four quintuple layers (QL) \nthickness a band gap is observed in the surface state spectrum, which we attribute to the \nhybridization of the top and bottom surfaces.  A t a film thickness of 3 QL the field effect  is able \nto reduce the gap  with incr eased gate field .  Comparing  an effective Hamiltonian model with \nstructure inversion asymmetry to the gap dependence on gating implies  that 3  QL Sb 2Te3 film is \ntopological ly non-trivial , and suggest s that a topological phase transition may occur if the \nelectric field applied to the film is large enough.  \nDespite many transport measurement s on gated TI devices  [22–26], making a gate \ntunable TI film that is accessible  to low temperature STM is  challenging, mainly because \nchalcogenide TI’s surfaces are environmentally sensitive and degrade  easily through  exposure to \nair during standard lithography  proce sses.  We overcome this problem by epitaxially growing TI \nfilms on pre -patterned SrTiO 3 (111) (STO) substrates mounted on special sample holders   [27].  \nThe sample s were  transferred in ultra-high vacuum  into an  STM right after growth, which \navoided any ex-situ processing.  The sample bias is applied through two top electrodes and a gate \nvoltage is applied to a back contact on the STO.  STM/STS and two -terminal transport \nmeasurements are both carried out in-situ, in a homemade STM operating at 5 K, which is \nconnected to the molecul ar beam  epitaxy (MBE) system  [28]. The sample s studied here are \nSb2Te3 thin film s grow n by co -depositing Sb and Te at a substrate temperature of 200  °C.  \nSamples were characterized with reflection -high-energy -electron diffraction during growth.   \n4 \n Fig. 1(a) shows the typ ical topography of a nominally 5  QL thick Sb 2Te3 film. Terraces \nwith thick nesses between 2  QL to 5  QL can be found on the same film , and their height \ndetermined by finding pinholes to the STO substrate level.   Figure 1(b) shows atomic resolution \nof the Te terminated (111) surface with lattice constant of 0.42  nm. The differential tunneling \nconductance , dI/dV, is measured by standard lock -in technique s, with a modulation frequency of \n≈500  Hz. Typical dI/dV  spectra on different layer thickness es are  shown in Fig. 1(c).  The \npronounced peaks (see arrows in Fig. 1(c)) are attributed to quantum well states (QWS) from \nbulk bands that undergo quantum confinement in a thin film geometry.   [29].  A closer inspection \nof the surface state  region, obtained at lower junction impedance,  shows a hybridized gap a t the \nsample bias ≈VB=0.25 V on 2  QL and 3  QL terraces [ Fig. 1(d)], with a width of ≈160 meV and \n80 meV, respectively.  The gap is absent for the 4 QL film and leaves a “V” shaped density of \nstates  that indicates an intact Dirac cone.  Similar results were  reported  on Sb 2Te3 film grown on \ngraphitized SiC  [30].  \nOur experiment al setup allows , for the first time , for a gate voltage VG to be applied to the \nSb2Te3 film during STM measurement s, without any ex-situ processing and exposure of the film \nto atmosphere .  To measure  the gating characteristics , we charged the sample (which can be \nthought of as  a TI-STO -gate capacitor) with a current source while measuring the change of VG.  \nFig. 1 (e) shows the total charge  density n and displacement field versus VG by integrating the \ncurrent over time.   n reaches about 4.5×1013/cm2 at VG = 200  V, which indicates a large gating \nability of STO at low temperature.  A two-terminal film resistance measurement versus VG is \nshown in Fig. 1(f), which displays a rapid increase at positive VG, as expected for gating a p-\ndoped semiconductor.    \n5 \n The gating effect on the local density of states (LDOS) is measured using the thin film \nQWS as a fiduciary mark in the tunneling spectra [ Fig. 1(c)].  The overall gati ng dependence of \nthe LDOS is shown in Figs. 2(a) and 2(b) for 2 QL and 3 QL films by focusing on a large sample \nbias range ( -0.4 V to 0.4 V) .  Increasing gate voltage is accompanied by the whole spectrum \nmoving to lower energies, wh ich is a clear signature of the shifting of the Fermi energy EF due to \nthe gate induced doping  of the Sb2Te3 film.  Note in the tunneling experiment EF is at VB = 0 V, \nand therefore the change in doping is observed by the shift of the electronic bands rela tive to \nzero sample bias .  The relative shifts of EF (with respect to VG=0) as a function of displacement \nfield are plotted in Fig. 2(c) for film thicknesses of 2  QL, 3  QL, and 4  QLs.  One can see that t he \ngating tu nability decrease s fast with increasing film thickness , approximately as \n2d  [Fig. 2(d)] .  \nFor the 2 QL film, the thinnest  case, we estimate the change in  surface carrier density , \ncorresponding to the measured shift in EF, only reach es to ¼ of the total charge density induced \nby the gate.  Therefore  the majority of the carriers are expected to be in the bulk  of the film , \nwhich screens the electric field reaching the top surface.   Subsequently, in these thin films an \noverall Fermi level shift and band bending will coexist through the film providing  a potential \nasymmetry between the top and bottom surfaces   [31].  Nevertheless, th e gate  tunability is still \nsufficient to deduce the topological character of the 3  QL f ilm, as described below.  \nNow we focus on the hybridization gaps that open due to the coupling of the top and \nbottom surfaces , and their response  to applied fields .  Figures 3(a) and 3(b) show the tunneling \nspectra of the surface state band gaps for 2 QL an d 3 QL terraces, respectively, as a fu nction of \nVG.  The gap size is measured by the peak positions in the second derivative d2I/dV2, which \nindicate the inflection points on either side of the surface state gap [see dashed line in Fig. 3(a)] .   \n6 \n A noticeable feature is that the surface states gap is rather constant at 2 QL, but varies \nconsiderably with VG for 3 QL .  Differences in the response of the gap  to fields , as observed \nbetween the 2  QL and 3  QL films, can occur due changes in the topology of the system .  Such \nchanges  can arise with variations  as small as 1  QL in film thickness   [18,19] .  The gap for the 3 \nQL film shows a linear dependence on displacement field [Fig. 3(d )].  This is reminiscent of bi -\nlayer graphene where the asymmetric potential controls the gap at the Dirac point  [32,33] .  The \ndifference here is that in a TI thin film, the inter -layer and spin orbit coupling play important \nroles.   \nTo understand how an electric field affects the surface  band structure, we refer to an \neffective Hamiltonian model that describes thin 3D TI films  as  [17,18] , \n \n22\n0ˆ ˆ ˆ ˆ ˆ ()2eff F y x x y z z xH k E Dk v k k Bk U             \n  (1) \nThe first three terms in Eq. (1) account for isolated TI surface states.  The fourth term describes \nthe inter -surface coupling, w hich induces a hybridized gap \n  and parabolic term  \n2()Bk .  \nˆ and \nˆ\n are Pauli matrices of electron spin and isospin  (i.e. bonding/anti -bonding states of the top and \nbottom surfaces) , respectively .  The asymmetric potential leading to structure inversion \nasymmetry (SIA) is introduced by adding an effective potential energy  \nU to \neffH .  The \nparameters \n0, , , ,F E D B v  and \n are material and thickness dependent , and \nU is the potential \ndifference  between the top and bottom surfaces induced by the gate electric field .  Solving for \nthe energy eigenvalues for t he effective Hamiltonian  in Eq. (1) gives rise to four surface state \nenergy bands  [17,18] ,  \n7 \n  \n2\n222\n102F E E Dk Bk U v k       \n  (2) \n \n2\n222\n202F E E Dk Bk U v k       \n  (3) \n \nwhere the \n() sign stands for the conduction  (valence ) band, and the 1 (2) stands for the inner \n(outer) branches of the bands .  Without the SIA term  \n( 0)U  these states consist  of spin \ndegenerate  conduction and valence  bands separated by a hybridization  gap   [Fig. 4(a)].  The \npresence of applied potential U leads to a Rashba -like splitting o f conduction and valence bands \n[Figs. 4(b)].  Moreover, from Eq s. (2) and  (3), the actual gap size also var ies with potential \nasymmetry [see Figs. 4, (a) -(d)].  This is  directly related to our observations  of the 3 QL Sb 2Te3 \ngap decreasing  with increasing applied gate potential [Figs. 3, (b ) and (d) ], which is examined \nbelow .   \nAs discussed in R efs.  [17] and  [18], the sign of \n/B  and the value of the SIA term \ndetermine if the system is a topologically trivial or non -trivial insulator .  If \n22BD  and ∆ and B \nhave the same sign, i.e. \n/0B , the system is  in the quantum spin hall state.  The SIA term \nU\n reduce s the energy gap [ Fig. 4(b)]  and eventually close it at the critical potential difference\n/2CFU v B\n [Fig. 4(c)] and then reopen it with increasing potential [Fig. 4(d)], leading to \na topological phase transition  as a function of applied electric field   [17,19] .  If ∆ and B have \nopposite sign, the system remain s gapped as a function of applied electric field.  Fig. 4(f) \ncompares these two cases, which we use below to determine the topological character of the 3  \nQL film.  We note that \nU  contributes to Eq. (2) and  (3) through  its absolute value , so the gap  \n8 \n variation  should be symmetric with respect to the gate voltage.  However, in Fig. 3(d) the gap \ndisplays a monotonic dependence on displacement field  within the range  investigated .  We \nexpect t his lack of symmetry is due to an initial band bending which exists even at VG = 0 [see \nschematic in Fig. 4(e)]  and is common for hetero -junctions .   \nWe can estimate th e initial bending direction from the thickness dependence of  the \nposition of the Dirac point  (or mid -gap energy position) .  As shown  by the dashed line  in Fig. \n1(d), the middle of the gap  moves closer to EF with incre asing thickness  from 2  QL to 4  QL, \nwhich implies  the top surface of the film is less doped at increased thicknesses.  The increased \ndoping for thinner films is probably due to an increased  number of  defects at the TI/STO \ninterface .  These defects are very effective acceptors  [29].  The higher hole doping at the \ninterface gives an upward band bending through the film .  The band bending is present at zero \napplied gate potential [Fig. 4(e)]  and will affect the surface band gap as if a gate field where \napplied  [31].  Applying a negative gate potential  leads to hole accumulation and further \nenhances the band bending intensity , and hence the potential asymmetry  [Fig. 4(e) top panel].  \nWe expect that the potential difference between the top and botto m 3 QL film is no less than the \ndifference of the EF shifts of the top surfaces of the 2 QL and 3  QL films, i.e.  45 meV at VG = \n200 V.  Therefore, i f the system is non -trivial , applying  a negative gate potential leads to a \ndecreas ed gap according to the above analysis  [Fig. 4(f)] .  This is what we observed in Fig 3(b)  \nfor 3 QL .  We model the spectra in Fig. 3(b ) for 3 QL by calculating the density of states  from \nEqs. (2) and ( 3), taking  \n2.5Fv eV Å, B = 15 eV Å2, and D = -5 eV Å2 from fits to \nphotoemission spectroscopy and density functional calculation results in  Ref.   [34].  The model  \nDOS curves are convoluted with a Gaussian ( with standard deviation σ=20 meV) to account for  \n9 \n instrumental broadening and are show n in Fig. 3(c).  The fitting parameters used in the DOS \ncalculation are:  = 0.16 eV, with  E0 varying  from 0.24  eV to 0.27  eV, and    varying  from  0.07 \neV to 0.12  eV, as VG changes from 200 V to -200 V .  The variation  of    is within the range of \ntotal potential as ymmetry we achieved by gating (≈  50 meV).  The simulated spectra in Fig. 3(c) \nare in good agreement with experiment [Fig. 3(b)].  The lack of an observed maximum in the \nexperimental gap measurements indicates  that the initial band bending at VG = 0 is  so strong that \neven VG = 200  V cannot flatten it.  From the model calculation  in Fig. 3(c), we observe  the gap \ndecreasing  with increased potential asymmetry in agreement with the experimental observations .  \nAs shown in Fig. 3(d), t he band gap s determined from the calculation show a trend quantitative ly \nsimilar to the  experiment  for the range of potential asymmetry chosen for the model calculation .  \nWe note that the experimental spectra have a different background than the model  DOS, which is \nlikely from contributions from bulk states.   \nFrom the model parameters,  ∆ and B have the same sign,  \n/0B , which indicates  a \ntopologically nontrivial QSH phase.  We  determine that the 3 QL Sb2Te3 system  is in a nontriv ial \ntopological state  by noting  that the band gap decreas es with negative applied gate voltage .  For \nthe initial p -type doping, such a decrease can only occur  if ∆ and B have the same sign [see Fig. \n4(f)].  The topological character of the system is typically characterized  by the Z2 invariant , \nwhere  Z2=1 signifies  the topological nontrivial phase  [1,2] .  Figure 4( g) summarizes the phase \ndiagram in terms of the expected Z2 invariant deduced from our results;  at VG=0, 3 QL Sb2Te3 is \nin a topologically nontrivial phase and th us should undergo a phase transition  with increasing \ngate field.  This conclusion is  similar to what has been predicted for 4 QL Sb 2Te3 thin films   [19].  \nGapless edge states are supposed to exist in  this regime   [13,17,19]  and s earching for  these edge  \n10 \n states will be a n interesting  subject for further studies.   We expect that the topological phase \ntransition should also be observable in future studies with more insulating material to obtain \nhigher electric fields .  \n \nAcknowledgements : We thank Mark Stiles, Paul Haney, Minsu ng Kim and Jisoon Ihm \nfor useful discussions .  We thank Steve Blankenship and Alan Band for technical assistance.  N. \nL., T. Z., and J. H. acknowledge support under the Cooperative Research Agreement between the \nUniversity of Maryland and the National Inst itute of Standards and Technology Center for \nNanoscale Science and Technology, Grant No. 70NANB10H193, through the University of \nMaryland. J. H. and Y. K. are partly supported by Korea Research Foundation through Grant No. \nKRF -2010 -00349.  \n   \n11 \n Figure Captions  \nFigure 1:  Characterization of Sb 2Te3 thin films. (a) STM topographic image, 150 nm × 150 nm, \nof nominally five -QL Sb 2Te3 film grown by MBE on SrTiO 3(111) .  (b) Atom ic resolution STM \nimage, 10 nm × 10 nm of Sb 2Te3 film.  The STM topographic height is shown  in a color scale \nfrom dark to bright covering a range of 5 nm for (a) and 0.14 nm for (b).  Tunneling parameters: \nI=30 pA, VB=1.5 V for (a) and VB =0.2 V for (b).  (c) -(d) dI/dV  spectra as a function of Sb 2Te3 \nfilm thickness for VG=0 V , for wide (c) and n arrow (d) energy ranges .  The narrow energy range \nspectra are obtained with lower tunneling impedance to boost signal to noise  in the band gap \nregion .  The curves are offset vertically for clarity.  The dashed line in (d) indicates the shift of \nthe mid -gap position toward the Fermi level with increased thickness.  (e) Total charge density \nand displacement field versus gate voltage for a 3 QL Sb 2Te3/SrTiO 3 device obtained by \nmeasuring the induced voltage as the device was charged at constant current.  (f) 3 QL Sb 2Te3 \nfilm two terminal resistance versus gate voltage.   \n \nFigure 2:  In-situ measur ement of electric field gating of MBE grown Sb2Te3 thin films.  dI/dV  \nversus VB spectra as a function of the indicated back gate voltage for a (a) 2 QL thick film, and \n(b) 3 QL thick film.   A shift of the QW peaks to higher energy is seen with increased gate \npotential, as indicated by the dashed lines which are guides to the eye.  (c) The relative shift of \nthe QW peak positions versus displacement fie ld as the gate potential is varied for  2, 3, and 4  QL \nfilms.  Error bars are one standard deviation uncertainties in determin ing the QW peak pos itions.  \nThe solid lines are linear fits to the data points.  (d) The gating tunability versus film thicknes s \nobtained from the slopes of the linear fits of the QW peak shifts in (c).  The error bars are one \nstandard deviation uncertainties from the linear fits.  The solid line is a polynomial  fit to the data \npoints . \n \nFigure 3:  Electric field effect of surface state  band gaps in MBE grown Sb2Te3 thin films.  dI/dV  \nversus VB spectra of the surface state gap as a function of the indicated back gate voltage for (a) \n2 QL Sb2Te3 film, and (b) 3 QL Sb2Te3 film. The curves are offset for clarity.  (c) Calculated \nsurface state DOS using the effective Hamiltonian in Eq. (1) and resulting dispersions in Eqs. (2) \nand (3) as a function of applied potential.  See text for model parameters.  (d) Measured surface \nstate band gap f rom the spectra in (b) of 3 QL Sb2Te3 thin films as a function of displacement \nfield (blue round symbols), and surface state band gap from the model DOS calculation  in (c) \nversus γ U (red square symbols).  The band gaps are determined from the peak minimum and \nmaximum in the 2nd deriv ative spectra (see green dashed curve in (a)) for both the experiment \nand model calculation .  The error bars are one standard deviation uncertainty determined from \nthe uncertainty in fitting the peaks.  The solid line is a linear fit to the data with a slo pe of 17.3 ± \n2.5 meV/V Å-1.  The error is one standard deviation uncertainty determined from the linear fit.  \n \nFigure 4:  Topological phase transitions in applied electric field.  (a) -(d) Surface state energy \nversus momentum dispersion calculated using Eqs. (2) and (3) showing the variation  of the band \ngap as a function of increasing structure inversion asymmetry potential.  See text for model \nparameters.  At the potential γ U=0.18 =γ UC the gap closes indicating a topological phase  \n12 \n transition.  (e) Schematic o f band bending through the TI film.  Th e diagram shows the \nconduction and valence  bands going through the film.  At each interface the surface state \nhyperbolic dispersions are indicated.   At VG=0 there is initial band bending indicating an initial \npotential asymmetry γ U0 which increases with negative gate voltage.  (f) Surface state b and gap s \ncalculated from the 3D TI model using Eqs. (2) and (3) with Δ=0.16 eV,  D=-5 eV Å2, vF=2.5 \neV Å , and Δ/B> 0 (blue; B=15 eV Å2), and Δ/B< 0 (red; B=-15 eV Å2). (g) Schematic of the Z 2 \ntopological invariant for 3 QL Sb2Te3 thin films as deduced from the analysis in this paper.  The \nschematic shows the initial potential asymmetry position, and the topological phase transition \nthat is approached with increasing negati ve gate voltage leading to a decrease  of the surface state \nband gap as seen in (a) to (c) and in the experimental data in Figs. 3, (b) and (d).   The grey bar \nindicates the approximate range of potential asymmetry probed by the current experiment  [see \nFig. 3(d)] .   \n   \n13 \n  \nReference s \n* Author to whom cor respondence should be addressed. joseph.stroscio@nist.gov  \n[1]  M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).  \n[2]  X.-L. Qi and S. -C. Zhang, Rev. Mod. Phys . 83, 1057 (2011).  \n[3]  C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).  \n[4]  B. A. Bernevig and S. -C. Zhang, Phys. Rev. Lett. 96, 106802 (2006).  \n[5]  H. Zhang, C. -X. Liu, X. -L. Qi, X. Dai, Z. Fang, and S. -C. Zhang, Nature Physics 5, 438 \n(2009). \n[6]  Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. \nCava, and M. Z. Hasan, Nat Phys 5, 398 (2009).  \n[7]  D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. \nCheckelsky, N . P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, \nand M. Z. Hasan, Nature 460, 1101 (2009).  \n[8]  Y. L. Chen, J. G. Analytis, J. -H. Chu, Z. K. Liu, S. -K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, \nX. Dai, Z. Fang, S. C. Zhang, I. R. Fi sher, Z. Hussain, and Z. -X. Shen, Science 325, 178 \n(2009).  \n[9]  P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. \nHasan, R. J. Cava, and A. Yazdani, Nature 460, 1106 (2009).  \n[10]  T. Zhang, P. Cheng, X. Chen, J. -F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, \nX. Xie, and Q. -K. Xue, Phys. Rev. Lett. 103, 266803 (2009).  \n[11]  Z. Alpichshev, J. G. Analytis, J. -H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and \nA. Kap itulnik, Phys. Rev. Lett. 104, 016401 (2010).  \n[12]  B. Zhou, H. -Z. Lu, R. -L. Chu, S. -Q. Shen, and Q. Niu, Phys. Rev. Lett. 101, 246807 \n(2008).  \n[13]  J. Linder, T. Yokoyama, and A. Sudbø, Phys. Rev. B 80, 205401 (2009).  \n[14]  C.-X. Liu, H. Zhang, B. Yan, X. -L. Qi, T. Frauenheim, X. Dai, Z. Fang, and S. -C. Zhang, \nPhys. Rev. B 81, 041307 (2010).  \n[15]  J. Li and K. Chang, Applied Physics Letters 95, 222110 (2009).  \n[16]  H.-Z. Lu, W. -Y. Shan, W. Yao, Q. Niu, and S. -Q. Shen, Phys. Rev. B 81, 115407 (2010).  \n[17]  W.-Y. Shan, H. -Z. Lu, and S. -Q. Shen, New Journal of Physics 12, 043048 (2010).  \n[18]  H. Li, L. Sheng, D. N. Sheng, and D. Y. Xing, Phys. Rev. B 82, 165104 (2010).  \n[19]  M. Kim, C. H. Kim, H. -S. Kim, and J. Ihm, PNAS 109, 671 (2012).  \n[20]  H. Li, L. Sheng,  and D. Y. Xing, Phys. Rev. B 85, 045118 (2012).  \n[21]  G. Liu, G. Zhou, and Y. -H. Chen, Applied Physics Letters 101, 223109 (2012).  \n[22]  J. Chen, H. J. Qin, F. Yang, J. Liu, T. Guan, F. M. Qu, G. H. Zhang, J. R. Shi, X. C. Xie, C. \nL. Yang, K. H. Wu, Y. Q.  Li, and L. Lu, Phys. Rev. Lett. 105, 176602 (2010).  \n[23]  J. G. Checkelsky, Y. S. Hor, R. J. Cava, and N. P. Ong, Phys. Rev. Lett. 106, 196801 \n(2011).  \n[24]  H. Steinberg, D. R. Gardner, Y. S. Lee, and P. Jarillo -Herrero, Nano Lett. 10, 5032 (2010).  \n[25]  D. Kong, Y. Chen, J. J. Cha, Q. Zhang, J. G. Analytis, K. Lai, Z. Liu, S. S. Hong, K. J. \nKoski, S. -K. Mo, Z. Hussain, I. R. Fisher, Z. -X. Shen, and Y. Cui, Nat Nano 6, 705 (2011).   \n14 \n [26]  D. Kim, S. Cho, N. P. Butch, P. Syers, K. Kirshenbaum, S. Adam, J. Pag lione, and M. S. \nFuhrer, Nat Phys 8, 459 (2012).  \n[27]  T. Zhang, N. Levy, J. Ha, Y. Kuk, and J. A. Stroscio, Phys. Rev. B 87, 115410 (2013).  \n[28]  Y. J. Song, A. F. Otte, V. Shvarts, Z. Zhao, Y. Kuk, S. R. Blankenship, A. Band, F. M. \nHess, and J. A. Stroscio, Review of Scientific Instruments 81, 121101 (2010).  \n[29]  Y. Jiang, Y. Y. Sun, M. Chen, Y. Wang, Z. Li, C. Song, K. He, L. Wang, X. Chen, Q. -K. \nXue, X. Ma, and S. B. Zhang, Phys. Rev. Lett. 108, 066809 (2012).  \n[30]  Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, and Q. -K. \nXue, Phys. Rev. Lett. 108, 016401 (2012).  \n[31]  D. Galanakis and T. D. Stanescu, Phys. Rev. B 86, 195311 (2012).  \n[32]  E. McCann, Phys. Rev. B 74, 161403 (2006).  \n[33]  T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, Science 313, 951 (2006).  \n[34]  C. Pauly, G. Bihlmayer, M. Liebmann, M. Grob, A. Georgi, D. Subramaniam, M. R. \nScholz, J. Sá nchez -Barriga, A. Varykhalov, S. Blügel, O. Rader, and M. Morgenstern, Phys. \nRev. B 86, 235106 (2012).  \n (a)\n(b)(c)\nQW\n(d)(e)\n(f)\nFigure 1.  Zhang et al.2 QL 3 QL(a) (b)\n(c) (d)\nFigure 2: Zhang et al.(a)\nVG = –100 V\nVG = 0 V\nVG = 30 V\nVG = 60 V\nVG = 100 VVG = –100 V\nVG = -50 V\nVG = 0 V\nVG = 50 V\nVG = 200 VVG = 100 VVG = 20 VVG = –20 V(d) (c) (b) 2 QL Sb2Te3\nGapd2I/d2V3 QL Sb2Te3Sb2Te3 Model DOS \nFigure 3.  Zhang et al.γU = –0.07 eVγU = –0.077 eVγU = –0.084 eVγU = –0.091 eVγU = –0.099 eVγU = –0.106 eVγU = –0.113 eVγU = –0.12 eVEnergy\nDistanceVG > 0VG < 0\nVG = 0\nBottom Surface\nSb2Te3STO InterfaceTop Surface\nSb2Te3 Vacuum InterfaceC\nV(a)\n(e) (g) (f)(d) (c) (b)\nγUcγU = 0 γU = –0.1 eV γU = –0.3 eV γU = –0.18 eV\nVG<0 VG>0γU0Experiment\nFigure 4.  Zhang et al."    },    {        "title": "1410.1425v1.Competing_Ultrafast_Energy_Relaxation_Pathways_in_Photoexcited_Graphene.pdf",        "content": "arXiv:1410.1425v1  [cond-mat.mes-hall]  3 Oct 2014Competing Ultrafast Energy Relaxation Pathways in Photoex cited Graphene\nS.A. Jensen,1,2Z. Mics,1I. Ivanov,1H.S. Varol,1D. Turchinovich,1\nF.H.L. Koppens,3,∗M. Bonn,1,∗and K.J. Tielrooij3,∗\n1Max Planck Institute for Polymer Research, Ackermannweg 10 , 55128 Mainz, Germany\n2FOM Institute AMOLF, Amsterdam, Science Park 104, 1098 XG Am sterdam, Netherlands\n3ICFO - Institut de Ci´ encies Fot´ oniques, Mediterranean Te chnology Park, Castelldefels (Barcelona) 08860, Spain\nFor most optoelectronic applications of graphene a thoroug h understanding of the processes that\ngovern energy relaxation of photoexcited carriers is essen tial. The ultrafast energy relaxation in\ngraphene occurs through two competing pathways: carrier-c arrier scattering – creating an elevated\ncarrier temperature – and optical phonon emission. At prese nt, it is not clear what determines the\ndominating relaxation pathway. Here we reach a unifying pic ture of the ultrafast energy relaxation\nby investigating the terahertz photoconductivity, while v arying the Fermi energy, photon energy,\nand fluence over a wide range. We find that sufficiently low fluenc e (<∼4µJ/cm2) in conjunction\nwith sufficiently high Fermi energy ( >∼0.1 eV) gives rise to energy relaxation that is dominated\nby carrier-carrier scattering, which leads to efficient carr ier heating. Upon increasing the fluence\nor decreasing the Fermi energy, the carrier heating efficienc y decreases, presumably due to energy\nrelaxation that becomes increasingly dominated by phonon e mission. Carrier heating through\ncarrier-carrier scattering accounts for the negative phot oconductivity for doped graphene observed\nat terahertz frequencies. We present a simple model that rep roduces the data for a wide range of\nFermi levels and excitation energies, and allows us to quali tatively assess how the branching ratio\nbetween the two distinct relaxation pathways depends on exc itation fluence and Fermi energy.\nGraphene is a promising material for, amongst others,\nphoto-sensing and photovoltaic applications [1], owing to\nits broadband absorption [2, 3], its high carrier mobility\n[4, 5] and the ability to create a photovoltage from\nheated electrons or holes [6]. It furthermore uniquely\nallows for electrical control of the carrier density and\npolarity [4]. To establish the potential and limitations\nof graphene-based optoelectronic devices, a thorough\nunderstanding of the ultrafast (sub-picosecond) primary\nenergy relaxation dynamics of photoexcited carriers is\nessential. For undoped graphene (with Fermi energy\nEF≈0), ultrafast energy relaxation through interband\ncarrier-carrier scattering was predicted [7] and observed\n[8, 9] to lead to multiple electron-hole pair excitation.\nFor doped graphene (with Fermi energy |EF|>0),\nultrafast energy relaxation through carrier-carrier in-\nteraction also plays an important role, with intraband\nscattering leading to carrier heating [10–13]. In addition,\nthe ultrafast energy relaxation was ascribed to optical\nphonon emission [14, 15], which reduces the carrier\nheating efficiency. Closely related, for undoped graphene\nthe sign of the terahertz (THz) photoconductivity is\npositive (see e.g. [16–19]), whereas for intrinsically\ndoped graphene the sign is negative, meaning that\nphotoexcitation gives rise to an apparent decrease of\nconductivity [11, 16, 17, 20–22]. This negative photo-\nconductivity was attributed to stimulated THz emission\n[20], and to a reduction of the intrinsic conductivity by\nenhanced scattering with optical phonons [21, 22] or\nby carrier heating [11, 16, 17]. Since the experimental\n∗Electronicaddress: klaas-jan.tielrooij@icfo.es, bonn@ mpip-mainz.mpg.de, frank.koppens@icfo.esparameters of all these studies differ strongly, different\nconclusions were drawn regarding the dominating energy\nrelaxation pathway and the origin of the sign of the THz\nphotoconductivity.\nHere, we reach a unifying picture of the energy\nrelaxation of photoexcited carriers in graphene by ex-\nperimentally studying the ultrafast energy relaxation of\nphotoexcited carriers for a large parameter space, where\nwe vary the Fermi energy and the fluence. We compare\nthe data with a simple model of carrier heating, which\nquantitatively reproduces the frequency-dependent THz\nphotoconductivity for a large range of Fermi energies\nand fluences with a single free fit parameter. This\nfit parameter is the carrier heating efficiency, i.e. the\nfraction of absorbed energy from incident light that\nis transferred to the electron system. Although the\nTHz photoconductivity is not a direct probe of the\ncarrier temperature, and the model parameters carry\nsome uncertainty, this approach allows us to iden-\ntify qualitatively how the ultrafast energy relaxation\nprocesses depend on excitation fluence and Fermi energy.\nWe study the ultrafast carrier-energy relaxation using\ntime-resolved THz spectroscopy experiments, where an\nexcitation pulse with a wavelength of 400, 800 or 1500\nnm (corresponding to a photon energy of E0= 3.10,\n1.55 or 0.83 eV, respectively) excites electron-hole pairs,\nand a low-energy terahertz (THz, f= 0.3−2 THz)\nprobe pulse interrogates the sample, where it interacts\nwith mobile carriers. An optical delay line controls the\nrelative time delay between excitation and probe pulses,2\nmaking it possible to determine the photo-induced\nchange in conductivity (the photoconductivity ∆ σ) as a\nfunction of time with a time resolution of ∼150 fs [23].\nWe use two different samples: The first sample, where\nthe graphene has a controllable Fermi energy, contains\nChemical Vapor Deposition (CVD) grown graphene\nwith an area of 1 cm2transferred onto a substrate that\nconsists of doped silicon, covered by a 300 nm thick\nlayer of SiO 2. The silicon serves as the backgate for\nelectrical control of the Fermi energy (carrier doping) of\nthe graphene sheet from EF= 0.3 eV down to ∼0.06\neV (the width of the neutrality region is ∼0.06 eV, see\nSupp. Info). We use weakly doped silicon (resistivity\nof 10-20 Ωcm) and sufficiently low pump energy (0.83\neV) to ensure that the photo-induced signal from the\nsilicon in the substrate (without graphene) is minimized\n(See Supp. Info). Two silver-pasted electrical contacts\nto the graphene sheet enable resistance measurements\nto retrieve the Fermi energy through the capacitive\ncoupling of the backgate (See Supp. Info). The second\nsample contains CVD grown graphene with an area\nof a square inch transferred onto SiO 2with a fixed\nFermi level of <0.15 eV (see Supp. Info). We carefully\nmeasure the fluence Fas described in Ref. [24] and\nuse the experimentally determined optical absorption of\ngraphene on SiO 2using standard UV-Vis spectrometry\nto obtain the absorbed fluence Fabs, and from this the\nnumber of absorbed photons Nexc=Fabs/E0, which is\nequal to the number of primary excited carriers.\nTo investigate the ultrafast (sub-picosecond) energy\nrelaxation dynamics of photoexcited carriers, we first\nexamine the temporal evolution of the photoconduc-\ntivity. In Fig. 1a we show the dynamics for a range\nof Fermi energies using the sample with controllable\nFermi energy, and in Fig. 1b for a range of fluences\nusing the sample with fixed Fermi energy. We find that\nall traces, except the one with Fermi energy within\nthe neutrality region width ( EF<0.06 eV), show\nnegative photoconductivity with a sub-picosecond rise,\nfollowed by a picosecond decay. Similar pump-probe\ndynamics have been observed before [11, 16–22] and\ncan be understood as follows: During the rise of the\nsignal three main processes take place: (i)the creation\nof initial electron-hole pairs; and subsequent ultrafast\nenergy relaxation through two competing relaxation\nchannels, namely (ii)carrier-carrier scattering and\n(iii)optical phonon emission. Thus, the peak signal\ncorresponds to a ’hot state’ with an elevated carrier\ntemperature Teland/or more energy in optical phonons\n[8, 10–15]. During the subsequent picosecond decay,\nthe ’hot state’ cools down to the same state as before\nphotoexcitation. The fraction of absorbed energy that\n– after the initial ultrafast energy relaxation – ends up\nin the electron system or in the phonon system depends\non the timescales associated with carrier heating and\nphonon emission, respectively.The ultrafast energy relaxation takes place during the\nfirst few hundred femtoseconds after photoexcitation,\ni.e. during the rise of the conductivity change. Figs. 1c\nand 1d show the normalized photoconductivity signals\nfor this time window. We will discuss the evolution\nof the signal amplitude as a function of Fermi energy\nand fluence later. First we note that the rise dynamics\nexhibit an intriguing effect: upon decreasing the Fermi\nenergy (i.e. the density of intrinsic carriers Nint) or\nincreasing the fluence (i.e. the density of primary excited\ncarriersNexc) the signal peak is reached at increasingly\nlater times. To quantify these results, we describe the\ndynamics using two rise times and an exponential decay\ntime. The two rise times allow for part of the conductiv-\nity change to occur with the (fixed) experimental time\nresolution and part with a (free) slower time scale. We\nthen examine the effective rise time τrise, which is the\namplitude-weighted average of the two (see Methods).\nThe insets of Fig. 1c and Fig. 1d show τriseas a function\nofNintandNexc, respectively. Indeed, for decreasing\nNintand increasing Nexcthe effective rise time increases\nfrom below 200 fs (limited by the experimental time\nresolution) up to 400 fs (for EF<0.1 eV).\nThe slowing down of the ultrafast energy relaxation\nof photoexcited carriers with decreasing Fermi energy\nis consistent with energy relaxation through intraband\ncarrier-carrier scattering (see bottom right inset in Fig.\n1c). The microscopic picture of this scattering process\nis shown for two different Fermi energies in the top left\ninset of Fig. 1c. Photoexcited carriers relax by exchang-\ning energy with intrinsic conduction band carriers that\nthus heat up. The amount of energy that is exchanged\nbetween photoexcited carriers and intrinsic conduction\nband carriers in each intraband carrier-carrier scattering\nevent is ∼EF[10]. Therefore, if EFdecreases, more\nenergy-exchangeevents are required for the photoexcited\ncarriers to complete their energy relaxation cascade and\ntherefore the relaxation time will increase (see bottom\nright inset of Fig. 1c). If the energy relaxation through\ncarrier-carrier scattering would slow down in such a\nway that the relaxation rate becomes comparable to the\nrate of other relaxation channels (e.g. optical phonon\nemission), we expect these channels to start contributing\nto the overall energy relaxation. This would lead to a\ndecrease in the fraction of energy that is transferred\nto the electron system, i.e. a reduced carrier heating\nefficiency.\nWe quantify the fraction of absorbed energy that leads\nto carrier heating by comparing our THz photoconduc-\ntivity data with the results of a simple thermodynamic\nmodel. In short, carrier heating leads to a broader\ncarrier distribution (higher Tel), which – in combination\nwith an energy-dependent scattering time [25, 26] –\nleads to the photo-induced change in THz conductivity\n(see Methods). The carrier heating is governed by the\namount of absorbed energy Fabsand the electronic heat3\ncapacity of graphene, which for a degenerate electron\ngas is given by Cel=αTel[27]. Here α=2πEF\n3¯h2v2\nFk2\nB, with\n¯h,vFandkBthe reduced Planck constant, the Fermi\nvelocity and Boltzmann’s constant, respectively. The\npossibility of controlling the Fermi energy of graphene\nthus allows for tunability of the heat capacity (see\nFig. 2a), which in turn determines the ’hot’ carrier\ntemperature T′\nelthat the system reaches. Figure 2b\nshows that T′\nelis equivalently determined by both the\nfluence and the Fermi energy.\nTo calculate the ’hot’ carrier temperature we\nuse a basic numerical approach, which produces a\nheat capacity in the degenerate regime that corre-\nsponds well with the analytical heat capacity (see\nFig. 2a), while remaining valid for non-degenerate\nelectron temperatures ( kBTel> EF). The numerical\napproach is based on the concept that before pho-\ntoexcitation there is a known amount of energy in\nthe electronic system: E1=/integraltext∞\n0dǫD(ǫ)ǫF(EF,Tel);\nand a known number of carriers in the conduction\nband:Nint=/integraltext∞\n0dǫD(ǫ)F(EF,Tel), where D(ǫ) is the\nenergy-dependent density of states and F(EF,Tel) is\nthe Fermi-Dirac distribution that depends on Fermi\nenergyEFand carrier temperature Tel. Due to optical\nexcitation, an amount of energy Fabsis absorbed in the\ngraphene and a fraction ηof this energy ends up in\nthe electronic system trough intraband carrier-carrier\nscattering. After intraband heating is complete, the\nsystem is then described by the following set of equa-\ntions:E2=E1+ηFabs=/integraltext∞\n0dǫD(ǫ)ǫF(E′\nF,T′\nel), and\n(conserving the number of conduction band carriers)\nNint=/integraltext∞\n0dǫD(ǫ)F(E′\nF,T′\nel). Here E′\nFandT′\nelare the\nchemical potential and the carrier temperature in the\n’hot state’, respectively. Carrier heating thus alters the\ncarrierdistribution, where we find by numerically solving\nthe equations for E2andNintthat the carrier temper-\nature increases and the chemical potential decreases\nby photoexcitation (see inset of Fig. 2c and Methods).\nThe photo-induced increase of carrier temperature,\nand the associated decrease in chemical potential were\nexperimentally confirmed recently [13].\nThe hot carrier distribution, with E′\nFandT′\nelcalcu-\nlated using the carrier heating model, directly leads to\nnegative THz photoconductivity (see Methods), which\nscales linearly with carrier temperature up to ∼2000\nK and then shows some saturation behavior (Fig. 2c).\nWe test the validity of our carrier heating model by\ncomparing its predictions for the frequency-resolved\nphotoconductivity with our experimental results for\nthe sample with fixed Fermi energy. In Fig. 3a we\nshow this comparison for a fluence of ∼12µJ/cm2\n(pump wavelength 800 nm, Nexc= 1·1012absorbed\nphotons/cm2). We compare the data (at the time\ndelay that corresponds to the signal peak) with the\nmodel result for a ground state Fermi energy of 0.11\neV and a scattering time proportionality constant of200 fs/eV (extracted from the Raman spectrum, THz\nconductivity measurements on the same sample without\nphotoexcitation, and the saturation value of the THz\nphotoconductivity at high fluence; see also Supp. Info)\nand find good agreement with a heating efficiency of η=\n0.75. The small discrepancies between data and model\ncan be ascribed to artifacts that arise from the temporal\nchange of the photoconductivity during the interaction\nwith the THz pulse [28], although we largely avoid these\nby moving the optical pump delay line simultaneously\nwith the THz probe delay line.\nThe overall agreement between data and model shows\nthat the observed negative THz photoconductivity of\nintrinsically doped graphene [11, 16, 17, 20–22] can be\nfully reproduced by considering intraband carrier heat-\ning, which reduces the thermally averaged conductivity\nof the intrinsic carriers. Despite the simplicity of the\nmodel, it can also explain the experimental results in\nRef. [21] using their experimental parameters, as well\nas the results in Ref. [20], by letting the environmental\ngas change the Fermi energy. These observations lead us\nto conclude that, despite some uncertainty in the deter-\nmination of the Fermi energy and the scattering time,\nthe model is suitable for obtaining reliable qualitative\nindications on how the carrier heating efficiency depends\non the Fermi energy and the fluence.\nWe now examine the ’high fluence’ regime using\nNexc= 8·1012absorbed photons/cm2(a fluence of\n∼100µJ/cm2) in Fig. 3b. This corresponds to the\nregime where the energy relaxation is slower than\nthe experimental time resolution (see Fig. 1d). Here\nwe find that we can only describe the data with a\nsignificantly reduced carrier heating efficiency of η≈0.5\n(keeping the ground state Fermi energy and scattering\ntime proportionality constant the same as in the low\nfluence regime). Combined, these results show that at\nsufficiently low fluence, a large fraction of the absorbed\nenergy ends up in the electron system, i.e. the ultrafast\nenergy relaxation occurs through efficient (and fast)\ncarrier-carrier scattering. However, upon increasing\nthe fluence (i.e. the carrier temperature), the relative\namount of energy transferred to the electron system\ndecreases, which means that carrier-carrier scattering\nbecomes less efficient (and slower) and other relaxation\nprocesses start to contribute.\nTo determine in more detail how the carrier heating\nefficiency depends on fluence, we study the peak pho-\ntoconductivity of the sample with fixed Fermi energy\nfor a large range of excitation powers, for both 800 nm\nand 400 nm excitation. We show the photoconductivity\nat the peak (when the ultrafast energy relaxation is\ncomplete) in Fig. 4a-b, together with the results of the\ncarrier heating model for the same parameters as in Fig.\n3a and a frequency of 0.7 THz. For low fluences (up to\n∼4µJ/cm2, corresponding to Nexc∼0.3·1012absorbed4\nphotons/cm2, 800 nm excitation, ∼2% absorption) the\nexperimental data are in agreement with the heating\nmodel with a fixed heating efficiency of η= 1. We notice\nthat the calculated photoconductivity shows saturation\nbehavior with Nexceven for constant η. This is related\nto the nonlinear dependence of the THz photoconduc-\ntivity on carrier temperature (Fig. 2c). Interestingly,\nat fluences above ∼4µJ/cm2the experimental photo-\nconductivity starts saturating and the model is only\nin agreement for a heating efficiency that gradually\ndecreases to ∼50% for the highest fluences applied here\n(Fig. 4a). These observations suggest that once a certain\ncarrier temperature ( ∼4000 K, see inset Fig. 4b) is\nreached, the heating efficiency decreases. Interestingly,\nthe experimental data for excitation with 400 nm light\nstart deviating from the model (with efficient heating)\natNexc∼0.15·1012absorbed photons/cm2(Fig. 4b),\ninstead of ∼0.3·1012absorbed photons/cm2in the case\nof excitation with 800 nm light. This is because each\n400 nm photon has twice the energy of a 800 nm photon.\nThus, in both cases the carrier heating efficiency starts\ndecreasing around the same carrier temperature.\nWe now determine how the heating efficiency depends\non the Fermi energy by measuring the peak photo-\nconductivity for the sample with controllable Fermi\nenergy as a function of both excitation power and Fermi\nenergy. The combined results (for excitation at 1500\nnm) are represented in Fig. 5a, where we show the\npeak photoconductivity as a function of gate voltage\nfor five different excitation powers. In Fig. 5b we show\nthe peak photoconductivity as a function of Nexcfor\nthree distinct gate voltages, corresponding to Nint≈\n1, 2 and 3 ·1012carriers/cm2, and compare these to\nthe results of the heating model. We find that for a\ndoping of Nint= 3·1012carriers/cm2(EF∼0.2 eV)\nthe data is in good agreement with the carrier heating\nmodel, using a scattering time proportionality constant\nof∼50 fs/eV and a carrier heating efficiency of η=1.\nHowever, for the lowest Nint, which corresponds to a\nFermi energy of ∼0.1 eV, we find a carrier heating\nefficiency of ∼20%, using the same energy-dependent\nscattering time. These results are in good agreement\nwith the observed slowdown of the rise dynamics with\ndecreasing Fermi energy in Fig. 1b.\nComparing the data and the heating model leads to\nthe following physical picture of the ultrafast energy re-\nlaxation in graphene: Until a certain carrier temperature\nis reached ( ∼4000 K), the ultrafast energy relaxation is\ndominated by carrier-carrier scattering, which leads to\nefficient and fast ( <150 fs) carrier heating. Once this\ncarriertemperature is reached, the relaxationslowsdown\nand the carrier heating efficiency decreases, as ultrafast\nenergy relaxation occurs through additional pathways\ninvolving optical phonon emission [14]. The reduction\nin heating efficiency that follows from the macroscopic\nheating model can be explained using the microscopicpicture of intraband carrier-carrier scattering, as put\nforward in Refs. [10, 11]. At increased electron temper-\natures, the (quasi-equilibrium) Fermi energy decreases\n(see Methods), which means that the electronic heat ca-\npacity decreases. It furthermore implies that the amount\nof energy that is exchanged in intraband carrier-carrier\nscattering events ( ∼EF) decreases. Therefore, energy\nrelaxation of a photoexcited carrier requires an increas-\ning number of intraband carier-carrierscattering cascade\nsteps. Thus for an increasingcarriertemperature, energy\nrelaxation through intraband carrierheating slows down.\nThe physical picture of carrier-temperature dependent\nultrafast energy relaxation of photoexcited carriers in\ngraphene unites the conclusions of a large fraction of\nthe existing literature on this topic. For example in Ref.\n[14], with excitation in the ’high fluence’ regime ( ∼1014\nabsorbed photons/cm2at 800 nm), it was concluded\nfrom the experimentally measured carrier temperature\nthat only part of the absorbed light energy ends up\nin the electronic system, whereas the rest couples to\noptical phonons. An ultrafast optical pump-probe study\nemploying a fluence of 200 µJ/cm2[15], also in the ’high\nfluence’ regime, similarly demonstrated optical phonon\nmediated relaxation, in addition to carrier heating. By\nmeasuring in the ’low fluence’ regime (a few µJ’s), Ref.\n[11] concluded that the ultrafast energy relaxation was\ndominated by carrier heating. Furthermore, two very\nrecent optical pump - THz probe studies [16, 17] both\nused anexcited carrierdensityof Nexc≈1·1012absorbed\nphoton/cm2at 800 nm and ascribe their observed nega-\ntiveTHzphotoconductivity(partially)tocarrierheating.\nIn conclusion, we provide a unifying explanation of\nthe ultrafast energyrelaxationofphotoexcited carriersin\ngraphene. For sufficiently low excitation power and suf-\nficiently high Fermi energy, the relaxation is dominated\nby carrier-carrier scattering, which leads to efficient gen-\neration of hot carriers. This regime typically persists up\nto a fluence of ∼4µJ/cm2(orNexc= 0.3·1012absorbed\n800 nm photons/cm2) for a Fermi energy of 0.1 eV. For\nlargerFermi energy, a higher fluence can be used without\nsignificant reduction in heating efficiency. In the case of\nlower Fermi energy and/or a higher fluence, the heating\nefficiency will decrease due to slower intraband carrier-\ncarrier scattering and additional energy relaxation chan-\nnels involving optical phonon emission. This opens up\nthe possibility to control the pathway of ultrafast energy\nrelaxation, i.e. the ability to tune the efficiency of energy\ntransfer from the primary excited carriers to electronic\nheat or to alternative degrees of freedom, such as lattice\nheat. Such tunability is useful for future applications, for\ninstance in the field of photodetection, where hot carriers\nare the dominant source of photocurrent generation [6].\nFinally, we note that the terrestrial solar radiation (on\nthe order of a pJ/cm2during a 10 ps timescale) corre-\nsponds to the ’low fluence’ regime with efficient carrier\nheating, which is therefore the relevant process to con-5\nsider for photovoltaic applications.\nMETHODS\nPhotoconductivity from carrier heating model\nWe use a numerical model based on carrier heating\nto calculate the complex photoconductivity of photoex-\ncited graphene. The frequency-dependent conductivity\nof graphene is generally given by [26]\nσ(ω) =e2v2\nF\n2/integraldisplay∞\n0dǫD(ǫ)τscatter(ǫ)\n1−iωτscatter(ǫ)dF(EF,Tel)\ndǫ,\n(1)\nwithethe elementary charge, vFthe Fermi velocity,\nD(ǫ) =2ǫ\nπ¯h2v2\nFthe density of states, ¯ hthe reduced Planck\nconstant, τscatter(ǫ) the energy-dependent momentum\nscattering time and F(EF,Tel) the Fermi-Dirac distribu-\ntion that is determined by the Fermi level and the carrier\ntemperature. For unexcited graphene we use the steady\nstate Fermi level EFand the ambient temperature Tel.\nWe furthermore use a scattering time that is determined\nby charged impurity scattering and increases linearly\nwith energy ǫ[25, 26]. After electron-hole pair excitation\nof graphene, photoexcited carriers can interact with the\nintrinsic carriers, leading to intraband thermalization\n(see also main text): the carrier temperature increases\ntoT′\neland the chemical potential decreases to E′\nF. The\nreason for the decrease of the chemical potential is the\nlinear scaling of the D(ǫ) with energy: a broader carrier\ndistribution (higher carrier temperature) would lead\nto an increased number of carriers in the conduction\nband, if the Fermi energy would be kept constant.\nTherefore, a higher carrier temperature leads to a lower\nFermi energy, as confirmed experimentally in Ref. [13].\nThe conductivity of photoexcited graphene σ′(ω) then\nfollows from Eq. 1, using the ’hot Fermi level’ and ’hot\ncarrier temperature’ with the photoconductivity given\nby ∆σ(ω) =σ′(ω)−σ(ω).\nOur model takes into account the effect of intraband\ncarrierheatingonthephotoconductivitythroughthecar-\nrier distribution, which is sufficient to explain the ob-\nservednegative photoconductivity [11, 16, 17, 20–22] and\nthe dependence on excitation power and Fermi energy.\nThe model does not explicitly include the effect of energy\nrelaxation to optical phonons on the photoconductivity,\nas proposed in Ref. [22]. This is justified, because for the\ngraphene used here (with an impurity-scattering-limited\nmobility below 2500 cm2/Vs) the effect of phonons on\nthe conductivity is negligibly small at low fluences [11].\nWe note that for very high fluences ( Nexc>1012ab-\nsorbed photons/cm2) and for phonon-scattering-limited\ngraphene with mobilities >10,000 cm2/Vs [29], this ef-\nfect likely plays a role, in addition to carrier heating.\nFurthermore, a more advanced model could include de-viations from linear scaling between the scattering time\nand the carrier energy [30], as well as changes in the\nDrude weight. The latter effect occurs when the valence\nand conduction band electrons no longer have separate\nthermal distributions and likely plays an important role\naround the Dirac point, where it correctly produces pos-\nitive photoconductivity [17].\nRise dynamics\nWe describe the time-resolved photoconductity with a\nphenomenological model that includes a rise step and an\nexponential decay step with time τdecay. In the low exci-\ntation power/high Fermi energy regime (highly efficient\ncarrier heating), the rise occurs within our instrument\nresponse function. For high excitation power/low Fermi\nenergythe risecontainsa veryfast component and aslow\ncomponent (reduced carrier heating efficiency). There-\nfore we model the rise dynamics with a ’fast’ rise compo-\nnent that is fixed at a value equal to the pulse duration\nofτpulse= 120–150fs (depending on the excitation wave-\nlength) and a variable ’slow’ rise component τslowwith\na longer pulse duration, and describe the time-resolved\nphotoconductivity by\n−∆σ(t) =A·Conv(e−t/τdecay,τpulse) +\nB·Conv(e−t/τdecay,τslow),(2)\nwhereAandBare the amplitudes correspond-\ning to the ’fast’ and the ’slow’ rise components and\nConv(y(t),τ) means taking the convolution of y(t) with\na Gaussian pulse of width τ. The effective rise time is\nthen given by\nτrise=A+B\nA/τpulse+B/τslow. (3)\nIn the fit, the free parameters are A,B,τslowand\nτdecay. The resulting effective rise time gives an indi-\ncation of how fast the initial energy relaxation of pho-\ntoexcited carriers takes place.\nACKNOWLEDGEMENTS\nWe would like to thank Justin Song, Leonid Levitov,\nSebastien Nanot and Enrique C´ anovas for useful discus-\nsions. KJT thanks NWO for a Rubicon fellowship. FK\nacknowledges support by the Fundacio Cellex Barcelona,\nthe ERC Career integration grant 294056 (GRANOP),\nthe ERC starting grant 307806 (CarbonLight) and\nsupport by the E. C. under Graphene Flagship (contract\nno. CNECT-ICT-604391).6\n[1] Bonaccorso et al., Graphene Photonics and Optoelec-\ntronics.Nature Phot. 4, 611 (2010)\n[2] J.M. Dawlaty et al. Measurement of the optical absorp-\ntion spectra of epitaxial graphene from terahertz to visi-\nble.Appl. Phys. Lett. 93, 131905 (2008)\n[3] L. Ren et al., Terahertz and Infrared Spectroscopy of\nGated Large-Area Graphene. Nano Lett. 12, 3711 – 3715\n(2012)\n[4] K.S. Novoselov et al., Electric field effect in atomically\nthin carbon films. Science306666 – 669 (2004)\n[5] L. Wang et al., One-Dimensional Electrical Contact to a\nTwo-Dimensional Material. Science342614-617 (2013)\n[6] N.M. Gabor et al., Hot Carrier-Assisted Intrinsic Pho-\ntoresponse in Graphene. Science334, 648-652 (2011)\n[7] Winzer, T. , Knorr, A.& Mali´ c, E. Carrier Multiplicatio n\nin Graphene. Nano Lett. 10, 4839–4843 (2010).\n[8] D. Brida et al., Ultrafast collinear scattering and carr ier\nmultiplication ingraphene. Nature Comm. 4, 1987 (2013)\n[9] T. Pl¨ otzing et al. Nano Lett. Article ASAP ,DOI\n10.1021/nl502114w (2014)\n[10] J.C.W. Song et al., Photoexcited carrier dynamics and\nimpact-excitation cascade in graphene. Phys. Rev. B 87\n,155429 (2013)\n[11] K.J. Tielrooij et al., Photoexcitation cascade and mul ti-\nple hot-carrier generation in graphene. Nature Phys. 9,\n248-252 (2013)\n[12] J.C. Johannsen et al., Direct view on the ultrafast carr ier\ndynamicsingraphene. Phys. Rev. Lett. 11, 027403 (2013)\n[13] I. Gierz et al., Snapshots of non-equilibrium Dirac car -\nrier distributions in graphene. Nature Mat. 12, 1119-1124\n(2013)\n[14] C.H. Lui et al., Ultrafast photoluminscence from\ngraphene. Phys. Rev. Lett. 105, 127404 (2010)\n[15] M. Breusing et al., Ultrafast nonequilibrium carrier d y-\nnamics in a single graphene layer. Phys. Rev. B 83,\n153410 (2011)\n[16] S.-F. Shi et al., Controlling Graphene Ultrafast Hot Ca r-\nrier Response from Metal-like to Semiconductor-like by\nElectrostatic Gating. Nano Lett. 14, 1578 – 1582 (2014)\n[17] A.J. Frenzel et al., Semiconducting-to-metallic phot ocon-\nductivity crossover and temperature-dependent Drude\nweight in graphene. Phys. Rev. Lett. 113, 056602 (2014)\n[18] J.H. Strait et al., Very Slow Cooling Dynamics of Pho-\ntoexcited Carriers in Graphene Observed by Optical-\nPump Terahertz-Probe Spectroscopy. Nano Lett. 11,\n4902-4906 (2011)\n[19] P.A. George et al., Ultrafast Optical-Pump Terahertz-\nProbe Spectroscopy of the Carrier Relaxation and Re-\ncombination Dynamics in Epitaxial Graphene. Nano\nLett.84248-4251 (2008)\n[20] C.J. Dochertyetal., Extremesensitivityofgraphenep ho-\ntoconductivity to environmental gases. Nature Comm. 3,\n1228 (2012)\n[21] A.J. Frenzel et al., Observation of suppressed teraher tz\nabsorption in photoexcited graphene. Appl. Phys. Lett.\n102, 113111 (2013)\n[22] G. Jnawali et al., Observation of a Transient Decrease\nin Terahertz Conductivity of Single-Layer Graphene In-\nduced by Ultrafast Optical Excitation. Nano Lett. 13,\n524–530 (2013)\n[23] R. Ulbricht et al., Carrier dynamics in semiconductorsstudied with time-resolved terahertz spectroscopy. Rev.\nMod. Phys. 83, 543 (2011)\n[24] J.J.H. Pijpers et al., Assessment of carrier-multipli cation\nefficiency in bulk PbSe and PbS. Nature Phys. 5, 811\n(2009).\n[25] T. Ando, Screening Effect and Impurity Scattering in\nMonolayer Graphene. J. Phys. Soc. Jpn. 75, 074716\n(2006)\n[26] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi,\nElectronic transport in two-dimensional graphene. Rev.\nMod. Phys. 83, 407 (2011)\n[27] C. Kittel, Introduction to Solid State Physics , Wiley and\nSons (2005)\n[28] E. Hendry et al., Interchain effects in the ultrafast pho to-\nphysics of a semiconducting polymer: THz time-domain\nspectroscopy of thin films and isolated chains in solution.\nPhys. Rev. B 71, 125201 (2005)\n[29] V. Perebeinos and P. Avouris, Inelastic scattering and\ncurrent saturation in graphene. Phys. Rev. B 81, 195442\n(2010)\n[30] E. H. Hwang, S. Adam, and S. Das Sarma, Carrier Trans-\nport in Two-Dimensional Graphene Layers. Phys. Rev.\nLett.98, 186806 (2007)7\nI. FIGURES\nTHz\n-0.5 0 0.501a\nc\nTime (ps)/c68/c115(norm.)THzTime (ps)/c45/c68/c115(e /h)2Decreasing Fermi Energy (eV) EF\ngatedrain\nsourceSiO2\nSigraphene\n-1 0 1 2 3\n/c116rise(ps)E (eV)F\n0.1 0.2\n0.20.5-10123\n-0.5 0 0.501b\ndIncreasing Fluence (10 /cm ) Nexc12 2\nTime (ps)/c68/c115(norm.)-1 0 1 2 3\nTime (ps)SiO\n2graphene\nN (10 /cm )exc12 2\n0 3 6 9\n0.10.304812\n/c45/c68/c115(e /h)2\n/c116rise(ps)0.267.6\n<0.060.21\nFIG. 1: Energy relaxation dynamics - dependence on Fermi energy and fluence. a)The time-resolved photoconductivity for\nnine different gate voltages ( V−VD= -70, -60, -50, -40, -30, -20, -15, -10 and 0 V), i.e. for decre asing Fermi energy EF\nor intrinsic carrier concentration Nint(sample with controllable Fermi energy). The photoconduct ivity is extracted from the\nmeasured changeinTHz transmission after photoexcitation with1500 nmlight (excitationpower corresponds to Nexc= 0.8·1012\nabsorbed photons/cm2), using the thin film approximation as in Ref. [11]. The inset shows the geometry of the sample, as\nexplained in the text. The photoconductivity is negative fo r all traces, except for the one closest to the Dirac point, wh ere\nit is positive. The solid lines are fits with an effective rise t imeτriseand an exponential decay time (see Methods). b)The\ntime-resolved photoconductivity for excitation with 800 n m light at six different excitation powers, corresponding to absorbed\nphoton densities of Nexc= 0.26, 0.50, 1.2, 2.5, 5.0 and 7.6 ·1012photons/cm2(sample with fixed Fermi energy). The geometry\nof the sample with a fixed Fermi energy of EF∼0.1 eV (see Supp. Info), is shown in the inset and explained in the text. The\nphotoconductivity is negative and increases for increasin g excitation power, as the carrier temperature increases. T he solid lines\nare fits with an effective rise time τriseand an exponential decay time (see Methods). c)The photoconductivity normalized\nto the peak of the signal (sample with controllable Fermi ene rgy), showing that decreasing the Fermi energy leads to slow er\nrise dynamics (i.e. slower energy relaxation). The fitted va lues forτriseas a function of Nintare shown in the inset, together\nwith the theoretically predicted energy relaxation time ba sed on Ref. [10] (scaled by a factor 1.8) and the experimental time\nresolution (horizontal dashed line). The schematics in the top left show the ultrafast energy relaxation of a photoexci ted carrier\nfor two different Fermi energies, according to the carrier he ating process described in Refs. [10, 11]. d)The photoconductivity\nnormalized to the peak of the signal (sample with fixed Fermi e nergy), showing that increasing the excitation power leads to\nslower rise dynamics. The fitted values for τriseas a function of Nexcare shown in the inset together with the experimental\ntime resolution (horizontal dashed line).8\nPump fluence ( J/cm ) /c1092Fermi energy (eV)a\nb\n0 0.5 1 1.5 2 2.5600 K900 K1200 K1500 K2200 K\nEF E’F\nTel= 300 K\nT’el= 2000 K\nT’el- (K)Tel2000 4000 6000 00THz\n/c45/c68/c115(a.u.)cC = /c97el el T\nHeating\nmodel\n0 0.1 0.2 0.300.20.4\nHeat capacity C ( J/m K)el/c1092\nFermi energy E (eV)F\n0.05\n0.15\n0.25\n50010002000\n150025003000T’ (K)el\nFIG. 2:Carrier heating model. a)The electronic heat capacity according to the analytical th eory (red line) and our numerical\nmodel (blue line), showing linear scaling with Fermi energy .b)Contour plot of the temperature of the carrier population,\nshowing thatdecreasing theFermi energy andincreasing the fluencehavean equivalenteffect onthe temperatureof theele ctrons\nafter photoexcitation and intraband thermalization. The c arrier temperature is calculated through a simple thermody namic\nmodel (see text). c)The calculated negative THz photoconductivity as a functio n of carrier temperature, showing a linear\ndependence up to ∼2000 K. The top left inset shows schematically the process of ultrafast energy relaxation of a photoexcited\ncarrier, which leads to an increase of the carrier temperatu re fromTeltoT′\nel. The inset on the bottom right shows the Fermi-\nDirac distribution of the conduction band carriers, where t he temperature increase leads to a decrease of the Fermi ener gy (to\nkeep the total number of conduction band carriers constant) .9\na\nb0.4 0.8 1.2 1.6/c68/c115(e /h)2\nReal\nFrequency (THz)Imag.\n-8-4048E = 1.55 eV0\n/c104= 0.75\nN = 110 /cmexc. 12 2\nE = 1.55 eV0\n/c104= 0.5\nN = 810 /cmexc. 12 2\n0.4 0.8 1.2 1.6/c68/c115(e /h)2\nFrequency (THz)-8-4048RealImag.\nFIG. 3: Comparison of experimental and theoretical photoconducti vity.a)The complex photoconductivity of the sample with\nfixed Fermi energy as a function of frequency for an excitatio n fluence corresponding to Nexc= 1·1012carriers/cm2, together\nwith the model result with a carrier heating efficiency of η= 0.75. The model (see also main text and Methods) describes\nthe photoconductivity after photoexcited carriers have th ermalized through intraband carrier-carrier scattering a nd directly\nyields the observed negative photoconductivity. b)The complex photoconductivity of the sample with fixed Fermi energy\nas a function of frequency for an excitation fluence correspo nding to Nexc= 8·1012carriers/cm2, together with the model\nresult with a carrier heating efficiency of η= 0.5. This shows that in this regime other energy relaxation channels contribute\nto the ultrafast energy relaxation. The insets in panels aandbschematically show the process of carrier heating through\nintraband scattering and the corresponding carrier distri butions for the ambient carrier temperature Tel(blue) and the elevated\ncarrier temperature T′\nelafter photoexcitation and carrier thermalization (red). T he error bars in panels aandbrepresent the\nexperimental (non-systematic) error bars (95 % confidence i nterval).10\na\nb0 5 10 15048/c104= 1/c104= 0.5\nN (10 /cm )exc12 2/c45/c68/c115(e /h)2\n0 0.4 0.8048\nN (10 /cm )exc12 2/c45/c68/c115(e /h)2\nE = 1.55 eV0E = 3.1 eV00 0.4 0.85000\n/c68T (K)E = 1.55 eV0\nN (10 /cm )exc12 20.1 1 10\n048\n/c104= 0.5\n/c104= 1N (10 /cm )exc12 2\n/c45/c68/c115(e /h)2\n0\nFIG. 4: Carrier heating efficiency - dependence on fluence. a)The peak photoconductivity of the sample with fixed Fermi\nenergy for a large range of excitation powers and an excitati on wavelength of 800 nm together with the model with η= 1\n(short dashed line) and a heating efficiency of η= 0.5 (long dashed line). The inset shows the same data on a log arithmic\nhorizontal scale. b)The peak photoconductivity of the sample with fixed Fermi ene rgy as a function of excitation power ( Nexc)\nfor excitation with 800 nm light (corresponding to an energy of the primary excited carrier of E0/2 = 0.75 eV) and 400 nm\nlight (E0/2 = 1.5 eV). The dashed lines correspond to the model with the s ame parameters (Fermi energy and scattering time\nproportionality constant) as in panel a, usingη= 1. The model describes the data well up to Nexc= 0.3·1012carriers/cm2for\n800 nm excitation and up to Nexc= 0.15·1012carriers/cm2for 400 nm excitation. The inset shows the carrier temperatu re\nthat is reached after excitation and thermalization by carr ier-carrier scattering, reaching ∼4000 K before heating becomes less\nefficient, for both excitation wavelengths. The symbol size o f the data in panels aandbrepresents the error.11\na\n/c45/c68/c115(e /h)2\nN (10 /cm )exc12 2bGate Voltage (V-V )D0 20 40 60-20246\n/c45/c68/c115(e /h)2\n0 2 4 60246\n/c104= 1\n/c104= 0.6\n/c104= 0.20 0.11 0.16 0.2Fermi energy (eV)\nEF~ 0.2 eV\nEF~ 0.17 eV\nEF~ 0.1 eVNexc= 5.7\nNexc= 3.8\nNexc= 1.9\nNexc= 0.24\nNexc= 0.14\nFIG. 5: Carrier heating efficiency – dependence on Fermi energy. a)The peak photoconductivity of the sample with\ncontrollable Fermi energy as a function of gate voltage for fi ve different excitation powers at excitation wavelength 150 0 nm.\nThe corresponding excitation density Nexcis given in units of 1012absorbed photons/cm2. At gate voltages away from the\nDirac point the photoconductivity is negative, while it cha nges sign upon approaching the Dirac point, in agreement wit h very\nrecent observations [16, 17]. b)The peak photoconductivity as a function of excitation powe r (Nexc) for three gate voltages,\ncorresponding to Nint= 1, 2 and 3 ·1012carriers/cm2(light to dark blue). The solid lines show the results of the m odel described\nin the text and the Methods section with a fixed heating efficien cy. For a decreasing number of intrinsic carriers (lower Fer mi\nenergy) the carrier heating efficiency decreases, in agreeme nt with the slowing down of the rise with decreasing Fermi ene rgy.\nThe symbol size of the data in panels aandbrepresents the error.12\nSUPPLEMENTARY MATERIAL\nA. Characterization of the sample with controllable Fermi e nergy\nIn order to translate the applied gate voltage Vginto the Fermi energy of the graphene sheet, we determine the\ncapacitive coupling of the weakly doped silicon backgate to the graph ene sheet. For this, we use a sample with the\nsame substrate as the sample used for the EF-dependent measurements, however with graphene shaped as a H all\nbar (see bottom right inset of Fig. S2a). We apply a current of I= 1µA in the x-direction and measure the Hall\nvoltageVHbetween two contacts in the y-direction. We measure the Hall voltage as a function of backgate v oltage\nfor az-directed magnetic field of both +0.6 T and -0.6 T, and use the differen ce between these two scans to account\nfor offsets (from device asymmetry, for example). From the Hall v oltage we extract the carrier density vs.backgate\nvoltage (see Fig. S1a) using\nNint=I·B\ne·VH, (S1)\nwithethe elementarycharge. This allowsus to extract the capacitanceo fthe backgate, wherewe find C= 6.3·10−5\nF/m2, which is somewhat smaller than the theoretical value for 300 nm oxid e:Cth= 11.6·10−5F/m2. The reason\nfor this could be the low density of carriers in the silicon or the large siz e of the graphene flake. We use the obtained\ncapacitive coupling to obtain the Fermi level corresponding to the a pplied gate voltages of the sample. It also allows\nus to obtain the device mobility, contact resistance and width of the neutrality point through the device resistance\nas a function of gate voltage. We find a contact resistance of 3.3 kΩ , a neutrality point width of ∆ = 58 meV, and a\n(lower bound) of the mobility of ∼300 cm2/Vs (see Fig. S1b). This is in reasonable agreement with the ∼700 cm2/Vs\nfound from the photoconductivity fits.\nGate Voltage (V )gNint(10 /cm12 2/c41\n-60 -40 -20 0 20 40 60-8-6-4-2\nx\nyz\n20 60 100 1400.411.6\nG (10 S/m)-4\nGate Voltage (V-V )Da b\nSample with controllable\nFermi energy\nC = 6.310 F/m. -5 2Hall bar sample\nC = 6.310 F/m. -5 2Fermi energy (eV)\n0.11 0.20 0.25 0.30\nFIG. S1: a)The number of intrinsic carriers as a function of gate voltag e, extracted from Hall measurements on a similar\ndevice as the one used in the Fermi-level dependent measurem ents, equipped with additional contacts in the Hall geometr y.b)\nThe device resistance as a function of ( Vg-VD) together with a fit using the determined capacitance, which yields the device\nmobility, contact resistance and neutrality region width.\nB. Characterization of the sample with fixed Fermi energy\nWe characterize the sample with fixed Fermi energy using Raman spe ctroscopy with pump wavelenth 532 nm (see\nan exemplary Raman trace in Fig. S2a). The width of the 2D peak of ∼33 cm−1shows that the graphene, grown by\nchemical vapor deposition, is predominantly monolayer. We also extr act the Fermi energy, by analyzing the G peak\nlocation and the ratio between the 2D and the G peak ( ∼1585 cm−1) and>2, respectively, for the trace shown in\nFig. S2). The Raman spectra of more than 100 traces on different lo cations on the sample yield an average G peak\nposition of ∼1585 cm−1, which indicates a Fermi energy of <0.15 eV [S1, S2]. To get the most realistic estimate for\nthe Fermi energy that corresponds to our optical pump - THz pro be experiment, the Raman spectra are taken under13\nPhoton counts (a.u.)\nRaman shift (cm )-11600 2000 2400 2800100020003000\n1560 1580 1600G peak:\n1585 cm-1\n2640 2680 27202D peak\nwidth:\n33 cm-1a b\n0.4 0.8 1.2 1.6 2048121620\nFrequency (THz)/c115(e /h)2Sample with fixed Fermi energy\n/c116/c47E\nEF\nF= 200 fs/eV\n= 0.155 eVSample with fixed Fermi energy\nFIG. S2: a)Raman spectrum of the sample with fixed Fermi energy. The inse ts show the G peak and the 2D peak. b)The\nsteady state conductivity spectrum of the sample with fixed F ermi energy\n(without photoexcitation), together with a Drude conducti vity fit (black solid line).\nvery similar environmental conditions (nitrogen flushing).\nWe further characterize the sample using THz conductivity measur ements, where we alternatively measure the\nsubstrate with and without graphene in the THz focus. This allows us to obtain the steady state conductivity\n(without photoexcitation). The results are shown in Fig. S2b toget her with a conductivity fit as explained in the\nMethods section. We obtain good agreement between data and mod el for a Fermi level of EF= 0.155 eV and a\nscattering time proportionality constant of 200 fs/eV, correspo nding to a mobility of ∼2300 cm2/Vs. We use this\nscattering time proportionality constant and a slightly lower Fermi e nergy (EF= 0.11 eV) to compare all optical\npump - terahertz probe data measured on this sample to the carrie r heating model. The slightly lower Fermi energy\nfollows from the photoconductivity at high fluence in Fig. 4a (given th e scattering time proportionality constant\nof 200 fs/eV), and could be due to spatial variation of the Fermi en ergy, photo-cleaning during the course of the\nexperiment, or due to a modified humidity during the experiment.\nC. Positive photoconductivity at the Dirac point and the Sil icon contribution\n/c68T/T (%)\nN (10 /cm )exc12 20 2 4 6-0.6-0.4-0.20\nGraphene,\nV = VDSilicon\nt = tpeak\n/c68T/T (%)\nGate voltage V-Vd0 20 40 60-0.6-0.4-0.20 a b\nSilicon\nt = tpeak\nFIG. S3: Pump-probe signal recorded at the pump-probe delay corresponding to the graphene signal peak, measured at the\nDirac point (blue) with negative change in transmission (po sitive conductivity) as a function of absorbed fluence (in th e\ngraphene sheet). The same measurement on the substrate with out graphene at the same pump-probe delay gives a quadratic\nsignal due to two photon absorption (red).\nWe use an excitation wavelength of 1500 nm for our pump-probe mea surements on the gated graphene sample to\navoid exciting electron-hole pairs in the silicon, which would obscure ou r signal. However, there is a small contribution\nof two-photon absorption that leads to THz photoconductivity in t he silicon, in addition to the photoconductivity of14\nthe graphene sheet. We find that for the lowest fluences the silicon signal is negligible. However due to the quadratic\nincrease with fluence, the silicon signal is not negligible at the highest fl uences, in cases where the Fermi energy,\nand therefore the graphene signal, is very low. In Fig. S3a we show t he pump probe signal that originates from the\nsubstrate without graphene and the pump probe signal for the su bstrate with graphene at very low Fermi energy. It\nis noteworthy to point out that the silicon contribution to the combin ed graphene-silicon signal is insignificant at gate\nvoltages far away from the Dirac point and that it does not change w ith gate voltage (see Fig. S3b). Only around\nthe Dirac point, and at the highest fluences, does part of the posit ive photoconductivity come from the graphene and\npart from the silicon. Importantly, at low fluence, the positive phot oconductivity that we observe stems completely\nfrom the graphene, showing the capability of tuning the Fermi ener gy close enough to the Dirac point to change the\nsign of the graphene THz photoconductivity. And even at the highe st employed fluence, the signal from graphene at\nthe Dirac point is still larger than the signal from the silicon substrat e. We corrected the fluence dependent results\npresented in Fig. 5 of the main paper for this small substrate contr ibution.\nSupplementary References\n[S1] H. Yan et al., Infrared spectroscopy of wafer-scale graphen e.ACS Nano 5, 9854–9860 (2011)\n[S2] A. Das et al., Monitoring dopants by Raman scattering in an electr ochemically top-gated graphene transistor.\nNature Nanotech. 3, 210–215 (2008)"    },    {        "title": "1810.13277v2.Many_body_filling_factor_dependent_renormalization_of_Fermi_velocity_in_graphene_in_strong_magnetic_field.pdf",        "content": "arXiv:1810.13277v2  [cond-mat.mes-hall]  13 Feb 2019Many-body filling-factor dependent renormalization of Fer mi velocity in graphene\nin strong magnetic field\nAlexey A. Sokolik1,2and Yurii E. Lozovik1,2,3,∗\n1Institute for Spectroscopy, Russian Academy of Sciences, 1 42190 Troitsk, Moscow, Russia\n2National Research University Higher School of Economics, 1 09028 Moscow, Russia\n3Dukhov Research Institute of Automatics (VNIIA), 127055 Mo scow, Russia\nWe present the theory of many-body corrections to cyclotron transition energies in graphene in\nstrong magnetic field due to Coulomb interaction, considere d in terms of the renormalized Fermi\nvelocity. A particular emphasis is made on the recent experi ments where detailed dependencies of\nthis velocity on the Landau level filling factor for individu al transitions were measured. Taking\ninto account the many-body exchange, excitonic correction s and interaction screening in the static\nrandom-phase approximation, we successfully explained th e main features of the experimental data,\nin particular that the Fermi velocities have plateaus when t he 0th Landau level is partially filled and\nrapidly decrease at higher carrier densities due to enhance ment of the screening. We also explained\nthe features of the nonmonotonous filling-factor dependenc e of the Fermi velocity observed in the\nearlier cyclotron resonance experimentwith disordered gr aphene bytakingintoaccount thedisorder-\ninduced Landau level broadening.\nI. INTRODUCTION\nMassless Dirac electrons in single-layer graphene offer\nan opportunity to study condensed-matter counterparts\nof relativistic effects and to achieve new regimes in quan-\ntum many-bodysystems [1–3]. Low-energyelectronic ex-\ncitations in this material obey the Dirac equation and\nmove with the constant Fermi velocity vF≈106m/s. In\na strong perpendicular magnetic field B, quantization of\nan electron kinetic energy in graphene results in the rel-\nativistic Landau levels [4]\nEn= sgn(n)vF/radicalbig\n2|n|Be/planckover2pi1/c, n= 0,±1,±2,...(1)\nUnlike usual Landau levels for massive electrons, the rel-\nativistic ones are not equidistant En∝/radicalbig\n|n|, scale as a\nsquare root of magnetic field, En∝√\nB, and obey the\nelectron-holesymmetry, En=−E−n. Relativisticnature\nofgrapheneLandaulevelswasfirstconfirmedbythe half-\ninteger quantum Hall effect [2], and direct observations\nof these levels using the scanning tunneling spectroscopy\nhad followed (see the review of experiments in [5]).\nAnother way to study Landau levels in graphene is to\ninduce electron interlevel transitions by an electromag-\nnetic radiation, typically in the infrared range. The se-\nlection rules for photon absorption[6] require∆ |n|=±1,\nimplying the intraband −n−1→ −n,n→n+ 1, and\ninterband −n−1→n(which will be referred to as T−\nn+1)\nand−n→n+1 (referred to as T+\nn+1) transitions. The\ninterband transitions T±\nn+1, which are more widely stud-\nied, have the energies\nEn+1−E−n=vF/radicalbig\n2Be/planckover2pi1/c/parenleftbig√n+√\nn+1/parenrightbig\n(2)\nin the ideal picture of massless Dirac electrons (1) in the\nabsence of interaction and disorder.\n∗Electronic address: lozovik@isan.troitsk.ruIn a series of cyclotron resonance measurements,\nmainly on epitaxial graphene, transition energies in very\ngoodagreementwith Eq.(2) werereported(see[7, 8] and\nreferences therein). However, the other experiments [9–\n12] demonstrated deviations from Eq. (2) due to many-\nbody effects and, possibly, disorder. Similar deviations\nwere discovered in magneto-Raman scattering for both\ncyclotron T±\nn+1[13] and symmetric interband −n→n\n[14–16] transitions. Indeed, the Kohn’s theorem [17],\nwhichprotectscyclotronresonanceenergiesofusualmas-\nsiveelectronsagainstmany-bodyrenormalizations,isnot\napplicable to graphene [18–29]. The observed energies of\nT±\nn+1can be described by the counterpart of Eq. (2)\nΩ±\nn+1=v∗\nF/radicalbig\n2Be/planckover2pi1/c/parenleftbig√n+√\nn+1/parenrightbig\n(3)\nwith the bare Fermi velocity vFreplaced by the renor-\nmalized velocity v∗\nF. While the former one, vF, should be\nclose to 0.85×106m/s, as indicated by fitting theoreti-\ncalcalculationstovariousexperimentaldataongraphene\n(see, e.g., [29–32]), the latter one, v∗\nF, range from 106m/s\nto 1.4×106m/s depending on carrier density, magnetic\nfield and substrate material [9–16]. The existing the-\nory describes renormalization of Fermi velocity in mag-\nnetic field with reasonable accuracy in the Hartree-Fock\n[15, 22–24] and static random-phase [26, 29, 33] approx-\nimations.\nIn two very recent experiments [12, 13], the energies of\nthe T±\nn+1transitions were measured with high accuracy\nas functions of the Landau level filling factor ν, that may\nprovide an especially deep insight into the many-body\nphysics of graphene in magnetic field. Unlike graphene\nwithout magneticfield, where v∗\nFdivergeslogarithmically\nupon approach to the charge neutrality point [3, 31, 34],\nhere it saturates to a finite value at ν→0, and, in the\nmost cases, has even a broad plateau in the range −2<\nν <2.\nIn this article, we calculate the energies of the T±\nn+1\ntransitions as functions of the filling factor νwith taking2\ninto account many-body effects. Our approach, which\nis described in Sec. II and Appendices A, B, and C,\ntakes into account the screening of the Coulomb inter-\naction as one of the key points, that contrasts with the\nmost calculations on this subject [15, 18–25, 28] based on\ntheHartree-Fockapproximationwithunscreenedinterac-\ntion. The screening allowed us to describe experimental\ndata on both Landau levels [35] and interlevel transition\nenergies [29] earlier, and provides improved understand-\ning to the filling-factor dependence of the observed v∗\nFin\nthis work as well.\nIn Sec. III we analyze the electron-hole asymmetry of\ntransition energies and the presence of plateaus at −2<\nν <2, following from the propertiesofinteraction matrix\nelements. In Sec. IV we present the results of numerical\ncalculations, which reproduce the main features of the\nexperimental the vF(ν) dependencies from Refs. [12, 13]:\na) the plateausin v∗\nFat−2<ν <2 when the 0th Landau\nlevelispartiallyfilled, b) the rapiddecreaseof v∗\nFat|ν|>\n2 with increasing carrier density, c) the decrease of v∗\nFat\nν= const at increasing magnetic field. We have found\ngood agreement between the experiments and the theory\nusing the bare Fermi velocity vF= 0.85×106m/s and\nrealisticvaluesofthedielectricconstant ε. Theintraband\ntransitionsn→n+1 and−n−1→nwerealsoanalyzed,\nandwepredicttheV-shapeddependence oftheirenergies\nonν.\nAdditionally, we have considered the nonmonotonous\ndependence v∗\nF(ν) for the T±\n1transition observed in [11]\nwith the maximum at ν= 0 and minima at ν=±2.\nTaking into account a disorder-induced broadening of\nLandau levels, we have explained this dependence with\ngood accuracy in Sec. V. Our conclusions are presented\nin Sec. VI.\nII. THEORETICAL APPROACH\nDynamical conductivity of graphene can be calculated\nusing the Kubo formula [36]\nσαβ(q,ω) =1\n/planckover2pi1ωS∞/integraldisplay\n0dtei(ω+iδ)t∝an}bracketle{t[jα(q,t),jβ(−q,0)]∝an}bracketri}ht,(4)\nwherejα(q,t) is theα-axis projection of the Fourier\ncomponent of the current density operator jα(q) =\nevF/integraltext\ndrΨ+(r)σαΨ(r)e−iqrevolving in time in the\nHeisenberg representation, Ψ( r) is the two-component\nfield operator for Dirac electrons, Sis the system area,\nandδ→+0.\nDiagrammatic representation of the conductivity,\nshown in Fig. 1(a), allows its calculation in terms of the\ncurrent vertex matrix Γ β, which would be equal to σβin\nthe absenceofinteractionand disorder. To find it, we use\nthe mean field approximation, where the excitonic ladder\n[Fig. 1(b)] for the vertex Γ βand the one-loop self-energy\ncorrections [Fig. 1(c)] for the single-particle Green func-\ntionsGare taken into account. Using the interaction,(a) σαβ=\n(b) = +\n(c) = +\n(d) = +\nFIG. 1: (a) Diagrammatic relationship (C2) between the cur-\nrent Green function and the vertex. (b), (c) Equations for,\nrespectively, the vertex function and the electron Green fu nc-\ntion in the mean-field approximation. (d) Coulomb interac-\ntion screening in the random-phase approximation.\nwhich is statically screened in the random-phase approx-\nimation [Fig. 1(d)], greatly simplify the calculations. If\nwe additionally neglect the mixing of different pairs of\nelectron and hole Landau levels, appearing in the exci-\ntonic ladder (which was shown to be weak under typical\nconditions with using the screened interaction [29]), the\noptical conductivity σαβ(ω)≡σαβ(0,ω) is (see the de-\ntails of calculations in Appendix C):\nσαβ(ω) =ie2v2\nF\nω/summationdisplay\nn1n2fn2−fn1\n/planckover2pi1ω−Ωn1n2+iδ\n×Tr[Φn1n2(0)σα]Tr/bracketleftbig\nΦ+\nn1n2(0)σβ/bracketrightbig\n.(5)\nHerefnis the occupation number (0 /lessorequalslantfn/lessorequalslant1) of the\nnth Landau level, and the matrix Φ n1n2(0), which is de-\nfined by (C5) and (A2), determines the selection rules\n|n1|=|n2|±1 for eachn2→n1transition. The resonant\ntransition energy Ω n1n2, whereσαβhas a pole, consists\nof the difference between the bare Landau level energies\nEn1−En2, the difference between electron self-energies\nΣn1−Σn2, and the excitonic correction ∆ E(exc)\nn1,n2(see the\nsimilar formula in [25]):\nΩn1n2=En1−En2+Σn1−Σn2+∆E(exc)\nn1n2.(6)\nIn the mean field approximation, the self-energy\nΣn=−/summationdisplay\nn′fn′∝an}bracketle{tnn′|V|n′n∝an}bracketri}ht, (7)\nas shown in Appendix B, is a sum of the exchange matrix\nelements\n∝an}bracketle{tnn′|V|n′n∝an}bracketri}ht= 2δn0+δn′0−2l2\nH\n2π/integraldisplay\ndqV(q)\n×/vextendsingle/vextendsinglesnsn′φ|n|−1,|n′|−1(aq)+φ|n||n′|(aq)/vextendsingle/vextendsingle2(8)3\nof the screened Coulomb interaction V(q) between the\nnthandallfilled n′thLandaulevels[15,18,19],wherethe\nfunctionsφnkare defined in (A2), and aq≡ −l2\nH[ez×q].\nThe excitonic correction\n∆E(exc)\nn1n2=−(fn2−fn1)∝an}bracketle{tn1n2|V|n1n2∝an}bracketri}ht(9)\nis the direct interaction matrix element\n∝an}bracketle{tn1n2|V|n1n2∝an}bracketri}ht= 2δn10+δn20−2l2\nH\n2π/integraldisplay\ndqV(q)\n×/braceleftBig\nφ∗\n|n1|−1,|n1|−1(aq)+φ∗\n|n1||n1|(aq)/bracerightBig\n(10)\n×/braceleftbig\nφ|n2|−1,|n2|−1(aq)+φ|n2||n2|(aq)/bracerightbig\nwith the minus sign, weighted with the difference of oc-\ncupation numbers of the final and initial levels.\nThe dynamically screened interaction in the random-\nphase approximation is [see Fig. 1(d)]\nV(q,iω) =vq\n1−vqΠ(q,iω), (11)\nwherevq= 2πe2/εqis the bare Coulomb interaction\nweakened by the surrounding medium with the dielec-\ntric constant ε, and\nΠ(q,iω) =g/summationdisplay\nnn′Fnn′(q)fn−fn′\niω+En−En′,(12)\nisthe polarizability(ordensityresponsefunction) ofnon-\ninteracting Dirac electrons [4, 37–41]. Here\nFnn′(q) = 2δn0+δn′0−2\n×/vextendsingle/vextendsinglesnsn′φ|n|−1,|n′|−1(aq)+φ|n||n′|(aq)/vextendsingle/vextendsingle2(13)\nis the form-factor of Landau level wave functions and\ng= 4 is the degeneracy of electron states by valleys and\nspin. The staticallyscreened interaction V(q) is obtained\nfrom (11), (12) by taking iω= 0.\nIn our model, there are three mechanisms leading to\ndependence of Ω n1n2on the filling factor νvia the occu-\npation numbers\nfn=\n\n0, ifν≤4n−2,\n(ν−4n+2)/4,if 4n−2<ν <4n+2,\n1, ifν≥4n+2,(14)\ni.e.: through exchange energies (7), excitonic corrections\n(9), and polarizability (12).\nNotethatthesum(7)overthefilledLandaulevels n′in\nthe valence band diverges at n′→ −∞, so we impose the\ncutoffn′/greaterorequalslant−ncto obtain finite results. The physical rea-\nson of thus cutoff is a finite actual number of Landau lev-\nels in the valence band, which can be found from the to-\ntal electron density: nc= 2π/planckover2pi1c/√\n3a2eH≈39600/B[T],\nwherea≈2.46˚A [29, 35].-6 -4 -2 0 2 4 61.051.101.151.20\n-4 -2 0 2 4 6v¤\nFv¤\nF\n(106m/s) (106m/s)\nº º º º0!10!1¡1!0¡1!0\n¡1!2¡1!2¡2!1¡2!1(a) (b) \nFIG. 2: Renormalized Fermi velocities v∗\nFfor (a) the T±\n1and\n(b) theT±\n2transitions calculated with the screened interac-\ntion atvF= 0.85×106m/s,ε= 3.27,B= 8T. Solid lines\nshow the velocities found from the weighted transition ener -\ngies (20).\nIII. ELECTRON-HOLE ASYMMETRY AND\nPLATEAUS AT −2< ν <2\nThe selection rule |n1|=|n2| ±1 for the interband\nn2→n1transitions implies n1,n2=n+1,−n(the T+\nn+1\ntransition)or n1,n2=n,−n−1(theT−\nn+1transition). In\nthe idealized Dirac model without interactions, the ener-\ngies of these transitions (2) are equal. However this is no\nlongerthecasewhenexchangeself-energiesaretakeninto\naccount. Any nonzero doping ν∝ne}ationslash= 0 introduces an asym-\nmetrybetweenΩ+\nn+1andΩ−\nn+1, atleast,inthemean-field\napproximation. Looking at (6) and taking into account\nthat∝an}bracketle{tn+1,−n|V|n+1,−n∝an}bracketri}ht=∝an}bracketle{t−n−1,n|V|−n−1,n∝an}bracketri}ht,\nwe have:\nΩ+\nn+1−Ω−\nn+1= Σn+1+Σ−n−1−Σn−Σ−n\n+(fn+fn+1−f−n−f−n−1)∝an}bracketle{tn+1,−n|V|n+1,−n∝an}bracketri}ht.(15)\nThe electron-hole asymmetry in graphene, which is in-\nduced by the exchange interaction in the absence of mag-\nnetic field and is similar in scale to our case, was found\nin [42].\nThe first line of (15) is a contribution of exchange self-\nenergies to the asymmetry. Let us separate the occu-\npation numbers fn′=f(0)\nn′+ ∆fn′on those of undoped\ngraphenef(0)\nn′and the doping-induced part ∆ fn′, and de-\nfine Σ(0)\nn=−/summationtext\nn′f(0)\nn′∝an}bracketle{tnn′|V|n′n∝an}bracketri}ht. Using (8) and (A6),\nand neglecting a difference of small matrix elements at\nn′≈ −nc, we get Σ(0)\nn+1+Σ(0)\n−n−1−Σ(0)\nn−Σ(0)\n−n= 0. Thus\nthe exchange energy contribution to (15) arises only at\nnonzero doping ν∝ne}ationslash= 0.\nThe second line of (15) corresponding to excitonic ef-\nfects is nonzeroonlywhen either ±nth or±(n+1)thlevel\nis partially filled, i.e. at 4 n−2<|ν|<4n+ 6. Since\nthe polarizability (12) and hence the screened interaction\nV(q) are even functions of ν, both parts of (15) change\nsign atν→ −ν, so\nΩ+\nn+1(ν) = Ω−\nn+1(−ν). (16)4\nThis is illustrated in Fig. 2, where the typical calculated\nv∗\nFare shown as functions of ν.\nThe caseof n= 0is the specialone. The explicit struc-\nture of the wave functions (A2) imply the following re-\nlationships connecting the matrix elements of direct and\nexchange interaction (valid even for non-Coulomb poten-\ntials):\n∝an}bracketle{t±1,0|V|±1,0∝an}bracketri}ht+∝an}bracketle{t±1,0|V|0,±1∝an}bracketri}ht=∝an}bracketle{t00|V|00∝an}bracketri}ht.(17)\nIn result, the doping-induced changes of exchange and\nexcitonic parts of (6) due to f0cancel each other at\n−2<ν <2, when the 0th level is partially filled. Addi-\ntionally, the polarizability (12) and hence V(q) are also\nunchanged in this range of ν, thus we expect plateaus in\nboth Ω±\n1:\nΩ+\n1(ν) = Ω−\n1(−ν) = const at −2<ν <2,(18)\nas seen in Fig. 2(a). For n∝ne}ationslash= 0 this is no longer the\ncase, although variations of Ω±\nn+1(ν) at−2< ν <2 are\ntypically very small [see Fig. 2(b)].\nIn experiments, Ω+\nn+1and Ω−\nn+1can be separated by\nobserving cyclotron resonant absorption of light with op-\nposite circular polarizations. Using linear polarization,\none can observe a mixture of these transitions with rela-\ntiveintensities I+\nn+1=f−n−fn+1andI−\nn+1=fn−f−n−1,\nequal to occupation number differences in final and ini-\ntial states. Assuming that experimental apparatus does\nnot resolve the individual lines Ω+\nn+1and Ω−\nn+1, we will\ncalculate the weighted transition energy\n∝an}bracketle{tΩn+1∝an}bracketri}ht=Ω+\nn+1I+\nn+1+Ω−\nn+1I−\nn+1\nI+\nn+1+I−\nn+1(19)\nand compare it with the experiments in the next section.\nFrom the particle-hole symmetry relationship fn(−ν) =\n1−fn(ν) we see that ∝an}bracketle{tΩn+1∝an}bracketri}htis even function of ν. At\n−2< ν <2,∝an}bracketle{tΩn+1∝an}bracketri}htis linear (via f0) and at the same\ntime even function of ν, so\n∝an}bracketle{tΩn+1∝an}bracketri}ht= const at −2<ν <2, (20)\nas seen in Figs. 2(a,b). Thus our model predicts plateaus\nin all weighted transition energies ∝an}bracketle{tΩn+1∝an}bracketri}htat−2<ν <2.\nSimilar conclusions about existence of the electron-hole\nasymmetry (15) and the conjugation property (16) were\nmade in the recent theoretical work [25], which considers\ntransition energies in the Hartree-Fock approximation.\nIV. CALCULATION RESULTS\nFirst we compare our calculations of the renormalized\nFermi velocities\nv∗\nF=∝an}bracketle{tΩn+1∝an}bracketri}ht/radicalbig\n2Be/planckover2pi1/c/parenleftbig√n+√n+1/parenrightbig (21)with the data of Ref. [12] where Ω 1...Ω6as functions of\nνwere measured at three magnetic fields B= 5,8,11T.\nWe fit the experimental points in three approximations:\n1) Hartree-Fock approximation, where the unscreened\nCoulomb potential vqis used in all calculations.\n2) Static random-phase approximation, where the po-\ntentialV(q) is screened (11) with using the polarizability\nof noninteracting electron gas in magnetic field.\n3) Self-consistent screening approximation, where the\npolarizability is multiplied by vF/v∗\nFto take into ac-\ncount weakening of the screening caused by many-body\nincrease of the energy differences En′−Enin denomi-\nnators of (12). Similarly to our previous works [29, 35],\nthis semi-phenomenological model is aimed to achieve a\nself-consistency between many-body renormalizations of\ntransitionenergiesandscreening. Usingtheiterativepro-\ncedure, we take v∗\nF, obtained on each step, to renormalize\nthe screening when calculating new v∗\nFon the next step.\nAbout 5-6 iterations are usually sufficient to achieve a\nconvergence.\nCalculations in our approach depend only on two pa-\nrameters: the bare Fermi velocity vFand the dielectric\nconstant of a surrounding medium ε. In principle, both\nvFandεcan be treated as fitting parameters. However\nvariation of vFin the range (0 .8÷0.95)×106m/s allows\ntoachievealmostequallygoodagreementwith theexper-\nimental data at slightly different ε, so a simultaneous fit-\nting of both parameters does not provide reliable results.\nTherefore we choose a specific value vF= 0.85×106m/s,\nwhichwasconcludedtobethemostsuitableonebasedon\ntheoretical fits of several experimental data on graphene\nboth in presence [29, 30] and absence [31, 32] of mag-\nnetic field. After that, the optimal dielectric constant\nof the surrounding medium εis the only adjustable pa-\nrameter in our model, and we find it by performing the\nleast square fitting of the experimental points for all n\nandBsimultaneously. Nevertheless it should be kept in\nmind that our fitting results can slightly change quanti-\ntatively with different choice of vF(although qualitative\nconclusions will be the same), and it could be promis-\ning to implement a renormalization-groupscheme for the\nLandau level data on graphene where all unobservable\nvariables like vFcan be excluded from the model.\nThe first line of Table I shows the optimal εused to\nfit the cyclotron resonance data of Ref. [12] where high-\nmobility graphene samples were encapsulated from both\nsides in hexagonal boron nitride monolayers and placed\nTABLE I: Dielectric constants of surrounding medium ε,\nwhich provide the best least-square fittings of the experime n-\ntal data from Refs. [12, 13] at vF= 0.85×106m/s in the three\napproximations for the interaction listed in Sec. IV.\nUnscreened Screened Self-consistent\nExperiment interaction interaction screening\nRussell et al. [12] 7.72 3.27 4.36\nSonntag et al. [13] 5.50 1.05 2.555\n-8 -4 0 4 8 -8 -4 0 4 81.051.101.151.20\n-8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8\n-8 -4 0 4 8 -8 -4 0 4 81.051.101.151.20\n-8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8\n-8 -4 0 4 81.051.101.151.20\n-8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8 -8 -4 0 4 8(a)B=5T (a)B=5T\n(b)B=8T (b)B=8T\n(c)B=11T (c)B=11Tº º\nº º\nº ºT1T1 T2T2 T3T3 T4T4 T5T5 T6T6\nT1T1 T2T2 T3T3 T4T4 T5T5 T6T6\nT1T1 T2T2 T3T3 T4T4 T5T5 T6T6v¤\nFv¤\nF\n(106m/s) (106m/s)v¤\nFv¤\nF\n(106m/s) (106m/s)v¤\nFv¤\nF\n(106m/s) (106m/s)\n2 4 6 800.010.020.030.04¢(106m/s) ¢(106m/s)(d)(d)\n\" \"\nFIG. 3: Renormalized Fermi velocities v∗\nFat (a)B= 5, (b) 8, and (c) 11T for the set of the Tn+1transitions ( −n→\nn+1/−n−1→n), taken from the experiment [12] (crosses), and calculated theoretically in the Hartree-Fock approximation\n(dashed lines), with taking into account the interaction sc reening (solid lines) and with the self-consistent screeni ng (dotted\nlines). The dielectric constants ε, used in each calculation, are listed in the first line of Tabl e I. Root mean square deviations\n(22) between the calculated and experimental v∗\nFare also shown (d) as functions of εin the three approximations.\non an oxidized silicon. Fig. 3 shows the experimental\npoints together with our calculations at these εin the\nthree approximations described above. The calculation\nwith the unscreened Coulomb interaction (Hartree-Fock\napproximation) demonstrates two significant drawbacks.\nFirst, the dielectric constant ε≈7.72 is unrealistically\nhigh, because in this approximation it should imitate the\ninteraction screening by Dirac electrons in graphene in\naddition to the screening by an external medium. Sec-\nond, thefalloffof v∗\nFat|ν|>2turnsouttobeinsufficient,\nbecause the increase of the screening strength (and, con-\nsequently, suppression of the upward renormalization of\nthe Fermi velocity) following the carrier density, is ab-\nsent here. For the T1transitions, the calculated v∗\nFeven\nincreases at |ν|>2, in contradiction with the experi-\nment, because the excitonic correction (9), which nor-\nmally decrease v∗\nF, become suppressed due to partial fill-\ning of 1th or −1th Landau level. The similar drawbacks\nof the Hartree-Fock approximations were mentioned in\nour previous works [29, 35].\nThe screening allows us to achieve much better agree-\nment with the experimental points at more realistic ε≈\n3.27, and the falloff of v∗\nFat|ν|>2 is reproduced very\nwell. The iterative calculations with the self-consistent\nscreening provide almost the same curves, but at some-\nwhat higher ε≈4.36. This distinction arises because the\nhigherεisneededtocompensatethescreeningweakeningcaused by an upward renormalization of energy denomi-\nnators in (12).\nOurcalculationswithtakingintoaccountthescreening\nare thus able to fit the data of Ref. [12] at three different\nBand for six resonances Tn+1simultaneously with the\nsingle adjustable parameter ε. We can explain both the\ndecrease of v∗\nFatν= 0 asBgets higher, the plateaus\nat|ν|<2, and the rapid falloff of v∗\nFat|ν|>2,n/greaterorequalslant1\ndue to increase of the screening strength. The exceptions\nare some inconsistencies of v∗\nFat specific resonances (T 2\nand T 6atB= 5T, T 5and T 6atB= 8T) and the\nlocal maxima at ν= 0 for the T 1transitions. Moreover,\nthe local minima of v∗\nFfor T2atν=±4, when the 1th\nor−1th Landau level is half-filled, which occur only at\nB= 8T and are absent in other fields, are not predicted\nby our approach.\nTo characterize an accuracy of our fitting, we present\nin Fig. 3(d) the root mean square deviation\n∆ =/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\ni=1[(v∗\nF)calc\ni−(v∗\nF)exp\ni]2\nN(22)\nof calculated renormalized Fermi velocities ( v∗\nF)calc\nifrom\nNexperimentalvalues( v∗\nF)exp\ni(hereit meansforallfields\nand all resonances at once, 260 points in total). As func-\ntions of the fitting parameter ε, ∆ reach rather sharp6\n-28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 2811.11.21.3\nv¤\nFv¤\nF\n(106m/s) (106m/s)\nº ºB=8TB=8T\n1)21)22)32)33)43)44)54)55)65)6-2)-1 -2)-1-3)-2 -3)-2-4)-3 -4)-3-5)-4 -5)-4 -6)-5 -6)-5\nFIG. 4: Renormalized Fermi velocities v∗\nFfor the intraband n→n+1 and −n−1→ −ntransitions calculated at B= 8T in\nthe Hartree-Fock approximation (dashed lines) and with tak ing into account the interaction screening (solid lines). T he self-\nconsistent iterative calculations are not shown because th eir results are close to those with the non-self-consistent screening.\nThe dielectric constants εare taken from the first line of Table I.\nminima at optimal εin each approximation. The mini-\nmal ∆ about 0 .015×106m/s are comparable to the ex-\nperimental uncertainties of ( v∗\nF)exp\ni[12], so the fitting can\nbe considered to be sufficiently accurate.\nFor completeness of the analysis, we can also consider\nthe intraband transitions n→n+1 and −n−1→ −n.\nSeveral examples calculated in the conditions of the ex-\nperiment [12] are presented in Fig. 4. Each n→n+ 1\n(−n−1→ −n) transition exists in the range 4 n−2<\nν <4n+6 (−4n−6< ν <−4n+2) of the filling fac-\ntors,andthetransitionenergiesareminimalat ν= 4n+2\n(ν=−4n−2). These minima arecaused by the excitonic\ncorrection (9), which is maximal when the initial Landau\nlevel is completely filled, and the final level is completely\nempty. We can also note that v∗\nFagain decreases with\nincreasing the doping level due to enhancement of the\nscreening, while the Hartree-Fock approximation misses\nthis effect and greatly overestimates the variations of v∗\nF\nvs.ν. The electron-hole asymmetry for these transitions\nis negligible.\nAnother experiment we analyze is Ref. [13] where\ngrapheneis suspended 160nmaboveoxidizedsilicon, and\nthe filling-factor dependence of the T 2transition energy\nwas measured at B= 3T by observing its avoided cross-\ningwiththephononenergyinRamanspectrum. InFig.5\nweplottheresultsofourcalculationsforthistransitionin\nthethreeapproximationsatoptimal εlistedinthesecond\nline of Table I. We observe the same regularities as in the\npreviouscase. TheHartree-Fockapproximationsrequires\noverestimated εand cannot explain the rapid falloff of v∗\nF\nat|ν|>2. At|ν|>6 we see even slight increase of v∗\nF\ndue to suppression of the excitonic correction when the\n2nd or−2nd Landau level start to be partially filled.\nIn contrast, with taking into account the screening we\nobtain the realistic εfor graphene suspended above the\noxidized silicon, and the falloff is well reproduced. Nev-\nertheless, the experimental points demonstrate an addi-\ntional maximum at ν= 0. This is not described by our\napproach,whichpredictsplateausat |ν|<2, asdiscussed\nin Sec. III.\nThe root mean square deviations (22), calculated for31 experimental points, are shown in the inset to Fig. 5\nand demonstrate pronounced minima at the optimal ε.\nThe minimal ∆ about 0 .015×106m/s, achieved with the\nscreened interaction, are comparable to the experimental\nuncertainties (0 .01÷0.05)×106m/s [13].\nV. LANDAU LEVEL BROADENING\nOne more experiment where the filling-factor depen-\ndent transition energy was measured is Ref. [11]. In this\nearlier work, graphene layer lied directly on an oxidized\nsilicon substrate and carrier mobility was one-two orders\n-8 -6 -4 -2 0 2 4 6 81.201.251.301.351.40\nv¤\nFv¤\nF\n(106m/s) (106m/s)\nº ºT2; B= 3T T2; B= 3T\n0 2 4 600.010.020.030.04\n\" \"¢ (106cm/s ) ¢ (106cm/s )\nFIG. 5: Renormalized Fermi velocity v∗\nFfor theT2transition\n(−1→2/−2→1) atB= 3T, taken from the experiment\n[13] (squares) and calculated theoretically in theHartree -Fock\napproximation (dashed lines), with taking into account the\ninteraction screening (solid lines) and with the self-cons istent\nscreening (dotted lines). The dielectric constants ε, used in\neach approximation, are listed in the second line of Table I.\nInset shows root mean square deviations (22) between the\ncalculated and experimental v∗\nFas functions of εin the three\napproximations.7\nof magnitude lower than in the aforementioned works\n[12, 13] due to charged impurities in the substrate. The\nT1cyclotron resonance was studied at B= 18T and the\nunusual W-shaped form of the transition energy vs. ν\nwas found with the local maximum at ν= 0 and two\nminima at integer Landau level fillings ν=±2.\nTo explain these results, we need to take into account\ndisorder, because at mobilities of several thousands of\ncm2/V·s, reported in [11], the disorder-induced Landau\nlevel widths ∼20meV become comparable with the en-\nergy scalee2/εlHof Coulomb interaction effects. The\nmain mechanism of disorder effect on the transition en-\nergies is the following. Assume that Landau levels are\nbroadened giving rise to Gaussian mini-bands in the den-\nsity of states, as shown in Fig. 6. At partial filling of\neach level, its mini-band is partially filled, so the aver-\nage energy of the filled (empty) electron states is lower\n(higher) than the band center where the unperturbed\nLandau level energy would be located. As a result, the\naverage transition energy increases due to Landau level\nbroadening in addition to interaction effects when either\ninitial or final level is partially filled ( ν∝ne}ationslash=±2 in our\ncase). The similar effect was discussed in [44] for a two-\ndimensional gas of massive electrons in the framework of\nself-consistent Born approximation.\nTo describe this effect, we assume the Gaussian spec-\ntral density ρn(E) = (√\n2πΓn)−1exp[−(E−En)2/2Γ2\nn]\nfor eachnth partially filled broadened level. Integrat-\ning it up to the Fermi level µand assuming low tem-\nperature, we find the occupation number fn, and, us-\ning (14), we get the relationship between νandµ:\nν= 4n+2Φ([µ−En]/√\n2Γn), where Φ is the error func-\ntion. The disorder-induced correction ∝an}bracketle{t∆Ωn∝an}bracketri}htto the tran-\nsition energy is a difference between the average energies\n(relative to the band centers) of empty states on a fi-\nnal Landau level and filled states on an initial level. It\nshould be additionally weighted according to (20), when\n−2< ν <2 and thus both transitions T±\n1are present,\nresulting in:\n∝an}bracketle{t∆Ωn∝an}bracketri}ht=\n\n/radicalbigg\n2\nπΓ−1e−(µ−E−1)2\n2Γ2\n−1\n3+ν/2,if−6<ν <−2,\n/radicalbigg\n2\nπΓ0e−(µ−E0)2\n2Γ2\n0,if−2<ν <2,\n/radicalbigg\n2\nπΓ1e−(µ−E1)2\n2Γ2\n1\n3−ν/2,if 2<ν <6.(23)\nThis dependence has a maximum at ν= 0 and minima\natν=±2 in accordance with the experiment [11].\nAnother effect of the disorder is the presence of inter-\nlevel transitions when any nth Landau level is partially\nfilled, which provide an extra contribution to the screen-\ning. In the simplest approximation, they lead to the po-\nlarizability of the Thomas-Fermi kind\nΠTF\nn(q) =−gFnn(q)ρn(µ), (24)E E\n¡1¡10 01 1\nT¡\n1T¡\n1¡6<º<¡2 ¡6<º<¡2\nE E\n¡1¡10 01 1\nT¡\n1T¡\n1¡2<º<2 ¡2<º<2\nT+\n1T+\n1E E\n¡1¡10 01 12<º<62<º<6\nT+\n1T+\n1\nFIG. 6: Broadened Landau levels n= 0,±1 (not in scale)\nand cyclotron transitions between them when these levels ar e\npartially filled.\nwhich was used in [43] to study Landau level broadening\nin graphene.\nWe use the self-consistent Born approximation for a\npolarizability in magnetic field, which was originally de-\nvelopedin[45,46]foratwo-dimensionalelectrongaswith\nshort-rangeimpurities. Inourwork,weassumethedisor-\ndertobelong-ranged,becausethemainoriginofdisorder\nin graphene on a SiO 2substrate are long-range charged\nimpurities [47]. Introducing the mean square ∝an}bracketle{tU2∝an}bracketri}htof the\nslowly varying disorder potential U(r), we get the follow-\ning polarizability of disordered graphene (see the similar\nformulas in [45, 46] obtained by summing an impurity\nladder in a polarization loop):\nΠD(q,iω) =g/summationdisplay\nnn′Fnn′(q)\n×T/summationdisplay\nǫGD\nn′(iǫ+iω)GD\nn(iǫ)\n1−∝an}bracketle{tU2∝an}bracketri}htGD\nn′(iǫ+iω)GDn(iǫ),(25)\nwhereGD\nn(iǫ) =/integraltext\ndEρn(E)/(iω−E+µ) is the Green\nfunction of electron on the nth Landau level in the pres-\nence of disorder. Instead of a half-elliptic spectral den-\nsity [43], which is known to be an artefact of the self-\nconsistent Born approximation [48], we use, as above,\nthe Gaussian function ρn(E).\nTaking the static limit iω→0 and switching in (25)\nfrom the frequency summation to an integration along\nthe branch cut at Im( iǫ) = 0, we get in the limit T→0:\nΠD(q,0) =−g\nπ/summationdisplay\nnn′Fnn′(q)\n×0/integraldisplay\n−∞ImGD\nn′(z+iδ)GD\nn(z+iδ)\n1−∝an}bracketle{tU2∝an}bracketri}htGD\nn′(z+iδ)GDn(z+iδ).(26)\nThis polarizability consists of two physically distinct\nparts. The first one is the contribution of interlevel tran-\nsitions with n∝ne}ationslash=n′. It does not differ too much from\nthan in a clean system (12) if the widths of Landau lev-\nels Γnare much smaller than interlevel separations. The\nsecond one is the contribution of intralayer transitions\nn=n′arising when the nth layer is partially filled. Tak-\ning the disorder strength to be equal to the Landau level8\nwidth/radicalbig\n∝an}bracketle{tU2∝an}bracketri}ht= Γn, as follows from calculations of GD\nwith the long-range disorder, we get the static polariz-\nability of disordered graphene:\nΠD(q,0)≈Π(q,0)−gFnn(q)0/integraldisplay\n−∞Im[GD\nn(z+iδ)]2\n1−Γ2n[GDn(z+iδ)]2(27)\nand use it in the following calculations.\nFig. 7 shows the examples of static polarizabilities cal-\nculated at half-fillings of 0th and ±1th Landau levels.\nIn a clean graphene, Π( q,0)∝q2atq→0 since the\nsystem becomes insulating in magnetic field, and the\nonly source of the screening are gapped interlevel tran-\nsitions. Disorder makes ΠD(q,0) nonzero at q→0 due\nto intralevel transitions. The Thomas-Fermi approxima-\ntion, by taking into account only the latter, provides a\nwrong short-wavelength asymptotic of the polarizability\nΠTF\nn(q), which should tend to the polarizability of un-\ndoped graphene Π( q,0) =−gq/16/planckover2pi1vF[1].\nWe calculated the renormalized Fermi velocity, corre-\nsponding to the weighted energy (20) of the T1transi-\ntion with taking into account the correction (23) and the\nscreening (27) in the disordered system. For comparison,\nwecarriedout the samecalculationsforthe clean system,\nas did in the previous section. The results of the fitting\nof experimental points from Ref. [11] are shown in Fig. 8,\nand the calculation parameters are listed in Table II.\nFor the values of ε, we observe the same regularities\nas noted in the previous section. These values are close\nto those obtained in our earlier analysis [29] of cyclotron\nresonance data for graphene on SiO 2. However the most\ndrastic effects come from inclusion of disorder: while in\n0 1 2 3 4 500.511.500.511.52\n~¦(q;0) ~¦(q;0)\n(b)º=§4 (b)º=§4\nqlHqlH~¦(q;0) ~¦(q;0)(a)º=0 (a)º=0\nFIG. 7: Dimensionless static polarizability of graphene in\nmagnetic field ˜Π(q,0) =−(2πvFlH/g)Π(q,0), where lH=/radicalbig\n/planckover2pi1c/eH, calculated (a) when the 0th Landau level is half-\nfilled,ν= 0, (b) when the 1th or −1th level is half-filled,\nν=±4. Solid lines: clean graphene (12), dashed lines: disor-\ndered graphene (27), dotted lines: the Thomas-Fermi approx -\nimation (24). Calculation parameters are vF= 0.85×106m/s,\nB= 18T, Γ 0= Γ±1= 20meV.v¤\nFv¤\nF\n(106m/s) (106m/s)\nº º(a)(a)\n(b)(b)T1; B= 18T T1; B= 18T\nT1; B= 18T T1; B= 18T\n-6 -4 -2 0 2 4 61.051.101.151.051.101.151.20\nv¤\nFv¤\nF\n(106m/s) (106m/s)\nFIG. 8: Renormalized Fermi velocity v∗\nFfor theT1transition\n(0→1/−1→0) atB= 18T, taken from the experiment\n[11] (squares) and calculated theoretically for (a) clean a nd\n(b) disordered graphene. The calculations are carried out i n\nthe Hartree-Fock approximation (dashed lines), with takin g\ninto account the interaction screening (solid lines) and wi th\ntheself-consistent screening(dottedlines). Thedielect riccon-\nstantsεused in each calculation are listed in Table II.\nthecleansystem v∗\nFhasthe plateauat |ν|<2andremain\nthe same (or slightly increases due to suppression of the\nexcitonic correction) at |ν|>2, in the disordered system\nit hasthe parabolic-likemaximumat ν= 0andthe sharp\nminima at ν=±2, just as the experiment shows. The\nvalues of Landau level widths Γ nobtained via the fitting\nprocedure (15 −25meV) look realistic, since they are\nclose to typical widths of spectral lines observed in the\nsame experiment [11] and in other works on graphene\non a SiO 2substrate [49]. The minimal value of the root\nmean square deviation (22) is about 0 .009×106m/s in\nthis case.\nTABLE II: First two lines: dielectric constants of surround ing\nmedium ε, which provide the best least-square fittings of the\nexperimental data from Ref. [11] at vF= 0.85×106m/s in\nthe three approximations for the interaction listed in Sec. IV\nfor clean or disordered system. For disordered system, the\nwidths of 0th and ±1st Landau levels are also specified in the\nlast two lines.\nUnscreened Screened Self-consistent\nSystem interaction interaction screening\nClean 7.26 2.82 3.85\nDisordered 9.24 4.95 5.74\nΓ0(meV) 22 25 23\nΓ±1(meV) 12 19 179\nVI. CONCLUSIONS\nWe present detailed calculations of the inter-Landau\nlevel cyclotron transition energies in graphene in strong\nmagnetic fields taking into account Coulomb interaction\nbetweenmasslessDiracelectrons. Calculatingtheoptical\nconductivity and solvingthe vertex equation in the static\nrandom-phase approximation with the excitonic ladder,\nwe found the many-body correctionsto the transition en-\nergies coming from the self-energy and excitonic effects.\nWe show that the cyclotron transition lines can be split\nin doped graphene for opposite circular polarizations be-\ncause of the electron-hole asymmetry of exchange self-\nenergies, although this splitting may be unobservable if\nthese lines are sufficiently wide or either a linearly po-\nlarized or unpolarized light is used. By this reason, we\ncalculate the weighted transition energy for both polar-\nizations at once and convert it to the renormalized Fermi\nvelocityv∗\nFfor each transition.\nThe dependence of v∗\nFon the Landau level filling fac-\ntorνisanalyzed. In the mean-field approximation, v∗\nF(ν)\nhas a plateau at −2< ν <2 due to a partial cancela-\ntion of the self-energy and excitonic effects and rapidly\ndecreases at |ν|>2 due to enhancement of the screen-\ning. Our calculations, carried out with the bare Fermi\nvelocityvF= 0.85×106m/s and with the dielectric con-\nstant ofsurroundings ε, treated as an adjustable parame-\nter, showed good agreement with two recent experiments\n[12, 13] on high-mobility graphene samples, when the\nscreening by graphene electrons is taken into account.\nThe obtained phenomenological εdescribe the external\ndielectric screening not only by an underlayingsubstrate,\nbut also by adjacent hexagonal boron nitride layers. The\nHartree-Fock approximation, which neglects the density-\ndependent screening by graphene electrons, fails to ex-\nplain the observed rapid decrease of v∗\nFat|ν|>2.\nOur calculations for the intraband transitions n→n+\n1 and−n−1→ −npredict the V-like dependence v∗\nF(ν)\nwith the minima at, respectively, ν= 4n+ 2 andν=\n−4n−2 caused by the excitonic effects. Existence of\nthese minima can be verified experimentally, although\nan accurate detection of the interband transition lines\ncan be challenging (but possible [6]) due to their much\nlower energies: even for the the highest magnetic fields\n20-30 T these energies are below 100 meV.We also describe the data of the earlier cyclotron res-\nonance experiment [11] with graphene sample on SiO 2,\nwhere carriermobility is much lower. In this casewe take\nintoaccountlong-rangedisorder,whichbroadensLandau\nlevels and thus shifts the resonant energy upward when\ninitial or final level is partially filled, and induces the in-\nterleveltransitionscontributingtothescreening. Assum-\ning the Gaussian spectral density for the 0th and ±1th\nbroadened Landau levels, we achieved good agreement\nwith the experiment and explained the main features of\nthev∗\nF(ν) dependence: the parabolic-like maximum at\nν= 0 and the sharp minima at ν=±2.\nAs shown, the combined action of exchange interac-\ntion, excitonic effects, interaction screening and disorder\nshould be taken into account when considering graphene\nin strong magnetic field. Our approach takes into con-\nsideration these factors and thus allowed us to explain\nmain features of the filling-factor dependent experimen-\ntal data [11–13], which would be hardly possible within\nthe Hartree-Fock approximation [15, 18–25, 28] where\nthe screening and Landau level broadeningare neglected.\nHowever, some issues remain to be clarified. In partic-\nular, the mean-field approach does not describe the Λ-\nshaped maxima of v∗\nFatν= 0 observed in [12, 13] for\nT1transitions, the minima at ν=±4 observed for T 2\natB= 8T in [12], and a possible splitting of the T 1\ntransition line observed in [12]. All these features go be-\nyond the mean-field theory for massless Dirac electrons\nand can be attributed to some unaccounted role of disor-\nder, finite size effects, Moire superlattice potential from\nadjacent boron nitride layers [50], Landau level splitting\n[4, 51] or electron dynamics on a partially filled level [52].\nNotethatassumptionofasubstrate-inducedbandgapal-\nlowed to explain some features of the experimental data\nof[12] in the recent work[25], so a further analysisin this\ndirection with considering possible symmetry breakings\nandgapformationin asystemofDiracelectronstogether\nwith the interaction, screening and disorder seems to be\npromising.\nAcknowledgments\nThe work was supported by the grant No. 17-12-01393\nof the Russian Science Foundation.\nAppendix A: Electron wave functions\nSimilarlyto[4,38–40], wedescribesingle-particlestatesofmasslesse lectronsinmagneticfield Husingthesymmetric\ngaugeA=1\n2[H×r]. In the absence of a valley splitting or intervalley transitions, it is suffi cient to consider the\nelectrons only in the Kvalley, where the Dirac Hamiltonian is\nH=vF/parenleftBig\np−e\ncA/parenrightBig\n·σ=/planckover2pi1vF√\n2\nlH/parenleftBigg\n0a\na+0/parenrightBigg\n. (A1)\nHerelH=/radicalbig\n/planckover2pi1c/|e|His the magnetic length (we assume e <0 in this section), a=lHp−//planckover2pi1−ir−/2lHanda+=\nlHp+//planckover2pi1+ir+/2lHare, respectively, lowering and raising operators obeying the comm utation relation [ a,a+] = 1, and10\np±= (px±ipy)/√\n2,r±= (x±iy)/√\n2.\nIntroducing the complementary set of ladder operators [4] b=lHp+//planckover2pi1−ir+/2lH,b+=lHp−//planckover2pi1+ir−/2lH, which\nobey [b,b+] = 1 and commute with a,a+, we can construct the states of a two-dimensional oscillator |φnk∝an}bracketri}ht=\n(a+)n(b+)k|φ00∝an}bracketri}ht/√\nn!k! with the wave functions in polar coordinates:\nφnk(r,ϕ) =i|n−k|\n√\n2πlH/radicalBigg\nmin(n,k)!\nmax(n,k)!e−r2/4l2\nH/parenleftbiggr√\n2lH/parenrightbigg|n−k|\nei(n−k)ϕL|n−k|\nmin(n,k)/parenleftbiggr2\n2l2\nH/parenrightbigg\n, (A2)\nwhereLm\nn(x) are the associated Laguerre polynomials. The eigenfunctions of t he Hamiltonian (A1) are [4, 18–21, 37–\n39, 41]\nψnk= (√\n2)δn0−1/parenleftBigg\nsnφ|n|−1,k\nφ|n|,k/parenrightBigg\n, (A3)\nand eigenvalues are (1). Here n= 0,±1,±2,...is the Landau level number, k= 0,1,2,...is the guiding center index\nresponsible for Landau levels degeneracy, sn≡sign(n), and we assume that φnk= 0 ifnorkis negative.\nThe bare electron Green function in the Matsubara representatio nG(r,r′,τ) =−∝an}bracketle{tTτΨ(r,τ)Ψ+(r′,0)∝an}bracketri}htcan be\nconstructed from (A3):\nG0(r,r′,iǫ) =/summationdisplay\nnkψnk(r)ψ+\nnk(r′)\niǫ−En+µ, (A4)\nwhereµis the chemical potential; note G0is the (2 ×2) matrix in the sublattice space.\nUsing the table integral Eq. 2.20.16.10 from [53], we can present (A2) in Cartesian coordinates as\nφnk(x,y) =in−k\n√\n2π/integraldisplay+∞\n−∞dteityϕn(lHt+x/2lH)ϕk(lHt−x/2lH), (A5)\nwhereϕn(x) =e−x2/2Hn(x)//radicalbig\n2nn!√πare the dimensionless eigenfunctions of quantum one-dimensional h armonic\noscillator,Hn(x) are Hermite polynomials. Then, using (A5), and orthonormality and completeness of the basis\n{ϕn(x)}, a lot of useful transformation rules for φnkcan be obtained, for example, the summation formula\n∞/summationdisplay\nk=0φn1k(r1)φ∗\nn2k(r2) =ei(r1r2ez)/2l2\nH\n√\n2πlHφn1n2(r1−r2) (A6)\nand the form-factor of Landau level wave functions (see also [40])\n/integraldisplay\ndreiqrφ∗\nn1n2(r)φn3n4(r) = 2πl2\nHφ∗\nn1n3(aq)φn2n4(aq),aq≡ −l2\nH[ez×q]. (A7)\nAppendix B: Exchange self-energies\nExchange self-energy acquired by an electron in the state ψnkis given by the usual Fock expression\nΣexch\nnk=−/summationdisplay\nn′k′fn′k′/integraldisplay\ndr1dr2v(r1−r2)ψ+\nnk(r1)ψn′k′(r1)ψ+\nn′k′(r2)ψnk(r2). (B1)\nAfter the Fourier transform of the Coulomb interaction v(r) = (2π)−2/integraltext\ndqvqeiqrand using (A6)–(A7), we get\nΣexch\nnk=−/summationdisplay\nn′fn′∝an}bracketle{tnn′|v|n′n∝an}bracketri}ht (B2)\nwith the exchange matrix elements of Coulomb interaction defined as\n∝an}bracketle{tnn′|v|n′n∝an}bracketri}ht= 2δn0+δn′0−2l2\nH\n2π/integraldisplay\ndqvq/vextendsingle/vextendsinglesnsn′φ|n|−1,|n′|−1(aq)+φ|n||n′|(aq)/vextendsingle/vextendsingle2. (B3)11\nWe assumed that the occupation numbers do not depend on k′,fn′k′≡fn′, and the resulting Σexch\nnkturns out to\nbe also independent on k, so the Landau level degeneracy is preserved. By replacing vqwith the statically screened\ninteraction V(q), as depicted in Fig. 1(c), we get the screened exchange energy ( 7)–(8), and the bare electron Green\nfunction (A4) becomes “dressed” with the interaction and turns in to\nG(r,r′,iǫ) =/summationdisplay\nnkψnk(r)ψ+\nnk(r′)\niǫ−En−Σn+µ. (B4)\nAppendix C: Vertex equation\nIntroducing the Green function for currents Gj\nαβ(r,r′,τ) =−∝an}bracketle{tTτΨ+(r,τ)σαΨ(r,τ)Ψ+(r′,0)σβΨ(r′,0)∝an}bracketri}ht, we can\nwrite the conductivity (4) as\nσαβ(q,ω) =ie2v2\nF\n/planckover2pi1ωS/integraldisplay\ndrdr′e−iq(r−r′)Gj\nαβ(r,r′,/planckover2pi1ω+iδ), (C1)\nwhereGj\nαβcan be calculated, as shown in Fig. 1(a), from the (2 ×2) vertex matrix:\nGj\nαβ(r,r′,iω) =T/summationdisplay\nε/integraldisplay\ndr1dr2Tr[σαG(r,r1,iǫ+iω)Γβ(r1,r2,r′,iǫ,iω)G(r2,r,iǫ)]. (C2)\nHere the sum is taken over the fermionic Matsubara frequencies ǫ=πT(2n+1).\nThe vertex equation in the mean-field (or ladder) approximation, de picted in Fig. 1(b), is written analytically as\nΓβ(r1,r2,r′,iǫ,iω) =δ(r1−r′)δ(r2−r′)σβ−T/summationdisplay\nǫ′/integraldisplay\ndr′\n1dr′\n2V(r1−r2,iǫ−iǫ′)\n×G(r1,r′\n1,iǫ′+iω)Γβ(r′\n1,r′\n2,r′,iǫ′,iω)G(r′\n2,r2,iǫ′). (C3)\nTo solve it, we can use the basis of magnetoexcitonic states of Dirac electrons in the symmetric gauge, which were\ndescribed earlier in [38] in slightly different notation:\nΨPn1n2(r1,r2) =1\n2πeiP(r1+r2)/2+i(r1r2ez)/2l2\nHΦn1n2(r1−r2−aP). (C4)\nHerePis the conserved magnetic momentum of the electron-hole pair and\nΦn1n2(r) =√\n2δn10+δn20−2/parenleftBigg\nsn1sn2φ|n1|−1,|n2|−1(r)sn1φ|n1|−1,|n2|(r)\nsn2φ|n1|,|n2|−1(r)φ|n1|,|n2|(r)/parenrightBigg\n(C5)\nis the matrix wave function of relative motion of electron and hole writ ten in the basis of their sublattices A,B. Using\n(A5), the unitary transformations between the magnetoexciton ic states and the states (A3) of individual electron and\nhole can be derived:\nψn1k1(r1)ψ+\nn2k2(r2) =l2\nH/integraldisplay\ndPφ∗\nk1k2(aP)ΨPn1n2(r1,r2),ΨPn1n2(r1,r2) =l2\nH/summationdisplay\nk1k2φk1k2(aP)ψn1k1(r1)ψ+\nn2k2(r2).(C6)\nProjecting the vertex matrix Γ βon the magnetoexcitonic states\nΓβ,Pn1n2(r′,iǫ,iω) =/integraldisplay\ndr1dr2Tr/bracketleftbig\nΨ+\nPn1n2(r1,r2)Γβ(r1,r2,r′,iǫ,iω)/bracketrightbig\n(C7)\nand using (B4), (C6), we get (C3) in the electron-hole pair (or magn etoexcitonic) representation:\nΓβ,Pn1n2(r′,iǫ,iω) = Γ(0)\nβ,Pn1n2(r′)−T/summationdisplay\nǫ′n′\n1n′\n2/integraldisplay\ndP′∝an}bracketle{tΨPn1n2|V(iǫ−iǫ′)|ΨP′n′\n1n′\n2∝an}bracketri}htΓβ,P′n′\n1n′\n2(r′,iǫ′,iω)\n(iǫ′+iω−En′\n1−Σn′\n1+µ)(iǫ′−En′\n2−Σn′\n2+µ).(C8)\nHere the bare vertex is Γ(0)\nβ,Pn1n2(r′) = (e−iPr′/2π)Tr/bracketleftbig\nΦ+\nn1n2(aP)σβ/bracketrightbig\n.12\nTo solve Eq. (C8), we use the static approximation V(r,iǫ−iǫ′) =V(r) and neglect the mixing of different\nelectron-hole pairs in the ladder diagrams, assuming n′\n1=n1,n′\n2=n2. Therefore the vertex matrix turns out to be\nindependent on a relative energy of electron and hole ǫ:\nΓβ,Pn1n2(r′,iω) =e−iPr′\n2πTr/bracketleftbig\nΦ+\nn1n2(aP)σβ/bracketrightbig/braceleftbigg\n1+∝an}bracketle{tn1n2|VP|n1n2∝an}bracketri}htfn2−fn1\niω+En2+Σn2−En1−Σn1/bracerightbigg−1\n.(C9)\nThe average interaction energies of magnetoexcitons are ∝an}bracketle{tn1n2|VP|n1n2∝an}bracketri}ht=/integraltext\ndrV(r−aP)Tr/bracketleftbig\nΦ+\nn1n2(r)Φn1n2(r)/bracketrightbig\n,\ntheir counterparts in usual 2D electron gas were extensively stud ied earlier [54]. Making the Fourier transform\nV(r) = (2π)−2/integraltext\ndqV(q)eiqrand using (A7), we obtain\n∝an}bracketle{tn1n2|VP|n1n2∝an}bracketri}ht=l2\nH\n2π/integraldisplay\ndqV(q)e−iqaPTr/bracketleftbig\nΦ+\nn1n1(aq)/bracketrightbig\nTr[Φn2n2(aq)]. (C10)\nThe Green function for currents (C2) can by found using (B4), (C 4), (C6), (C7), and (C9):\nGj\nαβ(r,r′,iω) =/summationdisplay\nn1n2/integraldisplaydP\n(2π)2eiP(r−r′)Tr[Φn1n2(aP)σα]Tr/bracketleftbig\nΦ+\nn1n2(aP)σβ/bracketrightbig\n(fn2−fn1)\niω+En2+Σn2−En1−Σn1+(fn2−fn1)∝an}bracketle{tn1n2|VP|n1n2∝an}bracketri}ht.(C11)\nSubstituting it in (C1) and taking P= 0 for optical transitions in (C10), we finally obtain (5)–(10).\n[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, The electronic properties of\ngraphene, Rev. Mod. Phys. 81, 109 (2009).\n[2] K.S.Novoselov, A.K.Geim, S.V.Morozov, D.Jiang, M.\nI. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A.\nFirsov, Two-dimensional gas of massless Dirac fermions\nin graphene, Nature 438, 197 (2005).\n[3] V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and\nA. H. Castro Neto, Electron-Electron Interactions in\nGraphene: Current Status and Perspectives, Rev. Mod.\nPhys.84, 1067 (2012).\n[4] M. O. Goerbig, Electronic properties of graphene in a\nstrong magnetic field, Rev. Mod. Phys. 83, 1193 (2011).\n[5] L.-J. Yin, K.-K. Bai, W. Wang, S.-Y. Li, Y. Zhang, and\nL.He, LandauquantizationofDiracfermions ingraphene\nand its multilayers, Front. Phys. 12, 127208 (2017).\n[6] M.L. Sadowski, G. Martinez, M. Potemski, C. Bergert,\nand W.A. De Heer, Magneto-spectroscopy of epitaxial\ngraphene, Int. J. Mod. Phys. B 21, 1145 (2007).\n[7] M. Orlita and M. Potemski, Dirac electronic states in\ngraphene systems: optical spectroscopy studies, Semi-\ncond. Sci. Technol. 25, 063001 (2010).\n[8] D. N. Basov, M. M. Fogler, A. Lanzara, F. Wang, and Y.\nZhang, Colloquium: Graphene spectroscopy, Rev. Mod.\nPhys.86, 959 (2014).\n[9] Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M.\nE. Schwartz, M. Y. Han, P. Kim, and H. L. Stormer,\nInfraredspectroscopy ofLandaulevels ofgraphene, Phys.\nRev. Lett. 98, 197403 (2007).\n[10] Z. Jiang, E. A. Henriksen, P. Cadden-Zimansky, L.-C.\nTung, Y.-J. Wang, P. Kim, and H. L. Stormer, Cyclotron\nresonance near the charge neutrality point of graphene,\nAIP Conf. Proc. 1399, 773 (2011).\n[11] E. A. Henriksen, P. Cadden-Zimansky, Z. Jiang, Z. Q.\nLi, L.-C. Tung, M. E. Schwartz, M. Takita, Y.-J. Wang,P. Kim, and H. L. Stormer, Interaction-induced shift of\nthe cyclotron resonance of graphene using infrared spec-\ntroscopy, Phys. Rev. Lett. 104, 067404 (2010).\n[12] B. J. Russell, B. Zhou, T. Taniguchi, K. Watanabe, and\nE. A. Henriksen, Many-Particle Effects in the Cyclotron\nResonance of Encapsulated Monolayer Graphene, Phys.\nRev. Lett. 120, 047401 (2018).\n[13] J. Sonntag, S. Reichardt, L. Wirtz, B. Beschoten, M. I.\nKatsnelson, F. Libisch, C. Stampfer, Impact of Many-\nBody Effects on Landau Levels in Graphene, Phys. Rev.\nLett.120, 187701 (2018).\n[14] S. Berciaud, M. Potemski, andC. Faugeras, Probingelec -\ntronic excitations inmono- topentalayergraphene bymi-\ncro magneto-Raman spectroscopy, Nano Lett. 14, 4548\n(2014).\n[15] C. Faugeras, S. Berciaud, P. Leszczynski, Y. Henni, K.\nNogajewski, M. Orlita, T. Taniguchi, K. Watanabe, C.\nForsythe, P. Kim, R. Jalil, A. K. Geim, D. M. Basko,\nand M. Potemski, Landau level spectroscopy of electron-\nelectron interactions in graphene, Phys. Rev. Lett. 114,\n126804 (2015).\n[16] C. Faugeras, M. Orlita, and M. Potemski, Raman scat-\ntering of graphene-based systems in high magnetic fields,\nJ. Raman Spectrosc. 49, 146 (2018).\n[17] W. Kohn, Cycloron resonance and de Haas-van Alphen\noscillations of an interacting electron gas, Phys. Rev.\n123, 1242 (1961).\n[18] A. Iyengar, J. Wang, H. A. Fertig, and L. Brey, Excita-\ntions from filled Landau levels in graphene, Phys. Rev.\nB75, 125430 (2007).\n[19] Yu. A. Bychkov and G. Martinez, Magnetoplasmon ex-\ncitations in graphene for filling factors ν/lessorequalslant6, Phys. Rev.\nB77, 125417 (2008).\n[20] R. Rold´ an, J.-N. Fuchs, and M. O. Goerbig, Spin-flip\nexcitations, spinwaves, andmagnetoexcitons ingraphene13\nLandau levels at integer filling factors, Phys. Rev. B 82,\n205418 (2010).\n[21] Y. Barlas, W.-C. Lee, K. Nomura, and A. H. MacDonald,\nRenormalized Landau levels and particle-hole symmetry\nin graphene, Int. J. Mod. Phys. B 23, 2634 (2009).\n[22] K. Shizuya, Many-body corrections to cyclotron reso-\nnance in monolayer and bilayer graphene, Phys. Rev. B\n81, 075407 (2010).\n[23] K. Shizuya, Many-body corrections to cyclotron reso-\nnance in graphene, J. Phys.: Conf. Ser. 334, 012046\n(2011).\n[24] L. A. Chizhova, J. Burgdorfer, and F. Libisch, Magneto-\noptical response of graphene: Probing substrate interac-\ntions, Phys. Rev. B 92, 125411 (2015).\n[25] K. Shizuya, Many-body effects on Landau-level spectra\nand cyclotron resonance in graphene, Phys. Rev. B 98,\n115419 (2018).\n[26] E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A.\nShovkovy,Coulomb interaction and magnetic catalysis in\nthe quantum Hall effect in graphene, Phys. Scr. T146,\n014018 (2012).\n[27] N. Menezes, V. S. Alves, and C. M. Smith, The influence\nof a weak magnetic field in the Renormalization-Group\nfunctionsof(2+1)-dimensional Diracsystems, Eur.Phys.\nJ. B89, 271 (2016).\n[28] K. Shizuya, Direct-exchange duality of the Coulomb in-\nteraction and collective excitations in graphene in a mag-\nnetic field, Int. J. Mod. Phys. B 31, 1750176 (2017).\n[29] A. A. Sokolik, A. D. Zabolotskiy, and Yu. E. Lozovik,\nMany-bodyeffectsofCoulombinteractiononLandaulev-\nels in graphene, Phys. Rev. B 95, 125402 (2017).\n[30] G. L. Yu, R. Jalil, B. Belle, A. S. Mayorov, P. Blake, F.\nSchedin, S. V. Morozov, L. A. Ponomarenko, F. Chiap-\npini, S. Wiedmann, U. Zeitler, M. I. Katsnelson, A. K.\nGeim, K. S. Novoselov, and D. C. Elias, Interaction phe-\nnomena in graphene seen through quantum capacitance,\nProc. Natl. Acad. Sci. USA 110, 3282 (2013).\n[31] D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Mo-\nrozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V.\nGrigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim,\nNature Phys. 7, 701 (2011).\n[32] L. Yang, J. Deslippe, C.-H. Park, M. L. Cohen, and S.\nG. Louie, Excitonic Effects on the Optical Response of\nGraphene and Bilayer Graphene, Phys. Rev. Lett. 103,\n186802 (2009).\n[33] I. A. Shovkovy and L. Xia, Generalized Landau level rep-\nresentation: Effect of static screening in the quantum\nHall effect in graphene, Phys. Rev. B 93, 035454 (2016).\n[34] J. Gonz´ alez, F. Guinea, and M. A. H. Vozmediano, Non-\nFermi liquid behavior of electrons in the half-filled honey-\ncomb lattice (A renormalization group approach), Nucl.\nPhys. B 424, 595 (1994).\n[35] A. A. Sokolik and Yu. E. Lozovik, Many-body renor-\nmalization of Landau levels in graphene due to screened\nCoulomb interaction, Phys. Rev. B 97, 075416 (2018).\n[36] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, AC\nconductivity of graphene: from tight-binding model to\n2+1-dimensional quantum electrodynamics, Int. J. Mod.\nPhys. B 21, 4611 (2007).\n[37] R. Rold´ an, J.-N. Fuchs, and M. O. Goerbig, Collec-\ntive modes of doped graphene and a standard two-\ndimensional electron gas in a strong magnetic field:\nLinear magnetoplasmons versus magnetoexcitons, Phys.\nRev. B80, 085408 (2009).[38] Yu. E. Lozovik and A. A. Sokolik, Influence of Landau\nlevel mixing on the properties of elementary excitations\ningrapheneinstrongmagnetic field, Nanoscale Res.Lett.\n7, 134 (2012).\n[39] R. Rold´ an, M. O. Goerbig, and J.-N. Fuchs, The mag-\nnetic field particle-hole excitation spectrum in doped\ngraphene andin astandard two-dimensional electron gas,\nSemicond. Sci. Technol. 25, 034005 (2010).\n[40] P. K. Pyatkovskiy and V. P. Gusynin, Dynamical polar-\nization of graphene in a magnetic field, Phys. Rev. B 83,\n075422 (2011).\n[41] G. Gumbs, A. Balassis, D. Dahal, and M. L. Glasser,\nThermal smearing and screening in a strong magnetic\nfield for Dirac materials in comparison with the two di-\nmensional electron liquid, Eur. Phys. J. B 89, 234 (2016).\n[42] A. Kretinin, G. L. Yu, R. Jalil, Y. Cao, F. Withers,\nA. Mishchenko, M. I. Katsnelson, K. S. Novoselov, A.\nK. Geim, and F. Guinea, Quantum capacitance mea-\nsurements of electron-hole asymmetry and next-nearest-\nneighbor hopping in graphene, Phys. Rev. B 88, 165427\n(2013).\n[43] C. H. Yang, F. M. Peeters, and W. Xu, Landau-\nlevel broadening due to electron-impurity interaction in\ngraphene in strong magnetic fields, Phys. Rev. B 82,\n075401 (2010).\n[44] T. Ando, Theory of Cyclotron Resonance Lineshape in\na Two-Dimensional Electron System, J. Phys. Soc. Jpn.\n38, 989 (1975).\n[45] T. AndoandY.Uemura, ScreeningEffect inaDisordered\nElectron System I. General Consideration of Dielectric\nFunction, J. Phys. Soc. Jpn. 30, 632 (1971).\n[46] T. Ando and Y. Uemura, Theory of Oscillatory gfactor\nin a MOS Inversion Layer under Strong Magnetic Fields,\nJ. Phys. Soc. Jpn. 37, 1044 (1974).\n[47] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi,\nElectronic transport in two-dimensional graphene, Rev.\nMod. Phys. 83, 407 (2011).\n[48] A. H. MacDonald, H. C. A. Oji, and K. L. Liu, Ther-\nmodynamic properties of an interacting two-dimensional\nelectron gas in a strong magnetic field, Phys. Rev. B 34,\n2681 (1986).\n[49] A. Luican, G. Li, and E. Y. Andrei, Quantized Landau\nlevel spectrum and its density dependence in graphene,\nPhys. Rev. B 83, 041405(R) (2011).\n[50] Z.-G. Chen, Z. Shi, W. Yang, X. Lu, Y. Lai, H. Yan,\nF. Wang, G. Zhang, and Z. Li, Observation of an in-\ntrinsic bandgap and Landau level renormalization in\ngraphene/boron-nitride heterostructures, Nature Com-\nmun.5, 4461 (2014).\n[51] Y. J. Song, A. F. Otte, Y. Kuk, Y. Hu, D. B. Torrance, P.\nN. First, W. A. de Heer, H. Min, S. Adam, M. D. Stiles,\nA. H. MacDonald, and J. A. Stroscio, High-resolution\ntunnelling spectroscopy of a graphene quartet, Nature\n467, 185 (2010).\n[52] D. M. Basko, P. Leszczynski, C. Faugeras, J. Binder, A.\nA. L. Nicolet, P. Kossacki, M. Orlita, and M. Potem-\nski, Multiple magneto-phonon resonances in graphene,\n2D Mater. 3, 015004 (2016).\n[53] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,\nIntegrals and Series. Volume 2. Special Functions, Gor-\ndon and Breach Science Publishers, 1986.\n[54] L. V. Butov, C. W. Lai, D. S. Chemla, Yu. E. Lozovik, K.\nL. Campman, and A. C. Gossard, Observation of Mag-\nnetically Induced Effective-Mass Enhancement of Quasi-14\n2D Excitons, Phys. Rev. Lett. 87, 216804 (2001)."    },    {        "title": "2205.01349v1.Observation_of_the_noise_driven_thermalization_of_the_Fermi_Pasta_Ulam_Tsingou_recurrence_in_optical_fibers.pdf",        "content": "Observation of the noise-driven thermalization of the Fermi-Pasta-Ulam-Tsingou\nrecurrence in optical \fbers\nGuillaume Vanderhaegen,1Pascal Szriftgiser,1Alexandre Kudlinski,1\nMatteo Conforti,1Andrea Armaroli,1and Arnaud Mussot1\n1University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Mol\u0013 ecules, F-59000 Lille, France\n(Dated: May 4, 2022)\nWe report the observation of the thermalization of the Fermi-Pasta-Ulam-Tsingou recurrence\nprocess in optical \fbers. We show the transition from a reversible regime to an irreversible one,\nrevealing a spectrally thermalized state. To do so, we actively compensate the \fber loss to make the\nobservation of several recurrences possible. We inject into the \fber a combination of three coherent\ncontinuous waves, which we call Fourier modes, and a random noise. We enhance the noise-driven\nmodulation instability process against the coherent one by boosting the input noise power level to\nspeed up the evolution to the thermalization. The distributions of the Fourier modes power along\nthe \fber length are recorded thanks to a multi-heterodyne time-domain re\rectometer. At low input\nnoise levels, we observe up to four recurrences. Whereas, at higher noise levels, the Fourier modes\nfade into the noise-driven modulation instability spectrum revealing that the process reached an\nirreversible thermalized state.\nI. INTRODUCTION\nThe study of the Fermi Pasta Ulam Tsingou (FPUT)\nparadox in the 50s [1] marked the birth of numerical sim-\nulations and of nonlinear physics [2]. These numerical ex-\nperiments aimed at studing the energy transfers between\nthe eigenmodes of a chain of oscillators. A linear coupling\nmakes the excited mode preserving its energy, while a\nnonlinear coupling enables energy exchanges. Fermi and\nco-workers were expecting an equidistribution of the en-\nergy into all the modes. But, against all odds, the system\nexhibited a reversible behavior, and, after a certain time,\nreturned to its initial state.\nThis recursive energy transfer is rather general and\ncan be observed in many other \felds of physics, where a\nweak nonlinear coupling exists between the system eigen-\nmodes. It is now conventionally denoted as FPUT re-\ncurrence process. Such behaviors were especially inves-\ntigated in focusing cubic media through the nonlinear\nstage of modulation instability (MI), as demonstrated\nin hydrodynamics [3, 4], bulk crystals optics [5], planar\nwaveguides [6], \fber optics [7{14] and also magnetic feed-\nback rings [15]. A recurrent regime is obtained by trig-\ngering these systems with a weak coherent seed aside the\npump located inside the MI gain band. During the prop-\nagation, the pump transfers energy to this input signal\nand to the symmetric idler wave generated by four wave\nmixing, the Fourier modes of the systems. Higher order\nFourier modes, also called harmonics, are then generated\nby four-wave mixing process up to a speci\fc distance be-\nfore the energy transfers reversed back and the system\nreturns to the initial excitation state [7{14].\nA major question was to \fgure out if these systems\nbeyond a single recurrence period could reach a thermal-\nized state, as \frstly expected by Fermi, Pasta, Ulam and\nTsingou. The limited performances of their MANIAC\ncomputer [2] didn't allow to pursue the study until this\nirreversible \fnal state. Indeed, the thermalization is along-process of energy equipartition between the eigen-\nmodes of the mechanical oscillators requiring lengthy cal-\nculations. Fortunately, the computational power of mod-\nern computers enabled the achievement of these complex\ncalculations. Hence, they revealed that after several re-\ncurrences, pseudo-recurrences appear before the system\neventually reaches a thermalized state [16{19]. By anal-\nogy, in \fber optics, the irreversible evolution towards an\nenergy distribution among the whole spectrum (not only\nharmonics of the signal, but any spectral components in-\ncluding noisy ones) after the recurrences breakdown, was\ncalled thermalization too [20]. In that case, the noise-\ndriven MI process enters into competition with the seeded\none, which becomes eventually the dominant e\u000bect [20].\nAside the FPUT recurrence process, this competition\nwas deeply investigated in the context of supercontinuum\ngeneration and rogue waves because MI is the triggering\nmechanism of these complex nonlinear processes [21{28].\nA coherent seed was used to either increase supercontin-\nuum coherence or inhibit rogue wave formation [29, 30].\nWhile the nonlinear stage of MI through the prism of\nFPUT recurrences in either noise-driven systems [25, 26]\nor coherently seeded ones [7{13] had already been stud-\nied, their competition and the route to a thermalized\nstate had never been reported experimentally. Indeed, for\na while, the recurrence number was limited to a single cy-\ncle [7{10, 12] or two periods [11, 13] due to the \fber loss.\nIndeed dissipation, while weak in optical \fbers, dramat-\nically increases the period of recurrence. This prevents\nthe study of the thermalization process that is expected\nto appear after many periods [16{18]. Recently, it was\npossible to overcome this issue in \fber optics thanks to\nan active loss compensation scheme through Raman am-\npli\fcation [11, 14, 31] or the use of a recirculating loop\n[26].\nIn this work, we investigate the competition between\nnoise-driven and seeded MI processes in the framework\nof multiple FPUT recurrences to report the observation\nof the thermalization process. As in Refs. [11, 14, 31],arXiv:2205.01349v1  [nlin.PS]  3 May 20222\nwe compensate almost perfectly the loss through Raman\nampli\fcation to get several recurrences periods and we\naccelerate the route to thermalization by increasing the\nnoise power level at the \fber input. It enables to \fnely\ncontrol the balance between both MI processes to clearly\nobserve the transition from a perfectly recurrent regime\nto an irreversible spectrally thermalized state. The pa-\nper is organized as follows. First, we provide numerical\nsimulations to illustrate the noise and coherently driven\nMI processes and an example of competition between\nboth. Second, we introduce our experimental setup,\nmade of a multi-heterodyne optical time domain re\rec-\ntometer (HODTR) system combined with an active loss\ncompensation scheme. Finally, the experimental results\nare displayed, compared to the NLSE numerics.\nII. NUMERICAL SIMULATION OF\nSPONTANEOUS AND SEEDED MI\nNumerical simulations rely on the integration of the\nnonlinear Schr odinger equation (NLSE):\n@E\n@z=\u0000\f2\n2@2E\nT2+i\rjEj2E (1)\nwhereEis the electrical \feld, zis the distance along\nthe \fber and Tis the retarded time of the frame moving\nat the group velocity. \f2is the group velocity dispersion\nwith\f2=\u000021 ps2km\u00001and\ris the nonlinear coe\u000ecient\nwith\r= 1:3 W\u00001km\u00001. This equation is solved numer-\nically using the split-step Fourier method [32]. The evo-\nlution of the noise-driven MI process as a function of the\n\fber length is illustrated in Fig. 1 with a typical example.\nThe input spectrum is shown in Fig. 1. (a). A monochro-\nmatic pump is surrounded by a white noise \roor with a\npower spectral density PSD =\u0000112 dBm/Hz. In all the\nnumerical simulation, the real and the imaginary part of\nthe input noise spectrum at each frequency bin are sam-\nples of two independent Gaussian random variables with\nzero mean and unit variance. The noise \roor is then ad-\njusted at the desired level by multiplying the amplitude\nof the noise bins with a constant factor.\nThe evolution of the spectrum along the propagation\ndistance is plotted in Fig. 1. (b) for a single shot of noisy\ninitial condition. At the beginning (up to 5 km), only the\nspectral components within the MI gain band (the cuto\u000b\nfrequency is !C=p\n4\rPP=j\f2j[32]) are ampli\fed. Then\nthe spectrum broadens due to four-wave mixing (FWM)\nbetween the unstable frequencies. Looking at the input\ntime pro\fle (Fig. 1. (c)) and its evolution along the \fber\nlength (Fig. 1. (d)), we notice the irregular appearance\nof high power pulses. We can even distinguish solitonic\nstructures similar to the Akhmediev breather (AB), the\nKuznetsov-Ma soliton (KM) and the Peregrine soliton\n(PS) as previously reported in [24{26]. To highlight the\ncontinuous spectral broadening during the propagation,\n00.51Power (W)\nSSE-100-500Rel. power\n|g12|\n0 5 10 15 20\nDistance (km)\n-40-30-20-1001020(a)\n00.20.40.6PP (W)(c)\n1234567Relative power (dBm)\nPower (W)\n(e)\n02468(50 dB/div)\n-150 -100 -50 0 50 100\nTime (ps)150\n-300 -200 -100 0 100 200\nFrequency (GHz)3000\n5\n10\n15\n20Distance (km)0\n5\n10\n15\n20Distance (km)(b) (d)FIG. 1. Numerical simulations of the dynamic of spontaneous\nMI. (a) Spectrum at the \fber input, and (b) power spectral\ndistribution over the whole \fber length. (c) Power time pro-\n\fles at the \fber input, and (d) temporal power distribution\nalong the \fber length. (e) Shannon spectral entropy SSE and\npower of the initial pump excitation as function of the \fber\ndistance. Parameters: group velocity dispersion \f2=\u000021\nps2km\u00001, nonlinear coe\u000ecient \r= 1:3 W\u00001km\u00001, input\npump power PP= 500 mW, noise PSD =\u0000112 dBm/Hz.\nwe calculate the Shannon spectral entropy (SSE) [33]:\nSSE =\u0000X\n!p!loge(p!) (2)\nwherep!is the fraction of total power contained in the\nfrequency bin !. The evolution of the SSE (solid blue\nline) together with the pump power along the \fber length\n(solid orange line) are plotted in Fig. 1. (e). The SSE\nis initially close to 1 and increases from about 5 km too,\nwhen spectral components are generated outside the MI\nband, to reach a maximum value around 7. Then the\nentropy slightly oscillates around this maximum during\nthe rest of the propagation. After dropping to almost\nzero, the pump power oscillates too, which is a signature\nof spontaneous FPUT recurrences [26]. We notice the\nSSE and the pump power are antiphased. This shows\nthe system entropy is maximum when the pump power\nis minimum and vice versa.\nA numerical simulation of coherently driven MI is pre-\nsented in Fig. 2. We used a set of three waves, a pump\nand two symmetric sidebands located close to the maxi-\nmum of the MI gain band [32] (Fig. 2. (a)). Fig. 2. (b)\nshows the evolution of the spectrum along the propaga-\ntion distance. On a \frst stage, the seeds are ampli\fed by\nthe pump and the spectrum is broadened with equidis-\ntant peaks into a triangular shaped frequency comb, at3\n00.51Power (W)\n0 5 10 15 20\nDistance (km)00.20.40.6PP + 2PS (W)\n0.511.522.53\n-40-30-20-1001020-100-500\nRelative power (dBm)\nPower (W)\nSSE\n02468Rel. power\n(50 dB/div)\n-100 -50 0 50 100 150\nTime (ps)0\n5\n10\n15\n20Distance (km)\n-200 -100 0 100 200 300\nFrequency (GHz)0\n5\n10\n15\n20Distance (km)(a)\n(b)(c)\n(d)\n(e)\nFIG. 2. Same as Fig. 1 but with seededed MI. The modulation\nwave is set at 12 dB below the pump wave. On (e) is plotted\nthe total power of initial 3 waves excitation. PSD\u0019 \u0000400\ndBm/Hz (numerical noise level).\nthe expense of the pump wave. New waves located out-\nside the gain band, called harmonics, are generated out-\nside the gain band due to multiple four-wave mixing\n(FWM) processes. Then the system reaches the max-\nimum pump saturation stage around 2 :5 km and the\nenergy transfer to the higher order modes is not pos-\nsible anymore. Therefore, the energy transfer direction\nreverses and \rows back to the pump wave before the\nsystem returns almost exactly to its initial state. This\ncorresponds to a \frst FPUT recurrence cycle. Then, this\ndynamics repeats itself periodically during the propaga-\ntion till the end of the \fber. In this case, we can distin-\nguish 4 recurrences. The temporal evolution along the\n\fber length in the time domain is depicted in Fig. 2.\n(d). The system is triggered with a weakly modulated\nCW (Fig. 2. (c)). We notice the emergence of maximum\ncompression points coinciding with the spectral broaden-\ning maxima, whose positions are accurately predicted by\nthe theory [34, 35]. The \u0019-shift between two consecutive\nrecurrences is due to the initial phase relation between\nthe initial three waves [11, 36]. As with spontaneous\nMI, we plot the SSE of the system in Fig. 2. (e) (solid\nblue line). The entropy is weak ( <2), oscillates periodi-\ncally, at the the FPUT period but doesn't expand. The\npower contained in the initial waves (orange solid line),\nthe pump and the modulation sidebands, is also periodic.\nThe minima of this curve corresponds to maximal spec-\ntral broadenings because a signi\fcant part of the total\npower is contained in the Fourier modes of order n>2.III. NUMERICAL SIMULATION OF\nNOISE-INDUCED THERMALIZATION\nWe now look at the competition between spontaneous\nand seeded MI, when the CW is initially modulated by\na coherent sinusoidal modulation with noise (Figs. 3. (a)\nand (c)). The spectral and time evolutions are presented\n-300 -200 -100 0 100 200\nFrequency (GHz)-100 -50 0 50 100 150\nTime (ps)0\n5\n10\n20Distance (km)\n15-100-500\n-50-40-30-20-1001020\n0 5 10 15 20\nDistance (km)00.20.40.6PP + 2PS (W)\n1234567 0\n5\n10\n20Distance (km)\n1500.51Power (W)(a)\n(b)(c)\n(d)Relative power (dBm)\nPower (W)\n0246SSE(e)8Rel. power\n(50 dB/div)\nFIG. 3. Same as in Fig. 1 and Fig. 2. Same 3 waves input pa-\nrameters as the seeded MI numerics but with the spontaneous\nMI numerics noise \roor, PSD =\u0000112 dBm/Hz.\nin Figs. 3. (b) and (d) respectively. At the beginning of\nthe propagation in the \fber, the dynamics is very similar\nto the one illustrated Fig. 2 with two clear recurrences.\nThis means the dynamics of the system is dominated by\nthe coherently driven MI process. However, by further\npropagating within the \fber, the noise level of spectral\ncomponent located within the MI gain band increases and\nthe noise driven MI becomes signi\fcant at the expense of\nthe Fourier modes. Higher Fourier orders progressively\ndisappear into the noise \roor. The dynamics is then sim-\nilar to the one illustrated in Fig. 1, with the appearance\nof irregular temporal pulses and the disappearance of the\nfourth recurrence. The evolutions of the SSE and the to-\ntal power contained in the central 3 waves (Fig. 3. (e))\nalso reveal this transition toward an irreversible state.\nThe SSE keeps increasing to 7, and the total power tends\nto zero, as for the spontaneous MI example (Fig. 1). A\nfully thermalized state showing a complete disappearance\nof the coherent waves could be asymptotically reached by\npropagating in a much longer \fber (not shown here for\nthe sake of clarity). The transition speed from sponta-\nneous and noise driven MI can be controlled by tuning\nthe input noise level. We will exploit experimentally this\nfeature for the observation of the of the system evolving\ntoward an irreversible thermalized state.4\nIV. EXPERIMENTAL SETUP\nThe setup is similar to the one used in [11, 31] and\na simpli\fed version is presented in Fig. 4. A 1550 nm\nFIG. 4. Simpli\fed sketch of the setup with \"spectrum boxes\".\nf1is the frequency of the main laser and f2of the local oscil-\nlator, phase-locked at 800 MHz from f1.fmis the modulation\nfrequency, set here at 38 :2 GHz. SM \fber: single mode \fber.\nWaveshaper: amplitude and phase programmable optical \fl-\nter.\nCW signal is generated from a single frequency laser\ndiode. To excite the modulation instability regime co-\nherently, the signal is \frst phase-modulated at the fre-\nquencyfm= 38:2 GHz, very close of the maximum gain\nfrequency ( fmax= 38:4 GHz). The signal is then time\nshaped with an intensity modulator into 50 ns square\npulses, which are short enough to avoid stimulated Bril-\nlouin scattering and allow optical time domain re\rectom-\netry (OTDR). These pulses are long enough to be con-\nsidered as a CW wave for the modulation instability time\nscale. To shape the signal into the desired 3 waves input,\nthe relative power between the Fourier modes are tuned\nwith a Waveshaper. The signal is then combined with\nsynchronized additional white noise pulses, from a power\ntunable white noise source. It is followed by an ampli-\n\fcation stage through an erbium doped \fber ampli\fer\n(EDFA) to tune the nonlinear length ( PP= 470 mW).\nAn acousto-optic modulator, which has a extinction ra-\ntio much higher than electro-optic modulators, is then\ncascaded to reject the inter-pulses noise and to control\nthe pulses sequence. The signal is injected into a 16 :8\nkm long single mode \fber (G-652) in which nonlinear\ninteractions will occur.\nEven if the dissipation value is low ( \u000b= 0:2 dB/km),\non tens of kilometers, the nonlinear dynamic is highly\nimpacted [31]. In order to keep the signal power almost\nconstant over the whole \fber length, allowing us to pre-\nserve the integrable NLSE dynamics, an active loss com-\npensation scheme was implemented. At the \fber output,\na backward 1450 nm laser is injected to amplify the sig-\nnal by Raman e\u000bect. The Raman pump is located 13 :2THz away from the ampli\fed signal to bene\ft from the\nmaximum Raman gain [32]. For these tens of ns duration\npulses, we can neglect the saturation of the ampli\fer and\nconsider \rat ampli\fed pulses [37].\nThe power distribution along the \fber of the main\nFourier modes are obtained by an OTDR combined with\na multi-heterodyne technique [11, 31]. At each location\nin the \fber, a tiny part of the signal is Rayleigh backscat-\ntered and combined with a frequency comb with repeti-\ntion frequency fm, shifted by 600 MHz from the main\nlaser (the AO modulator induces a 200 MHz shift on f1).\nThe independant recording of the beating between the\nmain Fourier modes allows to recover the relative evolu-\ntion of the power along the all \fber length.\nThe noise power is increased progressively to study the\nin\ruence of the input noise level on the thermalization of\nthe FPUT recurrences. For each initial noise level value,\nwe record the power distribution of the two main Fourier\nwaves, the pump and the signal (the initial excitation\nbeing symmetric, we consider that the idler wave evolves\nidentically to the signal) and the spectrum at the \fber\nend. Much details about the experimental setup can be\nfound in Ref. [31].\nV. EXPERIMENTAL RESULTS\nWe varied the initial noise power spectral density\n(PSD) from\u0000121:3 to\u000091:9 dBm/Hz. We recorded the\npump and signal waves power evolutions along the \fber\nlength as well as the output spectra. These results are\npresented on the left panel of Fig. 5. (a-g). Fig. 5. (a)\ndisplays the output spectra as a function of the initial\nnoise PSD. At low amplitude noise \roors, we clearly dis-\ncern the Fourier modes: the pump, the seeds (at \u000638:2\nGHz) and the \frst harmonics (at \u000676:4 GHz). Between\nthese sidebands, MI also ampli\fes noisy components. We\nnote spectral holes around signal and idler waves. This\nis due to multiple four-wave mixing processes involving\nnoise components and these waves [38] (see Fig. 5. (e)).\nBy increasing the input noise PSD, the higher order har-\nmonics intensity decreases at the expense of the inter-\nbands lobes. The \frst harmonics (at \u000676:4 GHz) are\neven not visible anymore from the ampli\fed noise \roor\nfrom about\u0000105 dBm/Hz. The seed (at \u000638:2 GHz) are\nalso lost in the noise around \u000095 dBm/Hz. At the same\ntime, the amplitude of the noise \roor at the \fber out-\nput increases with the initial noise level. Indeed, with a\nhigher input noise level, the pump saturates at a shorter\n\fber length. This happens at the expense of the coherent\ndynamic process of seeded MI. The pump is not power-\nful enough to pursue the power transfer with the other\nFourier modes anymore, the spontaneous MI process is\ntaking the lead.\nIn Figs. 5. (b-d) are represented the pump (solid blue\nline), the seed (dotted blue line) and the relative total\npower of the three central waves distributions along the\n\fber length for noise PSD of \u0000121:3,\u0000101:8 and\u000092:45\n-100 -50 0 50 100\nFrequency (GHz)\nRelative power (5 dB/div)\n-100 -50 0 50 100\nFrequency (GHz)-120\n-115\n-110\n-105\n-100\n-95-120\n-115\n-110\n-105\n-100\n-95Noise power spectral density (dBm/Hz)(a)Noise power spectral density (dBm/Hz)Experiments Numerics\nRelative power (10 dB/div)00.20.40.6Power (W)\n00.20.40.6Power (W)\n0 5 10 15\nDistance (km)00.20.40.6Power (W)\n010101\n00.20.40.6Power (W)\n00.20.40.6Power (W)\n0 5 10 15\nDistance (km)00.20.40.6Power (W)\n010101\nPP+ 2PS (rel.) PP + 2PS (rel.) PP + 2PS (rel.)PP+ 2PS (rel.) PP + 2PS (rel.) PP + 2PS (rel.)(b)\n(c)\n(d)\n-100 -50 0 50\nFrequency (GHz)Rel. power (10 dB/div) -100 -50 0 50\nFrequency (GHz)-100 -50 0 50 100\nFrequency (GHz)-100 -50 0 50\nFrequency (GHz)Rel. power (10 dB/div)(e) (f) (g)\n-100 -50 0 50\nFrequency (GHz)-100 -50 0 50\nFrequency (GHz)100(a') (b')\n(c')\n(d')\n(e') (f') (g')\nFIG. 5. (a), (a') Spectrum at the \fber end as a function of the initial power spectral density of noise. (b), (c), (d) and (b'),\n(c'), (d') Evolution of the pump (solid blue line), the signal (dotted blue line) and the sum of the central 3 waves PP+ 2PS\n(solid orange line) along the \fber distance. (e), (f), (g) and (e'), (f'), (g') Output spectra. The initial power spectral densities\nare respectively ( \u0000121:3;\u0000101:8;\u000092:4) dBm/Hz and ( \u0000121:3;\u0000106:4;\u000097:3) dBm/Hz. The left panel is corresponding to the\nexperimental recordings while the right panel to the numerics.\ndBm/Hz respectively. For the \frst case (PSD = \u0000121:3\ndBm/Hz, see Fig. 5 (b)), no additional noise has been\nadded and the noise level is the intrinsic noise of the\nsetup (lowest value). We observe 3 :5 recurrences with a\nvery small decrease of the 3 waves total power, as it has\nbeen observed in [14]. This process is then still highly\ncoherent and the seeded MI is dominating the nonlin-\near power transfers. When the initial noise \roor is in-\ncreased at\u0000101:8 dBm/Hz (Fig. 5 (c)), the power of the\nFourier modes decreases during the propagation. It in-\ndicates that a part of the power is located elsewhere in\nthe spectrum, mainly in between coherent components.\nFor instance at z= 7:4 km, (after 1.5 recurrences), only\nhalf of the power is located in the Fourier modes and\nit reaches 17% at the \fber end. We see in the output\nspectrum depicted in Fig. 5 (f) that only the three main\nFourier modes remain visible. The seeded MI process is\nnot longer e\u000bective and the spontaneous MI is not neg-\nligible anymore. In the last case, with a noise PSD of\n\u000092:4 dBm/Hz (Fig. 5 (d)), we can only distinguish one\nand half recurrences and the power of the central three\nwaves becomes lower than 50% before the system reaches\nthe \frst half recurrence. After one and half recurrence\n(about 5 km), the power of both the pump and the seed\nwaves drops drastically during the propagation to reach\nweak values, comparable to the noise \roor. At the \fber\noutput, the spectrum in Fig. 5. (g) has a triangular shape\nfrom which it is impossible to distinguish any coherent\nFourier modes. This shape is typical from parametrically\ndriven systems [39] and is the signature of the thermal-ization of the FPUT process in optical \fbers. With this\nhigh noise input level, all coherent states are hidden in\nthe noise components including the pump. The conse-\nquence is that a comeback to a coherent initial state is\nimpossible, the evolution being thus irreversible. This\nunambiguously proves the system reached a thermalized\nstate of the FPUT recurrence process in optical \fbers.\nThe results from numerical simulations are plotted in\nthe right panel of Fig. 5. The agreement between experi-\nments and numerics is very good. The experiments follow\nthe same dynamics than the one predicted through nu-\nmerics, the progressive coherence loss with the increase\nof the input noise. Remarkably, the disappearance of co-\nherent structures and the triangular shape of the output\nspectra for high input noise levels is very similar to ex-\nperiments (Fig. 5 (g) and (g')). This con\frms the system\nreached an irreversible thermalized state.\nVI. CONCLUSION\nWe reported the experimental observation of the ther-\nmalization of the FPUT process in optical \fbers. We\nshowed that the FPUT recurrences irreversibly disap-\npear due to a competition between noise driven and co-\nherently driven MI processes leading to an increase of\nthe system entropy. Our experiments were realized by\nmeans of a multi-HOTDR system to monitor the evolu-\ntion of the power along the \fber length. The observation\nof the route to the thermalisation is made possible with6\nan active loss compensation scheme to observe up to 4\nrecurrences and by adding noise at the \fber input to ac-\ncelerate the process. At high input noise level, we showed\nthat the system reaches an irreversible thermalized state,\nwith an output spectrum only made of noisy components.\nExperimental results were con\frmed by numerical simu-\nlations with an excellent agreement. These experimental\nworks revealed the evolution of a nonlinear system from a\ncoherent dynamics between its nonlinear modes toward\nan irreversible thermalized state. This behavior, origi-\nnally expected by Fermi, Pasta, Ulam and Tsingou in a\nnonlinearly coupled masses system, had never been re-\nported experimentally so far. We anticipate these resultsto be generalizable to all focusing cubic media and to\ncontribute to a better understanding of the dynamics of\nnoisy and coherently driven nonlinear systems and the\nformation of rogue waves.\nACKNOWLEDGMENTS\nThe present research was supported by the agence\nNationale de la Recherche (Programme Investissements\nd'Avenir, I-SITE VERIFICO); Ministry of Higher Edu-\ncation and Research; Hauts de France Council; European\nRegional Development Fund (Photonics for Society P4S)\nand the CNRS (IRP LAFONI).\n[1] E. Fermi, P. Pasta, S. Ulam, and M. Tsingou,\nCollected Papers of Enrico Fermi, edited by E. ed. Segr\u0013 e,\nVol. 2 (Univ. Chicago Press, Chicago, Illinois, 1955) pp.\n977{988.\n[2] M. Porter, N. Zabusky, B. Hu, and D. Campbell, Amer-\nican Scientist 97(2009), 10.1511/2009.78.214.\n[3] B. M. Lake, H. C. Yuen, H. Rungaldier, and W. E.\nFerguson, Journal of Fluid Mechanics 83, 49{74 (1977).\n[4] O. Kimmoun, H. C. Hsu, H. Branger, M. S. Li, Y. Y.\nChen, C. Kharif, M. Onorato, E. J. R. Kelleher, B. Ki-\nbler, N. Akhmediev, and et al., Scienti\fc Reports 6,\n28516 (2016).\n[5] D. Pierangeli, M. Flammini, L. Zhang, G. Marcucci, A. J.\nAgranat, P. G. Grinevich, P. M. Santini, C. Conti, and\nE. DelRe, Physical Review X 8, 041017 (2018).\n[6] C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley,\nand M. Chauvet, Journal of the Optical Society of Amer-\nica B19, 574 (2002).\n[7] G. Van Simaeys, P. Emplit, and M. Haelterman, Physi-\ncal Review Letters 87, 033902 (2001).\n[8] G. Van Simaeys, P. Emplit, and M. Haelterman, Journal\nof the Optical Society of America B 19, 477 (2002).\n[9] K. Hammani, B. Wetzel, B. Kibler, J. Fatome, C. Finot,\nG. Millot, N. Akhmediev, and J. M. Dudley, Optics Let-\nters36, 2140 (2011).\n[10] A. Bendahmane, A. Mussot, A. Kudlinski, P. Szriftgiser,\nM. Conforti, S. Wabnitz, and S. Trillo, Optics Express\n23, 30861 (2015).\n[11] A. Mussot, C. Naveau, M. Conforti, A. Kudlinski,\nF. Copie, P. Szriftgiser, and S. Trillo, Nature Photonics\n12, 303{308 (2018).\n[12] X. Hu, W. Chen, Y. Lu, Z. Yu, M. Chen, and Z. Meng,\nIEEE Photonics Technology Letters 30, 47{50 (2018).\n[13] J.-W. Goossens, H. Hafermann, and Y. Jaou en, Scienti\fc\nReports 9, 18467 (2019).\n[14] G. Vanderhaegen, P. Szriftgiser, A. Kudlinski, M. Con-\nforti, S. Trillo, M. Droques, and A. Mussot, Optics Ex-\npress28, 17773 (2020).\n[15] M. Wu and C. E. Patton, Physical Review Letters 98,\n047202 (2007).\n[16] Y. V. Lvov and M. Onorato, Physical Review Letters\n120, 144301 (2018).\n[17] G. Benettin, H. Christodoulidi, and A. Ponno, Journal\nof Statistical Physics 152, 195{212 (2013).[18] M. Onorato, L. Vozella, D. Proment, and Y. V. Lvov,\nProceedings of the National Academy of Sciences 112,\n4208{4213 (2015).\n[19] S. D. Pace and D. K. Campbell, Chaos: An Interdisci-\nplinary Journal of Nonlinear Science 29, 023132 (2019).\n[20] S. Wabnitz and B. Wetzel, Physics Letters A 378,\n2750{2756 (2014).\n[21] N. Akhmediev, J. Soto-Crespo, and A. Ankiewicz,\nPhysics Letters A 373, 2137{2145 (2009).\n[22] J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty,\nNature Photonics 8, 755{764 (2014).\n[23] S. Toenger, T. Godin, C. Billet, F. Dias, M. Erkintalo,\nG. Genty, and J. M. Dudley, Scienti\fc Reports 5, 10380\n(2015).\n[24] P. Suret, R. E. Koussai\f, A. Tikan, C. Evain, S. Ran-\ndoux, C. Szwaj, and S. Bielawski, Nature Communica-\ntions7, 13136 (2016).\n[25] M. N arhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre,\nJ.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and\nJ. M. Dudley, Nature Communications 7, 13675 (2016).\n[26] A. E. Kraych, D. Agafontsev, S. Randoux, and P. Suret,\nPhysical Review Letters 123, 093902 (2019).\n[27] A. Gelash, D. Agafontsev, V. Zakharov, G. El, S. Ran-\ndoux, and P. Suret, Physical Review Letters 123, 234102\n(2019).\n[28] J. M. Dudley, G. Genty, F. Dias, B. Kibler, and\nN. Akhmediev, Optics Express 17, 21497 (2009).\n[29] D. R. Solli, C. Ropers, and B. Jalali, Physical Review\nLetters 101, 233902 (2008).\n[30] J. M. Dudley, G. Genty, and B. J. Eggleton, Optics\nExpress 16, 3644 (2008).\n[31] C. Naveau, G. Vanderhaegen, P. Szriftgiser, G. Mar-\ntinelli, M. Droques, A. Kudlinski, M. Conforti, S. Trillo,\nN. Akhmediev, and A. Mussot, Frontiers in Physics 9,\n25 (2021).\n[32] G. Agrawal, Nonlinear Fiber Optics, 4th edition (Aca-\ndemic Press, 2007).\n[33] C. E. Shannon, The Bell System Technical Journal 27,\n379{423 (1948).\n[34] P. Grinevich and P. Santini, Physics Letters A 382,\n973{979 (2018).\n[35] C. Naveau, P. Szriftgiser, A. Kudlinski, M. Conforti,\nS. Trillo, and A. Mussot, Optics Letters 44, 5426 (2019).7\n[36] G. Vanderhaegen, P. Szriftgiser, C. Naveau, A. Kudlinski,\nM. Conforti, S. Trillo, N. Akhmediev, and A. Mussot,\nOptics Letters 45, 3757 (2020).[37] G. Vanderhaegen, P. Szriftgiser, M. Conforti, A. Kudlin-\nski, M. Droques, and A. Mussot, Optics Letters 46, 5019\n(2021).\n[38] K. Inoue and T. Mukai, Optics Letters 26, 869 (2001).\n[39] N. Akhmediev, A. Ankiewicz, J. Soto-Crespo, and J. M.\nDudley, Physics Letters A 375, 775{779 (2011)."    },    {        "title": "1012.5395v2.Decoherence_of_the_two_level_system_coupled_to_a_fermionic_reservoir_and_the_information_transfer_from_it_to_the_environment.pdf",        "content": "arXiv:1012.5395v2  [cond-mat.mes-hall]  1 Feb 2011Decoherence of the two-level system coupled to a fermionic r eservoir and the\ninformation transfer from it to the environment\nPei Wang∗\nWangcunlvtukuang, 255311 Zibo, P. R. China\n(Dated: November 8, 2018)\nWe study the evolution of reduced density matrix of an impuri ty coupled to a Fermi sea after the\ncoupling is switched on at time t= 0. We find the non-diagonal elements of the reduced density\nmatrix decay exponentially, and the decay constant is the im purity level width Γ. And we study the\ninformation transfer rate between the impurity and the Ferm i sea, which also decays exponentially.\nAnd the decay constant is kΓ withk= 2∼4. Our results reveal the relation between information\ntransfer rate and decoherence rate.\nPACS numbers: 03.65.Yz, 03.67.Mn, 73.23.-b\nI. INTRODUCTION\nCoherence is the fundamental characteristics of quan-\ntum systems. The state of an isolated system is repre-\nsented by a vector moving in Hilbert space. The state\nvector decides the mean value and probability distribu-\ntion of observables. But in reality truly isolated sys-\ntems do not exist. The systems we can observe are all\nopen systems, which will exchange energy, matter and\ninformation with the rest of the world. The state of an\nopen system is generally a mixed state and should be\nrepresented by a density matrix. Open systems show\nmany different features from isolated ones. One of the\nmost important is decoherence [1–3], i.e., the nondiago-\nnal elements of the density matrix in pointer state basis\nrapidly decay to zero. The decoherence can be modelled\nas a non-unitary process, which can be described by the\nmaster equation with a Lindblad decohering term [4, 5].\nThe decoherence marks the boundary between quantum\nworldandclassicalworld[6], andhasbeenintenselystud-\nied because of its role in quantum measurement theory\nand quantum computation [6, 7]. The two level sys-\ntems coupled to spin bath [12] and oscillator bath [3]\nare good examples for studying decoherence. Plenty of\nworks have been contributed to this subject (for a review\nsee Ref. [2, 6, 12]). Recently the two level system coupled\nto fermionic bath [13, 14] emerged before the explorers.\nIn this paper we consider a simple fermionic bath: the\nresonant level model.\nAdditionally there exists information transfer between\nan open system and its environment. The quantum mea-\nsurement is realized due to the information transfer from\nthe system to apparatus. Zurek noticed the relation\nbetween decoherence and information transfer from the\nsystem to the observer and environment [6]. This rela-\ntion is familiar to quantum mechanics community, as the\nwell known state vector collapse in measurement. And\nthe which-path experiments, which are designed to verify\ncomplementarityprinciple, giveastraightforwardpicture\n∗Electronic address: pei.wang@live.combetween decoherence and information about particle po-\nsitions. Generally speaking, any one unavoidably cause\ndecoherence of a system when trying to get the informa-\ntion of it.\nThe relation between information transfer and deco-\nherence can be understood in the language of state vec-\ntor. The system and environment compose an isolated\nsystem which is supposed to be in a pure state. If the\ncomposite system is all the time in a product state, the\nstate of the system we concern is always pure and its\nevolution can be described by a unitary transformation.\nAnd the system and environment share zero informa-\ntion. Otherwise the composite system evolves into an\nentangled state. The evolution of system we concern is\nnon-unitary,indicatingdecoherenceprocess. Atthesame\ntime the system and environment exchange information,\nincreasing their mutual information into a finite value.\nAn open system is all the time transfering informa-\ntion outside and suffering decoherence. The dynamics of\nentropy entanglement (the half mutual information) has\nbeen studied in different one dimensional systems [8–11].\nWhile thereis still little study about the relationbetween\ndecoherence rate and the information transfer rate in a\ndynamicalsystem. In this paper, wetryto usean exactly\nsolvable model to demonstrate this relation.\nII. TWO LEVEL SYSTEM COUPLED TO\nFERMIONIC RESERVOIR\nThe resonant level model is an exactly solvable model\nwhich has been widely used to describe the transport\nthroughquantum dot. The Hamiltonian ofresonantlevel\nmodel is\nˆH=/summationdisplay\nkǫkˆc†\nkˆck+ǫiˆd†ˆd+V/summationdisplay\nk(ˆc†\nkˆd+h.c.),(1)\nwhere ˆckandˆdare the single electron annihilation op-\nerators in the lead and at the impurity site respectively.\nWe set the Fermi energy of the lead to be zero. The\nresonant level model can be regarded as a two level sys-\ntem coupled to Fermi sea, if we use |1∝angbracketrightto represent the2\nstate when the impurity site is occupied by an electron\nand|0∝angbracketrightto represent the state when the impurity site is\nempty. We suppose at initial time the state of the impu-\nrity is (α|0∝angbracketright+β|1∝angbracketright). The decoherence will happen after\nwe switch on the coupling between the impurity and the\nFermi sea at time t= 0. The pointer states are |0∝angbracketrightand\n|1∝angbracketright. The non-diagonal elements of the reduced density\nmatrix of the impurity will decay to zero exponentially.\nThe simple way to get the elements of the reduced\ndensity matrix at arbitrary time is to calculate the ob-\nservables which can be related to them. We suppose the\nreduced density matrix is expressed as\nˆρi(t) =ρ00(t)|0∝angbracketright∝angbracketleft0|+ρ11(t)|1∝angbracketright∝angbracketleft1|\n+ρ01(t)|0∝angbracketright∝angbracketleft1|+ρ10(t)|1∝angbracketright∝angbracketleft0|.(2)\nAt time t= 0, we have ρ00(0) =|α|2,ρ11(0) =|β|2\nandρ01(0) =ρ∗\n10(0) =αβ∗. We notice the annihilation\noperator at the impurity site can be expressed as ˆd=\n|0∝angbracketright∝angbracketleft1| ⊗1F, where 1Fis the identity operator on the\nsubspace of the Fermi sea. Then it is easy to know\n∝angbracketleftˆd(t)∝angbracketright=ρ10(t). (3)\nSimilarly we have ˆd†=|1∝angbracketright∝angbracketleft0|⊗1Fandˆd†ˆd=|1∝angbracketright∝angbracketleft1|⊗1F,\nwhich indicate ρ01(t) =∝angbracketleftˆd†(t)∝angbracketrightandρ11(t) =∝angbracketleftˆd†(t)ˆd(t)∝angbracketright\nrespectively.\nWe could calculate the expectation value of ˆd(t) and\nˆd†(t)ˆd(t) by non-equilibrium Green’s function method\nor by diagonalizing the Hamiltonian (1) (for details see\nRef. [15]). Here wefollowthe latter way, and first express\nthe Hamiltonian in hybridization basis as\nˆH=/summationdisplay\nsǫsˆc†\nsˆcs, (4)\nwhere ˆcs=/summationtext\nkV\nǫs−ǫkBsˆck+Bsˆd. The transformation\ncoefficient is Bs=V√\n(ǫs−ǫi)2+Γ2, where Γ = ρπV2is\nthe linewidth of the impurity level and ρthe density of\nstates of the Fermi sea. The inverse transformation is\nˆd=/summationtext\nsBsˆcsand through this the time-dependent anni-\nhilation operator evaluates\nˆd(t) =e−iǫit−Γtˆd+V/summationdisplay\nke−iǫkt−e−iǫit−Γt\nǫk−ǫi+iΓˆck.(5)\nFor simplicity we set /planckover2pi1= 1 throughout the paper. Con-\nsidering the initial condition we then have ρ10(t) =\nα∗βe−iǫit−Γt,ρ01(t) =αβ∗eiǫit−Γt, and\nρ11(t) =V2/summationdisplay\nk|ei(ǫk−ǫi)t−e−Γt|2\n(ǫk−ǫi)2+Γ2θ(−ǫk)\n+e−2Γt|β|2. (6)\nThe non-diagonal elements ρ01(t) andρ10(t) decay ex-\nponentially, and the decay rated|ρ01(t)|\ndt=d|ρ10(t)|\ndt∝\nΓe−Γt. This is the effect of decoherence. Here the Fermi\nsea plays the role of environment.-8-40\n 0 0.5  1 1.5  2 2.5ln(T/Γ)\nTime t (1/ Γ)2\n1.5\n1\n0.5\n0T/Γ\nFIG. 1: Information transfer rate between the impurity and\nthe Fermi sea. The impurity level is at the resonant position ,\ni.e.,ǫi= 0. The top figure shows the time-dependentinforma-\ntion transfer rate. The bottom figure shows the logarithmic\ninformation transfer rate. We see clearly the exponential d e-\ncay ofT.\nIII. INFORMATION TRANSFER RATE\nNext we study the information transfer between the\nimpurity and the Fermi sea. In an open system, such\nas the impurity in resonant level model, the information\nwill broadcast to the environment. This is a stumbling\nblock in quantum computation, while a necessary pro-\ncess in quantum measurement where our purpose is to\nget the information of the system. The mutual informa-\ntion between the system and the environment Imeasures\nhow much information is shared between them. Its time\nderivative is a good measure of the information transfer\nrate, which is defined in this paper as\nT=1\n2dI\ndt. (7)\nHere we take the factor1\n2because the information trans-\nfer both from the impurity to the Fermi sea and vice\nversa. The composite system of the impurity and the\nFermi sea is in a pure state. According to Schmidt de-\ncomposition [7] we know the mutual information Iis two\ntimes of the entropy of the impurity, which can be got\nfrom the eigenvalues of the reduced density matrix. We\nfind at arbitrary time the information transfer rate is\nT=dλ+\ndt(lnλ−−lnλ+), (8)\nwhereλ±=1±/radicalbig\n1−4(ρ00ρ11−ρ01ρ10)\n2.\nThe non-diagonal elements of the reduced density ma-\ntrix decay exponentially, so does the information transfer\nrateT(see Fig. 1). The information transfer rate gradu-\nally decreases when the mutual information between the\nimpurity and the Fermi sea increases to maximum. We3\n-8-6-4-2 0 2\n 0  0.5  1  1.5  2ln(T/Γ)\nTime t (1/ Γ)|α|2=1\n|α|2=0\n|α|2=0.81\n|α|2=0.19\n|α|2=0.64\n|α|2=0.36\nFIG.2: Information transferrateatdifferent |α|2whenǫi= 0.\nThe decay constant of the information transfer rate depends\non|α|2.\nemploy Γ as the unit of informationtransfer rateand 1 /Γ\nthe unit of time. The decay rate of the function T(t) de-\npends on Γ. At large time scale we approximately have\nT∝Γe−kΓt, wherekis a dimensionless constant. Recall-\ning the fact that the decay rate of non-diagonal density\nmatrix elements is proportional to Γ e−Γt, we get the re-lation between decoherencerate and information transfer\nrate: they both decay exponentially, and their decay con-\nstants are proportional to each other.\nWe find at large time scale the information transfer\nratedoes notdepend on the position ofthe impurity level\nǫi, but only depends on the initial condition |α|2. Fig. 2\nshows the information transfer rate at different |α|2. The\ninformationtransferrateisproportionaltoΓ e−kΓt, where\nkdecreasesfrom 4 to 2 when |α|2decreases from 1 to 0 .5,\nand increases back to 4 when |α|2decreases from 0 .5 to\n0.\nIV. SUMMARY AND CONCLUSION\nWe study the decoherence of an impurity coupled to a\nFermi sea after the coupling is switched on at time t= 0.\nThe exponential decay of non-diagonal elements in the\nreduced density matrix of the impurity is observed. And\nthe decay constant is found to be Γ. We also study the\ninformation transfer rate between the impurity and the\nFermi sea. At large time scale the information transfer\nrate decays exponentially, and the decay constant is kΓ\nwithk= 2∼4 depending on the initial state of the im-\npurity (or |α|2). Our study reveals the relation between\ninformation transfer rate and decoherence rate.\n[1] W. H. Zurek, Physics Today 44, 36 (1991).\n[2] M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2004).\n[3] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.\nFisher, A. Garg, W. Zwerger, Rev. Mod. Phys. 59, 1\n(1987).\n[4] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J.\nMath. Phys. (N.Y.) 17, 821 (1976)\n[5] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).\n[6] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).\n[7] M. A. Nielsen, and I. L. Chuang, Quantum Computa-\ntion and Quantum Information (Cambridge Univ. Press,\nCambridge, 2000).\n[8] V. Eisler, and I. Peschel, J. Stat. Mech., P06005 (2007).\n[9] G. De Chiara, S. Montangero, P. Calabrese, R. Fazio, J.Stat. Mech., P03001 (2006).\n[10] J. Eisert, and T. J. Osborne, Phys. Rev. Lett. 97, 150404\n(2006).\n[11] P. Calabrese, and J. Cardy, J. Stat. Mech., P04010\n(2005).\n[12] N. Prokof’ev, and P. Stamp, Rep. Prog. Phys. 63, 669\n(2000).\n[13] R. M. Lutchyn, L. Cywinski, C. P. Nave, S. Das Sarma,\nPhys. Rev. B 78, 024508 (2008).\n[14] N. Yamada, A. Sakuma, H. Tsuchiura, J. Appl. Phys.\n101, 09C110 (2007).\n[15] P. Wang, M. Heyl, S. Kehrein, J. Phys.: Condens. Matter\n22, 275604 (2010)."    },    {        "title": "2101.10849v1.The_Central_Engines_of_Fermi_Blazars.pdf",        "content": "DRAFT VERSION JANUARY 27, 2021\nTypeset using L ATEXtwocolumn style in AASTeX63\nThe Central Engines of Fermi Blazars\nVAIDEHI S. P ALIYA ,1, 2A. D OM´INGUEZ ,3, 4M. A JELLO ,5A. O LMO -GARC´IA,6AND D. H ARTMANN5\n1Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital 263001, India\n2Deutsches Elektronen Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany\n3Institute of Particle and Cosmos Physics (IPARCOS), Universidad Complutense de Madrid, E-28040 Madrid, Spain\n4Department of EMFTEL, Universidad Complutense de Madrid, E-28040 Madrid, Spain\n5Department of Physics and Astronomy, Clemson University, Kinard Lab of Physics, Clemson, SC 29634-0978, USA\n6Departamento de F ´ısica de la Tierra y Astrof ´ısica, Universidad Complutense de Madrid (UCM, Spain) and Instituto de F ´ısica de Part ´ıculas y del Cosmos\n(IPARCOS)\n(Received January 27, 2021; Revised January 27, 2021; Accepted January 27, 2021)\nSubmitted to ApJS\nABSTRACT\nWe present a catalog of central engine properties, i.e., black hole mass ( MBH) and accretion luminosity\n(Ldisk), for a sample of 1077 blazars detected with the Fermi Large Area Telescope. This includes broad\nemission line systems and blazars whose optical spectra lack emission lines but dominated by the absorption\nfeatures arising from the host galaxy. The average MBHfor the sample ishlogMBH;allM\fi= 8:60and there\nare evidences suggesting the association of more massive black holes with absorption line systems. Our results\nindicate a bi-modality of Ldiskin Eddington units ( Ldisk/LEdd) with broad line objects tend to have a higher\naccretion rate ( Ldisk/LEdd>0.01). We have found that Ldisk/LEddand Compton dominance (CD, the ratio\nof the inverse Compton to synchrotron peak luminosities) are positively correlated at >5\u001bconfidence level,\nsuggesting that the latter can be used to reveal the state of accretion in blazars. Based on this result, we propose\na CD based classification scheme. Sources with CD >1 can be classified as High-Compton Dominated or HCD\nblazars, whereas, that with CD .1 are Low-Compton Dominated (LCD) objects. This scheme is analogous to\nthat based on the mass accretion rate proposed in the literature, however, it overcomes the limitation imposed\nby the difficulty in measuring LdiskandMBHfor objects with quasi-featureless spectra. We conclude that the\noverall physical properties of Fermi blazars are likely to be controlled by the accretion rate in Eddington units.\nThe catalog is made public at http://www.ucm.es/blazars/engines and Zenodo.\nKeywords: methods: data analysis — gamma rays: general — galaxies: active — galaxies: jets\n1.INTRODUCTION\nBlazars are a subclass of jetted active galactic nuclei\n(AGN) family that host relativistic jets closely aligned to\nthe line of sight to the observer. Due to their peculiar ori-\nentation, the emitted radiation is relativistically amplified\nthereby making blazars observable at cosmological distances\n(z >5). Typically, blazars are hosted by elliptical galaxies\nand powered by massive (10\u00188\u000010M\f) black holes (e.g.,\nUrry et al. 2000; Shaw et al. 2012). The multi-wavelength\nspectral energy distribution (SED) of a blazar exhibits a typ-\nCorresponding author: Vaidehi S. Paliya\nvaidehi.s.paliya@gmail.comical double hump structure. The low-energy bump peaks at\nradio-to-X-rays and is well explained with the synchrotron\nmechanism. On the other hand, the inverse Compton process\nis found to satisfactorily reproduce the high-energy bump lo-\ncated in the MeV-TeV enetgy range, considering the leptonic\nradiative models (e.g., Dermer et al. 2009; van den Berg et al.\n2019). The properties of the central engine, i.e., black hole\nmass (MBH) and accretion disk luminosity ( Ldisk), are found\nto significantly correlate with the broadband SED of blazars\n(Ghisellini et al. 2014; Paliya et al. 2017b, 2019, 2020a),\nindicating a close-connection between the accretion process\nand relativistic jets.\nHistorically, the beamed AGN have been classified based\non the appearance of the broad emission lines in their op-\ntical spectra. Blazars exhibiting strong and broad emissionarXiv:2101.10849v1  [astro-ph.HE]  26 Jan 20212 P ALIYA ET AL .\nlines (rest-frame equivalent width of EW >5˚A) are known\nas flat spectrum radio quasars or FSRQs. The objects show-\ning quasi-featureless spectra (EW <5˚A), on the other hand,\nbelong to the category of BL Lacertae or BL Lac sources\n(Stickel et al. 1991). It was predicted that the lack of broad\nemission lines in the optical spectra of the latter might be\ndue to Doppler boosted continuum swamping out any spec-\ntral lines even if they exist. However, stellar absorption fea-\ntures originated from the host galaxy are observed in many\nnearby (z<1) BL Lac sources, thus indicating that emission\nlines are intrinsically weak (Plotkin et al. 2011). To make\nthe case more complex, broad emission lines have been de-\ntected in the low jet activity states of many BL Lac objects\n(e.g., Vermeulen et al. 1995). Therefore, a blazar classifica-\ntion scheme based on the EW does not reveal the physical\ndistinction between FSRQs and BL Lac sources.\nGhisellini et al. (2011) proposed that the criterion to dis-\ntinguish FSRQs and BL Lac objects should be based on the\nbroad line region (BLR) luminosity ( LBLR). FSRQs are\nsources with a luminous BLR ( LBLR&10\u00003times Edding-\nton luminosity or LEdd) which, in turn, suggests a radiatively\nefficient accretion process ( Ldisk&1%LEdd). BL Lac\nsources, on the other hand, host a radiatively inefficient ac-\ncretion flow which fails to photo-ionize the BLR clouds (see\nalso, Sbarrato et al. 2014). The presence of broad and strong\nemission lines in the optical spectra of FSRQs and absence\nin BL Lacs, thus, can be explained in this more physically\nintuitive classification. However, one of the key parameters\nin this scheme, LEdd, requires the knowledge of the mass of\nthe central black hole. Moreover, an estimation of Ldisk, or\nequivalently mass accretion rate, remains a challenge for BL\nLac objects.\nIn the context of the \r-ray emitting beamed AGN, the re-\nsearch works focused on the central engine properties have\nso far remained concentrated on the broad emission lines\nblazars (cf. Ghisellini et al. 2014). Apart from a few individ-\nual works (e.g., Becerra Gonz ´alez et al. 2020), a population\nstudy on the central engine of BL Lac sources is still lacking.\nIn this work, we have attempted to address this outstanding\nissue by carrying out a detailed optical spectroscopic analy-\nsis of a large sample of blazars present in the Fermi Large\nArea Telescope (LAT) fourth source catalog data release 2\n(4FGL-DR2; Abdollahi et al. 2020). Furthermore, we have\nalso determined that Compton dominance (CD, the ratio of\nthe inverse Compton to synchrotron peak luminosities) can\nbe considered as a good proxy for the accretion rate in Ed-\ndington units. This parameter can be crucial for objects\nwhose optical spectroscopic analyses are tedious either due\nto telescope constraints, intrinsic source faintness, or lack of\nemission lines in their optical spectrum. As discussed above,\nthe latter could be due to a radiatively inefficient accretion\nprocess and/or Doppler boosted jet radiation swamping outTable 1. TheFermi blazar sample studied in this work.\nSample Classification Number of Sources\nEmission line blazars 258 (SDSS)\n416 (literature)\n674 (all)\nAbsorption line blazars 200 (SDSS)\n146 (literature)\n346 (all)\nBlazars with bulge magnitude from literature 47\nBlazars with MBHandLdiskfrom literature 10\nemission lines and thus does not allows us to infer the intrin-\nsic nature of the quasar. The CD, therefore, could be the key\nto connect the FSRQ-BL Lac dichotomy and a classification\nscheme based on this parameter may fully explain the diverse\nphysical properties of Fermi blazars.\nIn Section 2, we describe the Fermi blazar sample and the\ndetails of the optical spectroscopic data analysis procedures\nare provided in Section 3. We elaborate various techniques\nto determine MBHandLdiskin Section 4 and discuss them\nin Section 5. Section 6 and 7 are devoted to the physical\ninterpretation of the estimated central engine parameters and\nCD. We summarize our findings in Section 8. Throughout,\nwe adopt the flat cosmology parameters: h= \n \u0003= 0:7.\n2.SAMPLE\nWe have considered all \r-ray emitting blazars or blazar\ncandidates present in the 4FGL-DR2 catalog to prepare the\nbase sample of 5250 objects. Since the main goal of this\nwork is to derive MBHandLdiskfrom the optical spectrum,\nwe carried out an extensive search for the optical spectro-\nscopic information for as many blazars as possible. This was\ndone by: (i) cross-matching the 4FGL-DR2 catalog with the\n16th data release of the Sloan Digital Sky Survey (SDSS-\nDR16; Ahumada et al. 2020), (ii) searching the published\noptical spectrum of all remaining blazars in the literature us-\ning NASA Extragalactic Database and SIMBAD Astronom-\nical Database, and (iii) searching the published MBHand\nLdiskvalues in the literature for objects leftover after com-\npleting steps (i) and (ii). In all of the cases, we used the posi-\ntional information of the low-frequency counterpart of the \r-\nray source given in the 4FGL-DR2 catalog or in other works\nand visually examine the quality of the spectra. This exercise\nled to a collection of 458 SDSS spectra including 258 emis-\nsion line objects and 200 blazars showing prominent absorp-\ntion lines. From research articles, we were able to collect op-\ntical spectra of 562 blazars either in tabular format or simplyBLAZAR ’SCENTRAL ENGINE 3\nas plots which were digitized using WebPlotDigitizer1\n(Rohatgi 2020). Among them, 416 objects are broad emis-\nsion line systems and the optical spectra of 146 are domi-\nnated by absorption lines arising from the host galaxy. Fur-\nthermore, we were able to find host galaxy bulge magnitude\nfor 47 sources and MBHand/orLdiskvalues for 10 blazars\nfrom the literature. Altogether, our final sample consists of\n1020 objects with available optical spectra and 57 others with\nbulge magnitude or MBHandLdiskdirectly taken from the\nliterature. Note that emission line blazars mainly belong to\nFSRQ class of AGN, whereas BL Lacs dominate the ab-\nsorption line sample. However, since many of the BL Lac\nobjects have exhibited broad emission lines in their optical\nspectra taken during a low-jet activity state, i.e., akin to FS-\nRQs, we divide the whole sample in emission and absorption\nline systems, respectively, rather than distinguishing them as\nFSRQ/BL Lacs. Table 1 summarizes our blazar sample.\nThe redshift distribution of the sample is shown in Fig-\nure 1. Instead of directly using the redshift information from\nthe fourth catalog of the Fermi -LAT detected AGN (4LAC;\nAjello et al. 2020), we have visually inspected the optical\nspectrum of all of the sources to determine the accurate\nsource redshift. Though most of our measurements agree\nwith redshifts published in the 4LAC catalog, a small frac-\ntion (<5%) of sources do show differences (Figure 2, left\npanel). This is expected since 4LAC itself is a compilation of\ninformation taken from various databases. In the right panel\nof Figure 2, we show the optical spectrum of the blazar 4FGL\nJ1205.8+3321 as an illustration. The 4LAC catalog reports\nthe redshift of this object as z= 2:085which is likely to be\nphotometric in nature and probably adopted from Richards\net al. (2009). On the other hand, the SDSS-DR16 spectrum\nclearly shows many broad emission lines securing the spec-\ntroscopic redshift as z= 1:007.\nWe acquired optical spectra of a few ( <50) sources from\nthe 6-degree Field Galaxy Survey (Jones et al. 2009). Since\nthese spectra are not flux calibrated, we adopted the wave-\nlength dependent conversion factor reported in Chen et al.\n(2018) to convert counts spectra in energy flux units.\n3.OPTICAL SPECTROSCOPIC ANALYSIS\nA good fraction of sources in our sample has optical spec-\ntrum taken from the SDSS-DR16, however, we do not use the\nemission/absorption line measurements obtained with the au-\ntomatic SDSS data reduction pipeline. A more sophisticated\napproach was needed to decompose the broad and narrow\ncomponents of the emission lines, to model and subtract the\nhost galaxy emission, and to consider the Fe complexes since\nthese are likely not to be properly taken into account by the\nSDSS pipeline. Moreover, a significant number of blazars do\n1https://automeris.io/WebPlotDigitizer/\n0.03 0.1 0.3 12345\nRedshift0306090Number of sourcesemission line\nabsorption line\nallFigure 1. The redshift distribution of Fermi blazars (black) present\nin our sample. Red solid and blue dotted histograms refer to objects\nwhose optical spectra are dominated by broad emission lines and\nhost galaxy absorption features, respectively.\nnot have SDSS spectra and a separate spectroscopic analysis\nwas required. During the data analysis, every optical spec-\ntrum was individually analyzed rather than by running any\nautomatic pipeline.\n3.1. Emission Line Spectrum\nThere are 674 \r-ray sources in our sample whose opti-\ncal spectra exhibit at least one of the broad emission lines\nH\u000b, H\f, Mg II, and C IV. To derive the spectral parameters\nassociated with these lines, e.g., full-width at half-maximum\nor FWHM, and continuum luminosities at 1350 ˚A, 3000 ˚A,\nand 5100 ˚A (L1350,L3000, andL5100, respectively), we have\nadopted the publicly available software PyQSOFit2(Guo\net al. 2018). The tool applies the spectral models and tem-\nplates to data following a \u001f2-based fitting technique (see\nalso, Guo et al. 2019; Shen et al. 2019). Here we briefly\ndescribe the adopted steps.\nThe quasar spectrum was brought to rest-frame and cor-\nrected for Galactic reddening following the extinction curve\nfrom Cardelli et al. (1989) and dust map of Schlegel et al.\n(1998). This correction was applied only to the SDSS data.\nFor spectra taken from the published articles, we do not ap-\nply Galactic extinction correction since it is usually done as a\npart of the data reduction prior to the publication. The spec-\ntrum was decomposed into the quasar and host galaxy com-\nponents following the principal component analysis method\npresented in Yip et al. (2004a,b). We have considered a\npower-law and a third-order polynomial along with optical\n2https://github.com/legolason/PyQSOFit4 P ALIYA ET AL .\n00.5 11.5 22.5 33.5 44.5\nredshift (4LAC)00.511.522.533.544.5redshift (this work)\n4000 5000 6000 7000 8000 9000 10000\nWavelength ( ˚A)246810Fλ(10−17erg cm−2s−1˚A−1)\nMg IIC III\nHβ4FGL J1205.8+3321 ( z= 1.007)\nFigure 2. Left: A comparison of the redshift values reported in the 4LAC catalog and this work. The dashed line denotes the equality of the\nplotted quantities. Right: The SDSS-DR16 spectrum of the \r-ray source 4FGL J1205.8+3321 whose redshift reported in the 4LAC catalog is\nz= 2:085. However, based on the detection of various broad emission lines, its correct redshift is z= 1:007.\n4000 4500 5000 5500 6000 6500 7000\nWavelength ( ˚A, rest-frame)200030004000500060007000Fλ(10−17erg s−1cm−2˚A−1)\nHα+NII\nSII6718,6732Hβ\n[OIII]HeII4687HγCaII39344FGL J1229.0+0202 ( z= 0.158)\ndata\nline\nline br\nline na\nconti\n4500 5000 5500 6000 6500 7000\nWavelength ( ˚A, rest-frame)140160180200220Fλ(10−17erg s−1cm−2˚A−1)\nHα+NII\nSII6718,6732Hβ\n[OIII]HeII46874FGL J2202.7+4216 ( z= 0.0686)\ndata\nline\nline br\nline na\nconti\nFigure 3. The optical spectra of prototype blazars 3C 273 or 4FGL J1229.0+0202 (left) and BL Lacertae or 4FGL J2202.7+4216 modeled with\nPyQSOFit . The spectral data is shown with the black line. Broad and narrow components of the emission line are represented by red and\ngreen lines and the modeled continuum is plotted with the orange line. Blue line is the sum of all the components. Horizontal grey dahses at the\ntop of the plots denote the line-free wavelength regions selected to model the continuum emission. The data are adopted from Torrealba et al.\n(2012, for 3C 273) and Vermeulen et al. (1995, for BL Lacertae).\nand UV Fe IItemplates (Boroson & Green 1992; Vestergaard\n& Wilkes 2001) to fit the line-free continuum over the en-\ntire spectrum. The best fitted continuum was then subtracted\nfrom the spectrum to acquire a line-only spectrum which was\nthen used to extract the spectral properties of H \u000b, H\f, Mg II,\nand C IVemission lines.\nThe H\u000bline was fitted in the wavelength range\n[6400, 6800] ˚A. The broad component of H \u000bwas modeled\nwith 3 Gaussians (FWHM >1200 km s\u00001; Hao et al. 2005),\nwhereas, we adopted a single Gaussian with the FWHM up-\nper limit of 1200 km s\u00001to reproduce the narrow H \u000bline.\nFurthermore, the flux ratio of the [N II]\u0015\u00156549, 6585 doublet\nwas fixed to 3 (Shen et al. 2011).\nH\fline fitting was carried out in the wavelength range\n[4640\u00005100] ˚A. We used 2 Gaussians (FWHM >1200\nkm s\u00001) and a single Gaussian (FWHM <1200 km s\u00001) fit\nto model the broad and narrow components of the H \fline,\nrespectively. We also fitted the narrow [O III]\u0015\u00154959, 5007lines with 2 Gaussians and their flux ratio was allowed to\nvary. However, we tied the velocity and width of the narrow\nH\fline to the core [O III] component.\nWe fitted Mg IIand C IVemission lines in the wavelength\nrange [2700\u00002900] ˚A and [1500\u00001700] ˚A, respectively. The\nbroad component of both lines were modeled with 2 Gaus-\nsians (FWHM >1200 km s\u00001) and we used a single Gaus-\nsian function (FWHM <1200 km s\u00001) to fit their narrow\nemission line components.\nPyQSOFit determines the uncertainties in the derived pa-\nrameters employing a Monte-Carlo technique. In particular,\na random Gaussian fluctuation with a zero mean and disper-\nsion equals to the uncertainty measured at the given pixel\nwas added to the observed spectral flux at every pixel and\n50 mock spectra were created and fitted with the same strat-\negy as that adopted to model the real data. This exercise\nwas repeated 20 times and uncertainties were calculated asBLAZAR ’SCENTRAL ENGINE 5\nTable 2. The spectral parameters associated with the\nH\u000bemission line.\n4FGL name redshift FWHM LH\u000b\n[1] [2] [3] [4]\nJ0001.5+2113 0.439 1629 \u000684 42.942 \u00060.016\nJ0006.3 \u00000620 0.347 8991 \u00065494 42.782 \u00060.189\nJ0013.6+4051 0.256 2049 \u00061739 41.277 \u00060.205\nJ0017.5 \u00000514 0.227 2073 \u0006982 42.926 \u00060.090\nJ0049.6 \u00004500 0.121 4298 \u00063904 41.658 \u00060.117\nNOTE—Column information are as follows: Col.[1]: 4FGL\nname; Col.[2]: redshift; Col.[3]: FWHM of H \u000bline, in\nkm s\u00001; and Col.[4]: log-scale H \u000bline luminosity, in\nerg s\u00001. (This table is available in its entirety in a machine-\nreadable form in the online journal. A portion is shown here for\nguidance regarding its form and content.)\nTable 3. The spectral parameters associated with the H \femission line.\n4FGL name redshift L5100 ˚AFWHM LH\f\n[1] [2] [3] [4] [5]\nJ0001.5+2113 0.439 44.577 \u00060.003 2114 \u0006483 42.029 \u00060.054\nJ0006.3 \u00000620 0.347 44.757 \u00060.004 7051 \u00064699 42.004 \u00060.227\nJ0010.6+2043 0.598 44.867 \u00060.003 2593 \u0006207 43.047 \u00060.048\nJ0014.3 \u00000500 0.791 44.951 \u00060.008 1619 \u0006653 42.446 \u00060.060\nJ0017.5 \u00000514 0.227 44.347 \u00060.005 3294 \u0006487 42.302 \u00060.115\nNOTE—Column information are as follows: Col.[1]: 4FGL name; Col.[2]: redshift;\nCol.[3]: log-scale continuum luminosity at 5100 ˚A; Col.[4]: FWHM of H \fline,\ninkm s\u00001; and Col.[5]: log-scale H \fline luminosity, in erg s\u00001. (This table is\navailable in its entirety in a machine-readable form in the online journal. A portion\nis shown here for guidance regarding its form and content.)\nthe semi-amplitude of the range covering the 16th \u000084th per-\ncentiles of the parameter distribution from the trials.\nThe tool PyQSOFit is primarily developed to analyze the\nSDSS data and hence requires the pixel scale of the optical\nspectrum to be same as that of SDSS, i.e., 10\u00004in log-space3.\nSince a significant fraction of the optical spectra in our sam-\nple is collected from the literature and does not have the req-\nuisite binning, we rebinned them using a simple linear inter-\npolation to have the pixel scale same as that of the SDSS data.\nFurthermore, since we do not have the flux uncertainty mea-\nsurements in such cases, we conservatively assumed an un-\ncertainty of 25% of the measured fluxes, a typical value asso-\nciated with the ground-based optical observations (cf. Healey\net al. 2008; Shaw et al. 2012).\n3https://www.sdss.org/dr16/spectro/spectro basics/Table 4. The spectral parameters associated with the Mg IIemission line.\n4FGL name redshift L3000 ˚AFWHM LMgII\n[1] [2] [3] [4] [5]\nJ0001.5+2113 0.439 44.706 \u00060.002 1719 \u0006165 42.503 \u00060.032\nJ0004.4 \u00004737 0.880 45.345 \u00060.003 2965 \u0006913 42.885 \u00060.099\nJ0010.6+2043 0.598 44.799 \u00060.005 2222 \u000699 43.027 \u00060.017\nJ0011.4+0057 1.491 45.589 \u00060.003 3382 \u0006245 43.365 \u00060.045\nJ0013.6 \u00000424 1.076 44.898 \u00060.003 2383 \u0006221 42.818 \u00060.080\nNOTE—Column information are as follows: Col.[1]: 4FGL name; Col.[2]: red-\nshift; Col.[3]: log-scale continuum luminosity at 3000 ˚A; Col.[4]: FWHM of\nMgIIline, in km s\u00001; and Col.[5]: log-scale Mg IIline luminosity, in erg s\u00001.\n(This table is available in its entirety in a machine-readable form in the online\njournal. A portion is shown here for guidance regarding its form and content.)\nTable 5. The spectral parameters associated with the C IVemission line.\n4FGL name redshift L1350 ˚AFWHM LCIV\n[1] [2] [3] [4] [5]\nJ0004.3+4614 1.810 45.615 \u00060.001 2652 \u0006300 44.126 \u00060.031\nJ0011.4+0057 1.491 45.558 \u00060.001 4430 \u0006437 43.901 \u00060.014\nJ0016.2 \u00000016 1.577 45.445 \u00060.001 4648 \u0006216 43.809 \u00060.073\nJ0016.5+1702 1.721 45.304 \u00060.000 5722 \u0006275 43.849 \u00060.021\nJ0028.4+2001 1.553 45.499 \u00060.001 2631 \u0006128 43.927 \u00060.012\nNOTE—Column information are as follows: Col.[1]: 4FGL name; Col.[2]: red-\nshift; Col.[3]: log-scale continuum luminosity at 1350 ˚A; Col.[4]: FWHM of\nCIVline, in km s\u00001; and Col.[5]: log-scale C IVline luminosity, in erg s\u00001.\n(This table is available in its entirety in a machine-readable form in the online\njournal. A portion is shown here for guidance regarding its form and content.)\nIn Figure 3, we show an example of the emission line\nfitting done on the optical spectra of the prototype blazars\n3C 273 (4FGL J1229.0+0202, z= 0:158) and BL Lacertae\n(4FGL J2202.7+4216, z= 0:0686 ). The spectral parame-\nters derived for H \u000b, H\f, Mg II, and C IVemission lines are\nprovided in Table 2, 3, 4, and 5, respectively.\n3.2. Absorption Line Spectrum\nThe optical spectra of 346 \r-ray blazars show prominent\nabsorption lines, e.g., Ca H&K doublet, primarily originat-\ning from the stellar population in the host galaxy. It has been\nshown in various works that the mass of the central black\nhole significantly correlates with the stellar velocity disper-\nsion (\u001b\u0003, cf. Ferrarese & Merritt 2000; G ¨ultekin et al. 2009;\nKormendy & Ho 2013). Therefore, we have used the penal-\nized PiXel Fitting tool ( pPXF ; Cappellari & Emsellem 2004)\nto derive\u001b\u0003for blazars present in the sample. This software\nworks in pixel space and uses a maximum penalized like-\nlihood approach to calculate the line-of-sight velocity dis-6 P ALIYA ET AL .\n3040Fλ(10−17erg cm−2s−1)4FGL J1132.2-4736 ( z= 0.23) [σ∗= 287.99±6.97 km s−1]\n3500 3750 4000 4250 4500 4750 5000 5250 5500\nWavelngth ( ˚A, rest-frame)−101\n152025Fλ(10−17erg cm−2s−1)4FGL J2220.5+2813 ( z= 0.15) [σ∗= 261.88±10.52 km s−1]\n3500 4000 4500 5000 5500 6000 6500 7000\nWavelngth ( ˚A, rest-frame)−101\nFigure 4. The optical spectra of two Fermi -LAT detected blazars (black) along with the results of the stellar population synthesis done using\npPXF (red). In both plots, the bottom panel refers to the residual of the fit. The wavelength regions excluded from the fit to mask emission\nlines are highlighted with the grey shaded columns. The spectroscopic data of 4FGL J1132.2 \u00004736 is taken from Pe ˜na-Herazo et al. (2017),\nwhereas, that of 4FGL J2220.5+2813 is adopted from SDSS-DR16 (Ahumada et al. 2020).\ntribution (LOSVD) from kinematic data (Merritt 1997). A\nlarge set of stellar population synthesis models (adopted from\nVazdekis et al. 2010) with spectral resolution of FWHM = 2.5\n˚A and the wavelength coverage of [3525, 7500] ˚A (S ´anchez-\nBl´azquez et al. 2006) were used in this work. The pPXF code\nfirst creates a template galaxy spectrum by convolving the\nstellar population models with the parameterized LOSVD.\nTo mimic the non-thermal power-law contribution from the\ncentral nucleus, we further added a fourth-order Legendre\npolynomial to the model. The regions of bright emission\nlines were masked prior to the fitting. The model was then\nfitted on the rest-frame galaxy spectrum and \u001b\u0003and associ-\nated 1\u001buncertainty were derived from the best-fit spectrum.\nThe whole analysis was carried out on a case-by-case basis\nto achieve the best results. This is because the optical spectra\nof many sources reveal strong telluric lines which may over-\nlap with the absorption lines arising from the host galaxy and\nthus needed to be avoided. Therefore, although we attempted\na fit in the full [3525, 7500] ˚A wavelength range, it was not\nalways possible. The examples of this analysis are shown in\nFigure 4 and the derived \u001b\u0003values are provided in Table 6.\nUnlike broad line blazars where emission line luminosi-\nties can be used to infer the BLR luminosity, it is not pos-\nsible estimate the latter for absorption line systems since\nbroad emission lines are not detected. Therefore, we have\nderived 3\u001bupper limit in the H \f(or Mg II, depending on the\nsource redshift and wavelength coverage) line luminosity by\nadopting the following steps. The observed spectrum was\nfirst brought to the rest-frame and the host galaxy component\nwas subtracted using PyQSOFit . In the wavelength range\n[4700\u00005000] ˚A or [2650\u00002950] ˚A for H\for Mg II, respec-\ntively, we then fitted a power-law (reproducing continuum)\nplus a Gaussian function (mimicking the emission line) with\na variable luminosity while keeping the FWHM fixed to 4000\nkm s\u00001, a value typically observed in blazars (e.g., Shaw\net al. 2012). Based on the derived \u001f2, we considered the up-\nper limit to the line luminosity when \u001f2> \u001f2(99.7%), i.e.,\nat 3\u001bconfidence level. We show an example of the adopted\nmethod in Figure 5 and report the upper limits in Table 6.Table 6. The stellar velocity dispersion and line luminosity up-\nper limit measurements for 346 blazars.\n4FGL name redshift \u001b\u0003Lline;UL\n[1] [2] [3] [4]\nJ0003.2+2207 0.100 197.61 \u00067.87 40.34\nJ0006.4+0135 0.787 385.99 \u000661.79 42.41\nJ0013.9\u00001854 0.095 459.46 \u000619.50 40.87\nJ0014.2+0854 0.163 296.52 \u000610.53 40.97\nJ0015.6+5551 0.217 467.26 \u000622.05 41.65\nNOTE—Column information are as follows: Col.[1]:\n4FGL name; Col.[2]: redshift; Col.[3]: stellar velocity\ndispersion (\u001b\u0003) inkm s\u00001; and Col.[4]: 3 \u001bupper limit\nin H\fline luminosity (log-scale, in erg s\u00001), except for\nsources 4FGL J0006.4+0135, 4FGL J0204.0-3334, and\n4FGL J1146.0-0638, for which we quote the Mg IIline\nluminosity upper limit. (This table is available in its en-\ntirety in a machine-readable form in the online journal.\nA portion is shown here for guidance regarding its form\nand content.)\n4.BLACK HOLE MASS AND DISK LUMINOSITY\nMEASUREMENTS\n4.1. Emission Line Black Hole Mass\nWe have derived MBHof emission line blazars from the\nsingle-epoch optical spectra assuming that BLR is virialized,\nbroad line FWHM represents the virial velocity, and contin-\nuum luminosity can be considered as a proxy for the BLR\nradius. The virial MBHcan be calculated using the follow-\ning equation (Shen et al. 2011):\nlog\u0012MBH\nM\f\u0013\n=\u000b+\flog\u0012\u0015L\u0015\n1044erg s\u00001\u0013\n+2 log\u0012FWHM\nkm s\u00001\u0013\n(1)BLAZAR ’SCENTRAL ENGINE 7\n4700 4750 4800 4850 4900 4950 5000\nλ(Hz, rest-frame)036912Fλ(10−17erg cm−2s−1˚A−1)\n40.8 41 41.2\nlogLHβ[erg s−1]0246810 χ2−χ2\nmin4FGL J0837.3+1458 ( z= 0.1523)\nFigure 5. A demonstration of the technique adopted to derive the 3 \u001b\nupper limit on the emission line luminosity. Blue thin and black dot-\nted lines refer to the observed spectrum and power-law continuum,\nrespectively. We also plot Gaussian functions with variable line lu-\nminosity. The right panel shows the derived \u001f2as a function of the\nline (H\fin this case) luminosity. The vertical line corresponds to\nthe line luminosity beyond which \u001f2>\u001f2(99.7%).\nwhere\u0015L\u0015is the continuum luminosity at 5100 ˚A (for H\f),\n3000 ˚A (for Mg II), and 1350 ˚A (for C IV). The calibration co-\nefficients\u000band\fare taken from McLure & Dunlop (MD04;\n2004) for H\fand Mg IIand Vestergaard & Peterson (VP06;\n2006) for C IVlines and have the following values\n(\u000b;\f) =8\n><\n>:(0:672;0:61);H\f(MD04)\n(0:505;0:62);MgII(MD04)\n(0:660;0:53);CIV(VP06); (2)\nShen et al. (2011) also provided the following empirical\nrelation to estimate MBHfrom the H\u000bline FWHM and lu-\nminosity:\nlog\u0012MBH\nM\f\u0013\nH\u000b= 0:379 + 0:43 log\u0012LH\u000b\n1042erg s\u00001\u0013\n+ 2:1 log\u0012FWHM H\u000b\nkm s\u00001\u0013\n; (3)\nwhereLH\u000bis the total H \u000bline luminosity. For objects with\nmore than one MBHmeasurements, we have taken the geo-\nmetric mean.\nThe optical spectral continuum of blazars could be signif-\nicantly contaminated from the non-thermal jetted emission\nwhich can affect the MBHestimation. To address this issue,\nwe also derived MBHvalues using emission line parameters\nalone. This was done by replacing continuum luminosity in\nEquation 1 with the line luminosity and adopting the cali-\nbration coefficients ( \u000b;\f) as (1.63, 0.49), (1.70, 0.63), and\n(1.52, 0.46) for H \f, Mg II, and C IVlines, respectively, fol-\nlowing Shaw et al. (2012). We compared the masses cal-\nculated from two approaches in Figure 6. There are indi-\ncations that MBHcomputed from virial approach is slightly\nhigher than that calculated from H \fline alone (Figure 6, left\npanel) which could be due to contamination from the host\ngalaxy emission as also suggested by Shaw et al. (2012). ForMgIIand C IVlines, the overall impact of the jetted emis-\nsion on theMBHmeasurement is negligible. Therefore, the\nMBHvalues for emission line blazars computed from the tra-\nditional virial technique (Equation 1) are used in this work as\nthis may also allow a comparison study with other non-blazar\nclass of AGN. We report them in Table 7.\n4.2. Absorption Line Black Hole Mass\nWe used the following empirical relation to compute MBH\nfrom the measured stellar velocity dispersion (G ¨ultekin et al.\n2009)\nlog\u0012MBH\nM\f\u0013\n= (8:12\u00060:08)+(4:24\u00060:41) log\u0010\u001b\u0003\n200 km s\u00001\u0011\n;\n(4)\nThe estimated masses of 346 blazars are provided in Ta-\nble 8.\n4.3. Bulge Luminosity Black Hole Mass\nFrom the literature, we were able to obtain appar-\nent/absolute R- orK-band magnitudes of the host galaxy\nbulge for 47 blazars. The following equations waere used\nto deriveMBHfrom the bulge luminosity (Graham 2007)\nlog\u0012MBH\nM\f\u0013\n=(\n(\u00000:38\u00060:06)(MR+ 21) + (8:11\u00060:11)\n(\u00000:38\u00060:06)(MK+ 24) + (8:26\u00060:11)\n(5)\nwhereMRandMKare the absolute magnitudes of the host\ngalaxy bulge in R- andK-bands, respectively. The derived\nMBHvalues are reported in Table 9.\n4.4. Accretion Disk Luminosity\nFrom the emission line luminosities or 3 \u001bupper limits, we\ncomputed the BLR luminosity (or 3 \u001bupper limits) as fol-\nlows: we assigned a reference value of 100 to Ly \u000bemission\nand summed the line ratios (with respect to Ly \u000b) reported in\nFrancis et al. (1991) and Celotti et al. (1997, for H \u000b) giv-\ning the total BLR fraction hLBLRi= 555:77\u00185:6Ly\u000b.\nThe BLR luminosity can then be derived using the following\nequation\nLBLR=Lline\u0002hLBLRi\nLrel:frac:(6)\nwhereLlineis the emission line luminosity and Lrel:frac:is\nthe line ratio, 77, 22, 34, and 63 for H \u000b, H\f, Mg II, and\nCIVlines, respectively (Francis et al. 1991; Celotti et al.\n1997). When more than one line luminosity measurements\nwere available, we took their geometric mean to derive the\naverageLBLR and then calculated Ldiskfrom theLBLR as-\nsuming 10% BLR covering factor. The computed Ldiskval-\nues and 3\u001bupper limits are provided in Table 7 and 8, re-\nspectively.8 P ALIYA ET AL .\n7 8 9 1078910log Mass from lines alone (M ⊙)Hβ\n7 8 9 10\nlog Traditional virial mass (M ⊙)Mgii\n7 8 9 10Civ\nFigure 6. A comparison of the MBHvalues derived from the traditional virial technique using line FWHM and continuum luminosity with that\nestimated from the emission line alone. The dashed line represents the equality of the plotted quantities.\nTable 7. The central engine and SED properties of emission line blazars.\n4FGL name MBH Ldisk\u0017peak\nsyn\u0017Fpeak\n\u0017;syn\u0017peak\nIC\u0017Fpeak\n\u0017;IC CD\n[1] [2] [3] [4] [5] [6] [7] [8]\nJ0001.5+2113 7.54 \u00060.07 44.65\u00060.02 13.81 -11.97 20.64 -10.48 30.90\nJ0004.3+4614 8.36 \u00060.10 46.07\u00060.03 12.35 -12.49 21.35 -11.70 6.17\nJ0004.4\u00004737 8.28\u00060.27 45.10\u00060.10 13.01 -11.62 21.37 -11.24 2.40\nJ0006.3\u00000620 8.93\u00060.40 44.52\u00060.15 12.92 -11.09 19.66 -12.04 0.11\nJ0010.6+2043 7.86 \u00060.04 45.35\u00060.03 12.42 -12.12 22.60 -11.97 1.41\nNOTE—Column information are as follows: Col.[1]: 4FGL name; Col.[2]: log-scale mass\nof the central black hole, in M\f; Col.[3]: log-scale accretion disk luminosity, in erg s\u00001;\nCol.[4] and [5]: log-scale synchrotron peak frequency and corresponding flux, in Hz and\nerg cm\u00002s\u00001, respectively; Col.[6] and [7]: log-scale inverse Compton peak frequency\nand corresponding flux, in Hz and erg cm\u00002s\u00001, respectively; and Col.[8]: Compton dom-\ninance. (This table is available in its entirety in a machine-readable form in the online\njournal. A portion is shown here for guidance regarding its form and content.)\nTable 9 . The mass of the central black holes for 47\nblazars derived from the host galaxy bulge luminos-\nity.\n4FGL name redshift MBH\n[1] [2] [3]\nJ0037.8+1239 0.089 8.64 \u00060.14\nJ0050.7\u00000929 0.635 8.85 \u00060.18\nTable 9 continuedBLAZAR ’SCENTRAL ENGINE 9\nTable 8. The central engine and SED properties of absorption line blazars.\n4FGL name MBHLdisk\u0017peak\nsyn\u0017Fpeak\n\u0017;syn\u0017peak\nIC\u0017Fpeak\n\u0017;IC CD\n[1] [2] [3] [4] [5] [6] [7] [8]\nJ0003.2+2207 8.10 \u00060.11 42.74 15.15 -12.23 22.16 -12.91 0.21\nJ0006.4+0135 9.33 \u00060.33 44.62 15.96 -12.49 22.93 -12.65 0.69\nJ0013.9\u00001854 9.65\u00060.19 43.27 17.43 -11.39 23.95 -12.47 0.08\nJ0014.2+0854 8.85 \u00060.12 43.37 15.64 -12.21 22.25 -12.58 0.43\nJ0015.6+5551 9.68 \u00060.19 44.05 17.11 -11.61 24.89 -12.07 0.35\nNOTE—Column information are same as in Table 7 except Column 3 where 3 \u001b\nupper limits on the Ldiskare reported. (This table is available in its entirety in a\nmachine-readable form in the online journal. A portion is shown here for guidance\nregarding its form and content.)\nTable 9 (continued)\n4FGL name redshift MBH\n[1] [2] [3]\nJ0109.1+1815 0.145 9.09 \u00060.23\nJ0123.1+3421 0.272 8.68 \u00060.14\nJ0159.5+1046 0.195 8.51 \u00060.14\nJ0202.4+0849 0.629 8.85 \u00060.16\nJ0208.6+3523 0.318 8.62 \u00060.14\nJ0214.3+5145 0.049 8.55 \u00060.13\nJ0217.2+0837 0.085 8.40 \u00060.13\nJ0238.4\u00003116 0.233 8.73 \u00060.15\nJ0303.4\u00002407 0.266 8.95 \u00060.17\nJ0319.8+1845 0.190 8.66 \u00060.14\nJ0340.5\u00002118 0.233 8.29 \u00060.11\nJ0349.4\u00001159 0.188 8.60 \u00060.13\nJ0422.3+1951 0.512 8.57 \u00060.14\nJ0424.7+0036 0.268 8.76 \u00060.16\nJ0507.9+6737 0.416 9.00 \u00060.18\nJ0509.6\u00000402 0.304 8.88 \u00060.16\nJ0617.7\u00001715 0.098 8.83 \u00060.20\nJ0623.9\u00005259 0.513 8.85 \u00060.16\nJ0712.7+5033 0.502 8.49 \u00060.15\nJ0757.1+0956 0.266 8.53 \u00060.13\nJ0814.6+6430 0.239 8.23 \u00060.14\nJ1103.6\u00002329 0.186 9.01 \u00060.18\nTable 9 continuedTable 9 (continued)\n4FGL name redshift MBH\n[1] [2] [3]\nJ1136.4+7009 0.045 8.62 \u00060.14\nJ1217.9+3007 0.130 8.75 \u00060.15\nJ1257.2+3646 0.530 8.92 \u00060.19\nJ1315.0\u00004236 0.105 8.56 \u00060.13\nJ1359.8\u00003746 0.334 8.77 \u00060.16\nJ1501.0+2238 0.235 8.81 \u00060.16\nJ1517.7+6525 0.702 10.11 \u00060.31\nJ1535.0+5320 0.890 9.64 \u00060.24\nJ1548.8\u00002250 0.192 8.69 \u00060.16\nJ1643.5\u00000646 0.082 8.43 \u00060.14\nJ1728.3+5013 0.055 8.28 \u00060.11\nJ1748.6+7005 0.770 10.02 \u00060.30\nJ1757.0+7032 0.407 8.78 \u00060.18\nJ1813.5+3144 0.117 8.02 \u00060.11\nJ2005.5+7752 0.342 8.77 \u00060.16\nJ2009.4\u00004849 0.071 8.90 \u00060.17\nJ2039.5+5218 0.053 8.11 \u00060.11\nJ2042.1+2427 0.104 8.76 \u00060.16\nJ2055.4\u00000020 0.440 8.38 \u00060.15\nJ2143.1\u00003929 0.429 8.68 \u00060.15\nJ2145.7+0718 0.237 8.70 \u00060.15\nJ2252.0+4031 0.229 8.72 \u00060.15\nTable 9 continued10 P ALIYA ET AL .\nTable 9 (continued)\n4FGL name redshift MBH\n[1] [2] [3]\nJ2359.0\u00003038 0.165 8.67 \u00060.14\nNOTE—Column information are as follows:\nCol.[1]: 4FGL name; Col.[2]: redshift; and\nCol.[3]: log-scale black hole mass, in M\f.\n4.5. Literature Collection\nThere are 10 blazars in our sample whose optical spectra\ncould not be retrieved, however, their emission line pa-\nrameters or MBHvalues have been published (e.g., Bald-\nwin et al. 1981; Chen et al. 2015). We were able to col-\nlect/estimate both MBHandLdisk values for 10 such ob-\njects and report them in Table 7. These objects are: 4FGL\nJ0243.2\u00000550, 4FGL J0509.4+0542, 4FGL J0501.2 \u00000158,\n4FGL J0525.4\u00004600, 4FGL J0836.5 \u00002026, 4FGL\nJ1214.6\u00001926, 4FGL J1557.9 \u00000001, 4FGL J1626.0 \u00002950,\n4FGL J2056.2\u00004714, and 4FGL J2323.5 \u00000317.\nFinally, we tabulate references of all the research articles\nused to collect optical spectra or spectral parameters for 1077\nblazars in Appendix (Table 10).\n5.PROPERTIES OF THE CENTRAL ENGINE\nIn the left panel of Figure 7, we show the distribu-\ntion of the MBHvalues estimated from different meth-\nods described above. The average MBHfor the whole\npopulation ishlogMBH;allM\fi= 8:60. Considering\nemission and absorption line systems separately, we get\nhlogMBH;emiM\fi= 8:48andhlogMBH;absM\fi= 8:80,\nrespectively. The dispersion for all three distributions is sim-\nilar,\u00180.6, when fitted with a Gaussian function. Though the\nspread is large, there are tentative evidences hinting the as-\nsociation of more massive black holes with the absorption\nline systems, i.e., objects lacking broad emission lines or BL\nLacs. To understand it further, we compared MBHfor a sub-\nsample of blazars with 0:3\u0014z\u00140:7. This was done to\navoid any redshift dependent selection effect and to com-\npare a similar number of sources from emission (139) and\nabsorption line (110) blazar samples. In the middle panel\nof Figure 7, we show the histograms of MBHvalues derived\nfor the two populations in the opted redshift range. The av-\nerage masses for emission and absorption line blazars are\nhlogMBH;M\fi= 8:13 and 8:70, respectively. Though the\nspread is large ( \u001b\u00180.55), this finding hints that BL Lac ob-\njects do tend to host more massive black holes than broad\nline FSRQs. A Kolmogorov-Smirnov test was carried out to\ndetermine whether two populations are distinctively differ-\nent. The derived p-value is 1:9\u000210\u00008, suggesting that thenull-hypothesis of both samples belonging to the same source\npopulation can be rejected at >5\u001bconfidence level.\nThe observation of more massive black holes residing in\nBL Lac objects is likely to be connected with the cosmic\nevolution of blazars (B ¨ottcher & Dermer 2002; Cavaliere &\nD’Elia 2002; Ajello et al. 2014). According to the proposed\nevolutionary sequence, high-power ( Lbol:>1046erg s\u00001)\nblazars, primarily FSRQs, evolve to their low-power counter-\nparts, mainly BL Lacs, over cosmological timescales. Such\na transition is likely caused by a gradual depletion of the\ncentral environment by accretion onto the black hole. In\nother words, luminous broad emission line blazars evolve\nto low-luminosity objects exhibiting quasi-featureless spec-\ntra as their accretion mode changes from being radiatively ef-\nficient to advection dominated accretion flow (Narayan et al.\n1997). If so, one would expect the black holes hosted in the\nlatter to be more massive than that found in the former since\nthey would keep growing as more and more mass is dumped\nby accretion.\nWe show the variation of MBHas a function of the blazar\nredshift in the right panel of Figure 7. There is a little overlap\nsince all of the absorption line objects are located below z=\n0:85, whereas, a major fraction of the emission line sources\nare above it (see also Figure 1). There is a trend of more\nmassive black holes being located at higher redshifts among\nthe broad line sources. This is likely a selection effect since\nwe expect to detect only the most luminous systems, hence\nthe most massive black holes, at cosmological distances.\nThe left panel of Figure 8 shows the Ldiskdistribution for\nbroad line objects. For a comparison, we also plot the his-\ntogram of the 3 \u001bupper limits on the Ldisk measured for\nblazars lacking emission lines. As expected, strong emission\nline systems have more luminous accretion disks. The dif-\nference between the two populations becomes clearer when\nwe derive the Ldisk values in Eddington units. The re-\nsults are shown in the right panel of Figure 8 and reveal\na bi-modality. A major fraction of broad line objects have\nLdisk/LEdd&0:01, whereas,Ldiskof absorption line sys-\ntems have values .1% ofLEddwhich is expected to be even\nlower since the plotted quantity corresponds to 3 \u001bluminos-\nity upper limit. Interestingly, a low-level of accretion activ-\nity (Ldisk/LEdd.0:01) is noticed from a few emission line\nblazars. These are objects typically classified as BL Lacs,\nhowever, have revealed faint broad emission lines in their\noptical spectra taken during a low jet activity state (cf. Ver-\nmeulen et al. 1995).\n6.A COMPTON DOMINANCE BASED\nCLASSIFICATION\nA blazar classification based on Ldiskin units ofLEddis\nphysically intuitive (Ghisellini et al. 2011), however, extend-\ning it to lineless sources remained a challenge. If the opticalBLAZAR ’SCENTRAL ENGINE 11\n7 8 9 10\nlogMBH[M⊙]04080120160Number of sourcesemission line\nabsorption line\nall\n6 7 8 9 10\nlogMBH[M⊙, 0.3≤z≤0.7]081624Number of sourcesemission line\nabsorption line\n0.03 0.1 0.3 1 2345\nRedshift678910logMBH[M⊙]\nabsorption line\nemission line bulge luminosity\nFigure 7. Left: The histograms of the MBHfor broad emission line blazars (red filled), absorption line systems (blue dotted), and the whole\nsample (black empty) including MBHderived from the host galaxy bulge luminosity. Middle: The distributions of MBHfor blazars in the\nredshift range 0:3\u0014z\u00140:7. Right: The redshift dependence of the MBHderived from various methods as labeled.\n41 42 43 44 45 46 47 48\nlogLdisk[erg s−1]0306090120150180Number of sourcesemission line\nabsorption line (UL)\n-5 -4 -3 -2 -1 0\nlogLdisk/LEdd0306090120Number of sourcesemission line\nabsorption line (UL)\nFigure 8. Left: The luminosity of the accretion disk for emission line blazars is shown with the red filled histogram. The blue dotted histogram\nfor absorption line objects, on the other hand, refers to 3 \u001bupper limit in Ldisk. Right: Same as left but with Ldiskplotted in Eddington units.\nspectrum of a FSRQ is dominated by the non-thermal jet-\nted radiation which swamps out broad emission lines, it may\nnot be possible to characterize its state of accretion even if it\nhosts a luminous accretion disk. Therefore, it is necessary to\nidentify some other observational features which can indicate\nthe state of accretion or, in other words, show a correlation\nwithLdisk/LEdd. Below we show that Compton dominance\nor CD can be considered as one such parameter to reveal the\nphysics of the central engine in beamed AGN.\nAccording to the canonical picture of the blazar leptonic\nemission models, radio-to-optical-UV radiation is dominated\nby the synchrotron emission, though the big blue bump aris-\ning from the accretion disk has also been observed. If the\naccretion process is radiatively efficient, it will illuminate\nthe BLR and dusty torus enabling a photon-rich environ-\nment surrounding the jet. In such objects, the high-energy\nX- and\r-ray emission is mainly powered by the externalCompton mechanism (see, e.g., Sikora et al. 1994). Due\nto an additional beaming factor associated with the external\nCompton process (Dermer 1995), the bolometric jet emis-\nsion will be dominated by the high-energy \r-ray emission\nleading to the observation of a Compton dominated SED\n(e.g., Paliya 2015). On the other hand, the overall SED of\na blazar with an intrinsically low-level of accretion activity\n(Ldisk/LEdd<0:01), will be synchrotron dominated. This\nis because, the \r-ray emission in such objects originates pri-\nmarily via synchrotron self Compton mechanism (SSC; cf.\nFinke et al. 2008) since the environment surrounding the jet\nlacks the external photons needed for the external Compton\nprocess. To summarize, one can get a fair idea of the ac-\ncretion level of a blazar by estimating CD from its multi-\nfrequency SED. Having measured the Ldisk in Eddington12 P ALIYA ET AL .\n108101110141017102010231026\nν(Hz)10−1510−1410−1310−1210−11νFν(erg cm−2s−1)\n4FGL J0532.6+0732 ( z= 1.254)\n108101110141017102010231026\nν(Hz)10−1510−1410−1310−1210−11νFν(erg cm−2s−1)\n4FGL J0602.0+5315 ( z= 0.0522)\nFigure 9. The multi-frequency SEDs of two Fermi -LAT detected blazars (grey data points) when fitted with second-degree polynomials (left)\nand a synchrotron-SSC model (right). The fitted functions are shown with the blue dotted lines. In the right panel, the bump seen at IR-optical\nwavelengths arises from the host galaxy.\n0.01 0.1 1 10 100\nCompton dominance020406080100120Number of sourcesemission line\nabsorption line\nFigure 10. The Compton dominance histograms of the emission\n(red filled) and absorption line (blue dotted) blazars.\nunits for a large sample of blazars, we, therefore, next at-\ntempted determining their CD as described below.\n6.1. Compton Dominance Measurement\nWe collected broadband spectral data of blazars present in\nour sample from Space Science Data Center (SSDC) SED\nbuilder tool4. While doing so, we did not consider United\nStates Naval Observatory data and flux values reported by the\nCatalina Real-Time Transient Survey since their fluxes are\n4https://tools.ssdc.asi.it/SED/often outside the range of the other data taken at the same fre-\nquency. To improve the data coverage at X- and \r-ray bands,\nwe included the flux measurements from the 2ndSwift X-\nray Point Source catalog (2SXPS; Evans et al. 2020) and the\n4FGL-DR2 catalog (Abdollahi et al. 2020). The synchrotron\nand inverse Compton peak frequencies and corresponding\nflux values were then estimated by fitting a second-degree\npolynomial to both SED peaks using the built-in function\nprovided in the SSDC SED builder tool. In many blazars,\nthe soft X-ray spectrum is still dominated by the synchrotron\nemission and forms the tail-end of the low-energy peak. In\nsuch objects, it may not be possible to accurately constrain\nthe high-energy peak with only Fermi -LAT data. The multi-\nwavelength SEDs of such blazars, therefore, were fitted with\na synchrotron-SSC model assuming a log-parabolic electron\nenergy distribution (Tramacere et al. 2009). Furthermore,\nthe SEDs of powerful FSRQs often exhibit the accretion\ndisk bump at optical-UV energies and many high-frequency\npeaked BL Lacs show a host galaxy emission at IR-optical\nwavelengths. Such features were avoided during the fit. The\nexamples of this analysis are illustrated in Figure 9. The CD\nwas derived by taking the ratio of the high- and low-energy\npeak luminosities, which is equivalent to the ratio of their\npeak fluxes since it is essentially a redshift independent quan-\ntity. The estimated values are provided in Table 7 and 8 and\nwe plot the CD histograms for both emission and absorption\nline blazars in Figure 10. As can be seen, a bi-modality ap-\npears with the broad emission line blazars have an average\nCD>1 (hCDemii= 5:85), whereas, absorption line objects\nhave less Compton dominated SED ( hCDabsi= 0:46). This\nfinding is in line with the hypothesis of broad line blazars be-\ning more Compton dominated sources compared to lineless\nBL Lac objects as discussed above.BLAZAR ’SCENTRAL ENGINE 13\n-4 -3 -2 -1 0\nlogLdisk/LEdd0.1110100Compton dominance\nabsorption line\nemission line\nFigure 11. The Compton dominance as a function of the Ldiskin Eddington units. For the sake of clarity, uncertainties associated with the\nMBHvalues were not propagated while calculating Ldisk/LEdd.\n6.2. Compton Dominance and Ldisk/LEdd\nIn Figure 11, we show the distribution of CD as a function\nofLdiskin Eddington units. A positive correlation is appar-\nent with highly accreting objects tend to have larger CD. To\nquantify the strength of the correlation, we adopted the astro-\nnomical survival analysis package (ASURV; Lavalley et al.\n1992) which takes into account the upper/lower limits in the\ndata (Isobe et al. 1986). The derived Spearman’s correlation\ncoefficient is \u001a= 0:76with probability of no correlation or\nPNC<1\u000210\u000010indicating a positive correlation with high\n(>5\u001b) confidence. This strengthens our argument that CD\ncan be considered as a good proxy for the accretion rate in\nblazars. Furthermore, a linear regression analysis carried out\nusing ASURV software resulted in the following empirical\nrelation connecting the two variables\nlog CD = (1 :16\u00060:03)+(0:74\u00060:02) log\u0012Ldisk\nLEdd\u0013\n:(7)Figure 11 reveals a clear distinction of two groups in re-\ngions defined by CD>1,Ldisk/LEdd>0.01 for broad line\nblazars and CD<1,Ldisk/LEdd<0.01 for absorption line\nsystems. The two populations overlap smoothly for interme-\ndiate values of CD and Ldiskin Eddington units. Based on\nthis finding, we propose that blazars can be classified as high-\nCompton dominated (HCD) with CD >1 and low-Compton\ndominated (LCD) sources for CD <1. Moreover, blazars\nhaving CD>1,Ldisk/LEdd>0.01 should be identified as\nFSRQs and those with CD<1,Ldisk/LEdd<0.01 as BL\nLacs. One good example is the blazar 4FGL J0238.6+1637\nor AO 0235+164 ( z= 0:94) which is historically classified\nas a BL Lac object (Spinrad & Smith 1975). For this object,\nwe have found Ldisk/LEdd= 0.04 and CD = 2.69, thus in-\ndicating an underlying radiatively efficient accretion. There-\nfore, this source can be identified as a FSRQ and as a HCD\nblazar (see also Ghisellini et al. 2011).\nThe above mentioned classification scheme is analogous\nto that based on LBLR/LEddproposed by Ghisellini et al.14 P ALIYA ET AL .\n(2011). However, it overcomes the limitation imposed by\nthe difficulty in measuring LBLR andMBHfor objects with\nquasi-featureless spectra. Another advantage of a CD based\nclassification is that one can easily get an idea about the\nintrinsic physical nature of blazars just by measuring the\nrelative dominance of the high-energy peak in their multi-\nfrequency SEDs. The proposed scheme is useful keep-\ning in mind the ongoing and next-generation surveys, e.g.,\nVery Large Array Sky Survey, Dark Energy Survey, e-\nROSITA , All-sky MeV Energy Gamma-ray Observatory, and\nCherenkov Telescope Array, which will provide unprece-\ndented broadband coverage of the blazar SEDs.\nBased on Figure 11, we argue that the region of CD.1,\nLdisk/LEdd>0.01 could be populated with strong line AGN\nhaving misaligned jets, e.g., broad line radio galaxies. This\nis due to the fact that the external Compton mechanism pro-\nducing X- to \r-ray emission is highly sensitive to the Doppler\nboosting. Therefore, as the viewing angle increases, the de-\ncrease in the high-energy peak flux is much more rapid than\nthe synchrotron peak flux, thereby effectively reducing CD.\nHowever, since the BLR emission is likely to be isotropic,\ntheLdiskof the source does not change. At the same time,\nthe region of CD&1,Ldisk/LEdd<0.01 may remain largely\nforbidden due to low-level of accretion and hence a jet envi-\nronment starved of the seed photons needed for the external\nCompton process.\nRecently, a minority of sources, so-called masquerading\nBL Lacs, have been identified challenging our current under-\nstanding of the FSRQ/BL Lac division. It has been proposed\nthat these objects are intrinsically FSRQs, i.e., radiatively ef-\nficient accreting systems. However, since their optical spec-\ntra lack emission lines due to strongly amplified jetted radi-\nation contamination, they are classified as BL Lac sources\n(Giommi et al. 2013; Padovani et al. 2019). Furthermore,\nmany of such objects exhibit high-frequency peaked SEDs\n(Padovani et al. 2012). This was explained by arguing the \r-\nray emitting region to be located outside BLR where a rela-\ntively weak cooling environment allows jet electrons to attain\nvery high-energies (Ghisellini et al. 2012). In the context of\nour work, physics of such blue FSRQs can be explained as\nfollows: the Doppler boosting effect that swamps out broad\nemission lines by enhancing synchrotron radiation should\namplify the inverse Compton emission by an even larger fac-\ntor if the primary mechanism to produce \r-rays is external\nCompton. If so, the ratio of the SED peak luminosities, i.e.,\nCD, cannot be significantly lower than one (see, e.g., Ghis-\nellini et al. 2012). This supports our claim that radiatively ef-\nficient accreting systems should have a Compton dominated\nSED. Based on Figure 11, it can be understood that sources\nlying around CD\u00191 andLdisk/LEdd\u00190.01 may belong to\nthe family of masquerading BL Lacs.\n12 13 14 15 16 17 18\nlogνpeak\nsyn[Hz]−5−4−3−2−10logLdisk/LEdd\nabsorption line\nemission lineFigure 12. This diagram illustrates the variation of the Ldisk in\nEddington units as a function of the synchrotron peak frequency for\nFermi blazars studied in this work.\nWe stress that our physical interpretation of the obtained\nCD andLdisk/LEdd correlation may not be extendable to\nblazar flares which can be extremely complex and diverse\n(e.g., Rajput et al. 2020). This is because we used multi-\nwavelength SEDs generated using all archival observations\nto compute CD hence reflect the average activity state of\nsources. Similarly, optical spectra too were mostly taken dur-\ning average/low activity state of sources and may not be dur-\ning any specific flaring episodes. Therefore, the derived cor-\nrelation of CD and Ldisk/LEddrefers to the average physical\nproperties of the blazar population. Extending this work to\nconsider blazar flares is beyond the scope of this paper since\neven one source behaves differently during different flares\n(e.g., Chatterjee et al. 2013; Paliya et al. 2015, 2016).\n7.A SEQUENCE OF Ldisk/LEdd\nThe observed anti-correlation between the synchrotron\npeak luminosity and corresponding peak frequency, i.e.,\nblazar sequence, has remained one of the crucial topics of\nresearch in jet physics. This phenomenon has been ar-\ngued to have a physical origin (e.g., Ghisellini et al. 1998)\nand also criticized due to possible selection effects of miss-\ning high-frequency peaked, luminous blazars (cf. Padovani\n2007). A few other works have proposed the Doppler boost-\ning/viewing angle to be the driving factor of the observed\nsequence (e.g., Nieppola et al. 2008; Meyer et al. 2011; Fan\net al. 2017; Keenan et al. 2020). Recently, the identification\nof the firstz > 3BL Lac object (Paliya et al. 2020c) has\nalso suggested that a population of luminous, high-frequency\npeaked blazars may exist which is against the prediction of\nthe blazar sequence (Ghisellini et al. 2017).\nIn Figure 12, we show the variation of the MBHnormal-\nizedLdiskas a function of the synchrotron peak frequency for\nobjects studied in this work. Note that Ldiskvalues are de-BLAZAR ’SCENTRAL ENGINE 15\nrived from the optical spectroscopic emission line fluxes or\nupper limits and therefore are free from the Doppler boost-\ning effects. In other words, Figure 12 provides a glimpse of\nthe intrinsic physical behavior of beamed AGN population.\nA strong anti-correlation of the plotted quantities is evident\nwhich is confirmed from the Spearmann’s test done using\nASURV giving \u001a=\u00000:75with PNC<1\u000210\u000010. This\nanti-correlation can be understood as follows.\nPopulation studies focused on the average activity state\nof blazars have revealed the blazar-zone to lie at a distance\nof a few hundreds/thousands of Schwarzschild radii from\nthe central black hole (e.g., Paliya et al. 2017a). Consid-\nering the BLR/torus radius- Ldiskrelationship (e.g., Tavec-\nchio & Ghisellini 2008), strong line blazars hosting a lu-\nminous accretion disk have a large BLR/torus and there-\nfore the emission region is typically located within it. The\nrelativistic electrons present in the jet would loose energy\nmainly by interacting with ambient photons before they could\nreach to high-energies, leading to the observation of a low-\nfrequency peaked SED. In radiatively inefficient systems, on\nthe other hand, Ldiskhas a low-value implying a relatively\nsmall BLR/torus radius and hence the emission region would\nbe located farther out from it. In the absence of a strong pho-\nton field, jet electrons can attain high-energies making the\nobserved SED to be high-frequency peaked. Moreover, Fig-\nure 12 also suggests that the shift of the SED peaks to higher\nfrequencies occurs with the depletion of the central environ-\nment by accretion, i.e., with the evolution of HCD blazars to\ntheir LCD counterparts (B ¨ottcher & Dermer 2002; Cavaliere\n& D’Elia 2002).\nFinally, since Ldisk/LEdd positively correlates with CD\n(Figure 11), one would also expect an anti-correlation of the\nlatter with the synchrotron peak frequency similar to Fig-\nure 12. Such an anti-correlation has already been reported\nby Finke (2013).\n8.SUMMARY\nIn this work, we have presented a catalog of the central\nengine properties, i.e., MBHandLdisk, for a sample of 1077\n\r-ray detected blazars. We summarize our results below.\n1. The average MBHfor the whole blazar population is\nhlogMBH;allM\fi= 8:60. There are evidences indi-cating black holes residing in absorption line systems\nto be more massive compared to emission line blazars.\n2. The distribution of Ldisk/LEdd reveals a clear bi-\nmodality with emission line sources tend to have a\nlarger accretion rate ( Ldisk/LEdd>0.01) compared to\nblazars whose optical spectra are dominated by the ab-\nsorption lines arising from the host galaxy. Similar re-\nsults were obtained for CD where emission line blazars\nexhibit a more Compton dominated SED than absorp-\ntion line sources.\n3. There is strong positive correlation between\nLdisk/LEdd and CD, suggesting that the latter can\nbe used to determine the state of accretion activity in\nbeamed AGNs.\n4. Based on their position in Ldisk/LEdd-CD plane,\nblazars can be classified as High-Compton Domi-\nnated (HCD, CD >1) and Low-Compton Dominated\n(LCD, CD<1) objects. Our results suggests that\nthis scheme is analogous to that predicted based on\nLBLR/LEdd, however, can be extended to objects\nlacking high-quality optical spectroscopic measure-\nments. We also propose that blazars having CD>1,\nLdisk/LEdd>0.01 should be identified as FSRQs and\nCD<1,Ldisk/LEdd<0.01 as BL Lacs.\n5.Fermi blazars show a significant anti-correlation be-\ntweenLdisk/LEdd and synchrotron peak frequency.\nBeing free from Doppler boosting related effects, the\nobserved trend has a physical origin.\n6. The overall findings reported in this work directly\npoints to a scenario in which the average physical prop-\nerties of blazars are mainly controlled by the accretion\nrate in Eddington units.\nWe make the catalog publicly available online at Zenodo\n(doi: 10.5281/zenodo.4456365)5.\nSoftware: ASURV (Lavalley et al. 1992), Astropy (As-\ntropy Collaboration et al. 2013, 2018), pPXF (Cappellari &\nEmsellem 2004), PyQSOFit (Guo et al. 2018).\nREFERENCES\nAbdollahi, S., Acero, F., Ackermann, M., et al. 2020, ApJS, 247,\n33, doi: 10.3847/1538-4365/ab6bcb\n5The fits file is also available at http://www.ucm.es/blazars/enginesAcosta-Pulido, J. A., Agudo, I., Barrena, R., et al. 2010, A&A,\n519, A5, doi: 10.1051/0004-6361/200913953\nAfanas’Ev, V . L., Dodonov, S. N., Moiseev, A. V ., et al. 2005,\nAstronomy Reports, 49, 374, doi: 10.1134/1.1923547\n—. 2006, Astronomy Reports, 50, 255,\ndoi: 10.1134/S106377290604001916 P ALIYA ET AL .\nAhumada, R., Allende Prieto, C., Almeida, A., et al. 2020, ApJS,\n249, 3, doi: 10.3847/1538-4365/ab929e\nAjello, M., Romani, R. W., Gasparrini, D., et al. 2014, ApJ, 780,\n73, doi: 10.1088/0004-637X/780/1/73\nAjello, M., Ghisellini, G., Paliya, V . S., et al. 2016, ApJ, 826, 76,\ndoi: 10.3847/0004-637X/826/1/76\nAjello, M., Angioni, R., Axelsson, M., et al. 2020, ApJ, 892, 105,\ndoi: 10.3847/1538-4357/ab791e\nAliu, E., Aune, T., Beilicke, M., et al. 2011, ApJ, 742, 127,\ndoi: 10.1088/0004-637X/742/2/127\n´Alvarez Crespo, N., Massaro, F., Milisavljevic, D., et al. 2016a, AJ,\n151, 95, doi: 10.3847/0004-6256/151/4/95\n´Alvarez Crespo, N., Masetti, N., Ricci, F., et al. 2016b, AJ, 151,\n32, doi: 10.3847/0004-6256/151/2/32\nAstropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013,\nA&A, 558, A33, doi: 10.1051/0004-6361/201322068\nAstropy Collaboration, Price-Whelan, A. M., Sip ˝ocz, B. M., et al.\n2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f\nBade, N., Beckmann, V ., Douglas, N. G., et al. 1998, A&A, 334,\n459. https://arxiv.org/abs/astro-ph/9803204\nBade, N., Fink, H. H., Engels, D., et al. 1995, A&AS, 110, 469\nBaker, J. C., Hunstead, R. W., Kapahi, V . K., & Subrahmanya,\nC. R. 1999, ApJS, 122, 29, doi: 10.1086/313209\nBaldwin, J. A., Wampler, E. J., & Burbidge, E. M. 1981, ApJ, 243,\n76, doi: 10.1086/158568\nBaldwin, J. A., Wampler, E. J., & Gaskell, C. M. 1989, ApJ, 338,\n630, doi: 10.1086/167224\nBecerra Gonz ´alez, J., Acosta-Pulido, J. A., & Clavero, R. 2020,\nMNRAS, 494, 6036, doi: 10.1093/mnras/staa1144\nBechtold, J., Dobrzycki, A., Wilden, B., et al. 2002, ApJS, 140,\n143, doi: 10.1086/342489\nBoisse, P., & Bergeron, J. 1988, A&A, 192, 1\nBoroson, T. A., & Green, R. F. 1992, ApJS, 80, 109,\ndoi: 10.1086/191661\nB¨ottcher, M., & Dermer, C. D. 2002, ApJ, 564, 86,\ndoi: 10.1086/324134\nBruni, G., Panessa, F., Ghisellini, G., et al. 2018, ApJL, 854, L23,\ndoi: 10.3847/2041-8213/aaacfb\nButtiglione, S., Capetti, A., Celotti, A., et al. 2009, A&A, 495,\n1033, doi: 10.1051/0004-6361:200811102\nCappellari, M., & Emsellem, E. 2004, PASP, 116, 138,\ndoi: 10.1086/381875\nCarangelo, N., Falomo, R., Kotilainen, J., Treves, A., & Ulrich,\nM. H. 2003, A&A, 412, 651, doi: 10.1051/0004-6361:20031519\nCardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245,\ndoi: 10.1086/167900\nCavaliere, A., & D’Elia, V . 2002, ApJ, 571, 226,\ndoi: 10.1086/339778\nCelotti, A., Padovani, P., & Ghisellini, G. 1997, MNRAS, 286, 415Chai, B., Cao, X., & Gu, M. 2012, ApJ, 759, 114,\ndoi: 10.1088/0004-637X/759/2/114\nChatterjee, R., Fossati, G., Urry, C. M., et al. 2013, ApJL, 763,\nL11, doi: 10.1088/2041-8205/763/1/L11\nChen, S., Berton, M., La Mura, G., et al. 2018, A&A, 615, A167,\ndoi: 10.1051/0004-6361/201832678\nChen, Y . Y ., Zhang, X., Zhang, H. J., & Yu, X. L. 2015, MNRAS,\n451, 4193, doi: 10.1093/mnras/stv658\nChomiuk, L., Strader, J., Landt, H., & Cheung, C. C. 2013, The\nAstronomer’s Telegram, 4777, 1\nChu, Y .-q., Zhu, X.-f., & Butcher, H. 1986, ChA&A, 10, 196,\ndoi: 10.1016/0275-1062(86)90005-6\nColless, M., Dalton, G., Maddox, S., et al. 2001, MNRAS, 328,\n1039, doi: 10.1046/j.1365-8711.2001.04902.x\nde Menezes, R., Amaya-Almaz ´an, R. A., Marchesini, E. J., et al.\n2020, Ap&SS, 365, 12, doi: 10.1007/s10509-020-3727-5\nDermer, C. D. 1995, ApJL, 446, L63, doi: 10.1086/187931\nDermer, C. D., Finke, J. D., Krug, H., & B ¨ottcher, M. 2009, ApJ,\n692, 32, doi: 10.1088/0004-637X/692/1/32\nDesai, A., Marchesi, S., Rajagopal, M., & Ajello, M. 2019, ApJS,\n241, 5, doi: 10.3847/1538-4365/ab01fc\ndi Serego Alighieri, S., Danziger, I. J., Morganti, R., & Tadhunter,\nC. N. 1994, MNRAS, 269, 998, doi: 10.1093/mnras/269.4.998\nDrinkwater, M. J., Webster, R. L., Francis, P. J., et al. 1997,\nMNRAS, 284, 85\nDunlop, J. S., Peacock, J. A., Savage, A., et al. 1989, MNRAS,\n238, 1171, doi: 10.1093/mnras/238.4.1171\nEvans, I. N., & Koratkar, A. P. 2004, ApJS, 150, 73,\ndoi: 10.1086/379649\nEvans, P. A., Page, K. L., Osborne, J. P., et al. 2020, ApJS, 247, 54,\ndoi: 10.3847/1538-4365/ab7db9\nFalomo, R., Kotilainen, J., Pursimo, T., et al. 1997, A&A, 321, 374\nFalomo, R., Scarpa, R., Treves, A., & Urry, C. M. 2000, ApJ, 542,\n731, doi: 10.1086/317044\nFalomo, R., Treves, A., Scarpa, R., Paiano, S., & Land oni, M.\n2017, MNRAS, 470, 2814, doi: 10.1093/mnras/stx1411\nFan, J. H., Yang, J. H., Xiao, H. B., et al. 2017, ApJL, 835, L38,\ndoi: 10.3847/2041-8213/835/2/L38\nFerrarese, L., & Merritt, D. 2000, ApJL, 539, L9,\ndoi: 10.1086/312838\nFinke, J. D. 2013, ApJ, 763, 134,\ndoi: 10.1088/0004-637X/763/2/134\nFinke, J. D., Dermer, C. D., & B ¨ottcher, M. 2008, ApJ, 686, 181,\ndoi: 10.1086/590900\nFoschini, L., Berton, M., Caccianiga, A., et al. 2015, A&A, 575,\nA13, doi: 10.1051/0004-6361/201424972\nFrancis, P. J., Hewett, P. C., Foltz, C. B., et al. 1991, ApJ, 373, 465,\ndoi: 10.1086/170066\nFricke, K. J., Kollatschny, W., & Witzel, A. 1983, A&A, 117, 60BLAZAR ’SCENTRAL ENGINE 17\nGhisellini, G., Celotti, A., Fossati, G., Maraschi, L., & Comastri,\nA. 1998, MNRAS, 301, 451,\ndoi: 10.1046/j.1365-8711.1998.02032.x\nGhisellini, G., Righi, C., Costamante, L., & Tavecchio, F. 2017,\nMNRAS, 469, 255, doi: 10.1093/mnras/stx806\nGhisellini, G., Tavecchio, F., Foschini, L., & Ghirlanda, G. 2011,\nMNRAS, 414, 2674, doi: 10.1111/j.1365-2966.2011.18578.x\nGhisellini, G., Tavecchio, F., Foschini, L., et al. 2012, MNRAS,\n425, 1371, doi: 10.1111/j.1365-2966.2012.21554.x\nGhisellini, G., Tavecchio, F., Maraschi, L., Celotti, A., & Sbarrato,\nT. 2014, Nature, 515, 376, doi: 10.1038/nature13856\nGioia, I. M., Wolter, A., Mullis, C. R., et al. 2004, A&A, 428, 867,\ndoi: 10.1051/0004-6361:20041426\nGiommi, P., Padovani, P., & Polenta, G. 2013, MNRAS, 431, 1914,\ndoi: 10.1093/mnras/stt305\nGorham, P. W., van Zee, L., Unwin, S. C., & Jacobs, C. 2000, AJ,\n119, 1677, doi: 10.1086/301289\nGraham, A. W. 2007, MNRAS, 379, 711,\ndoi: 10.1111/j.1365-2966.2007.11950.x\nG¨ultekin, K., Richstone, D. O., Gebhardt, K., et al. 2009, ApJ, 698,\n198, doi: 10.1088/0004-637X/698/1/198\nGuo, H., Liu, X., Shen, Y ., et al. 2019, MNRAS, 482, 3288,\ndoi: 10.1093/mnras/sty2920\nGuo, H., Shen, Y ., & Wang, S. 2018, PyQSOFit: Python code to fit\nthe spectrum of quasars. http://ascl.net/1809.008\nHalpern, J. P., Chen, V . S., Madejski, G. M., & Chanan, G. A.\n1991, AJ, 101, 818, doi: 10.1086/115725\nHalpern, J. P., Eracleous, M., & Forster, K. 1997, AJ, 114, 1736,\ndoi: 10.1086/118602\nHalpern, J. P., Eracleous, M., & Mattox, J. R. 2003, AJ, 125, 572,\ndoi: 10.1086/345796\nHao, L., Strauss, M. A., Tremonti, C. A., et al. 2005, AJ, 129,\n1783, doi: 10.1086/428485\nHealey, S. E., Romani, R. W., Cotter, G., et al. 2008, ApJS, 175,\n97, doi: 10.1086/523302\nHenstock, D. R., Browne, I. W. A., Wilkinson, P. N., & McMahon,\nR. G. 1997, MNRAS, 290, 380, doi: 10.1093/mnras/290.2.380\nIsobe, T., Feigelson, E. D., & Nelson, P. I. 1986, ApJ, 306, 490,\ndoi: 10.1086/164359\nJones, D. H., Read, M. A., Saunders, W., et al. 2009, MNRAS,\n399, 683, doi: 10.1111/j.1365-2966.2009.15338.x\nJunkkarinen, V . 1984, PASP, 96, 539, doi: 10.1086/131374\nKeenan, M., Meyer, E. T., Georganopoulos, M., Reddy, K., &\nFrench, O. J. 2020, arXiv e-prints, arXiv:2007.12661.\nhttps://arxiv.org/abs/2007.12661\nKlindt, L., van Soelen, B., Meintjes, P. J., & V ¨ais¨anen, P. 2017,\nMNRAS, 467, 2537, doi: 10.1093/mnras/stx218\nKormendy, J., & Ho, L. C. 2013, ARA&A, 51, 511,\ndoi: 10.1146/annurev-astro-082708-101811Koss, M., Trakhtenbrot, B., Ricci, C., et al. 2017, ApJ, 850, 74,\ndoi: 10.3847/1538-4357/aa8ec9\nKotilainen, J. K., Hyv ¨onen, T., & Falomo, R. 2005, A&A, 440,\n831, doi: 10.1051/0004-6361:20042548\nLandoni, M., Falomo, R., Treves, A., et al. 2013, AJ, 145, 114,\ndoi: 10.1088/0004-6256/145/4/114\nLandoni, M., Paiano, S., Falomo, R., Scarpa, R., & Treves, A.\n2018, ApJ, 861, 130, doi: 10.3847/1538-4357/aac77c\nLandt, H., Padovani, P., Perlman, E. S., et al. 2001, MNRAS, 323,\n757, doi: 10.1046/j.1365-8711.2001.04269.x\nLaurent-Muehleisen, S. A., Kollgaard, R. I., Ciardullo, R., et al.\n1998, ApJS, 118, 127, doi: 10.1086/313134\nLavalley, M. P., Isobe, T., & Feigelson, E. D. 1992, in Bulletin of\nthe American Astronomical Society, V ol. 24, Bulletin of the\nAmerican Astronomical Society, 839–840\nLawrence, C. R., Zucker, J. R., Readhead, A. C. S., et al. 1996,\nApJS, 107, 541, doi: 10.1086/192375\nLondish, D., Croom, S. M., Heidt, J., et al. 2007, MNRAS, 374,\n556, doi: 10.1111/j.1365-2966.2006.11165.x\nMarcha, M. J. M., Browne, I. W. A., Impey, C. D., & Smith, P. S.\n1996, MNRAS, 281, 425, doi: 10.1093/mnras/281.2.425\nMarchesini, E. J., Andruchow, I., Cellone, S. A., et al. 2016, A&A,\n591, A21, doi: 10.1051/0004-6361/201527632\nMarchesini, E. J., Pe ˜na-Herazo, H. A., ´Alvarez Crespo, N., et al.\n2019, Ap&SS, 364, 5, doi: 10.1007/s10509-018-3490-z\nMarziani, P., Sulentic, J. W., Zamanov, R., et al. 2003, ApJS, 145,\n199, doi: 10.1086/346025\nMasetti, N., Mason, E., Landi, R., et al. 2008, A&A, 480, 715,\ndoi: 10.1051/0004-6361:20078901\nMassaro, F., Landoni, M., D’Abrusco, R., et al. 2015, A&A, 575,\nA124, doi: 10.1051/0004-6361/201425119\nMcLure, R. J., & Dunlop, J. S. 2004, MNRAS, 352, 1390,\ndoi: 10.1111/j.1365-2966.2004.08034.x\nMerritt, D. 1997, AJ, 114, 228, doi: 10.1086/118467\nMeyer, E. T., Fossati, G., Georganopoulos, M., & Lister, M. L.\n2011, ApJ, 740, 98, doi: 10.1088/0004-637X/740/2/98\nMezcua, M., Chavushyan, V . H., Lobanov, A. P., & Le ´on-Tavares,\nJ. 2012, A&A, 544, A36, doi: 10.1051/0004-6361/201117724\nMorton, D. C., Savage, A., & Bolton, J. G. 1978, MNRAS, 185,\n735, doi: 10.1093/mnras/185.4.735\nMurdoch, H. S., Hunstead, R. W., & White, G. L. 1984,\nProceedings of the Astronomical Society of Australia, 5, 341,\ndoi: 10.1017/S1323358000017173\nNarayan, R., Garcia, M. R., & McClintock, J. E. 1997, ApJL, 478,\nL79, doi: 10.1086/310554\nNieppola, E., Valtaoja, E., Tornikoski, M., Hovatta, T., & Kotiranta,\nM. 2008, A&A, 488, 867, doi: 10.1051/0004-6361:200809716\nNilsson, K., Pursimo, T., Heidt, J., et al. 2003, A&A, 400, 95,\ndoi: 10.1051/0004-6361:2002186118 P ALIYA ET AL .\nNkundabakura, P., & Meintjes, P. J. 2012, MNRAS, 427, 859,\ndoi: 10.1111/j.1365-2966.2012.21953.x\nOlgu ´ın-Iglesias, A., Le ´on-Tavares, J., Kotilainen, J. K., et al. 2016,\nMNRAS, 460, 3202, doi: 10.1093/mnras/stw1208\nPadovani, P. 2007, Ap&SS, 309, 63,\ndoi: 10.1007/s10509-007-9455-2\nPadovani, P., Giommi, P., & Rau, A. 2012, MNRAS, 422, L48,\ndoi: 10.1111/j.1745-3933.2012.01234.x\nPadovani, P., Oikonomou, F., Petropoulou, M., Giommi, P., &\nResconi, E. 2019, MNRAS, 484, L104,\ndoi: 10.1093/mnrasl/slz011\nPaiano, S., Falomo, R., Franceschini, A., Treves, A., & Scarpa, R.\n2017a, ApJ, 851, 135, doi: 10.3847/1538-4357/aa9af4\nPaiano, S., Falomo, R., Treves, A., Franceschini, A., & Scarpa, R.\n2019, ApJ, 871, 162, doi: 10.3847/1538-4357/aaf6e4\nPaiano, S., Falomo, R., Treves, A., & Scarpa, R. 2020, MNRAS,\n497, 94, doi: 10.1093/mnras/staa1840\nPaiano, S., Landoni, M., Falomo, R., Treves, A., & Scarpa, R.\n2017b, ApJ, 844, 120, doi: 10.3847/1538-4357/aa7aac\nPaiano, S., Landoni, M., Falomo, R., et al. 2017c, ApJ, 837, 144,\ndoi: 10.3847/1538-4357/837/2/144\nPaliya, V . S. 2015, ApJ, 804, 74, doi: 10.1088/0004-637X/804/1/74\nPaliya, V . S., Ajello, M., Cao, H. M., et al. 2020a, ApJ, 897, 177,\ndoi: 10.3847/1538-4357/ab9c1a\nPaliya, V . S., B ¨ottcher, M., Olmo-Garc ´ıa, A., et al. 2020b, ApJ,\n902, 29, doi: 10.3847/1538-4357/abb46e\nPaliya, V . S., Diltz, C., B ¨ottcher, M., Stalin, C. S., & Buckley, D.\n2016, ApJ, 817, 61, doi: 10.3847/0004-637X/817/1/61\nPaliya, V . S., Marcotulli, L., Ajello, M., et al. 2017a, ApJ, 851, 33,\ndoi: 10.3847/1538-4357/aa98e1\nPaliya, V . S., Sahayanathan, S., & Stalin, C. S. 2015, ApJ, 803, 15,\ndoi: 10.1088/0004-637X/803/1/15\nPaliya, V . S., Stalin, C. S., Ajello, M., & Kaur, A. 2017b, ApJ, 844,\n32, doi: 10.3847/1538-4357/aa77f5\nPaliya, V . S., Koss, M., Trakhtenbrot, B., et al. 2019, ApJ, 881,\n154, doi: 10.3847/1538-4357/ab2f8b\nPaliya, V . S., Dom ´ınguez, A., Cabello, C., et al. 2020c, ApJL, 903,\nL8, doi: 10.3847/2041-8213/abbc06\nPe˜na-Herazo, H. A., Marchesini, E. J., ´Alvarez Crespo, N., et al.\n2017, Ap&SS, 362, 228, doi: 10.1007/s10509-017-3208-7\nPe˜na-Herazo, H. A., Massaro, F., Chavushyan, V ., et al. 2019,\nAp&SS, 364, 85, doi: 10.1007/s10509-019-3574-4\nPe˜na-Herazo, H. A., Amaya-Almaz ´an, R. A., Massaro, F., et al.\n2020, A&A, 643, A103, doi: 10.1051/0004-6361/202037978\nPerlman, E. S., Padovani, P., Giommi, P., et al. 1998, AJ, 115,\n1253, doi: 10.1086/300283\nPerlman, E. S., Stocke, J. T., Schachter, J. F., et al. 1996, ApJS,\n104, 251, doi: 10.1086/192300\nPietsch, W., Bischoff, K., Boller, T., et al. 1998, A&A, 333, 48.\nhttps://arxiv.org/abs/astro-ph/9801210Piranomonte, S., Perri, M., Giommi, P., Landt, H., & Padovani, P.\n2007, A&A, 470, 787, doi: 10.1051/0004-6361:20077086\nPita, S., Goldoni, P., Boisson, C., et al. 2014, A&A, 565, A12,\ndoi: 10.1051/0004-6361/201323071\nPlotkin, R. M., Markoff, S., Trager, S. C., & Anderson, S. F. 2011,\nMNRAS, 413, 805, doi: 10.1111/j.1365-2966.2010.18172.x\nPunsly, B., & Kharb, P. 2017, MNRAS, 468, L72,\ndoi: 10.1093/mnrasl/slx024\nPursimo, T., Ojha, R., Jauncey, D. L., et al. 2013, ApJ, 767, 14,\ndoi: 10.1088/0004-637X/767/1/14\nRaiteri, C. M., Villata, M., Capetti, A., et al. 2007, A&A, 464, 871,\ndoi: 10.1051/0004-6361:20066599\nRajput, B., Stalin, C. S., & Sahayanathan, S. 2020, MNRAS, 498,\n5128, doi: 10.1093/mnras/staa2708\nRector, T. A., Stocke, J. T., Perlman, E. S., Morris, S. L., & Gioia,\nI. M. 2000, AJ, 120, 1626, doi: 10.1086/301587\nRicci, F., Massaro, F., Landoni, M., et al. 2015, AJ, 149, 160,\ndoi: 10.1088/0004-6256/149/5/160\nRichards, G. T., Myers, A. D., Gray, A. G., et al. 2009, ApJS, 180,\n67, doi: 10.1088/0067-0049/180/1/67\nRohatgi, A. 2020, Webplotdigitizer: Version 4.3.\nhttps://automeris.io/WebPlotDigitizer\nS´anchez-Bl ´azquez, P., Peletier, R. F., Jim ´enez-Vicente, J., et al.\n2006, MNRAS, 371, 703,\ndoi: 10.1111/j.1365-2966.2006.10699.x\nSandrinelli, A., Treves, A., Falomo, R., et al. 2013, AJ, 146, 163,\ndoi: 10.1088/0004-6256/146/6/163\nSbarrato, T., Padovani, P., & Ghisellini, G. 2014, MNRAS, 445,\n81, doi: 10.1093/mnras/stu1759\nSbarufatti, B., Ciprini, S., Kotilainen, J., et al. 2009, AJ, 137, 337,\ndoi: 10.1088/0004-6256/137/1/337\nSbarufatti, B., Falomo, R., Treves, A., & Kotilainen, J. 2006a,\nA&A, 457, 35, doi: 10.1051/0004-6361:20065455\nSbarufatti, B., Treves, A., Falomo, R., et al. 2005, AJ, 129, 559,\ndoi: 10.1086/427138\n—. 2006b, AJ, 132, 1, doi: 10.1086/503031\nSchlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500,\n525, doi: 10.1086/305772\nShaw, M. S., Filippenko, A. V ., Romani, R. W., Cenko, S. B., & Li,\nW. 2013a, AJ, 146, 127, doi: 10.1088/0004-6256/146/5/127\nShaw, M. S., Romani, R. W., Cotter, G., et al. 2012, ApJ, 748, 49,\ndoi: 10.1088/0004-637X/748/1/49\n—. 2013b, ApJ, 764, 135, doi: 10.1088/0004-637X/764/2/135\nShen, Y ., Richards, G. T., Strauss, M. A., et al. 2011, ApJS, 194,\n45, doi: 10.1088/0067-0049/194/2/45\nShen, Y ., Hall, P. B., Horne, K., et al. 2019, ApJS, 241, 34,\ndoi: 10.3847/1538-4365/ab074f\nSikora, M., Begelman, M. C., & Rees, M. J. 1994, ApJ, 421, 153,\ndoi: 10.1086/173633BLAZAR ’SCENTRAL ENGINE 19\nSmith, P. S., Montiel, E., Rightley, S., et al. 2009,\narXiv:0912.3621. https://arxiv.org/abs/0912.3621\nSowards-Emmerd, D., Romani, R. W., & Michelson, P. F. 2003,\nApJ, 590, 109, doi: 10.1086/374981\nSowards-Emmerd, D., Romani, R. W., Michelson, P. F., &\nUlvestad, J. S. 2004, ApJ, 609, 564, doi: 10.1086/421239\nSpinrad, H., & Smith, H. E. 1975, ApJ, 201, 275,\ndoi: 10.1086/153883\nSteidel, C. C., & Sargent, W. L. W. 1991, ApJ, 382, 433,\ndoi: 10.1086/170732\nStickel, M., Fried, J. W., & Kuehr, H. 1988, A&A, 191, L16\n—. 1989, A&AS, 80, 103\n—. 1993a, A&AS, 98, 393\nStickel, M., & Kuehr, H. 1993, A&AS, 100, 395\n—. 1994, A&AS, 105, 67\nStickel, M., Kuehr, H., & Fried, J. W. 1993b, A&AS, 97, 483\nStickel, M., & Kuhr, H. 1993, A&AS, 101, 521\nStickel, M., Padovani, P., Urry, C. M., Fried, J. W., & Kuehr, H.\n1991, ApJ, 374, 431, doi: 10.1086/170133\nTavecchio, F., & Ghisellini, G. 2008, MNRAS, 386, 945,\ndoi: 10.1111/j.1365-2966.2008.13072.x\nThompson, D. J., Djorgovski, S., & de Carvalho, R. 1990, PASP,\n102, 1235, doi: 10.1086/132758\nTitov, O., Jauncey, D. L., Johnston, H. M., Hunstead, R. W., &\nChristensen, L. 2011, AJ, 142, 165,\ndoi: 10.1088/0004-6256/142/5/165\nTitov, O., Pursimo, T., Johnston, H. M., et al. 2017, AJ, 153, 157,\ndoi: 10.3847/1538-3881/aa61fd\nTitov, O., Stanford, L. M., Johnston, H. M., et al. 2013, AJ, 146,\n10, doi: 10.1088/0004-6256/146/1/10\nTorrealba, J., Chavushyan, V ., Cruz-Gonz ´alez, I., et al. 2012,\nRMxAA, 48, 9. https://arxiv.org/abs/1107.3416Tramacere, A., Giommi, P., Perri, M., Verrecchia, F., & Tosti, G.\n2009, A&A, 501, 879, doi: 10.1051/0004-6361/200810865\nUrry, C. M., Scarpa, R., O’Dowd, M., et al. 2000, ApJ, 532, 816,\ndoi: 10.1086/308616\nvan den Berg, J. P., B ¨ottcher, M., Dom ´ınguez, A., & L ´opez-Moya,\nM. 2019, ApJ, 874, 47, doi: 10.3847/1538-4357/aafdfd\nVandenbroucke, J., Buehler, R., Ajello, M., et al. 2010, ApJL, 718,\nL166, doi: 10.1088/2041-8205/718/2/L166\nVazdekis, A., S ´anchez-Bl ´azquez, P., Falc ´on-Barroso, J., et al. 2010,\nMNRAS, 404, 1639, doi: 10.1111/j.1365-2966.2010.16407.x\nVermeulen, R. C., Ogle, P. M., Tran, H. D., et al. 1995, ApJL, 452,\nL5, doi: 10.1086/309716\nVermeulen, R. C., Taylor, G. B., Readhead, A. C. S., & Browne,\nI. W. A. 1996, AJ, 111, 1013, doi: 10.1086/117847\nVestergaard, M., & Peterson, B. M. 2006, ApJ, 641, 689,\ndoi: 10.1086/500572\nVestergaard, M., & Wilkes, B. J. 2001, ApJS, 134, 1,\ndoi: 10.1086/320357\nWallace, P. M., Halpern, J. P., Magalh ˜aes, A. M., & Thompson,\nD. J. 2002, ApJ, 569, 36, doi: 10.1086/339286\nWhite, R. L., Becker, R. H., Gregg, M. D., et al. 2000, ApJS, 126,\n133, doi: 10.1086/313300\nWilkes, B. J., Wright, A. E., Jauncey, D. L., & Peterson, B. A.\n1983, Proceedings of the Astronomical Society of Australia, 5, 2\nWolter, A., Ruscica, C., & Caccianiga, A. 1998, MNRAS, 299,\n1047, doi: 10.1046/j.1365-8711.1998.01834.x\nYip, C. W., Connolly, A. J., Szalay, A. S., et al. 2004a, AJ, 128,\n585, doi: 10.1086/422429\nYip, C. W., Connolly, A. J., Vanden Berk, D. E., et al. 2004b, AJ,\n128, 2603, doi: 10.1086/42562620 P ALIYA ET AL .\nTable 10. References used in this work.\n4FGL name Reference 4FGL name Reference 4FGL name Reference\nJ0001.5+2113 Ahumada et al. (2020) J0833.4-0458 Paliya et al. (2020a) J1417.9+2543 Ahumada et al. (2020)\nJ0003.2+2207 Ahumada et al. (2020) J0833.9+4223 Ahumada et al. (2020) J1417.9+4613 Ahumada et al. (2020)\nJ0004.3+4614 Sowards-Emmerd et al. (2003) J0836.5-2026 Fricke et al. (1983) J1418.4+3543 Ahumada et al. (2020)\nJ0004.4-4737 Shaw et al. (2012) J0836.9+5833 Ahumada et al. (2020) J1419.5+3821 Ahumada et al. (2020)\nJ0006.3-0620 Stickel et al. (1989) J0837.3+1458 Ahumada et al. (2020) J1419.8+5423 Ahumada et al. (2020)\nNOTE—(This table is available in its entirety in a machine-readable form in the online journal. A portion is shown here for guidance\nregarding its form and content.)\nReferences —Acosta-Pulido et al. (2010); Afanas’Ev et al. (2005, 2006); Ahumada et al. (2020); Ajello et al. (2016); Aliu et al. (2011);\n´Alvarez Crespo et al. (2016a,b); Bade et al. (1995, 1998); Baker et al. (1999); Baldwin et al. (1981, 1989); Becerra Gonz ´alez et al.\n(2020); Bechtold et al. (2002); Boisse & Bergeron (1988); Bruni et al. (2018); Buttiglione et al. (2009); Carangelo et al. (2003);\nChai et al. (2012); Chomiuk et al. (2013); Chu et al. (1986); Colless et al. (2001); Desai et al. (2019); Drinkwater et al. (1997);\nDunlop et al. (1989); Evans & Koratkar (2004); Falomo et al. (1997, 2000, 2017); Foschini et al. (2015); Fricke et al. (1983); Gioia\net al. (2004); Gorham et al. (2000); Halpern et al. (1991, 1997, 2003); Healey et al. (2008); Henstock et al. (1997); Jones et al.\n(2009); Junkkarinen (1984); Klindt et al. (2017); Koss et al. (2017); Kotilainen et al. (2005); Landoni et al. (2013, 2018); Landt et al.\n(2001); Laurent-Muehleisen et al. (1998); Lawrence et al. (1996); Londish et al. (2007); Marcha et al. (1996); Marchesini et al. (2016,\n2019); Marziani et al. (2003); Masetti et al. (2008); Massaro et al. (2015); Mezcua et al. (2012); Morton et al. (1978); Murdoch et al.\n(1984); Nilsson et al. (2003); Nkundabakura & Meintjes (2012); Olgu ´ın-Iglesias et al. (2016); Padovani et al. (2019); Paiano et al.\n(2017a,b,c, 2019, 2020); Paliya et al. (2019, 2020a,b); Pe ˜na-Herazo et al. (2017, 2019, 2020); Perlman et al. (1996, 1998); Pietsch\net al. (1998); Piranomonte et al. (2007); Pita et al. (2014); Punsly & Kharb (2017); Pursimo et al. (2013); Raiteri et al. (2007); Rector\net al. (2000); Ricci et al. (2015); Sandrinelli et al. (2013); Sbarufatti et al. (2005, 2006a,b, 2009); Shaw et al. (2012, 2013a,b); Smith\net al. (2009); Sowards-Emmerd et al. (2003, 2004); Steidel & Sargent (1991); Stickel & Kuehr (1993, 1994); Stickel & Kuhr (1993);\nStickel et al. (1988, 1989, 1993a,b); Thompson et al. (1990); Titov et al. (2011, 2013, 2017); Torrealba et al. (2012); Urry et al. (2000);\nVandenbroucke et al. (2010); Vermeulen et al. (1995, 1996); Wallace et al. (2002); White et al. (2000); Wilkes et al. (1983); Wolter\net al. (1998); de Menezes et al. (2020); di Serego Alighieri et al. (1994)\nAPPENDIX\nACKNOWLEDGMENTS\nWe are grateful to the journal referee for constructive crit-\nicism. VSP is thankful to A. Desai, R. de Menezes, T. Pur-\nsimo, and J. Strader for providing optical spectra of a few\nblazars in tabular format and H. Guo and M. Cappellari for\nuseful discussions over the application of PyQSOFit and\npPXF tools, respectively. VSP’s work was supported by the\nInitiative and Networking Fund of the Helmholtz Associa-\ntion. A.D. acknowledges the support of the Ram ´on y Ca-\njal program from the Spanish MINECO. A.O.G. acknowl-\nedges financial support from the Spanish Ministry of Science,\nInnovation and Universities (MCIUN) under grant numbers\nRTI2018-096188-B-I00, and from the Comunidad de Madrid\nTec2Space project S2018/NMT-4291.\nFunding for the Sloan Digital Sky Survey IV has been pro-\nvided by the Alfred P. Sloan Foundation, the U.S. Depart-ment of Energy Office of Science, and the Participating Insti-\ntutions. SDSS-IV acknowledges support and resources from\nthe Center for High-Performance Computing at the Univer-\nsity of Utah. The SDSS web site is www.sdss.org.\nSDSS-IV is managed by the Astrophysical Research Con-\nsortium for the Participating Institutions of the SDSS Col-\nlaboration including the Brazilian Participation Group, the\nCarnegie Institution for Science, Carnegie Mellon Univer-\nsity, the Chilean Participation Group, the French Participa-\ntion Group, Harvard-Smithsonian Center for Astrophysics,\nInstituto de Astrof ´ısica de Canarias, The Johns Hopkins\nUniversity, Kavli Institute for the Physics and Mathemat-\nics of the Universe (IPMU) / University of Tokyo, the Ko-\nrean Participation Group, Lawrence Berkeley National Lab-\noratory, Leibniz Institut f ¨ur Astrophysik Potsdam (AIP),\nMax-Planck-Institut f ¨ur Astronomie (MPIA Heidelberg),BLAZAR ’SCENTRAL ENGINE 21\nMax-Planck-Institut f ¨ur Astrophysik (MPA Garching), Max-\nPlanck-Institut f ¨ur Extraterrestrische Physik (MPE), National\nAstronomical Observatories of China, New Mexico State\nUniversity, New York University, University of Notre Dame,\nObservat ´ario Nacional / MCTI, The Ohio State University,\nPennsylvania State University, Shanghai Astronomical Ob-\nservatory, United Kingdom Participation Group,Universidad\nNacional Aut ´onoma de M ´exico, University of Arizona, Uni-\nversity of Colorado Boulder, University of Oxford, Univer-\nsity of Portsmouth, University of Utah, University of Vir-\nginia, University of Washington, University of Wisconsin,\nVanderbilt University, and Yale University.\nThis research has made use of data obtained through the\nHigh Energy Astrophysics Science Archive Research Cen-\nter Online Service, provided by the NASA/Goddard Space\nFlight Center. This research has made use of the NASA/IPAC\nExtragalactic Database (NED), which is operated by the Jet\nPropulsion Laboratory, California Institute of Technology,\nunder contract with the National Aeronautics and Space Ad-\nministration. Part of this work is based on archival data, soft-\nware or online services provided by the Space Science Data\nCenter (SSDC). This research has made use of NASA’s As-\ntrophysics Data System Bibliographic Services.\nThis research made use of Astropy,6a community-\ndeveloped core Python package for Astronomy (Astropy Col-\nlaboration et al. 2013, 2018).\n6http://www.astropy.org"    },    {        "title": "2107.07950v1.Band_gap_measurements_of_monolayer_h_BN_and_insights_into_carbon_related_point_defects.pdf",        "content": "Band gap measurements of monolayer h-BN and\ninsights into carbon-related point defects\nRicardo Javier Pe~ na Rom\u0013 an,\u0003,yF\u0013 abio J. R. Costa ,yAlberto Zobelli,zChristine\nElias,{Pierre Valvin,{Guillaume Cassabois,{Bernard Gil,{Alex Summer\feld,x\nTin S. Cheng,xChristopher J. Mellor,xPeter H. Beton,xSergei V. Novikov,xand\nLuiz F. Zagonel\u0003,y\nyInstitute of Physics \\Gleb Wataghin\", Department of Applied Physics, State University of\nCampinas-UNICAMP, 13083-859, Campinas, Brazil\nzUniversit\u0013 e Paris-Saclay, CNRS, Laboratoire de Physiques des Solides, 91405, Orsay,\nFrance\n{Laboratoire Charles Coulomb, UMR5221 CNRS-Universit\u0013 e de Montpellier, 34095\nMontpellier, France\nxSchool of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK\nE-mail: rikrdopr@i\f.unicamp.br; zagonel@unicamp.br\nThis is the version of the article before peer review or editing, as submitted by an author\nto IOP 2D Materials.\nFind the published version at: https://doi.org/10.1088/2053-1583/ac0d9c\nCite as: Ricardo Javier Pe~ na Rom\u0013 an et al 2021 2D Mater. 8 044001\n1arXiv:2107.07950v1  [cond-mat.mes-hall]  16 Jul 2021Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nAbstract\nBeing a \rexible wide band gap semiconductor, hexagonal boron nitride (h-BN) has\ngreat potential for technological applications like e\u000ecient deep ultraviolet (DUV) light\nsources, building block for two-dimensional heterostructures and room temperature sin-\ngle photon emitters in the UV and visible spectral range. To enable such applications,\nit is mandatory to reach a better understanding of the electronic and optical properties\nof h-BN and the impact of various structural defects. Despite the large e\u000borts in the\nlast years, aspects such as the electronic band gap value, the exciton binding energy\nand the e\u000bect of point defects remained elusive, particularly when considering a single\nmonolayer.\nHere, we directly measured the density of states of a single monolayer of h-BN epi-\ntaxially grown on highly oriented pyrolytic graphite, by performing low temperature\nscanning tunneling microscopy (LT-STM) and spectroscopy (STS). The observed h-BN\nelectronic band gap on defect-free regions is (6 :8\u00060:2) eV. Using optical spectroscopy\nto obtain the h-BN optical band gap, the exciton binding energy is determined as being\nof (0 :7\u00060:2) eV. In addition, the locally excited cathodoluminescence and photolumi-\nnescence show complex spectra that are typically associated to intragap states related\nto carbon defects. Moreover, in some regions of the monolayer h-BN we identify, us-\ning STM, point defects which have intragap electronic levels around 2.0 eV below the\nFermi level.\nIntroduction\nHexagonal boron nitride (h-BN) is a layered compound that is isomorphous with graphite.\nIn its bulk form, h-BN is formed from monolayers composed of boron and nitrogen atoms\nin a hexagonal sp2covalent lattice that are organized vertically by van der Waals (vdW)\ninteractions.1{3With an optical band gap of about 6 eV, and an indirect-to-direct band\ngap crossover in the transition from bulk to monolayer,4{6h-BN shows very bright deep\nultraviolet (DUV) emission7{9and defect mediated emission from the DUV all the way\nPage 2Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nto the near-infrared.10{14In particular, point defects have been observed to act as single-\nphoton sources.15{23Such properties place bulk h-BN and its monolayer form in the spotlight\nfor many potential applications, including DUV light emitting devices,24{26dielectric layers\nfor two-dimensional (2D) heterostructures27{29and room temperature (RT) single photon\nemitters (SPEs) for quantum technologies.30{34Similarly to other technologically-relevant\nsemiconductors, the successful application of h-BN depends on the understanding and control\nof its electronic and optical properties. However, partially due to its wide band gap and\nchallenging large area synthesis, many fundamental electronic and optical properties of h-\nBN remain elusive. Moreover, the morphology, the electronic properties and the optical\nemission of structural point defects are complex and are currently poorly understood.\nGiven the wide band gap of h-BN and its insulating character, studying its morpho-\nlogical, electronic and optical properties can be tricky, particularly for the case of a single\nmonolayer. The band structure of bulk and monolayer h-BN have been reported by Angle\nResolved Photoemission Spectroscopy (ARPES).35,36However, ARPES cannot resolve the\nconduction band structure. Electron Energy Loss spectroscopy (EELS) has already been\napplied to measure the band gap on h-BN nanotubes and h-BN monolayers, but EELS mea-\nsures the optical band gap, similarly to optical absorption.37,38Therefore, scanning tunneling\nspectroscopy (STS) becomes an appropriate approach to probe both the valence band and\nconduction band edges as well as for the determination of the electronic band gap.39,40Still,\nso far, STS has not provided a clear measurement of the electronic band gap for monolayer\nh-BN or intra-gap states related to defects. The structure of individual defects in h-BN has\nbeen explored in previous works by means of scanning tunneling microscopy (STM)41and\nhigh resolution transmission electron microscopy,42but, despite these e\u000borts, a clear correla-\ntion between electronic levels and light emission related to defects has not been established\nyet. Optically, the properties of few-layer h-BN samples have recently been investigated by\ncathodoluminescence (CL) measurements. However, the thinnest h-BN samples for which\na CL signal could be collected were six43and three44monolayers \rakes. Consequently, the\nPage 3Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nelectronic band gap value and the exciton binding energy are unaddressed experimentally\neven now, alongside the CL emission in the monolayer and electronic signatures of defect\nstates. Finally, the direct character of the band gap and optical transitions in the DUV in\nmonolayer h-BN have been recently reported,6but the optical transition in the near UV and\nvisible spectral range have not yet been extensively explored.\nHere, we show that monolayer h-BN epitaxially grown on highly oriented pyrolytic\ngraphite (HOPG) is a model system for the study of the morphological, electronic, and\noptical properties of h-BN monolayers. The weak sample-substrate interaction enables, for\nthe \frst time, the determination of the electronic band gap value by means of STS measure-\nments. Correlation between tunneling spectroscopy with DUV optical spectroscopy enables\na value for the exciton binding energy in monolayer h-BN to be estimated. Additionally,\nstructural point defects are observed by STM images. Luminescence due to defects in a\nbroad emission range is observed for the \frst time using CL in monolayer h-BN. Moreover,\ninsights on the optical signatures of defects are obtained from the interesting contrast be-\ntween in situ photoluminescence (PL) and CL. These results indicate that monolayer h-BN\ndoes not form signi\fcant interface states with HOPG, so that h-BN exhibits its fundamental\nelectronic and optical properties.\nResults and discussion\nSample and methods descriptions\nIn Figure 1, a general description of the h-BN sample investigated in this work is displayed.\nAs shown in Figure 1(a), h-BN and HOPG have an in-plane hexagonal structure with similar\nlattice parameters. The small lattice mismatch makes them highly compatible and appro-\npriate for the epitaxial growth of vertical heterostructures. Additionally, the surfaces of\nthese kinds of materials are naturally passivated without any dangling bonds. Therefore, in\nvdW epitaxial heterostructures, an atomically sharp interface is obtained, where chemical\nPage 4Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nbonds are absent and only vdW interactions are present between the sample and the sub-\nstrate.45{49The region at the interface is usually referred to as vdW gap, such as illustrated\nin Figure 1(a). Since the vdW interaction is weak, it is expected that each material in the\nheterostructure preserves most of its electronic properties. This was revealed in recent works\nby ARPES measurements. Sediri et al.50proved that when a monolayer h-BN is epitaxially\ngrown on graphene (Gr), the electronic structure of Gr remains una\u000bected under the pres-\nence of h-BN. In the same way, Pierucci et al.36demonstrated in monolayer h-BN/HOPG\nthat the electronic properties of h-BN are not perturbed signi\fcantly by the substrate. This\nmeans that in vdW heterostructures, the sample and the substrate are electronically decou-\npled, and there are no doping e\u000bects or charge transfer. All these e\u000bects make the vdW\nheterostructure of monolayer h-BN on HOPG an important model system for the study of\nthe fundamental properties of h-BN, and for applications when conductivity is relevant as\nin STM/STS and light emitting diodes.\nFigure 1: (a) Top view of the in-plane atomic structure of h-BN and HOPG. Side view\nof the monolayer h-BN/HOPG van der Waals heterostructure. AFM (b) topographic and\n(c) phase images of the epitaxial monolayer h-BN/HOPG. (d) Height pro\fle of a single\nmonolayer h-BN.\nPage 5Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nThe h-BN layers were grown on HOPG substrates by high-temperature plasma-assisted\nmolecular beam epitaxy (PA-MBE).6,51{53Figures 1(b) and (c) show large area topographic\nand phase atomic force microscopy (AFM) images, respectively, of the h-BN layer on the\nHOPG substrate, acquired with the amplitude-modulated tapping mode in ambient con-\nditions. As observed, the HOPG surface is almost totally covered by h-BN, where the\nwhite regions in the phase image correspond to uncovered HOPG surface areas. The AFM\ntopographic image in Figure 1(b) shows that the sample is predominantly composed by\nlarge terraces of monolayer h-BN with some small bilayer regions and some thick regions\naround the HOPG grain boundaries and surface steps, which appear as bright (dark) fea-\ntures in the topography (phase) zoomed image (see also Figure S1). The height pro\fle in\nFigure 1(d), taken along the white line on the h-BN/HOPG surface of Figure 1(b), shows\nthat the step height is approximately 0.4 nm, in agreement with the expected thickness of\na single monolayer h-BN. This morphology is consistent with previous reported results on\nsimilar samples.6,51{53\nThe surface morphology and electronic properties of the monolayer h-BN were investi-\ngated locally by means of STM and STS measurements performed under ultra-high vacuum\n(UHV) conditions at cryogenic temperatures using a modi\fed RHK PanScan FlowCryo mi-\ncroscope. An STM image revealing the smooth sample surface is shown in Figure 2(a), as\nobserved previously by AFM (Figure S1). Considering only the STM image, it is di\u000ecult\nto know if the imaged area corresponds to the h-BN or to the HOPG surface because they\nare morphologically very similar. Despite being isomorphic, h-BN and HOPG have strik-\ning di\u000berent electronic structures (wide band gap semiconductor and zero-gap conductor,\nrespectively) and therefore very di\u000berent local density of states (LDOS) to be measured by\nSTS. The gray curve in Figure 2(b) is a typical STS spectrum obtained in the region of\nFigure 2(a). This dI/dV curve shows an electronic band gap of \u00186 eV, de\fned by the low\ndi\u000berential conductance range from \u0018-3 V to\u0018+3V, con\frming that the scanned region\ncorresponds to a h-BN covered surface. Considering the general morphology of the h-BN\nPage 6Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nFigure 2: (a) STM image of the monolayer h-BN/HOPG surface (0.8 nA, 0.2 V, 80 K).\nScale bar of 30 nm. (b) STS curves at 80 K (The feedback loop was disabled at -6.2 V, -150\npA.). Schematic illustration of the experimental setup for (c) in-situ PL/Raman and (d)\nSTM-CL measurements in the STM con\fguration. The spectra shown here are uncorrected\nraw data.\ngrown on HOPG observed by AFM and the fact the the majority of the sample is covered by\nmonolayers, the h-BN thickness in smooth regions as that one in Figure 2(a) is a considered\nto be a monolayer. In other regions with similar surface morphology, the STS curves exhibit\nthe characteristic parabolic shape around the Fermi level of the LDOS of graphite,54,55as in\nthe blue plot in Figure 2(b).\nThe STM was adapted to include a high numerical aperture light collection and injection\nsystem with optimized transmission. This system uses an o\u000b-axis parabolic mirror inserted\naround the tunneling junction and in situ PL and Raman spectroscopies can be performed\nusing the light injection and collection system as illustrated in Figure 2(c). In this exper-\nimental con\fguration, the STM tip is retracted from the sample surface and the light of a\ngreen (532 nm) laser diode is injected into the STM junction under UHV conditions. Addi-\ntionally, STM-induced light emission or the CL response of the sample can be investigated\nPage 7Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nusing the setup, see Figure 2(d). For recording the CL signal, the STM is operated in \feld\nemission mode using an external voltage source of (0-500) V for the local excitation of the\nsample by electron bombardment (STM-CL). The PL/Raman and CL spectra, shown in\nFigure 2(c) and (d), correspond to the spectroscopic raw data that need to be corrected for\nthe proper interpretation, as explained in the Supplementary materials (SM). This setup has\nalso been previously used to investigate WSe 2and MoS 2monolayers.56,57\nElectronic band gap, optical band gap and exciton binding energy\nSince monolayer h-BN is an atomically thin and a wide band gap material, access to its\nelectronic structure is always a highly challenging experimental and theoretical task. Within\nthe last twenty years, the h-BN electronic band gap has been calculated a number of times\nwithin the GW approximation, while the optical response has been calculated using the\nBethe-Salpeter equation.58{61In this approximation, the Green's function for the quasi-\nparticles are evaluated within the screened Coulomb potential in a way to consider many-\nbody problems of the interacting electrons.62However, non self-consistent GW calculations\n(G0W0), which start from DFT orbitals and perform the calculation only once, typically\nprovide underevaluated values for the electronic band gap and, in the case of h-BN, a rigid\nshift of the optical spectra is required for a direct comparison with experiments. Only\nvery recently, self-consistent GW calculations, which update interactively both the wave\nfunctions and the eigenvalues, managed a good agreement with synchrotron ellipsometry\nexperiments for bulk h-BN.63In the case of free-standing monolayer h-BN and using the\nGW 0approach, which updates interactively the eigenvalues only in the Green's function, the\nelectronic band gap has been evaluated as 8.2 eV and the related optical gap at 6.1 eV.60,64\nSlightly higher values might be expected within the full self-consistent GW approach. On the\nother hand, at the moment, there is no experimental evidence with e\u000bective measurements\nof the electronic band gap in both bulk or monolayer h-BN. A particular reason for the lack\nof such results is the technical challenge regarding the accomplishment of experiments for\nPage 8Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nDUV wavelengths. The electronic properties and band structure of epitaxial monolayer h-\nBN on graphite have been studied recently by ARPES measurements.36This study reported\nimportant information about band alignment, Fermi level and Valence Band (VB) positions.\nHowever, it is noteworthy that ARPES measurements only resolve \flled states,35,36,65,66and\ntherefore the description of the whole band structure and electronic band gap of monolayer\nh-BN are still open questions.\nOne technique able to probe both \flled and empty states, with a complete information\nabout the LDOS, doping e\u000bects and charge transfer is scanning tunneling spectroscopy or\nSTS.40,67{71Previous STM/STS measurements on h-BN have not given a clear answer for\nthe electronic band gap. The main reason for that is because most of the STM/STS studies\nhave been carried out on h-BN samples grown directly on metallic substrates.72Nowadays,\nit is well documented that the growth of a 2D material on a metallic substrate leads to a\nband gap renormalization due to strong interactions with the substrate, which is related to\nthe dielectric screening by the metal and/or the formation of additional interface electronic\nstates, including hybridization among others.57,73{76These e\u000bects reduce the electronic band\ngap value to about 3 eV for h-BN on Ru(0001)75and on Re(0001)77surfaces, to\u00184 eV\nfor h-BN on Au(001),78and to\u00185 eV for the case of h-BN on Rh(111),79,80Ir(111),81and\nCu(111)82substrates. In addition, performing scanning tunneling experiments on h-BN can\nbe complex due to its expected insulating character. For instance, Wong, et al used Gr\nas conducting layer on top of bulk h-BN.41Using the Gr as conducting electrode a\u000bects\nconsiderably STM images and make STS spectra di\u000ecult to interpret, particularly with\nrespect to the h-BN band gap. Interestingly, when h-BN is grown on Gr/Cu(111), electronic\nstates of the metallic substrate are observed on the Gr and also on the h-BN layer.76In this\ncase, intragap states due to the copper support are measured by STS and the h-BN band\ngap is not observed. For vdW epitaxial heterostructures, the sample-substrate interaction is\nnot an issue because, as mentioned above, there is a sharp interface between the monolayer\nh-BN and the HOPG substrate governed by weak vdW bonds (vdW gap). Therefore, h-BN\nPage 9Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\non HOPG can be considered as being electronically decoupled from the substrate, and is\nnot expected to demonstrate intragap states coming from graphite, but only e\u000bect of the\ndielectric environment surrounding the monolayer.\nFigure 3(a) and (b) shows STM and STS results acquired at 80 K in the monolayer h-\nBN/HOPG. A STM image of a smooth and defect-free region of the sample surface together\nwith an insert showing a moir\u0013 e structure with with a periodicity of approximately 2.4 nm are\ndisplayed in Figure 3(a). This moir\u0013 e pattern corresponds to a local rotation of approximately\n5.8 °between the monolayer h-BN and the HOPG surface in a particular region, as recently\nobserved on similar samples.51The apparent height pro\fle of the moir\u0013 e is presented in\nFigure 3(b). Detailed studies of the moir\u0013 e structure in this system were reported in recent\nworks.51,83\nThe LDOS of the monolayer h-BN has been investigated by performing STS measure-\nments on the defect-free region at di\u000berent tip positions. The tunneling processes that probe\nboth empty and \flled states by STS in the monolayer h-BN are illustrated in Figure 3(c)\nand (d), respectively. The alignment between the Fermi level in h-BN and HOPG at 0 V\nwas demonstrated in the results depicted in Figure 2(b). Thus, when a positive sample bias\nranging from 0 V to +6 V is applied, the Fermi level in the tip is shifted with respect to the\nFermi level in h-BN/HOPG, and, as a consequence, electrons tunnel from the tip to empty\nstates of the sample, such as shown in Figure 3(c). The red curve in the \fgure represents the\nmeasured dI/dV for positive biases, which is a direct measurement of the local empty DOS.\nIt can be observed that the LDOS increases from biases above \u00183.0 V, which means that\nelectrons are tunneling only for states in the Conduction Band (CB) of h-BN. Below \u00183.0\nV the curve is \rat (the tunneling current is equal to \u00180 pA), because there are no available\nelectronic states inside the h-BN electronic band gap for electrons to tunnel to. The same\nprocess happens when the polarity of the bias is changed to negative values in order to\nprobe \flled states, as illustrated in Figure 3(d). It is worthwhile to note that, during typical\nSTS measurements, the voltage ramps while the current is measured but the tip-sample dis-\nPage 10Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nFigure 3: (a) STM image of a smooth and defect-free monolayer h-BN region (0.66 nA, 3.8\nV, 80 K). Scale bar of 30 nm. Inset: moir\u0013 e structure (-0.3 nA, -1.0 V, 16 K). Scale bar of\n5 nm. (b) Height pro\fle associated to the moir\u0013 e. Schematic illustration of the tunneling\nprocess for (c) positive and (d) negative biases. The red curve corresponds to the LDOS\nmeasured by STS as dI/dV. The blue arrow indicates the tunneling direction. (e) I-V and\ndI/dV typical curves obtained on a smooth and defect-free monolayer h-BN region and at 80\nK (The feedback loop was disabled at -6.2 V, -150 pA). The electronic band gap and band\nedges are indicated with the values obtained of a statistical analysis of 480 individual curves.\nGlobal measurements of the (f) optical absorption at RT and (g) PL at 10 K. (h) Energy\nlevels diagram proposed for defect-free monolayer h-BN. In the \fgure VBM, E g, Eoptand\nEbcorrespond to the valence band maximum, the electronic band gap, the optical band gap\nand the exciton binding energy, respectively.\ntance is not adjusted. Therefore, the tip-sample distance is set by the tunneling current and\nsample bias prior to disabling the tip-sample distance feedback loop and starting the STS\nPage 11Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nmeasurement (stable tunneling parameters). Here, we used typically -150 pA at -6.2 V (see\nMethods section for more details) as the stable condition prior to starting the acquisition\nof the scanning tunneling spectra in order to ensure reproducible and reliable spectra. If\nthe tunneling current is set to slightly higher or lower values (from -50 to -300 pA) before\ndisabling the feedback loop, the obtained spectra are compatible to those measured using\n-150 pA, see Figure S2.\nIt is relevant to mention that direct tunneling from or to the HOPG is possible even\nthrough the monolayer h-BN. In fact, this e\u000bect can be observed when STS curves were\nrecorded stabilizing the tunneling junction at high tunneling currents ( \u0015600 pA) with -6.2\nV of sample bias, as discussed in the SM, see Figure S2. A possible explanation for this\nobservation is that for a \fxed sample bias and higher stabilization currents, the tip-sample\ndistance becomes shorter, which means that the tunneling barrier is narrower, and then the\nprobability of the direct tunneling from or to the substrate is not negligible. This leads\nto the measurement of an apparent reduced band gap, because states of the HOPG near\nthe band edges of h-BN are observed in the STS curve for h-BN, as can be seen in Figure\nS2(b). The direct tunneling from/to the HOPG substrate for short tip-sample distance\nin STS measurements on vdW epitaxy MoS 2/HOPG and WSe 2/HOPG samples have been\ndemonstrated by Chiu et al.84In that work, it was also demonstrated that the LDOS of\nHOPG can be measured by tunneling electrons through the TMD layer if the stabilizing\nbias for the STS acquisition lays inside the band gap region of the TMD. For wide band\ngap materials such as h-BN, using appropriate tunnel junction conditions before starting the\nSTS is therefore crucial.\nFigure 3(e) shows typical STS results obtained from a smooth and defect-free h-BN\nsample surface region. The top panel of the \fgure shows the I-V curve plotted together with\nthe dI/dV tunneling spectrum. The electronic band gap of the h-BN sample is de\fned by the\nbias range at which the current and dI/dV are close to their background values. The band\nedges and band gap values were extracted from the dI/dV curve in logarithmic scale,68,85\nPage 12Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nas represented in the bottom panel of Figure 3(e). The values shown in the \fgure were\nobtained following the method and the statistical analysis explained in the SM, Figures S3\nand S4, which considers a total number of 480 individual dI/dV curves similar to Figure 3(e)\ntaken at di\u000berent tip positions. The \fnal results are a Valence Band Maximum (VBM)\nand a Conduction Band Minimum (CBM) positioned at ( \u00003:4\u00060:1) eV and (3 :4\u00060:1)\neV, respectively, for monolayer h-BN on HOPG. This results in an electronic band gap of\n(6:8\u00060:2) eV. Also, the Fermi level is in the middle of the band gap, indicating that the\ndefect-free monolayer h-BN regions, as shown in Figure 3(a), are undoped and therefore\nthere is no charge transfer with the HOPG. ARPES found the VBM position close to the\n\u0018-2.8 eV on a similar h-BN sample.36This ARPES result for the VBM is slightly di\u000berent\nthan the value measured by STS (-3.4 eV). This di\u000berence is possibly caused by some degree\nof p-type doping on the sample and/or some level of hole excess coming from a insu\u000ecient\ncompensation of the photo-emitted electrons. On 3-monolayer thick h-BN samples, ARPES\ndetermined the VBM at -3.2 eV, which, considering the band structure for 3-monolayer thick\nh-BN and the expected electronic band gap opening for a monolayer h-BN, agrees well with\nthe results presented here.86From the experimental point of view, STS is not sensitive to the\ndirect or indirect nature of the electronic band gap. Nonetheless, recently, it was con\frmed\nthe recombination of direct excitons in monolayer h-BN,6which suggests that this is a direct\nband gap material.\nThe electronic band gap value found here for monolayer h-BN is larger, as expected from\nsimulations,61than values for bulk single crystal and thin \flms of h-BN, as obtained indi-\nrectly by photocurrent spectroscopy, which results in an electronic band gap of about 6.45\neV.87,88Regarding theoretical predictions for free-standing monolayer h-BN, GW 0calcula-\ntions indicate a band gap of 8.2 eV at the K point of the Brillouin zone with an exciton\nbinding energy of 2.1 eV, giving an optical band gap of 6.1 eV.60,64This value is close to the\noptical onset of 6.3 eV measured by EELS on a free-standing monolayer.38Simulations for\nmonolayer h-BN on top of graphene have shown a strong renormalization of the electronic\nPage 13Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nband gap, of the order of 1 eV, due to strong screening e\u000bects of the substrate.89,90The\nsimulations also show that the electronic band gap renormalization of monolayer h-BN is\nbasically the same on top of graphene and on top of graphite.89This last scenario is the\nperfect model for our system of monolayer h-BN on HOPG and gives a strong support for\nthe band gap value measured here by STS.\nOptical absorption and PL were performed in the DUV to determine the optical (ex-\ncitonic) band gap of monolayer h-BN on HOPG (see details in SM). Figure 3(f) and (g)\nshow measurements of the optical absorption and PL (spot-size of 200 \u0016m and 50\u0016m, re-\nspectively) obtained at RT and at 10 K, respectively. The optical absorption spectrum was\ncalculated from spectroscopic ellipsometry data. The PL in the DUV employs a 6.4 eV\nlaser obtained from the fourth harmonic of a Ti:Sa oscillator. A systematic study on the\ncomplex DUV absorption and emission of high-temperature PA-MBE h-BN on HOPG can\nbe found in recent reports.6,52The spectrum in Figure 3(f) shows an abrupt change in the\nabsorption coe\u000ecient that suggests an optical band gap of above 5.7 eV. Furthermore, the\nPL in Figure 3(g) shows the direct fundamental exciton transition at 6.1 eV (optical band\ngap) in monolayer h-BN, besides other emission lines associated with transitions in few-layer\nand defective regions around 5.9 eV and 5.6 eV, respectively. These results are in excellent\nagreement with previous experiments by Elias et al.6Moreover, this value for the optical\nband gap is consistent with recent simulation and indicate, as expected, an invariance of the\nexciton emission.60,64,68,90\nThe knowledge of both the electronic band gap and the optical band gap is essential\nto the design of applications in optoelectronics and light emitting devices employing large\nexciton binding energy materials, like the monolayer h-BN. Considering the h-BN electronic\nband gap found by STS and the energy observed for the exciton recombination, the obtained\nexciton binding energy is (0 :7\u00060:2) eV. The comparison between this experimental evaluation\nand the theoretical predicted value of 2.1 eV for the free-standing monolayer, suggest a\nstrong renormalization, of about 1.4 eV, of the exciton binding energy due to substrate\nPage 14Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nscreening e\u000bects.64Moreover, the observed renormalization of the exciton binding energy is\nexpected to be the same on the electronic band gap, which nearly leads to an invariance\nof the optical onset.68,90This renormalization is in close agreement with simulations for\nmonolayer h-BN on top of graphite.89The similar shift on the electronic band gap and on the\nexciton biding energy agrees well with simulations for monolayer h-BN and with experiments\nand simulations for monolayer MoSe 2.68,90Based on all the spectroscopic data, Figure 3(h)\nsummarizes the energy levels determined here for a defect-free region of monolayer h-BN on\nHOPG. In Figure 3(h), the h-BN electronic band gap is indicated as (6 :8\u00060:2) eV, while\nthe h-BN optical band gap is 6.1 eV. This leads to a Frenkel exciton with a binding energy\nestimated as (0 :7\u00060:2) eV.\nElectronic Structure and Light Emission related to point defects\nThe wide electronic band gap in h-BN allows the observation of several optical transitions\ninvolving di\u000berent intragap states associated with defects.10,12,13,91Therefore, light emission\nin h-BN may be dominated by structural defects. Point defects have a particular interest\ndue to their characteristic quantum emission at RT in a broad range of wavelengths.16{18,20\nDi\u000berent types of point defects have been proposed as being the sources of the single photon\nemission observed in h-BN.14,91,92However, their morphological and electronic signatures\nhave not been measured directly. Furthermore, a direct correlation with the optical emission\nis missing. Even though STM is a tool able to identify atomic defects in 2D materials,56,93{96\nSTM/STS measurements of point defects in monolayer h-BN have not been reported yet.\nThe properties of individual point defects have been investigated only by simulated STM\nimages in recent theoretical works.97{100Here some results obtained on defective regions of\nthe sample are presented in order to help acquiring a better understanding of the impact of\npoint defects in the properties of monolayer h-BN.\nFigure 4(a) shows a STM image of the monolayer h-BN/HOPG surface, where point\ndefects are revealed. These defects are bright spots of about 1 to 2 nm of diameter, according\nPage 15Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nFigure 4: (a) STM image of bright point defects in monolayer h-BN (0.8 nA, 0.8 V, 80 K).\nScale bar of 30 nm. (b) Heigth pro\fle of some individual defects. (c) Typical STS curve\nobtained on the defective region at 80 K (The feedback loop was disabled at -6.2 V, -150\npA). (d) In-situ PL and STM-CL spectra in which Phonon Side-Bands (PSB) are observed\nnear Zero Phonon Loss (ZPL) peaks. (e) In-situ Raman spectrum at RT. Inset: Lorenztian\n\ftting of the peak at 1350 cm\u00001.\nto the full width at half maximum in the height pro\fles of Figure 4(b), see also Figure S5.\nDefects in h-BN on HOPG were already observed by Summer\feld et al.51using conductive\nAFM and STM measurements, and were attributed to defects in the HOPG substrate,\ncreated by the active nitrogen plasma irradiation damage during the sample growth process.\nIn another study, bright point defects relating to possible carbon impurities were imaged\nby STM in Gr capped bulk h-BN.41In order to inspect the electronic signature of the\npoint defects found here, STS curves were acquired. Figure 4(c) displays a typical tunneling\nspectrum recorded when the STM tip is paced close to such a defect. By plotting the STS\ncurve in logarithmic scale, it can be noted that the bands onset are not linear, in contrast\nPage 16Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nto the ones observed in defect-free regions (see Figure 3(e) and Figure S3(c)). The curve\nin Figure 4(c) presents three main resonances at \u0018\u00003:5 V,\u0018\u00002:0 V and\u0018+3:5 V. By\ncomparing with the results for defect-free regions, the resonances at \u00063:5 V are located on\ntop of the CB and VB edges, respectively, indicating that such defects have energy levels\nclose to the band edges. Besides the spectra on Figure 4(c), the electronic level near -2.0\nV was observed in several other tunneling spectra obtained on regions with point defects,\nsee Figure S5. Therefore, this energy level is attributed to the observed defects. The fact\nthat the STS curves show roughly the expected band gap for h-BN even on defective regions\nsuggests that the scanned regions corresponds to a h-BN covered area and the observed point\ndefects are in the monolayer h-BN.\nA good starting point to interpret the imaged defects and the STS results is to consider\nthat in the system of monolayer h-BN on HOPG, the atoms of Boron (B), Nitrogen (N)\nand Carbon (C) are present. Thus, substitutional defects such as C Band C N, carbon dimer\ndefects like C BCN, or carbon anti-sites vacancies of the types V BCN, and V NCBare possi-\nble, and also are the main point defect candidates to be considered. Indeed, the random\ndistribution of C atoms in the h-BN lattice is energetically possible,101and on the other\nhand, the high temperature and plasma assisted growth process can create vacancies during\nthe sample preparation.19,51From the electronic point of view, band structure calculations\nshow that for the LDOS in monolayer h-BN the states near the VBM are concentrated in\nthe N sites, while the states near the CBM are in the B sites.101,102Impurities occupying\nN or B sites could act as acceptor-like or donor-like impurities, respectively. There are sev-\neral theoretical reports exploring the doping e\u000bects of C impurities in single layers of h-BN.\nIn those works, it is demonstrated that C Band C Ndefects induce n-doping and p-doping,\nrespectively.97,99,103,104In particular, C Ndefects introduce an acceptor state localized at 2.0\neV below the Fermi level and between 1-2 eV above the VBM.97,103,105The density of this\ndefect state depends on the C concentration,101and the energy position can be shifted due\nto local strain e\u000bects,97,106,107and also by band bending e\u000bects.93More recently, it also has\nPage 17Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nbeen predicted that the C BCNdimer introduces a stable and neutral intragap state localized\nat\u00180.8 eV above the VBM.108,109\nIt is important to point out that our STM results are consistent with the predictions\nmade by Fujimoto et al.97and by Haga et al.99,100in their works on simulated STM images of\nindividual point defects in h-BN and Gr vertical heterostructures. In those reports, the bright\nappearance of C Npoint defects were simulated for negative tunneling biases. The calculated\nSTM images of C-doped monolayer h-BN show that C atoms produce a redistribution of\nthe local electron density around the defects, being the electron density in C Nhigher than\nin C B, which results in STM images where C Ndefects look as bright spots when compared\nwith the image of C B.97,99Moreover, the local electron density, and as a consequence, the\nsize of the bright spots related to C Ndefects can be a\u000bected by the stacking and moir\u0013 e\npattern between the h-BN and Gr.99,100Defects imaged with negative bias can be observed\nin Figure S5. These theoretical considerations indicate that the defects identi\fed by STM\nare consistent with defects involving C on N sites, i. e. C N, CBCNand V BCN, even if other\nkinds of defect can not be excluded. Nonetheless, it is noteworthy that a detailed study of the\nmorphological and electric structure of individual point defects in h-BN is required, which\ncan be achieved by performing STM/AFM imaging and STS measurements at cryogenic\ntemperatures, such as what has been done in the case of point defects in single layers of\ntransition metal dichalcogenides.110{113\nIn order to obtain more insights on the observed defects, in situ PL and STM-CL ex-\nperiments were carried out at 100 K and 300 K, using the experimental setup described in\nFigure 2. The raw luminescence data were presented in Figure 2(c) and (d), but for the\nproper interpretation, the spectra were corrected following the procedure explained in SM\nand the results are presented in Figure 4(d). The PL spectra show a main peak at 2.10\neV, which is best resolved at 100 K, and two shoulders around 1.96 eV and 1.81 eV. These\ntransitions are related to the zero-phonon line (ZPL) and two phonon replicas or phonon\nside bands (PSB) of carbon-related defects, which behave as single photon sources.22Simu-\nPage 18Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nlations suggest V BCNor C BVNas defects at the origin of this emission.22,106The width of\nthe PL peaks is larger than that typically observed in thick h-BN \rakes, but in agreement\nwith reports on monolayers.15Also in the PL spectra, two sharp peaks are observed and\nlabeled with a red star symbol. These peaks correspond to the Raman response depicted\nfully in Figure 4(e), where the stronger peaks around 1580 cm\u00001and 2700 cm\u00001are the G\nand 2D Raman bands of HOPG, respectively.114In addition, an asymmetric and weak peak\nis observed at 1350 cm\u00001, and as shown in the inset \fgure, this peak has two contributions.\nOne contribution at 1341 cm\u00001associated to the D Raman mode of HOPG and another one\nat 1362 cm\u00001corresponding to the E 2gRaman mode of monolayer h-BN, in agreement with\nrecent reported results.53\nThe CL spectra in the bottom panel of Figure 4(d) show a broad luminescence that\nranges from the near infrared to the UV. As in PL, the peaks are better resolved at 100\nK than at 300 K. Again, the optical transitions associated to carbon-related defects are\nobserved with the ZPL at 2.10 eV and the PSB at 1.81 eV, respectively, as seen in PL.\nAt higher energies, the CL spectra present a series of peaks at 3.90 eV, 3.70 eV and 3.43\neV, which have been reported as being phonon replicas of a deep well-known carbon-related\ndefect level at 4.1 eV.105,115Unfortunately, the emission at 4.1 eV has not been resolved here\ndue to the transmission function of the setup. The spectra only could be corrected for the\ninstrument response function up to 300 nm (4.1 eV). However, the PSB were observed, which\nmeans that the transitions at 4.1 eV were also present in the emitted spectra. Interestingly,\nthe light emission at 4.1 eV has been reported as being an optical transition related to\nthe single photon emission.16Recent literature suggests that this emission is associated\nwith C BCNdefects, carbon dimers, but other defects like C Nare also considered in some\ncases.108,109,116,117The most intense features observed in the CL spectra of Figure 4(d) are\nthe peaks at 3.08, 2.65 and 2.37 eV, with shifts of 430 meV and 280 meV between them.\nSuch peaks possibly have distinct origins related to the several possible defects, particularly\nin the presence of carbon.116Emissions in this energy range have been observed in bulk\nPage 19Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nh-BN by performing CL and PL experiments, and are associated to carbon21,105impurities\nor nitrogen vacancy type-centers.118,119\nConclusions\nWe used low temperature UHV STM together with an optimized light collection and injection\ndevice to study the morphological, electronic, and optical properties of a monolayer h-BN\nepitaxially grown on HOPG. The STM images reveal h-BN regions free of defects and regions\nwith point defects. An electronic band gap of (6 :8\u00060:2) eV was determined by performing\nSTS measurements on defect-free monolayer h-BN. This result, combined with the h-BN\noptical band gap of 6.1 eV from the exciton transition, leads to a binding energy for the\nFrenkel exciton of (0 :7\u00060:2) eV. Additionally, bright point defects were observed in h-\nBN by STM imaging. The STS indicates an acceptor level around -2 eV related to the\npresence of the observed defects. PL and CL have shown the emission typically associated\nto carbon-related defects at 2.1 eV. Besides that, emissions at 3.08 eV and photon side band\npossibly associated to an emission at 4.1 eV were observed by CL on monolayer h-BN. These\nresults indicate the simultaneous presence of more than one kind of carbon-related defect.\nWe consider that the \fndings presented in this work could help in the understanding of the\nfundamental properties of monolayer h-BN, as well as in the identi\fcation at the atomic level\nof sources responsible for the SPEs in h-BN samples. Moreover, h-BN on HOPG represents\nan excellent platform to study individual defects with respect to their morphology, electronic\nand optical properties.\nPage 20Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nMethods\nSample preparations\nMonolayer h-BN was grown on HOPG substrate by the high-temperature plasma-assisted\nmolecular beam epitaxy (PA-MBE) method.6,51{53This sample preparation method allows to\nproduce monolayer and few-layer h-BN with atomically \rat surfaces and monolayer control\nof the sample thickness. The h-BN thickness and coverage can be controlled by substrate\ntemperature, boron:nitrogen \rux ratio and growth time. In particular, the sample investi-\ngated here was grown at a substrate temperature of about 1390\u000eC with a high-temperature\ne\u000busion Knudsen cell for boron and a standard Veeco radio-frequency plasma source for ac-\ntive nitrogen. More details about the sample growth conditions and the MBE system can\nbe found elsewhere.53,120{122\nSTM/STS\nSTM and STS measurements were performed under ultra-high vacuum (UHV) conditions\nat low-temperatures using a modi\fed RHK PanScan FlowCryo microscope. This STM was\nadapted to receive a high numerical aperture light collection and injection system with\noptimized transmission. In this setup, imaging can be associated to electronic and optical\nspectroscopies as illustrated in Figure 2.\nPrior to STM/STS measurements, the h-BN sample was annealed at 773 K under a base\npressure of 1.9x10\u000010mbar for about six hours, in order to eliminate surface impurities and\ncontaminants. STM images were acquired in the constant-current mode using a grounded\ntungsten tip and a bias voltage applied to the h-BN sample. Tungsten tips were prepared\nby electrochemical etching with a NaOH solution. After preparation, the tips were inserted\nin the UHV chamber to be subjected to thermal annealing and argon sputtering. This\nprocedure allows the removal tungsten oxide layers. For the acquisition of STS curves,\ndi\u000berential conductance (dI/dV) measurements at 80 K were carried out using the lock-in\nPage 21Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\ntechnique, with a bias modulation of 80 mV of amplitude and 800 Hz of frequency.\nTo perform STS measurements, the tunnel junction was stabilized at a sample bias outside\nthe electronic band gap of the monolayer h-BN. In most measurements, we stabilized the\ntip at -150 pA and -6.2 V on a speci\fc tip position before disabling the tip-sample distance\nfeedback loop. Using these parameters, the tunnel current is e\u000bectively related to states on\nthe h-BN. Each tunneling spectrum was recorded by turning o\u000b the tip distance feedback\nloop and ramping the bias from -6.0 V to 6.0 V in steps of 50 mV and a dwell time of 50 ms.\nThis dwell time is needed to avoid overestimating the electronic band gap due to low signal\nto noise ratio near the band edges. A detailed description of the STS analysis is presented\nin the SM, Figures S3 and S4.\nIn situ PL, Raman and CL\nPL and Raman were excited with a laser diode of 532 nm. The spot size of the light focused\nby the mirror on the sample surface is \u00182\u0016m. The PL/Raman signal emitted by the sample\nis collected by the mirror and sent as parallel rays towards an optical setup outside the UHV\nSTM chamber.\nCL, also called STM-CL, was performed by operation of the STM in \feld emission mode.\nIn this case, the STM tip was retracted by approximately 150 nm from the h-BN sample\nsurface. A high bias voltage, between 150-180 V, was applied to the sample, which caused\na \feld emission currents of 5-10 \u0016A. In this STM operation mode, the spatial (lateral)\nresolution is roughly similar to the tip-to-sample distance.45Therefore, the obtained CL\nspectra refer to a region of about one hundred nanometers wide of the monolayer h-BN.\nAcknowledgement\nThis work was supported by the Funda\u0018 c~ ao de Amparo \u0012 a Pesquisa do Estado de S~ ao Paulo\n(FAPESP) Projects 14/23399-9 and 18/08543-7. This work at Nottingham was supported by\nPage 22Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nthe Engineering and Physical Sciences Research Council UK (Grant Numbers EP/K040243/1\nand EP/P019080/1). We also thank the University of Nottingham Propulsion Futures Bea-\ncon for funding towards this research. PHB thanks the Leverhulme Trust for the award\nof a Research Fellowship (RF-2019-460). This work was \fnancially supported by the net-\nwork GaNeX (ANR-11-LABX-0014), the ZEOLIGHT project (ANR-19-CE08-0016), and the\nBONASPES project (ANR-19-CE30-0007).\nSupplementary materials available\nAFM measurements and sample surface morphology. Tunneling parameters and STS. Elec-\ntronic band gap determination and statistical analysis. Optical absorption. STM/STS of\npoints defects. Correction in the luminescence spectra.\nReferences\n(1) Adachi, S. Optical Constants of Crystalline and Amorphous Semiconductors: Numer-\nical Data and Graphical Information ; Springer US: Boston, MA, 1999; pp 127{136.\n(2) Alem, N.; Erni, R.; Kisielowski, C.; Rossell, M. D.; Gannett, W.; Zettl, A. Atomically\nthin hexagonal boron nitride probed by ultrahigh-resolution transmission electron mi-\ncroscopy. Phys. Rev. B 2009 ,80, 155425.\n(3) Bhimanapati, G.; Glavin, N.; Robinson, J. In 2D Materials ; Iacopi, F., Boeckl, J. J.,\nJagadish, C., Eds.; Semiconductors and Semimetals; Elsevier, 2016; Vol. 95; pp 101 {\n147.\n(4) Watanabe, K.; Taniguchi, T.; Kanda, H. Direct-bandgap properties and evidence for\nultraviolet lasing of hexagonal boron nitride single crystal. Nature Materials 2004 ,3,\n404{409.\nPage 23Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(5) Cassabois, G.; Valvin, P.; Gil, B. Hexagonal boron nitride is an indirect bandgap\nsemiconductor. Nature Photonics 2016 ,10, 262{266.\n(6) Elias, C.; Valvin, P.; Pelini, T.; Summer\feld, A.; Mellor, C. J.; Cheng, T. S.; Eaves, L.;\nFoxon, C. T.; Beton, P. H.; Novikov, S. V.; Gil, B.; Cassabois, G. Direct band-gap\ncrossover in epitaxial monolayer boron nitride. Nature Communications 2019 ,10,\n2639.\n(7) Watanabe, K.; Taniguchi, T.; Miya, K.; Sato, Y.; Nakamura, K.; Niiyama, T.;\nTaniguchi, M. Hexagonal boron nitride as a new ultraviolet luminescent material and\nits application|Fluorescence properties of hBN single-crystal powder. Diamond and\nRelated Materials 2011 ,20, 849{852.\n(8) Kubota, Y.; Watanabe, K.; Tsuda, O.; Taniguchi, T. Deep Ultraviolet Light-Emitting\nHexagonal Boron Nitride Synthesized at Atmospheric Pressure. Science 2007 ,317,\n932{934.\n(9) Valvin, P.; Pelini, T.; Cassabois, G.; Zobelli, A.; Li, J.; Edgar, J. H.; Gil, B. Deep\nultraviolet hyperspectral cryomicroscopy in boron nitride: Photoluminescence in crys-\ntals with an ultra-low defect density. AIP Advances 2020 ,10, 075025.\n(10) Museur, L.; Feldbach, E.; Kanaev, A. Defect-related photoluminescence of hexagonal\nboron nitride. Phys. Rev. B 2008 ,78, 155204.\n(11) Bourrellier, R.; Amato, M.; Galv~ ao Tizei, L. H.; Giorgetti, C.; Gloter, A.; Heggie, M. I.;\nMarch, K.; St\u0013 ephan, O.; Reining, L.; Kociak, M.; Zobelli, A. Nanometric Resolved\nLuminescence in h-BN Flakes: Excitons and Stacking Order. ACS Photonics 2014 ,\n1, 857{862.\n(12) Schu\u0013 e, L.; Stenger, I.; Fossard, F.; Loiseau, A.; Barjon, J. Characterization methods\ndedicated to nanometer-thick hBN layers. 2D Materials 2016 ,4, 015028.\nPage 24Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(13) Vuong, T. Q. P.; Cassabois, G.; Valvin, P.; Ouerghi, A.; Chassagneux, Y.; Voisin, C.;\nGil, B. Phonon-Photon Mapping in a Color Center in Hexagonal Boron Nitride. Phys.\nRev. Lett. 2016 ,117, 097402.\n(14) Gil, B.; Cassabois, G.; Cusco, R.; Fugallo, G.; Artus, L. Boron nitride for excitonics,\nnano photonics, and quantum technologies. Nanophotonics 2020 ,9, 3483{3504.\n(15) Tran, T. T.; Bray, K.; Ford, M. J.; Toth, M.; Aharonovich, I. Quantum emission from\nhexagonal boron nitride monolayers. Nature Nanotechnology 2016 ,11, 37{41.\n(16) Bourrellier, R.; Meuret, S.; Tararan, A.; St\u0013 ephan, O.; Kociak, M.; Tizei, L. H. G.;\nZobelli, A. Bright UV Single Photon Emission at Point Defects in h-BN. Nano Letters\n2016 ,16, 4317{4321, PMID: 27299915.\n(17) Tran, T. T.; Elbadawi, C.; Totonjian, D.; Lobo, C. J.; Grosso, G.; Moon, H.; En-\nglund, D. R.; Ford, M. J.; Aharonovich, I.; Toth, M. Robust Multicolor Single Photon\nEmission from Point Defects in Hexagonal Boron Nitride. ACS Nano 2016 ,10, 7331{\n7338, PMID: 27399936.\n(18) Tran, T. T.; Zachreson, C.; Berhane, A. M.; Bray, K.; Sandstrom, R. G.; Li, L. H.;\nTaniguchi, T.; Watanabe, K.; Aharonovich, I.; Toth, M. Quantum Emission from\nDefects in Single-Crystalline Hexagonal Boron Nitride. Phys. Rev. Applied 2016 ,5,\n034005.\n(19) Xu, Z.-Q. et al. Single photon emission from plasma treated 2D hexagonal boron\nnitride. Nanoscale 2018 ,10, 7957{7965.\n(20) Tan, Q.-H.; Xu, K.-X.; Liu, X.-L.; Guo, D.; Xue, Y.-Z.; Ren, S.-L.; Gao, Y.-F.;\nDou, X.-M.; Sun, B.-Q.; Deng, H.-X.; Tan, P.-H.; Zhang, J. Ultraviolet to Near-\ninfrared Single Photon Emitters in hBN. arXiv:1908.06578 [cond-mat, physics:physics]\n2019 , arXiv: 1908.06578.\nPage 25Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(21) Hayee, F.; Yu, L.; Zhang, J. L.; Ciccarino, C. J.; Nguyen, M.; Marshall, A. F.;\nAharonovich, I.; Vu\u0014 ckovi\u0013 c, J.; Narang, P.; Heinz, T. F.; Dionne, J. A. Revealing\nmultiple classes of stable quantum emitters in hexagonal boron nitride with correlated\noptical and electron microscopy. Nature Materials 2020 ,19, 534{539.\n(22) Mendelson, N. et al. Identifying carbon as the source of visible single-photon emission\nfrom hexagonal boron nitride. Nature Materials 2020 ,20, 321{328.\n(23) Mart\u0013 \u0010nez, L. J.; Pelini, T.; Waselowski, V.; Maze, J. R.; Gil, B.; Cassabois, G.;\nJacques, V. E\u000ecient single photon emission from a high-purity hexagonal boron ni-\ntride crystal. Phys. Rev. B 2016 ,94, 121405.\n(24) Watanabe, K.; Taniguchi, T.; Niiyama, T.; Miya, K.; Taniguchi, M. Far-ultraviolet\nplane-emission handheld device based on hexagonal boron nitride. Nature Photonics\n2009 ,3, 591{594.\n(25) Watanabe, K.; Taniguchi, T. Hexagonal Boron Nitride as a New Ultraviolet Lumines-\ncent Material and Its Application. International Journal of Applied Ceramic Technol-\nogy2011 ,8, 977{989.\n(26) Jiang, H. X.; Lin, J. Y. Hexagonal boron nitride for deep ultraviolet photonic devices.\nSemiconductor Science and Technology 2014 ,29, 084003.\n(27) Hui, F.; Pan, C.; Shi, Y.; Ji, Y.; Grustan-Gutierrez, E.; Lanza, M. On the use of two\ndimensional hexagonal boron nitride as dielectric. Microelectronic Engineering 2016 ,\n163, 119{133.\n(28) Wang, J.; Ma, F.; Sun, M. Graphene, hexagonal boron nitride, and their heterostruc-\ntures: properties and applications. RSC Adv. 2017 ,7, 16801{16822.\n(29) Wu, L. et al. InSe/hBN/graphite heterostructure for high-performance 2D electronics\nand \rexible electronics. Nano Research 2020 ,13, 1127{1132.\nPage 26Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(30) Wrachtrup, J. Single photons at room temperature. Nature Nanotechnology 2016 ,11,\n7{8.\n(31) Aharonovich, I.; Englund, D.; Toth, M. Solid-state single-photon emitters. Nature\nPhotonics 2016 ,10, 631{641.\n(32) Aharonovich, I.; Toth, M. Quantum emitters in two dimensions. Science 2017 ,358,\n170{171.\n(33) Kim, S.; Duong, N. M. H.; Nguyen, M.; Lu, T.-J.; Kianinia, M.; Mendelson, N.;\nSolntsev, A.; Bradac, C.; Englund, D. R.; Aharonovich, I. Integrated on Chip Platform\nwith Quantum Emitters in Layered Materials. Advanced Optical Materials 2019 ,7,\n1901132.\n(34) Lukishova, S. G. Room-Temperature Single-Photon Sources: State of the Art. OSA\nQuantum 2.0 Conference. 2020; p QW6B.3.\n(35) Henck, H.; Pierucci, D.; Fugallo, G.; Avila, J.; Cassabois, G.; Dappe, Y. J.; Silly, M. G.;\nChen, C.; Gil, B.; Gatti, M.; Sottile, F.; Sirotti, F.; Asensio, M. C.; Ouerghi, A. Direct\nobservation of the band structure in bulk hexagonal boron nitride. Phys. Rev. B 2017 ,\n95, 085410.\n(36) Pierucci, D.; Zribi, J.; Henck, H.; Chaste, J.; Silly, M. G.; Bertran, F.; Le Fevre, P.;\nGil, B.; Summer\feld, A.; Beton, P. H.; Novikov, S. V.; Cassabois, G.; Rault, J. E.;\nOuerghi, A. Van der Waals epitaxy of two-dimensional single-layer h-BN on graphite\nby molecular beam epitaxy: Electronic properties and band structure. Applied Physics\nLetters 2018 ,112, 253102.\n(37) Arenal, R.; St\u0013 ephan, O.; Kociak, M.; Taverna, D.; Loiseau, A.; Colliex, C. Electron\nEnergy Loss Spectroscopy Measurement of the Optical Gaps on Individual Boron\nNitride Single-Walled and Multiwalled Nanotubes. Phys. Rev. Lett. 2005 ,95, 127601.\nPage 27Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(38) Liu, Z.; Tizei, L. H. G.; Sato, Y.; Lin, Y.-C.; Yeh, C.-H.; Chiu, P.-W.; Terauchi, M.;\nIijima, S.; Suenaga, K. Postsynthesis of h-BN/Graphene Heterostructures Inside a\nSTEM. Small 2015 ,12, 252{259.\n(39) Feenstra, R. M. Scanning tunneling spectroscopy. Surface Science 1994 ,299-300 ,\n965{979.\n(40) Feenstra, R. M. Tunneling spectroscopy of the (110) surface of direct-gap III-V semi-\nconductors. Phys. Rev. B 1994 ,50, 4561{4570.\n(41) Wong, D.; Velasco, J.; Ju, L.; Lee, J.; Kahn, S.; Tsai, H.-Z.; Germany, C.;\nTaniguchi, T.; Watanabe, K.; Zettl, A.; Wang, F.; Crommie, M. F. Characterization\nand manipulation of individual defects in insulating hexagonal boron nitride using\nscanning tunnelling microscopy. Nature Nanotechnology 2015 ,10, 949{953.\n(42) Jin, C.; Lin, F.; Suenaga, K.; Iijima, S. Fabrication of a Freestanding Boron Nitride\nSingle Layer and Its Defect Assignments. Phys. Rev. Lett. 2009 ,102, 195505.\n(43) Schu\u0013 e, L.; Berini, B.; Betz, A. C.; Pla\u0018 cais, B.; Ducastelle, F.; Barjon, J.; Loiseau, A.\nDimensionality e\u000bects on the luminescence properties of hBN. Nanoscale 2016 ,8,\n6986{6993.\n(44) Hern\u0013 andez-M\u0013 \u0010nguez, A.; L ahnemann, J.; Nakhaie, S.; Lopes, J. M. J.; Santos, P. V.\nLuminescent Defects in a Few-Layer h-BN Film Grown by Molecular Beam Epitaxy.\nPhys. Rev. Applied 2018 ,10, 044031.\n(45) S\u0013 aenz, J. J.; Garc\u0013 \u0010a, R. Near \feld emission scanning tunneling microscopy. Applied\nPhysics Letters 1994 ,65, 3022{3024.\n(46) Koma, A. Van der Waals epitaxy for highly lattice-mismatched systems. Journal of\nCrystal Growth 1999 ,201-202 , 236{241.\nPage 28Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(47) Novoselov, K. S.; Mishchenko, A.; Carvalho, A.; Castro Neto, A. H. 2D materials and\nvan der Waals heterostructures. Science 2016 ,353.\n(48) Walsh, L. A.; Hinkle, C. L. van der Waals epitaxy: 2D materials and topological\ninsulators. Applied Materials Today 2017 ,9, 504{515.\n(49) Choi, J.-H.; Cui, P.; Chen, W.; Cho, J.-H.; Zhang, Z. Atomistic mechanisms of van der\nWaals epitaxy and property optimization of layered materials. WIREs Computational\nMolecular Science 2017 ,7, e1300.\n(50) Sediri, H.; Pierucci, D.; Hajlaoui, M.; Henck, H.; Patriarche, G.; Dappe, Y. J.;\nYuan, S.; Toury, B.; Belkhou, R.; Silly, M. G.; Sirotti, F.; Boutchich, M.; Ouerghi, A.\nAtomically Sharp Interface in an h-BN-epitaxial graphene van der Waals Heterostruc-\nture. Scienti\fc Reports 2015 ,5, 16465.\n(51) Summer\feld, A.; Kozikov, A.; Cheng, T. S.; Davies, A.; Cho, Y.-J.; Khlobystov, A. N.;\nMellor, C. J.; Foxon, C. T.; Watanabe, K.; Taniguchi, T.; Eaves, L.; Novoselov, K. S.;\nNovikov, S. V.; Beton, P. H. Moir\u0013 e-Modulated Conductance of Hexagonal Boron Ni-\ntride Tunnel Barriers. Nano Letters 2018 ,18, 4241{4246, PMID: 29913062.\n(52) Vuong, T. Q. P.; Cassabois, G.; Valvin, P.; Rousseau, E.; Summer\feld, A.; Mel-\nlor, C. J.; Cho, Y.; Cheng, T. S.; Albar, J. D.; Eaves, L.; Foxon, C. T.; Beton, P. H.;\nNovikov, S. V.; Gil, B. Deep ultraviolet emission in hexagonal boron nitride grown by\nhigh-temperature molecular beam epitaxy. 2D Materials 2017 ,4, 021023.\n(53) Cho, Y.-J.; Summer\feld, A.; Davies, A.; Cheng, T. S.; Smith, E. F.; Mellor, C. J.;\nKhlobystov, A. N.; Foxon, C. T.; Eaves, L.; Beton, P. H.; Novikov, S. V. Hexagonal\nBoron Nitride Tunnel Barriers Grown on Graphite by High Temperature Molecular\nBeam Epitaxy. Scienti\fc Reports 2016 ,6, 34474.\n(54) Zhou, S. Y.; Gweon, G.-H.; Graf, J.; Fedorov, A. V.; Spataru, C. D.; Diehl, R. D.;\nPage 29Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nKopelevich, Y.; Lee, D.-H.; Louie, S. G.; Lanzara, A. First direct observation of Dirac\nfermions in graphite. Nature Physics 2006 ,2, 595{599.\n(55) Castellanos-Gomez, A.; Wojtaszek, M.; Arramel,; Tombros, N.; van Wees, B. J. Re-\nversible Hydrogenation and Bandgap Opening of Graphene and Graphite Surfaces\nProbed by Scanning Tunneling Spectroscopy. Small 2012 ,8, 1607{1613.\n(56) Pe~ na Rom\u0013 an, R. J.; Auad, Y.; Grasso, L.; Alvarez, F.; Barcelos, I. D.; Zagonel, L. F.\nTunneling-current-induced local excitonic luminescence in p-doped WSe2 monolayers.\nNanoscale 2020 ,12, 13460{13470.\n(57) do Amaral, G. M.; da Costa Tonon, I.; Pe~ na Rom\u0013 an, R. J.; de Oliveira Plath, H.;\nTaniguchi, T. M.; de Lima, L. H.; Zagonel, L. F.; Landers, R.; de Siervo, A. Epitaxial\ngrowth, electronic hybridization and stability under oxidation of monolayer MoS2 on\nAg(111). Applied Surface Science 2021 ,538, 148138.\n(58) Blase, X.; Rubio, A.; Louie, S. G.; Cohen, M. L. Quasiparticle band structure of bulk\nhexagonal boron nitride and related systems. Physical Review B 1995 ,51, 6868{6875.\n(59) Arnaud, B.; Leb\u0012 egue, S.; Rabiller, P.; Alouani, M. Huge Excitonic E\u000bects in Layered\nHexagonal Boron Nitride. Physical Review Letters 2006 ,96.\n(60) Wirtz, L.; Marini, A.; Rubio, A. Excitons in Boron Nitride Nanotubes: Dimensionality\nE\u000bects. Physical Review Letters 2006 ,96.\n(61) Paleari, F.; Galvani, T.; Amara, H.; Ducastelle, F.; Molina-S\u0013 anchez, A.; Wirtz, L.\nExcitons in few-layer hexagonal boron nitride: Davydov splitting and surface local-\nization. 2D Materials 2018 ,5, 045017.\n(62) Aryasetiawan, F.; Gunnarsson, O. TheGWmethod. Reports on Progress in Physics\n1998 ,61, 237{312.\nPage 30Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(63) Art\u0013 us, L.; Feneberg, M.; Attaccalite, C.; Edgar, J. H.; Li, J.; Goldhahn, R.; Cusc\u0013 o, R.\nEllipsometry Study of Hexagonal Boron Nitride Using Synchrotron Radiation: Trans-\nparency Window in the Far-UVC. Advanced Photonics Research 2021 ,2, 2000101.\n(64) Hunt, R. J.; Monserrat, B.; Z\u0013 olyomi, V.; Drummond, N. D. Di\u000busion quantum Monte\nCarlo andGW study of the electronic properties of monolayer and bulk hexagonal\nboron nitride. Phys. Rev. B 2020 ,101, 205115.\n(65) Yang, H.; Liang, A.; Chen, C.; Zhang, C.; Schroeter, N. B. M.; Chen, Y. Visualiz-\ning electronic structures of quantum materials by angle-resolved photoemission spec-\ntroscopy. Nature Reviews Materials 2018 ,3, 341{353.\n(66) Zhang, Y. et al. Direct observation of the transition from indirect to direct bandgap\nin atomically thin epitaxial MoSe 2. Nature Nanotechnology 2014 ,9, 111{115.\n(67) Zhang, C.; Johnson, A.; Hsu, C.-L.; Li, L.-J.; Shih, C.-K. Direct Imaging of Band\nPro\fle in Single Layer MoS2 on Graphite: Quasiparticle Energy Gap, Metallic Edge\nStates, and Edge Band Bending. Nano Letters 2014 ,14, 2443{2447, PMID: 24783945.\n(68) Ugeda, M. M.; Bradley, A. J.; Shi, S.-F.; da Jornada, F. H.; Zhang, Y.; Qiu, D. Y.;\nRuan, W.; Mo, S.-K.; Hussain, Z.; Shen, Z.-X.; Wang, F.; Louie, S. G.; Crommie, M. F.\nGiant bandgap renormalization and excitonic e\u000bects in a monolayer transition metal\ndichalcogenide semiconductor. Nature Materials 2014 ,13, 1091{1095.\n(69) Huang, Y. L.; Chen, Y.; Zhang, W.; Quek, S. Y.; Chen, C.-H.; Li, L.-J.; Hsu, W.-\nT.; Chang, W.-H.; Zheng, Y. J.; Chen, W.; Wee, A. T. S. Bandgap tunability at\nsingle-layer molybdenum disulphide grain boundaries. Nature Communications 2015 ,\n6, 6298.\n(70) Zhang, C.; Chen, Y.; Johnson, A.; Li, M.-Y.; Li, L.-J.; Mende, P. C.; Feenstra, R. M.;\nShih, C.-K. Probing Critical Point Energies of Transition Metal Dichalcogenides: Sur-\nPage 31Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\nprising Indirect Gap of Single Layer WSe2. Nano Letters 2015 ,15, 6494{6500, PMID:\n26389585.\n(71) Krane, N.; Lotze, C.; Franke, K. J. Moir\u0013 e structure of MoS2 on Au(111): Local\nstructural and electronic properties. Surface Science 2018 ,678, 136{142, Surface\nStructure and Dynamics { in Honor of Karl-Heinz Rieder.\n(72) Auw arter, W. Hexagonal boron nitride monolayers on metal supports: Versatile tem-\nplates for atoms, molecules and nanostructures. Surface Science Reports 2019 ,74,\n1{95.\n(73) Bruix, A.; Miwa, J. A.; Hauptmann, N.; Wegner, D.; Ulstrup, S.; Gr\u001cnborg, S. S.;\nSanders, C. E.; Dendzik, M.; Grubi\u0014 si\u0013 c \u0014Cabo, A.; Bianchi, M.; Lauritsen, J. V.; Khaje-\ntoorians, A. A.; Hammer, B.; Hofmann, P. Single-layer MoS 2on Au(111): Band gap\nrenormalization and substrate interaction. Phys. Rev. B 2016 ,93, 165422.\n(74) Dendzik, M.; Bruix, A.; Michiardi, M.; Ngankeu, A. S.; Bianchi, M.; Miwa, J. A.;\nHammer, B.; Hofmann, P.; Sanders, C. E. Substrate-induced semiconductor-to-metal\ntransition in monolayer WS 2.Phys. Rev. B 2017 ,96, 235440.\n(75) Zhang, Q.; Chen, Y.; Zhang, C.; Pan, C.-R.; Chou, M.-Y.; Zeng, C.; Shih, C.-K.\nBandgap renormalization and work function tuning in MoSe2/hBN/Ru(0001) het-\nerostructures. Nature Communications 2016 ,7, 13843.\n(76) Pan, M.; Liang, L.; Lin, W.; Kim, S. M.; Li, Q.; Kong, J.; Dresselhaus, M. S.;\nMeunier, V. Modi\fcation of the electronic properties of hexagonal boron-nitride in\nBN/graphene vertical heterostructures. 2D Materials 2016 ,3, 045002.\n(77) Qi, Y.; Han, N.; Li, Y.; Zhang, Z.; Zhou, X.; Deng, B.; Li, Q.; Liu, M.; Zhao, J.; Liu, Z.;\nZhang, Y. Strong Adlayer{Substrate Interactions \\Break\" the Patching Growth of h-\nBN onto Graphene on Re(0001). ACS Nano 2017 ,11, 1807{1815, PMID: 28110522.\nPage 32Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(78) Zhang, Z. et al. Direct Chemical Vapor Deposition Growth and Band-Gap Character-\nization of MoS2/h-BN van der Waals Heterostructures on Au Foils. ACS Nano 2017 ,\n11, 4328{4336, PMID: 28333441.\n(79) Brihuega, I.; Michaelis, C. H.; Zhang, J.; Bose, S.; Sessi, V.; Honolka, J.; Alexander\nSchneider, M.; Enders, A.; Kern, K. Electronic decoupling and templating of Co nan-\nocluster arrays on the boron nitride nanomesh. Surface Science 2008 ,602, L95{L99.\n(80) Herrera-Reinoza, N.; dos Santos, A. C.; de Lima, L. H.; Landers, R.; de Siervo, A.\nAtomically Precise Bottom-Up Synthesis of h-BNC: Graphene Doped with h-BN Nan-\noclusters. Chemistry of Materials 2021 ,33, 2871{2882.\n(81) Liu, M.; Li, Y.; Chen, P.; Sun, J.; Ma, D.; Li, Q.; Gao, T.; Gao, Y.; Cheng, Z.;\nQiu, X.; Fang, Y.; Zhang, Y.; Liu, Z. Quasi-Freestanding Monolayer Heterostructure\nof Graphene and Hexagonal Boron Nitride on Ir(111) with a Zigzag Boundary. Nano\nLetters 2014 ,14, 6342{6347, PMID: 25268563.\n(82) Li, Q.; Zou, X.; Liu, M.; Sun, J.; Gao, Y.; Qi, Y.; Zhou, X.; Yakobson, B. I.; Zhang, Y.;\nLiu, Z. Grain Boundary Structures and Electronic Properties of Hexagonal Boron\nNitride on Cu(111). Nano Letters 2015 ,15, 5804{5810.\n(83) Thomas, J.; Bradford, J.; Cheng, T. S.; Summer\feld, A.; Wrigley, J.; Mellor, C. J.;\nKhlobystov, A. N.; Foxon, C. T.; Eaves, L.; Novikov, S. V.; Beton, P. H. Step-\row\ngrowth of graphene-boron nitride lateral heterostructures by molecular beam epitaxy.\n2D Materials 2020 ,7, 035014.\n(84) Chiu, M.-H.; Zhang, C.; Shiu, H.-W.; Chuu, C.-P.; Chen, C.-H.; Chang, C.-Y. S.;\nChen, C.-H.; Chou, M.-Y.; Shih, C.-K.; Li, L.-J. Determination of band alignment\nin the single-layer MoS 2 /WSe 2 heterojunction. Nature Communications 2015 ,6,\n7666.\nPage 33Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(85) Bradley, A. J.; M. Ugeda, M.; da Jornada, F. H.; Qiu, D. Y.; Ruan, W.; Zhang, Y.;\nWickenburg, S.; Riss, A.; Lu, J.; Mo, S.-K.; Hussain, Z.; Shen, Z.-X.; Louie, S. G.;\nCrommie, M. F. Probing the Role of Interlayer Coupling and Coulomb Interactions\non Electronic Structure in Few-Layer MoSe2 Nanostructures. Nano Letters 2015 ,15,\n2594{2599, PMID: 25775022.\n(86) Zribi, J.; Khalil, L.; Avila, J.; Chaste, J.; Henck, H.; Oehler, F.; Gil, B.; Liu, S.;\nEdgar, J. H.; Giorgetti, C.; Dappe, Y. J.; Lhuillier, E.; Cassabois, G.; Ouerghi, A.;\nPierucci, D. Structural and electronic transitions in few layers of isotopically pure\nhexagonal boron nitride. Physical Review B 2020 ,102.\n(87) Museur, L.; Brasse, G.; Pierret, A.; Maine, S.; Attal-Tretout, B.; Ducastelle, F.;\nLoiseau, A.; Barjon, J.; Watanabe, K.; Taniguchi, T.; Kanaev, A. Exciton optical\ntransitions in a hexagonal boron nitride single crystal. physica status solidi (RRL) {\nRapid Research Letters 2011 ,5, 214{216.\n(88) Uddin, M. R.; Li, J.; Lin, J. Y.; Jiang, H. X. Probing carbon impurities in hexagonal\nboron nitride epilayers. Applied Physics Letters 2017 ,110, 182107.\n(89) H user, F.; Olsen, T.; Thygesen, K. S. Quasiparticle GW calculations for solids,\nmolecules, and two-dimensional materials. Phys. Rev. B 2013 ,87, 235132.\n(90) Guo, C.; Xu, J.; Ping, Y. Substrate e\u000bect on excitonic shift and radiative lifetime of\ntwo-dimensional materials. Journal of Physics: Condensed Matter 2021 ,33, 234001.\n(91) Sajid, A.; Ford, M. J.; Reimers, J. R. Single-photon emitters in hexagonal boron\nnitride: a review of progress. Reports on Progress in Physics 2020 ,83, 044501.\n(92) Zhang, J.; Sun, R.; Ruan, D.; Zhang, M.; Li, Y.; Zhang, K.; Cheng, F.; Wang, Z.;\nWang, Z.-M. Point defects in two-dimensional hexagonal boron nitride: A perspective.\nJournal of Applied Physics 2020 ,128, 100902.\nPage 34Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(93) Edelberg, D. et al. Approaching the Intrinsic Limit in Transition Metal Diselenides\nvia Point Defect Control. Nano Letters 2019 ,19, 4371{4379, PMID: 31180688.\n(94) Addou, R.; Colombo, L.; Wallace, R. M. Surface Defects on Natural MoS2. ACS\nApplied Materials & Interfaces 2015 ,7, 11921{11929, PMID: 25980312.\n(95) Ziatdinov, M.; Dyck, O.; Li, X.; Sumpter, B. G.; Jesse, S.; Vasudevan, R. K.;\nKalinin, S. V. Building and exploring libraries of atomic defects in graphene: Scan-\nning transmission electron and scanning tunneling microscopy study. Science Advances\n2019 ,5.\n(96) Zhussupbekov, K.; Ansari, L.; McManus, J. B.; Zhussupbekova, A.; Shvets, I. V.;\nDuesberg, G. S.; Hurley, P. K.; Gity, F.; \u0013O Coile\u0013 ain, C.; McEvoy, N. Imaging and\nidenti\fcation of point defects in PtTe2. npj 2D Materials and Applications 2021 ,5,\n14.\n(97) Fujimoto, Y.; Saito, S. E\u000bects of strain on carbon donors and acceptors in hexagonal\nboron nitride monolayers. Phys. Rev. B 2016 ,93, 045402.\n(98) Fujimoto, Y. Formation and Physical Properties of h-BN Atomic Layers: A First-\nPrinciples Density-Functional Study. Advances in Materials Science and Engineering\n2017 ,2017, 1{6.\n(99) Haga, T.; Fujimoto, Y.; Saito, S. STM visualization of carbon impurities in sandwich\nstructures consisting of hexagonal boron nitride and graphene. Japanese Journal of\nApplied Physics 2019 ,58, SIIB03.\n(100) Haga, T.; Fujimoto, Y.; Saito, S. Electronic structure and scanning tunneling mi-\ncroscopy images of heterostructures consisting of graphene and carbon-doped hexag-\nonal boron nitride layers. Phys. Rev. B 2019 ,100, 125403.\nPage 35Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(101) Jamr\u0013 oz, A.; Majewski, J. A. Morphology, Ordering, Stability, and Electronic Struc-\nture of Carbon-Doped Hexagonal Boron Nitride. physica status solidi (b) 2019 ,256,\n1800554.\n(102) Galvani, T.; Paleari, F.; Miranda, H. P. C.; Molina-S\u0013 anchez, A.; Wirtz, L.; Latil, S.;\nAmara, H.; Ducastelle, F. m. c. Excitons in boron nitride single layer. Phys. Rev. B\n2016 ,94, 125303.\n(103) Berseneva, N.; Gulans, A.; Krasheninnikov, A. V.; Nieminen, R. M. Electronic struc-\nture of boron nitride sheets doped with carbon from \frst-principles calculations. Phys.\nRev. B 2013 ,87, 035404.\n(104) Attaccalite, C.; Bockstedte, M.; Marini, A.; Rubio, A.; Wirtz, L. Coupling of excitons\nand defect states in boron-nitride nanostructures. Phys. Rev. B 2011 ,83, 144115.\n(105) Katzir, A.; Suss, J. T.; Zunger, A.; Halperin, A. Point defects in hexagonal boron ni-\ntride. I. EPR, thermoluminescence, and thermally-stimulated-current measurements.\nPhys. Rev. B 1975 ,11, 2370{2377.\n(106) Sajid, A.; Thygesen, K. S. VNCB defect as source of single photon emission from\nhexagonal boron nitride. 2D Materials 2020 ,7, 031007.\n(107) Grosso, G.; Moon, H.; Lienhard, B.; Ali, S.; Efetov, D. K.; Furchi, M. M.; Jarillo-\nHerrero, P.; Ford, M. J.; Aharonovich, I.; Englund, D. Tunable and high-purity room\ntemperature single-photon emission from atomic defects in hexagonal boron nitride.\nNature Communications 2017 ,8, 705.\n(108) Mackoit-Sinkevi\u0014 cien_ e, M.; Maciaszek, M.; Van de Walle, C. G.; Alkauskas, A. Carbon\ndimer defect as a source of the 4.1 eV luminescence in hexagonal boron nitride. Applied\nPhysics Letters 2019 ,115, 212101.\nPage 36Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(109) Jara, C.; Rauch, T.; Botti, S.; Marques, M. A. L.; Norambuena, A.; Coto, R.;\nCastellanos- \u0013Aguila, J. E.; Maze, J. R.; Munoz, F. First-Principles Identi\fcation of\nSingle Photon Emitters Based on Carbon Clusters in Hexagonal Boron Nitride. The\nJournal of Physical Chemistry A 2021 ,125, 1325{1335, PMID: 33554602.\n(110) Zheng, Y. J.; Chen, Y.; Huang, Y. L.; Gogoi, P. K.; Li, M.-Y.; Li, L.-J.; Tre-\nvisanutto, P. E.; Wang, Q.; Pennycook, S. J.; Wee, A. T. S.; Quek, S. Y. Point\nDefects and Localized Excitons in 2D WSe2. ACS Nano 2019 ,13, 6050{6059, PMID:\n31074961.\n(111) Schuler, B. et al. How Substitutional Point Defects in Two-Dimensional WS2 Induce\nCharge Localization, Spin{Orbit Splitting, and Strain. ACS Nano 2019 ,13, 10520{\n10534, PMID: 31393700.\n(112) Barja, S. et al. Identifying substitutional oxygen as a proli\fc point defect in monolayer\ntransition metal dichalcogenides. Nature Communications 2019 ,10, 3382.\n(113) Schuler, B.; Cochrane, K. A.; Kastl, C.; Barnard, E. S.; Wong, E.; Borys, N. J.;\nSchwartzberg, A. M.; Ogletree, D. F.; de Abajo, F. J. G.; Weber-Bargioni, A. Elec-\ntrically driven photon emission from individual atomic defects in monolayer WS2.\nScience Advances 2020 ,6.\n(114) Z\u0013 olyomi, V.; Koltai, J.; K urti, J. Resonance Raman spectroscopy of graphite and\ngraphene. physica status solidi (b) 2011 ,248, 2435{2444.\n(115) Pelini, T.; Elias, C.; Page, R.; Xue, L.; Liu, S.; Li, J.; Edgar, J. H.; Dr\u0013 eau, A.;\nJacques, V.; Valvin, P.; Gil, B.; Cassabois, G. Shallow and deep levels in carbon-\ndoped hexagonal boron nitride crystals. Phys. Rev. Materials 2019 ,3, 094001.\n(116) Weston, L.; Wickramaratne, D.; Mackoit, M.; Alkauskas, A.; Van de Walle, C. G.\nNative point defects and impurities in hexagonal boron nitride. Phys. Rev. B 2018 ,\n97, 214104.\nPage 37Pena Roman et al 2021 2D Material 8 044001 doi:10.1088/2053-1583/ac0d9c\n(117) Linder alv, C.; Wieczorek, W.; Erhart, P. Vibrational signatures for the identi\fcation\nof single-photon emitters in hexagonal boron nitride. Phys. Rev. B 2021 ,103, 115421.\n(118) Shevitski, B.; Gilbert, S. M.; Chen, C. T.; Kastl, C.; Barnard, E. S.; Wong, E.;\nOgletree, D. F.; Watanabe, K.; Taniguchi, T.; Zettl, A.; Aloni, S. Blue-light-emitting\ncolor centers in high-quality hexagonal boron nitride. Phys. Rev. B 2019 ,100, 155419.\n(119) Berzina, B.; Korsaks, V.; Trinkler, L.; Sarakovskis, A.; Grube, J.; Bellucci, S. Defect-\ninduced blue luminescence of hexagonal boron nitride. Diamond and Related Materials\n2016 ,68, 131{137.\n(120) Cheng, T.; Summer\feld, A.; Mellor, C.; Khlobystov, A.; Eaves, L.; Foxon, C.; Be-\nton, P.; Novikov, S. High-Temperature Molecular Beam Epitaxy of Hexagonal Boron\nNitride with High Active Nitrogen Fluxes. Materials 2018 ,11, 1119.\n(121) Cheng, T. S.; Summer\feld, A.; Mellor, C. J.; Davies, A.; Khlobystov, A. N.; Eaves, L.;\nFoxon, C. T.; Beton, P. H.; Novikov, S. V. High-temperature molecular beam epitaxy\nof hexagonal boron nitride layers. Journal of Vacuum Science & Technology B 2018 ,\n36, 02D103.\n(122) Wrigley, J.; Bradford, J.; James, T.; Cheng, T. S.; Thomas, J.; Mellor, C. J.;\nKhlobystov, A. N.; Eaves, L.; Foxon, C. T.; Novikov, S. V.; Beton, P. H. Epitaxy\nof boron nitride monolayers for graphene-based lateral heterostructures. 2D Materials\n2021 ,8, 034001.\nPage 38"    },    {        "title": "1205.3600v1.Ab_initio_Low_Energy_Model_of_Transition_Metal_Oxide_Heterostructure_LaAlO3_SrTiO3.pdf",        "content": "arXiv:1205.3600v1  [cond-mat.str-el]  16 May 2012Typeset with jpsj2.cls <ver.1.2> Full Paper\nAb-initio Low-Energy Model of Transition-Metal-Oxide Heterostruct ure\nLaAlO 3/SrTiO 3\nMotoaki Hirayama∗,1,3TakashiMiyake,2,3and Masatoshi Imada1,3\n1Department of Applied Physics, University of Tokyo, 7-3-1 H ongo, Bunkyo-ku, Tokyo 113-8656, Japan\n2Nanosystem Research Institute, AIST, Tsukuba 305-8568, Ja pan\n3CREST, JST, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\nWe develop the multi-scale ab-initio scheme for correlated electrons (MACE) for transition-\nmetal-oxide heterostructures, and determine the paramete rs of the low-energy effective model.\nBy separating Ti t2gbands near the Fermi level from the global Kohn-Sham (KS) ban ds\nof LaAlO 3/SrTiO 3which are highly entangled with each other, we are able to cal culate the\nparameters of the low-energy effective model of the interfac e with the help of constrained\nrandom phase approximation (cRPA). The on-site energies of the Tit2gorbitals in the 1st-\nlayer is about 650 meV lower than those in the 2nd-layer. In th e 1st-layer, the transfer integral\nof the Ti t2gorbital is nearly the same as that of the bulk SrTiO 3, while the effective screened\nCoulomb interaction becomes about 10 percent larger than th at of the bulk SrTiO 3. The\ndifferences of the parameters from the bulk SrTiO 3reduce rapidly with increasing distance\nfrom the interface. Our present versatile method makes it po ssible to derive effective ab-initio\nlow-energy models and allows studying interfaces of strong ly correlated electron systems from\nfirst principles.\nKEYWORDS: first-principles calculation, effective Hamilto nian, downfolding, constrained RPA method,\ncorrelated-electron systems, heterostructure, interfac e, two-dimensional electron systems\n1. Introduction\nIn recent years, interfaces of strongly correlated\nelectrons have been under intense investigations. Es-\npecially, transition-metal-oxide heterostructure SrTiO 3\n(STO)/LaAlO 3(LAO) has received a lot of attention,\nbecause of its remarkable transport properties.1The in-\nterface of the SrTiO 3/LaAlO 3shows metallic conductiv-\nity, although the bulk materials of each transition-metal-\noxide, SrTiO 3and LaAlO 3, are band insulators. The\nTiO2-terminated ( n-type) interfaces show the metallic\nconductivity,whenthe thicknessofdepositedLAOlayers\nis thicker than four unit cells,2but the SrO-terminated\n(p-type)interfacesareinsulatingforanyLAOthickness.3\nThe origin of this conductivity at the n-type interface\nremains a debated issue. An intrinsic electronic effect,\nnamely the polar discontinuity, and an extrinsic atomic\neffect, atomic vacancies, were proposed to play key roles\nat high carrier concentrations. When the polar LAO lay-\nersaredepositedontheTiO 2-terminatedsubstrateSTO,\nan electric potential along the [001] direction diverges as\nthe LAO thickness increases. The instability of this elec-\ntric potential is suppressed, if 1 /2 doped electron or a\ncorresponding atomic charge vacancy per unit cell ex-\nist at the interface.3,4Besides such transport properties,\nthe existence of superconductivity and magnetic order\nare also reported at the SrTiO 3/LaAlO 3interfaces.5,6\nThese unique transport properties are expected to offer\na useful functionality in the possible applications.\nThe Local Density Approximation (LDA) calculations\nfor transition-metal-oxide heterostructures have already\nbeen performed by several groups. Park et al.inves-\ntigated LaAlO 3/SrTiO 3for three kinds of superlattice\n∗E-mail: hirayama@solis.t.u-tokyo.ac.jpstructures with n-type,p-type, and both types of in-\nterfaces.7By taking into account the relaxation of lat-\ntice, Ishibashi and Terakura investigated the influence of\nLaAlO 3thicknesson the carrierdensity at the interface.8\nThe LDA, however, often fails to capture correlation ef-\nfectsintransition-metal-oxides.Tostudynovelelectronic\nphases in transition-metal-oxide heterostructures, in ad-\ndition to the carrier doping from the electronic recon-\nstruction, we should treat the correlation effects beyond\nthe LDA, because the correlation effect is, on general\ngrounds, expected to be enhanced at interfaces because\nof the effective reduction of the spatial dimensionality.\nThe LDA+U is useful for the strongly correlated ma-\nterials, especially for insulators. Pentcheva and Pickett\ntried to explain the insulating behavior of the p-type\nLAO/STO interface by introducing the on-site U, since\nthe LDAgivesametallic state.9However,in the LDA+U\nmethod, there is no established ab-initio way to estimate\nU, althoughthe value of Ustronglyaffects the calculated\nresults of the low-energy physics. Furthermore, owing to\nthe experimental difficulties, Uhas to be cited from the\nexperimental values of the bulk, not of the interface, es-\npecially in the case where thick LAO layersare on a STO\nsubstrate. Furthermore, the single-particle theory often\ncollapses in strongly correlated materials, even with the\nsuitableU.\nTo overcome such serious problems, a combined\nmethod of the multi-scale ab-initio scheme for corre-\nlated electrons (MACE) is very useful.10With the con-\nstrained random phase approximation (cRPA), we can\ncalculate not only the transfer integral but also the\nscreened Coulomb interaction without relying on any ex-\nperimentalparameters.Correlationeffectscanbetreated\naccurately by solving the obtained ab-initio effective\n12 J. Phys. Soc. Jpn. Full Paper Author Name\nmodel using low-energysolverssuch asvariationalMonte\nCarlo,11dynamical mean field theory (DMFT)12,13\nand path-integral renormalization group (PIRG).14,15\nThe MACE has already been applied to a wide vari-\nety of materials; semiconductor,16transition-metal,17–19\ntransition-metal-oxide,19–23molecular organic conduc-\ntors24and Fe-based layered superconductor.25,26So far,\nthe MACE has been applied to the bulk material. Since\ninterfaces have opened a new avenue of research, it is\nhighly desired to develop a methodology which is able to\ntreatthe interfacefrom the samefootingofthe MACEto\nunderstand the electron correlation effect from the first\nprinciples.\nIn this paper, we apply the ab-initio downfolding\nmethod to transition-metal-oxide heterostructures, and\ndetermine an ab-initio low-energy effective Hamiltonian\nof the SrTiO 3/LaAlO 3. We derive the effective model\nfrom the conduction bands near the Fermi level origi-\nnatedmainlyfromthe Ti t2gorbital.The on-siteenergies\nof the Tit2gorbitals in the 1st-layer is about 650 meV\nlower than those in the 2nd-layer. In the 1st-layer, the\ntransfer integral is nearly the same as that of the bulk\nSrTiO 3, while the effective screened Coulomb interaction\nbecomes about 10 percent larger than that of the bulk\nSrTiO 3. The parameters of the 2nd-layer are similar to\nthose of the bulk SrTiO 3.\nIn Sec.2 we describe our method. Section 3 describes\nthe band structure and the derived effective model of the\nSrTiO 3/LaAlO 3. We also present an effective model of\nSrTiO 3forcomparison.Section 4 is devoted to summary.\n2. Methods\nTo construct the effective low-energy model of the\nheterostructure with well defined parameters, we treat\nthe screening by the high-energy parts working on the\nlow-energy parts starting from the band structure of\nLAO/STO, and renormalize the high-energy parts into\nthe low-energy parts. This downfolding procedure is in-\ntroduced by Aryasetiawan et al.27and Solovyev et al.17\nThe low-energy effective model obtained from the down-\nfolding procedure offers a starting point for studies on\nlow-energy physics.\nWe consider an extended Hubbard Hamiltonian de-\nscribing low-energy electronic properties of the interface\nof LAO/STO,\nH=/summationdisplay\nσ/summationdisplay\nij/summationdisplay\nnmtmn(Ri−Rj)aσ†\ninaσ\njm\n+1\n2/summationdisplay\nσρ/summationdisplay\nij/summationdisplay\nnm/braceleftbigg\nUmn(Ri−Rj)aσ†\ninaρ†\njmaρ\njmaσ\nin\n+Jmn(Ri−Rj)/parenleftbig\naσ†\ninaρ†\njmaρ\ninaσ\njm+aσ†\ninaρ†\ninaρ\njmaσ\njm/parenrightbig/bracerightbigg\n,\n(1)\nwhereaσ†\nin(aσ\nin) is a creation (annihilation) operator of\nan electron with spin σin then-th orbital, which is de-\nfinedbyamaximallylocalizedWannierfunctioncentered\nat theRi-th unit cell.28,29Especially, in this paper, we\nconstruct a model for all Ti t2gorbitals, not only at theinterface but also in the bulk STO region, in the super-\ncell of LAO/STO. With thick LAO and STO layers, this\n3-dimensional model corresponds to the semi-infinite 3-\ndimensional model for (vacuum)-(LaAlO 3)N-(SrTiO 3)∞\nwhere the LAO layers are deposited on the substrate\nbulk STO, because the inter-supercell screening effect is\nrather weaker than the intra-one. Therefore, the param-\neters in this calculation may be used not only for super-\nlattices but also for a semi-infinite interface LAO/STO.\nThe parameters of the Ti t2gorbitals away from the in-\nterface recover the bulk STO nature in both the super-\ncell and semi-infinite interface, because the polarizations\naway from the bulk STO region contribute little to the\nscreening to such t2gorbitals. In fact, in the present cal-\nculation, we show that the parameters in the 2nd STO\nlayer from the interface nearly converge to those of the\nbulk STO. With thick LAO layers, the parameters of\nthis 3-dimensional model are also similar to those of a 2-\ndimensional model for LAO/STO superlattice, where all\nof the inter-supercell screening effects are renormalized.\nWe derive the effective model in the following way:\nFirst, we calculate the whole band structure of the\ntransition-metal-oxide heterostructure in the framework\nof density functional theory (DFT). We then choose\nthe target low-energy band around the Fermi level and\nconstruct the maximally-localized Wannier functions28\nin the low-energy Hilbert space. The transfer integral,\nwhich definesthe one-bodypartofthe low-energymodel,\nisobtainedasthematrixelementoftheKohn-Sham(KS)\nHamiltonian HKS,\ntmn(R) =/angbracketleftφ0m|HKS|φRn/angbracketright, (2)\nwhereφRnis then-th Wannier function centered at the\ncellR.\nNext, we renormalize the effect from the high-energy\nspace (rspace) into the target low-energy space ( d\nspace), and evaluate effective electron interaction by us-\ning the cRPA. We divide the polarization\nP(r,r′;ω) =occ/summationdisplay\niunocc/summationdisplay\njψi(r)ψ∗\ni(r′)ψ∗\nj(r)ψj(r′)\n×/bracketleftBig1\nω−ǫj+ǫi+iδ−1\nω+ǫj−ǫi−iδ/bracketrightBig\n(3)\ninto the polarization Pdthat includes only the d-dtran-\nsitions and the rest of the polarization Pr. The screened\nCoulomb interaction Wis given by\nW= [1−vP]−1v\n= [1−WrPd]−1Wr,(4)\nwhere we define the partially screened Coulomb interac-\ntionWrthat does not include the screening arising from\nthe polarization from the d-dtransitions as\nWr= [1−vPr]−1v. (5)\nThe screened Coulomb matrix is defined by\nWr(R1n,R2n′,R3m,R4m′;ω)J. Phys. Soc. Jpn. Full Paper Author Name 3\n=/integraldisplay\nd3rd3r′φ†\nR1n(r)φR2n′(r)\n×Wr(r,r′;ω)φ†\nR3m(r′)φR4m′(r′).(6)\nThe effective Coulomb interaction Uand the exchange\ninteraction Jare given by\nUnm(R) =Wr(0n,0n,Rm,Rm;0),(7)\nJnm(R) =Wr(0n,0m,Rm,Rn;0).(8)\nIn LaAlO 3/SrTiO 3, the Tit2gstates are entangled\nwith the La 4 fstates. We disentangle the Ti t2gbands\nusing the recently developed disentangling technique.30\nIn the LDA, the La 4 flevel in LAO/STO is located\nnear the Fermi level. However, in the real material, the\nLa 4fbands are expected to be at higher energy, and, as\naconsequence,donotstronglyscreentheinteractionsbe-\ntween the electrons in the t2gbands. To calculate the pa-\nrametersofthe low-energymodel with ahigher accuracy,\na better way is to take into account the correlation effect\nof the La 4fbands by the GW approximation (GWA) as\na preconditioning before determining the effective model\nfor Ti 3d t2gorbitals. The detail of the GWA for the\ndisentangled 4 fbands is explained in Appendix.\nComputational conditions are as follows. We calcu-\nlate the band structures of the bulk STO and LAO,\nand the LAO/STO heterostructure based on the DFT-\nLDA.31,32The calculations are carried out with the pro-\ngrambasedonthe full-potentiallinearmuffin-tin orbitals\n(FP-LMTO) method.33Because the LAO are deposited\nto fit the substrate STO, the lattice constants of the de-\nposited LAO layers in the direction horizontal to the in-\nterfacearesameasthoseofthesubstrateSTO,whilethat\nin the direction perpendicular to the interface slightly\nchanges depending on the carrier density at the inter-\nface. Therefore, in this study, the structure is fixed as\ncubic, and the lattice constants for the bulk and the het-\nerostructureperpendiculartothe[001]stackingdirection\nare fixed to 3 .905˚A,which correspondsto the experimen-\ntal lattice constant of the bulk SrTiO 3. In the LDA cal-\nculations, 8 ×8×8k-point sampling is employed for the\nbulk, and 8 ×8×2k-point sampling is employed for the\nLAO/STO to represent electronic structures of the sys-\ntem. The muffin-tin (MT) radii are: RMT\nTi= 2.50 bohr,\nRMT\nSr= 2.10bohr,RMT\nAl= 1.90 bohr,RMT\nLa= 1.60 bohr,\nandRMT\nO= 1.6 bohr. The angular momentum cut off is\ntaken atl= 4 for all the sites. In the cRPA and the GW\ncalculations, 3 ×3×3k-point sampling is employed for\nthe bulk, and 3 ×3×1k-point sampling is employed for\nthe LAO/STO.\n3. Results\n3.1 band structure and density of state\nFirst, we show the band structure and the density of\nstates of the insulators, bulk SrTiO 3and LaAlO 3. The\nupper panels of Figs. 1 and 2 show the band structure\nand the density of states , respectively. At room temper-\nature, SrTiO 3has the cubic perovskite structure, and\nbecomes tetragonal below 105 K. SrTiO 3has a high di-\nelectric constant at low temperatures because of the na-\nture of quantum paraelectricity.34In this calculation, theFig. 2. (color online) Density of states of bulk STO and LAO,\nand LAO/STO obtained by LDA. Energy is measured from the\nFermi level.\nstructure is fixed as cubic, and the lattice parameters\nare fixed at 3 .905˚A of the bulk SrTiO 3. The lower three\nconduction bands are derived from the t2gorbital of Ti\nsites, where the octahedral crystal field of O2−partially\nbreaks the 5-fold symmetry of the 3 dorbitals into the\nlower orbitals of the t2gand the higher orbital eg. The\ncalculated band gap is 2 .1 eV and the band width of\nthet2gband is 2.8 eV. In the experiment, the band gap\nis 3.3 eV.35In the strongly correlated materials, such\nunderestimation of the band gap causes to overestimate\nthe screening effect from the high-energy bands to the\nlow-energy bands in the cRPA. In the LAO/STO, the\nunderestimation of the energy levels of La 4 fbands is a\nmajor problem which are entangled with Ti t2gbands in\nthe LDA as we will show. LaAlO 3has the rhombohedral\nperovskite structure at room temperature, and becomes\ncubicabove821K.TocomparewiththeLAOlayerinthe\nheterostructure, the structure and the lattice parameters\nare fixed at those of the cubic SrTiO 3in this calculation.\nThe lowernarrowconductionbandsofLAOat ∼4eVare\nderived from the 4 forbital of La sites. The La 5 dbands\nare hybridized with the La 4 fbands at the Γ-point. In\nthe experiment, the band gap between the La 4 fand the\nO 2pis 5.6 eV.36In this calculation, however, the band\ngap is 3.4 eV because the LDA underestimates the value\nof the band gap. This band-gap problem in the LDA is4 J. Phys. Soc. Jpn. Full Paper Author Name\nFig. 1. Electronic band structures obtained by the LDA. The z ero energy corresponds to the Fermi level. Upper left panel: Electronic\nband of STO. Upper right panel: Electronic band of LAO. Lower panels: Electronic bands of LAO1 .5STO3.5. The right one is the\nenlarged view around the Fermi level. Wave functions of the s tates indicated by arrows are displayed in Fig. 4.\nFig. 3. (color online) Schematic picture of the model of the\ntransition-metal-oxide heterostructure LAO1 .5STO3.5. The sys-\ntem is modeled by a periodically repeated supercell contain ing\ntwo interfaces. The both interfaces are LaO-TiO 2. Owing to neu-\ntrality of the electrons and atoms, these systems are formal ly\ncharged by −eelectron per supercell.\nimproved dramatically with the GWA. We will show a\nresult in the GWA later.\nNext,weshowthe LDAresultsofthetransition-metal-\noxide heterostructure LAO1 .5STO3.5. We refer to the\nTiO2layer at the interface as the 1st-layer and the TiO 2\nlayerinthe bulk regionofthe SrTiO 3asthe 2nd-layer.Inthis paper, we refer to the heterostructure -(LaAlO 3)1-\nLaO/TiO 2-(SrTiO 3)3- as LAO1 .5STO3.5 (see Fig. 3).\nThis heterostructure has two crystallographically equiv-\nalentn-type interface, and has about 1 /2 carrierelectron\nat eachn-type interface, because LAO1 .5+1STO3.50has\na positive charge in the ionic limit. In terms of the po-\nlar discontinuity, these models of the heterostructures\nare the cases where the instabilities of the potential di-\nvergence are completely suppressed.3,4The lower pan-\nels of Fig. 1 show the band structure of the n-type\nLAO1.5STO3.5 heterostructure. The energy bands are\nrather degenerate because there are two crystallograph-\nically equivalent n-type interfaces. The LAO1 .5STO3.5\nhas the Fermi surface around the Γ point. The density of\nstates ofthe LAO/STOcomes from nearly superimposed\nstates of the bulk LAO and STO (see Fig. 2).\nFigures 4 shows the isosurface contours of selected\nBloch functions of the conduction bands at the Γ-point.\nThe panel (a) of Fig. 4 is the isosurface contour of the\nlowest conduction band (see Fig. 1 (d)). This band is\noriginated mainly from the Ti dxyin the 1st-layer. The\npanel (b) is the isosurface contour of the Ti dyzband,\nwhich spreads in the direction horizontal to the inter-\nface. The panel (c) and (d) are the isosurface contours\nof the Tidxyband in the 2nd-layer and the La 4 fband,\nrespectively. The Ti t2gand La 4forbitals are spatially\nclose and, in the LDA level, energetically close, so thatJ. Phys. Soc. Jpn. Full Paper Author Name 5\nFig. 4. (color online)Isosurface contours of the Bloch func tions of\nLAO1.5STO3.5 conduction bands at Γ-point[000]. The structure\ncorresponds to Fig. 3, for example, the panel A represents th e\nTidxyat the interface. Value of the isosurface is set to ±0.05\nbohr−3/2. Each energy level is measured from the Fermi level\nand is depicted as arrows in Fig. 1.\nthe Tit2gand La 4forbitals are hybridized at the in-\nterface (see Fig. 4). The Bloch functions transfer to the\nbulk region with increasing those energy levels. There is\na positive crystal field from the polar perovskite LaAlO 3\nto the non-polar substrate SrTiO 3as compared with the\ncase with only Sr2+for the bulk SrTiO 3. This crystal\nfield vanishes in the bulk region of SrTiO 3, because the\nnegative field from the doped electrons compensates this\npositive field. From these reasons, the energy levels of\nthe orbitals in the 1st-layer are lower than that in the\n2nd-layer, and the gap between the valence and conduc-\ntion bands in LAO1 .5STO3.5 is smaller than that of the\nbulk STO. Such effect of compensation and confinement\nbecomes strong as the polar perovskite LaAlO 3becomes\nthicker. This tendency is also seen in Fig. 5 which shows\nthe partial density of states at the Ti sites of dorbitals\nand the O sites of porbitals in the TiO 2layer of the\nbulk STO and LAO/STO obtained by the LDA. Here-Fig. 5. (color online) Partial densities of states of Ti dand Opat\nTiO2layers of bulk STO and LAO/STO obtained by the LDA.\nEnergy is measured from the Fermi level.\nafter, for instance, a Wannier orbital of the xyorbital in\nthe 1st-layer is denoted by 1 xy. Pronounced peak shifts\nof DOS to the lower energy are found for the sites in the\n1st-layer.\nConsidering the experimental band gap of the bulk\nLaAlO 3, we note that the location of the energy of the\nLa 4fbands in the LaAlO 3/SrTiO 3is too low in the\nLDA. The La 4 fbands screen and hybridize with the Ti\nt2gbands weaker in the real material. To calculate the\nparameter of the low-energymodel with a high accuracy,\na better way is to take into account the correlation effect\nof La separately by the GWA. To calculate the self ener-\ngies of the La 4 fbands, we first construct the maximally\nlocalized Wannier function of La 4 ffrom a linear com-\nbination of the target low-energy KS-bands. We choose\nthe energy window from −1 eV to 3 eV for the Wannier\nfunctions. We find, however, that the screened Coulomb\ninteraction of the low-energy bands derived from the Ti\nt2gis not sensitive to the choice of the energy window if\nthe window exceeds a certain width but is not too wide.\nNext, we calculate the self-energy corrections of the La\n4fbands by the 1-shot GW scheme. Figure 6 shows the\nband structure after the 1-shot GW corrections for the6 J. Phys. Soc. Jpn. Full Paper Author Name\nFig. 6. (color online) Electric band structures with 1-shot GW\nself-energy for 4 fbands. Energy is measured from the Fermi\nlevel.\nFig. 7. (color online) Disentangled d-bands having strong t2g\ncharacter ((blue) dashed line) and diagonalized r-bands (solid\nline).\n4fbands.The energylevel of4 fbands israisedbyabout\n1.5 eV with GW. This value is consistent with the ex-\nperimental result ofthe LAO.36In the following sections,\nwe show this self-energy effect on the effective model pa-\nrameters.\n3.2 Wannier function and transfer integral\nWe calculate the transfer integral tof the Tit2gor-\nbital to determine the 1-body part of low-energy effec-\ntive Hamiltonian for the heterostructure. First, we con-\nstruct 6×2 maximally localized Wannier functions hav-\ning strong Ti t2gcharacters, from the linear combination\nof the target low-energy KS-bands. We choose −1-3 eV\nas the energy window for the Wannier functions. Figure\n7 shows the disentangled t2gbands and the rest bands,\nand Fig. 8 shows the isosurfaces of Wannier functions of\nthedxyanddyzorbitals.\nWe show in Table I the transfer integrals tof the bulk\nSTO calculated in the LDA. In the tables and this sub-\nsection,mandnspecify symmetries of the Ti t2gor-\nbitals. The values of on-site energies are listed in the\ncolumn for ( Rx,Ry,Rz) = (0,0,0). The values of the\non-site energies are the same for all the t2gorbitals be-\ncause of the cubic crystal symmetry. The major values\nof the nearest hopping between the same symmetries\nFig. 8. (color online) Isosurface of the maximally localize d Wan-\nnier functions ±0.05 a.u. for the dxyanddyzorbitals in the\n1st-layer.\n(Rx,Ry,Rz) = (1,0,0) are 299 meV. These values are\nconsistent with the values in the literature in nearest-\nneighbor tight-binding models of the transition-metal-\noxide interfaces ( t∼0.3 eV).37The nearest hopping for\nthe perpendicular directions of the orbitals symmetry is\n−39 meV, which is 13 percent of these of the main direc-\ntions. The next nearest hopping ( Rx,Ry,Rz) = (1,1,0)\nis 35 percent of the nearest neighbor hopping.\nNext, we show in Table II the transfer integrals tof\nthe LAO1.5STO3.5 calculated in the LDA and the GW,\nrespectively. The values of on-site energy in the 1st-layer\nt1m,1m(0,0,0) are about 650 meV lower than those in\nthe 2nd-layer t2m,2m(0,0,0), mainly because the dipole\nmoment of LaO1+-AlO1−\n2stabilizes the energy of the t2g\norbitals in the 1st-layer. The hybridization between the\nLa 4fand Tit2gorbitals also slightly stabilizes the en-\nergy of the t2gorbitals. These layer dependent potential\nlocalizes carriers at the interface. Actually, in the exper-\niment, the transition into the 2D superconducting state,\nnamely the Berezinskii-Kosterlitz-Thouless transition is\nseen at the interface of the LAO/STO.5The value of\nthe on-site energy of 1 xy,t1xy,1xy(0,0,0), is lower than\nthat of the other 1 t2gdue to the crystal field and the\nhybridization with the LAO layer. In the GWA, this dif-\nference of the on-site energy at the 1st-layer is 16 meV\nlarger than that in the LDA, because the hybridizations\nof the La 4forbitals with the Ti t2gorbitals in the 1st-\nlayer, especially with 1 yzand 1zx, become weaker. In\nthe 2nd-layer, the dipole moment of LaO1+-AlO1−\n2have\nlittle effect, and the values of on-site energy partially\nrecover the bulk STO nature. The hopping parameters\nare similar to those of the bulk STO. The major val-\nues of the nearest hopping between the same symmetries\nt1m,1m(1,0,0) are about 0 .3 eV. The nearest hopping of\nthe 1xyis smaller than that of the bulk STO. The near-\nest hopping of the 1 yzand 1zxare nearly the same as\nthose of the bulk STO. The main difference of the hop-\nping parametersfrom the bulk STOis seen in the nearest\nhopping for the perpendicular directions of the orbitals\nsymmetryt1yz,1yz(1,0,0). Such hoppings are −59 meV,\nwhich are about twice as large as those of the bulk STO.J. Phys. Soc. Jpn. Full Paper Author Name 7\nIn the LAO/STO, therefore, the carriers tend to spread\nin the direction horizontal to the interface compared to\nthe bulk STO. As with the on-site energy, the hopping\nparameters in the 2nd-layer t2m,2m(1,0,0) are nearly the\nsame as those of the bulk STO. In the GWA, the hy-\nbridization between the La 4 fandt2gorbitals in the\n1st-layer, especially 1 yzand 1zx, becomes weaker. As a\nresult, the splitting of on-site energies of t2gorbitals be-\ncomes larger, and the hopping parameter between 1 zx-\n2zxbecomes smaller.\n3.3 screened Coulomb interaction\nNext, we calculate the on-site screened Coulomb in-\nteractionUmn(0,0,0) and the on-site screened exchange\ninteraction Jmn(0,0,0) of the Ti t2gorbital to determine\nthe 2-body part of the low-energy effective Hamiltonian\nfor the heterostructure. The screened Coulomb interac-\ntion is computed with the matrix elements in the max-\nimally localized Wannier basis in the framework of the\ncRPA (see eq. (6)).\nIn the top of Table III, we show the effective on-site\nCoulomb interaction UandJfor the bulk STO. As with\nthe transfer integrals, mandndenote symmetries of the\nt2gorbitalsofTiinthetablesandthissubsection.Theef-\nfective on-site Coulomb interaction Ubetween the same\norbitals is 3 .76 eV, while the bare on-site Coulomb in-\nteraction is 14 .27 eV. The on-site Coulomb interaction is\nreduced about to 1 /4 by the polarizations without the d-\ndcontributions. The effective on-site screened exchange\ninteraction Jis 0.46 eV.\nWe show the effective on-site Coulomb interaction U\nandJof LAO/STO calculated with cRPA in the mid-\ndle and bottom of the Table III. Here, Ubetween the\nsame orbitals are 3 .4-3.6 eV from the LDA and 3 .7-4.0\neV from the GWA, while that of the bulk STO is 3 .76\neV. In both the LDA and GWA, U1xy,1xyis the largest\namong on-site screened Coulomb interactions in the 1st-\nlayer, andU1yz,1yzandU1zx,1zxare 4-5 percent smaller\nthanU1xy,1xy. In the GWA, UandJat the 1st-layer\nbecome larger than those of the bulk STO, while these\nare smaller in the LDA. This is because the La bands\nare raised away from the Fermi level by the self-energy\ncorrection,thepolarizationsbetweenLa4 fandTit2gor-\nbitals become weaker compared to the LDA bands, and,\nas a result, the screening effect becomes weaker. The on-\nsite screened Coulomb interaction Umnand the on-site\nscreened exchange interaction Jmnsatisfy the equation\nUmn=Umm−2Jmnin thebulk STO.Onthe otherhand,\nat the interface of the LAO/STO, these parameters do\nnot satisfy this equation because of the inversion sym-\nmetry breaking. In the 2nd-layer, UandJrecover the\nbulk STO nature. Although, similarly in the 1st-layer,\nU2xy,2xyis the largest on-site screened Coulomb inter-\naction in the 2nd-layer, the difference between U2xy,2xy\nand the others of t2gis only about 1 percent. Because the\nelectrons confined at the interface have low dimensional-\nity, the correlation becomes effectively stronger than the\nbulk STO, even if the transfer integrals at the interface\nare similar to those of the bulk STO.4. Summary\nIn this paper, we have determined the parameters of\nthe low-energy effective model of LaAlO 3/SrTiO 3by\nthe MACE. As with many interfaces, LaAlO 3/SrTiO 3\nhas a complex band structure where the bands of both\nthe interface and bulk regions are highly entangled with\neach other. By the disentangling scheme using the max-\nimally localized Wannier function,30we disentangle the\nlow-energy part having the strong characters of Ti t2g\norbital from the global KS-bands, and thus enable to\ncalculate the parameters in the effective Hamiltonian of\nLaAlO 3/SrTiO 3by the cRPA. The parameters in this\nstudy offer not only the superlattice model but also\nthe semi-infinite interface model (vacuum)-(LaAlO 3)N-\n(SrTiO 3)∞, because the screening effect from the inter-\nsupercell rapidly decreases with distance and thus the\nsuperlattice model corresponds to the semi-infinite in-\nterface model in the limit of thick LAO and STO. The\nparameters have anisotropies and a layer dependence.\nThe on-site energies in the 1st-layer are 650 meV higher\nthan those in the 2nd-layer, which causes localization of\nthe carriers at the interface. In the 1st-layer, while the\ntransfer integral of the t2gorbital is similar to that of\nthe bulk SrTiO 3, the screened Coulomb interaction Uof\nthet2gorbital becomes 10 percent larger than that of\nthe bulk SrTiO 3. In the bulk region of LaAlO 3/SrTiO 3,\nthe parameters of the t2gorbitals recover the values\nand the symmetry of the bulk SrTiO 3. The obtained\n3-dimensional parameters constitute the low-energy ef-\nfective models of LaAlO 3/SrTiO 3, either in semi-infinite\nstructures in one of the directions or in supercell struc-\ntures. The resultant low-energy effective model offers a\nfirm and quantitative basis when one wishes to solve the\neffective model by using accurate low-energy solvers in\nthe future.\nRecently, an ab-initio dimensional downfolding\nscheme, which downfolds a 3-dimensional model to\na lower-dimensional model in real space, has been\nformulated.39By applying the dimensional downfolding\nscheme to the 3-dimensionalmodel ofLAO/STO,we can\nalso obtain a 2-dimensional model of a single LAO/STO\nsupercell. In the dimensional downfolding, one is able\nto expect that the weakness of inter-layer (inter-chain)\ncouplings justify the RPA type perturbative treat-\nment.10In the present case, however, the interface layer\nis not particularly weakly coupled with the other layers.\nNevertheless, the layers far away from the interface\ndo not join in the low-energy excitations if the bulk is\ninsulating as in the present case. Therefore, the gapped\nexcitations in the bulk part can be safely downfolded\ninto the metallic low-energy excitation near the interface\nin the same spirit of the cRPA. This enables to derive\neffective low-energy models solely for the interface part.\nOf course, we may need to keep the bands of not one\nbut several layers near the interface in the low-energy\nmodels. This dimensional downfolding is a challenging\nfuture issue.8 J. Phys. Soc. Jpn. Full Paper Author Name\nTable I. Transfer integrals for the t2gorbitals of the Ti sites in the bulk STO, tmn(Rx,Ry,Rz), where mandndenote symmetries of\nt2gorbitals. Units are given in meV.\nSTO\nPPPPPP(m,n)R/bracketleftbig\n0,0,0/bracketrightbig /bracketleftbig\n1,0,0/bracketrightbig /bracketleftbig\n1,1,0/bracketrightbig\n(xy,xy) 2430 −299−109\n(xy,yz) 0 0 0\n(xy,zx) 0 0 0\n(yz,yz) 2430 −39 7\n(yz,zx) 0 0 9\n(zx,zx) 2430 −299 7\nTable II. Transfer integrals for t2gorbitals of Ti sites in LAO1 .5STO3.5 calculated in LDA and GWA level, tmn(Rx,Ry,Rz), where m\nandndenote layers and symmetries of t2gorbitals. Units are given in meV. The parameters calculated from the LDA band are listed\nas “LDA”, and the parameters calculated from the LDA band wit h the self-energy correction of the La 4 fare listed as “GW”.\nLAO/STO LDA GW\nPPPPPP(m,n)R/bracketleftbig\n0,0,0/bracketrightbig /bracketleftbig\n1,0,0/bracketrightbig /bracketleftbig\n1,1,0/bracketrightbig/bracketleftbig\n0,0,0/bracketrightbig /bracketleftbig\n1,0,0/bracketrightbig /bracketleftbig\n1,1,0/bracketrightbig\n(1xy,1xy)2606 −289−101 2607 −288−104\n(1xy,1yz) 0−7−1 0−4 2\n(1xy,1zx) 0 0 −1 0 0 2\n(1xy,2xy) −42 6 −9−42 6 −9\n(1xy,2yz) 0 10 6 0 9 6\n(1xy,2zx) 0 0 6 0 0 6\n(1yz,1yz)2617 −57 −22634 −59 −5\n(1yz,1zx) 0 0 −6 0 0 −9\n(1yz,2xy) 0 8 8 0 8 8\n(1yz,2yz)−296 3 −6−299 4 −5\n(1yz,2zx) 0 0 8 0 0 8\n(1zx,1zx)2617 −302 −22634 −298 −5\n(1zx,2xy) 0 0 8 0 0 8\n(1zx,2yz) 0 0 8 0 0 8\n(1zx,2zx)−296 −98 −6−299 −98 −5\n(2xy,2xy)3268 −298−109 3264 −298−109\n(2xy,2yz) 0−3 0 0−3 0\n(2xy,2zx) 0 0 0 0 0 0\n(2yz,2yz)3262 −39 6 3262 −39 6\n(2yz,2zx) 0 0 8 0 0 7\n(2zx,2zx)3262 −297 6 3262 −297 6\nAcknowledgment\nMH would like to thank Kazuma Nakamura, Takahiro\nMisawa, Youhei Yamaji, Hiroshi Shinaoka, and Ryota\nWatanabeforusefuladvicesandfruitfuldiscussions.This\nwork has been supported by Grant-in-Aid for Scientific\nResearch from MEXT Japan under the grant numbers\n22104010 and 22340090. This work has also been finan-\ncially supported by MEXT HPCI Strategic Programsfor\nInnovative Research (SPIRE) and Computational Mate-\nrials Science Initiative (CMSI).Appendix: GWA for disentangled band\nThe quasiparticle energies and wave functions are ob-\ntained by solving\n(T+Vext+VH)ψnk(r)+/integraldisplay\ndr′Σ(r,r′;Enk)ψnk(r′)\n=Enkψnk(r),\n(A·1)\nwhereTis the kinetic energy operator, and Vextis the\nexternal potential, VHis the Hartree potential. If we con-\nsider only the diagonal parts of the self-energy Σ, eq.\n(A·1) is reduced to\nEnk=ǫLDA\nnk−/angbracketleftnk|VLDA\nex|nk/angbracketright+/angbracketleftnk|Σ(Enk)|nk/angbracketright.(A·2)\nThe self-energy operator must be estimated at the quasi-\nparticle energy Enk. This is done by expanding the ma-J. Phys. Soc. Jpn. Full Paper Author Name 9\nTable III. Effective Coulomb interaction ( U)/exchange ( J) interactions between the two electrons for all the combina tions of Ti- t2g\norbitals in the bulk STO and LAO1 .5STO3.5, respectively (in eV). Especially, for the combinations b etween the orbitals in the same\nTi sites, the matrix elements represent the effective on-sit e Coulomb interaction. The band structures of LAO1 .5STO3.5 are calculated\nin the LDA and GWA.\nSTO U J\nxy yz zx xy yz zx\nxy 3.76 2.81 2.81 xy 0.46 0.46\nyz 2.81 3.76 2.81 yz0.46 0.46\nzx 2.81 2.81 3.76 zx0.46 0.46\nLAO/STO(LDA) U J\n1xy1yz1zx2xy2yz2zx 1xy1yz1zx2xy2yz2zx\n1xy 3.65 2.63 2.63 0.49 0.55 0.55 1 xy 0.46 0.46 0.00 0.00 0.00\n1yz 2.63 3.48 2.58 0.55 0.65 0.63 1 yz0.46 0.44 0.00 0.01 0.00\n1zx 2.63 2.58 3.48 0.55 0.63 0.65 1 zx0.46 0.44 0.00 0.00 0.01\n2xy 0.49 0.55 0.55 3.62 2.67 2.67 2 xy0.00 0.00 0.00 0.46 0.46\n2yz 0.55 0.65 0.63 2.67 3.59 2.65 2 yz0.00 0.01 0.00 0.46 0.46\n2zx 0.55 0.63 0.65 2.67 2.65 3.59 2 zx0.00 0.00 0.01 0.46 0.46\nLAO/STO(GW) U J\n1xy1yz1zx2xy2yz2zx 1xy1yz1zx2xy2yz2zx\n1xy 4.00 2.98 2.98 0.72 0.79 0.79 1 xy 0.46 0.46 0.00 0.00 0.00\n1yz 2.98 3.83 2.93 0.78 0.89 0.87 1 yz0.46 0.44 0.00 0.01 0.00\n1zx 2.98 2.93 3.83 0.78 0.87 0.89 1 zx0.46 0.44 0.00 0.00 0.01\n2xy 0.72 0.78 0.78 3.80 2.84 2.84 2 xy0.00 0.00 0.00 0.46 0.46\n2yz 0.79 0.89 0.87 2.84 3.76 2.83 2 yz0.00 0.01 0.00 0.46 0.46\n2zx 0.79 0.87 0.89 2.84 2.83 3.76 2 zx0.00 0.00 0.01 0.46 0.4610 J. Phys. Soc. Jpn. Full Paper Author Name\nFig. A·1. Real and imaginary parts of the matrix elements of the\nself-energy Σ for a 4 fWannier band in LAO1 .5STO3.5 at Γ-\npoints. Solid and dashed lines represent the real and imagin ary\nparts of the Σ, respectively.\ntrix elements ofthe self-energy operatorto the first order\nin the energy around ǫnk. Then the quasiparticle energy\nis obtained explicitly;\nEnk=ǫnk+Znk(ǫnk)(ǫLDA\nnk−ǫnk+∆Σnk(ǫnk)),(A·3)\nwhere the self-energy is given by\n∆Σnk= Σnk−VLDA\nex (A·4)\nandZnkis the renormalization factor\nZnk= (1−∂∆Σnk(ǫnk)\n∂ω)−1. (A·5)\nIfǫnkcoincides with ǫLDA\nnk, then eq. (A ·3) is simplified;\nEnk=ǫLDA\nnk+Znk(ǫLDA\nnk)∆Σnk(ǫLDA\nnk).(A·6)\nWe show the real and imaginary parts of the self-\nenergy for a 4 fWannier band calculated by the 1-shot\nGW approximation in Fig. A ·1. The real and imaginary\npartsofthe self-energystronglyoscillateabovethe Fermi\nlevelEF= 0 eV, especially at 2-3 eV near the La 4 f\nbands in the LDA level. This lack of smoothness of the\nself-energy Σ is obtained in the “1-shot” correction. The\nself-energy Σ should hopefully be calculated in a self-\nconsistent procedure of the Hedin’s set of coupled equa-\ntions.38In this calculation, we have calculated the self-\nenergy correction by one shot without iteration to save\nthe computational cost. Considering the unstable behav-\nioroftheself-energyneartheLa4 fbands,weexpandeq.\n(A·2) aroundEF= 0 eV and then quasiparticle energies\nare approximated in the first order as follows;\nEnk=Znk(0)(ǫLDA\nnk+∆Σnk(0)).(A·7)\nThe approximation eq. (A ·6) is equivalent to eq. (A ·7),\nwhich is justified if the linearity of self-energy Σ in\nthe low-energy region is eventually recovered in the\nself-consistent accurate estimate. The smoothness of Σ\nshould be eventually obtained after the self-consistent\ncalculation of the GWA. In this calculation, we average\nZnk(0) and ∆Σ nk(0) in terms of the band index and thek-points;\nEnk=/angbracketleftZnk(0)/angbracketrightnk(ǫLDA\nnk+/angbracketleft∆Σnk(0)/angbracketrightnk).(A·8)\nSince we are interested in the La 4f level measured\nfrom the Fermi level, the self-energy correction to the\nLa 4f level is corrected by subtracting the correction to\nthe Fermi level. The latter is approximately evaluated by\nthe GW self-energy for the bottom of the t2gband us-\ning eq.(A ·7). Figure 6 shows the band structure thus ob-\ntained after the 1-shot GW corrections for the 4 fbands.\n1) A. Ohtomo and H. Y. Hwang: Nature (London) 427(2004)\n423.\n2) S. Thiel, G. Hammer, A. Schmehl, C. W. Schneider, and J.\nMannhart: Science 313(2006) 1942.\n3) N. Nakagawa, H. Y. Hwang, and D. A. Muller: Nat. Mater. 5\n(2006) 204.\n4) M. Hirayama and M. Imada: J. Phys. Soc. Jpn. 79(2010)\n034704.\n5) N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis ,\nG. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S.\nR¨ uetschi, D.Jaccard, M.Gabay, D.A.Muller, J.-M.Triscon e,\nand J. Mannhart: Science 317(2007) 1196.\n6) A. Brinkman, M. Huijben, M. V. Zalk, J. Huijben, U. Zeitler ,\nJ. C. Maan, W. G. Van Der Wiel, G. Rijders, D. H. A. Blank,\nand H. Hilgenkamp: Nat. Mater. 6(2007) 493.\n7) M. S. Park, S. H. Rhim, and A. J. Freeman: Phys. Rev. B 74\n(2006) 205416.\n8) S. Ishibashi and K. Terakura: J. Phys. Soc. Jpn. 77(2008)\n104706.\n9) R. Pentcheva and W. E. Pickett: Phys. Rev. B 74(2006)\n035112.\n10) M.Imada, and T.Miyake: J.Phys.Soc.Jpn. 79(2010) 112001.\n11) D.Tahara and M.Imada: J.Phys.Soc.Jpn. 77(2008) 114701.\n12) W. Metzner and D. Vollhardt: Phys. Rev. Lett. 62 (1989) 32 4.\n13) A.Georges, G.Kotliar, W.Krauth, and M.J.Rosenberg: Re v.\nMod. Phys. 68(1996) 13.\n14) M.Imada, and T.Kashima: J.Phys.Soc.Jpn. 69(2000) 2723.\n15) T. Mizusaki and M. Imada: Phys. Rev. B 74(2006) 014421.\n16) K. Nakamura, Y. Yoshimoto, R. Arita, S. Tsuneyuki, and M.\nImada: Phys. Rev. B 77(2008) 195126.\n17) I.V.Solovyev, and M.Imada: Phys.Rev.B 71(2005) 045103.\n18) I. V. Solovyev: Phys. Rev. B 73(2006) 155117.\n19) T.Miyake, and F. Aryasetiawan: Phys. Rev. B 77(2008)\n085122.\n20) E. Pavarini, S. Biermann, A. Poteryaev, A. I. Lichtenste in,\nA. Georges, and O. K. Andersen: Phys. Rev. Lett. 92(2004)\n176403.\n21) Y. Imai, I. V. Solovyev, and M. Imada: Phys. Rev. Lett. 95\n(2005) 176405.\n22) Y. Imai and M. Imada: J. Phys. Soc. Jpn. 75(2006) 094713.\n23) Y.Otsuka and M.Imada: J.Phys.Soc.Jpn. 75(2006) 124707.\n24) K. Nakamura, Y. Yoshimoto, T. Kosugi, R. Arita, and M.\nImada: J. Phys. Soc. Jpn. 78(2009) 083710.\n25) K. Nakamura, R. Arita, and M. Imada: J. Phys. Soc. Jpn. 77\n(2008) 093711.\n26) T. Miyake, K. Nakamura, R. Arita, and M. Imada: J. Phys.\nSoc. Jpn. 79(2010) 044705.\n27) F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Bie r-\nmann, and A. I. Lichtenstein: Phys. Rev. B 70(2004) 195104.\n28) I.Souza, N.Marzari,and D.Vanderbilt: ibid65(2001) 035109.\n29) N. Marzari and D. Vanderbilt: Phys. Rev. B 56(1997) 12847.\n30) T. Miyake, F. Aryasetiawan, and M. Imada: Phys. Rev. B 80\n(2009) 155134.\n31) P. Hohenberg and W. Kohn: Phys. Rev. 136(1964) B864.\n32) W. Kohn and L. S. Sham: Phys. Rev. 140(1965) A1133 .\n33) M. Methfessel, M. van Schilfgaarde, and R. A. Casali; in L ec-\nture Notes in Physics, edited by H. Dreysse (Springer-Verla g,\nBerlin, 2000), Vol. 535.J. Phys. Soc. Jpn. Full Paper Author Name 11\n34) K. A. Muller and H. Burkard: Phys. Rev. B 19(1979) 3593.\n35) K.van Benthem, C.Elsasser, and R.H.French: J.Appl.Phy s.\n90(2001) 6156.\n36) S.-G. Lim, S. Kriventsov, T. N. Jackson, J. H. Haeni, D. G.\nSchlom, A. M. Balbashov, R. Uecker, P. Reiche, J. L. Freeouf,and G. Lucovsky: J. Appl. Phys. 91(2002) 4500.\n37) S. Okamoto, and A. J. Millis: Phys. Rev. B 70(2004) 075101.\n38) L. Hedin: Phys. Rev. 139(1965) A796.\n39) K. Nakamura, Y. Yoshimoto, Y. Nohara, and M. Imada: J.\nPhys. Soc. Jpn. 79(2010) 123708."    },    {        "title": "1804.09389v2.Nonsymmorphic_cubic_Dirac_point_and_crossed_nodal_rings_across_the_ferroelectric_phase_transition_in_LiOsO__3_.pdf",        "content": "arXiv:1804.09389v2  [cond-mat.mtrl-sci]  30 Apr 2018Nonsymmorphic cubic Dirac point and crossed nodal rings acr oss the ferroelectric\nphase transition in LiOsO 3\nWing Chi Yu∗,1XiaotingZhou∗,1,2,†Feng-ChuanChuang,3ShengyuanA. Yang,4,‡Hsin Lin,1,5,6,§and Arun Bansil7\n1Center for Advanced 2D Materials and Graphene Research Cent er,\nNational University of Singapore, Singapore 117546, Singa pore\n2Physics Division, National Center for Theoretical Science , Hsinchu 30013, Taiwan\n3Department of Physics, National Sun Yat-Sen University, Ka ohsiung 804, Taiwan\n4Research Laboratory for Quantum Materials, Singapore Univ ersity of Technology and Design, Singapore 487372, Singapo re\n5Department of Physics, National University of Singapore, S ingapore 117546, Singapore\n6Institute of Physics, Academia Sinica, Taipei 11529, Taiwa n\n7Department of Physics, Northeastern University, Boston, M assachusetts 02115, USA\nCrystalline symmetries can generate exotic band-crossing features, which can lead to unconven-\ntionalfermionic excitationswithinterestingphysicalpr operties. WeshowhowacubicDiracpoint—a\nfour-fold-degenerate band-crossing point with cubic disp ersion in a plane and a linear dispersion in\nthe third direction—can be stabilized through the presence of a nonsymmorphic glide mirror sym-\nmetry in the space group of the crystal. Notably, the cubic Di rac point in our case appears on a\nthreefold axis, even though it has been believed previously that such a point can only appear on\na sixfold axis. We show that a cubic Dirac point involving a th reefold axis can be realized close\nto the Fermi level in the non-ferroelectric phase of LiOsO 3. Upon lowering temperature, LiOsO 3\nhas been shown experimentally to undergo a structural phase transition from the non-ferroelectric\nphase to the ferroelectric phase with spontaneously broken inversion symmetry. Remarkably, we\nfind that the broken symmetry transforms the cubic Dirac poin t into three mutually-crossed nodal\nrings. There also exist several linear Dirac points in the lo w-energy band structure of LiOsO 3, each\nof which is transformed into a single nodal ring across the ph ase transition.\nI. INTRODUCTION\nIn the past decade, the study of topological phases\nof matter has become one of the most active areas of\nresearch in condensed matter physics [ 1–4]. Extending\nthe earlier work on gapped topological phases, such as\ntopological insulators [ 5–16] and topological supercon-\nductors [ 17–27], the focus has shifted recently to gap-\nless phases, especially the so-called topological semimet-\nals (TSMs) [ 28–33].\nTSMs arecharacterizedby protected band-crossingsin\nmomentum space, which can be zero-dimensional nodal\npoints or one-dimensional nodal lines. Around these\nband-crossings, electrons behave drastically differently\nfrom the conventional Schr¨ odinger fermions. For exam-\nple, the low-energy electrons in three-dimensional (3D)\nDirac [34–38] and Weyl semimetals [ 39–42] resemble the\nrelativistic Dirac and Weyl fermions with linear disper-\nsion, allowing investigation of related exotic phenomena,\nwhich have previously belonged to the domain of high-\nenergy physics, in a desktop materials setting. In fact,\nthe family of nodal-line semimetals [ 43–55] harbors an\neven richer topological structure, and supports nodal\nchains [56,57], crossing nodal lines [ 58–60] as well as\nHopf links [ 61–66], which have no counterparts in high-\n∗These authors contributed equally to this work.\n†Electronic address: physxtzhou@gmail.com\n‡Electronic address: shengyuan yang@sutd.edu.sg\n§Electronic address: nilnish@gmail.comenergy physics. In addition to these novel bulk fermionic\nexcitations, TSMs also possess exotic topological surface\nstates in the form of Fermi arcs in Dirac/Weyl semimet-\nals and drumhead surface states in nodal-loop semimet-\nals [27,32,39].\nNonsymmorphic crystalline symmetries, i.e. sym-\nmetries which involve a fractional lattice translation,\ncan generate TSMs with exotic type of band crossing\nfeatures. For example, a glide mirror or a two-fold\nscrew axis can give rise to an hourglass dispersion [ 67],\nhourglass loop/chain [ 54–57], and 2D spin-orbit Dirac\npoints [68,69], which are robust against spin-orbit cou-\npling (SOC) effects.\nHere, we show that the glide mirror symmetry, com-\nbined with symmorphic three-fold rotation, inversion,\nand time-reversalsymmetries, can generatea cubic Dirac\npoint. This cubic Diracpointis four-folddegeneratewith\nan associated band dispersion, which is cubic in a plane\nand linear in the third direction normal to the plane.\nNotably, as a result of the nonsymmorphic symmetry,\nthe cubic Dirac point here appears on a threefold axis,\neven though it has been believed previously that such\na point can only appear on a sixfold axis [ 70,71]. We\nfurther show that when the inversion symmetry is bro-\nken, this cubic Dirac point transforms into three crossed\nnodal rings.\nThrough first-principles electronic structure computa-\ntions, weshowthatLiOsO 3providesamaterialsplatform\nfor realizing the novel topological physics outlined in the\npreceding paragraph. LiOsO 3is the first example of a\nferroelectric metal [ 72]. In a recent experiment, Shi et\nal.report that LiOsO 3undergoes a ferroelectric phase2\ntransition at a critical temperature T∗≈140 K [73], at\nwhich the material transforms from a centrosymmetric\nR¯3cstructure to a non-centrosymmetric R3cstructure,\nwhile remaining metallic. LiOsO 3thus realizes the sce-\nnario proposed in Ref. [ 72]. Notably, a related material,\nHgPbO 3, has been suggested recently to host a ferroelec-\ntric Weyl semi-metal phase [ 74].\nOur analysis shows that the non-ferroelectric R¯3c\nstructure satisfies the symmetry requirements for hosting\na cubic Dirac point. Also, when the inversion symmetry\nbreaks at the ferroelectric transition, we find that the\ncubic Dirac point gives rise to three crossed nodal rings\nin the ferroelectric R3cstructure. The low-energy band\nstructure of LiOsO 3further shows the presence of several\nlinear Dirac nodes, and we show that these nodes turn\ninto nodal loops during the ferroelectricphase transition.\nOur study thus not only discovers a new mechanism for\ngeneratingacubicDiracpoint, butitalsooffersapromis-\ning new materials platform for exploring the interplay\nbetween structural transitions, ferroelectricity, and novel\ntopological fermions.\nII. COMPUTATIONAL METHODS\nOurab-initio calculations were performed by using\ntheall-electronfull-potentiallinearizedaugmentedplane-\nwave code WIEN2k [ 75]. Generalized gradient ap-\nproximation (GGA) of Perdew, Burke and Ernzerhof\n(PBE) [76] was employed for the exchange-correlation\npotential. Experimental lattice constants [ 73] were used\nin the calculations and the internal structure was op-\ntimized. The self-consistent iteration process was re-\npeated until the charge, energy and force converged to\nless than 0.0001e, 0.00001 Ry, and 1 mRy/a.u, respec-\ntively. SOC was included by using the second variational\nprocedure [ 77]. The muffin-tin radii were set to 1.47,\n1.84, and 1.51 a.u. for the Li, Os, and O atoms, respec-\ntively.Rmin\nMTKmax= 7 and ak-point mesh of 4000 in the\nfirst Brillouin zone were used. An effective tight-binding\nmodel was constructed via projection onto the Wannier\norbitals [ 78–82] for carrying out topological analysis of\nthe electronic spectrum.We used the Os dorbitals with-\nout performing the maximizing localization procedure.\nSurface states were obtained using the surface Green’s\nfunction technique [ 83] applied on a semi-infinite slab.\nIII. CRYSTAL STRUCTURE AND SYMMETRY\nThenon-ferroelectricphaseofLiOsO 3hasarhombohe-\ndral structure with space group R¯3c(No.167) [Fig. 1(a)].\nThe Os atom is located at the center between two Li\natomsandalsoatthesymmetricpositionoftwoOplanes,\nwhich preserves inversion symmetry P. Upon lowering\nthe temperature below T∗, the two Os atoms are dis-\nplaced along the [111] direction, resulting in the loss of\nX\nL\n\u0001T\n\u0000\nKM(111) surface\nQOsLi\nO\n(c)\n (b) (a)\nFIG.1: CrystalstructuresofLiOsO 3(a)Thenon-ferroelectric\nphase and (b) the ferroelectric phase. The insets are\nschematic drawings of the oxygen planes (red lines) and the\nlocation of the Os and Li atoms relative to these planes. (c)\nBrillouin zone (BZ) ofLiOsO 3and theprojected (111) surface\nBZ.\ninversion center [Fig. 1(b)] [73]. The crystal symmetry is\nthus lowered to R3c(No.161) in the ferroelectric phase.\nWe emphasize that the key physics underlying the cu-\nbic Dirac point and its transformations in LiOsO 3is con-\ntrolledbysymmetry, andnotby thedetails ofthe specific\nmaterial involved. The important symmetries are as fol-\nlows. TheR¯3cspace group of the non-ferroelectric phase\ncan be generated by three symmetry elements: the inver-\nsionP, the three-fold rotation along [111]-direction ( z-\naxis)c3z, and the glide mirror with respect to the ( ¯110)\nplane (xz-plane)/tildewiderMy={My|1\n21\n21\n2}, which is a nonsym-\nmorphic symmetry. In the ferroelectric phase, when Pis\nbroken, the generators of the remaining R3cspace group\nreduce toc3zand/tildewiderMyonly. Lattices in both phases are\nbipartite: there are two sublattices (denoted as A and B)\nwith a relative displacement of (1\n2,1\n2,1\n2).Pand/tildewiderMymap\nbetween the two sublattices ( A↔B), whereasC3zmaps\nwithin each sublattice ( A(B)↔A(B)). The system pre-\nserves the time-reversal symmetry T, since no magnetic\nordering has been observed in either phase [ 73].\nIV. BAND STRUCTURES WITHOUT SOC\nWithout the SOC, the band structures of non-\nferroelectric and ferroelectric phases of LiOsO 3are sim-\nilar as seen in Fig. 2. Both phases exhibit two-fold de-\ngenerate (four-fold if spin-degeneracy is counted) nodal\nlines passing through the L and T points of the BZ per-\npendicular to the glide mirror planes. These features of\nthe band structure can be understood by noting that the\nL and T are time-reversal invariant momentum (TRIM)\npoints, which also reside on the glide mirror planes /tildewiderMi\n(i= 1,2,3)(the othertwoglidemirrorsarerelatedto /tildewiderMy\nbyC3z), see Fig. 1(c). It is easily seen that the line pass-\ning through an L point and normal to the corresponding\nmirror plane /tildewiderMiis invariant under the anti-unitary sym-3\n(b)\n(f)(a)\n(c)\n(d)\n(e)\nFIG. 2: Electronic band structure of LiOsO 3in the absence\nof SOC for (a) the non-ferroelectric phase and, (b) the ferro -\nelectric phase. (c)-(f) Zoom-in images of the band crossing s\nat the L and T symmetry points marked by red circles in (a)\nand (b).\nmetry operation T/tildewiderMi, so that\n(T/tildewiderMi)2=e−ikL\nz=−1, (1)\nwhere we have used T2= +1 for a spinless system and\nkL\nzis thekz-component of L. The T/tildewiderMisymmetry thus\nguarantees a two-fold Kramers-like degeneracy on this\nline. A similar analysis applies to the T point: Since\nT lies on the intersection of three glide mirrors, three\nsuch nodal lines pass through T. Moreover, since /tildewiderMi+1=\nC3z/tildewiderMiC−1\n3z, [C3z,/tildewiderMy]/ne}ationslash= 0,whichleadsto[ /tildewiderMi,/tildewiderMj/negationslash=i]/ne}ationslash= 0.\nFor an eigenstate |ψ0/an}bracketri}htat T, the three states |ψi=1,2,3/an}bracketri}ht=\nT/tildewiderMi|ψ0/an}bracketri}htmust be orthogonal to each other, indicating\na four-fold (eight-fold if counting spin) degeneracy at T.\nThis analysis is in accord with our DFT band structure\nresults.V. BAND STRUCTURES WITH SOC\nA. Non-ferroelectric phase: linear and cubic Dirac\npoints\nThe band structure including SOC is of greater inter-\nest because the low-energy states mainly arise from the\nOs-5dorbitalswith strongSOCeffects. When theSOCis\nturnedon, theband-crossingsinthenon-ferroelectric R¯3c\nphase evolve from 1D nodal lines into 0D Dirac points,\neachwithfour-folddegeneracy. Fig. 3(a)showsthatclose\nto Fermi level, now we have a linear Dirac node at each\nL point (there are three inequivalent L points in the BZ)\n[Fig.3(c) and Fig. 4(a,b)]. The original eight-fold degen-\neracy at T splits under SOC into two Dirac points, where\none is linear [Fig. 3(d)] and the other is cubic [Fig. 3(e)\nand Fig. 4(d,e)]. Hence, we may call this phase as a\nmulti-type Dirac semimetal.\nThe cubic Dirac point, around which the dispersion\nis linear along one direction ( kz-axis) and cubic along\nthe other two directions, has been rarely reported in real\nmaterials[ 70]. Inpreviouswork,itwasbelievedthatsuch\na Dirac point can only appear on a sixfold axis [ 70,71]\nand is possible only for two space groups (No. 176 and\nNo. 192) [ 70]. Our results clearly demonstrate that it\ncan also occur on a threefold axis in the presence of the\nadditional nonsymmorphic (glide mirror) symmetry. In\nthe following, we shall show that the cubic Dirac point\nis indeed protected by the glide mirror together with the\nsymmorphic crystal symmetries.\nThe following points may be noted in connection with\nsymmetry considerations. (1) In the non-ferroelectric\nphase, the presence of both TandPsymmetries forces\nthe two-fold spin-degeneracy of each band. Hence the\ncrossingbetween the bandsmust be at least 4-folddegen-\nerate (i.e. Dirac type). (2) All states at T and L symme-\ntry points must be degenerate quadruplets. This reason\nis that Kramers degeneracy at TRIM points requires the\neigenvalues of Pto be paired as (1 ,1) or (−1,−1). In\ncontrast, at T and L,\n[P,/tildewiderMy]/ne}ationslash= 0, (2)\nimplying that the degeneratestates generatedby /tildewiderMywill\nhavePeigenvalues paired as (1 ,−1), which guarantees\nthe four-fold degeneracy at T and L points. (3) The na-\nture of the Dirac point (linear or cubic) strongly relies on\nthe additional rotational symmetry C3z, which is present\nat T but not at L, as we will explicitly demonstrate be-\nlow.\nAs we already pointed out, the low-energy bands are\nmainly derived from the Os-5 dorbitals. Under trigonal\nprismatic coordination, the Os d-orbitals split into two\ngroups:A1g(dz2) andEg{(dx2−y2,dxy),(dxz,dyz)}. Un-\nder SOC, these states can be combined into the follow-\ning spin-orbit-coupled symmetry-adapted basis, keeping\nin mind that each unit cell contains two Os atoms, one4\n0.050.10.150.550.60.65-0.075-0.07-0.0650.060.070.080.09-0.12-0.1-0.08\n0.540.560.58\n(b)(f)\n(g)\n(h)\n( \u0002 \u0003\n\u0004 \u0005 \u0006\n(d)\n(e)\nFIG. 3: Electronic band structure for (a) the non-ferroelec tric\nphase and (b) the ferroelectric phase where SOC is included.\n(c)-(h) Zoom-in images of the band crossings at the L and T\npoints marked by red circles in (a) and (b).\nin each sublattice:\nφτ\n0,s=|dτ\nz2,s/an}bracketri}ht, (3)\nφτ\n±,s=1√\n2(sinλ|dτ\nxz,s±isdτ\nyz,s/an}bracketri}ht+cosλ|dτ\nx2−y2,s∓is2dτ\nxy,s/an}bracketri}ht),\n(4)\nwhereτlabels the two sublattices A and B, slabels the\nspin, andλis a normalization coefficient.\nAt the T point, Kramers degeneracy requires that the\nfour basis functions φτ\ni,swith a fixed i= 0,±are cou-\npled together into a quadruplet. Consider first the two\nquadruplet basis Ψ 0/−= (φA\n0/−,↑,φA\n0/+,↓,φB\n0/−,↑,φB\n0/+,↓),\nfor which the symmetry operations at T take the follow-\ning representations:\nT=τ3⊗s2K ,P=−τ2,\n/tildewiderMy=e−iqz/2τ1⊗s2, C3z=τ0⊗e∓iπ\n3s3.(5)\nHereτandsare Pauli matrices acting on the sublattice\nand spin spaces, respectively, and qis the wave-vector\nmeasured from T. These expressions fix the k·pHamil-\ntonian at T expressed using Γ matrices, which we define\nhere asγ1=τ3s1,γ2=τ3s2,γ3=τ1s0,γ4=τ2s0, and\nγ5=τ3s3. Then, to the lowest order, we find that\nHlinear\nT(q) =α1(qxγ1±qyγ2)+α2qzγ3+α3qzγ5,(6)\n(g)(a)\u0007 \b \t\n\n \u000b \f\n(d) (e)\r \u000e \u000f\nlinear + cubic Dirac\nlinear Dirac\nT\nLT\nL(h)\nFIG. 4: Top and middle panels show the dispersions around\nthe L and T points. (a), (b), (d) and (e) are for the non-\nferroelectric phase, while (c) and (f) are for the ferroelec tric\nphase. (g, h) Schematic diagrams showing the band-crossing s\nin the non-ferroelectric (g) and ferroelectric (h) phases.\nwhereα’s arethe expansioncoefficients, andthe sign ±is\nfor the two basis Ψ 0/−. This model describes the linear\nDirac points at T, which can be viewed as consisting\nof two Weyl points with Chern numbers 1 and −1. As\nexpected, the total Chern number for a closed surface\nsurrounding the Dirac points vanishes.\nAs for the other quadruplet basis, Ψ +=\n(φA\n+,↑,φA\n−,↓,φB\n+,↑,φB\n−,↓), the representations of the\nsymmetry operations are the same as in Eq. ( 5) except\nthatC3z=−τ0⊗s0. Due to this different transformation\nbehavior of the basis under C3z, the related effective\nHamiltonian is different from ( 6), where the diagonal\nterms proportional to τ0s0have been dropped:\nHcubic\nT(q) =/parenleftbiggh11h12\nh†\n12−h11/parenrightbigg\n, (7)\nwhere\nh11= [c1(q3\n++q3\n−)+b1(q+q−qz)+a1qz]s1\n+ic2(q3\n+−q3\n−)s2\n+[c3(q3\n++q3\n−)+b2(q+q−qz)+a2qz]s3,\nh12= [c4(q3\n++q3\n−)+b3(q+q−qz)+a3qz]s0.\nHereq±=qx±iqy,andai,bi,andciaretheexpansionco-\nefficients. We find that the band-crossing at T described\nby Eq. (7) to be a cubic Dirac point, with cubic disper-\nsion in theqx-qyplane. The diagonal blocks describe two\ntriple-Weyl fermions with Chern numbers ±3. The cubic\nDirac point can be viewed as being composed of the two\ntriple-Weyl points.5\n(b)\n(f)(a)\n(c)\n(d)\n(e)\nFIG. 5: Accidental band crossing points along the Γ-T sym-\nmetry line. (a) Band structure with SOC for the non-\nferroelectric phase. (b) Band structure with SOC for the fer -\nroelectric phase. (c,d) Zoom-in images of the band-crossin g\npoints marked by green circles in (a). (e,f) Zoom-in images o f\nthe band-crossing points in (b). Each band in (c,d) is two-fo ld\ndegenerate, so that the crossing points are four-fold degen er-\nate. The crossing points in (e,f) lie between a non-degenera te\nanda two-fold degenerate band, so thatthese points are trip ly\ndegenerate.\nWe emphasize that when onlysymmorphic symmetries\nare considered, cubic Dirac points require the presence\nof a six-fold axis. [ 70,71] Our case, however, involves a\nnonsymmorphicglidemirror,whichplaysacrucialrolein\nrealizing the cubic Dirac point on a three-fold axis. Our\nanalysis indicates that a sufficient condition for realizing\na cubic Diracpoint is the presenceofP, T, aglide mirror,\nand a c3-axis within the mirror. P, T, and the glide\nmirror only protect a four-fold degeneracy; the presence\nof an additional three-fold rotation then yields the cubic\nin-plane dispersion.\nA similar analysis for the L point at (0 ,0,1\n2) leads to\nthe following effective Hamiltonian:\nHL(q) =/summationdisplay\ni=1,3,5(βx,iqx+βz,iqz)γi+βyqyγ2.(8)\nHereqis measured from L, and the β’s are the expansion\ncoefficients. This demonstrates that the crossings at L\nare linear Dirac points.\nThe linear and cubic Dirac points discussed above are\nessential in the sense that their appearance at the high-symmetryTRIMpointsTandLismandatedbythesym-\nmetry operations of the system. In addition, we note the\npresenceofaccidentalDiracpoints, whichappearinpairs\nalongtheprimaryrotationaxisΓ-T.Therearethreesuch\npairs around the Fermi level, one of these is seen to be\nof type-II with an over-tilted dispersion. Zoom-in images\nof these Dirac points are shown in Figs. 5(a,c,d).\nB. Ferroelectric phase: crossed nodal rings\nIn going across the ferroelectric R3cphase transition,\nthe loss of inversion symmetry induces profound changes\nin the band structure, see Fig. 3(b). The spin-degeneracy\nof the bands gets lifted, and the Dirac points in the non-\nferroelectric phase become unstable. We find that each\nlinear Dirac point at L develops into a nodal ring on the\nglide mirror plane, which encloses the L point [Figs. 3(f)\nand4(c)]. Also, the cubicDiracpoint atTistransformed\ninto three mutually-crossed nodal rings, each lying on a\nglide mirror plane [Figs. 3(h) and4(f)]. This topological\nphase maythus be called a crossed-nodal-ringsemimetal.\nThe transformations of these band-crossings across the\nphase transition are illustrated in Fig. 4(g,h).\nThe occurrence of the preceding nodal rings in the\nband structure is also essential in that it is solely de-\ntermined by symmetry considerations. Note that the\nglide eigenvalues are ±iat Γ1∈ {Γ,X}, and±1 at\nΓ2∈ {T,L}. The presence of There requires that the\nKramers pairs at the above TRIM points carry complex-\nconjugated eigenvalues, i.e.(i,−i) at Γ 1; and (1,1) or\n(−1,−1)at Γ 2. Hence, alonganypathconnecting Γ 1and\nΓ2in the glide plane, the evolution of the glide eigen-\nvalues drives a switching of Kramers partners. During\nthis switching, two bands with opposite glide eigenvalues\nmust produce a crossing. As this argument holds for any\nin-plane path, a nodal loop separating Γ 1and Γ 2must\nappear. This symmetry analysis highlights the impor-\ntance of the nonsymmorphic glide mirror in producing\nthese essential band-crossings.\nThe transformations described above can also be cap-\ntured in the effective models. For example, at the T\npoint, starting from the model of Eq. 7for the cubic\nDirac point, the leading order perturbation δτ1s2when\nthePis broken, so that a minimal model may be ex-\npressed as\nHCNR\nT(q) =Hcubic\nT(q)+δτ1s2. (9)\nIt is straightforward to verify that this model gives three\nmutually-crossed nodal rings like the DFT calculations.\nSimilarly, the nodal ring around the L point can be de-\nrived from the model of Eq. 8as follows\nHNR\nL(q) =HL(q)+δ′τ1s2. (10)\nNote that the linear Dirac points at T are still pre-\nserved [Figs. 3(g)], which can be attributed to the non-\ncommutativity of C3zand/tildewiderMyin the correspondingbasis.6\nThe three pairs of accidental Dirac points residing on the\nprimary rotational axis are, however, turned into triply-\ndegenerate nodes as shown in Fig. 5(b,e,f).\nUnlike Weyl points, the essential Dirac points may or\nmay not give rise to nontrivial surface states [ 71]. In\nFig.6, we plot the surface spectrum of the (111)-surface\nfor both the non-ferroelectric and ferroelectric phases.\nBothphasesareseentosupportsurfacebandsconnecting\nthe two surface-projected L points.\n(b)\n(d)\n(a) \n(c)\nFIG. 6: Surface states of LiOsO 3in the non-ferroelectric (up-\nper panel) and ferroelectric (lower panel) phases in the pro -\njected spectra of the (111) surface. SOC is included. (a) and\n(c) Constant energy surfaces at the band-crossing energy at\nL. (d) and (e) Dispersions along the kx-direction at ky= 0.5.\nVI. CONCLUSION\nWe emphasize a number of points in closing. Firstly,\nthe Dirac nodes at the TRIM points and the multiple\nnodal rings surrounding these points discussed here are\nessential band-crossings in the sense that the presence of\nthese features is solely dictated by the nonsymmorphic\nspace group symmetries (plus T) of the system. Their\nexistence is thus guaranteed as long as these symmetries\nare preserved. Our analysis of symmetry considerations\nunderlying cubic Dirac points and crossed nodal rings\nwill provide a useful guide in searching for exotic band\ncrossings in realistic material systems more generally.\nSecondly, our analysis clearly indicates that LiOsO 3\nwill provide a useful platform for exploring exotic topo-\nlogical phases and their interplay with ferroelectric or-\ndering. The unique advantages of LiOsO 3are as follows.\n(i) The essential band-crossings are close to the Fermi\nlevel, so that the associated topological physics will be\nreflected in various electronic properties. (ii) The twotopological phases that coexist in the same material are\nconnected via a ferroelectric phase transition, which is\ntunable by varying temperature [ 73]. And, (iii) the ma-\nterial has already been realized experimentally and its\nferroelectric phase transition has been observed. The in-\nterestingtransformationsintheband-crossingsacrossthe\nphase transition, which we have predicted here, could be\ndetected by ARPES experiments.\nThirdly, we have identified the first case of an ex-\nperimentally realized material, which harbors a cubic\nDirac point in a 3D material. Such a Dirac point has\nbeen predicted previously only in quasi-1D molybdenum\nmonochalcogenide compounds [ 70], where an experimen-\ntal verification is still lacking. Our work offers a new\nrouteforrealizingcubicDiracpoints,extendingtherange\nof materials in search of cubic Dirac fermions.\nFinally, our study opens a new pathway for exploring\na variety of novel effects associated with the various non-\ntrivial band-crossings. For example, a linear Dirac point\nmayexhibitnegativemagnetoresistance[ 84,85], aspecial\nmagnetic oscillation frequency driven by surface Fermi\narcs [86], and an artificial gravity field via strain modu-\nlation[87]. AcubicDiracpointcanexhibitunusualquan-\ntuminterferencecontributionstomagneto-transport[ 88],\nstronger screening of interactions, and possible presence\nof continuous quantum phase transitions driven by inter-\nactions [89]. A nodal ring may yield strong anisotropy in\nelectricaltransport[ 47], unusualopticalresponse[ 90]and\ncircular dichroism [ 91], and possible surface magnetism\nand superconductivity [ 92].\nACKNOWLEDGMENTS\nThe workat National University ofSingapore wassup-\nported by the Singapore National Research Foundation\nunder the NRF fellowship Award No. NRF-NRFF2013-\n03. The work at Northeastern University was supported\nby the US Department of Energy (DOE), Office of Sci-\nence, Basic Energy Sciences grant number DE-FG02-\n07ER46352, and benefited from Northeastern Univer-\nsity’s Advanced Scientific Computation Center and the\nNational Energy Research Scientific Computing Center\nthroughDOE grantnumberDE-AC02-05CH11231. FCC\nand XZ acknowledge support from the National Cen-\nter for Theoretical Sciences. Work at Singapore Uni-\nversity of Technology and Design is supported by Sin-\ngapore MOE Academic Research Fund Tier 2 (Grant\nNo. MOE2015-T2-2-144). FCC also acknowledges sup-\nport from the Ministry of Science and Technology of\nTaiwan under Grants Nos. MOST-104-2112-M-110-002-\nMY3 and the support under NSYSU-NKMU JOINT RE-\nSEARCH PROJECT #105-P005 and #106-P005. He is\nalsogratefultotheNationalCenterforHigh-performance\nComputing for computer time and facilities.7\n[1] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. 88,\n021004(2016).\n[2] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n[3] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[4] B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys.\n8, 337 (2017).\n[5] Kane, C. L., and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[6] Kane, C. L., and E. J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n[7] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science\n314, 1757 (2006).\n[8] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,\n106803 (2007).\n[9] J. E. Moore, and L. Balents, Phys. Rev. B 75, 121306\n(2007).\n[10] R. Roy, Phys. Rev. B 79, 195322 (2009).\n[11] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buh-\nmann, L.W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Sci-\nence318, 766 (2007).\n[12] D.Hsieh, D.Qian, L. Wray,Y.Xia, Y.S.Hor, R.J. Cava,\nand M. Z. Hasan, Nature (London) 452, 970 (2008).\n[13] D. Hsieh et al., Nature (London) 460, 1101 (2009).\n[14] D. Hsieh et al., Phys. Rev. Lett. 103, 146401 (2009).\n[15] Y. L. Chen et al., Science 325, 178 (2009).\n[16] Y. Xia et al., Nat. Phys. 5, 398 (2009).\n[17] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-\nwig, Phys. Rev. B 78, 195125(2008).\n[18] A. Kitaev, AIP Conf. Proc. 1134, 22 (2008).\n[19] S. B. Chung, and S. C. Zhang, Phys. Rev. Lett. 103,\n235301 (2009).\n[20] X. L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys.\nRev. Lett. 102, 187001 (2009).\n[21] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C.\nZhang, Nature Phys. 5, 438 (2009).\n[22] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407\n(2008).\n[23] X. L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B\n82, 184516 (2010).\n[24] L. Santos, T. Neupert, C. Chamon, and C. Mudry, Phys.\nRev. B81, 184502 (2010).\n[25] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,\nPhys. Rev. Lett. 104, 040502 (2010).\n[26] A. Das et al., Nature Phy. 8, 887 (2012).\n[27] S. A. Yang, H. Pan, and F. Zhang, Phys. Rev. Lett. 113,\n046401 (2014).\n[28] M. Z. Hasan, S.-Y. Xu, and G. Bian, Phys. Scripta 2015,\n014001 (2015).\n[29] L. Balents, Physics 4, 36 (2011).\n[30] A. M. Turner and A. Vishwanath, Beyond band insula-\ntors: Topology of semimetals and interacting phases (El-\nsevier, New York, 2013).\n[31] S. A. Yang, SPIN 6, 1640003 (2016).\n[32] C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B\n25, 117106 (2016).\n[33] A. A. Burkov, arXiv:1704.06660.\n[34] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J.\nMele, and A. M. Rappe, Phys. Rev. Lett. 108, 140405\n(2012).[35] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H.\nWeng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320\n(2012).\n[36] Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Phys.\nRev. B88, 125427 (2013).\n[37] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,\nD. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai,\nZ. Hussain, and Y. L. Chen, Science 343, 864 (2014).\n[38] T. R. Chang et al., Phys. Rev. Lett. 119, 026404 (2017).\n[39] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.\nSavrasov, Phys. Rev. B 83, 205101 (2011).\n[40] B. Singh, A. Sharma, H. Lin, M. Z. Hasan, R. Prasad,\nand A. Bansil, Phys. Rev. B 86, 115208 (2012).\n[41] H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai,\nPhys. Rev. X 5, 011029 (2015).\n[42] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G.\nChang, B. Wang, N. Alidoust, G. Bian, M. Neupane,\nC. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan,\nNat. Commun. 6, 7373 (2015).\n[43] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev.\nB84, 235126 (2011).\n[44] M. Phillips and V. Aji, Phys. Rev. B 90, 115111 (2014).\n[45] C.-K. Chiu and A. P. Schnyder, Phys. Rev. B 90, 205136\n(2014).\n[46] H. Weng, Y. Liang, Q. Xu, R. Yu, Z. Fang, X. Dai, and\nY. Kawazoe, Phys. Rev. B 92, 045108 (2015)\n[47] K. Mullen, B. Uchoa, and D. T. Glatzhofer, Phys. Rev.\nLett.115, 026403 (2015).\n[48] Y. Chen, Y. Xie, S. A. Yang, H. Pan, F. Zhang, M. L.\nCohen, and S. Zhang, Nano Lett. 15, 6974 (2015).\n[49] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Phys. Rev. B\n92, 081201 (2015).\n[50] R. Yu, H. Weng, Z. Fang, X. Dai, and X. Hu, Phys. Rev.\nLett.115, 036807 (2015).\n[51] Y. Kim, B. J. Wieder, C. L. Kane, and A. M. Rappe,\nPhys. Rev. Lett. 115, 036806 (2015).\n[52] R. Li, H. Ma, X. Cheng, S. Wang, D. Li, Z. Zhang, Y. Li,\nand X.-Q. Chen, Phys. Rev. Lett. 117, 096401 (2016).\n[53] S. Li, Z.-M. Yu, Y. Liu, S. Guan, S.-S. Wang, X. Zhang,\nY. Yao, and S. A. Yang, Phys. Rev. B 96, 081106 (2017).\n[54] Y. Chen, H.-S. Kim, and H.-Y. Kee, Phys. Rev. B 93,\n155140 (2016).\n[55] S. Li, Y. Liu, S.-S. Wang, Z.-M. Yu, S. Guan, X.-L.\nSheng, Y. Yao, and S. A. Yang, Phys. Rev. B 97, 045131\n(2018).\n[56] T. Bzdusek, Q. Wu, A. Ruegg, M. Sigrist, and A. A.\nSoluyanov, Nature 538, 75 (2016).\n[57] S.-S. Wang, Y. Liu, Z.-M. Yu, X.-L. Sheng, and S. A.\nYang, Nat. Commun. 8, 1844 (2017).\n[58] M. Zeng, C. Fang, G. Chang, Y.-A. Chen, T. Hsieh, A.\nBansil, H. Lin, and L. Fu, arXiv:1504.03492.\n[59] S. Kobayashi, Y. Yamakawa, A. Yamakage, T. Inohara,\nY. Okamoto, and Y. Tanaka, Phys. Rev. B 95, 245208\n(2017).\n[60] X.-L. Sheng, Z.-M. Yu, R. Yu, H. Weng, and S. A. Yang,\nJ. Phys. Chem. Lett. 8, 3506 (2017).\n[61] G. Chang, S.-Y. Xu, X. Zhou, S.-M. Huang, B. Singh, B.\nWang, I. Belopolski, J. Yin, S. Zhang, A. Bansil, H. Lin,\nand M. Z. Hasan, Phys. Rev. Lett. 119, 156401 (2017).\n[62] C. Zhong, Y. Chen, Z.-M. Yu, Y. Xie, H. Wang, S. A.\nYang, and S. Zhang, Nat. Commun. 8, 15641 (2017).8\n[63] W. Chen, H.-Z. Lu, and J.-M. Hou, Phys. Rev. B 96,\n041102 (2017).\n[64] Z. Yan, R. Bi, H. Shen, L. Lu, S.-C. Zhang, and Z. Wang,\nPhys. Rev. B 96, 041103 (2017).\n[65] P.-Y. Chang and C.-H. Yee, Phys. Rev. B 96, 081114\n(2017).\n[66] M. Ezawa, Phys. Rev. B 96, 041202 (2017).\n[67] Z. Wang, A. Alexandradinata, R. J. Cava, and B. A.\nBernevig, Nature 532, 189 (2016).\n[68] S. M. Young and C. L. Kane, Phys. Rev. Lett. 115,\n126803 (2015).\n[69] S. Guan, Y. Liu, Z.-M. Yu, S.-S. Wang, Y. Yao, and S.\nA. Yang, Phys. Rev. Materials 1, 054003 (2017).\n[70] Q. Liu and A. Zunger, Phys. Rev. X 7, 021019 (2017).\n[71] B.-J. Yang and N. Nagaosa, Nat. Commun. 5, 4898\n(2014).\n[72] P. W. Anderson and E. I. Blount, Phys. Rev. Lett. 14,\n217 (1965); Phys. Rev. Lett. 14, 532(1965).\n[73] Y. Shi et al., Nat. Mater. 12, 1024 (2013).\n[74] R. Li, Y. Xu, J. He, S. Ullah, J. Li, J.-M. Liu, D. Li, C.\nFranchini, H. Weng, and X.-Q. Chen, arXiv:1610.07142.\n[75] K. Schwarz and P. Blaha, Comput. Mater. Sci. 28, 259\n(2003).\n[76] J. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996).\n[77] A. H. MacDonald, W. E. Pickett, and D. D. Koelling, J.\nPhys. C: Solid State Phys. 13, 2675 (1980).\n[78] A. A. Mostofi, J. R. Yates, G. Pizzi, Y. S. Lee, I. Souza,\nD. Vanderbilt, and N. Marzari, Comput. Phys. Commun.\n185, 2309(2014)\n[79] N. Marzari and D. Vanderbilt, Phys. Rev. B 56,22(1997).\n[80] I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B\n65, 035109 (2001).\n[81] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D.\nVanderbilt, Rev. Mod. Phys. 84, 1419 (2012)\n[82] J. Kuneˇ s, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, an d\nK. Held, Comput. Phys. Commun. 181, 1888 (2010).\n[83] M. P. L. Sancho, J. M. L. Sancho, J. Rubio, J. Phys. F\nMet. Phys. 15, 851(2000).\n[84] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M.\nHirschberger, W. Wang, R. Cava, and N. P. Ong, Science\n350, 413 (2015).\n[85] Y. Gao, S. A. Yang, and Q. Niu, Phys. Rev. B 95, 165135\n(2017).\n[86] P. J. W. Moll, N. L. Nair, T. Helm, A. C. Potter, I.\nKimchi, A. Vishwanath, and J. G. Analytis, Nature 535,\n266 (2016).\n[87] S. Guan, Z.-M. Yu, Y. Liu, G.-B. Liu, L. Dong, Y. Lu,\nY. Yao, and S. A. Yang, npj Quantum Materials 2, 23\n(2017).\n[88] X. Dai, H.-Z. Lu, S.-Q. Shen, and H. Yao, Phys. Rev. B\n93, 161110 (2016).\n[89] B. Roy, P. Gaswami, and V. Juriˇ ci´ c, Phys. Rev. B 95,\n201102 (2017).\n[90] S. Ahn, E. J. Mele, and H. Min, Phys. Rev. Lett. 119,\n147402 (2017).\n[91] Y. Liu, S. A. Yang, and F. Zhang, Phys. Rev. B 97,\n035153 (2018).\n[92] T. T. Heikkila and G. E. Volovik, JETP Lett. 93, 59\n(2011)."    },    {        "title": "1004.2934v2.Low_Energy__Coherent__Stoner_like_Excitations_in_CaFe___2__As___2__.pdf",        "content": "arXiv:1004.2934v2  [cond-mat.mtrl-sci]  29 Jun 2010Low Energy, Coherent, Stoner-like Excitations in CaFe 2As2\nLiqin Ke1, M. van Schilfgaarde1, J.J.Pulikkotil2, T. Kotani3, and V.P.Antropov2\n1School of Materials, Arizona State University\n2Ames Laboratory, IA 50011 and\n3Tottori University, Tottori, Japan\n(Dated: October 9, 2018)\nAbstract\nUsing linear-response density-functional theory, magnet ic excitations in the striped phase of\nCaFe2As2are studied as a function of local moment amplitude. We find a n ew kind of excitation:\nsharp resonances of Stoner-like (itinerant) excitations a t energies comparable to the N´ eel temper-\nature, originating largely from a narrow band of Fe dstates near the Fermi level, and coexist with\nmore conventional (localized) spin waves. Both kinds of exc itations can show multiple branches,\nhighlighting the inadequacy of a description based on a loca lized spin model.\n1Magnetic interactions are likely to play a key role in mediating supercon ductivity in\nthe recently discovered family of iron pnictides; yet their characte r is not yet well under-\nstood. In particular, whether the system is best described in term s of large, local magnetic\nmoments centered at each Fe site, in which case elementary excitat ions are collective spin\nwaves (SWs) called magnons, or is itinerant (elementary excitations characterized by sin-\ngle particle electron-hole transitions) is a subject of great debate . This classification also\ndepends on the energy scale of interest. The most relevant energ y scale in CaFe 2As2and\nother pnictides ranges to about twice the N´ eel temperature, 2 TN≈40 meV. Unfortunately,\nneutron scattering experiments have focused on the character of excitations in the 150-200\nmeV range [1, 2], much larger than energy that stabilizes observed m agnetism or supercon-\nductivity. Experiments in Refs. [1, 2], while very similar, take complet ely different points\nof view concerning the magnetic excitations they observe. There is a similar dichotomy in\ntheoretical analyses of magnetic interactions[3, 4]. Model descrip tions usually postulate a\nlocal-moments picture [5]. Most ab initio studies start from the local spin-density approxi-\nmation (LSDA) to density functional theory. While the LSDA traditio nally favors itinerant\nmagnetism (weak on-site Coulomb correlations), practitioners str ongly disagree about the\ncharacter of pnictides; indeed the same results are used as a proo f of both localized and\nitinerant descriptions [3, 4].\nThe dynamic magnetic susceptibility (DMS), is the central quantity t hat uniquely char-\nacterizes magnetic excitations. It can elucidate the origins of magn etic interactions and\ndistinguish between localized and itinerant character. However, th e dynamical linear re-\nsponse is very difficult to carry out computationally; studies to date been limited to a few\nsimple systems. Here we adapt an all-electron linear response techn ique developed recently\n[6, 7] to calculate the transverse DMS, χ(q,ω). Besides SW excitations seen in neutron\nscattering, we find low energy particle-hole excitations at low q, and also at high q. In\nstark contrast to conventional particle-hole excitations in the St oner continuum, they can\nbe sharply peaked in energy (resonances) and can be measured.\nThe full transverse DMS χ(r,r′,q,ω) is a function of coordinates randr′(confined to\nthe unit cell). It is obtained from the non-interacting susceptibility χ0(r,r′,q,ω) via the\nstandard relation [8]\nχ=χ0[1−χ0I]−1(1)\nIis the exchange-correlation kernel. When computed within the time- dependent LSDA\n2−1−0.5 00.5 112344\nM=0.4\nM=1.1(a)N(E)\nE (eV) 0.2 0.6 11.400.511.522.5N(EF)\nM(µB)(b)\nlocalized itinerant\n00.51.01.502468M=0.4\nM=0.8\nM=1.1\n(c)Imχ0(q=0)\nω (eV)\nFIG. 1: ( a)N(E), in units of eV−1per cell containing one Fe atom. Data are shown for M=0.4µB\nand 1.1µB. Both kinds of transitions are reflected in peaks in χ0(q=0,ω), shown in panel ( c) for\nM=0.4, 0.8, and 1.1 µB. (b)N(E) at the Fermi level EFas a function of moment M. The blue\nvertical bar denotes the experimental moment, and also appr oximately demarcates the transition\nfrom itinerant to localized behavior. N(EF) in the nonmagnetic case is shown as a dotted line. ( c)\nbare susceptibility χ0(q=0,ω) in the same units. The text discusses the significance of the arrows\nin panels ( a) and (c).\n(TDLDA) Iis local: I=I(r)δ(r−r′) [8].χ0can be obtained from the band structure\nusing the all-electron methodology we developed [6, 7]. To reduce the computational cost\nwe calculate Im χ0and obtain Re χ0from the Kramers Kronig transformation [6]. Im χ0\noriginates from spin-flip transitions between occupied states at kand unoccupied states at\nk+q: it is a k-resolved joint density of states Ddecorated by products Pof four wave\nfunctions [7]\nD(k,q,ω) =f(ǫ↑\nk)(1−f(ǫ↓\nq+k))δ(ω−ǫ↓\nq+k+ǫ↑\nk) (2)\nImχ+−\n0=/integraldisplay\ndωd3kP(r,r′,k,q)×D(k,q,ω) (3)\nEven with the Kramers-Kronig transformation, Eq. (3) poses a hu ge computational burden\nfor the fine frequency and kresolution required here. We make a simplification, map-\npingχ0onto the local magnetization density which is assumed to rotate rigid ly. The full\nχ0(r,r′,q,ω) simplifies to the discrete matrix χ0(R,R′,q,ω) associated with pairs of mag-\nnetic sites ( R,R′) in the unit cell; and I(r) simplifies to a diagonal matrix IRR. In Ref. [7]\n3we show that we need not compute Iexplicitly but can determine it from a sum rule. I\ncan be identified with the Stoner parameter in models. We have found that for Fe and Ni\nthese approximations yield results in rather good agreement with th e full TDLDA results,\nand expect similar agreement here. We essentially follow the procedu re described in detail\nin Ref. [7], and obtain χ(q,ω) as a 4×4 matrix corresponding to the four Fe sites in the unit\ncell. To make connection with neutron experiments, spectra are ob tained from the matrix\nelement/summationtext\nR,R′< eiq·R|χ(R,R′,q,ω)|eiq·R′>. For brevity we omit indices RR′henceforth.\nAswe will see, thecharacter of χ0(q,ω)can largelybeunderstood fromstates near EF, so\nwe begin by analyzing the ground state band and magnetic structur e of the low-temperature\n(striped) phase of CaFe 2As2within the LSDA. Results for tetragonal and orthorhombic\nstriped structures are very similar, suggesting that the slight diffe rence in measured aand\nblattice parameters plays a minor role in the description of magnetic pr operties. Experi-\nmental lattice parameters were employed[9]. CaFe 2As2has an internal parameter zwhich\ndetermines the Fe-As distance RFe−As. Here we treat zas a parameter whos main effect is\nto control the magnetic moment M, which in turn strongly affects the character of magnetic\ninteraction, as we will show.\nIn Fig. 1( a), the density of states N(E) is shown over a 2 eV energy window for a small\nand large moment case ( M=0.4µBand 1.1µB). Particularly of note is a sharp peak near\nEF, of width ∼50 meV. This narrow band is found to consist almost entirely of major ity-\nspin Fedxyanddyzorbitals. The peak falls slightly below EFforM=0.4µBand slightly\nabove for M=1.1µB. Thus, as RFe−Asis smoothly varied so that Mchanges continuously,\nthis peak passes through EF. As a result N(EF) reaching a maximum around M=0.8µB\n(Fig. 1(b)). This point also coincides with the LSDA minimum-energy value of RFe−As.\nThat this unusual dependence originates in the magnetic part of th e Hamiltonian can be\nverified by repeating the calculation in the nonmagnetic case. As Fig. 1(b) shows, the\nnonmagnetic N(EF) is large, and approximately independent of RFe−As. In summary, the\nmagnetic splitting produces a pseudogap in N(E) for large M; the pseudogap shrinks as\nMdecreases and causes a narrow band of Fe dstates to pass through EF, creating a sharp\nmaximum in N(EF) nearM=0.8µB. As we will show below that moment demarcates a\npoint of transition from itinerant to localized behavior (see also Ref. [13]).\nNext we turn to magnetic excitations. In the standard picture of m agnons in metals,\nImχ0(ω) is significant only for frequencies exceeding the magnetic (Stoner ) splitting of the d\n4bands,ǫ↓\nd−ǫ↑\nd=IM(cfEq. 2). In such cases Im χ0is small at low qand low energies and well\ndefined magnons appear at energies near |1−IReχ0|= 0 (cfEq. 1). As Im χ0increases, it\ninitially broadens the (formerly sharp) SW spectrum ¯ ω(q); as it becomes large the spectrum\ncan become incoherent, or (Stoner) peaks can arise from Im χ0, possibly enhanced by small\n1−IReχ0. Fig. 1(c) shows Im χ0(q=0,ω) on a broad energy scale. The picture we developed\nforN(E) leads naturally to a classification of Stoner transitions into three m ain types.\n(1) Excitations between the (largely) Fe dxzstates centered near −0.5 eV to dstates\ncentered near 1 eV (for M=1.1µB). The magnetic splitting of these states matches well\nwith the usual Stoner splitting IMin localized magnets, and scales with M. (I≈1 eV in\n3dtransition metals.) These high-energy transitions are depicted by a large red arrow in\nFig. 1(a) forM=1.1µB, and by vertical arrows in Fig. 1( c) showing χ0(q=0,ω) forM=0.4,\n0.8, and 1.1 µB. They are too high in energy to be observed by neutron measureme nts.\n(2) Excitations from dxztodxyanddyzstates just above the psueudogap, depicted by\na small red arrow in Fig. 1( a), and slanting arrows in Fig. 1( d). These are the excita-\ntions probably detected in neutron measurements [1, 2]. This pseud ogap is well defined for\nM≥1.1µB, but is modified as Mdecreases, which leads to the following:\n(3) Near M=0.8µB, the narrow dband passes through EF, opening up channels, not\npreviously considered, for low-energy, particle-hole transitions w ithin this band. When M\nreaches 0.4 µB, this band has mostly passed through EFand the pseudogap practically\ndisappears.\nWhat makes the pnictide systems so unusual is that Im χ0is already large at very low\nenergies ( ∼10 meV) once the sharp peak in N(E) approaches EF. One of our central find-\nings is that this system undergoes a transition from localized to a coexistence of localized\nand itinerant behavior asMdecreases from M/greaterorsimilar1.1µBtoM≈0.8µB. Moreover, the itiner-\nant character is of an unusual type: elementary excitations are m ostly single particle-hole\nlike: they can be well defined in energy and q. Those represent coherent excitations. The\ndependence of N(EF) onMis not only responsible for them, but also may explain the\nunusual linear temperature-dependence of paramagnetic susce ptibility, and the appearance\nof a Lifshitz transition with Co doping [10].\nFig. 2 focuses on the AFM line, q=[H00]2π/a. Panel ( a) shows the full Im χ(ω) for\nM=1.1µBfor several q-points spanning the entire line, 0 <H<1. At low q, peaks ¯ωinχ\nare sharp; and ¯ ωdepends on Hin the expected manner (¯ ω∝H). ¯ωreaches a maximum\n501002003004000.1\n0.3\n0.50.70.8\n1.0\nω(meV)(a)\n00.20.40.60.80100200300\nHω\n(b)\n0100200300400M=1.1\nM=0.8\nM=0.4\nω (meV)(c)\n00.20.40.60.8100.10.20.30.40.5M=1.1\nM=0.8\nM=0.4Γ/ω\nH(d)\nFIG. 2: Im χ(q,ω) along the AFM axis, q=[H,0,0]2π/a. (a) Imχ(ω) for various HandM=1.1µB.\nExtracted from Im χ(H,ω) are peak positions ¯ ω(H) (meV) and HWHM Γ. ( b) and (d) depict ¯ ω\nand the ratio Γ /¯ω, for the three moments shown in the key. Panel ( b) also depicts, as solid lines,\ncontours |1−IReχ0|= 0inthe( ω,H) plane, for M=1.1and0.4 µB. Panel ( c) showsIm χ(H=0.9,ω)\nfor the three moments.\nnearH=1/2 (Fig. 2( b)), for all three values of M. The peaks broaden with increasing\nq; nevertheless we can associate them with magnons, because they coincide closely with\nvanishing |1−IReχ0|and Γ is not too large. The magnon character is preserved for most\nqat all moments, as Fig. 2( b) shows: but for H>0.8 the peaks are strongly broadened,\nespecially for small M. ¯ωis in good agreement with neutron data of Ref.[2], except that\nneutron data are apparently smaller than large Mcalculations predict [11].\nOne experimental measure of the validity of the local-moment pictur e is the ratio of half-\nwidth at half-maximum (HWHM) Γ to ¯ ω, shown in Fig. 2( d). For large Mand most of q,\nΓ/¯ω∼0.15-0.18. For intermediate and small M, Γ/¯ω≥0.2 for a wide diapason of q. This\nis significant: it reflects the increasing Stoner character of the ele mentary excitations. Were\nthere an abrupt transition into a conventional Stoner continuum a s argued in Ref. [2], it\nwould be marked by an abrupt change in Γ /¯ω. This is not observed; yet damping appears\nto increase with energy and q, reflecting normal metallic behavior.\nSpectra for H=0.9 (Fig. 2( c)) adumbrate two important findings of this work. When\nM=0.8µB, a sharp peak in χ(ω) appears near 10 meV. There is a sharp peak in χ0(ω) at\n¯ω≈10 meV also, classifying this as a particle-hole excitation originating fr om the narrow d\nband depicted in Fig. 1. Being well defined in energy it is coherent, ana logous to a SW,\n6only with a much larger Γ /¯ω. Yet it is strongly enhanced by collective interactions, since\n|1−IReχ0|ranges between 0.1 and 0.2 for ω<100meV. This new kind of itinerant excitation\nwill be seen at many values q, typically at small q. The reader many note the sharp rise\nand fall in Γ /¯ωat small qin Fig. 2( d). This anomaly reflects a point where a SW and a\nparticle-hole excitation coalesce to the same ¯ ω.\nReturning to H=0.9 when M=1.1, there is a standard (broadened) SW at 200 meV; cf\ncontours in Fig. 2( b). Asecond, low-energy excitation can be resolved near 120 meV. This\nis no corresponding zero in the denominator in Eq. (1), but no stron g peak in χ0(ω), either.\nThis excitation must be classified as a hybrid intermediate between St oner excitations and\nSWs. Only a single peak remains when M=0.4µBand the peak in N(E) has mostly passed\nthroughEF(Fig. 1).\n010020030040013\n5 7911\nω(meV)(a)\n00.20.40.60.80100200300\nKω\n(b)\nFIG. 3: Im χ(q,ω) along the FM axis, q=[0,K,0]2π/a. (a) Imχ(ω) atK= 1/12, 3/12, 5/12, 7/12,\n9/12, 11/12, for M=0.8µB. Strong Stoner excitations can be seen for K=1/12 and 3/12 near\n¯ω∼10-20 meV. ( b): Contours |1−IReχ0|= 0 in the ( ω,K) plane, for M=1.1µBand 0.8µB,\nanalogous to Fig. 2( b), and dominant peak positions ¯ ωobtained by a nonlinear least-squares fit of\none or two gaussian functions to χ(ω) over the region where peaks occur.\nAlong the FM line q=(0,K,0)2π/a,χ(ω) is more complex, and more difficult to interpret.\nAtM=1.1µBsharp, well defined collective excitations are found at low q, and broaden with\nincreasing q. There is a reasonably close correspondence with the zeros of |1−IReχ0|and\npeaks in χ, as Fig. 3( b) shows. The M=0.8µBcase is roughly similar, except that for K>0.4\nexcitations cannot be described by a single peak. Note that for fixe dq, propagating spin\nfluctuations (characterized by peaks at ¯ ω) can exist at multiple energies — the magnetic\nanalog of the dielectric function passing through zero and sustainin g plasmons at multiple\nenergies. Peaks in χ(ω) can broaden as a consequence of this; the K=7/12, 9/12 and 11/12\ndata of Fig. 3( a) are broadened in part by this mechanism as distinct from the usual one\n(intermixing of Stoner excitations). Second, consider how ∂¯ω/∂qchanges with MforK>0.5\n7(Fig. 3(b)). When M=1.1µB, ¯ωincreases monotonically with K. ForM=0.8µB, ¯ωhas a\ncomplex structure but apparently reaches a maximum before Kreaches 1. That ∂¯ω/∂q\nchanges sign is significant: it marks the disappearance of magnetic f rustration between the\nferromagnetically aligned spins at lowmoments, andthe emergence o f stable FMorder along\n[010], characteristic of local-moment behavior (see also Ref.[12]). Ex perimentally, Ref. [2]\nreports∂¯ω/∂q>0 forK>0.5. While our calculations provide a clear physical interpretation,\nwe note significant differences in the moment where we observe this e ffect (M=0.8-1.1µB)\nand the effective spin ( S=0.2) used in Ref.[2].\n0501001500.1\n0.2\n0.3\n0.4\n0.5\nω(meV)(a)\n00.10.20.30.4Lω\n050100(b)\nFIG. 4: Im χ(q,ω) along the caxis,q=[0,0,L]2π/c. (a) Imχ(ω) at values of Llisted in the key,\nforM=0.8µB. WhenM=1.1µB(not shown) the spectrum is well characterized by a single sh arp\npeak at any L. WhenM=0.8 or 0.4 µBthe SW peak remains, but a low-energy Stoner excitation\ncoexists with it. The latter are most pronounced for larger Lbut can be resolved at every L. Peak\npositions (single for M=1.1µB, double for M=0.8 and 0.4 µB) are shown in ( b).\nElementary excitations along the caxis,q=(0,0,L)2π/c, bring into highest relief the\ntransition from pure local-moment behavior (collective excitations) , to one where coherent\nitinerant Stoner and collective excitations coexist. Collective excita tions are found for all\nLand allM. ForM=1.1µB, a single peak is found: excitations are well described by the\nHeisenberg model with weak damping. Comparing Fig. 4( b) to Figs. 2( b) and 3( b), it\nis apparent that ¯ ωrises much more slowly along Lthan along HorK, confirming that\ninterplane interactions are weak. When Mdrops to 0.8 µB, a second-low energy peak at\n¯ω′emerges at energies below 20 meV, for small qalong [0,K,0], and for all qalong [0,0, L],\ncoexisting with the collective excitation. Why these transitions are a bsent for M=1.1µBand\nare so strong at M=0.8µBcan be understood in terms of roughly cylindrical Fermi surface\natk=(1/2,0,kz). Single spin-flip transitions between occupied states ǫ↑\nkand an unoccupied\nstatesǫ↓\nk+qseparated by ∼10 meV are responsible for this peak (see Eq. 2). They originate\nfrom “hot spots” where N(ǫ↑\nk) andN(ǫ↓\nk+q) are both large.\n8From the SW velocities , (∂¯ω/∂q)q=0, anisotropy of the exchange couplings can be de-\ntermined [4, 12, 14]. We find strong in-plane and out-of-plane anisot ropies, and pre-\ndictvb/va=0.55 and find vc/va=0.35, where va=490meV ·˚A. Neutron scattering experi-\nments [14, 15] have measured with vc/va−bto be∼0.2-0.5.\nIn summary, we broadly confirm the experimental findings of Refs. [1, 2], that there is\na spectrum of magnetic excitations of the striped phase of CaFe 2As2, which for the most\npart are weakly damped at small qand more strongly damped at large q. A new kind of\nexcitation was found, which originates from single particle-hole tran sitions within a narrow\nband of states near EF, renormalized by a small denominator |1−IReχ0|(Eq. (1)). They\nappear when the Fe moment falls below a threshold, at which point the narrow band passes\nthroughEF. The character of itineracy is novel: excitations occur at low-ener gy, at energy\nscales typically below TN, and can be sharply peaked, which should make them accessible\nto experimental studies. The distinction between the two kinds of e xcitations is particularly\nobservable in the anomalous dependence of Γ /¯ωonqin Fig. 2( d). Collective spin-wave-like\nexcitations also are unusual: multiple branches are found at some q. Finally, at higher q\nwe find that the SW velocity changes sign, which is also observed expe rimentally at similar\nvalues of M.\nOverall a picture emerges where localized and itinerant magnetic car riers coexist and\ninfluence each other. This description falls well outside the framewo rk of a local-moments\nmodel such as the Heisenberg hamiltonian. The low-energy particle- hole excitations are\nstrongly affected by Fe-As distance (and thus by lattice vibrations ) while the SWs are much\nless so. While phonons and Stoner excitations are generally consider ed separately, these\nfindings suggest that they strongly influence each other.\nThis work was supported by ONR, grant N00014-07-1-0479 and by DOE contract DE-\nFG02-06ER46302. Work at the Ames Laboratory was supported b y DOE Basic Energy\nSciences, Contract No. DE-AC02-07CH11358.\n[1] S.O. Diallo et al., Phys. Rev. Lett. 102, 187206 (2009).\n[2] J. Zhao et al., Nature Physics 5, 555 (2009).\n[3] M. D. Johannes, I.I. Mazin. Phys. Rev. B 79, 220510 (2009) ; I.I. Mazin, J. Schmalian. Physica\n9C, 469, 614 (2009).\n[4] Z. P. Yin et al., Phys. Rev. Lett. 101, 047001 (2008).\n[5] Chen Fang et al., Phys. Rev. B 77, 224509 (2008); F. Kruger , S. Kumar, J. Zaanen and J.\nvan den Brink. Phys. Rev.B 79, 054504 (2009); C. Lee, W. Lin an d W. Ku. arXiv.0905.3782;\nR. R. P. Singh. arxiv.0903.4408.\n[6] T. Kotani, M. van Schilfgaarde, S. V. Faleev, Phys. Rev. B 76, 165106 (2007).\n[7] T. Kotani and M. van Schilfgaarde, J. Phys. Cond. Matt. 20 , 295214 (2008).\n[8] R. Martin, Electronic Structure , Cambridge University Press (Cambridge, 2004).\n[9] A.I. Goldman et al., Phys. Rev. B 78, 100506 (2008).\n[10] N. Ni et al., Phys. Rev. B 80, 024511 (2009); E. D. Mun, S. L . Bud’ko, N. Ni, P. C. Canfield.\narXiv:0906.1548.\n[11] The origin of the discrepancy in ¯ ωwith experiment is not clear. Our calculation contains\nseveral important approximations (local potential, linea r response, rigid spin).\n[12] J J Pulikkoti et. al, Supercond. Sci. Technol. 23, 054012 (2010).\n[13] G. Samolyuk and V.P.Antropov. Phys. Rev. B 79, 052505 (2 009).\n[14] R. J. McQueeney et al., Phys. Rev. Lett. 101, 227205 (200 8).\n[15] J. W. Lynn and P. Dai. Physica C469, 469 (2009).\n10"    },    {        "title": "1508.00624v1.High_field_response_of_gated_graphene_at_THz_frequencies.pdf",        "content": "High \feld response of gated graphene at THz frequencies\nHadi Razavipour, Wayne Yang, Abdeladim Guermoune, Michael Hilke, and David G. Cooke\u0003\nDepartment of Physics, McGill University, Montr\u0013 eal, Qu\u0013 ebec, Canada\nIbraheem Al-Naib and Marc M. Dignam\nDepartment of Physics, Engineering Physics and Astronomy, Queen's University, Kingston, Ontario, Canada\nFran\u0018 cois Blanchard\nD\u0013 epartement de g\u0013 enie \u0013Electrique, \u0013Ecole de technologie sup\u0013 erieure, Montr\u0013 eal, Qu\u0013 ebec, Canada\nHassan A. Hafez, Xin Chai, Denis Ferachou, and Tsuneyuki Ozaki\nINRS-EMT, Advanced Laser Light Source, Varennes, Qu\u0013 ebec, Canada\nPierre L\u0013 evesque and Richard Martel\nD\u0013 epartement de Chimie, Universit\u0013 e de Montr\u0013 eal, Montr\u0013 eal, Qu\u0013 ebec, Canada\nWe study the Fermi energy level dependence of nonlinear terahertz (THz) transmission of gated\nmulti-layer and single-layer graphene transferred onto sapphire and quartz substrates. The two\nsamples represent two limits of low-\feld impurity scattering: short-range neutral and long-range\ncharged impurity scattering, respectively. We observe an increase in the transmission as the \feld\namplitude is increased due to intraband absorption bleaching starting at \felds above 8 kV/cm. This\ne\u000bect arises from a \feld-induced reduction in THz conductivity that depends strongly on the Fermi\nenergy. We account for intraband absorption using a free carrier Drude model that includes neutral\nand charged impurity scattering as well as optical phonon scattering. We \fnd that although the\nFermi-level dependence in the monolayer and \fve-layer samples is quite di\u000berent, both exhibit a\nstrong dependence on the \feld amplitude that cannot be explained on the basis of an increase in\nthe lattice temperature alone. Our results provide a deeper understanding of transport in graphene\ndevices operating at THz frequencies and in modest kV/cm \feld strengths where nonlinearities\nexist.\nPACS numbers: 78.67.Wj, 72.20.Ht, 78.67.-n, 72.80.Vp\nI. INTRODUCTION\nThe well known linear electronic dispersion of graphene\nnaturally gives rise to unusual optoelectronic properties,1\npotentially useful for high-speed modulators and the\nother active components. While it has been touted as\nan important material for terahertz (THz) frequency\ndevices,2charge transport properties at THz frequencies\nunder \feld conditions commonly found in devices, on the\norder 10's kV/cm, are poorly understood. This is in part\ndue to a lack of control over the extrinsic properties of the\ndevices and graphene's sensitivity to its environment.3\nThe electrodynamic response of doped graphene has pre-\nviously been studied in the low \feld, linear regime under\ngated conditions4for frequencies ranging from near dc5\nto THz,6{10mid IR11and optical regimes.12Under in-\ntense \felds, however, charge carriers can acquire signi\f-\ncant momentum and for Fermi energies ( Ef) close to the\ncharge neutrality point (CNP), a complicated interplay\nbetween intra- and inter-band transitions can occur.13A\nbetter understanding of these nonlinear interactions and\ntheir density dependence is needed for the future device\ndesign, which is the aim of this study. For this purpose,\nwe study the THz electrodynamic response of graphene\nunder high \feld conditions and controlled Fermi level\nthrough a gate voltage VGapplied to an ionic gel topgate which is transmissive to THz light (see Fig. 1).\nThe development of e\u000ecient nonlinear optical methods\nfor the generation of intense THz pulses14has made it\npossible to time resolve the high \feld response of charge\nand spin excitations in materials.15In graphene, due to a\nstrong interband dipole matrix element that diverges at\nthe charge neutrality (Dirac) point, the THz light-matter\ninteraction can be easily pushed into the nonperturba-\ntive regime even by modest electric \felds on the order\nof several 10's kV/cm.16,17As the thermalization time of\naccelerated hot carriers is noted to be exceedingly short\nin graphene,18these nonlinearities can be quanti\fed as\nan elevated temperature of the electronic subsystem.23\nRecent works have demonstrated a strong THz \feld-\ninduced transparency in room-temperature graphene19in\nboth doped19{22and optically excited graphene.24This\nphenomenon has been described by ultrafast intra-band\ncarrier thermalization on the order of 30 fs,18leading\nto a transient elevated carrier temperature on the order\nof 2000 K and subsequent optical phonon emission dur-\ning cooling. In this model, the non-equilibrium heating\nand generation of optical phonons subsequently enhances\nthe carrier scattering, suppressing the \feld-driven current\nand thus THz absorption in graphene. Other sources of\ncarrier scattering that can potentially play an important\nrole in the THz response of graphene include long-rangearXiv:1508.00624v1  [cond-mat.mes-hall]  3 Aug 20152\nFIG. 1. (a) Schematic of the graphene sample on a substrate\nwith gold four-point-probe contacts and a side gate contact.\nThe broadband THz pulses traverse the sample at normal\nincidence. (b) The mechanism for ionic gating of graphene is\nshown in the inset, illustrating the accumulated Debye layer\nunder a gate voltage V G.\nimpurity scattering, due to the presence of the charged\nimpurities in the substrate, short-range scattering, due\nto neutral impurities or disorder in the graphene itself,\nand carrier-carrier scattering. While impurity scattering\ndominates at low lattice temperatures, optical phonon\nscattering becomes more dominant at higher tempera-\ntures (T > 700 K).25\nSaturable absorption due to interband contributions\nhas also been observed in the visible regime due to the\nphase-space \flling e\u000bects as well as carrier heating. At\nTHz frequencies, the threshold for an interband response\nis determined by the carrier density, as the carriers must\nbe excited at twice the quasi-Fermi energy to undergo\na transition. In this work, we study the Fermi energy\ndependence of the nonlinear THz transmission of gated\ngraphene and show that although the nonlinear response\nis similar in di\u000berent samples with di\u000berent doping lev-\nels, it does not appear that this e\u000bect can be described\nby heating of the lattice alone. The paper is organized\nas follows. In Sec. II, we discuss the possible scattering\nmechanisms and their e\u000bects on THz transmission as a\nbasis for the model used to describe the data. The results\nfor both \fve-layer and single-layer graphene samples are\npresented in Sec. III. Finally, the conclusions are sum-\nmarized in Sec. IV.II. THEORETICAL MODEL\nIn this section, we discuss the model that we use to\naccount for the e\u000bects of tuning the Fermi energy on\nthe transmission of THz light. Using this model, we\nextract the dependence of the scattering time on the\nFermi energy and the lattice and electron temperature.\nThere are four potentially important sources of carrier\nscattering in graphene: short-range impurity scattering,\nlong-range impurity scattering, phonon scattering, and\ncarrier-carrier scattering.\nStrictly speaking the carrier dynamics should be mod-\neled using a full dynamic model that includes the nonlin-\near response arising from the linear band dispersion and\ninterband transitions.13We \fnd however, in agreement\nwith our previous work,24that for the samples and \feld\nstrengths considered here, there is essentially no intrin-\nsically nonlinear response and a linear Drude model is\nsu\u000ecient. The nonlinear response of the graphene is in-\ncorporated in the model by employing a scattering time\nthat depends on the THz \feld amplitude. Thus, in what\nfollows, we employ the Drude model.\nFirst, we need to relate the transmission level to the\nconductivity of graphene. It can be easily shown that the\ntransmission through the graphene sample normalized to\nthe transmitted \feld through the substrate is given by6\nT(!) =TG+S\nTS=1 +n\n1 +n+Zo\u001b(!)N; (1)\nwhereTG+SandTSare the transmission through the\nsample and the substrate only, respectively, nis the re-\nfractive index of the substrate, \u001b(!) is the complex ac\nsheet conductivity of a single layer of graphene and Nis\nthe number of graphene layers in the sample. The con-\nductivity, dominated by the intra-band response at THz\nfrequencies is given by a simple Drude model11\n\u001b(!) =D\n(1=\u001c\u0000i!); (2)\nwhere!is the angular frequency, \u001cis the total carrier\nmomentum scattering time, and Dis the Drude weight\ngiven by9,10\nD=2e2kBTe\n(\u0019~2)ln\u0014\n2cosh\u0016f\n2kBTe\u0015\n; (3)\nwhereeis the electron charge, kBis the Boltzmann\nconstant,Teis the electron temperature, ~is the re-\nduced Planck's constant, and \u0016fis the chemical poten-\ntial. For simplicity, the chemical potential is considered\nto be equal to the Fermi energy at room temperature.\nHowever, as the electron temperature rises, we adjust\nthe chemical potential so that the net charge on the\ngraphene is unchanged as the temperature rises. The\nadjusted chemical potential along with the correspond-\ning temperature are used then to calculate the Drude3\nweight. Using Eqs. (1-3), we can extract the total ef-\nfective scattering time \u001cfor any measured transmission\nlevel for each gate voltage and THz \feld amplitude. The\ngate voltage and \feld amplitude dependence of the scat-\ntering time can therefore be directly obtained from the\nTHz transmission data once the electron temperature is\ndetermined.\nWe now turn to the model that we employ for the\nscattering time used in the Drude model. We present\na simple model of the e\u000bects of the lattice temperature\nand the carrier density on the scattering times based on\nthree scattering mechanisms: short-range neutral impu-\nrity scattering, charged long-range impurity scattering\nand scattering due to absorption of optical phonons. Al-\nthough there can be contributions to the scattering rate\ndue to interactions with acoustic phonons, these are usu-\nally considered to be relatively small at the temperatures\nconsidered here.29We do not include carrier-carrier scat-\ntering due to the di\u000eculties in accurately modelling its\ne\u000bects and because, for similar Fermi energies and THz\n\feld amplitudes, it has been shown that carrier-carrier\nscattering can be neglected without qualitatively chang-\ning the results.26We will return to this issue when dis-\ncussing our \feld-dependent results.\nThe expression for the total scattering rate is thus\ngiven by\n1\n\u001c=1\n\u001cimp+1\n\u001cop; (4)\nwhere\u001cimpis the impurity scattering time \u001copis the opti-\ncal phonon scattering time. The impurity scattering rate\nis given by\n1\n\u001cimp=1\n\u001csr+1\n\u001cCoul+1\n\u001cml: (5)\nIn this expression, 1 =\u001csris the short-range neutral impu-\nrity scattering rate, which has been shown theoretically\nto be linearly dependent on the energy of the carriers,27;\nin our model, we take this rate to be proportional to the\naverage energy per carrier with respect to the Dirac point\nK, which is given by\nEav=P\nkf\u001aee(K+k) +\u001ahh(K+k)g~vFk\nP\nkf\u001aee(K+k) +\u001ahh(K+k)g;(6)\nwhere\u001aee(\u001ahh) is the free electron (hole) population den-\nsity, andvFis the Fermi velocity. The second term in the\nimpurity scattering rate is, \u001cCoul, which is due to charged\nlong-range impurities. This rate has been shown to be\ninversely proportional to the square root of the carrier\ndensity (due to carrier screening)28. Finally,\u001cmlis an\nadditional gate-voltage-independent scattering time that\nwe \fnd is necessary in order to model the bias depen-\ndence of the scattering time in our multi-layer sample.\nThe physical origin of this term is not clear, but it may\nbe due to scattering from the ionic gel or is perhaps arisesdue to a correction term in the relationship between the\ngate voltage and the Fermi energy. Thus, the \fnal ex-\npression for the impurity scattering rate is\n1\n\u001cimp=csrEav+cCoul\nEeff+1\n\u001cml; (7)\nwherecsrandcCoulare constants that are experimentally\ndetermined and are dependent on the impurity densities,\nandEeffis the e\u000bective Fermi energy that corresponds\nto the square root of the carrier density and is given by\nD\u0019~2=e2.\nWe now turn to the scattering due to optical phonons.\nThere are two potential types of optical phonons that can\ncontribute to scattering: optical phonons in the graphene\nitself and polar surface phonons in the substrate.41,42\nWe \fnd, using the parameters in Ref. 41 that, due to\nthe weak coupling strength, the scattering rate contribu-\ntion from the surface phonons in both substrate materials\nis much smaller than those due to the graphene optical\nphonons. Thus, in what follows, we neglect the phonons\nin the substrate and include only the contribution from\nthe phonons in the graphene itself.29\nPrevious studies have shown that energy transfer from\nelectrons to phonons takes place on a timescale of about\n100 fs.30,31Thus, in principle, if there are electrons\nthat have been driven by the \feld more than an optical\nphonon energy above any vacant states, one would ex-\npect emission of optical phonons over the pulse duration,\nwhich would increase the lattice temperature. However,\nwithout performing a detailed simulation that tracks the\nenergies of individual electrons, it is very di\u000ecult to accu-\nrately include this contribution to the scattering. Thus,\nfor simplicity, we only include the process of phonon ab-\nsorption, but not phonon emission.\nThe optical phonon scattering rate due to optical\nphonon emission in the graphene is given by\n1\n\u001cop=D2\no\n(2~2v2\nF\u001bm!o)(Eav+~!o)\n(e~!o=kBTl\u00001); (8)\nwhere ~!o= 147 meV is the graphene optical phonon\nenergy,\u001bm= 7:6\u000210\u00008g/cm2is the 2D mass density\nof graphene29,Do= 5\u0002109eV/cm is the deformation\npotential for the optical phonons32andTlis the lattice\ntemperature. Various values for the optical phonon en-\nergy and deformation potential have been reported in the\nliterature29,32{34. We employ those found in Ref. 24, al-\nthough, as we discuss in the following sections, we have\nconsidered the e\u000bect of modifying the deformation po-\ntential on the results. We \fnally note that in all cases we\nassume equality of the carrier and lattice temperatures\nthat appear respectively in the Drude weight (Eq. (3))\nand the optical phonon scattering rate (Eq. (8)).4\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\nIn order to investigate the e\u000bect of the substrates and\nnumber of graphene layers on the transport properties\nof graphene to the intense THz \feld, we fabricated two\ndi\u000berent samples treated in the next two sections. The\n\frst is a \fve layer graphene sample, fabricated by trans-\nferring sequentially monolayer graphene onto a sapphire\nsubstrate, and the second a monolayer sample on a quartz\nsubstrate. The substrates were patterned with 5/100 nm\nthickness of Cr/Au (prior to transfer) for the electrical\nfour-point-probe measurements as shown in Fig. 1. The\ngate was created by placing the \ffth contact at the edge\nof the substrate so that it is in contact with the ionic gel\nfor electrochemical doping. The ionic gel was fabricated\nusing the method proposed by Ref. 35 and transferred to\nthe sample under an inert glovebox environment. A uni-\nform layer of ionic gel with a thickness of less than 50 \u0016m\nwas spin coated on the graphene sample as well as the\nbare substrate to serve as a reference. Upon applying the\ngate voltage, mobile ions are transferred to the surface of\nthe graphene to form a Debye layer with a thickness on\nthe order of 1 nm.36This type of solid polymer electrolyte\ngate is much thinner than the conventional 300 nm SiO 2\nback gate, thus a much higher gate capacitance can be\nachieved. With the electrolyte ionic gating, high doping\nlevels on the order of 5 \u00021013cm\u00002are possible.36The\ngating response of the graphene is measured in terms of\nthe electrical current driven from the source to the drain\nelectrode through the graphene channel as a function of\nVG. In order to estimate the Fermi energy as a function of\ngate voltage relative to the charge neutrality point volt-\nage (VCNP), we use the following equation from Ref. 10\nfor a similar geometry and ionic gel, where\nEf= 0:346p\n(VG\u0000VCNP) [eV]: (9)\nThe case of VG=VCNP needs to be treated separately\nbecause, in agreement with other researchers,37we \fnd\nthat due to spatial inhomogeneity of the charge density\nacross the graphene sample an e\u000bective Efmust be de-\ntermined at the nominal CNP gate voltage, in a range of\n\u0006100 meV.10\nThe high-\feld time-domain THz spectrometer was de-\nveloped based on tilted-pulse front optical recti\fcation in\nlithium niobate,14with peak THz \feld amplitudes lim-\nited to 70 kV/cm in this work. The bandwidth of the\nTHz pulses was\u00183 THz and was detected using electro-\noptic sampling in a 300 \u0016m thick (110)-cut GaP crystal.\nThe emitted THz pulses were collected and focused by a\nset of o\u000b-axis parabolic mirrors to a 450 \u0016m spot size as\ndetermined by an uncooled microbolometer camera. The\ngraphene sample was positioned carefully at the THz fo-\ncus for high-\feld transmission measurements at normal\nincidence. The amplitude of the THz pulses was varied\nusing a pair of wire-grid polarizers prior to the sample in\nthe collimated section of the beam. The measurements\nwere performed by sweeping the graphene gate voltage\nFIG. 2. (a) Four-point-probe measurements of the 5-layer\nsample dc resistance with the charge neutrality point indi-\ncated by the resistance peak. (b) The Raman data (with\nthe G and 2D peaks) of single layer of CVD grown graphene\nbefore stacking to form the 5 layers.\nand registering the value of the transmitted peak THz\nelectric \feld. No THz pulse reshaping was observed as\nthe gate voltage varies from \u00002:5V(below the Dirac\npoint) to 2 :5V, thus all information on the nonlinear\ntransmission can be quanti\fed by monitoring the THz\npulse peak \feld transmission.\nA. Five-layer sample\nLarge-area, single-layer graphene was fabricated by\nchemical vapor deposition (CVD) on Cu-foil using stan-\ndard techniques. Poly(methyl methacrylate) (PMMA) is\nused to transfer the graphene to a c-cut sapphire sub-\nstrate with refractive index of 3.31 and the PMMA layer\nwas removed. The process was repeated \fve times so that\nthe \fve-layer graphene sample was obtained, with the\nlayers electrically decoupled from one another. Fig. 2(a)\nshows the sheet resistance of graphene as a function of\ngate voltage. The CNP is speci\fed as the resistance\nmaximum which occurs in this case at VG=\u00000:8V.\nWe note that hysteresis was observed in these resistance\nmeasurements when sweeping the gate voltage, which is\ncommonly observed in ionic gel gates.38,39We therefore\nonly perform measurements on the positive sweep of the\ngate voltage for repeatability. The Debye absorption of5\nFIG. 3. For the \fve-layer sample: (a) The THz pulse peak\n\feld transmission through the gated graphene sample for the\n\feld strengths indicated, normalized to the transmission at\nthe Dirac point at VG\u0000VCNP = 0. (b) The \feld dependence\nof the THz peak \feld transmission jtj=EG+S\nESat various\ngate voltages. Eincis the incident THz electric \feld at the\ngraphene sample position.\nthe ionic gel is located at lower frequencies than THz,\nproducing a transparent top gated graphene sample. All\nmeasurements in this work are performed at room tem-\nperature. Fig. 2(b) shows the Raman spectroscopy of one\nof the \fve layers of graphene before stacking. The ratio\nbetween the 2D and G peak values is 2.8 which indicates\nhigh quality single layer graphene.40\nFig. 3(a) shows the variation of the transmitted peak\nTHz electric \feld (normalized to its maximum value at\nthe Dirac point) as a function of gate voltage sweep\nfor di\u000berent incident peak electric \felds in the range\nof 2 kV/cm to 70 kV/cm. The THz modulation in-\nduced by the gate voltage sweep (Fermi level change in\ngraphene) is a strongly nonlinear function of the peak\nelectric \feld value. At low \felds, gate-induced THz mod-\nulation is more than 31% whereas for higher \feld ampli-\ntudes the modulation decreases and saturates at a value\nof 7:4% above 60 kV/cm. As in previous work on un-\ngated graphene, a saturable power transmission function\ncan be used to phenomenologically describe the behavior\nobserved in Fig. 3(b).19The power transmission is de-\n\fned asjTj=\u0010\nEsamp\nEref\u00112\n, whereEsamp is the peak THz\nelectric \feld transmitted through the ionic gel, 5 layers\nFIG. 4. For the \fve layer sample: (a) THz power transmission\nde\fned asjTj=E2\nsamp\nE2\nref, curve \ftted by Eq. 10 for two\ndi\u000berent gate voltages, one at the Dirac point ( VG\u0000VCNP =\n0V, red line) and the other one at a n-doped Fermi level\ncorresponding to ( VG\u0000VCNP =\u00002:4V, blue line). (b-d) Fit\nparameters Esat,Tlin, andTnsextracted from Eq. (10) for\npower transmission curves as a function of gate voltage.\nof graphene and the substrate, and Erefis the peak \feld\ntransmitted through the ionic gel and the substrate only.\nThe model is given by:19\nT(Einc) =Tnsln[1 +Tlin\nTns(eEinc2=Esat2\u00001)]\nEinc2=Esat2(10)\nwhereEinc,Esat,TlinandTnsare the incident peak elec-\ntric \feld, the saturation electric \feld, the linear power\ntransmission coe\u000ecient and the nonsaturable power\ntransmission coe\u000ecient, respectively.\nIn Fig. 4(a) we present the measured transmitted\npower as a function of the incident \feld along with the\n\ft using Eq. (10) for VG=VCNP (corresponding to\nthe CNP, red line) and VG\u0000VCNP =\u00002:4V(i.e.\nhighly the n-doped case, blue line). The saturation on-\nset of the power transmission, jTj, occurs at a lower\nEinc= 9:7 kV/cm at the CNP relative to what occurs\nfor higher doping, as demonstrated by Fig. 4(b). Tlin\n[Fig. 4(c)] and Tns[Fig. 4(d)] are higher at the CNP\nthan at other Fermi levels, overall expected due to the\nlarger conductivity of the doped graphene. The satura-\ntion behavior for the curves in Fig. 4(a) is explained by a\ndecrease in carrier mobility as the THz excitation redis-\ntributes carriers within the conduction band.19There-\nfore, the decrease in Esatwhen the gate voltage ap-\nproaches the Dirac point can be attributed to an increase\nin the scattering rate when the Fermi level is tuned to the\nCNP, as we shall discuss shortly.\nWe now use the model presented in Section II to ex-\ntract the scattering time as a function of the Fermi energy\nand the incident THz \feld. While the Fermi energy may6\nnot be precisely the same in all 5 layers of the graphene\nsample, for simplicity we use the same value for the \fve\nlayers. Similarly, we assume that the scattering time is\nthe same in all layers. We estimate Effrom gate voltages\nother than the CNP voltage using Eq. 9. To determine\nthe Fermi energy for VG=VCNP, we \frst use the model\nof Section II to extract the \ftting parameters, csr,cCoul\nand\u001cmlin Eq. 7 for the transmission data for the lowest\nTHz \feld amplitude of 2 kV/cm. Taking the lattice and\nelectron temperatures at this lowest \feld to be 300 K, we\nobtaincsr=49 ps\u00001eV\u00001,cCoul = 0 and1\n\u001cml=7.3 ps\u00001.\nThe data as well as the \ft are shown in Fig. 5(a). We now\nextract an e\u000bective Fermi energy at the VG=VCNP of\napproximately 90 meV by extrapolating the \ftting curve.\nThis value is in good agreement with other studies that\nused the same mechanism to tune the Fermi energy.10We\n\fnd from our \ft that the contribution of charged impu-\nrities is negligible and that at this temperature and \feld\nstrength, the contribution of the optical phonons is also\nvery small. Thus, the dominant contribution to scatter-\ning at this lowest \feld appears to arise from short-range\nimpurities, which results in the approximately hyperbolic\nrelationship between \u001candEfseen in the \fgure.27\nWe now turn to the dependence of the scattering time\non the incident \feld for di\u000berent Fermi energies. Rather\nthan using our temperature-dependent model of scatter-\ning, we \frst assume that the electron and phonon tem-\nperatures do not change with the incident THz \feld am-\nplitude and simply extract the net scattering time as a\nfunction of the incident \feld strength to two di\u000berent\nFermi levels. This is plotted in Fig. 5(b). As can be\nseen, for both Fermi energies, the scattering time de-\ncreases dramatically with increasing \feld amplitude. We\nsee that the decrease in the scattering time with \feld is\nmuch larger for the low Fermi energy case. This would\nbe expected if the increased scattering is due to a mech-\nanism that has a rate that is roughly proportional to\nthe Fermi energy and that would increase at high \felds.\nThere are two likely candidates: optical phonon scatter-\ning and carrier-carrier scattering.\nWe \frst consider optical phonon scattering. When the\n\feld is higher, it heats the graphene up considerably; this,\nin turn results in a larger optical phonon population and\na shorter scattering time. To test the viability of the\noptical phonon mechanism, we employed our model that\nincludes the e\u000bects of electron and lattice temperature.\nUsing the impurity scattering rates found at the lowest\n\feld and requiring that the electron and phonon temper-\natures are equal, we were able to obtain good agreement\nwith the experimental data for the transmission by using\nthe temperature as a \ftting parameter. We found that at\nthe highest \felds, the temperature required was approx-\nimately 1550 K. To determine if such temperatures are\nphysically sensible, we compared the experimental result\nfor the energy absorbed per unit area to the increase in\nthe electron and phonon populations per unit area. We\nfound that the required increase in the system energy\nwas more than an order of magnitude greater than theabsorbed THz energy for the 70 kV/cm \feld. Most of\nthe increase in the energy of the system when the tem-\nperature increases to such levels resides in the optical\nphonons, not the electrons. Because there is not uni-\nversal agreement as the values of the optical phonon en-\nergy and deformation potential, we tried adjusting these\nparameters. However, we could not \fnd any values of\nthese parameters that yielded anything approaching en-\nergy balance at all \felds and Fermi energies for one set of\nparameters. We conclude from this, that for this sample\nat least, the decrease in the scattering time with increas-\ning \feld does not arise entirely from an increase in the\noptical phonon population due to an increase in the lat-\ntice temperature.\nAnother possible mechanism for the \feld dependent\nscattering is carrier-carrier scattering. This scattering\nrate increase rapidly as the carrier density increases (as\nobserved). Moreover, it also increases with the energy of\nthe carriers, which will occur when the \feld amplitude\nis high. Modeling carrier-carrier scattering with energy\ndependence would require a sophisticated Monte-Carlo\nor density matrix simulation. Thus, although we cannot\nrule out this mechanism, it is beyond the scope of the\ncurrent work.\nWe now examine the results for our monolayer\ngraphene sample to see how its transmission depends on\nthe Fermi energy and the THz \feld amplitude.\nB. Single-layer sample\nA single-layer graphene sample was prepared by chem-\nical vapor deposition on Cu-foil and transferred onto a\nz-cut quartz substrate with a refractive index n= 1:96.\nThe sheet resistance of this sample as a function of VG\nwas determined by a two terminal measurement shown in\nFig. 6(a) with the CNP indicated again by the resistance\npeak. Electron (hole) doping is induced by changing VG\nabove (below) the CNP voltage VCNP and Eq. 9 is used\nto calculate the Fermi level energy. Fig. 6(b) shows the\nRaman spectra for the single layer graphene sample, with\na 2D/G peak ratio of 1.78 con\frming high quality mono-\nlayer graphene.\nFig. 7(a) shows the normalized THz peak electric \feld\namplitude transmitted through the graphene when the\nFermi-level is swept from -1 to 0.5 V. Again, our data\ndoes not show any phase change in the THz waveform\ntransmitted through the graphene by increasing the in-\ncident THz \feld and so the conductivity change can be\ncompletely quanti\fed by monitoring the peak transmis-\nsion. For the lowest THz peak electric \feld strength of 1.5\nkV/cm, a signi\fcant 20% reduction in the peak transmis-\nsion was observed at V G- VCNP =\u00001 V or Ef=\u0000340\nmeV. As expected, the increased doping leads to higher\nconductivity and therefore absorption of the THz pulse,\nhowever, increasing the incident THz \feld leads to a re-\nduction in the doping-induced absorption as in the 5-\nlayer sample.21,24As in the 5-layer case, a continuous7\nFIG. 5. (a) The impurity scattering time ( \u001cimp) versus Fermi\nenergy, and (b) total scattering time versus incident \feld am-\nplitude at the CNP and at the highest Fermi level of 442 meV\nfor the \fve-layer sample. Note that in (a), the value for Ef\nthe \frst point is determined via a \ft to the curve.\nreduction in \feld-induced transmission modulation oc-\ncurs as the gate voltage approaches the charge neutrality\npoint. This behavior is best seen in Fig. 7(b), showing\nthe peak transmission as a function of the THz peak \feld\nstrength for various applied V G. The THz \feld-induced\ntransparency (the percentage increase in transmission)\nis more pronounced for high doping levels. As with the\n\fve-layer sample, we \fnd that the induced nonlinearity is\nmainly due to an increase in the carrier scattering when\nthe THz \feld is increased.21,24\nWe extract the scattering behavior by using the model\nof Section II. We extract the low-\feld scattering parame-\nters in the same way as was done in the last section taking\nthe temperature to be room temperature. The transmis-\nsion data at the lowest \feld amplitude can be well repro-\nduced by the theory only if we consider both short-range\nneutral and long-range charged impurity scattering in ad-\ndition to optical phonon scattering. We \fnd that the best\n\ft is obtained for csr=30ps\u00001eV\u00001,cCoul=3.63ps\u00001eV\nand and\u001cml=0. The data as well as the \ft are shown in\nFig. 8(a). Note that the increase in the scattering time\nwith Fermi energy indicates that long range charged im-\nFIG. 6. (a) Two terminal measurements of the resistance\nand (b) the Raman data (with the G and 2D peaks) of the\nmonolayer sample.\npurity scattering is dominant at this \feld. The presence\nof long range scattering in this sample may be linked\nwith the direct contact of the graphene with the SiO 2\nsubstrate as opposed to the 5 layer sample, where 4 of\nthe layers are physically removed from the substrate and\nthe e\u000bects of charged impurities on carriers in the up-\nper layers are partially screened by carriers in the lower\nones. Finally, using the extrapolation of the \ft to low\nFermi energy, we extract and Fermi energy of 88 meV\nfor the bias V G= VCNP.\nWe now use the same constant-temperature model of\nthe previous section to determine the net scattering time\nas a function of the incident \feld strength for the \fve dif-\nferent Fermi levels and plot these times in Fig. 8(b). As\nin the \fve-layer case, the scattering time decreases dra-\nmatically with increasing \feld amplitude for all Fermi en-\nergies. However, in contrast to the results of the previous\nsection, we \fnd that the decrease in the scattering time\nwith the \feld is much smaller when V G= VCNP. One\nreason for this di\u000berence is that in the monolayer, the\nlow-\feld scattering time is the shortest when the Fermi\nenergy is the lowest, thus the e\u000bect of any additional scat-\ntering channels is smaller. At the highest \felds, just as\nin the \fve-layer sample, the scattering time is the largest\nwhen VG= VCNP. This seems to indicate again that8\nFIG. 7. (a) The normalized peak transmission as a function of\nthe gate voltage for various THz peak electric \feld strengths\nand (b) the dependence of the peak transmission on the THz\npeak electric \feld for various doping levels for the monolayer\nsample.\nthe \feld-dependent scattering rate is larger when the car-\nrier density (or perhaps average carrier energy) is larger.\nHowever, the trend is not the same for the intermediate\nFermi energies, where at the highest \feld, the scattering\nrate is larger when the Fermi energy is smaller (with the\nexception of the CNP result). It appears that the result\nat the CNP is somewhat of an anomaly in this sample.\nAlthough, the Fermi energy dependence of the \feld-\ndependent scattering component is not as clear as in\nthe \fve-layer sample, we again examined where optical-\nphonon scattering is a possible mechanism. We modeled\nthe optical phonon scattering by taking the temperature\nto be a \ftting parameter. Again, although we could re-\nproduce the observed transmission curves, the tempera-\ntures required were very large. The temperatures at the\nhighest \feld ranged from 1300 K for the highest Fermi en-\nergy to almost 3000 K at the CNP. Again, such huge tem-\nperatures require that the energy deposited in the system\nis a factor of 3 to 10 larger than the energy that is experi-\nmentally found to be absorbed by the graphene. As in the\n\fve-layer case, we tried adjusting the phonon energy and\ndeformation potential, but for all combinations that we\nFIG. 8. (a) The impurity scattering time ( \u001cimp) versus Fermi\nenergy at the lowest incident THz \feld of 1.5 kV/cm, and (b)\nthe THz \feld dependence of (a) the carrier scattering rate.\nNote that in (a), the value for Efthe \frst point is determined\nvia a \ft to the curve.\ntried we could not achieve anything approaching energy\nbalance at all \felds and Fermi energies. We thus conclude\nthat for this sample as well, optical phonons cannot be\nthe major contributor to the increase in the scattering\nrate with increasing \feld. In this sample, however, due\nto the more complex dependence of the \feld-dependent\nscattering contribution on the Fermi energy, it is not as\nclear that carrier-carrier scattering is a likely candidate.\nIV. CONCLUSION\nWe have studied nonlinear THz light-matter interac-\ntion for both monolayer graphene on quartz and a stack\nof 5 monolayers of graphene on sapphire under the con-\ntrolled Fermi level conditions. The two samples exhibited\nopposite dependencies on the Fermi energy for low THz\n\felds. The monolayer sample scattering rate was domi-\nnated by charged impurities, while the \fve layer sample\nscattering was neutral impurity dominated. This di\u000ber-\nence is likely explained as arising from di\u000berent prepara-\ntion recipes as well as di\u000berences in the substrates.\nWe also observed a strong increase in the THz trans-9\nmission with increasing \feld that occurs for THz peak\nelectric \felds as low as 8 kV/cm. This increase appears\nto result from a decrease in the carrier scattering times\nwith increasing \feld. Our analyses indicate that opti-\ncal phonon absorption is not likely the main source of\nthis \feld-induced increase in the scattering rate. The\nsource of this increase remains uncertain. For the \fve\nlayer sample, the most promising mechanism is perhaps\ncarrier-carrier scattering, but for the monolayer sample\nthe mechanism is even more unclear.\nThis work points to the importance of future modelling\nof full carrier and lattice dynamics in graphene excited by\nintense THz pulses to determine the main source of the\nstrongly nonlinear response. The interaction of high \feldTHz pulses with graphene is signi\fcantly dependent on\nFermi energy. Moreover, given the variability in process-\ning of graphene \flms, we show that it is entirely possible\nthat di\u000berent impurity scattering mechanisms dominate\nin di\u000berent samples, even though the \feld dependence is\nsimilar for a \fxed Fermi energy. Thus any comprehen-\nsive study must be under controlled gated conditions. We\nanticipate that our \fndings will be important for future\ndesigns of graphene devices operating at THz frequencies,\nin\ruencing performance due to nonlinearities arising at\nthe kV/cm operating \felds.\nThe authors gratefully acknowledge funding support\nfrom NSERC, FRQNT and CFI.\n1K. S. A. Novoselov, A. K. Geim, S. V. B. Morozov,\nD. Jiang, M. I. V. Grigorieva, S. V. Dubonos, A. A. Firsov,\nNature 438, 197-200 (2006).\n2T. Otsuji, S. A. B. Tombet, A. Satou, H. Fukidome,\nM. Suemitsu, E. Sano, V. Popov, M. Ryzhii, V. Ryzhii,\nJ. Phys. D: Appl. Phys. 45, 303001 (2012).\n3C. J. Docherty, C. T. Lin, H. J. Joyce, R. J. Nicholas,\nL. M. Herz, L. J. Li, M. B. Johnston, Nat. Commun. 3,\n1228 (2012).\n4F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crom-\nmie, Y. R. Shen, Sience 320, 206-209 (2008).\n5Y. M. Lin, K. A. Jenkins, A. Valdes-Garcia, J. P. Small,\nD. B. Farmer, P. Avouris, Nano Lett. 9, 422-426 (2009).\n6J. M. Dawlaty, S. Shivaraman, J. Strait, P. George,\nM. V. S. Chandrashekhar, F. Rana, M. G. Spencer, D. Vek-\nsler, Y. Chen, App. Phys. Lett. 93, 131905 (2008).\n7J. D. Buron, D. H. Petersen, P. B\u001cggild, D. G. Cooke,\nM. Hilke, J. Sun, E. Whiteway, P. F. Nielsen, O. Hansen,\nA. Yurgens, Nano Lett. 12, 5074-5081 (2012).\n8J. L. Tomaino, A. D. Jameson, J. W. Kevek, M. J. Paul,\nA. M. van der Zande, R. A. Barton, P. L. McEuen,\nE. D. Minot, Y. S. Lee, Opt. Expr. 19, 141-146 (2011).\n9A. J. Frenzel, C. H. Lui, Y. C. Shin, J. Kong, N. Gedik,\nPhys. Rev. Lett. 113, 056602 (2014).\n10S. F. Shi, T. T. Tang, B. Zeng, L. Ju, Q. Zhou, A. Zettl,\nF. Wang, Nano Lett. 14, 1578-1582 (2014).\n11J. Horng, C. F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao,\nH. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. Ron\nShen, F. Wang, Phys. Rev.B 83, 165113 (2011).\n12K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich,\nT. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).\n13I. Al-Naib, J. E. Sipe, M. M. Dignam, Phys. Rev. B 90,\n245423 (2014).\n14H. Hirori, A. Doi, F. Blanchard, K. Tanaka, Appl. Phys.\nLett. 98, 091106 (2011).\n15A. Pashkin, A. Sell, T. Kampfrath, R. Huber, New J. Phys.\n15, 065003 (2013).\n16P. Bowlan, E. Martinez-Moreno, K. Reimann, T. Elsaesser,\nM. Woerner, Phys. Rev. B. 89, 041408 (2014).\n17J. A. Crosse, X. Xu, M. S. Sherwin, R. B. Liu, Nat. Com-\nmun. 5, 4854 (2014).\n18J. C. Johannsen, S. Ulstrup, F. Cilento, A. Crepaldi,\nM. Zacchigna, C. Cacho, I. .C. E. Turcu, E. Springate,\nF. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Gri-oni, P. Ho\u000bmann , Phys. Rev. Lett. 111, 027403 (2013).\n19H. Y. Hwang, N. C. Brandt, H. Farhat, A. L. Hsu, J. Kong,\nK. A. Nelson, J. Phys. Chem. B 117, 15819-15824 (2013).\n20H. A. Hafez, I. Al-Naib, K. Oguri, Y. Sekine, M. M. Dig-\nnam, A. Ibrahim, D. G. Cooke, S. Tanaka, F. Komori,\nH. Hibino, T. Ozaki, AIP Advances 4, 117118 (2014).\n21M. J. Paul, Y. C. Chang, Z. J. Thompson, A. Stickel,\nJ. Wardini, H. Choi, E. D. Minot, B. Hou, J. A. Nees,\nT. B. Norris, Y. S. Lee, New J. Phys. 15, 085019 (2013).\n22M. J. Paul, B. Lee, J. L. Wardini, Z. J. Thompson,\nA. D. Stickel, A. Mousavian, H. Choi, E. D. Minot,\nY. S. Lee, Appl. Phys. Lett. 105, 221107 (2014).\n23Z. Mics, K. J. Tielrooij, K. Parvez, S. A. Jensen, I. Ivanov,\nX. Feng, K. M ullen, M. Bonn, D. Turchinovich, Nat. Com-\nmun. 6:7655 doi: 10.1038/ncomms8655 (2015).\n24H. A. Hafez, I. Al-Naib, M. M. Dignam, Y. Sekine,\nK. Oguri, F. Blanchard, D. G. Cooke, S. Tanaka, F. Ko-\nmori, H. Hibino, T. Ozaki, Phys. Rev. B 91, 035422 (2015).\n25W. S. Bao, S. Y. Liu, X. L. Lei, C. Wang, J. Phys. Condens.\nMatter. 21, 305302 (2009).\n26I. Al-Naib, M. Poschmann, M. M. Dignam, Phys. Rev. B\n91, 205407 (2015).\n27E. H. Hwang, S. D. Sarma, Phys. Rev. B 77, 195412 (2008).\n28S. Das Sarma, S. Adam, E. H. Hwang, E. Rossi, Rev. Mod.\nPhys. 83, 407 (2011).\n29T. Fang, A. Konar, H. Xing, D. Jena, Phys. Rev. B 84,\n125450 (2011).\n30M. Breusing, C. Ropers, T. Elsaesser, Phys. Rev. Lett.\n102, 086809 (2009).\n31C. H. Lui, K. F. Mak, J. Shan, T. F. Heinz, Phys. Rev.\nLett. 105, 127404 (2010).\n32N. Sule, I. Knezevic, J. Appl. Phys. 112, 053702 (2012).\n33J. H. Strait, H. Wang, S. Shivaraman, V. Shields,\nM. Spencer, F. Rana, Nano Lett. 11, 4902-4906, (2011).\n34E. H. Hwang, S. D. Sarma, Phys. Rev. B 79, 165404 (2009).\n35C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng,\nB. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segal-\nman, S. G. Louie, F. Wang, Nature 471, 617-620 (2011).\n36A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S. K. Saha,\nU. V. Waghmare, K. S. Novoselov, H. R. Krishnamurthy,\nA. K. Geim, A. C. Ferrari, A. K. Sood, Nature Nanotech.\n3, 210-215 (2008).\n37J. Martin, N. Akerman, G. Ulbricht, T. Lohmann,\nJ. H. Smet, K. Von Klitzing, A. Yacoby, Nat. Phys. 4,10\n144-148, (2008).\n38S. K. Lee, H. Y. Jang, S. Jang, E. Choi, B. H. Hong, J. Lee,\nS. Park, J. H. Ahn, Nano Lett. 12, 3472-3476 (2012).\n39J. Ye, M. F. Craciun, M. Koshino, S. Russo, S. Inoue,\nH. Yuan, H. Shimotani, A. F. Morpurgo, Y. Iwasa, Proc.\nNatl. Acad. Sci. 108, 13002-13006 (2011).40D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen,\nC. Hierold, L. Wirtz, Nano Lett. 7, 238-242 (2007).\n41K. Aniruddha, F. Tian, J. Debdeep, Phys. Rev. B 82,\n115452 (2010).\n42S. Berciaud, M. Y. Han, K. F. Mak, L. E. Brus, P. Kim,\nT. F. Heinz, Phys. Rev. Lett. 104, 227401 (2010)."    },    {        "title": "0912.3700v1.Magnetic_impurity_transition_in_a__d_s__wave_superconductor.pdf",        "content": "arXiv:0912.3700v1  [cond-mat.supr-con]  18 Dec 2009physica statussolidi, 10 November2018\nMagnetic impurity transition in a\nd+s- wave superconductor\nL. S.Borkowski*\nQuantum Physics Division, Facultyof Physics,A.Mickiewic z University, Umultowska 85, 61-614 Poznan, Poland\nPACS74.20.Rp,74.25.Dw,74.62.Dh,74.72.-h\n∗Corresponding author: e-mail lech.s.borkowski@gmail.com\nWe considerthe superconductingstate of d+ssymme-\ntry with finite concentration of Anderson impurities in\nthe limit ∆s/∆d≪1. The model consists of a BCS-\nlike term in the Hamiltonian and the Andersonimpurity\ntreatedintheself-consistentlarge- Nmeanfieldapprox-\nimation.Increasing impurity concentration or lowering the ratio\n∆s/∆ddrives the system through a transition from a\nstate with two sharppeaksat low energiesandexponen-\ntially small density of states at the Fermi level to one\nwithN(0)≃(∆s/∆d)2. This transition is discontinu-\nous if the energyof the impurity resonanceis the small-\nest energyscale inthe problem.\nCopyrightlinewillbe provided by the publisher\n1 Introduction The order parameter symmetry of\nhigh-Tccompounds is believed to be predominantly of d-\nwave type. However a subdominant s-wave componentof\nthe order parameter may also occur, especially in materi-\nals with orthorombic distortions. One of such compounds\nis YBCO, exhibiting strong structural distortion and a\nsubstantial anisotropy in the London penetration depth in\nthea-bplane.[1] Raman scattering[2] provided evidence\nfor a5%admixture of s-wave component while thermal\nconductivity measurements in rotating magnetic field[3]\nplaced an upper limit of 10%. Angle-resolved photoemis-\nsion spectroscopy[4] on monocrystalline YBCO gave the\nratio of 1.5 for gap amplitudes in the aandbdirections in\ntheCuO2. Measurementsof a Josephson current between\nmonocrystalline YBa 2Cu3O7ands-wave Nb showed that\nthe obtained anisotropy could be explained by a 83%\nd-wave with a 17%s-wave component.[5] In another ex-\nperiment on YBCO/Nb junction rings the stodgap ratio\ninoptimallydopedYBCO wasestimatedtobe 0.1.[6]\nInelasticneutronscatteringonmonocrystallineandun-\ntwinned samples of YBCO lead to magnetic susceptibili-\ntieswithintensitiesandlineshapesbreakingthetetragon al\nsymmetry.[7,8,9,10] It was shown that these data may be\ninterpretedwithinananisotropicbandmodelwithanorder\nparameterofmixed dandssymmetry.[11]\nSuperconductingstates with mixed symmetry are also\nconsidered in other classes of compounds, e.g. in the re-\ncentlydiscoveredferropnictides.[12]The effects of dilute concentrations of magnetic and\nnonmagnetic point defects on a BCS superconductor of\npuresord-wavesymmetrywereintensivelystudiedinthe\npast andarewell known.Ina d-wave superconductorwith\nlines of order parameter nodes any amount of disorder in-\nducesanonzerodensityofstatesattheFermilevel.Inan s-\nwavesystemonlymagneticimpuritieschangetheresponse\nof the superconducting state. For sufficiently strong cou-\nplingbetweentheimpurityandtheconductionbandbound\nstatesmayappearin theenergygap.[13]\nIn an earlier work on nonmagneticimpuritiesin a d+\ns-wave superconductor it was shown that in the unitary\nlimit a nonzero density of states (DOS) at the Fermi level\nappears above certain critical impurity concentration, de -\npendingonthesizeofan s-wavecomponent.[14]However\nthelow-energyDOSismostlyfeaturelesssincethe s-wave\ncomponentpreventsabuildupofstatesduetononmagnetic\nscattering. In contrast the presence of magnetic impuritie s\nin asuperconductorwithan s-wavecomponentmayresult\ninsharppeaksinthelow-energyDOSforsmallconcentra-\ntion of defects, provided the energy scale associated with\ntheimpurityresonanceissmall.\n2 Model and Results We consider the order param-\neterofd+ssymmetryona cylindricalFermisurface.\n∆s+eiθ∆d(ˆk), (1)\nCopyrightlinewillbe provided by the publisher2 L. S.Borkowski: Magnetic impuritytransition ina d+s-wave superconductor\nwhere∆sand∆dareamplitudesof s-andd-wavecompo-\nnentrespectively.We assume θ= 0and∆s≪∆d.\nThesuperconductoristreatedinaBCSapproximation.\nThe magnetic scatterer is modelled as an Anderson impu-\nritytreatedwithintheslavebosonmeanfieldapproach.[15]\nThe low energy physics is dominated by the presence of\nstronglyscatteringimpurityresonance.Theself-consist ent\nself-energyequationsdescribingthe interplaybetweenth e\nsuperconductingandmagneticdegreesoffreedomhavethe\nfollowingform,\n/tildewideω=ω+nN\n2πN0Γ¯ω\n(−¯ω2+ǫ2\nf), (2)\n/tildewide∆=∆s+nN\n2πN0Γ¯∆\n(−¯ω2+ǫ2\nf),(3)\n¯ω=ω+Γ/angbracketleft/tildewideω\n/parenleftBig\n/tildewide∆2(k)−/tildewideω2/parenrightBig1/2/angbracketright,(4)\n¯∆=Γ/angbracketleft/tildewide∆\n/parenleftBig\n/tildewide∆2(k)−/tildewideω2/parenrightBig1/2/angbracketright, (5)\nwhere/tildewideω(¯ω),/tildewide∆(¯∆)is the renormalized frequency and or-\nder parameter of conduction electrons (impurity) respec-\ntively.\nWe assume ∆d/D= 0.01,where2Disthebandwidth\nof the conductionelectron band. In the equationsabove Γ\nis the hybridization energy between the impurity and the\nconduction band and ǫfis the resonant level energy. Here\nwe assume constant density of states in the normal state\nN0= 1/2Dand do the calculations for a nondegenerate\nimpurity, N= 2. Brackets denote average over the Fermi\nsurface.\nInitial results for the density of states of this system\nwere presented in an earlier paper.[16] For small impurity\nconcentration n,such that ∆sand∆dare notsignificantly\naffected, there are two peaks located symmetrically near\nthe gap center, provided ǫf≪Γ≪∆s. For larger n\nthe two peaksmerge into one peak centrally located at the\nFermienergy.\nFig. 1 shows the dependence of the density of states\nN(0)atEFas a function of impurity concentration.\nAt small n,N(0)is exponentially small, N(0)/N0∼\n(∆s/∆d)exp(−α(∆s/∆d)2/n), whereαis a numerical\nfactor. The critical concentration n0for the discontinuous\ntransition to N(0)/N0≃∆s/∆dis a quadratic func-\ntion of∆s/∆d. Forn > n 0,N(0)approaches DOS of a\nd-wavesuperconductorinboththemagnitudeanditsfunc-\ntional dependence on n. These relations are valid when\nǫf≪Γ,∆s.\nQualitatively similar scaling of N(0)as a function of\n∆s/∆din presence of nonmagnetic impurities in the uni-\ntarylimitwasobtainedinref.[14].Thedifferencebetween\nmagnetic and nonmagnetic defetcs in a superconductor-8-6-4-2 0\n-6 -5 -4 -3 -2log10(N(0)/N0)\nlog10n∆s+∆d(k)εf=10-4\n∆s/∆d=0.001\n0.01\n0.020.050.1\nFigure 1 Logarithm of the density of states at the Fermi\nlevel as a function of impurity concentration for several\nvaluesoftheratio ∆s/∆d0.Theresonantimpuritylevel ǫf\nisclose totheFermilevel. Γ/Disfixedat 0.001.\n-3-2-1\n-6-5-4-3-2-1log10(ωpeak)\nlog10n∆s+∆d(k)∆s/∆d=0.01\nεf=010-42x10-45x10-410-3\nFigure 2 Position of the peak in the conduction electron\nDOS as a functionof impurityconcentrationfor d+ssu-\nperconductor for several values of the resonant level en-\nergy. Energy is scaled by half of the conduction electron\nbandwidth DandΓ/D= 0.001. The suppression of ∆s\nand∆dwasnottakenintoaccount.\nwith∆s/∆d≪1is the character of the transition from\nthe exponentially small N(0)to finiteN(0)in the strong\nscattering limit. In contrast to nonmagneticimpurities, t he\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3\ntransition caused by resonantly scattering magnetic impu-\nrities is discontinuous. and there are two sharp peaks of\nN(ω)on both sides of the transition.[16] The position of\npeaks as a function of impurity concentration for differ-\nent values of ǫfis shown in Fig 2. These peaks strongly\nalter the low-energyand low-frequencyresponse and may\nbe detected in thermodynamicor transport measurements.\nThe increase of ∆sshifts the resonances towards EFand\nmakesthemmorenarrow.\nThed-wave component of the superconducting order\nparameter is more sensitive to pair breaking by magnetic\nimpuritiesin the limit T≪TK, whereTK=/radicalBig\nΓ2+ǫ2\nf,\nthanthes-wavepart.Dependingontherelativesize of ∆d\nand∆sthere maybeanotherimpuritytransitionforlarger\nn, when the order parameter nodes disappear due to van-\nishing of the d-wave component and a full gap opens up.\nTheimpuritypeakisthensplitand N(0)fallstozeroagain.\nHower a detailed description of this possibility requires a\ncareful analysis involving four energy scales: ∆d,∆s,Γ,\nandǫf.\nCan such transition be observed experimentally? The\ntransition may be tuned either by varying impurity con-\ncentrationor the ratio ∆s/∆d. The density of states of the\nnormal state at EF,N0, also has an effect. It appears in\nequations2 and3. One shouldbear in mind,however,that\nboth the superconducting transition temperature and the\nimpurityresonanceenergyscale TKareexponentiallysen-\nsitivetochangesof N0.Experimentsconductedatverylow\ntemperatures may give different signatures of low-energy\nbehavior depending on the location of the system on the\nphasediagramrelativetotheimpuritycriticalpoint.\nInad+s-superconductorinthelimitofhighconcentra-\ntionofpointdefectsthe d-wavecomponentvanisheswhile\nthes-wavepartoftheorderparameterremainsunaffected.\nThe situation is different in presence of magnetic scatter-\ners.If the s-wave componentissmall andthe energyscale\nof the resonance due to impurity scattering is at most of\ntheorder ∆s, increasingimpurityconcentrationmaydrive\n∆s→0while∆dwill be reducedandfinite. Thisfollows\nfrom the fact that the largest pair breaking occurs when\nthe energy scales of the impurity resonance and the order\nparameter are comparable. If ∆d≫TK, the rate of sup-\npressionof ∆dwithincreasing nissmall.\nThed+ssuperconductorwith a subdominant s-wave\ncomponent doped with magnetic impurities has a differ-\nent phase diagram in the T-nplane compared to the same\nsuperconductor with nonmagnetic defects. If the impurity\nresonance scale TKis comparable to ∆sand∆s≪∆d,\nthes-wavecomponentmayvanishat T≃TK.\n3 Conclusions TheT= 0impurity transition dis-\ncussedinthisworkmaybedetectedinthelowtemperature\nlimit. While the slave boson mean field formalism used in\nthispapercannotbeappliedattemperaturesexceeding TK,\nwe may also qualitatively describe the expected behavior\nof the system as a function of temperature. At finite butsmallimpurityconcentrationtheremaybeevenfourphase\ntransitionsasafunctionoftemperature:normalto d-wave,\nd-wave to d+s-wave,d+stod-wave, and d-wave to\nd+s-wave.Dueto thedifferenceofmagnitudesthesmall\n∆smay be driven to zero faster with increasing nthan\n∆d. Nonmonotonic behavior around T∼TKis a conse-\nquenceofstrongscatteringby theresonancestate forming\non the impurity site. As T→0, the impurity scattering\nbecomes weaker and the s-wave component may appear\nagain.Whetherthisparticularscenarioisrealized,depen ds\non the relative size of energyscales: ∆s/∆d,∆s/TK, and\nǫf/Γ.\nAcknowledgements Some of the computations were per-\nformed in the Computer Center of the Tri-city Academic Com-\nputer Network inGdansk.\nReferences\n[1] D. N. Basov, R. Liang, D. A. Bonn, W. A. Hardy,\nB. Dabrowski, M. Quijada, D. B. Tanner, J. P. Rice,\nD. M. Ginsberg, and T. Timusk, Phys. Rev. Lett. 74, 598\n(1995).\n[2] M. F. Limonov, A. I. Rykov, S. Tajima and A. Yamanaka,\nPhys.Rev. Lett. 80, 825 (1998)\n[3] H.Aubin, K. Behnia, M. Ribault, R.Gagnon, andL. Taille-\nfer,Phys.Rev. Lett. 782624.\n[4] D.H.Lu,D.L.Feng, N.P.Armitage,K.M.Shen, A.Dam-\nascelli, C. Kim, F. Ronning, Z.-X. Shen, D. A. Bonn,\nR. Liang, W. N. Hardy, A. I. Rykov, and S. Tajima, Phys.\nRev. Lett. 86, 4370 (2001).\n[5] H. J. H. Smilde, A. A. Golubov, Ariando, G. Rijnders,\nJ. M. Dekkers, S. Harkema, D. H. A. Blank, H. Rogalla,\nandH. Hilgenkamp, Phys. Rev. Lett. 95, 157001 (2005).\n[6] J. R. Kirtley, C. C. Tsuei, Ariando, C. J. M. Verwijs,\nS.Harkema, and H.Hilgenkamp, Nat.Phys. 2, 198 (2006).\n[7] H. A. Mook, P. C. Dai, F. Dogan, and R. D. Hunt, Nature\n404, 729 (2000).\n[8] C. Stock, W. J. L. Buyers, R. Liang, D. Peets, Z. Tun,\nD. Bonn, W. N. Hardy, and R. J. Birgeneau, Phys. Rev. B\n69, 014502 (2004).\n[9] C. Stock, W. J. L. Buyers, R. A. Cowley, P. S. Clegg,\nR. Coldea, C. D. Frost, R. Liang, D. Peets, D. Bonn,\nW. N. Hardy, and R. J. Birgeneau, Phys. Rev. B 71, 024522\n(2005).\n[10] V.Hinkov,S.Pailhes,P.Bourges,Y.Sidis,A.Ivanov,A .Ku-\nlakov, C. T. Lin, D. P. Chen, C. Bernhard, and B. Keimer,\nNature430, 650 (2004).\n[11] A.P.Schnyder, D.Manske, C.Mudry, andM.Sigrist,Phys .\nRev. B73, 224523 (2006).\n[12] V. Mishra, G. Boyd, S. Graser, T. Maier, P. J. Hirschfeld ,\nand D. J. Scalapino, Phys. Rev. B 79, 094512 (2009), and\nreferences therein.\n[13] L. S. Borkowski and P. J. Hirschfeld, J. Low Temp. Phys.\n96, 185 (1994), and references therein.\n[14] H.Kim,and E.J.Nicol, Phys.Rev. B 5213576 (1995)\n[15] L.S.Borkowski, Phys. Rev. B 78020507 (2008).\n[16] L.S.Borkowski, J.Phys.:Conf.Series 150,052023 (2009).\nCopyrightlinewillbe provided by the publisher"    },    {        "title": "1003.2452v1.Highlights_from_Fermi_GRB_observations.pdf",        "content": "arXiv:1003.2452v1  [astro-ph.HE]  12 Mar 2010Highlights from Fermi GRB observations\nJonathan Granot\non behalf of the Fermi LAT and GBM collaborations\nCentre for Astrophysics Research, University of Hertfords hire,\nCollege Lane, Hatfield AL10 9AB, UK\nAbstract\nThe Fermi Gamma-Ray Space Telescope has more than doubled\nthe number of Gamma-Ray Bursts (GRBs) detected above 100 MeV\nwithin its first year of operation. Thanks to the very wide energy\nrange covered by Fermi’s Gamma-ray Burst Monitor (GBM; 8 keV to\n40 MeV) and Large Area Telescope (LAT; 25 MeV to >300 GeV)\nit has measured the prompt GRB emission spectrum over an unprece -\ndentedlylargeenergyrange(from ∼8 keVto ∼30 GeV). HereIbriefly\noutline some highlights from Fermi GRB observations during its first\n∼1.5 yr ofoperation, focusing onthe promptemission phase. Interes t-\ning new observations are discussed along with some of their possible\nimplications, including: (i) What can we learn from the Fermi-LAT\nGRB detection rate, (ii) A limit on the variation of the speed of light\nwith photon energy(for the firsttime beyond the Planckscale fora lin-\near energy dependence from direct time of arrival measurements ), (iii)\nLower-limits on the bulk Lorentz factor of the GRB outflow (of ∼1000\nfor the brightest Fermi LAT GRBs), (iv) The detection (or in other\ncases, lackthereof) ofa distinct spectralcomponent athigh (an d some-\ntimes also at low) energies, and possible implications for the prompt\nGRB emission mechanism, (v) The later onset (and longer duration)\nof the high-energy emission ( >100 MeV), compared to the low-energy\n(<∼1 MeV) emission, that is seen in most Fermi-LAT GRBs.\n1 Pre-Fermi high-energy GRB observations\nThe Energetic Gamma-Ray Experiment Telescope (EGRET) on-b oard the\nCompton Gamma-Ray Observatory (CGRO; 1991 −2000) was the first to\ndetect high-energy emission from GRBs. EGRET detected only five GRBs\nwith its Spark Chambers (20MeV to 30GeV) and a few GRBs with it s\nTotal Absorption Shower Counter (TASC; 1 −200 MeV). Nevertheless,\n1these events already showed diversity. The most prominent e xamples are\nGRB 940217, with high-energy emission lasting up to ∼1.5 hr after the\nGRB including an 18 GeV photon after ∼1.3 hr, [1] and GRB 941017 which\nhad adistinct high-energy spectral component [2] detected upto∼200 MeV\nwithνFν∝ν. This high-energy spectral component had ∼3 times more\nenergy and lasted longer ( ∼200 s) than the low-energy (hard X-ray to soft\ngamma-ray) spectral component (which lasted several tens o f seconds), and\nmay be naturally explained as inverse-Compton emission fro m the forward-\nreverse shock system that is formed as the ultra-relativist ic GRB outflow\nis decelerated by the external medium [3, 4]. Nevertheless, better data are\nneeded in order to determine the origin of such high-energy s pectral compo-\nnents more conclusively. The Italian experiment Astro-riv elatore Gamma a\nImmagini LEggero (AGILE; launched in 2007) has detected GRB 080514B\nat energies up to ∼300 MeV, and the high-energy emission lasted longer\n(>13 s) than the low-energy emission ( ∼7 s) [5]. Below are some highlights\nof Fermi GRB observations so far and what they have taught us.\n2 LAT GRB detection rate: what can it teach us?\nDuring its first 1 .5 yr of routine operation, from Aug. 2008 to Jan. 2010, the\nLAT has detected 14 GRBs, corresponding to a detection rate o f∼9.3 yr−1.\nTable 1 summarizes their main properties. While at least 13 o f the 14 LAT\nGRBs had ≥10 photons above 100 MeV, 4 were particularly bright in the\nLAT with ≥1 photon above 10 GeV, ≥10 photons above 1 GeV, and\n≥100 photons above 100 MeV. This corresponds to a bright LAT GR B\n(as defined above) detection rate of ∼2.7 GRB/yr (with a rather large\nuncertainty due to the small number statistics). There were also 11 GRBs\nwith≥1 photon above 1 GeV, corresponding to ∼7.3 GRB/yr. These\ndetection rates are compatible with pre-launch expectatio ns [6] based on a\nsample of bright BATSE GRBs for which the fit to a Band spectrum over\nthe BATSE energy range (20 keV to 2 MeV) was extrapolated into the LAT\nenergy range (see Fig. 1). The agreement is slightly better w hen excluding\ncases with a rising νFνspectrum at high energies (i.e. a high-energy photon\nindexβ >−2).1This suggests that, on average, there is no significant\nexcess or deficit of high-energy emission in the LAT energy ra nge relative to\nsuch an extrapolation from lower energies. As described in §5, however, in\nindividual cases we do have evidence for such an excess. The o bserved LAT\n1Such a hard high-energy photon index may be an artifact of the limited energy range\nof the fit to BATSE data, or may be affected by poor photon statis tics at>∼1 MeV.Figure1: LAT GRBdetection rates(colorellipses)superposedonto pofpre-launch\nexpected rates based on the extrapolation of a Band spectrum fit from the BATSE\nenergy range [6]. The ellipses’ inner color indicates the minimal photon energy\n(green, yellow and cyan correspond to 0.1, 1 and 10 GeV, respectiv ely), while their\nhight indicates the uncertainty ( ±N1/2/1.5yr)on the correspondingLAT detection\nrate (N/1.5yr) due to the small number ( N) of detected GRBs.\nGRB detection rate implies that, on average, only about ∼10−20% of the\nenergy that is radiated during the prompt GRB emission phase is channeled\ninto the LAT energy range, suggesting that in most GRBs the hi gh-energy\nradiative output does not significantly affect the total energ y budget. Short\nGRBs, however, appear to be different in this respect (see §7 and Fig. 3).\n3 Limits on Lorentz Invariance Violation\nSome quantum gravity models allow violation of Lorentz inva riance, and in\nparticular allow the photon propagation speed, vph, to depend on its energy,\nEph:vph(Eph)∝ne}ationslash=c, where c≡lim\nEph→0vph(Eph). The Lorentz invariance\nviolating (LIV) part in the dependence of the photon momentu m,pph, onlong number of HE emission extra highest\nGRB θLAT or events above starts lasts spec. energy z\nshort 0.1GeV 1GeV later longer comp. (GeV)\n080825C ∼60◦long ∼10 0 ? yes no ∼0.6 —–\n080916C 49◦long 145 14 yes yes ?∼13∼4.35\n081024B 21◦short ∼10 2 yes yes ? ∼3—–\n081215A ∼86◦long —– — ? ? — —– —–\n090217 ∼34◦long ∼10 0 no no no ∼1—–\n090323 ∼55◦long ∼20 >0? yes ? ? 3.57\n090328 ∼64◦long ∼20 >0? yes ? ?0.736\n090510 ∼14◦short >150 >20yes yes yes ∼310.903\n090626 ∼15◦long ∼20 >0? yes ? ? —–\n090902B 51◦long >200 >30yes yes yes ∼331.822\n090926 ∼52◦long >150 >50yes yes yes ∼202.1062\n091003A ∼13◦long ∼20 >0? ? ? ?0.8969\n091031 ∼22◦long ∼20 >0? ? ?∼1.2 —–\n100116A ∼29◦long ∼10 3 ? ? ?∼2.2 —–\nTable 1: Summary of the 14 GRBs detected by the LAT between August 2008 and\nJanuary 2010 – its first 1.5 years of routine operation follow ing Fermi’s launch on 11 June\n2008;θLATis the angle from the LAT boresight at the time of the GBM GRB tr igger.\nits energy, Eph, can be expressed as a power series,\np2\nphc2\nE2\nph−1 =∞/summationdisplay\nk=1sk/parenleftBigg\nEph\nMQG,kc2/parenrightBiggk\n, (1)\nintheratioof Ephandatypicalenergyscale MQG,kc2forthekthorder,which\nis expected to be up to the order of the Planck scale, MPlanck= (¯hc/G)1/2≈\n1.22×1019GeV/c2, wheresk∈ {−1,0,1}. Since we observe photons of\nenergy well below the Planck scale, the dominant LIV term is a ssociated\nwith the lowest order non-zero term in the sum, of order n= min{k|sk∝ne}ationslash= 0},\nwhich is usually assumed to be either linear ( n= 1) or quadratic ( n= 2).\nThe photon propagation speed is given by the corresponding g roup velocity,\nvph=∂Eph\n∂pph≈c/bracketleftBigg\n1−snn+1\n2/parenleftBigg\nEph\nMQG,nc2/parenrightBiggn/bracketrightBigg\n. (2)\nNote that sn= 1 correspondsto thesub-luminal case ( vph< cand apositive\ntime delay), while sn=−1 corresponds to the super-luminal case ( vph> c\nand a negative time delay). Taking into account cosmologica l effects [9], this\ninduces a time delay (or lag) in the arrival of a high-energy p hoton of energy\nEh, compared to a low-energy photon of energy El(emitted simultaneously\nat the same location), of\n∆t=sn(1+n)\n2H0(En\nh−En\nl)\n(MQG,nc2)n/integraldisplayz\n0(1+z′)n\n/radicalbig\nΩm(1+z′)3+ΩΛdz′.(3)Here we concentrate on our results for a linear energy depend ence (n= 1).\nWe have applied this formula to the highest energy photon det ected in\nGRB 080916C, with an energy of Eh= 13.22+0.70\n−1.54GeV, which arrived at t=\n16.54 s after the GRB trigger (i.e. the onset of the El∼0.1 MeV emission).\nSinceitis hardtoassociate thehighestenergy photonwitha particular spike\ninthelow-energy lightcurve, wehave madetheconservative assumptionthat\nit was emitted anytime after the GRB trigger, i.e. ∆ t≤t, in order to obtain\na limit for the sub-luminal case ( sn= 1):MQG,1>0.1MPlanck. This was\nthe strictest limit of its kind [7], at that time.\nHowever, the next very bright LAT GRB, 090510, was short and h ad\nvery narrow sharp spikes in its light curve (see Fig 2), thus e nabling us to\ndo even better [8]. Our main results for GRB 090510 are summar ized in\nTable 2. The first 4 limits are based on a similar method as desc ribed above\nfor GRB 080916C, using the highest energy photon, Eh= 30.53+5.79\n−2.56GeV,\nand assuming that its emission time thwas after the start of a relevant lower\nenergy emission episode: th> tstart. These 4 limits correspond to different\nchoices of tstart, which are shown by the vertical lines in Fig. 2. We con-\nservatively used the low end of the 1 σconfidence interval for the highest\nenergy photon ( Eh= 28 GeV) and for the redshift ( z= 0.900). The most\nconservative assumption of this type is associating tstartwith the onset of\nany detectable emission from GRB 090510, namely the start of the small\nprecursor that GBM triggered on, leading to ξ1=MQG,1/MPlanck>1.19.\nHowever, it is highly unlikely that the 31 GeV photon is indee d associated\nwith the small precursor. It is much more likely associated w ith the main\nsoft gamma-ray emission, leading to ξ1>3.42. Moreover, for any reasonable\nemission spectrum, the emission of the 31 GeV photon would be accompa-\nnied by the emission of a large number of lower energy photons , which\nwould suffer a much smaller time delay due to LIV effects, and woul d there-\nfore mark its emission time. We could easily detect such phot ons in energies\nabove 100 MeV, and therefore the fact that significant high-e nergy emission\nis observed only at later times (see Fig. 2) strongly argues t hat the 31 GeV\nphoton was not emitted before the onset of the observed high- energy emis-\nsion. One could choose either the onset time of the emission a bove 100 MeV\norabove1 GeV, whichcorrespondto ξ1>5.12, andξ1>10.0, respectively.2\n2We note that there is no evidence for LIV induced energy dispe rsion that might be\nexpected if indeed the 31 GeV photon was emitted near our choi ces fortstart, together\nwith lower energy photons, as can be expected for any reasona ble emission spectrum. This\nis evident from the lack of accumulation of photons along the solidcurves in panel (a) of\nFig. 2, at least for the first 3 tstartvalues, and provides support for these choices of tstart\n(i.e. that they can indeed serve as upper limits on a LIV induc ed energy dispersion).tstartlimit on Reason for choice of Elvalid lower limit on\n(ms)|∆t|(ms) tstartor limit on ∆ t (MeV) forsnMQG,1/MPlanck\n−30<859 start of any observed emission 0.1 1 >1.19\n530 <299 start of main <1MeV emission 0.1 1 >3.42\n630 <199 start of>100 MeV emission 100 1 >5.12\n730 <99 start of >1 GeV emission 1000 1 >10.0\n— <10 association with <1MeV spike 0.1 ±1 >102\n— <19 if 0.75GeVγis from 1stspike 0.1 ±1 >1.33\n|∆t\n∆E|<30ms\nGeVlag analysis of all LAT events — ±1 >1.22\nTable2: Lower-limits ontheQuantumGravity(QG)mass scale associa ted withapossible\nlinear (n= 1) variation ofthespeedoflight with photonenergy, thatw e canplace from the\nlack of time delay (of sign sn) in the arrival of high-energy photons relative to low-ener gy\nphotons, from our observations of GRB 090510 (from [8]).\nThe5thand 6thlimits inTable2aremorespeculative, as they relyon the\nassociation ofanindividualhigh-energyphotonwithapart icularspikeinthe\nlow-energy light curve, on top of which it arrives. While the se associations\nare not very secure (the chance probability is roughly ∼5−10%), they are\nstill most likely, making the corresponding limits interes ting, while keeping\nthis big caveat in mind. The allowed emission time of these tw o high-energy\nphotons, if these associations are real, is shown by the two t hin vertical\nshadedregions inFig. 2. Forthe31 GeV photonthisgives alim itofξ1>102\nfor either sign of sn.\nThe last limit in Table 2 is based on a different method, which is com-\nplementary and constrains both signs of sn. It relies on the highly variable\nhigh-energy light curve, with sharp narrow spikes, which wo uld be smeared\nout if there was too much energy dispersion (of either sign). We have used\nthe DisCan method [10] to search for linear energy dispersio n within the\nLAT data (the actual energy range of the photons used was 35 Me V to\n31 GeV)3during the most intense emission interval (0.5–1.45s). Thi s ap-\nproach extracts dispersion information from all detected L AT photons and\ndoes not involve binning in time or energy. Using this method we obtained\na robust lower limit of ξ1>1.22 (at the 99% confidence level).\nOur most conservative limits (the first and last limits in Tab le 2) rely on\nvery different and largely independent analysis, yet still gi ve a very similar\nlimit:MQG,1>1.2MPlanck. This lends considerable support to this result,\nandmakes itmorerobustandsecurethanforeachofthemethod sseparately.\n3We obtain similar results even if we use only photons below 3 G eV or 1 GeV.0 0.5 1 1.5 2Counts/bin \n050 100 150 \nCounts/s \n05000 10000 15000 -0.03 \nGBM NaIs \n0 0.5 1 1.5 2Counts/bin \n050 100 150 200 \nCounts/s \n05000 10000 15000 20000 GBM BGOs Counts/bin \n020 40 \nCounts/s \n02000 4000 LAT \n(All events) Counts/bin \n024\nCounts/s \n0200 400 LAT \n(> 100 MeV) \nTime since GBM trigger (May 10, 2009, 00:22:59.97 UT) (s) -0.5 0 0.5 1 1.5 2Counts/bin \n0123\nEnergy [GeV] 2\n1510 20 LAT \n(> 1 GeV) 0.63 0.73 0.73 Energy (MeV) \n10 210 310 410 \n0\n0(0.26–5 MeV) (8–260 keV) (a) \n(b) \n(c) \n(d) \n(e) \n(f) 0.53 \nFigure 2: Light curves of GRB 090510 at different energies (for det ails see [8]).4 Lower limits on the bulk Lorentz factor\nThe GRB prompt emission typically has very large isotropic e quivalent lu-\nminosities ( L∼1050−1053erg s−1), significant short time scale variability,\nand typical photon energies>∼mec2(in the source cosmological frame).\nThis would result in a huge optical depth to pair production ( γγ→e+e−)\nat the source, which would thermalize the spectrum and thus b e at odds\nwith the observed non-thermal spectrum, unless the emittin g material was\nmoving toward us relativistically, with a bulk Lorentz fact or Γ≫1. This\n“compactness” argument has been used to derive lower-limit s, Γmin, on the\nvalue of Γ, which were typically ∼102and in some cases as high as a few\nhundred (see [11] and references therein). However, the pho tons that pro-\nvided the opacity for these limits were well above the observ ed energy range,\nso there was no direct evidence that they actually existed in the first place.\nWith Fermi, however, we adopt a more conservative approach o f relying\nonly on photons within the observed energy range. Under this approach,\nΓmin<∼(1+z)Eph,max\nmec2≈200(1+z)/parenleftbiggEph,max\n100MeV/parenrightbigg\n, (4)\nwhereEph,maxis the highest observed photon energy, so that setting a larg e\nΓminrequires observing sufficiently high-energy photons.\nThe main uncertainty in deriving Γ minis usually the exact choice for\nthe variability time, tv. Other uncertainties arise from those on the spectral\nfit parameters, or on the degree of space-time overlap betwee n the high-\nenergy photon and lower energy target photons, in cases wher e there is\nmore than one spectral component without conclusive tempor al correlation\nbetween their respective light curves. Finally, the fact th at our limits rely\non a single high-energy photon also induces an uncertainty, as it might still\nescape from an optical depth of up to a few. However, in most ca ses the\nsecond or third highest-energy photons help to relax the affec t this has on\nΓmin(as the probability that multiple photons escape from τγγ>1 rapidly\ndecreases with the number of photons). Thus, we have derived reasonably\nconservative Γ minvalues for 3 of the brightest LAT GRBs: Γ min≈900 for\nGRB 080916C [7], Γ min≈1200 for GRB 090510 [12], and Γ min≈1000\nfor GRB 090902B [13]. This shows that short GRBs (such as 0905 10) are\nas highly relativistic as long GRBs (such as 080916C or 09090 2B), which\nwas questioned before the launch of Fermi [14]. Since our hig hest values of\nΓminare derived for the brightest LAT GRBs, they are susceptible to strong\nselection effects. It might be that GRBs with higher Γ tend to be brighter\nin the LAT energy range (e.g. by avoiding intrinsic pair prod uction [15]).5 Delayed onset and a distinct high-energy spec-\ntral component\nA delayed onset of thehigh-energy emission ( >100 MeV) relative tothelow-\nenergy emission (<∼1 MeV) appears to be a very common feature in LAT\nGRBs. It clearly appears in all 4 of the particularly bright L AT GRBs,\nwhile in dimmer LAT GRBs it is often inconclusive due to poor p hoton\nstatistics near the onset time. The time delay, tdelay, appears to scale with\nthe duration of the GRB ( tdelay∼several seconds in thelong GRBs 080916C\nand 090902B, while tdelay∼0.1−0.2 s in the short GRBs 090510 and\n081024B, though with a smaller significance for the latter).\nOnly 3 LAT GRBs so far have shown clear ( >5σ) evidence for a distinct\nspectral component. However, these GRBs are the 3 brightest in the LAT,\nwhile the next brightest GRB in the LAT (080916C) showed a hin t for\nan excess at high energies. This suggests that such a distinc t high-energy\nspectral component is probably very common, but we can clear ly detect it\nwith high significance only in particularly bright LAT GRBs, since a large\nnumberofLATphotonsisneededinordertodetectitwith >5σsignificance.\nThe distinct spectral component is usually well fit by a hard p ower-\nlaw that dominates at high energies. In GRB 090902B a single p ower-law\ncomponent dominates over the usual Band component both at hi gh energies\n(above∼100 MeV) and low energies (below ∼50 KeV; see lower panel\nof Fig. 3). There is also marginal evidence that the high-ene rgy power-law\ncomponent in GRB 090510, which dominates above ∼100 MeV, might also\nappear at the lowest energies (below a few tens of keV).\nBoth the delayed onset and distinct spectral component rela te to and\nmay help elucidate the uncertain prompt GRB emission mechan ism. The\nmain two competing classes of models are leptonic and hadron ic origin.\nLeptonic: the high-energy spectral component might be inverse-Compt on\nemission, and in particular synchrotron-self Compton (SSC ) if the usual\nBand component is synchrotron. In this case, however, it may be hard to\nproduce the observed tdelay> tv, wheretvis the width of individual spikes\nin the lightcurve ( tdelay<∼tvmight occur due to the build-up of the seed\nsynchrotron photon field in the emitting region over the dyna mical time).\nMoreover, the gradual increase in the photon index βof the distinct high-\nenergy power-law spectral component is not naturally expec ted in such a\nmodel, and the fact that it is different than the Band low-energ y photon\nindex as well as the excess flux (above the Band component) at l ow energies\nare hard to account for in this type of model.Hadronic: tdelaymight be identified with the acceleration time, tacc, of pro-\ntons(or heavier ions)toveryhighenergies(at whichtheylo osemuchoftheir\nenergy on a dynamical time, e.g. via proton synchrotron [16] , in order to\nhave a reasonable radiative efficiency). If the observed high -energy emission\n(and in particular thedistinct high-energy spectral compo nent) also involves\npaircascades (e.g. inverse-Compton emissionbysecondary e±pairs[17]pro-\nduced in cascades initiated by photo-hadronic interaction s) then it might\ntake some additional time for such cascades to develop. Such an origin for\ntdelay(∼tacc), however, requires the high-energy emission to originate from\nthe same physical region over times > tdelay, and implies high-energy emis-\nsion rise and variability times tv>∼tacc∼tdelay, due to the stochastic nature\nof the acceleration process (while tv< tdelayis usually observed). The grad-\nual increase in βis not naturally expected in hadronic models, though it\nmight be mimicked by a time-evolution of a high-energy Band- like spectral\ncomponent [16]. For GRB 090510 a hadronicmodel requires ato tal isotropic\nequivalent energy >102times larger than that observed in gamma-rays [17],\nwhich may pose a serious challenge for the progenitor of this short GRB.\nThe excess flux at low energies that is observed in GRB 090902B (and the\nhint for such an excess in GRB 090510) may be naturally explai n in this\ntype of model by synchrotron emission from secondary pairs [ 17, 13].\nAltogether, hadronicmodelsseemtofaresomewhatbetter, h owever both\nleptonic and hadronic models still face many challenges, an d do not yet\nnaturally account for all of the Fermi observations.\n6 Long-lived high-energy emission\nIn most LAT GRBs thehigh-energy ( >100 MeV) emission lasts significantly\nlonger than the low-energy (<∼1 MeV) emission. While the high-energy\nemissionusuallyshowssignificantvariabilityduringthep rompt(low-energy)\nemission phase, in some cases showing temporal correlation with the low-\nenergy emission, the longer lived emission is typically tem porally smooth\nand consistent with a power-law flux decay (of ∼t−1.2−t−1.5) with a LAT\nphoton index corresponding to a roughly flat νFν.\nIt is most natural to interpret the prompt high-energy emiss ion as the\nhigh-energy counterpart of the prompt soft gamma-ray emiss ion, from the\nsame emission region, especially when there is temporal cor relation between\nthe low and high-energy light curves, and sometimes even fro m the same\nspectral component (as appears to be the case for GRB 080916C ). However,\nwhen there is no such temporal correlation, an origin from a d ifferent emis-sion region is possible. The longer lived smooth power-law d ecay phase is\nmore naturally attributed to the high-energy afterglow, fr om the forward\nshock that is driven into the external medium. An afterglow o rigin has been\nsuggested in some cases for the whole LAT emission [20, 15], i ncluding dur-\ning the prompt soft gamma-ray emission stage, however in thi s scenario it is\ngenerally hard to explain the sharp spikes in the LAT lightcu rve during the\nprompt phase. It is easier to test the origin of this long live d high-energy\nemission when there is good multi-wavelength coverage of th e early after-\nglow emission (e.g., in X-ray and/or optical), such as for GR B 090510 [18].\nProducing particularly high-energy photons is challengin g for a synchrotron\norigin, both during the prompt emission [7], and even more so during the\nafterglow (e.g. [21]), as it requires a very high bulk Lorent z factor and up-\nstream magnetic field, in addition to a very efficient shock acc eleration (e.g.\na 33 GeV photon observed in GRB 090902B after 82 s, well after t he end of\nthe prompt emission [13], requires Γ >1500).\n7 High-energy emission of long versus short GRBs\nSo far, 2 (12) out of the 14 LAT GRBs are of the short (long) dura tion\nclass. This implies that ∼14% of LAT GRBs are short, with a large un-\ncertainty due to the small number statistics, which is consi stent with the\n∼20% short GRBs detected by the GBM. As can be seen from Table 1, the\nhigh-energy emission properties of short and long GRBs appe ar to be rather\nsimilar. They can both produce very bright emission in the LA T energy\nrange (090510 vs. 080916C, 090902B and 090926), with a corre spondingly\nhigh lower-limit on the bulk Lorentz factor (Γ min∼103), as well a distinct\nspectral component (090510 vs. 090902B and 090926). Both sh ow a delayed\nonsetandlonerlivedhigh-energyemission, comparedtothe low-energyemis-\nsion. However, the delay in the onset of the high-energy emis sion appears\nto roughly scale with the duration of the GRB, being ∼0.1−0.2 s for\nshort GRBs and several seconds for long GRBs. This is especia lly intrigu-\ning when comparing GRBs 080916C and 090510, which had a compa rable\nisotropic equivalent luminosity (of several 1053erg s−1), suggesting another\nunderlying cause for the difference in the time delay (e.g. [22 ]).\nAnotherinterestingpotential difference, whichstillneeds tobeconfirmed\n(as there are only 2 short LAT GRBs so far, and possible select ion effects),\nis that short GRBs appear to have a comparable energy output a t high and\nlow photon energies, while long GRBs tend to radiate a smalle r fraction of\ntheir energy output at high photon energies (see upper panel of Fig. 3).Figure 3: Top panel : the fluence at high (0.1–10 GeV) versus low (20keV –\n2MeV) energies (from [19]), for 4 long (080825C, 080916C, 090217 , 090902B) and\n2 short (081024B,090510)duration LAT GRBs. The diagonallines in dicate high to\nlow energy fluence ratios of 1%, 10%, and 100%. Bottom panel : the best fit time-\nintegrated νFνspectra for the same GRBs, two of which (090510, 090902B) show a\ndistinct spectral component, well described by a hard power-law, in addition to the\nusual Band spectral component. The colored shaded regions indic ate the energy\nranges used for calculating the fluences that are displayed in the top panel .JG gratefully acknowledges a Royal Society Wolfson Researc h Merit Award.\nReferences\n[1] Hurley, K., et al. 1994, Nature, 372, 652\n[2] Gon´ zalez, M. M., et al. 2003, Nature, 424, 749\n[3] Granot, J., & Guetta, D. 2003, ApJ, 598, L11\n[4] Pe’er, A. A., & Waxman, E. 2004, ApJ, 613, 448\n[5] Giuliani, A., et al. 2008, A&A, 491, L25\n[6] Band, D. L. , et al. 2009, ApJ, 701, 1673\n[7] Abdo, A. A., et al., 2009, Science, 323, 1688\n[8] Abdo, A. A., et al., 2009, Nature, 462, 331\n[9] Jacob, U., & Piran, T. 2008, JCAP, 01, 031\n[10] Scargle, J. D., Norris, J. P., & Bonnell, J. T. 2008, ApJ, 673, 972\n[11] Lithwick, Y., & Sari, R. 2001, ApJ, 555, 540\n[12] Abdo, A. A., et al., 2010, submitted to ApJ\n[13] Abdo, A. A., et al., 2009, 706, L138\n[14] Nakar, E. 2007, Phys. Rep., 442, 166\n[15] Ghisellini, G., Ghirlanda, G., Nava, L., & Celotti, A. 2 010, MNRAS in\npress (doi:10.1111/j.1365-2966.2009.16171.x; arXiv:09 10.2459)\n[16] Razzaque, S., Dermer, C. D., & Finke, J. D. 2009, preprin t\n(arXiv0908.0513)\n[17] Asano, K., Guiriec, S., & M´ esz´ aros, P. 2009, ApJ, 705, L191\n[18] De Pasquale, M., et al. 2010, ApJ, 709, L146\n[19] Abdo, A. A., et al. 2010, ApJ in press (arXiv:1002.3205)\n[20] Kumar, P., & Barniol Duran, R. 2009, MNRAS, 400, L75\n[21] Li, Z., & Waxman, E. 2006, ApJ, 651, 328\n[22] Toma, K., Wu, X. F., & Meszaros, P., 2010, preprint (arXi v:1002.2634)"    },    {        "title": "1912.01332v2.Electronic_and_Magnetic_Structure_of_Infinite_layer___textrm_NdNiO__2___Trace_of_Antiferromagnetic_Metal.pdf",        "content": "arXiv:1912.01332v2  [cond-mat.str-el]  4 Dec 2019Electronic and Magnetic Structure of Infinite-layer NdNiO 2:\nTrace of Antiferromagnetic Metal\nZhao Liu,1Zhi Ren,2W. Zhu,2Z. F. Wang,1and Jinlong Yang3,∗\n1Hefei National Laboratory for Physical Sciences at the Micr oscale,\nCAS Key Laboratory of Strongly-Coupled Quantum Matter Phys ics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n2Institute of Natural Sciences, Westlake Institution of Adv anced Study\nand School of Science, Westlake University, Hangzhou 30002 4, China\n3Hefei National Laboratory for Physical Sciences at the Micr oscale,\nSynergetic Innovation Center of Quantum Information and Qu antum Physics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\nThe recent discovery of Sr-doped infinite-layer nickelate NdNiO 2[D. Li et al. Nature 572, 624\n(2019)] offers an exciting platform for investigating uncon ventional superconductivity in nickelate-\nbased compounds. In this work, we present a first-principles calculations for the electronic and\nmagnetic properties of undoped parent NdNiO 2. Intriguingly, we found that: 1) the paramagnetic\nphase has complex Fermi pockets with 3D characters near the F ermi level; 2) by including electron-\nelectron interactions, 3d-electrons of Ni tend to form (π,π,π)antiferromagnetic ordering at low\ntemperatures; 3) with moderate interaction strength, 5d-electrons of Nd contribute small Fermi\npockets that could weaken the magnetic order akin to the self -doping effect. Our results provide\na plausible interpretation for the experimentally observe d resistivity minimum and Hall coefficient\ndrop. Moreover, we elucidate that antiferromagnetic order ing inNdNiO 2is relatively weak, arising\nfrom the small exchange coupling between 3d-electrons of Ni and also hybridization with 5d-electrons\nof Nd.\nSince the discovery of high-temperature (high- Tc)\nsuperconductivity in cuprates1, extensive effort has\nbeen devoted to investigate unconventional supercon-\nductors, ranging from non-oxide compounds2,3to iron-\nbased materials4,5. Exploring high- Tcmaterials could\nprovide a new platform to understand the fundamental\nphysics behind high- Tcphenomenon, thus is quite valu-\nable. Very recently, the discovery of superconductivity\nin Sr-doped NdNiO 26potentially raises the possibility\nto realize high- Tcin nickelate family7,8.\nOne key experimental observation for the infinite-\nlayerNdNiO 2is that its resistivity exhibits a minimum\naround 70 K and an upturn at a lower temperature6.\nAt the same time that the resistivity reaches minimum,\nthe Hall coefficient drops towards a large value, sig-\nnalling the loss of charge carriers6. Interestingly, no\nlong-range magnetic order has been observed in pow-\nder neutron diffraction on NdNiO 2when temperature\nis down to 1.7 K6. This greatly challenges the exist-\ning theories, since it is generally believed that mag-\nnetism holds the key to understand unconventional\nsuperconductivity9–12. Therefore, it is highly desirable\nto study the magnetic properties of undoped parent\nNdNiO 2and elucidate its experimental indications.\nIn this work, the electronic and magnetic properties\nofNdNiO 2are systemically studied by first-principles\ncalculations combined with classical Monte Carlo cal-\nculations. Firstly, the paramagnetic (PM) phase is\nstudied. Its Fermi surface includes one large sheet and\ntwo electron pockets at Γand A point, respectively.\nThis can be described by a three-band low-energy effec-\ntive model that captures the main physics of exchange\ncoupling mechanism. Then, the magnetic properties\nare studied by including Hubbard U and (π,π,π)anti-\nferromagnetic (AFM) ordering is confirmed to be the\nmagnetic ground state. Most significantly, the Fermi\nsurface of AFM phase is simpler than that of PMphase, demonstrating an interaction induced elimina-\ntion of Fermi pockets. Before NdNiO 2enters corre-\nlated insulator, it is a compensated metal with one\nsmall electron pocket formed by dxyorbital of Nd and\nfour small hole pockets formed by dz2orbital of Ni.\nThe estimated phase transition temperature ( TN) from\nPM phase to (π,π,π)AFM phase is 70 ∼90 K for\nmoderate interaction strength of U = 5 ∼6 eV.\nThrough these studies, we identify two key mes-\nsages that are distinguishable from the cuprates: 1)\nNdNiO 2is dominated by the physics of Mott-Hubbard\ninstead of charge-transfer; 2) effective exchange cou-\npling parameters are about one-order smaller than\nthose of cuprates. In this regarding, supposed that\nthe ground state is magnetic, our calculations demon-\nstrate(π,π,π)AFM ordering is energetically favor-\nable. Moreover, our results provide a natural under-\nstanding of two experimental observations. First, 3d-\nelectrons of Ni tend to form AFM ordering around 70\n∼90 K, coinciding with the minimum in resistivity and\nthe drop in Hall coefficient. Second, the (π,π,π)AFM\nordering could be weak (compared with cuprates), be-\ncause of the small effective exchange coupling and the\nhybridization with itinerant 5 d-electrons of Nd. This\ncould be the reason why AFM ordering is missing in\nprevious study, which calls for more careful neutron\nscattering measurements on NdNiO 2.\nThe first-principle calculations are carried out with\nthe plane wave projector augmented wave method\nas implemented in the Vienna ab initio simulations\npackage (VASP)13–15. The Perdew-Burke-Ernzerhof\n(PBE) functionals of generalized gradient approxima-\ntion (GGA) is used for PM phase16. To incorpo-\nrate the electron-electron interactions, DFT + U is\nused for AFM phase, which can reproduce correctly\nthe gross features of correlated-electrons in transi-\ntion metal oxides17–19. The 4 felectrons of Nd3+2\nFigure 1: (a) Atomic structure of tetragonal NdNiO 2and first Brillouin zone. The red and blue lines (labels) deno tes the\nfirst Brillouin zone of PM phase and (π,π,π)AFM phase, respectively. The high symmetric line for band ca lculation are\nΓ(0, 0, 0)-X(0.5, 0, 0)-M(0.5, 0.5, 0)- Γ-Z(0, 0, 0.5)-R(0.5, 0, 0.5)-A(0.5, 0.5, 0.5)-Z. (b) Orbita l resolved band structure of\nPM phase. The dxy,dz2,dxz+yzanddx2−y2of Nd (spin-up Ni, spin-down Ni) are marked by black, red, blu e and pick\nfilled circles. The pzandpx+yof O are marked by red, blue and pick filled circles. The size of circles represents the orbital\nweights. (c) Perspective view of Fermi surfaces. LEP and HEP denotes light and heavy electron pocket, respectively. (d)\nThe comparison between first-principles and Wannier-fittin g bands around the Fermi-level. (e) Top and side view of three\nmaximally localized Wannier functions.\nare expected to display the local magnetic moment\nasNd3+inNd2CuO420and are treated as the core-\nlevel electrons. The Hubbard U (0 ∼8 eV) term\nis added to 3 delectrons of Ni. The energy cut-\noff of 600 eV, and Monkhorst-Pack kpoint mesh of\n11×11×11and18×18×30is used for PM and\nAFM phase, respectively. The maximally localized\nWannier functions (WFs) are constructed by using\nWannier90 package23,24. The structure of infinite-layer\nNdNiO 2is shown in Fig. 1(a), including NiO2layers\nsandwiched by Nd, which can be obtained from the\nperovskite NdNiO 3with reduction of apical O atoms\nincdirection21,22. Due to apical O vacancies, the lat-\ntice constant in cdirection shrinks (smaller than adi-\nrection) and the space group becomes P4/mmm . The\nexperimental lattice constant a=b= 3.92 Å and c=\n3.28 Å are used in our calculations.\nFirstly, We present the band structure of PM phase\nwithout Hubbard U. The orbital resolved band struc-\nture of PM phase is shown in Fig. 1(b). Compar-\ning with typical cuprates CaCuO 225, two significant\ndifferences are noted: 1) there is a gap ∼2.5 eV be-\ntween2porbitals of O and 3dorbitals of Ni. According\nto Zaanen-Sawatzky-Allen classification scheme26, this\nindicates that the physics of NdNiO 2is close to Mott-\nHubbard rather than charge-transfer; 2) there are two\nbands crossing the Fermi level, in which one is mainly\ncontributed by dx2−y2orbital of Ni (called pure-band)\nand the other one has a complicated orbital composi-\ntions (called mixed-band). In kz= 0 plane, the mixed-\nband is mainly contributed by dz2orbital of Nd andNi. The dispersion around Γpoint is relatively small,\ncalled heavy electron pocket (HEP). In kz= 0.5 plane,\nthe mixed-band is mainly contributed by dxy(dxz,dyz\nanddz2) orbital of Nd (Ni). The dispersion around\nA point is relatively large, called light electron pocket\n(LEP). As a comparison, one notices that there is only\none pure-band crossing the Fermi level in CaCuO 225.\nThe Fermi surface of PM phase is shown in Fig. 1(c).\nThere is a large sheet contributed by the pure-band, as\nthe case in CaCuO 225. This Fermi surface is obviously\ntwo-dimensional (2D), because of the weak dispersion\nalongΓ-Z. In addition, there are two electron pock-\nets residing at Γand A point, respectively, showing a\nfeature of three-dimensional (3D) rather than 2D (see\nlabels HEP and LEP in Fig. 1(c)). Therefore, the\n3D metallic state will be hybridized with the 2D cor-\nrelated state in NiO2plane, suggesting NdNiO 2to be\nan \"oxide-intermetalic\" compound27,28.\nThe existence of mixed-band also reflects the inher-\nent interactions between Nd 5dand Ni3delectrons.\nTo explore the low energy physics of NdNiO 2, a three-\nband model consisting of Ni dx2−y2, Nddz2and Nd\ndxyorbitals is constructed by Wannier90 package. As\nshown Fig. 1(d), one can see the good agreement be-\ntween first-principles and Wannier-fitting bands near\nthe Fermi level. The corresponding three maximally\nlocalized WFs are shown in Fig. 1(e), demonstrat-\ning the main feature of dz2(WF1) and dxy(WF2)\norbital of Nd, and dx2−y2(WF3) orbital of Ni. How-\never, these WFs still have some derivations from stan-\ndard atomic orbitals, that is, WF1 and WF2 are mixed3\nFigure 2: (a) Illustration of six collinear spin configurati ons. The while and black ball represents local up and down spi n\nmoment, respectively. The four exchange coupling paramete rs are indicated by the blue arrows. (b) Energy comparison\nfor the six collinear spin configurations with different valu es of Hubbard U. Energy of (π,π,π)AFM is set to zero.\nwithdz2orbital of Ni, and WF3 is mixed with px/yor-\nbital of O in the NiO2plane. According to the classi-\ncal Goodenough-Kanamori-Anderson rules29–31, these\nderivations (or hybridizations) will give clues for the\nmagnetic properties.\nTo determine the magnetic ground state of NdNiO 2,\nsix collinear spin configurations are taken into account\nin a2×2×2supercell, that is, AFM1 with q =\n(π,π,π), AFM2 with q =(π,π,0), AFM3 with q\n=(0,0,π), AFM4 with q =(π,0,0), AFM5 with q\n=(π,0,π)and FM with q= (0, 0, 0), as shown in\nFig. 2(a). Within all Hubbard U ranges, we found\nthat AFM1 configuration always has the lowest en-\nergy, as shown in Fig. 2(b), indicating a stable (π,π,π)\nAFM phase with respect to electron-electron interac-\ntions and is in accordance with random phase approx-\nimation treatment32. This can be attributed to the\nspecial orbital distributions around the Fermi level.\nThe intralayer NN exchange coupling is the typical\n180◦typed Ni-O-Ni superexchange coupling, that is,\nthe coupling between dx2−y2orbital of Ni is mediated\nbypx/yorbital of O (see WF3), preferring a (π,π)\nAFM phase in NiO2plane. The interlayer NN ex-\nchange coupling is due to the superexchange between\nthe Nidz2orbitals mediated by Nd dz2orbital as shown\nin WF1, preferring a (π,π)AFM phase between NiO2\nplanes. Therefore, the superexchange coupling results\nin a stable (π,π,π)AFM phase in NdNiO 2. Moreover,\nthe magnetic anisotropy is further checked by includ-\ning the spin-orbit coupling (SOC). We found that the\nspin moment prefers along cdirection with the mag-\nnetic anisotropic energy of ∼0.5 meV/Ni. Thus, the\ntiny SOC effect can be safely neglected in the follow-\ning phase transition temperature calculations.\nIn cuprates, the Fermi surface is unstable with\nelectron-electron interactions, making its parent phase\nto be an AFM insulator. However, this is apparently\nnot the case in NdNiO 2, because of the extra electron\npockets and the inherent interaction between Nd 5d\nand Ni3delectrons. At U=0 eV, there are two elec-\ntron pockets at Γpoint and two hole pockets along\nX-R direction as shown in Fig. 3(a)-(b). Physically,the origin of these four pockets can be easily under-\nstood through the comparison of orbital resolved band\nstructures between PM phase (Fig. 1(b)) and (π,π,π)\nAFM phase (Fig. 4). Because of the Zeeman field on\nNi, its spin-up and -down bands are split away from\neach other. The original pure-band ( dx2−y2orbital of\nNi) in PM phase becomes partially occupied in spin-\nup channel (forming two hole pockets) and totally un-\noccupied in spin-down channel. Hence, the two hole\npockets in AFM phase are inherited from large sheet\nin PM phase, showing a 2D character with neglectable\ndispersion along Γ-Z direction. For the electron pock-\nets atΓpoint, the heavier one is mainly contributed\nbydz2orbital of Nd and Ni, so it comes from the HEP\natΓpoint of PM phase. While for the lighter one, it\ncomes from the LEP at A point of PM phase which\nis folded into the Γpoint of (π,π,π)AFM phase [see\nFig. 1(a)]. The orbital composition can also be used\nto check this folded band, which is contributed by dxy\norbital of Nd, dxz/yz (dz2) orbital of spin-down (-up)\nNi.\nThese pockets have a different evolution with the\nincreasing value of Hubbard U. For electron pockets,\nthe heavier one is very sensitive to Hubbard U and\ndisappears at U = 1 eV. Meanwhile the lighter one\ndoesn’t appear until U = 6 eV. In addition, the orbital\ncomponents of lighter electron pockets are purified by\nelectron-electron interaction and it mainly contributed\nbydxyof Nd in the large U limit as shown in Fig.\n4. The case for hole pockets is rather complicated.\nFirstly, the bands of hole pockets become flat with the\nincreasing value of Hubbard U. Secondly, the original\nhole pockets formed by dx2−y2orbital of Ni gradually\ndisappear, meanwhile, a new hole pocket formed by\ndz2orbital of Ni appears along Γ-M as shown in Fig.\n3(d). At U = 6 eV, NdNiO 2is a compensated metal\nwith a small electron pocket at Γpoint and four hole\npockets along Γ-M as displayed in Fig. 3(e). Further\nincreasing the value of Hubbard U, the system enters\nan AFM insulator, just like cuprates. Therefore, the\nmetal-to-insulator phase transition point is near U ∼6\neV. If Hubbard U is less than 6 eV, NdNiO 2is an AFM4\nFigure 3: (a) Band structures of (π,π,π)AFM phase with different values of Hubbard U. (b)-(e) Perspec tive view of the\nFermi surfaces of (π,π,π)AFM with U = 0, 2, 4 6 eV. The two hole pockets in (b)-(d) are dege nerated and can not be\ndistinguished from this picture. The high symmetry k-point s in (e) are labelled to guide the eye. (f) Schematic diagram\nof major self-doping channel at U = 0 and U = 6 eV. The red/blue c olor represents dorbital of Nd/Ni respectively.\nmetal with relatively small amount of holes that are\nself-doped27,32–35intodorbitals of Ni. Interestingly,\nthere is an orbital shift from 3dx2−y2orbital of Ni at\nU = 0 eV to 3dz2orbital of Ni at U = 6 eV in the\nNiO2plane, as depicted in Fig. 3(f). We speculate\nthat this orbital shift may change the paradigm after\ndoping8,36. Moreover, without the Hubbard U, the 2p\norbital of O is far away from the Fermi level, just like\nthe case of PM phase. However, with the increasing\nvalue of Hubbard U, the gap between 3dorbital of\nNi and2porbital of O gradually decreases (Fig. 4),\ndemonstrating an evolution from Mott-Hubbard metal\nto charge-transfer insulator.\nIn order to quantitatively describe such a phe-\nnomenon, the phase transition temperature is further\ncalculated. For (π,π,π)AFM phase, the magnetic mo-\nmentum of Niincreases from 0.58 µB(U = 0 eV) to\n1.04µB(U = 8 eV) and becomes gradually saturated,\nas shown in Fig. 5(a). This is also consistent with\nthe fact that dx2−y2orbital of Ni is closer to single\noccupation with the increasing value of Hubbard U.\nTherefore, Niis spin one half ( S= 1/2) in infinite-layer\nNdNiO 2, just like the case in cuprates. To extract the\nexchange coupling parameters of J1,J2,J3andJ4(as\nlabelled in Fig. 2(a)), the total energy of five AFM\nconfigurations obtained from DFT + U calculations\nare mapped onto the Heisenberg spin Hamiltonian. In\nthe2×2×2supercell, there are 8 Ni atoms and the\ntotal energy of different AFM configurations are:EAFM1=E0−16J1S2−8J2S2+16J3S2+32J4S2\nEAFM2=E0−16J1S2+8J2S2+16J3S2−32J4S2\nEAFM3=E0+16J1S2−8J2S2+16J3S2−32J4S2\nEAFM4=E0+8J2S2−16J3S2+32J4S2\nEAFM5=E0−8J2S2−16J3S2−32J4S2(1)\nwhereE0is the reference energy without magnetic or-\nder. The calculated exchange coupling parameters as\na function of Hubbard U are shown in Fig. 5(b). We\nwould like to make several remarks here: 1) the NN\nintralayer exchange coupling ( J1), mediated by dx2−y2\norbital of Ni, demonstrates a 1/Ulaw; 2) the NN in-\nterlayer exchange coupling ( J2) is∼10 meV with lit-\ntle variation. The positive value of J2indicates an\nAFM coupling between NiO2planes; 3) the next NN\nintralayer exchange coupling ( J3), mediated by dxy\norbital of Nd, is comparable to J1at large value of\nHubbard U, which is dramatically different to that in\ninfinite-layer SrFeO 237–39. This large value could be\nattributed to the relative robustness of lighter electron\npocket and orbital purification; 4) the next NN inter-\nlayer exchange coupling ( J4) is∼0 meV, indicating\nthe validity of our Hamiltonian up to the third NN;\n5)J1,J2andJ3have the same strength at large U,\nsuggesting NdNiO 2is a 3D magnet rather than 2D\nmagnet.\nBased on the above exchange coupling parameters,\nthe phase transition temperature ( TN) is calculated by5\nFigure 4: Orbital resolved band structures of (π,π,π)AFM phase with different values of Hubbard-U. The first, secon d,\nthird and forth row represents Nd, spin-up Ni, spin-down Ni a nd O, respectively. The filled circles with different colors\nhave the same meaning as those in Fig. 1. The name of dorbitals in the AFM supercell has been aligned to that of unit\ncell.\nclassical Monte Carlo method in a 12×12×12super-\ncell based on the classical spin Hamiltonian:\nH=/summationdisplay\n/angbracketlefti,j/angbracketrightJij/vectorSi·/vectorSj (2)\nwhere the spin exchange parameters Jijhave been de-\nfined above. First, we calculate the specific heat ( C)\nafter the system reaches equilibrium at a each given\ntemperature ( T), as shown in Fig. 5(c). Then, TNis\nextracted from the peak position in the curve of C(T),\nas shown in Fig. 5(c). For U = 1 eV, TNis as high as220 K, which can be ascribed to the large value of J1.\nWith the increasing value of Hubbard U, TNgradually\ndecreases and becomes ∼70 K at U = 6 eV. To fur-\nther check the effect of interlayer exchange coupling on\n3D magnet, an additional Monte Carlo calculation is\nperformed without J2andJ4. As shown in Fig. 5(e),\ntheC(T)vsTplot shows a broaden peak at a lower\ntemperature. Since Mermin-Wagner theorem prohibit\nmagnetic order in 2D isotropic Heisenberg model at\nany nonzero temperatures40, the broad peak in C(T)\nvsTplot implies the presence of short-range order.6\nFigure 5: The Hubbard U dependence of (a) the magnetic moment um ofNiin(π,π,π)AFM phase, (b) the exchange\ncoupling parameters, (c) specific heat ( C) vsTwith four J’s, (d) the estimated TNwith four J’s, (e) specific heat ( C) vs\nTwithJ1,J3only and (f) the estimated TNwithJ1,J3only. The shaded region in (d) highlights the possible TNof 70\n∼90 K with a reasonable U = 5 ∼6 eV.\nRegarding the small drop of TN(about 30 K in Fig.\n5(f)), our MC simulations indicate that the weak in-\nteractions between NiO2planes.\nAlthough the exact value of Hubbard U cannot be\ndirectly extracted from the first principles calculations,\nits value range can still be estimated based on similar\ncompounds. The infinite-layer nickelates are undoubt-\nedly worse metals compared to elemental nickel with U\n∼3 eV41, which can be considered as a lower bound of\nHubbard U. The Coulomb interaction in infinite-layer\nnickelates should be smaller than that in the charge-\ntransfer insulator NiO with U ∼8 eV17, which can be\nconsidered as a upper bound of Hubbard U. Therefore,\na reasonable value of Hubbard U in NdNiO 2will be-\ntween 3 eV and 8 eV. In the following discussions, we\nuse U = 5 ∼6 eV27,42to draw our conclusions: 1) with\nthe decreasing of temperature, there is a phase tran-\nsition from PM phase to (π,π,π)AFM phase near 70\n∼90 K; 2) the exchange-coupling parameters are ∼10\nmeV, which is one order smaller than cuprates43–47and\nresults in a low TNcompared with cuprates; 3) the self-\ndoing effect from 5dorbital of Nd and 3dorbital of Ni\nmay screen the local magnetic momentum in dx2−y2\norbital of Ni, which gives a small magnetic momen-\ntum less than 1 µBand makes the long-range AFM\norder unstable6,21,22,48; 4) the Fermi surface of PM\nphase is quite large with two 3D-liked electron pock-\nets, while the Fermi surface of (π,π,π)AFM phase\nis quite small with one 3D-liked electron pocket and\nfour 2D-liked hole pocket. Therefore, there could ex-\nist a crossover from normal metal to bad AFM metal\naroundTN∼70−90K, which provides a plausible\nunderstanding of minimum of resistivity and Hall co-efficient drop in infinite-layer NdNiO 26. We envision\nthat our calculations will intrigue intensive interests\nfor studying the magnetic properties of high quality\ninfinite-layer NdNiO 2samples.\nLastly, we would like to make some remarks on the\nexisting experiments. Some recent experiments fail to\nfind bulk superconductivity in NdNiO 2systems, and\nthe parent samples show strong insulating behaviors49.\nThe insulating behavior could be attributed to strong\ninhomogenious disorder or improper introduction of\nH during the reaction with CaH 250. Especially, it is\nworth noting, the experiments cannot rule out the pos-\nsibility of weak AFM ordering, due to the presence of\nNi impurities in their samples. (Actually this prob-\nlem has been pointed out before21,22.) The strong fer-\nromagnetic order from elemental Ni would dominate\nover and wash out the weak signal of AFM ordering\nfromNdNiO 2as we suggested in this work. In this re-\ngarding, the upcoming inelastic neutron scattering on\nhigh-quality samples is highly desired.\nW.Z. thanks Chao Cao for sharing their unpub-\nlished DMFT results, and thanks Filip Ronning, H. H.\nWen, G. M. Zhang for helpful discussion. This work\nwas supported by NSFC (No. 11774325, 21603210,\n21603205, 21688102), National Key Research and De-\nvelopment Program of China (No. 2017YFA0204904,\n2016YFA0200604), Anhui Initiative in Quantum Infor-\nmation Technologies (No. AHY090400), Fundamen-\ntal Research Funds for the Central Universities and\nthe Start-up Funding from Westlake University. We\nthank Supercomputing Center at USTC for providing\nthe computing resources.\n∗E-mail: jlyang@ustc.edu.cn1J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189-1937\n(1986).\n2J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani\nand J. Akimitsu, Nature 410, 63-64 (2001).\n3A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Kseno-\nfontov and S. I. Shylin, Nature 525, 73-76 (2015).\n4Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura,\nH. Yanagi, T. Kamiya and H. Hosono, J. Am. Chem.\nSoc.128, 10012-10013 (2006).\n5Q.-Y. Wang et al., Chin. Phys. Lett. 29, 037402 (2012).\n6D. Liet al., Nature 572, 624-627 (2019).\n7V. I. Anisimov, D. Bukhvalov and T. M. Rice, Phys.\nRev. B 59, 7901-7906 (1999).\n8K.-W. Lee and W. E. Pickett, Phys. Rev. B 70, 165109\n(2004).\n9A. L. Patrick, N. Nagaosa and X.-G. Wen, Rev. Mod.\nPhys.78, 17-85 (2006).\n10B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida\nand J. Zaanen, Nature 518, 179-186 (2015).\n11P. Dai, J. Hu and E. Dagotto, Nat. Phys. 8, 709-718\n(2012).\n12P. Dai, Rev. Mod. Phys. 87, 855-896 (2015).\n13M. P. Teter, M. C. Payne and D. C. Allan, Phys. Rev.\nB40, 12255 (1989).\n14G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169\n(1996).\n15P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).\n16J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev.\nLett.77, 3865-3868 (1996).\n17V. I. Anisimov, J. Zaanen and O. K. Anderson, Phys.\nRev. B 44, 943-954 (1991).\n18S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys and A. P. Sutton, Phys. Rev. B 57, 1505\n(1998).\n19A. Rohrbach, J. Hafner and G. Kresse, J. Phys.: Con-\ndens. Matter 15, 979-996 (2003).\n20H. Casalta, P. Bourges, M. d’Astuto, D. Petitgrand and\nA. Ivanov, Phys. Rev. B 57, 471-475 (1998).\n21M. A. Hayward, M. A. Green, M. J. Rosseinsky and J.\nSloan, J. Am. Chem. Soc. 121, 8843-8854 (1999).\n22M. A. Hayward and M. J. Rosseinsky, Solid State Sci.\n5, 839-850 (2003).\n23A. A. Mostofi, J. R. Yats, Y.-S. Lee, I. Souza, D. Van-\nderbilt and N. Marzari, Comput. Phys. Commun. 178,\n685 (2008).\n24N. Marzari, A. A. Mostofi, J. R. Yats, I. Souza and D.\nVanderbilt, Rev. Mod. Phys. 84, 1419-1475 (2012).\n25A. S. Botana and M. R. Norman, arXiv: 1908.10946\n(2019).26J. Zaanen, G. A. Sawatzky and J. W. Allen, Phys. Rev.\nLett.55, 418-421 (1985).\n27M. Hepting et al., arXiv: 1909.02678 (2019).\n28Y. Gu, S. Zhu, X. Wang, J. Hu and H. Chen, arXiv:\n1911.00814 (2019).\n29J. B. Goodenough, Phys. Rev. 100, 564 (1955).\n30P. W. Anderson, Phys. Rev. 115, 2 (1959).\n31J. Kanamori, J. Phys. Chem. Solids 10, 87-89 (1959).\n32X. Wu, D. Di Sante, T. Schwemmer, W. Hanke, H. Y.\nHwang, S. Raghu and R. Thomale, arXiv: 1909.03015\n(2019).\n33Y. Nomura, M. Hirayama, T. Tadano, Y. Yoshimoto, K.\nNakamura and R. Arita, arXiv: 1909.03942 (2019).\n34H. Sakakibara, H. Usui, K. Suzuki, T. Kotani, H. Aoki,\nand K. Kuroki, arXiv: 1909.00060 (2019).\n35G.-M. Zhang, Y.-F. Yang and F.-C. Zhang, arXiv:\n1909.11845 (2019).\n36F. Lechermann, arXiv: 1911.11521 (2019).\n37Y. Tsujimoto et al., Nature 450, 1062 (2007).\n38M. A. Hayward and M. J. Rosseinsky, Nature 450, 960\n(2007).\n39H. J. Xiang, S.-H. Wei and M.-H. Whangbo, Phys. Rev.\nLett.100, 167207 (2008).\n40N. D. Mermin and H. Wagner, Phys. Rev. Lett. 100,\n1133-1136 (1966).\n41F. Aryasetiawan, K. Karlsson, O. Jepsen and U. Schön-\nberger, Phys. Rev. B 74, 125106 (2006).\n42M. Cococcioni and S. de Gironcoli, Phys. Rev. B 17,\n1133 (1966).\n43P. Bourges, H. Casalta, A. S. Ivanov and D. Petitgrand,\nPhys. Rev. Lett. 79, 4906 (1997).\n44R. Coldea, S. M. Hayden, G. Aeppli, T. G. Perring, C.\nD. Frost, T. E. Mason, S.-W. Cheong and Z. Fisk, Phys.\nRev. Lett. 86, 5377 (2001).\n45L. Braicovich et al., Phys. Rev. Lett. 102, 167401\n(2009).\n46G. Blumberg, R. Liu, M. V. Klein, W. C. Lee, D. M.\nGinsberg, C. Gu, B. W. Veal and B. Dabrowski, Phys.\nRev. B 49, 13295 (1994).\n47G. Blumberg, P. Abbamonte, M. V. Klein, W. C. Lee,\nD. M. Ginsberg, L. L. Miller and A. Zibold, Phys. Rev.\nB53, R11930 (1994).\n48Y. Fu et al., arXiv: 1911.03177 (2019).\n49Q. Li, C. He, S. Jin, X. Zhu, Y. Zhang and H.-H. Wen,\narXiv: 1911.02420 (2019).\n50L. Si, W. Xiao, J. Kaufmann, J. M. Tomczak, Y. Lu, Z.\nZhong and K. Held, arXiv: 1911.06917 (2019)."    },    {        "title": "1611.10062v2.Optical_properties_of_the_optimally_doped_Ca___8_5__La___1_5___Pt__3_As__8___Fe___2__As___2_____5__single_crystal.pdf",        "content": "Optical properties of the optimally doped Ca 8:5La1:5(Pt3As8)(Fe 2As2)5single crystal\nYu-il Seo1, Woo-Jae Choi1, Shin-ichi Kimura2, Yunkyu Bang3,\u0003and Yong Seung Kwon1y\n1Department of Emerging Materials Science, Daegu Gyeongbuk Institute\nof Science and Technology (DGIST), Daegu 711-873, Republic of Korea\n2FBS and Department of Physics, Osaka University, Suita 565-0871, Japan\n3Department of Physics, Chonnam National University, Kwangju 500-757, Republic of Korea\n(Dated: August 6, 2021)\nWe have measured the reflectivity of the optimally doped Ca 8:5La1:5(Pt3As8)(Fe 10As10) single crystal ( Tc\n= 32.8K) over the broad frequency range from 40 cm\u00001to 12000cm\u00001and for temperatures from 8K to 300\nK. The optical conductivity spectra of the low frequency region ( <1;000cm\u00001) in the normal state (80 K\n< T\u0014300 K) is well fitted with two Drude forms, which indicates the presence of multiple bands at the\nFermi level. Decreasing temperature below 80 K, this low frequency Drude spectra develops pseudogap (PG)\nhump structure at around \u0019100cm\u00001and continuously evolves into the fully opened superconducting (SC)\ngap structure below Tc. Theoretical calculations of the optical conductivity with the preformed Cooper pair\nmodel provide an excellent description of the temperature evolution of the PG structure above Tcinto the SC\ngap structure below Tc. The extracted two SC gap sizes are \u0001S= 4.9meV and\u0001L= 14.2meV , suggesting\nCa8:5La1:5(Pt3As8)(Fe 10As10) as a multiple gap superconductor with a mixed character of the weak coupling\nand strong coupling superconductivity.\nPACS numbers: 74.25.nd,74.20.Rp,74.70.Xa\nI. INTRODUCTION\nRecently discovered Fe-pnictide compounds,\nCa10(Pt3As8)(Fe 2As2)5(so-called 10-3-8 compound)\nand Ca 10(Pt4As8)(Fe 2As2)5(so-called 10-4-8 compound)1–4,\nadded another new class to the family of the Fe-based\nsuperconductors (IBSs). They are showing the prototype\nbehavior of the subtle balance and competition between\nmagnetism and superconductivity commonly observed in\nother IBSs. Both compounds share similar overall crystal\nstructure, consisting of tetrahedral FeAs planes sandwiched\nbetween the planar Pt nAs8(n= 3;4) intermediary layers.\nHowever, in fine details, there exist also distinct differences\nbetween them.\nThe parent 10-4-8 compound has a tetragonal crystal struc-\nture and is metallic. It becomes superconducting (SC) with-\nout doping with the maximum Tcof\u001838K1,5, and doping\nof electrons only suppresses the SC transition temperature4,6.\nOn the other hand, the parent 10-3-8 compound has a triclinic\ncrystal structure. Sturzer et al.6reported that this compound\nis semiconducting and becomes antiferromagnetically ordered\nbelowTN\u0019120K without any further reduction in the crys-\ntal symmetry. Superconductivity can be induced by doping\nof electrons: for instance, La substitution at the Ca site, or\nPt substitution at the Fe site. The maximum Tcfor each sub-\nstitution can be\u001832K and\u001815K, respectively6. How-\never, Neupane et al.7reported that the undoped 10-3-8 com-\npound has the multiple Fermi surfaces like other typical FeAs-\nsuperconducting compounds and becomes superconducting at\nTc\u00188Kwithout doping.\nThe band structure calculations8suggested that these dif-\nferences arise from the increased metallicity of the PtAs lay-\ners. And recent angle resolved photoemission spectroscopy\n(ARPES) experiment further suggested that this difference be-\ntween two compounds could be developed by the number of\nthe band-edge singularities9. In order to study this subtle re-lation between the electronic structure and superconductivity\nin these materials, various experiments have already been per-\nformed: such as transport10, pressure effect11, ARPES7, mag-\nnetic force microscope (MFM)12, upper critical field ( Hc2)13,\nnuclear magnetic resonance (NMR)14, and penetration depth\n\u0015(T)15measurements, etc. However, the IR spectroscopy ex-\nperiment has not yet been carried out with these compounds.\nThis technique is particularly a useful tool to directly study\nthe low energy dynamics of the correlated materials with tem-\nperature variation, and, therefore, it is able to investigate the\nsystematic development of superconductivity with tempera-\nture and any non-trivial evolution of the correlation effects, if\nexists.\nIn this paper, we have measured the IR reflectivity and an-\nalyzed the optical properties of the optimally La-doped 10-\n3-8 compound, Ca 8:5La1:5(Pt3As8)(Fe 2As2)5single crystal\n(Tc\u001932.8 K). In view of the phase diagram with electron dop-\ning of the previous study10, our compound is located far from\nthe antiferromagnetic (AFM) quantum critical point (QCP)\nwhereTNgoes to zero in the phase diagram. However, we\nspeculate that our compound is located near another non-\nmagnetic QCP in the universal phase diagram, from the ob-\nservations of (1) the maximum Tcwith doping variation, and\n(2) theT-linear resistivity data above Tcup to 300 K. It is\ninteresting to note that these QCP behaviors far away from\nthe AFM QCP are similar to the case of the cuprate super-\nconductors. Then, most interestingly, we have also found the\npseudogap (PG) behavior in the optical conductivity of our\nsample up to the temperature about three times higher ( \u001880\nK) thanTc(\u001933 K). With the observation that our compound\nis far away from the AFM QCP, it is logical to conclude that\nthe PG behavior of the optimal doped 10-3-8 compound is\nnot related to the AFM correlation. We have employed the\npreformed Cooper pair model to calculate the real part of the\noptical conductivity \u001b(!;T)and successfully reproduced the\nPG feature far above Tcand its consistent evolution to the SCarXiv:1611.10062v2  [cond-mat.supr-con]  28 Feb 20172\n05 01 001 502 002 503 00050100150C\na8.5La1.5(Pt3As8)(Fe2As2)5 \n ρ(µΩ⋅cm)T\n(K)Tc=35.2 KT\nc,zero=32.8K\nFIG. 1: (Color online) Temperature dependence of the DC electrical\nresistivity\u001a(T)of the Ca 8:5La1:5(Pt3As8)(Fe 2As2)5single crystal.\ngap structure below Tc.\nCombining the previous reports of the PG observations in\nseveral IBS compounds and cuprate superconductors, our ob-\nservation establishes that the PG phenomena induced by a SC\ncorrelation is a generic phenomena of the strongly correlated\nunconventional superconductors.\nII. EXPERIMENTAL DETAILS\nHigh quality single crystals of\nCa8:5La1:5(Pt3As8)(Fe 2As2)5were grown by a Bridge-\nman method with a sealed molybdenum crucible at 1250\u000e\nC. Before performing the Bridgeman method, we made\nthe precursors of CaAs, LaAs and FeAs in advance. Fig-\nure 1 shows the in-plane electrical resistivity \u001a(T)of the\nCa8:5La1:5(Pt3As8)(Fe 2As2)5single crystal. We do not see\nany noticeable anomaly in \u001a(T)aroundT0\nN\u0019120 K (the\nNeel temperature of the undoped parent 10-3-8 compound)\nand below due to the spin density wave (SDW) transition.\nThe SC transition begins at Tc=35.2 K and zero resistivity\noccurs atTc;zero = 32:8K, respectively. These transition\ntemperatures are about 7-9 K higher than the one with similar\nnominal doping of La ( x= 1:45) in the previously reported\nwork10, indicating that our sample is a substantially higher\nquality single crystal.\nAnother noticeable feature is that \u001a(T)shows theT-linear\ntemperature dependence above Tcup to 300 K. This linear-\nin-Tbehavior in \u001a(T)is commonly observed in other opti-\nmally doped IBSs16–19and is a signature that the sample is\nnear the QCP. This is consistent with the phase diagram of\nRef.10, where the maximum SC Tcwas also located with the\nLa doping ” xsc\nmax”\u00191:5. Moreover the phase diagram of\nRef.10shows that the magnetic critical doping ” xmag\nc”, where\nTN!0, is near\u00190:3, which is far apart from the optimaldopingxsc\nmax for the maximum SC Tc. Therefore, in view\nof the phase diagram of Ref.10and our result, we speculate\nthat there exist two QCPs in the La-doped 10-8-3 compound:\none is the QCP due to magnetic fluctuations where TN!0,\nand the other is the QCP due to unknown quantum fluctuations\nwhere the SC Tcbecomes maximum. The former type of QCP\nis more common with many IBSs as well as most of heavy\nfermion superconductors, while the latter type of QCP is well\nrepresented in the high- Tccuprate superconductors. There-\nfore, clarifying the nature of the QCP in the 10-3-8 compound\nwill be particularly interesting in connection with the mystery\nof the high-Tccuprate superconductors.\nIn order to investigate the electronic structure as well as the\nSC properties of Ca 8:5La1:5(Pt3As8)(Fe 2As2)5, we have mea-\nsured the optical reflectivity R(!;T)of this single crystal in\na broad frequency range from 40 to 12000 cm\u00001and for vari-\nous temperatures from 8 to 300 K. We used a Michelson-type\nrapid-scan Fourier spectrometer (Jasco FTIR610). In particu-\nlar, we used a specially designed feedback positioning system\nto drastically reduce the overall uncertainty level. The uncer-\ntainty of our data was maintained less than 0.5 %. For more\ndetails of experimental methods of reference setting and con-\ntrol of data uncertainty, we refer to our previous paper20.\nIII. RESULTS AND DISCUSSIONS\nA. Reflectivity and Optical Conductivity\nFigure 2 shows the reflectivity spectra R(!;T)of\nCa8:5La1:5(Pt3As8)(Fe 2As2)5single crystal at several differ-\nent temperatures from 8 K to 300 K. The main panel shows\nthe full measured data for frequencies from 50 cm\u00001to 12000\ncm\u00001in log scale, and the inset shows the close-up view of the\ndata from 0 cm\u00001to 300cm\u00001in linear scale; here the data\nfrom 0cm\u00001to 60cm\u00001(dotted lines) are not the measured\nones but extrapolated ones as explained below (although we\nhave data from 40 cm\u00001, we didn’t use the data of 40 cm\u00001-\n60cm\u00001because the data in this part are not uniformly clean).\nIn the normal state (the data sets for T\u001540K),R(!)\napproaches to unity at zero frequency and increases as de-\ncreasing temperature in far-infrared region, showing a typi-\ncal metallic behavior of Ca 8:5La1:5(Pt3As8)(Fe 2As2)5. Upon\nentering the SC phase (the data sets below 40 K), the low fre-\nquency reflectivity turns up quickly, and reaches the flat unity\nresponse below\u001880cm\u00001reflecting the SC gap opening. As\nthe temperature increases toward Tcthe flat unity response\nshrinks to the lower frequency region as the SC gap size is\ndecreasing. The shape of the flat part in R(!)at low frequen-\ncies and its temperature dependence are clear signatures of the\nfully opened SC gap. However, the most interesting develop-\nment of the reflectivity spectra in Fig.2 occurs in the data sets\nfor 80 K (cyan color) and 40 K (purple color), which are still\nin the normal state. Both data sets show a curious suppression\nof spectra in the low frequency region around 100 cm\u00001(see\nthe inset of Fig.2).\nFor more convenient analysis, we converted our reflectivity\ndataR(!)into the real part of the optical conductivity \u001b1(!)3\nby the Kramers-Kronig (KK) transformation. For extrapola-\ntion of the data for the KK-transformation, at low frequencies,\nwe used the Hagen-Rubens extrapolation formula for the nor-\nmal state and the form ( 1\u0000A!4) below the gap in the SC\nstate. For high frequencies above 12000 cm\u00001, the standard\n1=!4form plus a constant reflectivity are used up to 40 eV.\n1001 0001 00000.40.50.60.70.80.91.00\n5 00.920.940.960.981.001\n001 502 002 503 00 \n ReflectivityF\nrequency (cm-1) \n Ca8.5La1.5(Pt3As8)(Fe2As2)5F\nrequency (cm-1) \n Reflectivity 8 K \n17 K \n28 K \n40 K \n80 K \n130 K \n240 K \n300 K\nFIG. 2: (Color online) Reflectivity spectra R(!)of the\nCa8:5La1:5(Pt3As8)(Fe 2As2)5single crystal at several temperatures.\nThe inset shows the enlarged view of spectra below 300 cm\u00001; dot-\nted lines are the part from extrapolation as explained in the main text.\n02 004 006 008 001 00002000400060008000F\nrequency (cm-1)\n02 0004 0006 0008 0001 0000100015002000250030003500F\nrequency (cm-1)σ1 (Ω-1cm-1) \n Ca8.5La1.5(Pt3As8)(Fe2As2)5 \n  \n8K \n17K \n28K \n40K \n80K \n130K \n240K \n300Kσ1(Ω−1cm-1)\nFIG. 3: (Color online) Real part of optical conductivity \u001b1(!)of\nCa8:5La1:5(Pt3As8)(Fe 2As2)5single crystal for 8, 17, 28, 40, 80,\n130, 240, and 300 K, respectively. The pseudogap feature of the 40K\nand 80 K data is indicated inside the dotted circle. The inset shows\nthe same data \u001b1(!)in wider view up to 10,000 cm\u00001.\nFigure 3 shows these results of \u001b1(!), obtained from the\ndata of Fig.2, in the frequency range from 0 to 1000 cm\u00001\nfor various temperatures. The optical conductivity spectra at\n130, 240 and 300 K – which are in normal state – show a verybroad Drude peak centered at != 0 and then monotonically\ndecrease until the interband transition starts appearing around\n1000cm\u00001(see the inset of Fig.3). The interband transition\nspectra continues to form a broad hump around 6000 cm\u00001;\nwhile the peak position of the hump is almost temperature in-\ndependent, there exists an interesting spectral weight transfer\nwith temperature variation which will be analyzed in section\nIII.B.\nAs the temperature decreases down to 80 K (cyan color) and\n40 K (blue color) – which are still in normal state by trans-\nport/thermodynamic measurements – (1) the Drude part of\nspectra rapidly sharpens, and then (2) a peculiar hump struc-\nture around 80-150 cm\u00001appears on top of the smooth Drude\nspectra. The former is the expected evolution due to the for-\nmation of coherent quasiparticles as temperature decreases.\nThe striking feature is the latter, namely, the appearance of the\nshoulder-like hump structure far above Tcon top of the Drude\nresponse in the frequency range of 80-150 cm\u00001, indicating\nan incomplete gap – hence called PG – formation over the\nFermi surfaces or in a part of the Fermi surfaces. Then below\nTc, i.e., at 28 K, 17 K and 8 K, \u001b1(!)shows a clean open-\ning of the SC energy gap with the absorption edge at about 80\ncm\u00001and a peak position at around 150 cm\u00001. Comparing\nthis SC gap structure at around 80-150 cm\u00001belowTc, the\nhump structure around 80-150 cm\u00001, aboveTcat 40 K and\n80 K, appears to be a continuous evolution of the SC correla-\ntion belowTc.\nThe observation of the PG features in the IBSs with the\noptical spectroscopy measurements is now quite common.\nFor example, the underdoped ( x\u00190:12;0:2)21and the\nslightly underdoped ( x\u00190:3)20Ba1\u0000xKxFe2As2have re-\nported the similar behavior in their optical measurements:\nnamely, the hump structures in the optical conductivity \u001b1(!)\naboveTc(about three times of Tcfor all cases) in the same\nenergy scale as the SC gap energy, and their continuous evo-\nlution into the SC gap. These authors concluded that this\nhump structure is not related with a magnetic correlation but\nrather connected to the SC gap, hence possibly a precur-\nsor of the preformed Cooper pairs. The optimally Co-doped\nBa(Fe 0:92Co0:08)As 222also has shown a similar hump struc-\nture at 30 K ( Tc=22.5 K) and its evolution into the SC gap be-\nlowTc; however, the authors of Ref.22advocated, as its origin,\nthe impurity bound state or a low-energy interband transition,\nwith which we do not agree.\nMore recently, Surmach et al.23have performed a com-\nprehensive measurements of muon-spin relaxation ( \u0016SR),\ninelastic neutron scattering (INS), and NMR on Pt-doped\n(CaFe 1\u0000xPtxAs) 10Pt3As8(Tc= 13 K) – the same 10-3-8 com-\npound we studied in this paper but with Pt doping on Fe sites.\nThey found a PG behavior in the 1=T1data of75As NMR be-\nlowT\u0003\u001945K (about three times higher than Tc). These\nauthors concluded from a combination of measurements of\n\u0016SR, INS, together with NMR that this PG behavior is likely\nto be associated with the preformed Cooper pairs. There-\nfore, our observation of a similar PG behavior in the optimally\ndoped Ca 8:5La1:5(Pt3As8)(Fe 2As2)5compound has strength-\nened the universal nature of the PG phenomena originating\nfrom the preformed Cooper pairs or a precursor effect from4\n05 01 001 502 002 503 000.00.51.01.505 01 001 502 002 503 000.60.81.0(\na)c\nutoff freq. (cm-1) \n SW(T)/SW(300K) \n1000 \n500 \n200(b) \n SW(T)/SW(300K) \n1000 \n2000 \n4000 \n5000 \n7000 \n10000cutoff freq. (cm-1)\n05 01 001 502 002 503 000.00.51.01.52.0c\nutoff freq. (cm-1) \n SW(T)/SW(300K)T\n(K) 200 \n100 \n 80 \n 60 \n 40(c)\nFIG. 4: (Color online) Temperature dependence of the normalized\n”partial” spectral weights SW(T;!c)=SW (300K;!c)of the opti-\ncal conductivity data in Fig.3 with different cutoff frequency !c. (A)\ndata with!c\u00151000cm\u00001; (B) data with !c\u00141000cm\u00001; and (C)\ndata with finer variation of cutoff frequencies for !c\u0014200cm\u00001:\n200, 100, 80, 60, and 40 cm\u00001, respectively.\nthe SC correlation in the 10-3-8 compound.\nB. Temperature Dependence of Spectral Weight Transfer\nTo further understand the temperature evolution of the opti-\ncal conductivity shown in Fig.3, we have analyzed the temper-\nature dependence of the ”partial” spectral weight SW(T;!c)\nwith different cutoffs defined as\nSW(T;!c) =Z!c\n0+\u001b1(!;T)d!; (1)\n0 2000 4000 6000 8000 10000100015002000250030003500\nFrequency (cm-1)1 (-1cm-1)\n  FIG. 5: (Color online) A schematic illustration of the spectral weight\ntransfer with temperature variation. The spectral density in mid-\nfrequency range ( \u0018500cm\u00001< ! < \u00185;000cm\u00001) is contin-\nuously depleted with decreasing temperature. This depleted spectral\nweight is roughly divided by !\u0003\u00181;000cm\u00001(black vertical line),\nand the spectral weight below !\u0003and the spectral weight above !\u0003\nare separately conserved (of course, this separate conservation rule\nholds only approximately).\nwhere!cis a cutoff frequency. Figure 4 shows the results of\nSW(T;!c), normalized by SW(T= 300K;!c), that reveal\nnon-trivial information of the correlated electron system.\nFig.4(A) shows that the total spectral weight up to 10,000\ncm\u00001is constant on changing temperature from Tcto 300 K\nconfirming that the sum rule is satisfied24. However, with low-\nering cutoff frequency !c, the sum rule is being deviated, as\nit should be, but in a non-trivial way. First, the normalized\n”partial” spectral weight SW(T;!c)=SW (300K;!c)with\n1;000cm\u00001< !c<10;000cm\u00001is monotonically de-\ncreasing with lowering temperature down to Tc. This means\nthat the spectral weight below the cutoff frequency !cis trans-\nferred to the higher frequency region above the cutoff fre-\nquency!cwhen temperature decreases, which is an opposite\nbehavior from a standard Drude type metallic state. Second,\nthe rate of this spectral weight transfer is not monotonously\nincreasing with lowering the cutoff frequency !c; the de-\nceasing rate increases to the maximum when the cutoff fre-\nquency is lowered to !c= 4;000cm\u00001(blue triangles) and\nthen it becomes weaker with lowering cutoff frequencies to\n!c= 2;000cm\u00001;and1;000cm\u00001. Third, in particular,\nwhen!c= 1;000cm\u00001, the partial sum rule is ”almost” re-\ncovered again: namely, the spectral weight below !\u0003\nc= 1;000\ncm\u00001(red circles) is separately conserved with respect to the\ntemperature variation up to Tc.\nFig.4(B), on the other hand, shows the similar plots of\nSW(T;!c)=SW (300K;!c)with!c\u0014!\u0003\nc(= 1;000cm\u00001).\nIt clearly shows that the partial spectral weight is ”increasing”5\n– not decreasing –with decreasing temperature. This is an op-\nposite behavior to the ones in Fig.4(A).\nThis complicated spectral weight transfer of SW(T;!c)\nwith temperature variation is summarized in Fig.5. It shows\nthat the spectral weight transfer of \u001b1(!)with temperature\nvariation is roughly divided into two parts: one part below\n!\u0003\nc= 1;000cm\u00001and the other part above !\u0003\nc= 1;000\ncm\u00001, and each part separately conserves the spectral weight.\nThe physical meaning of it is that the correlated electrons are\ndivided into the low energy itinerant part (Drude spectra) and\nthe high energy localized part (Lorentzian spectra). At high\ntemperatures, a large portion of spectra in the intermediate\nenergy range – in this particular compound, in between 500\ncm\u00001and 5000cm\u00001– is undetermined, hence remains in-\ncoherent. Lowering temperature, these incoherent spectra are\ncontinuously depleted and transferred either into the low fre-\nquency Drude part or into the high frequency Lorentzian part:\nthe final fate of the incoherent spectra was roughly predeter-\nmined by the frequency !\u0003\nc= 1;000cm\u00001– which is the\ndeepest valley point of the \u001b1(!)spectra in Fig.5 – in this\nparticular 10-3-8 compound. The sharpening of the Drude\nspectra with decreasing temperature is well understood as a\ndevelopment of coherent quasiparticles. The spectra transfer\nof the high frequency localized part should be associated with\nthe strong correlation effects of the local interactions such as\nHubbard interactions ( U;U0) and Hund’s couplings ( J)25–27.\nFinally, in order to scrutinize the development of hump\nstructure around 80-150 cm\u00001with temperature variation, we\nanalyzed the low frequency spectra transfer in Fig.4(C), with\nfiner variation of cutoff frequencies for !c\u0014200cm\u00001: 200,\n100, 80, 60, and 40 cm\u00001, respectively. The main observa-\ntion is that there is a sudden change of the slopes of the spec-\ntral weight transfer with the cutoff frequency !cbelow and\nabove!c= 100cm\u00001; the slopes suddenly increase when\n!c\u0014100cm\u00001. This means that decreasing temperature be-\nlow\u0018130K, there appears another drain of spectral weight\ntransfer to the region in between 100cm\u00001\u0000200cm\u00001be-\nsides the narrowing of Drude spectra. This another drain of\nspectral weight transfer is the formation of ”hump” structure\naround 80cm\u00001\u0000150cm\u00001as seen with the 40K and 80K\ndata in Fig.3.\nC. Drude-Lorentz model fitting for \u001bexp\n1(!;T >T c)in the\nnormal state\nWe have shown that the total spectra of the normal state op-\ntical conductivity \u001b1(!)consists of two spectral parts – Drude\npart and Lorentz part – and each part conserves their spectral\nweights: this separate conservation of sum rule will be con-\nfirmed once more in this section. Therefore we tried to fit the\ndata with the standard Drude-Lorentz model\n\u001b1(!) =1\n4\u0019RehX\nj!2\np;j\n1\n\u001cD;j\u0000i!+X\nkSk!\n!\n\u001cL;k+i(!2\nL;k\u0000!2)i\n(2)\nwhere!p;jand1\n\u001cD;jare the plasma frequency and the scatter-\ning rate for the j-th free carrier Drude band, respectively, andSk,!L;kand1\n\u001cL;kare the spectral weight, the Lorentz oscil-\nlator frequency, and the scattering rate of the k-th oscillator,\nrespectively.\n1001 0001 000002000400060008000100001\n001 0001 00000200040006000800010000Lorentz 3Drude 2 \n  \ndata (T=130 K) \nTotal \nSum of Drudes \nSum of LorentzesCa8.5La1.5(Pt3As8)(Fe2As2)5σ1 (Ω-1cm-1)F\nrequency (cm-1)Drude 1L\norentz 1L\norentz 2(a)(\nb)D\nrude 2Ca8.5La1.5(Pt3As8)(Fe2As2)5F\nrequency (cm-1)σ1 (Ω-1cm-1) \n  \n data (T=40 K) \nTotal \nSum of Drudes \nSum of LorentzesD\nrude 1\nFIG. 6: (Color online) Typical Drude-Lorentz model fittings of the\noptical conductivity \u001b1(!)in the normal state: (A) 130 K data; (B)\n40 K data. Both data were fitted with two Drude forms and three\nLorentz oscillators. The Lorentz oscillator parts and Drude-2 form do\nnot change much with temperatures, but the Drude-1 form becomes\nsubstantially sharpened with decreasing temperature.\nFigure 6(A) shows that the 130 K data of \u001b1(!)and its\nDrude-Lorentz model fitting. It is well fitted with two Drude\nterms (one narrow and the other broad) and three Lorentz os-\ncillator terms. Here two Drude terms show the presence of\nmultiple bands in Ca 8:5La1:5(Pt3As8)(Fe 2As2)5, as in many\nIBS compounds, e.g., LiFeAs compound28. The data of \u001b1(!)\nof 300 K and 240 K are also well fitted with the Drude-Lorentz\nmodel with the similar parameters: the spectral weights,\n!2\np;j=1;2andSk=1;2;3, and the Lorentz oscillator frequencies\n!L;k=1;2;3are the same, but only the scattering rates,1\n\u001cD;j=1;2\nand1\n\u001cL;k=1;2;3, need to be adjusted (see Fig.7).\nHowever, the 40 K and 80 K data are different. While the\nLorentzian part can be fitted as above only with the scattering\nrates1\n\u001cL;k=1;2;3adjusted, the low frequency Drude part has an\nextra hump structure around \u0018100cm\u00001, as seen in Fig.3,6\n600080001000012000 \nDrude band 2 \nDrude band 1 \n ωp,j=1,2 (cm-1)(\na)0\n5 01 001 502 002 503 003006009001200T\n(K)(b) \n1/τD,j=1,2 (cm-1)x\n 5\nFIG. 7: (Color online) (a) The plasma frequency of the Drude bands,\n!p1(black solid square), and !p2(red solid circle). (b) The cor-\nresponding scattering rates, 1=\u001cD1(black solid square), and 1=\u001cD2\n(red solid circle).\nhence can not be fitted by smooth Drude spectra. Neverthe-\nless, it was shown in the previous section that the low fre-\nquency spectra of \u001b1(!;T)for! < 1;000cm\u00001separately\nconserve the spectral weight with temperature variation (see\nthe pink inverse triangle data in Fig.4). Therefore, we tried to\ncontinue to use the same Drude-Lorentz model to fit the 40 K\nand 80 K data.\nFigure 6(B) shows the fitting result of the 40 K data. The\nLorentzian part is fitted well as in the case of T= 130 K,\nconfirming the separate sum rule, but the Drude part show a\nclear deviation. Comparing the experimental data (black solid\nline) and the model fitting (red solid line), it shows that the\nhump structure (extra spectral peak) around !\u0018100cm\u00001\nis formed by draining a spectral density from the lower fre-\nquency part ( ! < 50cm\u00001) of Drude spectra. Combining\nwith the separate sum rule of the lower frequency \u001b1(!;T)\nfor! < 1;000cm\u00001, this hump spectra should be due to a\npartial gapping in the Drude spectra. If it were from an addi-\ntional inter-band transition29, the sum rule should have been\nviolated. Impurity bound state scenario is also unrealistic be-\ncause any potential scattering cannot form a bound state inside\na Drude spectra without first forming a deep gap.\nTo summarize Fig.6, (1) the spectral weight of the hump\nstructure around !\u0018100cm\u00001is drained from the Drude\nspectra with a partial gapping in them; (2) this hump struc-\nture at 80 K and 40 K continuously evolve into the SC gap\nstructure below Tc. We then conclude that the most plausible\nscenario for the hump structure around !\u0018100cm\u00001is due\nto the preformed Cooper pairs.\nIn Fig.7, we show the temperature dependence of the fit-\nting parameters of the Drude spectra of the normal state\noptical conductivity \u001b1(!), i.e., plasma frequencies of the\nDrude bands, !p;i=1;2, and the corresponding scattering\nrates, 1=\u001cD;i=1;2, for all measured temperatures. It is in-TABLE I: Fitting parameters of Lorentz Oscillators\n!L;k=1;2;3TSk 1=\u001cL;k\n(cm\u00001)(K) (cm\u00001) (cm\u00001)\n40K 14224 5700Lorentz #1 2734130K 14224 5744\n40K 16811 8371Lorentz #2 5000130K 16812 10000\n40K 42977 15000Lorentz #3 8500130K 43319 15854\nteresting to note the T-linear scattering rates 1=\u001cD;1(T)\nof the narrow Drude band in Fig.7(b) and the T-linear\nresistivity data \u001aDC(T)in Fig.1, consistently indicating\nthat the optimal doped Ca 8:5La1:5(Pt3As8)(Fe 2As2)5is lo-\ncated near the QCP, as discussed in the Introduction sec-\ntion. A similar behavior was also observed with the op-\ntimally doped Ba 0:6K0:4Fe2As2by optical spectroscopy30,\nindicating the proximity to the QCP. However, the case of\nBa0:6K0:4Fe2As2is an antiferromagnetic QCP30, while our\ncase of Ca 8:5La1:5(Pt3As8)(Fe 2As2)5is a non-magnetic QCP,\nlocated far away from the AFM QCP10. Finally, Table I shows\nthe fitting parameters for the Lorentz oscillators of Fig.6: the\n40K and 130K data.\nD.\u001bexp\n1(!;T <T c)in the SC state\nThe three data set of \u001bexp\n1(!;T < Tc)forT=8 K, 17 K,\nand 28 K, respectively, shown in Fig.3, clearly shows clean\nopening of a full gap, indicating an s-wave pairing. Combin-\ning with the presence of two Drude terms in the low frequency\npart of\u001bexp\n1(!;T > Tc)in the normal state, as shown in the\nprevious section, it is natural to assume two s-wave gaps in\nthe SC state. Indeed, the line shape of the 8 K data (black\nsolid line) of \u001bexp\n1(!)appears to have two gaps with different\nsizes, \u00011and\u00012. Therefore, we use the generalized Mattis-\nBardeen formula for two band superconductor to fit the 8 K\ndata of\u001b1(!). The spectral weight of two band and its scat-\ntering rate were taken as fitting parameters for optimal fitting.\nThe results are shown in Fig.8.\nThe black solid line in Fig. 8 shows the SC optical con-\nductivity after subtracting the Lorentz oscillator contribution\nat high frequencies, which is almost independent of tempera-\nture, from the optical conductivity of 8 K. The data shows the\nabsorption edge at about 80 cm\u00001, proportional to the first\nSC gap (\u00182\u00011), and a weak kink at about 220 cm\u00001indicat-\ning a second SC gap ( \u00182\u00012). With the generalized Mattis-\nBardeen model31, the SC gap sizes (the scattering rates) were\ndetermined to be \u00011= 4.9meV (1=\u001c1= 13:1meV ) and\n\u00012= 14.2meV (1=\u001c2= 37:1meV ), respectively. The ra-\ntios of gap magnitude to Tc,R= 2\u0001=kBTc, were evalu-\nated as 3.6 and 10.2, respectively. These mixed values of\nR, compared to the value of RBCS = 3:5, suggest that the\nsuperconductivity in Ca 8:5La1:5(Pt3As8)(Fe 2As2)5could be7\n05 01 001 502 002 503 003 504 004 50050010001500200025003000 \nσ8K - σinterband \nσS (Δ1= 4.9meV, 1/τ1=13.1meV) \nσL (Δ2= 14.2meV, 1/τ2=100meV) \nσS + σL\n51 01 52 02 53 0540056005800600062006400 Ωp,s (cm-1)T\n (K)250260270280290λ (nm) \n σ1 (Ω-1cm-1)F\nrequency (cm-1)\nFIG. 8: (Color online) Mattis-Bardeen model fitting for the 8 K data\n(black solid line) of \u001bexp\n1(!)after the Lorentz oscillator contribution\nsubtracted. The data are decomposed into two Mattis-Bardeen terms\n(blue dashed line, and purple dash dotted line). The sum of the two\nMattis-Bardeen terms is plotted as a red solid line. The inset repre-\nsents the SC plasma frequency \np;S(T)(black solid square) and the\ncorresponding penetration depth \u0015(T)(red open square).\na mixture of weak coupling and strong coupling SC states.\nTo our knowledge, there is not yet an experimental report of\nthe gap sizes of 10-3-8 compound. However, our gap values\nare consistent with the gap values of a typical FeAs-SC com-\npounds with similar Tc; e.g., Ba 0:6K0:4Fe2As2(Tc= 37K;\n\u00011\u00186meV; \u00012\u001812meV )32.\nThe relative sizes of the plasma frequency of each band\nare 1 (small gap band) to 2 (large gap band), consistent with\nthe normal state Drude weights of two band. The scattering\nrate of the small gap band 1=\u001c1= 13:1meV\u0018104cm\u00001\nis slightly larger than the value of 1=\u001cD1\u001875cm\u00001at the\nnormal state, while the scattering rate of the larger gap band\n1=\u001c2= 100meV\u0018800cm\u00001is smaller than the value\nof1=\u001cD2\u00181100cm\u00001; the former is physically not very\nconsistent while the latter is more reasonable. This inconsis-\ntency comes from the difference of two conductivity formula –\nMattis-Bardeen model31and the Drude model – which are not\ncontinuously connected. Considering this, the small inconsis-\ntency of the estimated scattering rates of the narrow band is in\nan understandable range.\nIn the inset, we also calculated the condensation strength\n(the SC plasma frequency \np;S(T)) from the missing spec-\ntral weight and the corresponding penetration depth \u0015(T)us-\ning the London formula \n2\np;S(T) =c2=\u00152(T). Having only\nthree data points below Tc(8K, 17K, and 28K, respectively),\nwe cannot extract much about the temperature dependence of\nthese quantities. However, the overall trend of temperature\nvariation – a flatter behavior at lower temperatures (i.e., 0<\nT <Tc=2) and a rapid collapse near Tc(i.e.,Tc=2<T <Tc)\n– is very consistent with the temperature dependence for a s-\n01 002 003 004 005 0004000800012000σ1 (Ω-1cm-1)F\nrequency (cm-1)Ca8.5La1.5(Pt3As8)(Fe2As2)5 \n  \n8K \n18K \n28K \n40K \n80K \nTheoryFIG. 9: (Color online) Theoretical calculation of \u001btheo\n1(!;T >T c)\nby the incoherent preformed Cooper pair model with T= 40 K,\noverlayed on the experimental data of \u001bexp\n1(!;T)forTbelow and\naboveTc= 32:8K.\nwave SC gap(s). Also the absolute magnitude of the low tem-\nperature penetration depth \u0015(T= 8K)\u0018250nm\u00180:25\u0016m\nbelongs to the standard range ( 0:2\u0016m<\u0015ab(0)<0:4\u0016m) of\ntypical FeAs-superconductors33.\nE.\u001bexp\n1(!;Tc<T\u001480K): Pseudo Gap and Preformed\nCooper Pair Model\nIn order to understand the data of \u001bexp\n1(!;T)aboveTc\nbut below 80 K, we adopted the phase incoherent preformed\nCooper pair model34. In the ordinary SC state below Tc, the\nreal part of the optical conductivity is calculated by a standard\nKubo formula with sum of two bands (a= 1;2)as follows,\n\u001b1(!;T) =\u0000P\naIm\u0003a;xx(!)\n!(3)\nwith\nIm\u0003a;xx(!) =\u0019e2Z\nd2kd!0v2\na;x(k)\u0002\nTr[\u0016Aa(k;!+!0)\u0016Aa(k;!0)][f(!+!0)\u0000f(!)](4)\nwheref(!)is the Fermi-Dirac distribution function, and va;x\nis the Fermi velocity along the x-direction of the quasiparti-\ncles of the band ” a”. And \u0016Aa(k;!)is the 2\u00022spectral density\nmatrix of Nambu green’s function of the band ” a” in the SC\nstate defined as \u0016Aa(k;!) =\u0000Im\u0016Ga(k;!)=\u0019with\n\u0016Ga(k;!) =~!\u001c0+\u0018a(k)\u001c3+ \u0001a\u001c1\n~!2\u0000\u00182a(k)\u0000\u00012a(5)\nwhere \u0001ais the SC gap of the band ” a” and\u001ciare Pauli ma-\ntrices. The damping rate \u0000ais included in ~!=!+i\u0000a8\nto fit the overall line shape, and without knowing va;x, the\ncombined factor v2\na;xR\nd2kare taken as the parameter of the\nrelative spectral weight of the each band N0\na=1;2.\nTo simulate the phase fluctuations, we followed the recipe\nof Franz and Millis35and averaged the Nambu green’s func-\ntion Eq.(5) with Doppler shift energy \u0011in the quasiparticle\nexcitations as\n~\u0016G(k;!) =Z\nd\u0011P(\u0011)\u0016G(k;!\u0000\u0011) (6)\nwhere the probability distribution of \u0011is given by P(\u0011) =p\n2\u0019We\u0000\u00112=2WwithW\u00193:48\u000bv(T=Tc)\u00012.\u000bvis a param-\neter derived from the XY-model and was estimated \u00190:009\nin the high-Tccuprates by Franz and Millis35, for example.\nHere, we take the whole Was a fitting parameter and chose\nWa= 0:06\u00012\na, which corresponds to T\u00191:4Tc, for the best\nfitting of the 40 K optical conductivity data. The results are\nplotted in Fig.9 by open red circle symbols. We used the same\ngap values of \u00011= 4:9meV and\u00012= 14:2meV of Mattis-\nBardeen model fitting in the SC state for 8 K data, and ad-\njusted the scattering rates as 1=\u001c1= 13meV and1=\u001c2= 100\nmeV , with cutoffs for ! < 2\u0001a=1;2, respectively. The over-\nall fitting is reasonably good and, in particular, it demonstrates\nthat the hump structure around 100cm\u00001is qualitatively and\nquantitatively reproduced by the fluctuations of the incoherent\nSC OPs \u0001a=1;2.\nFinally, we have a remark on a possibility of a parasitical\ndomain of undoped 10-3-8 phase as an origin for the hump\nstructure above Tcstudied here. Although this kind of possi-\nbility of domain mixture always exists, we think this possibil-\nity is very low because: (1) the hump structure ( \u0018100cm\u00001)\nhas too small energy scale to be from the magnetic (SDW) or-\ndering withTNeel= 120K, because, for example, BaFe 2As2\nhas a SDW ordering at 138K (a similar Neel temperature) but\nthe optical data of BaFe 2As236shows that double humps ap-\npear at 360cm\u00001and890cm\u00001, respectively – far higher than\n\u0018100cm\u00001; (2) our hump structure at ( \u0018100cm\u00001) contin-\nuously evolves from the 80K and 40K data ( >Tc) to the 28K,\n17K, and 8K data ( < Tc) of the SC gap structure. For this\nhump structure to be from a magnetic gap, we need a unusual\naccidental coincidence to have the almost same gap energy\nscale both for the magnetic gap and the SC gap.\nIV . CONCLUSIONS\nWe have measured the optical reflectivity R(!;T)of the\noptimally doped Ca 8:5La1:5(Pt3As8)(Fe 2As2)5single crystal\n(Tc= 32.8 K) for frequencies from 40 cm\u00001to 12,000 cm\u00001\nand for temperatures from 8 K to 300 K. In the normal state\nforT=130 K, 240 K, and 300 K, the optical conductivity\ndata\u001b1(!;T)are well fitted by the Drude-Lorentz model with\ntwo Drude forms and three Lorentz oscillators. We have also\nfound that (1) despite a huge variation of the spectral weight\nredistribution with temperature variation, the f-sum rule is\nsatisfied; (2) also, divided by !\u0003\u00181;000cm\u00001, the spectral\nweights above and below !\u0003are separately conserved, whichsuggests that the original bare conduction electrons are split\ninto the low energy itinerant carriers (Drude spectra) and the\nhigh energy localized carriers (Lorentzian oscillators) due to\nthe strong correlation effects.\nIn the SC state for T=28 K, 17 K, and 8 K, the \u001b1(!;T <\nTc)data show a clean gap opening at \u001880cm\u00001and the\nsecond gap structure at \u0018220cm\u00001. The\u001b1(!;T = 8\nK)data is well fitted with two SC gaps, \u00011= 4:9meV\nand\u00012= 14:2meV , consistent with the presence of two\nDrude bands observed in the normal state data. From the es-\ntimated fraction of the condensate spectral weight below Tc,\nwe have estimated that only about \u001829%37of the conduction\nband carriers participates in the SC condensation indicating\nthat Ca 8:5La1:5(Pt3As8)(Fe 2As2)5is in dirty limit.\nMost interestingly, the \u001b1(!;T = 40;80K)data, in the\nintermediate temperature region above Tcbut below\u001880\nK, showed a PG-like hump structure at the exactly same\nenergy scale as the SC gap energy on top of the smooth\nDrude-like spectra. We have demonstrated that this PG-like\nhump structure can be consistently fitted with the preformed\nCooper pair model using the same SC gap values of \u00011;2\nof the SC state. Our work showed that the optimally doped\nCa8:5La1:5(Pt3As8)(Fe 2As2)5is a multigap superconductor\nhaving a clear PG formed by incoherent preformed Cooper\npairs up to 80 K (about three times of Tc), therefore added\none more case to the list of the IBSs with the PG due to the\nSC correlation20,21.\nAcknowledgments\nYSK was supported by the NRF grant\n2015M2B2A9028507 and 2016R1A2B4012672. YKB\nwas supported by NRF Grant 2016-R1A2B4-008758.\nAppendix A: DC limit of conductivity \u001b1(T;!)\nIn Fig.A.1, we show the DC limits of \u001b1(T;!)in the nor-\nmal state for 40, 80, 130, 240, and 300 K, respectively, over-\nlayed with the DC conductivities directly obtained from the\n\u001aDC(T)data of Fig.1 as \u001bDC(T) = 1=\u001aDC(T). The agree-\nment at the DC limit is excellent for all temperatures which\ndemonstrates the quality of our reflectivity data R(T;!)\n(Fig.2) and the faithfulness of the KK-transformation.\nFor the low frequency region below 60cm\u00001, we use the\nHagen-Rubens extrapolation formula RHR(T;!) = 1\u0000\n2q\n2\u000f0!\n\u001bDC, where the only free parameter \u001bDC(T)can be sub-\nstituted by the DC resistivity data of Fig.1 as \u001bDC(T) =\n1=\u001aDC(T). However, to make the best smooth connection\nbetweenRHR(T;!)and our measured data R(T;!), in the\nregion of 40\u000060cm\u00001(our data exist from 40cm\u00001), we al-\nlow some adjustment of the value \u001bDCin the Hagen-Rubens\nformula. The necessary adjustments were less than 10% for\nall temperatures as demonstrated in Fig.A.1, which shows the\nexcellent agreement between \u001b1(T;!!0)and\u001bDC(T) =\n1=\u001aDC(T).9\n02 004 006 008 001 00005000100001500020000\n-100 1 02 03 04 05 05000100001500020000F\nrequency (cm-1)σ1 (Ω-1cm-1) \n C\na8.5La1.5(Pt3As8)(Fe2As2)5σ1 (Ω-1cm-1) \n F\nrequency (cm-1) 40K \n80K \n130K \n240K \n300K \n40K \n80K \n130K \n240K \n300K\nFIG. A.1: (Color online) The same plot of Fig.3, \u001b1(!), but for wider\ny-axis range, to show the DC limit of \u001b1(!). The data are shown only\nfor the normal state for 40, 80, 130, 240, and 300 K, respectively, and\noverlayed with the DC conductivities (solid square symbols) directly\nobtained from the data of Fig.1 as \u001bDC(T) = 1=\u001aDC(T).\n\u0003ykbang@chonnam.ac.kr\nyyskwon@dgist.ac.kr\n1N. Ni, J. M. Allred, B. C. Chan, and R. J. Cava, Proc. Natl. Acad.\nSci.108, 45 (2011)\n2S. Kakiya, K. Kudo, Y . Nishikubo, K. Oku, E. Nishibori, H. Sawa,\nT. Yamamoto, T. Nozaka, and M. Nohara, J. Phys. Soc. Jpn. 80,\n093704 (2011)\n3T. Sturzer, G. Derondeau, and D. Johrendt, Phys. Rev. B 86,\n060516 (2012)\n4A. Sapkota, G. S. Tucker, M. Ramazanoglu, W. Tian, N. Ni, R. J.\nCava, R. J. McQueeney, A. I. Goldman, and A. Kreyssig, Phys.\nRev. B 90, 100504 (2014)\n5C. Loehnert, T. Sturzer, M. Tegel, R. Frankovsky, G. Friederichs,\nand D. Johrendt, Angew. Chem. Int. Ed. 50, 9195 (2011).\n6T. Sturzer, G. M. Friederichs, H. Luetkens, A. Amato, H. H.\nKlauss, F. Winter, R. Pottgen, and D. Johrendt, J. Phys.: Condens.\nMatter 25, 122203 (2013).\n7M. Neupane, C. Liu, S. Y . Xu, Y . J. Wang, N. Ni, J. M. Allred, L.\nA. Wray, N. Alidoust, H. Lin, R. S. Markiewicz, A. Bansil, R. J.\nCava, and M. Z. Hasan, Phys. Rev. B 85, 094510 (2012)\n8I. R. Shein and A. L. Ivanovskii, arXiv:1108.0495 (2011)\n9S. Thirupathaiah, T. Sturzer, V . B. Zabolotnyy, D. Johrendt, B.\nBuchner, and S. V . Borisenko, Phys. Rev. B 88, 140505 (2013)\n10N. Ni, W. E. Straszheim, D. J. Williams, M. A. Tanatar, R. Pro-\nzorov, E. D. Bauer, F. Ronning, J. D. Thompson, and R. J. Cava,\nPhys. Rev. B 87, 060507 (2013)\n11P. Gao , L. Sun , N. Ni , J. Guo , Q. Wu , C. Zhang , D. Gu , K.\nYang , A. Li , S. Jiang , R. J. Cava , and Z. Zhao, Adv. Mater. 26,\n2346 (2014)\n12J. Kim, F. Ronning, N. Haberkorn, L. Civale, E. Nazaretski, N.\nNi, R. J. Cava, J. D. Thompson, and R. Movshovich, Phys. Rev. B\n85, 180504 (2012)\n13E. Mun, N. Ni, J. M. Allred, R. J. Cava, O. Ayala, R. D. McDon-\nald, N. Harrison, and V . S. Zapf, Phys. Rev. B 85, 100502 (2012)\n14T. Zhou, G. Koutroulakis, J. Lodico, N. Ni, J. D. Thompson, R.J. Cava and S. E. Brown, J. Phys.: Condens. Matter 25, 122201\n(2013)\n15K. Cho, M. A. Tanatar, N. Ni and R. Prozorov, Supercond. Sci.\nTechnol. 27, 104006 (2014)\n16Y . Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.\nChem. Soc. 130, 3296 (2008)\n17X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and D. F. Fang,\nNature. 453, 761 (2008)\n18Z. A. Ren, G. C. Che, X. L. Dong, J. Yang, W. Lu, W. Yi, X.\nL. Shen, Z. C. Li, L. L. Sun, F. Zhou, Europhys. Lett. 83, 17002\n(2008)\n19F. C. Hsu, J. Y . Luo, K. W. Yeh, T. K. Chen, T. W. Huang, P. M.\nWu, Y . C. Lee, Y . L. Huang, Y . Y . Chu, D. C. Yan, and M. K. Wu,\nProc. Natl. Acad. Sci. 105, 38 (2008)\n20Y . S. Kwon, J. B. Hong, Y . R. Jang, H. J. Oh, Y . Y . Song, B. H.\nMin, T. Iizuka, S. Kimura, A. V . Balatsky, and Y . Bang, New J.\nPhys. 14, 063009 (2012)\n21Y . M. Dai, B. Xu, B. Shen, H. H. Wen, J. P. Hu, X. G. Qiu, and R.\nP. S. M. Lobo, Phys. Rev. B 86, 100501 (2012)\n22R. P. S. M. Lobo, Y . M. Dai, U. Nagel, T. Room, J. P. Carbotte,\nT. Timusk, A. Forget, and D. Colson, Phys. Rev. B 82, 100506\n(2010)\n23M. A. Surmach, F. Bruckner, S. Kamusella, R. Sarkar, P. Y .\nPortnichenko, J. T. Park, G. Ghambashidze, H. Luetkens, P. K.\nBiswas, W. J. Choi, Y . I. Seo, Y . S. Kwon, H.-H. Klauss, and D.\nS. Inosov, Phys. Rev. B 91, 104515 (2015)\n24M. Tinkham, and R. A. Ferrell, Phys. Rev. Lett. 2, 331 (1959).\n25N. L. Wang, W. Z Hu, Z. G Chen, R. H. Yuan, G. Li, G. F. Chen\nand T. Xiang, Jour. of Phys: Conden. Matt. 24, 294202 (2012).\n26A. A. Schafgans, S. J. Moon, B. C. Pursley, A. D. LaForge, M. M.\nQazilbash, A. S. Sefat, D. Mandrus, K. Haule, G. Kotliar, and D.\nN. Basov, Phys. Rev. Lett., 108, 147002 (2012).\n27B. Xu et al. Chin. Phys. B 23, 087401 (2014).\n28B. H. Min, J. B. Hong, J. H. Yun, T. Iizuka, S. Kimura, Y . Bang,\nand Y . S. Kwon, New J. Phys. 15, 073029 (2013)10\n29E. van Heumen, Y . Huang, S. de Jon, A.B. Kuzmenko, M.S.\nGolden, and D. van der Marel, Europhys. 90, 37005 (2010).\n30Y . M. Dai, B. Xu, B. Shen, H. Xiao, H. H. Wen, X. G. Qiu, C. C.\nHomes, and R. P. S. M. Lobo Phys. Rev. Lett. 111, 117001 (2013).\n31W. Zimmermann, E. H. Brandt, M. Bauer, E. Seider, and L. Gen-\nzel, Physica C 183, 99 (1991).\n32K. Nakayama et al., Europhysics Letters 85, 67002 (2009).\n33R. Prozorov, V . G. Kogan, Rep. Prog. Phys. 74, 124505 (2011).\n34V . J. Emery and S. A. Kivelson, Nature 374, 434 (1995).\n35M. Franz and A. J. Millis, Phys. Rev. B 5814572 (1998).36W. Z. Hu, J. Dong, G. Li, Z. Li, P. Zheng, G. F. Chen, J. L. Luo,\nand N. L. Wang, Phys. Rev. Lett. 101, 257005 (2008).\n37This condensate spectral weight is calculated from the total Drude\nspectral weight of the 40K data (Fig.6(B)) and the missing spectral\nweight of the 8K data (Fig.8). This number 29% is smaller than\nthe number \u001840% which can be read with the data (red circles)\nwith the cutoff frequency 1000cm\u00001in Fig.4(b). This is because\nthe Drude spectra have long tail beyond !>1000cm\u00001."    },    {        "title": "1505.04770v2.Van_Hove_Singularity_as_the_Driver_of_Pseudogap_Physics_in_Cuprate_High__T__Superconductors.pdf",        "content": "Condensation Bottleneck Driven by a Hidden Van Hove Singularity as the Origin of\nPseudogap Physics in Cuprate High- TcSuperconductors\nR.S. Markiewicz, I.G. Buda, P. Mistark, and A. Bansil1\n1Physics Department, Northeastern University, Boston MA 02115, USA\nWe propose a new approach to understand the origin of the pseudogap in the cuprates. The near-\nsimultaneous softening of a large number of different q-bosons yields an ordering bottleneck, wherein\nthe growth of magnetic correlations with decreasing temperature is anomalously slow, leading to\nextended ranges of short-range order. This effect is not tied to the Fermi level, but driven by a\nVan Hove singularity (VHS) nesting that is strongest near a temperature TVHS that scales as the\nenergy separation between the Fermi energy and the energy of the VHS peak. By identifying TVHS\nas the pseudogap onset temperature T∗, we explain many characteristic features of the pseudogap,\nincluding the observed transport anomalies and the termination of the pseudogap phase when EVHS\ncrosses the Fermi level. The condensation bottleneck (CB) provides a new pathway for understanding\nstrong correlation effects in the cuprates. We find that LSCO lies close to an anomalous disorder-free\nspin glass quantum critical point, where the frustration is due to the CB. To study this point we\ndevelop an approach to interpolate between different cuprates.\nPACS numbers:\nI. INTRODUCTION\nEvidence is growing that the ‘pseudogap phase’ found in cuprates is actually home to one or more competing phases,\nincluding a variety of stripe, spin-, or charge-density wave (S/CDW) phases.1-12The CDW phase, in particular,\nhas stimulated considerable interest13-22. Many of these phases, including superconductivity, seem to appear at\ntemperatures well below the pseudogap temperature T∗, so the exact relation between the pseudogap and these other\nphases remains elusive. Indeed, the real puzzle is understanding why the pseudogap phase bears so little resemblance\nto a conventional phase transition. Here we demonstrate that the pseudogap phenomenon arises from a condensation\nbottleneck (CB), where a large number of density waves (DWs) with similar q-vectors attempt to soften and condense\nat the same time, leading to anomalously low transition temperatures and extended ranges of short-range order,\ncharacteristic of the pseudogap phase. We note that many treatments of correlation effects – e.g., path-integral\napproaches – begin by finding a mean-field order parameter at some particular q-vector – a form of random-phase\napproximation (RPA) – and then include the effect of fluctuations. Then if several instabilities compete, the one\nwith lowest free energy wins out. In this case, the CB represents a particularly strong correlation effect, insofar as\none cannot solve the problem one- qat a time, but must account for the q-competition from the start, to account\nfor the entropy associated with the q-manifold. We note that RPA-type approximations extend well beyond simple\nmean-field theories, arising in spin-fermion models and in Hertz-Millis23,24type of quantum critical theories when the\nStoner denominator is approximated by a T-independent Ornstein-Zernicke25form, and in all cases they miss the\nessential and strongly T-dependent mode-coupling physics.\nSimilar effects have arisen in the past. In Overhauser’s theory of CDWs in alkali metals,26theq-vectors for\nall points on a spherical Fermi surface (FS) become unstable simultaneously and cannot be handled one at a time.\nThese effects are often referred to as phonon entropy27, or more generally as boson entropy, and can be analyzed via\nvertex corrections which take proper account of mode coupling28,29. In excitonic theories, the various DW instabilities\nrepresent condensation of some electronic boson at a single q-vector, with triplet excitons corresponding to SDWs and\nsinglet excitons to CDWs30. Summing only bubble and ladder diagrams reproduces the RPA results with BCS-like\nratios of gap to critical temperature, 2∆ /kBTc. Going beyond ladders (e.g., Bethe-Salpeter equation) incorporates\nmode coupling with enhanced ∆ /Tcratios (compare Refs. [31] and [32]; for a review of excitonic insulators, see\nRef. [30]). A particularly interesting analogy is provided by the Bose condensation of excitons. The mean-field\ntransition is found to correspond to the temperature at which excitons are formed. However, since the excitons are\nlocalized in real space, they are greatly spread in q. When fluctuations are included, the real transition lies at a much\nlowerT, when all excitons condense into the lowest qstate. Here we develop a similar theory for the cuprates via a\nself-consistent renormalization calculation of the vertex corrections. We find that the CB not only drives pseudogap\nphysics, but can potentially lead to a novel disorder-free spin-glass phase. Similar effects are likely to be present in\nmany other families of correlated materials.\nWe find that the phase diagram of La 2−xSrxCuO 4(LSCO) is different from that of other cuprates. To explore\nthe transition between them, we simplify the first principles dispersions to equivalent ‘reference family’ dispersions,\ndepending only on the three nearest neighbor hopping parameters, t,t/prime, andt/prime/prime(Supplementary Material Section I).arXiv:1505.04770v2  [cond-mat.supr-con]  28 Mar 20162\nWe find that all cuprates fall along a single cut in t/prime/t−t/prime/prime/t-space, namely t/prime/prime=−t/prime/2, with LSCO characterized by\na smaller value of t/prime.\nII. RESULTS\nA. Bosonic VHS and Origin of the Susceptibility Plateau\nA proper understanding of the CB involves two different issues: firstly, since Fermi surface (FS) nesting typically\nsingles out only a few discrete q-vectors, what causes many q–vectors to soften together? Secondly, a formalism needs\nto be developed to handle the effects of bottleneck. Both of these issues are addressed here. Remarkably, in cuprates\nwe find that the underlying source of strong mode coupling can be traced back to the Lindhard susceptibility χ0.\nIn a typical calculation of classical or quantum phase transitions23,24, material parameters are introduced via χ0,\nwhich is used to define an interacting susceptibility χ. In our calculation, χ0is calculated from density functional\ntheory (DFT) bands corrected by a GW self-energy [Methods Section], and χis the resulting RPA susceptibility. The\nphase transition then corresponds to the vanishing of the denominator of χat frequency ω= 0 for some momentum q.\nTo properly incorporate fluctuation effects, this ‘Stoner denominator’ is then typically reduced to Ornstein-Zernicke\n(OZ)25form,\nχ(q,ω)∼1\nq2+ξ−2+i(ω/ωc)z, (1)\nin terms of various deviations from the critical point (in q,ω, and a ‘tuning parameter’ which is proportional to ξ−2,\nwhereξis the correlation length). Here zis a dynamic exponent and qis measured from the Qwhere the susceptibility\nhas a peak. An important result of our calculation is that in cuprates, this OZ form must be treated carefully, since\nthe coefficients of the deviation parameters can be strong functions of temperature and doping. In particular, we\nfind that the susceptibility inverse curvature (coefficient of q2) can diverge due to a competition between conventional\nFermi-surface nesting and Van Hove singularity (VHS) nesting.\nWhereas many properties of a Landau Fermi liquid are determined by their values at the Fermi level, the suscep-\ntibility is an exception, having both bulk and Fermi surface contributions. While the FS part contributes a ridge\ntoχ0that is a map of the FS at q= 2kF34(wherekFis the Fermi wave vector), in the cuprates there is also an\nimportant bulk contribution, which provides a smoothly varying background, peaking at ( π,π) and giving rise to the\nnear-(π,π)-plateau in the susceptibility. This peak can shift the balance of the FS nesting to q-vectors closer to the\npeaks, and in special cases can lead to commensurate nesting away from the FS nesting vector, at exactly ( π,π).\nMoreover, as Tincreases, coherent FS features are washed out, leaving behind only the commensurate bulk contribu-\ntion. This peak is a bosonic VHS (b-VHS), the finite- qanalog of the Van Hove excitons found in optical spectra35,\nbut present in the intraband susceptibility. However, it is a “hidden” b-VHS. Despite the fact that it is pinned to\nzero energy independent of doping or hopping parameters , it is hidden in the sense that the effective density-of-states\n(DOS) exactly at the b-VHS peak almost always vanishes. Some consequences of this are discussed in Supplementary\nMaterials Section II.\nThe imaginary part of the susceptibility can be thought of as the DOS of electronic bosons, electron-hole (e-h)\npairs, which may become excitons when a Coulomb interaction is turned on. If the renormalized dispersion of a single\nelectron is/epsilon1kwith wave vector k, then an e-h pair at wave vector qhas a dispersion ωq(k) =/epsilon1k+q−/epsilon1k=−2/epsilon1q−(k),\nwhere/epsilon1q±(k) = (/epsilon1k±/epsilon1k+q)/2, and a Pauli blocking factor ∆ fk,q=f(/epsilon1k+q)−f(/epsilon1k). Then the corresponding pair DOS\nisDq(ω) =/summationtext\nk∆fk,qδ(ω−ωq(k)) =χ/prime/prime\n0(q,ω). For LSCO, the dominant pairs are those at q=Q≡(π,π), the AF\nnesting vector. The associated dispersion ωQ(π,π), plotted in Fig. 1(a), resembles the electronic dispersion /epsilon1k, but\nwith an important distinction: it depends only on /epsilon1Q−(k), whereas all of the hopping terms that shift the electronic\nVHS away from half filling ( t/prime,t/prime/prime) are contained in /epsilon1Q+(k), i.e. the b-VHS is pinned at ω= 0. Since χ/prime/prime\n0is an odd\nfunction of ω,χ/prime/prime\n0(Q,ω = 0) = 0. However, excitonic instabilities depend on a Stoner criterion, and hence on\nχ/prime\n0(q,ω= 0) = 2/integraldisplay∞\n0dω/prime\nπχ/prime/prime\n0(q,ω/prime)\nω/prime. (2)\nThe FS contribution thus arises from ω/prime∼0, while bulk contributions arise from peaks in the bosonic DOS, such\nas the b-VHS peak in χ/prime/prime\n0near (π,π). However, while the b-VHS is pinned at ω= 0, Fig. 1(a), its weight vanishes\natT= 0, due to the Pauli blocking factor, ∆ fk,Q= 0 neark= (π,0) atT= 0, Fig. 1(b). Finite Trestores weight,\noptimally near 1000K, although ∆ falways vanishes exactly at ( π,0). In the Supplementary Materials, Section II, we\ndeconvolve the near-( π,π) susceptibility to show that it is a superposition of bulk and Fermi surface features. The3\n1.0\n0.5\n0.0 | ∆f |\n1.0ka(b)\nΓ (π,0) (π,π)1.0\n0.0\n-1.0ωQ (eV)\n3 2 1 0ka(a)\nΓ Γ (π,0) (π,π)\n2000\n1000\n0T (K)\n0.20 0.15 0.10 0.05 0.00\nx0TVHS(π,π−δ)(π,π)\n  5 1(c)\nFIG. 1: Coherent-incoherent crossover at the VHS. (a) Pair dispersion, ωQ, as a function of k, and (b) the corresponding\nweight ∆f, atT= 0K (red), 500K (blue), and 2000K (violet) for LSCO ( x= 0). (c) Phase diagram for LSCO, showing\nVHS-related crossovers and regions where the susceptibility peaks at different q-vectors. Here x0is the doping at T= 0\n(plots are at constant EF), but for qualitative purposes we can assume x/similarequalx0. Dominant fluctuations are at commensurate\n(π,π) (white shaded region) or incommensurate ( π,π−δ) (green shaded region); crossovers are TVHS (red short-dashed line),\nDOS peak (green long-dashed line), coherent-incoherent crossover (pink shaded region), Tγ(violet dot-dashed curve), and the\nposition of the ( π,π) peak vs doping (light blue dot-dashed line) – the approximate electron-hole symmetry point.\nbulk feature is dominated by the b-VHS, while the FS feature smears out with increasing Tin a coherent-incoherent\ncrossover, pink shaded region in Fig. 1(c).\nTo demonstrate the close similarity to VHS effects, we also plot two characteristic features of the VHS. The crossover\nscales with both kBTVHS =EF−EVHS, dark red dashed curve in Fig. 1(c), and with Tγ(violet dot-dashed line), the\ntemperature at which the Sommerfeld constant γ=dS/dT , has a peak. Since the Fermi function evolves smoothly\nwithT, we wxpect sharp Fermi surface features to wash out as Tis increased, but why should this coherent-incoherent\ncrossover move to lower Tas doping is increased? Because a source of entropy, the b-VHS, is moving closer to EF. To\ndemonstrate the connection with entropy, we look at the Sommerfeld constant γ. We calculate γfrom the electronic\ndispersion, assumng a paramagnetic phase, to avoid compications arising from phase transitions. At T= 0,γis\nproportional to the DOS, and hence diverges when the VHS crosses EF. At finite Tthe peak in γrepresents the\nexcess entropy associated with mode coupling. The other curves in Fig. 1(c) will be considered in the Discussion\nSection below.\nB. Strong Mode Coupling Leads to an Extended Range of Short-range Order\nIn conventional mode-coupling calculations33,36-38, the OZ parameters are assumed to be T- and doping-\nindependent, and the resulting physics becomes quite simple. For 2D materials, the Mermin-Wagner (MW) theorem39\nis satisfied, and the mean-field transition at Tmfturns into a pseudogap onset at T∗∼Tmf, with a crossover to long-\nrange order when interlayer coupling is strong enough – in short, not much changes from the mean-field results.\nHowever, when we incorporate realistic susceptibilities we find that this OZ form fails to properly account for the\nstrong mode-coupling effects, leading to dramatically different results. In particular, the entropic effects are encoded\nin a strong T-dependence of the model parameters, which greatly slows down the growth of correlations. This in\nturn leads to an extended T-range of short-range fluctuations, which is typical of pseudogap physics. We find that,\nparticularly at high T, the doping dependence reflects the evolution of the anomalous VH scattering. We note that\nwhile the OZ form is used in quantum critical theory23,24, it is explicitly stated that it is to be used only in a limited\nT-range, and only in the absence of FS nesting24, both of which are violated here.\nMode coupling modifies the bare Coulomb interaction U, producing a vertex-renormalized effective U,Usp= ΓU\nwith Γ = 1/(1 +λ), whileλis found self-consistently from [see Methods section below]\nλ= ΓTA 0/integraldisplay∞\nUcdX−N−(X−)\nX−−Usp. (3)\nwhereA0= 12u/χ 0(Q0,0),χ0(Q0,0) =maxq[χ/prime\n0(q,0)],uis the mode-coupling parameter, and Uc= 1/χ0(Q0,0).\nHere,χ−1\n0, denoted by X−, is the variable of integration and we introduce a corresponding susceptibility density of\nstates (SDOS) N−. In Section III of Supplementary Materials, we show that Eq. 3 is closely related to excitonic Bose\ncondensation, with λproportional to the effective number of bosons.4\n10-1100101102103104105106ξ/a\n2345678\n1002345678\n10002\nT (K)(b)\nIII\nI II4\n3\n2\n1\n0N-\n2.5 2.0 1.5 1.0X- (eV)T= 5K50K250K500K2000K(a)3\n2\n1\n00.94 0.93 0.92\n1 100 ξ/a\n5x10-34321\n1/T (1/K)X2 (c)\nFIG. 2: Structures in LSCO correlation length. (a) SDOS for undoped LSCO at several temperatures; light-blue dotted\nline gives OZ form of SDOS. (b) Corresponding temperature dependence of correlation length ξth. Forx= 0, the red solid line\nwith filled dots is ξthcalculated from the solid lines in (a), while the thin green dot-dashed line is based on the dashed lines in\n(a), showing that ξthis relatively insensitive to the structures away from threshold. In contrast, the blue dashed line is based\non the shape of the SDOS at T= 2000K, but shifted and renormalized to match the SDOS at lower T, illustrating sensitivity\nto the leading edge structure. The black dotted line illustrates the scaling ξth∝T−1/2. (c) Calculated ξthreplotted for x= 0\n(red solid line with filled circles) compared with ξ(blue filled circles) and with experiment (green dot-dot-dashed line)40; the\nblue dotted line is twice ξth.\nFigure 2 illustrates the profound effects that strong mode coupling has in LSCO, as well as the complete inability\nof the OZ approximation to capture this physics. The SDOS, Fig. 2(a), contains VHS-like features characteristic of\nconventional DOSs. However, the singular behavior of Eq. 3 involves only features near threshold, Uc=min(X−),\nwhich evolve strongly with T, see inset to Fig. 2(a). For T > 0, the threshold behavior is always a step at Uc,\nindicative of a parabolic peak in χ0with curvature inversely proportional to the step height. Thus for a qualitative\nunderstanding of the evolution of ξ(Eq. 1) with Twe can assume an OZ form of χ, but with a strongly T-dependent\nstep height A2(T). This allows a threshold correlation length ξth(red solid lines in Figs. 2(b) and (c)) to be defined\nfrom Eq. 1 as the inverse half-width in q, assuming that the curvature ( ∝A−1\n2) isq-independent. In reality, the\ncurvature increases with q, and the correlation length ξobtained from the renormalized susceptibility half-width is\ntypically a factor of 2 larger. Fig. 2(c) compares the x= 0 values of ξth(solid red line) and ξ(filled blue circles)\nvs 1/T; the values of ξlie on the blue dotted curve, which represents twice ξth, and are in good agreement with\nexperiment (green dot-dot-dashed line)40.\nFigure 2(b) illustrates how strong mode coupling slows down the correlation length divergence. Undoped LSCO\n(red solid line) shows two regions I and III of exponential growth of ξthwith decreasing T, separated by an anomalous\nregion II where ξthactually decreases with decreasing T. We will not discuss the MW-like divergence at low T(region\nIII). In the high- Tlimit (region I), the leading-edge parabolic curvature is quite small, and if it were T-independent, as\nin the OZ approximation, the growth in ξthwould follow the blue dashed line (Eq. 11 below), but thermal broadening\ncauses the curvature to decrease with increasing T, leading to the faster growth of the red solid line. The anomalous5\nFIG. 3: t’ dependence of correlation bottleneck. (a)ξthfor the minimal reference family at several values of t/prime/t[see\nlegend] and t/prime/prime= 0. (b-d): SDOS at three values of t/prime/t= -0.21 (b) [as in Fig. 2(b)], -0.27 (c), and -0.35 (d). All curves are\nshifted to line up the ( π,π) data at ∆ X−=X−−X−(π,π) = 0. (e)ξthfor the PA reference family at several values of t/prime/t\n[see legend] and t/prime/prime=−t/prime/2.\nbehavior in region II will be discussed next.\nC. Exploring parameter space\nTo understand the origin of the CB in region II, it is necessary to explore hopping parameter space away from the\nphysical cuprates. To do this, we introduce the notion of reference states: states with simplified hopping parameters\n(onlyt,t/prime, andt/prime/primenonzero) but which match the phase diagram of the real cuprates in a well-defined way [Supplemen-\ntary Material Section I]. This allows us to tune the system between LSCO and Bi2212, and explore phase space beyond\nthese limits. We study two important cuts in t/prime/t−t/prime/prime/tspace, a minimal cut ( t/prime/prime= 0) and the Pavarini-Andersen\n[PA] cut (t/prime/prime=−t/prime/2) – the latter seems to best capture the physics of the cuprates. By tuning t/primewe unveil the origin\nof the CB as a localization-delocalization crossover tied to the crossover from ( π,π)- to FS-nesting. As a byproduct,\nwe gain insight into why LSCO is so different from other cuprates, and how cuprates evolve from the pure Hubbard\nlimit (t/prime= 0).\nFigure 3(a) shows the T-evolution of ξthfor several values of t/primealong the minimal cut, including the data of Fig. 2(b).\nFort/prime/t >−0.17, the system is characterized by commensurate (C) ( π,π) order with a finite Neel temperature\nTN∼1000K [the correlation length grows so rapidly that interlayer correlations will drive a transition to full 3D\norder]. Similarly, for t/prime/t<−0.345 there is incommensurate (I) ( π,π−δ) order with TNabout a factor of 10 smaller.\nBut for intermediate t/prime/tthe C-I transition is highly anomalous, with correlation length orders of magnitude smaller\nthan expected. Figure 3(b) shows that a similar evolution follows along the PA cut in parameter space. The reason\nfor this anomalous behavior can be seen by looking at the leading edge SDOS in the crossover regime, shown for three\nvalues oft/primealong the minimal cut in Figs. 3(c)-(e). For ease in viewing, these curves have been shifted to line up\nthe SDOS at ( π,π) at allT. It is seen that the anomalous collapse of ξthis associated with a rapid growth of the\nstep height A2, culminating in a near-divergence at t/prime\nc=−0.27t, where the leading edge curvature goes to zero. This\ndivergence coincides with the C-I crossover of the leading edge SDOS [Supplementary Materials Section I]. Here many\ndifferentq-vectors compete simultaneously, frustrating the divergence of any particular mode. This is the electronic\nanalog of McMillan’s phonon entropy: if many phonons are simultaneously excited, the transition is suppressed to\nanomalously low temperatures. Note that ξthdrops by 9 orders of magnitude at T= 200Kwhent/prime/tchanges from\n-0.17 to -0.27, then grows by a similar amount at 100K when t/prime/tchanges from -0.27 to -0.345. [Over this same\nrange, the true ξwill be frozen at the value corresponding to the half-width of the ( π,π) plateau.] The green shaded\nregion in Fig. 3(c) shows that the range of the anomalous growth II of ξin Fig. 3(a) coincides with the range of rapid6\ngrowth of the SDOS leading edge; such behavior is absent if a T-independent OZ form (black dashed line) is assumed.\nThe divergence of the SDOS at T= 0 signals that the system at t/prime=−0.27tis highly anomalous, with an infinitely\ndegenerate ground state . This represents an anomalous form of spin glass arising in the absence of disorder, with the\nfrustration arising from strong mode coupling. Note also that when the C-I transition is at T > 300K,ξ(T) has a\nsharp downward cusp at the transition, while above the transition ξ<a is strongly suppressed.\nFrom the relationship between ξthand the step height A2, Eq. 11 below, we see that a diverging A2will cause\nξth→0. The divergence arises when the susceptibility at ( π,π) crosses over from a maximum to a local minimum.\nIn the latter case, the maximum intensity of χ(q,0) is spread along a ‘ring’ in q-space surrounding ( π,π), so that\nthe inverse susceptibility resembles a ‘Mexican hat’. Approximating the χ−1-dispersion by a Mexican hat form\nχ−1\n0−A2q2+A4q4leads to a threshold divergence N−∼(χ−1−Uc)−1/2, Fig. 2(f). Note that the SDOS-divergence\nresembles a 1D VHS in the conventional DOS, even though here the 1-D direction is the radial direction away from\n(π,π). The divergence of χalong a ring can be thought of as a 2D analog of Overhauser’s effect26. Similar effects\nare found in Bi2201 [Supplementary Materials Section IV]. Finally, for t/primenear but greater than t/prime\nc, the divergence is\navoided, but proximity to the C-I transition leads to the rapid growth of a narrow peak in the leading-edge SDOS,\nFig. 2(e). This in turn can cause an approximate power-law growth of ξth∝T−1/2(black dotted line in Fig. 2(b)),\nas calculated in the Methods section below, although the sharp turn-on of structure in the SDOS actually leads to an\nanomalous decrease in ξthwith decreasing T.\nIII. DISCUSSION\nA. Cuprate Pseudogap\nFigure 2(g) captures an essential aspect of pseudogap physics: an extended regime of phase space where correlations\nremain only short ranged. The reason that the C-I transition is so anomalous is that it is also an incoherent-coherent\ntransition, with the bare susceptibility for small |t/prime|dominated, not by FS nesting, but by a broad peak at ( π,π)\nassociated with a b-VHS, crossing over to conventional (coherent) FS nesting only for t/prime< t/prime\ncor for larger x. Mode\ncoupling associated with self-consistent vertex corrections leads to a regime where the local and extended behaviors\nbecome strongly entangled, leading to a collapse of the magnetic correlation length and very slow growth of magnetic\nfluctuations.\nWhile the CB was explained by tuning parameter space, the results bear a striking resemblance to the temperature\nand doping evolution of the pseudogap phase. Experimentally, both LSCO and YBCO display C ( π,π) long-range\nAFM order at x= 0, withTNdropping rapidly with doping below x= 0.03, crossing over into a regime of I ( π,π−δ)\nfluctuations and low- Tspin glass behavior. Given the proximity of LSCO to the C-I transition, finding a quantitative\nmodel of its doping phase diagram may prove difficult. We can however note one plausible scenario. It has been\npredicted38that a large Hubbard Uwill renormalize t/primeto smaller values at half filling, although the magnitude of the\neffect is debated. If doping screens U, and restores a larger |t/prime|, it could drive LSCO across the anomalous regime II,\ncausingTNto drop by an order of magnitude. Moreover, Fig. 2(g) shows that the C-I transition involves a highly\ndisordered regime separating two well-ordered phases. Since ordering tends to lower the free energy of the electronic\nsystem, the disordered regime may represent a state of high free energy, and doping across this region can lead to a\nregime of [nanoscale] phase separation (NPS), which in LSCO is manifest as the stripe phase. Hence, a model of the\nC-I transition, incorporating effects of NPS and disorder, could provide a good description of LSCO.\nThis broad T- and doping-regime of only short-range SDW order is a key characteristic of pseudogap physics.\nIndeed,TVHS is typically close to the measured pseudogap temperature T∗in most cuprates. Identifying T∗with\nTVHS can explain a number of puzzling features. In particular, it has been found that the pseudogaps in Bi2201 and\nBi2212 terminate when TVHS→0, i.e., at the conventional VHS,43-47similar to the crossovers seen in Fig. 1(c), as\nwell as in Fig. 1 of the Supplementary Materials.\nThe presence of an excitonic VHS contribution to the incoherent susceptibility has a number of further consequences\nfor cuprate physics. First, the VHS onset near TVHS combined with the C-I crossover at Tcoh∼TVHS/3 can explain\nthe anomalous transport properties found near the pseudogap. Thus, the pink shaded region in Fig. 1(c) shows the\ncrossover from incoherent susceptibility dominated by the VHS at high- Tto coherent, FS-dominated susceptibility\nat low-T. This parallels transport, where for T > T∗, the resistivity ρvaries linearly with T,9behavior expected\nnear a VHS46. For lower T, the resistivity is mixed, but ρ∼T2, as expected for a coherent Fermi liquid, is found\nbelow aTcoh< TVHS. This picture bears a resemblance to the Barzykin-Pines model of the cuprate pseudogap,47\nidentifying TVHS andTVHS/3 withT∗andT∗/3 in their model. The underlying physics of the incoherent-to-coherent\ncrossover in their model is related to Kondo lattice physics48, with the VHS peak standing in for the Kondo resonance\n(see Supplementary Materials Section V). This raises the question of whether a similar mode-coupling calculation in\nheavy-fermion compounds could lead to a similar anomalous entanglement at the f-electron C-I transition.7\nThe excitonic instability should be maximal when the susceptibility is approximately electron-hole (e-h) symmetrical\nin doping, and at T= 0 this happens when the VHS is at the Fermi level.34At finiteT, the e-h symmetrical point\ncontinues to coincide with a peak in the ( π,π) susceptibility at doping xp, butxpshows a remarkably rapid evolution\nwithTtowards half-filling (light-blue dot-dashed line in Fig. 1(c)), nearly coinciding with the coherent-incoherent\ncrossover. We interpret this temperature dependence as follows: The parameters t/primeandt/prime/primeare relevant perturbations\nshifting the VHS from the pure Hubbard model value at x= 0 where t/prime=t/prime/prime= 0. When kBT >|t/prime|, these\nperturbations become irrelevant, so that the effective VHS becomes electron-hole symmetric at x= 0. Lastly, the\nlower branch of the ( π,π)−(π,π−δ) commensurate-incommensurate transition (green shaded region in Fig. 1(c)) also\nscales, following TVHS very closely in this simple ( t−t/primeonly) model. Indeed, at T= 0, this transition corresponds\nto the Fermi energy falling off of the ( π,π) plateau, providing another indication that the effective VHS is shifting\ntowards half-filling as Tincreases. In Supplemental Material Section II we show that this is another consequence of\nPauli blocking.\nThe rapid thermal evolution of the ( π,π)-VHS peak should be contrasted with the much smaller change in conven-\ntional VHS effects found near Γ. In particular, the DOS peak is dominated by near-FS physics, and hence displays a\nmuch weaker doping dependence, green long-dashed line in Fig. 1(c).\nAn additional consequence is that in cuprates with small t/primehopping, such as LSCO, the anomalous VHS suscep-\ntibility can dominate even at T= 0, leading to strong deviations from FS-nesting. Thus, in most of Fig. 1(c), the\npeak susceptibility is associated with VHS nesting at q= (π,π), and only in an intermediate T-regime is FS nest-\ning is found at ( π,π−δ) (green shaded region). The resulting VHS-FS nesting competition plays a strong role in\nunderdoped LSCO, associated with retrograde correlation length change with T, region II of Fig. 2(b), which is a\nsignal of proximity to a novel disorder-free spin-glass QCP. Notably, in LSCO commensurate ( π,π) order disappears\nrapidly by∼2% doping, being replaced by incommensurate magnetic fluctuations and low- Tspin-glass effects, while\nneutron scattering has found that doped LSCO is close to a magnetic QCP.49The very different situation in most\nother cuprates is discussed in Sections I, IV of Supplementary Materials.\nB. Strong coupling physics\nWe recall that our self-energy formalism50is able to reproduce most spectral features of the insulating cuprates\nin terms of a ( π,π) ordered phase. We reproduce not only the photoemission dispersions, limited to the lower\nHubbard bands, but optical and x-ray spectra which depend sensitively on the Mott gap. In a related 3-band model,\nwe reproduced the Zhang-Rice result that the first doped holes are predominantly oxygen character, and our overall\ndispersions at half filling agree with [subsequent] DMFT results at least as well as DMFT results from different groups\nagree. [See further Supplementary Materials VI.] The problem with our earlier calculations is that they predict long-\nrange (π,π) AFM order at too high T. The new self-consistent renormalization calculations have only short range\norder, in which case the upper and lower Hubbard band dispersion is reproduced, with a broadening ∼1/ξ.33\nAn antiferromagnet is considered to be weakly coupled if the Neel temperature TNincreases with U, and strongly\ncoupled ifTNdecreases with increasing U[TN∼J= 4t2/U]. Since this is a finite Tcriterion, it is sensitive to the\nboson entropy effects we have been discussing. Indeed, for large Uthe system is nearly localized and the physics\nis again reminiscent of Bose condensation. The simultaneous softening of many magnetic modes can be readily\ndemonstrated from a simple Hartree-Fock model of an antiferromagnet. For arbitrary q, the resulting gap for large U\nis\n∆E=/radicalBig\nU2+/epsilon12\n−/similarequalU+J˜/epsilon12\n−, (4)\nwhere/epsilon1−= (/epsilon1k−/epsilon1k+q)/2 =−2t˜/epsilon1−. Thus, the difference in energy between any two spin configurations is a quantity\nof orderJ, so that when T∼J, the system gains entropy by mixing different q-states, and long-range AF order is\ndestroyed.33While this effect can be recognized in HF, only a theory that properly accounts for the mode competition\ncan resolve it. The present mode coupling model properly accounts for this effect via the vertex renormalization\nfactorλ, which is an integral over the RPA χ. This can be seen from Fig. 2(c), where the measured ξin LSCO\n(green dot-dot-dashed line)40is compared to our calculation (filled blue dots). The experimental data are shown only\nabove 300K, since at lower Tinterlayer coupling drives a transition to long-range order. In the Heisenberg model,\nξis a function only of T/J, and these data have been used to measure the exchange J. Hence, our mode coupling\ncalculation successfully reproduces this strong coupling result, as was found earlier for electron-doped cuprates33. The\nabove results suggest that the C-I, incoherent-coherent transition is also a localization-delocalization transition, with\nthe mode coupling associated with small effective hopping.\nFor more insight into the strong coupling limit, we note that in the two-particle self-consistent approach51,λis8\ndetermined by a sum rule involving double occupancy, which is fixed by assuming\nUsp/U=<n↑n↓>/(n/2)2,\nwhich leads to a saturation of UspasU→∞ (or<n↑n↓>∼1/U). In our calculation, this saturation arises naturally,\nsinceUspcan never exceed Uc.\nQuantum critical points (QCPs) in strongly correlated materials are often discussed in terms of deconfined QCPs\n(DQCPs), involving a non-Landau transition between two types of competing order, where a new form of excitation\nemerges exactly at the DQCP. This has been refined for cuprates into an underlying competition between Mott physics\nand Fermi liquid physics, masked by a low-energy order parameter of the FL52. This is an apt description of the\ncurrent results, with the Mott physics evolving into a low- TAFM and the FL to a spin-density wave (SDW), and a\nspin glass at the DQPT. This confirms the finding from dynamical cluster approximation calculations of the strong\nrole of the VHS in Mott physics.53\nC. Conclusions\nIn conclusion, we find that the pseudogap is driven by tendencies toward magnetic order at or near q= (π,π).\nStrong mode coupling effects drive the characteristic anomalies of the pseudogap, in particular the existence of a broad\ndoping- and temperature-range of only short- to intermediate-range order. Finally, the underlying cause of the strong\nmode coupling is localization in particular the fact that the extended-to-localized transition is not a simple crossover\nbut a competition between FS-dominated and non-FS dominated (VHS-dominated) physics. This competition across\na manifold of q-states leads to a condensation bottlneck, which requires a new formalism for dealing with multi- q\nmode softening. The close similarity to heavy Fermion physics should be noted.\nThis can be restated slightly differently. The original ( t-only) Hubbard model is particularly difficult to solve,\nas three separate instabilities are simultaneously present at half filling: the Mott instability, FS nesting, and VHS\nnesting. A finite t/primeshifts the latter two to finite doping. For large |t/prime|the three instabilities become well separated,\nbut for small|t/prime|the two nesting instabilities overlap and compete, leading to frustration and pseudogap physics.\nSignificantly, we approach the problem from the intermediate-coupling side, suggesting that the full crossover from\nhalf-filling to large doping could be explored.\nIn turn, this sheds light on strong correlation effects in the cuprates, in particular Mott vs Slater physics. The\nlocalization associated with Mott physics has several manifestations. One is that coherent hopping is restricted to\nshorter range. That is consistent with stronger effects found at smaller t/prime. Particularly for Mott physics, a second\naspect is the breakdown of hybridization. The Cu-O hybridization in the cuprates spoils Mott physics [half filling does\nnot imply one electron on each Cu]. Zhang-Rice singlet formation can be considered as a breakdown of hybridization\n[pure oxygen states at the top of the lower magnetic band, pure Cu at the bottom of the upper magnetic band],\nand is found more generally for ( π,π) AFM fluctuations near half filling in a three-band model50. Hence the strong\nevolution with t/primefrom localized to extended physics, with only LSCO close to Mott physics.\nRice et al.54recently noted: “The need for theoretical methods to handle ... short range correlations ... is a key\nchallenge for the future.” Along this line, the present approach appears to explain a number of anomalous features\nassociated with the pseudogap, including a coherent crossover associated with a competition between two DW orders\n(see Supplementary Materials Section IV). In fact, we have incorporated an important ingredient for strong-coupling\ncalculations: a model of short-range AF order from which a t−Jmodel can be derived.\nIn Anderson’s RVB picture, he envisaged a regime of Mott physics where the FS played a neglible role; instead,\nmost experimental studies find clear evidence for a well-defined FS, with competing phases associated with FS nesting,\nand a QCP associated with FS reconstruction. Our results have a strong bearing on these competing scenarios: in\nparameter space ,t/primeis a relevant parameter tuning the system away from quasilocalized physics near t/prime= 0, where the\nFS has a negligible role, into a delocalized regime dominated by FS physics. The crossover is marked by a regime of\nstrong disorder, where the correlation length remains small down to very low T. Most families of cuprates lie on the\ndelocalized side of this crossover, and hence are most easily understood from a FL-type picture. LSCO appears to\nbe on the localized side of the crossover, but close to it, consistent with experiment.49We note that the disorder line\nmay be hard to directly access: since ordering transitions tend to lower the free energy, the line will be a high-energy\nstate with low energy states to either side of it, suggesting an instability to nanoscale phase separation.\nThe VHS has been predicted to play a significant role in many materials, particularly in lower dimensional systems\nwhereχdiverges, but clear evidence for this remains sparse. Thus, VH nesting was introduced as a possible cause\nof CDWs in dichalcogenides55, but this interpretation remains disputed. Even in VO 2, the striking metal-insulator\ntransition has been found to be driven by large phonon entropy.56Clearly, one problem is that the VHS has been\nassumed to play a role only when it is near the Fermi level, whereas Fig. 1(c) demonstrates that its influence extends9\nover a much wider doping range. For those who think condensed matter physics lies on the surface of the Fermi sea,\nthe VHS is the iceberg, lurking.\nIt will be interesting to examine how the present results are modified by effects of disorder and interlayer coupling.\nAn important issue is the extent to which the total number of QPs is conserved. More specifically, what is the relation\nbetween the number of incoherent electrons (1 −Z)Nand the effective number of excitons Neff? Proximity to a\nspin glass phase could explain the failure of recent attempts to create an ‘artificial cuprate from related nickelate\ncompounds, and may suggest a path to improved analog states.\nIV. METHODS\nOur calculation is a form of many-body perturbation theory (MBPT) based on Hedin’s scheme. The scheme\ninvolves four elements: electrons are described by Green’s functions Gwith DFT-based dispersions renormalized by a\nself-energy Σ; electronic bosons [electron-hole pairs] are described by a spectral weight [susceptibility] χrenormalized\nby vertex corrections Γ. Neglecting vertex corrections, the self energy can be calculated as a convolution of Gand\nW=U2χ, the GW approximation. This approach has been used to solve the energy gap problem in semiconductors,\nwhere the Γ correction leads to excitons via the solution of a Bethe-Salpeter equation, and in extending DMFT\ncalculations to incorporate more correlations [e.g., DMFT+GW, etc.]. Our approach here is to extend our previous\nGW calculations [quasiparticle-GW or QPGW50] to include vertex corrections.\nIn QPGW we introduce an auxiliary function GZ=Z/(ω−/epsilon1QP\nk), where the dressed, or QP dispersion is /epsilon1QP\nk=\nZ/epsilon1DFT\nk, and/epsilon1DFT\nk is the bare, or DFT dispersion. GZbehaves like the Green’s function of a Landau-type QP –\na free electron with renormalized parameters that describes the low-energy dressed electronic excitations. However,\nthis is a non-Fermi liquid type QP, since the frequency-integral of Im(GZ) isZand not unity. That is, the Z-QP\ndescribes only the coherent part of the electronic dispersion, and is not in a 1:1 correspondence with the original\nelectrons. The importance of such a correction can be readily demonstrated. Since a Z-QP has only the weight Z\nof a regular electron, the susceptibility [a convolution of two Gs] is weaker by a factor of Z2than an ordinary bare\nsusceptibility. To match this effect in the Stoner criterion requires introducing an effective Ueff=ZU. In contrast,\nMBPT calculations in semiconductors typically set the GW-corrected Green’s function to G−1\nGW=ω−/epsilon1QP\nk, where/epsilon1QP\nk\nis the average GW-renormalized dispersion57, thereby missing the reduced spectral weight of the low-energy, coherent\npart of the band.\nWe work in a purely magnetic sector, with a single Hubbard Ucontrolling all fluctuations; there is a competition\nin this model between near-nodal (NNN) and antinodal nesting (ANN) which mimics the SDW-CDW competition in\ncuprates, sharing the same nesting vectors22,34.\nThe self-consistent parameter λis found from a Matsubara sum of the susceptibility\nλ=A0T\nN/summationdisplay\nq,iωnχ(q,iωn), (5)\nwhereNis the number of q-points,\nχ(q,iωn) =χ0(q,iωn)\n1 +λ−Uχ0(q,iωn), (6)\nand the summation in Eq. 5 can be transformed:\nT\nN/summationdisplay\nq,iωnχ(q,iωn) =/integraldisplayd2qa2\n4π2/integraldisplay∞\n0dω\nπcoth(ω\n2T)χ/prime/prime(q,ω+iδ)\n/similarequalˆλ1+ˆλ2T, (7)\nwithathe in-plane lattice constant,\nˆλ1=/integraldisplayd2qa2\n4π2/integraldisplay∞\n0dω\nπχ/prime/prime(q,ω), (8)\nˆλ2=/integraldisplayd2qa2\n4π2χ/prime(q,0). (9)\nThe term ˆλ1introduces a small, nonsingular correction33to Eq. 5, which we neglect.10\nFIG. 4: Origin of SDOS features. (a) Inverse susceptibility χ−1\n0(q) for Bi2201 at doping x= 0. Note that Uc=min(χ/prime−1\n0)\n(circled region). Curves are at T= 10 (red) and 100K (blue). (b) Corresponding susceptibility density of states (SDOS)\nN−(X−), plotted horizontally, corresponding to the susceptibility of (a). (c,d) Blowups of circled regions in (a,b).\nIn order to explore the role of susceptibility plateaus for realistic band dispersions, the self-consistency equation is\nevaluated numerically. For this purpose, we note that since χ−1\n0has dimensions of energy, a plot of χ−1\n0(q,0) resembles\na dispersion map. Hence we can define a SDOS:\n/integraldisplayd2qa2\n4π2=/integraldisplay\nN−(X−)dX−. (10)\nFigure 4 illustrates how integrating over the inverse susceptibility, Fig. 4(a), leads to the SDOS, Fig. 4(b), at x=\n0 where the NNN plateau is dominant. Here we use hopping parameters appropriate to the DFT dispersion of\nBi2Sr2CuO 6+x(Bi2201),t= 419.5,t/prime=−108.2 andt/prime/prime= 54.1 eV (Supplementary Materials Section IV). This yields\na doping phase diagram that is qualitatively similar to that of most cuprates, except LSCO. By comparing Figs 4(a)\nand 4(b), one can see how features in χ−1\n0translate into features in N−. Thus, the intense, flat-topped peak in N−at\nsmall values of χ−1\n0represents the NNN plateau. Its broad leading edge (smaller χ−1\n0) is controlled by anisotropy of\nthe plateau edge between ( π,π−δ) and (π−δ,π−δ), while the sharp trailing edge corresponds to the local maximum\nofχ−1\n0at (π,π). For the ANN peak, its leading edge scarcely leaves any feature in N−, but a local maximum translates\ninto a large peak in N−. Finite temperature, T= 100K(blue line) rounds off the cusp in χ−1\n0, Fig. 4(c), leading to a\nstep inN−, Fig. 4(d), but otherwise has little effect.\nOnceN−has been calculated, Eq. 3 can be evaluated numerically. The singular part of the integral is treated\nanalytically, and the remainder numerically, with uandUapproximated as constants, u= 0.8eV−1, whileU= 2 eV\nunless otherwise noted. It is convenient to fix the SDOS at some temperature T/prime, and then solve Eq. 3 for T(ξth,T/prime),\nwith self-consistency requiring T(ξth,T/prime) =T/prime.\nWhen the OZ form of χis assumed, N−becomes a constant, which we denote Na, and Eq. 3 leads to long range\norder atT= 0 only, with correlation length ξth/a= 1/√4πNaδgiven by\nξthqc=eT2/T, (11)\nwhereδ=Uc−Usp,Uc= 1/χ0(Q0,0),T2=πA2λ/6Γua2,Aiis the coefficient of qi, andqcis a wave number cutoff.\nThe origin of the anomalous region II for undoped LSCO, Fig. 2(b), can be readily understood from Fig. 2(a),\nwhere forT < 1500Kthere is excess SDOS weight near Uc, leading to a strong peak (inset) as T→0. This feature\nrepresents the development of the ( π,π)-plateau. For the small- t/primematerials, the susceptibility on the plateau remains\nparabolic, but with a small curvature and a sharp cutoff. This leads to an additional contribution to the SDOS of the\nformN−=NpifX−≤Xp. Then the integral of Eq. 3 becomes\nI= (Na−Nbδ)ln(Xc\nδ+ 1) +NbXc+Npln(Xp\nδ+ 1). (12)\nwhereXcis the maximum of X−. The last term, which we denote Ip, has two distinct limits. If Xp<< δ , then\nIp→NpXp/δ. This is equivalent to assuming that the plateau has a flat top, in which case N−can be represented as11\naδ-function,N−=NpXpδ(X−−Uc), withNpXpthe excess height of Nat threshold, proportional to the area of the\nplateau atT= 0. AsTdecreases,δalso decreases, and the opposite limit for Ipbecomes appropriate, δ <<Xp, in\nwhich case Ip=Npln(Xp/δ). From Eq. 3, Ip∼1/δtranslates into ξth∼1/T1/2, dotted line in Fig. 2(b). The actual\nanomaly roughly follows this line, but the correlation length growth is actually reversed.\nThe present GZ−WZ−ΓZmodel is the simplest which captures the essential physics of the pseudogap. For a\nmore quantitative comparison with experiment, two additional problems must be solved. First, an extension to fully\nself-consistent G−W−Γ may be needed to capture the splitting of the saddle VHS peak into the VHS exciton and\nthe residual continuum part. Secondly, the term in Eq. 1 proportional to ωzmust be included to describe quantum\nfluctuations. Since we are here primarily interested in the opening of the pseudogap at higher temperatures, we\nignored it in our analysis. Finally, it will be interesting to explore if excitonic superfluidity exists, and how it is\nrelated to high- Tcsuperconductivity.\nV. REFERENCES\n1. Kivelson, S.A. Bindloss, I.P. Fradkin, E. Oganesyan, V. Tranquada, J.M. Kapitulnik, A. & Howald, C. How to\ndetect fluctuating order in the high-temperature superconductors. Rev. Mod. Phys. 75,1201-1241 (2003).\n2. Vojta, M. Lattice symmetry breaking in cuprate superconductors: stripes, nematics, and superconductivity.\nAdv. Phys. 58,699-820 (2009).\n3. Wu, T. et al. Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa 2Cu3Oy.\nNature 477, 191-194 (2011).\n4. Ghiringhelli, G. et al. Long-range incommensurate charge fluctuations in (Y,Nd)Ba 2Cu3O6+x.Science 337,\n821-825 (2012).\n5. Achkar, A.J. et al. Distinct charge orders in the planes and chains of ortho-III-ordered YBa 2Cu3O6+δsuper-\nconductors identified by resonant elastic x-ray scattering. Phys. Rev. Lett. 109, 167001 (2012).\n6. Chang, J. et al. Direct observation of competition between superconductivity and charge density wave order in\nYBa 2Cu3O6.67.Nature Phys. 8,871-876 (2012).\n7. LeBoeuf, D. et al. Thermodynamic phase diagram of static charge order in underdoped YBa 2Cu3Oy.Nature\nPhys. 9,79-83 (2013).\n8. Blackburn, E. et al. X-say diffraction observations of a charge-density-wave order in superconducting ortho-II\nYBa 2Cu3O6.54single crystals in zero magnetic field. Phys. Rev. Lett. 110, 137004 (2013).\n9. Doiron-Leyraud, N. et al. Hall, Seebeck, and Nernst coefficients of underdoped HgBa 2CuO 4+δ: Fermi-surface\nreconstruction in an archetypal cuprate superconductor. Phys. Rev. X 3,021019 (2013).\n10. Comin, R. et al. Charge order driven by Fermi-arc instability in Bi 2Sr2xLaxCuO 6+δ.Science 343, 390-392\n(2014).\n11. da Silva Neto, E.H. Ubiquitous interplay between charge ordering and high-temperature superconductivity in\ncuprates. Science 343, 393-396 (2014).\n12. Fujita, K. et al. Simultaneous transitions in cuprate momentum-space topology and electronic symmetry break-\ning. Science 344, 612-616 (2014).\n13. Metlitski, M.A. & Sachdev, S. Quantum phase transitions of metals in two spatial dimensions. II. Spin density\nwave order. Phys. Rev. B 82,075128 (2010).\n14. Wang, Y. & Chubukov, A.V. Charge-density-wave order with momentum (2Q,0) and (0,2Q) within the spin-\nfermion model: continuous and discrete symmetry breaking, preemptive composite order, and relation to pseu-\ndogap in hole-doped cuprates. Phys. Rev. B 90,035149 (2014).\n15. Efetov, K.B. Meier, H. & P´ epin, C. Pseudogap state near a quantum critical point. Nature Phys. 9,442-446\n(2013).\n16. Meier, H. P´ epin, C. Einenkel, M. & Efetov, K.B. Cascade of phase transitions in the vicinity of a quantum\ncritical point. Phys. Rev. B 89,195115 (2014).12\n17. La Placa, R. & Sachdev, S. Bond order in two-dimensional metals with antiferromagnetic exchange interactions.\nPhys. Rev. Lett. 111, 027202 (2013).\n18. Hayward, L.E. Hawthorn, D.G. Melko, R.G. & Sachdev, S. Angular fluctuations of a multicomponent order\ndescribe the pseudogap of YBa 2Cu3O6+x.Science 343, 1336-1339 (2014).\n19. Bulut, S. Atkinson, W.A. & Kampf, A.P. Spatially modulated electronic nematicity in the three-band model of\ncuprate superconductors. Phys. Rev. B 88,155132 (2013).\n20. Allais, A. Bauer, J. & Sachdev, S. Density wave instabilities in a correlated two-dimensional metal. Phys. Rev.\nB90,155114 (2014).\n21. Fujita, K. et al. Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates.\nProc. Nat. Acad. Sci. of the USA 111, E3026-E3032 (2014).\n22. Markiewicz, R.S. Lorenzana, J. Seibold, G. & Bansil, A. Gutzwiller charge phase diagram of cuprates, including\nelectron-phonon coupling effects. New Journal of Physics 17,023074 (2015).\n23. Hertz, J.A. Quantum critical phenomena. Phys. Rev. B 14,1165-1184 (1976).\n24. Millis, A.J. Effect of a nonzero temperature on quantum critical points in itinerant fermion systems. Phys. Rev.\nB48,7183-7196 (1993).\n25. Ornstein, L. S. and Zernike, F. Accidental deviations of density and opalescence at the critical point of a single\nsubstance. Proc. Acad. Sci. Amsterdam 17,793-806 (1914).\n26. Overhauser, A.W. Exchange and correlation instabilities of simple metals. Phys. Rev. 167, 691-698 (1968).\n27. McMillan, W.L. Microscopic model of charge-density waves in 2HTaSe 2.Phys. Rev. B 16,643-650 (1977).\n28. Motizuki, K. & Suzuki, N. Structural Phase Transitions in Layered Transition-Metal Compounds (Reidel, Dor-\ndrecht, 1986).\n29. Yoshiyama, H. Takaoka, Y. Suzuki, N. & Motizuki, K. Effects on lattice fluctuations on the charge-density-wave\ntransition in transition-metal dichalcogenides. J. Phys. C 19,5591-5606 (1986).\n30. Halperin, B.I. & Rice, T.M. The excitonic state at the semiconductor-semimetal transition. in Solid State\nPhysics , Vol. 21, ed. Seitz, F. Turnbull, D. & Ehrenreich H. (New York, Academic) pp. 115-192.\n31. Bronold, F.X., & Fehske, H. Possibility of an excitonic insulator at the semiconductor-semimetal transition.\nPhys. Rev. B 74,165107 (2006).\n32. Cˆ ot´ e, R., & Griffin, A. Excitonic modes in a Bose-condensed electron-hole gas in the pairing approximation.\nPhys. Rev. B 37,4539-4551 (1988).\n33. Markiewicz, R.S. Mode-coupling model of Mott gap collapse in the cuprates: Natural phase boundary for\nquantum critical points Phys. Rev. B 70,174518 (2004).\n34. Markiewicz, R.S. Lorenzana, J. Seibold, G. & Bansil, A. Gutzwiller magnetic phase diagram of the cuprates.\nPhys. Rev. B 81,014509 (2010).\n35. Phillips, J.C. Ultraviolet absrption of insulators III: fcc alkali halides Phys. Rev. 136, A1705 (1964).\n36. Andergassen, S. Caprara, S. Di Castro, C. & Grilli, M. Anomalous isotopic effect near the charge-ordering\nquantum criticality. Phys. Rev. Lett. 87,056401 (2001).\n37. Nagaosa, N. Quantum Field Theory in Condensed Matter Physics , (Berlin, Springer, 1999).\n38. Yamada, K. Electron Correlation in Metals , (Cambridge, University Press, 2004).\n39. Mermin N.D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional\nisotropic Heisenberg models. Phys. Rev. Lett. 17,1133-1136 (1966).\n40. Birgeneau, R.J. et al. Magnetism and magnetic fluctuations in La 2−xSrxCuO 4for x=0 (2D antiferromagnet),\n0.04 (3D spin glass) and x=0.15 (superconductor). J. Phys. Chem. Solids 56,1913-1920 (1995).13\n41. Kusko, C. Markiewicz, R.S. Lindroos, M. & Bansil, A. Fermi surface evolution and collapse of the Mott pseu-\ndogap in Nd 2xCexCuO 4δ.Phys. Rev. B 66,140513(R) (2002).\n42. Das, T. Markiewicz, R.S. & Bansil, A. Optical model-solution to the competition between a pseudogap phase\nand a charge-transfer-gap phase in high-temperature cuprate superconductors. Phys. Rev. B 81,174504 (2010).\n43. Piriou, A. Jenkins, N. Berthod, C. Maggio-Aprile, I. & Fischer, Ø. First direct observation of the Van Hove\nsingularity in the tunnelling spectra of cuprates. Nature Communications 2,221 (2011).\n44. Nieminen, J. Suominen, I. Das, T. Markiewicz, R.S.& Bansil, A. Evidence of strong correlations at the van Hove\nsingularity in the scanning tunneling spectra of superconducting Bi 2Sr2CaCu 2O8+δsingle crystals. Phys. Rev.\nB85,214504 (2012).\n45. Benhabib, S. et al. Collapse of the normal state pseudogap at a Lifshitz transition in Bi 2Sr2CaCu 2O8+δcuprate\nsuperconductor. Phys. Rev. Lett. 114,147001 (2015).\n46. Buhmann, J.M. Ossadnik, M. Rice, T.M. & Sigrist, M. Numerical study of charge transport of overdoped\nLa2xSrxCuO 4within semiclassical Boltzmann transport theory. Phys. Rev. B 87,035129 (2013).\n47. Barzykin, V. & Pines, D. Universal behavior and a two-fluid description of the cuprate superconductors. Adv.\nPhys. 58,1-65 (2009).\n48. Curro, N. Fisk, Z. & Pines, D. Scaling and the magnetic origin of emergent behavior in correlated electron\nsuperconductors. MRS Bulletin 30,442-446 (2005).\n49. Aeppli, G. Mason, T.E. Hayden, S.M. Mook, H.A. & Kulda, J. Nearly singular magnetic fluctuations in the\nnormal state of a high-T ccuprate superconductor Science 278,1432-1435 (1997).\n50. Das, T. Markiewicz, R.S. & Bansil, A. Intermediate coupling model of the cuprates. Advances in Physics 63,\n151-266 (2014).\n51. Vilk, Y.M. & Tremblay, A.-M.S. Non-perturbative many-body approach to the Hubbard model and single-\nparticle pseudogap. J. Phys. I France 7,1309-1368 (1997).\n52. Senthil, T. Vishwanath, A. Balents, L. Sachdev, S. & Fisher, M.P.A. Deconfined quantum critical points Science\n303, 1490-1494 (2004).\n53. Chen, K.-S. Meng, Z.Y. Pruschke, T. Moreno, J. & Jarrell, M. Lifshitz transition in the two-dimensional Hubbard\nmodel Phys. Rev. B 86,165136 (2012).\n54. Rice, T.M. Yang, K.-Y. & Zhang, F.C. A phenomenological theory of the anomalous pseudogap phase in\nunderdoped cuprates. Rep. Prog. Phys. 75,016502 (2012).\n55. Rice T.M. & Scott, G.K. New mechanism for a charge-density-wave instability Phys. Rev. Lett. 35,120-123\n(1975).\n56. Budai, J.D. et al. Metallization of vanadium dioxide driven by large phonon entropy Nature 515, 535539 (2014).\n57. Bechstedt, F., Many-Body Approach to Electronic Excitations (Springer, Berlin, 2015).\nVI. ACKNOWLEDGEMENTS\nThis work is supported by the US Department of Energy, Office of Science, Basic Energy Sciences grant number DE-\nFG02-07ER46352, and benefited from Northeastern University’s Advanced Scientific Computation Center (ASCC)\nand the allocation of supercomputer time at NERSC through grant number DE-AC02-05CH11231.\nVII. AUTHOR CONTRIBUTIONS\nR.S.M., I.G.B., P.M., and A.B. contributed to the research reported in this study and the writing of the manuscript.14\nVIII. ADDITIONAL INFORMATION\nThe authors declare no competing financial interests.SUPPLEMENTARY MATERIAL for\nCondensation Bottleneck Driven by a Hidden Van Hove Singularity as the Origin of\nPseudogap Physics in Cuprate High- TcSuperconductors\nR.S. Markiewicz, I.G. Buda, P. Mistark, and A. Bansil\nPACS numbers:\nI. GENETIC ALGORITHM FOR CUPRATES\nA. A note on normalization conventions\nIn this paper susceptibilities are calculated at in-\ntermediate coupling level, using Z-corrected first-\nprinciples dispersions to describe the coherent part of\nthe quasiparticle-GW dressed carriers.1The results can\nbe expressed in two different ways. If the susceptibility\nχ0is calculated based on the density functional theory\n(DFT) dispersion /epsilon10(k), then the phase diagrams are ex-\npressed in terms of renormalized temperatures TZ=T/Z\nand Hubbard U’s,UZ=ZU, whereZis the QPGW\nrenormalization factor. This is used in the Supplemen-\ntary Material. For convenience in comparing with ex-\nperiment, in the main text we calculated χbased on\nthe dressed dispersion /epsilon1Z=Z/epsilon10, assuming a doping-\nindependent Z= 0.5.\nAlso, when referring to Figures or Equations in the\nmain text, we will use a leading ‘M’, as in Fig. M1.\nB. Cuprate algorithm\nTo understand differences between cuprates, and to\ngauge their proximity to the pure Hubbard model, it is\nnecessary to add a third axis to the usual T- and doping-\nphase diagrams. Here we introduce a technique to reduce\ndifferences in band structure to the variation of a single\nhopping parameter.\nWhile DFT has been liberating in that it provides a\nparameter-free starting point for studies of materials, it\ncan also be constraining, in that each material is defined\nas a single point in material parameter space, and it is\nhard to define connections between the members of a\nfamily of materials, such as cuprates or pnictides. What\nis needed is a form of materials genome, with an algo-\nrithm defining the relatedness of materials, the evolution\nfrom one material to another, and the probable route for\nextending the family to new materials. Here we propose\nsuch an algorithm for the cuprates – a ‘cuprate genome’.\nThe cuprates are particularly simple, in that their prop-\nerties are dominated by a single CuO 2band crossing the\nFermi level, so their parameter space consists of hopping\nparameters, traditionally t,t/prime,t/prime/prime, ..., which can be deter-\nmined, e.g., by a Wannier-type downfolding of the DFT\nbands. In fact, since tsets an energy scale, the band\ndispersion is determined by the ratios t/prime/t,t/prime/prime/t, ... In\n6000\n4000\n2000\n0\n0.4 0.2 0.0(b)\nΓ\n(δ,δ)(π,π)\n(π,π−δ)ring\n6000\n4000\n2000\n0\n0.4 0.2 0.0 x0(π,π−δ)\n(δ,δ)ring(π,π)\n(π−δ,π−δ)\nΓ(f)2000\n1500\n1000\n500\n0TZ (K)\n0.2 0.1 0.0(π,π)\n(π,π−δ)(a)\n2000\n1000\n0TZ (K)\n0.2 0.1 0.0x0(e)\n(π,π)\n(π,π−δ)2000\n1000\n0TZ (K)\n0.2 0.1 0.0(π,π)\n(π,π−δ)(c)6000\n4000\n2000\n0\n0.4 0.2 0.0(d)\n(δ,δ)(π,π)\n(π,π−δ)ring(π−δ,π−δ)FIG. 1: Fluctuation fingerprints of cuprates. Maps ofq-\nvectors of largest susceptibility for DFT-based model of LSCO\n(c) and Bi2201 (d) compared to t−t/prime−t/prime/primereference phase a\n[t/prime/prime= 0] witht/prime= -0.23t(a) or -0.43t(b), and reference phase\nb [t/prime/prime=−0.5t/prime] witht/prime= -0.12t(e) or -0.258 t(f). Recall, from\nFig. M1(c), that x0is the doping at T= 0, but we assume\nx/similarequalx0.\ngenetics, a family would be defined as a group lying on\na low-dimensional surface in parameter space. The sim-\nplest genetics is associated with changes of a single gene,\ncalled Mendelian. Similarly, the simplest material family\nwould lie along a single line in material space. We show\nthat the cuprates correspond to such a Mendelian family,\nby introducing reference families depending only on t/prime/t,\nt/prime/prime/t, and show that the cuprates lie along a particular\ncut in this 2D-space.\nTo do this, we introduce a definition of equivalence:\nideally, two materials are equivalent if they have the same\nphase diagram. However, the real phase diagrams are\ndefined by interactions beyond the current scope of DFTarXiv:1505.04770v2  [cond-mat.supr-con]  28 Mar 20162\n4000\n2000\n0\n0.4 0.2 0.0\nx0(d)\n(δ,δ)(π,π)\n(π,π−δ)ring(π−δ,π−δ)2000\n1000\n0TZ (K)\n0.2 0.1 0.0x0(π,π)\n(π,π−δ)(c)(b) UZ=1.1eV\n1.0eV\n0.75eV\n0.67eV 0.50eV\n0.50eV1.0eV2.0eV\n1.4eV\n1.1eV\n0.75eV(a)\nFIG. 2: Mean-field phase diagrams of cuprates , for\nDFT-based models of LSCO (a) and Bi2201 (b). Black dots\nindicate mean field phase boundaries for different UZvalues,\nincludingUZ= 2 (down triangles), 1 (diamonds), or 0.5 eV\n(up triangles).\ntheories. The best we can do is to say that they would\nhave the same phase diagrams in the context of DFT.\nFor weak and intermediately correlated materials, the\nordered phases are generally determined from the Stoner\ncriterion, and hence correlated with peaks in the Lind-\nhard susceptibility at ω= 0, which can readily be calcu-\nlated.\nFigure 1 is a map of the q-vectors associated with the\nlargest susceptibilities, calculated for a variety of disper-\nsions. The left-hand frames (a), (c), (e), refer to LSCO,\nthe right-hand frames (b), (d), (f), to Bi2201. The mid-\ndle row (c), (d), uses the full DFT dispersion, compared\nto two reference families, with ( a) cuts defined by t/prime/prime= 0\n(top row, (a), (b)) or ( b) the Pavarini-Andersen cut2,\nt/prime/prime=−t/prime/2 (bottom row, (e), (f)). We find that refer-\nence family bsemi-quantitatively reproduces the phase\ndiagram for both cuprates. On the other hand, setting\nt/prime/prime= 0 works well for LSCO, Fig. 1(a), but fails for Bi2201\nFig. 1(b). We further find that a reasonable choice of\nt/prime/tis the one which matches the Van Hove singularity\n(VHS) doping of the full DFT dispersion. In these figures\nwe fix an energy scale by taking t= 0.4195 eV, appropri-\nate for LSCO3. We note that both reference families have\nthe additional desirable feature that they evolve from the\nstate witht/prime= 0 – i.e., the original Hubbard model.\nThis confirms our earlier finding that LSCO, Fig. 1(c),\nhas a very different phase diagram from most other\ncuprates, here represented by Bi2201, Fig. 1(d).4In the\nphase labeled ‘ring’, the susceptibility is approximately\nconstant along a ring of q-vectors surrounding ( π,π). We\nnote in particular that antinodal nesting (red-shaded re-\ngion) emerges naturally in most cuprates at higher dop-\ning, but is absent in LSCO. Shown also in Fig. 1 are\nTVHS (dark red dashed line) and the shift of the DOS-\npeak (thin green dashed line), discussed in the main text\nfor LSCO.\nIn Fig. 2, we replot the DFT fluctuation maps, adding\nthe mean-field phase boundaries, based on a Stoner cri-terion,\nUZχ0(q,ω= 0) = 1. (1)\nIn all cases, there is a critical value of U=U0below\nwhich there is no phase transition at x= 0. ForU <U 0,\nthe phase boundary is dome-shaped, but with a peak\nshifted away from the T= 0 value of xVHS to a value\nclose to the line xVHS(T), red-dashed line. For U >U 0\nthe phase boundary rapidly evolves to a strong coupling\nform where the highest Ttransition occurs near half fill-\ning. This is consistent with our finding in the main text\nthat there is an apparent shift of the VHS toward x= 0\nasTincreases. In all cases, at sufficiently high TFS\nnesting is washed out and the dominant instability is at\nq= (π,π).\nAn advantage of using the genetic algorithm is that it\nlets us systematically vary parameter space, to study the\nevolution from one cuprate to another, as in Fig. M2,\nor to extrapolate the parameter space to better under-\nstand the origin of various phases. Figure 3(a) shows\nthe cuts in t/prime−t/prime/prime-space associated with reference fami-\nliesa(t/prime/prime= 0) [red line] and b(t/prime/prime=−t/prime/2) [blue line],\nwhere the light-green shaded segments show the range\nover which ξis significantly depressed, Fig. M2. The\ndotted lines indicate ground state transitions at x= 0,\nfrom (π,π) to (π,π−δ) to (δ,δ) order. The densities-\nof-states (DOS) shown for the two families in Figs. 3(b)\nand 3(c) provide insight into the origins of the crossovers.\nFor reference family athe DOS reveals two crossovers as-\nsociated with a change from 2D to 1D nesting – note that\nthe DOS at t/prime=−t/2 shows the characteristic 1 //radicalbig\n(E)\ndivergence expected for a 1D system. Near t/prime=−0.5t,\nthe DOS has two peaks, one at the saddle-point VHS\nfound in 2D materials, the other at the leading edge, as\nfound in 1D systems. For t/prime<−0.46t, the leading edge\nis larger, suggestive of weakly-coupled chains, while for\nt/prime>−0.28tthe leading edge peak turns into a local min-\nimum. This quasi-1D behavior is suggestive of the ne-\nmatic phase found in cuprates. Remarkably, t/prime=−0.28t\napproximately coincides with the C-I transition found in\nFig. M2 to lead to a severe ξdepression. This explains\nthe extended T-range of ANN [( δ,δ)] nesting found in\nFig. 1(b), which is not characteristic of cuprates.\nThe origin of the 1D-behavior can be easily under-\nstood. For reference family a, the electronic dispersion\nhas the form E=−2t(cx+cy)−4t/primecxcy→− 2twhen\nt/prime=−0.5tand eitherkxorky= 0. [Here ci=cos(kia),\ni=x,y.] That is, at the VHS the FS reduces to just\nthexandyaxes. This explains the strong role of t/prime/prime,\nwhich breaks up this perfect nesting, and shifts the dop-\ning of optimal ANN nesting away from the VHS-doping.\nThis causes the FS to split into two pockets, with the\nVHS corresponding to the topological transition where\nthe pockets split, inset to Fig. 3(b). The onset of this\ntransition lies close to the doping for Bi2201 (light blue\nDOS cur).3\n0.25\n0.20\n0.15\n0.10\n0.05\n0.00t''/t\n-0.5 -0.4 -0.3 -0.2 -0.1 0.0\nt'/tLSCOBi2201(π,π)(π,π−δ)(δ,δ) (a)5\n4\n3\n2\n1DOS (eV-1)\n-1.2 -0.8 -0.4 0.0\nEF (eV)(b)\n12\n8\n4\n0DOS (eV-1)\n-0.8 -0.4 0.0 0.4EF (eV)(c)1.0\n0.8\n0.6-1.4 -1.2 -1.01.0\n0.5\n0.0kya/π\n1.0 0.5 0.0kxa/π(d)\nΓ (π,0)(0,π)\nFIG. 3: Reference families of the cuprates. (a) Refer-\nence cuts in t/prime−t/prime/prime-space. Dotted lines indicate ground state\ntransitions at x= 0. Light green line segments indicate the\nranges over which ξis significantly depressed, Fig. M2, with\nmaximal depression indicated by yellow diamonds. Violet tri-\nangles [green squares] = reference states for LSCO [Bi2201];\ncolored circles indicate t/prime-values for the corresponding DOSs\nin frame (b) [upper line] or (c) [lower line]. (b) DOS for\nseveral values of t/primefor reference family b(t/prime/prime=−t/prime/2). As\nVHS moves from right to left, t/prime/t= 0 (red curve), -0.1 (or-\nange), -0.2 (yellow-green), -0.25 (green), -0.258 (light blue),\n-0.3 (blue), -0.4 (violet), and -0.5 (black). Inset: correspond-\ning Fermi surfaces at VHS. (c) DOS for several values of t/primefor\nreference family a(t/prime/prime= 0). As VHS moves from right to left,\nt/prime/t= 0 (red curve), -0.3 (orange), -0.4 (yellow-green), -0.45\n(green), -0.47 (light blue), -0.48 (blue), -0.49 (violet), and -0.5\n(black). Inset: leading edge of DOS, near band bottom for\n(from bottom to top) t/prime/t= -0.22, -0.27, -0.28, -0.29, -0.30, in\nregime where it crosses over from a local minimum to a local\nmaximum.\nII. COMPETITION BETWEEN FS AND VHS\nNESTING\nA. Origin of FS and VHS nesting\nHere we attempt to separate the two contributions to\nthe susceptibility, from bulk and Fermi surface, and to\nclarify the origin of the bulk contribution. To our knowl-\nedge there is no standard way to do this, so we have\ndeveloped simple schemes.\nConsider the expression for the ω= 0 bare susceptibil-ity that enters into the Stoner criterion,\nχ0(q) =/summationdisplay\nkf(/epsilon1k+q)−f(/epsilon1k)\n/epsilon1k−/epsilon1k+q. (2)\nThe denominator is largest if /epsilon1k=/epsilon1k+q, but in this case\nthe numerator vanishes unless /epsilon1kis at the Fermi level.\nEven in this case, the term contributes little weight to\nthe susceptibility integral unless the q-shifted FSs are\ntangent, which is generally satisfied if q= 2kF, giving\nrise to a folded map of the FS – the ridge of FS nesting.\nThis can be seen by comparing a map of χ/prime\n0, Fig. 4(a),\nwith the corresponding map of χ/prime/prime\n0, Fig. 4(b). The plot is\nappropriate for a related cuprate, Bi 2Sr2CuO 6(Bi2201),\natx= 0.13 andω= 5 meV, showing in the process\nthat VH nesting also produces a significant effect in\nthe Bi-cuprates. Due to the energy δ-function,χ/prime/prime\n0con-\ntains mainly the FS contribution5. [The extra feature\nat Γ is the DOS contribution.] One sees a very simi-\nlar ‘inverted- γ’ shaped ridge in both χ/prime\n0andχ/prime/prime\n0, which\nis aq= 2kFmap of the Fermi surface4. In Fig. 4(d),\nwe plotχ/prime\n0(red line) and χ/prime/prime\n0(blue line) along a cut\nΓ→(π,0)→(π,π)→Γ in momentum space. It can\nbe seen that, away from Γ, the peaks in both compo-\nnents fall at the same q-values, and the relative weights\nof the peaks is comparable. Thus, the anti-nodal peaks\nnear Γ are stronger than the near-( π,π) peaks. However,\nχ/prime\n0is largest on the near-( π,π) peaks, since the FS con-\ntribution is riding on a large peak associated with the\nbulk contribution to the susceptibility.\nThis bulk contribution arises as follows. Even when\n/epsilon1k/negationslash=/epsilon1k+q, there will be a finite contribution to the sus-\nceptibility, reduced by the factor 1 /(/epsilon1k−/epsilon1k+q), as long\nas the two energies lie on opposite sides of the Fermi\nlevel (forT= 0). This contribution will be especially\nsignificant near a VHS, where there is a large local DOS\n(LDOS). Indeed, this contribution will remain large even\nasxis doped away from xVHS, since while the LDOS on\none side ofEFdecreases, that on the other side increases,\nso that the product remains little changed until the VHS\npeak is quite far from EF. This bulk part provides a\nsmoothly varying background, with hills that can shift\nthe balance of the FS nesting, and in special cases can\nlead to commensurate nesting away from the FS nest-\ning vector. Here we try to quantify the dominant role of\nVHS nesting in creating this background, thereby pro-\nviding a deeper understanding of the VHS vs FS nesting\ncompetition.\nB. Separating FS and VHS nesting contributions\nTwo approaches are taken to eliminate the FS nest-\ning features to see the background susceptibility by it-\nself. First, raising T(violet long-dashed line in Fig. 4(d))\nwashes out the sharp FS features first, while leaving the\nbackground features behind. Secondly, in frame (c), we\nplot the difference between χ/prime\n0and suitably scaled quan-\ntity proportional to χ/prime/prime\n0; this is also plotted as the green4\n200\n100\n0\n200 100 0ω = 5meV(b)\n200\n100\n0\n200 100 0(a)\nΓ (π,0)(0,π) (π,π)\n200\n100\n0\n200 100 0(c)0.4\n0.2\n0.0χ0', χ0'' (eV-1)\n600 400 200 0Γ Γ (π,0) (π,π)(d)\nFIG. 4: Extracting the bulk χ0contribution. (a) Sus-\nceptibilityχ/prime\n0of Bi2201, x= 0.13, plotted in first Brillouin\nzone, atT= 0,ω= 5 meV. Color scale from minimum (blue)\nto maximum (white). (b) As in (a), but for χ/prime/prime\n0. (c) Difference\nplot of (a) and scaled version of (b). (d) Cuts of suscepti-\nbility from earlier frames plotted along high symmetry lines,\nwith red curve representing data in frame (a), blue curve for\nframe (b), and green curve for frame (c). Shown also is the\ncurve corresponding to frame a, but at T= 1000K (violet\nlong-dashed line) scaled by 1.4 and shifted by -0.045 eV−1to\nmatch other curves near ( π,π)..\ncurve in frame (d). Since χ/prime\n0andχ/prime/prime\n0are Kramers-Kr¨ onig\ntransforms, they have different lineshapes, but since the\nFS map is nearly one-dimensional, a large part of it is still\ncancelled in taking the difference. From frame (d) it can\nbe seen that both approaches lead to similar results for\nthe bulk susceptibility, and that this contribution pro-\nvides the dominant contribution to virtually the whole\n(π,π) plateau structure.\nC. Susceptibility Analysis\nFinally, we demonstrate the connection between the\nbulk contribution to χ0and the (π,π) bosonic-VHS (b-\nVHS), by deconvolving the bare susceptibility, Eq. 2, into\nits various/epsilon1kcomponents. To understand this procedure,\nrecall that the VHS in the electronic dispersion arises\nwhen the Fermi surface crosses the Brillouin zone bound-\nary near (π,0) and related points, crossing from hole-like\nto electron-like. The electron-hole pairs associated with\nthe VHS come in two kinds: intra-VHS pairs, where both\ncarriers lie near the same VHS, so their relative momen-\ntum is near q= (0,0) = Γ, and inter-VHS pairs, where\n-300-200-1000100\n200 100 0(b)\n-150-100-50050εk (meV)\n200 100 0(a)\n-300-200-1000100\n200 100 0(d)\n-150-100-50050εk (meV)\n200 100 0(c)\n-300-200-1000100\n200 100 0qa(f)\n(π,π/2) (π/2,π/2) (π,π)-150-100-50050εk (meV)\n200 100 0qa(e)\n(π,π/2) (π,π) (π/2,π/2)FIG. 5: Coherent-incoherent crossover at x=0: role\nof VHS nesting. Deconvolution of the susceptibility for\nk-states in the ( π,0)-patch at several temperatures T. Con-\ntribution to the susceptibility of χ0,2(q,/epsilon1k), Eq. 3, is plot-\nted alongq-cuts near ( π,π) with the color scale ranging from\nwhite for a large contribution to blue for negligible contribu-\ntion for:T= 0 (a), 25K (b), 50K (c), 250K (d), 500K (e), and\n2000K (f).\nan electron near ( π,0) and a hole near (0 ,π) have rel-\native momentum near ( π,−π). The latter pairs carry\nmore weight, forming a b-VHS near q= (π,π). Thus, to\nconfirm that the susceptibility peak at ( π,π) is assiciated\nwith the b-VHS, we analyze Eq. 2 for χ0, assuming that\nthe electron is associated with /epsilon1kand the hole near /epsilon1k+q.\nWe must demonstrate that, when q∼(π,π), the domi-\nnant contribution to χ0hasknear the VHS at ( π,0) [or\n(0,π)] and/epsilon1knearEVHS.\nWe divide the Brillouin zone into four patches, around\nthe Γ, (π,0), (0,π), and (π,π) points, and calculate sep-\narately the susceptibility due to each patch. In LSCO,\nthe plateau is predominantly composed of equal contri-\nbutions from the (0 ,π) and (π,0) patches, as expected\nfor a VHS. We then explore the distribution of values\nof/epsilon1kwhich contribute to the susceptibility in the ( π,0)\npatch atx= 0, Fig. 5, rewriting Eq. 2 as\nχ0(q) =/summationdisplay\ni=1,4/summationdisplay\n/epsilon1kχ0,i(q,/epsilon1k), (3)\nwhereiruns over the four quadrants Γ, ( π,0), (0,π), and\n(π,π).\nThere is a clear crossover from a dominant near-FS\ncontribution ( /epsilon1k= 0) at low Tto a dominant contri-5\nbution at high- Tcoming from the VHS, caused by the\nbreakdown of Pauli blocking, Fig. M1(b). At high T,\nthe dominant contribution lies along a flat line centered\nat energy proportional to kBTVHS =EF−EVHS. The\nsame crossover is evident in the susceptibility, as the co-\nherent, FS-related features present at low Tare washed\nout, leaving only a featureless (incoherent) ( π,π) plateau\nat highT, Fig. 6. As doping increases, the same pattern\nis repeated, but with the flat line moving toward EFas\nTVHS decreases, and the crossover from coherent to inco-\nherent shifts to lower T, consistent with the pink shaded\nregion in Fig. M1(c). The boundaries of this shaded re-\ngion areTVHS/2 andTVHS/3, whereTVHS is plotted as\nthe dark red dashed curve in Fig. M1(c).\nWhile this explains what happens at T > TVHS, the\nquestion remains why the broad peak at ( π,π) persists\nto lowerT[e.g., Figs. 1(a,c,e) at x=0]. To understand\nthis we first look at what happens when Pauli block-\ning is ostensibly absent – when EF=EVHS, Fig. 6(b).\nWhile the VHS peak is clearly present at T= 0 (vi-\nolet line), finite temperature does not lead to simple\nthermal broadening but actually to a split peak. This\nis a thermally-induced Pauli blocking effect: for |/epsilon1k|,\n|/epsilon1k+q|< T,fk=fk+q= 1/2 and ∆f= 0. Since the\n(π,π) susceptibility is pinned at ω= 0, the Pauli block-\ning turns on first at ( π,π) and then spreads at higher T.\nThis has a number of consequences: (1) As xshifts to-\nward half filling, the peak splitting turns on at higher Ts,\nFig. 6(a) – this is the lower branch of the ( π.π)−(π,π−δ)\ncrossover, plotted in Fig. M1(c). (2) This is the origin of\nthe ring phase, discussed above in relation to Figs. 1(d,f),\nwhere the peak susceptibility shifts from ( π,π) to a ring\nof states encircling ( π,π). (3) Finally, since the integral\noffis fixed by the net doping, when a thermal blocking\nturns on, some other states will be unblocked at nearby\nenergies. Thus, the persistence of the net ( π,π) peak at\nT < TVHS still reflects the underlying presence of the\nVHS, but is more and more controlled by states away\nfrom (π,π) and closer to EF. This evolution can be\nclearly traced in Fig. 5. It is in this sense that we can\nrefer to this VHS as hidden.\nThe low-Tfadeout of the b-VHS is controlled by the\nPauli blocking factor,\n∆fk,Q=sinhx−\ncoshx−+ coshx+, (4)\nwherex−=/epsilon1Q,−(k)/kBT,x+= (/epsilon1Q,+(k)−EF)/kBT.\nWhen the electronic VHS is at the Fermi level, x+=\n0 and ∆fk,Q = tanhx−/2, while at lower doping\nthe b-VHS is exponentially suppressed, cosh x+∼\nexp(TVHS/T)/2, which equals 1 when T∼TVHS/ln(2).\n1.5\n1.0\n0.5\n0.0\n600 400 200 0(b)\nΓ Γ (π,0) (π,π)1.0\n0.5\n0.0χ0' (eV-1)\n600 400 200 0(a)FIG. 6: Lindhard susceptibility of LSCO near the\nVHS doping. χ/prime\n0(q,ω= 0) is plotted for a series of tem-\nperatures (from top to bottom) T= 0 (violet), 10 (blue –\nframe (b) only), 25, 50, 100, 200, 400, 1000 (green), 2000,\nand 4000K at doping x= 0.18 (a) and 0.20 = xV HS (b).\nIII. INTERPRETATION OF\nSELF-CONSISTENCY\nA. Relation to Bose-Einstein Condensation\nIn the conventional picture of Bose-Einstein conden-\nsation (BEC), the total number of bosonic modes must\nsatisfy\nN=/integraldisplay\ndωnBE(ω)DB(ω), (5)\nwherenBEis the BE distribution function and DBis the\nbosonic density of states. As Tdecreases, the bosonic\nchemical potential µmust adjust to maintain the equal-\nity. However, the largest µcan be is zero, for which\nnBE→nP, the Planck distribution, and the RHS of\nEq. 5 takes a fixed value. For lower Tthe equation\ncan only be satisfied by putting a macroscopic number of\nbosons into the lowest energy mode.\nIn the present situation, DB=/summationtext\nqχ/prime/prime, andnPcan be\napproximated as ∼T/ω, in which case Eq. 5 becomes\nN/similarequalT/summationdisplay\nq/integraldisplay\ndωχ/prime/prime(q,ω)\nω=πT/summationdisplay\nqχ/prime(q,ω= 0).(6)\nComparing this to Eq. 5 in the main text, self-consistency\nhas produced a number Neff=Nπλ/A 0of excitons\nwhich can subsequently Bose condense. Whereas for 3D\nBEC the macroscopic occupation of the lowest bosonic\nmode must be added by hand, in 2D the self-consistent\ncondition automatically causes the occupation of that\nmode to diverge as T→0.6\nB. Entropy and Dissipation\nAs originally noted by McMillan, anomalously large ra-\ntios of 2∆/kBTcfor any phase transition are generically\nassociated with a dominance of bosonic entropy at finite\nT, associated with low-lying bosonic fluctuations. But\nfrom the fluctuation-dissipation theorem, the net fluctu-\nations at a given frequency ωare equal to the dissipation\nD(ω) at that frequency,\nD(ω) =/summationdisplay\nqχ/prime/prime(q,ω)(nBE(ω) + 1/2)\n→T/summationdisplay\nqχ/prime/prime(q,ω)\nω, (7)\nso the right-hand side of Eq. 6 is just the total dissipa-\ntion at temperature T. In an ordered phase near T= 0,\nthe dominant fluctuations will be associated with Gold-\nstone modes, but to show this in a calculation requires\nextending the present results to full self-consistency be-\ntween self energy and vertex corrections, as in a parquet\ncalculation.\nIV. MODE COUPLING IN BI2201\nIn Bi2201 and most other cuprates, a larger ratio |t/prime/t|\nexpands the ( π,π)-plateau, leading to a rich phase dia-\ngram, Fig. 2(b). In this case the b-VHS is dominant only\nat highT, while FS-related susceptibility features cause\na crossover of the ( π,π) susceptibility from a maximum\nat highTto a local minimum at lower T. This leads to\na different kind of strong mode coupling at intermediate\nT, best exemplified by the ring phase, Figs. 1(c,d). In\nthis phase, the leading-edge SDOS has a ‘Mexican hat’-\nlike divergence [Main Section II.C], Fig. 7(a) near 2000K,\nleading to a slow growth in ξ∼T−1, Fig. 7(c). Near the\nonset of ring order, TZ= 3250K, the correlation length\nhas a nonmonotonic T-dependence, similar to that seen\nin LSCO, Fig. M2(b).\nWhile many pseudogap-like features are similar to\nLSCO, Bi2201 has some novel properties associated with\nthe ANN instability, which becomes a CDW when charge\norder is included in the model. Since the ANN suscepti-\nbility peak has a much larger intrinsic curvature, mode\ncoupling effects are weak. Hence the ANN susceptibility\npeak grows rapidly at low temperature, and as doping\nincreases, it can surpass the ( π,π)-peak, thereby sharply\ncutting off the growth of near-( π,π) fluctuations. This\nis illustrated in Fig. 7 for x= 0.20. Notably, since this\npeak is not associated with strong mode coupling it has a\nweak SDOS with a very strong T-dependence, Figs. 7(b)\nand 7(d), crossing the NNN peak near TZ= 200K. The\nlarge subsequent growth of the ANN correlation length\nwould ensure the rapid establishment of a true long-range\n3D ANN order. However, corresponding to the large T-\ndependence of χANN there is a similar strong disorder\ndependence – particularly in the presence of short-range\n0.11101001000ξ/a\n1002 3456\n10002 34\nTZ (K)(c)\n(π,π)ring(π,π−δ)(δ,δ)1.0\n0.8\n0.6\n0.4\n0.2\n0.0\n1.30 1.25 1.20 1.15 1.10X- (eV)(b) 12\n8\n4\n0N- (eV-1)\n1.60 1.50 1.40 1.30X- (eV)(π,π) (π,π−δ)\n(δ,δ)ring(a)\n0.8\n0.7χ (eV)\n4000 2000 0 TZ (K)(d)FIG. 7: Structures in Bi2201 correlation lengths. Tem-\nperature dependence of: (a) SDOS N−fort−t/prime−t/prime/primereference\nstate of Bi2201, x= 0.20. Data sets are offset for clarity;\nfrom highest to lowest, TZ= 4000 (red), 2000 (violet), 1000\n(blue), 750 (green), 500 (brown), and 200K (light blue). (b)\nContinuation of SDOS in (a) to lower TZ, showing evolution\nof leading edge. From top to bottom: TZ= 200 (light blue),\n100 (red), 50 (blue), and 10K (green). For each curve, a\npartial SDOS near the ANN peak is superposed to show the\nSDOS associated with the ANN peak whose onset is marked\nby arrows. (c) Correlation length vs TZ. Blue dotted line\nrepresentsξ∼T−1. (d) Susceptibility of χ(q) peak vsTZfor\nqnear the (π,π)-plateau (red line) or for ANN nesting (blue\nline).\nnear-(π,π) order, which should cause the ANN corre-\nlation length to saturate. Nevertheless, the transition\nshould be manifest as a coherent-incoherent crossover in\nthe overdoped regime, which shifts to higher Tas dop-\ning increases. Remarkably, just such a feature has been\nfound in ARPES studies.6Combining this crossover with\ntheT∗crossover at lower doping produces the V-shaped\nboundary of the ‘strange metal’ phase associated with\nlinear-in-Tresistivity.\nV. VHS EXCITONIC INSTABILITY AND\nKONDO-LIKE PHYSICS\nThe physics of semimetals is strongly influenced by the\nsimultaneous presence of both holes and electrons, and\nthe attractive interaction between them. In lightly doped\nsemimetals this can lead to exciton condensation, an ex-\ntension of the excitonic instability found in small gap\nsemiconductors7,8. [See Ref. 9 for a recent experimental\nexample.] While screening modifies excitons in semimet-\nals and doped semiconductors, excitonic effects do not\ndisappear, but shift from δ-function bound excitons to\nexcitonic resonances with a finite width, which persist to\nhigh dopings and show strong, even resonant interaction\nwith phonons10. Indeed, Mahan has proposed that ex-\nciton formation in a doped semiconductor is resonantly\nenhanced by the presence of a sharp Fermi surface, in7\nanalogy with the edge singularity in x-ray spectra.11\nThe strong effect of excitons in doped semiconductors\nraises new questions about the role of VHSs in metals.\nAt a saddle-point VHS in a two-dimensional (2D) metal,\nor anM1VHS in a 3D metal12, the topology change of\nthe Fermi surface (FS) leads to a large coexisting popu-\nlation of electrons and holes. Indeed, resonant interband\nexcitons associated with the M1VHS (VH excitons) are\nknown to dominate the optical spectra of metals. Sim-\nilarly, in cuprates, an intraband Van Hove exciton can\narise near an electronic saddle-point VHS13,14. Just as in\nconventional exciton Bose condensation, the mean-field\ntransition corresponds to the Twhere [quasi]excitons\nform, but since these are nearly localized, there is a large\nspread inq, leading to incoherent VHS nesting. The real\ntransition lies at much lower T, when all pairs condense\ninto the lowest energy q-state. This excitonic VHS con-\ntinues to play a role even when the electronic VHS is\naway fromEF, leading to a peak in χ/prime\n0when either Tor\nωis∼EF−EVHS. This is due to scattering between\nstates near the VHS and states near EF, and is similar\nto Kondo physics, with the VHS states playing the role\noff-electrons.\nTo clarify the connection to Kondo physics, we write\na variational ground state for the singlet exciton as\n|ψG>=/productdisplay\nk,σ/parenleftBig\nuk+vkc†\nk+Q,σck,σ/parenrightBig\n|F(π,π)>, (8)\nwherec†\nk,σ[ck,σ] are creation [annihilation] operators for\nan electronic state of momentum kand spinσ, and\nuk,vkare variational parameters. Here |F(π,π)>is a\nfilled Fermi sea of the first ( π,π)-superlattice Brillouin\nzone, and the excitonic order parameter is ∆ CDW =<\nc†\nk+Q,↑ck,↑+c†\nk+Q,↓ck,↓+c.c. > . This model is a direct\ngeneralization of Yosida’s variational calculation of the\nscreening of a Kondo impurity to a single-band Kondo\nlattice model.15,16Moreover, to form a localized exci-\nton wave-packet, the restriction q=Qin Eq. 8 must\nbe dropped, and many q-vectors summed over, account-\ning for the bosonic entropy of the model – indeed the\nSDOS quantifies the number of near- Qstates that con-\ntribute to the entropy. Since the Fermi velocity vF= 0\nat a VHS, these excitons are nearly localized. This rein-\nforces the connection of the coherent-incoherent crossover\nto Kondo physics17: in Main Fig. M2(b), region I corre-\nsponds to the buildup of a Kondo lattice of nearly lo-\ncalized spins, while region II is a regime of crossover to\nitinerant physics.\nVI. RELATION TO OTHER CALCULATIONS\nA comparison between our QPGW model and other\n‘DFT+’ calculations was given in Ref. 1; here we dis-cuss this comparison specifically in regard to pseudogap\nphysics. A number of groups are working on MBPT-\ntype extensions of DFT calculations. However, DMFT\nand its cluster extensions are limited by a very low res-\nolution in momentum q. In particular, this means that\neven if they capture Mermin-Wagner physics, they are\ntypically limited to ξ < 7a18[Fig. 50 of Ref. 1]. Fur-\nthermore, in averaging over large patches in momentum\nspace, they lose the ability to resolve ordinary nesting\ninstabilities, such as the ANN instability responsible for\ncuprate CDWs [Fig. 49 of Ref.1]. On the other hand,\nthe fact that they can see the pseudogap, exen when av-\neraging over a quarter of the Brillouin zone is consistent\nwith our finding that the pseudogap is spread over many\nseparate modes in q.\nHowever, some DMFT extensions [dynamic vertex\napproximation19–21, dual fermion22,23, and one-particle\nirreducible approaches24] seek to calculate the two-\nparticle vertex, which allows them to calculate the\nmomentum-dependent self energy, and these approaches\nare able to reproduce Mermin-Wagner physics and vertex\ncorrections. We note in particular that a recent dynam-\nical cluster approximation (DCA)25calculation sorted\nout the contribution of various channels of two-particle\nscattering to the self energy (‘fluctuation diagnostics’),\nand found that spin fluctuations are the origin of the\npseudogap, while charge and pairing fluctuations play a\nmarginal role, consistent with our results. This has also\nbeen found experimentally26.\nWe further note that the parquet equations27,28could\nbe quite useful, if the recently discovered divergences25\ncan be overcome. The mutual self-consistency between\nthe self-energy and vertex corrections will be necessary\nto extend the current results to low temperatures.\nFinally, in many strong-coupling calculations, it\nis implicitly assumed that Fermi surface nesting is\nirrelevant29, and there is a special role for commensu-\nrate (π,π) magnetic order. Thus, in the Yang-Rice-\nZhang phenomenological model30, the nesting is tied to\nthe magnetic Brillouin zone boundary. The commensu-\nrate phase we have found provides a clear DFT-based\nexplanation for that effect, as well as a clear picture of\nhow this evolved into a more Fermi-liquid-like phase at\nhigher doping or larger t/prime.\nAfter our paper was posted [arXiv:1505.04770], we be-\ncame aware of another paper31that also finds that strong\nbosonic fluctuations over a ‘quasidegenerate distribution\nof 2pFordering wave vectors’ drives the pseudogap, al-\nthough they find it is predominantly in the charge sector,\nforming a ‘resonant Peierls excitonic state’.8\n1Das, T. Markiewicz, R.S. & Bansil, A. Intermediate cou-\npling model of the cuprates. Advances in Physics 63,151-\n266 (2014).\n2Pavarini, E. Dasgupta, I. Saha-Dasgupta, T. Jepsen, O.\n& Andersen O.K. Band-structure trend in hole-doped\ncuprates and correlation with T cmax.Phys. Rev. Lett. 87,\n047003 (2001).\n3Markiewicz, R.S. Sahrakorpi, S. Lindroos, M. Lin, H. &\nBansil, A. One-band tight-binding model parametrization\nof the high- Tccuprates including the effect of kzdispersion\nPhys. Rev. B 72,054519 (2005).\n4Markiewicz, R.S. Lorenzana, J. Seibold, G. & Bansil, A.\nGutzwiller magnetic phase diagram of the cuprates. Phys.\nRev. B 81,014509 (2010).\n5Johannes, M.D. Mazin, I.I. & Howells, C.A. Fermi-surface\nnesting and the origin of the charge-density wave in NbSe 2\nPhys. Rev. B 73,205102 (2006).\n6Chatterjee, U. et al. Electronic phase diagram of high-\ntemperature copper oxide superconductors. Proc. Nat.\nAcad. Sci. 108, 9346-9349 (2011).\n7Kohn, W. in Many Body Physics , edited by C. DeWitt and\nR. Balian (Gordon & Breach, New York, 1968), p. 351.\n8Halperin, B.I. & Rice, T.M. The excitonic state at\nthe semiconductor-semimetal transition. in Solid State\nPhysics , Vol. 21, ed. Seitz, F. Turnbull, D. & Ehrenreich\nH. (New York, Academic) pp. 115-192.\n9Zhu, Z. McDonald, R.D. Shekhter, A. Ramshaw,\nB.J. Modic, K.A. Balakirev, F.F. Harrison, N. Tun-\nable excitonic insulator in quantum limit graphite\narXiv:1508.03645 .\n10Shokhovets, S. K¨ ohler, K. Ambacher, O. & Gobsch, G.\nObservation of Fermi-edge excitons and exciton-phonon\ncomplexes in the optical response of heavily doped n-type\nwurtzite GaN Phys. Rev. B 79,045201 (2009).\n11Mahan, G.D. Excitons in degenerate semiconductors Phys.\nRev.153, 882-889 (1967); Excitons in metals: infinite hole\nmass Phys. Rev. 163, 612-617 (1967).\n12Bassani, F. & Pastori Parravicini, G. Electronic States and\nOptical Transitions in Solids. Pergamon Press (1975).\n13Markiewicz, R.S. Van Hove excitons and High-T csuper-\nconductivity (IV) Properties of the excitons. Physica C\n168, 195-204 (1990).\n14Onufrieva, F. & Pfeuty, P. Quantum critical point asso-\nciated with the electronic topological transition in a two-\ndimensional electron system as a driving force for anoma-\nlies in underdoped high-Tc cuprates. Phys. Rev. B 61, 799-\n820 (2000).\n15Yosida, K. Bound state due to the sd exchange interaction.\nPhys. Rev. 147, 223-227 (1966).\n16Mahan, G.D. Many-Particle Physics , 3d Ed., (New York,\nKluwer, 2000).\n17Curro, N. Fisk, Z. & Pines, D. Scaling and the magnetic\norigin of emergent behavior in correlated electron super-conductors. MRS Bulletin 30,442-446 (2005).\n18Maier, T.A. Jarrell, M. Schulthess, T.C. Kent, P.R.C. &\nWhite, J.B. Systematic study of d-wave superconductivity\nin the 2D repulsive Hubbard model Phys. Rev. Lett. 95,\n237001 (2005).\n19Toschi, A. Katanin, A.A. & Held, K. Dynamical vertex ap-\nproximation: A step beyond dynamical mean-field theory\nPhys. Rev. B 75,045118 (2007).\n20Katanin, A.A. Toschi, A. & Held, K. Comparing pertinent\neffects of antiferromagnetic fluctuations in the two- and\nthree-dimensional Hubbard model Phys. Rev. B 80,075104\n(2009).\n21Toschi, A. Rohringer, G. Katanin, A.A. & Held, K. Ab ini-\ntio calculations with the dynamical vertex approximation\nAnn. Phys. (N.Y.) 523, 698-705 (2011).\n22Rubtsov, A.N. Katsnelson, M.I. & Lichtenstein, A.I. Dual\nfermion approach to nonlocal correlations in the Hubbard\nmodel Phys. Rev. B 77,033101 (2008).\n23Hafermann, H. Li, G. Rubtsov, A.N. Katsnelson, M.I.\nLichtenstein, A.I. & Monien, H. Efficient perturbation\ntheory for quantum lattice models Phys. Rev. Lett. 102,\n206401 (2009).\n24Rohringer, G. Toschi, A. Hafermann, H. Held, K. Anisi-\nmov, V.I. & Katanin, A.A. One-particle irreducible func-\ntional approach: A route to diagrammatic extensions of\nthe dynamical mean-field theory Phys. Rev. B 88,115112\n(2013).\n25Gunnarsson, O. Schfer, T. LeBlanc, J.P.F. Gull, E. Merino,\nJ. Sangiovanni, G. Rohringer, G. & Toschi, A. Fluctuation\ndiagnostics of the electron self-energy: origin of the pseu-\ndogap physics Phys. Rev. Lett. 114, 236402 (2015).\n26Badoux, S. et al. Change of carrier density at the pseu-\ndogap critical point of a cuprate superconductor Nature ,\npublished online doi:10.1038/nature16983.\n27Tam, K.-M. Fotso, H. Yang, S.-X. Lee, T.W. Moreno, J.\nRamanujam, J. & Jarrell, M. Solving the parquet equa-\ntions for the Hubbard model beyond weak coupling Phys.\nRev. E 87,013311 (2013).\n28Li, G. Wentzell, N. Pudleiner, P. Thunstrm, P. & Held, K.\nEfficient implementation of the parquet equations – role of\nthe reducible vertex function and its kernel approximation\narXiv:1510.03330 .\n29Anderson, P. W. The resonating valence bond state in\nLa2CuO 4and superconductivity Science 235, 1196-1198\n(1987).\n30Yang, K.-Y. Rice, T.M. & Zhang, F.-C. Phenomenologi-\ncal theory of the pseudogap state Phys. Rev. B 73,174501\n(2006).\n31Kloss, T. Montiel, X. & P’epin, C. Resonant Peierls\nexcitonic state as a new mechanism for the pseudo-\ngap andT-linear resistivity in cuprate superconductors\narXiv:1510.3038 ."    },    {        "title": "2405.10222v1.Kramers_nodal_line_in_the_charge_density_wave_state_of_YTe__3__and_the_influence_of_twin_domains.pdf",        "content": "Kramers nodal line in the charge density wave state of YTe 3and\nthe influence of twin domains\nShuvam Sarkar1∗, Joydipto Bhattacharya2,3, Pramod\nBhakuni1, Pampa Sadhukhan1, Rajib Batabyal1,\nChristos D. Malliakas4, Marco Bianchi5, Davide Curcio5, Shubhankar\nRoy6, Arnab Pariari7, Vasant G. Sathe1, Prabhat Mandal7, Mercouri G.\nKanatzidis4,8, Philip Hofmann5, Aparna Chakrabarti2,3, Sudipta Roy Barman1†\n1UGC-DAE Consortium for Scientific Research,\nKhandwa Road, Indore 452001, Madhya Pradesh, India\n2Theory and Simulations Laboratory,\nRaja Ramanna Centre for Advanced Technology,\nIndore 452013, Madhya Pradesh, India\n3Homi Bhabha National Institute, Training School Complex,\nAnushakti Nagar, Mumbai 400094, Maharashtra, India\n4Department of Chemistry, Northwestern University, Evanston, 60208, Illinois, USA\n5Department of Physics and Astronomy,\nInterdisciplinary Nanoscience Center (iNANO),\nAarhus University, 8000 Aarhus C, Denmark\n6Vidyasagar Metropolitan College, 39 Sankar Ghosh Lane, Kolkata 700006, India\n7Saha Institute of Nuclear Physics, HBNI,\n1/AF Bidhannagar, Kolkata 700 064, India and\n8Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA\n1arXiv:2405.10222v1  [cond-mat.mtrl-sci]  16 May 2024Abstract\nRecent studies have focused on the relationship between charge density wave (CDW) collective\nelectronic ground states and nontrivial topological states. Using angle-resolved photoemission and\ndensity functional theory, we establish that YTe 3is a CDW-induced Kramers nodal line (KNL)\nmetal, a newly proposed topological state of matter. YTe 3is a non-magnetic quasi-2D chalcogenide\nwith a CDW wave vector ( qCDW) of 0.2907c∗. Scanning tunneling microscopy and low energy\nelectron diffraction revealed two orthogonal CDW domains, each with a unidirectional CDW and\nsimilar qCDW. The effective band structure (EBS) computations, using DFT-calculated folded\nbands, show excellent agreement with ARPES because a realistic x-ray crystal structure and twin\ndomains are considered in the calculations. The Fermi surface and ARPES intensity plots show\nweak shadow bands displaced by qCDWfrom the main bands. These are linked to CDW modulation,\nas the EBS calculation confirms. Bilayer split main and shadow bands suggest the existence of\ncrossings, according to theory and experiment. DFT bands, including spin-orbit coupling, indicate\na nodal line along the Σ line from multiple band crossings perpendicular to the KNL. Additionally,\ndoubly degenerate bands are only found along the KNL at all energies, with some bands dispersing\nthrough the Fermi level.\nI. INTRODUCTION\nUnderstanding of the coupling between collective electronic ground states such as charge\ndensity wave (CDW) or superconductivity and non-trivial topology has become an exciting\nresearch frontier in the field of condensed matter physics [1–10]. The intricate interplay of\nCDW and the non-trivial band topology often gives rise to various exotic topological phases\nsuch as an axion insulator state [3], Kramers nodal line (KNL) metal [1], quantum spin-Hall\ninsulator [4], fractional Chern insulator states [8], eightfold fermionic quasiparticle [9] and\nmanipulation of topologically protected states [5, 10], to mention a few.\nCDW is typically observed in layered materials exhibiting quasi-one dimensional (quasi-\n1D) or quasi-two dimensional (quasi-2D) structures. Due to the associated modification in\nlattice symmetry, CDW can drive topological phase transitions [1, 3, 7, 10]. For instance, we\nhave recently demonstrated that CDW induced inversion symmetry breaking gives rise to a\nKNL [11] in the non-magnetic layered rare-earth tritelluride, LaTe 3[1]. KNLs are new type\n2of two-fold degenerate nodal lines that always connect two time reversal invariant momenta\n(TRIM) of the Brillouin zone (BZ) and are robust under spin-orbit coupling (SOC) [11]. Xie\net al. have proposed that all non-centrosymmetric achiral crystal symmetries with sizable\nSOC, when coupled with time reversal (TR) symmetry, should host this KNL state [11].\nBesides our recent work on LaTe 3[1], experimental evidence of KNL is limited to ruthenium\nsilicides [12] and SmAlSi [13].\nYttrium tritelluride (YTe 3) is another member of the RTe 3series (R= rare earth) that\nis non-magnetic and exhibits an incommensurate CDW state at ambient temperature [14–\n17]. As in other RTe 3compounds, it has a structure consisting of R-Te1 corrugated blocks\nsandwiched between the Te2-Te3 bilayers along the long axis ( y) [Fig. 1(a)]. The Te\nbilayers in YTe 3host the CDW [14, 15]. The CDW transition temperature (T CDW) of 334 K\nhas been determined from resistivity measurements [18]. The magnetization measurements\nconducted on YTe 3indicate that the diamagnetic susceptibility remains unchanged up to\nthe room temperature [14]. The non-magnetic nature of YTe 3indicates the existence of TR\nsymmetry. YTe 3exhibits an orthorhombic structure in the non-CDW state above T CDW,\ncharacterized by the Cmcm space group [14]. However, in contrast to the other members\nof RTe 3series (R: La-Tm), whose detailed x-ray crystallography study in the CDW state is\nreported in literature [17], the structure of YTe 3has not been studied. An angle-resolved\nphotoemission spectroscopy (ARPES) investigation on YTe 3[15] reported a k-dependent\n(in-plane) variation of the CDW gap, where the gap decreases as kxincreases. Their study\nsuggests a smaller CDW gap and consequently a smaller gapped region in the Fermi surface\n(FS) in YTe 3compared to LaTe 3. An interesting characteristic of YTe 3is that it exhibits\nsuperconductivity with a transition temperature of :3 K with 8% Pd intercalation, which\nin turn, inhibits the CDW state [19].\nThe existence of the CDW and resemblance of the physical properties with LaTe 3mo-\ntivated us to perform a comprehensive study of YTe 3. At the first step, the structure of\nYTe 3in the CDW phase was solved by x-ray crystallography. Two mutually orthogonal\nincommensurate CDW domains with similar CDW wave vector ( qCDW) values observed in\nscanning tunneling microscopy (STM) and low energy electron diffraction (LEED) signifi-\ncantly modifies the ARPES intensity plots and the Fermi surface. Effective band structure\n(EBS) calculations based on density functional theory (DFT) considering the twin domains\nand using a realistic structure of the CDW state determined by x-ray crystallography pro-\n3vide excellent agreement with ARPES, which shows faint bilayer split shadow bands that\nshow potential crossing with the main bands. DFT calculations show formation of KNL\nalong the Σ line characterized by doubly degenerate bands along the KNL and crossings\nperpendicular to it.\nII. METHODS\nA. Experimental:\nSingle crystals of YTe 3with residual resistivity ratio [RRR, ρ(300 K)/ ρ(2 K)] of :32 were\ngrown using the tellurium flux method [20]. High-purity Y and Te were mixed in a molar\nratio of 1:39. This mixture was sealed under high vacuum in a crucible and heated at 900◦C\nfor 10 h, and subsequently cooled slowly to 600◦C in 4 days. Excess Te was separated using\na high-temperature centrifuge, resulting in gold-colored, plate-like YTe 3crystals.\nSingle-crystal x-ray diffraction data for YTe 3were collected at 100 K with the use\nof graphite-monochromatized MoK αradiation ( λ= 0.71073 ˚A) STOE IPDS diffractome-\nter. The collection of intensity data as well as cell refinement and data reduction were\ncarried out with the use of the program X-Area. An analytical absorption correction\nwas performed (X-Shape within X-Area) and the modulated structure was refined with\nJANA2006 [21]. Atomic coordinates of the atoms in the subcell and initial values of their\nmodulation functions were determined by the charge-flipping method [22, 23]. The distortion\n(positional or displacement parameter) of a given atomic parameter x4in the subcell was ex-\npressed by a periodic modulation function p(x4) in a form of a Fourier expansion p(k+x4)=\nPm\nn=1Asnsin[2πqn(k+x4)]+Pm\nn=1Acncos[2πqn(k+x4)], where Asnis the sinusoidal coef-\nficient of the given Fourier term, Acnthe cosine coefficient, nthe number of modulation\nwaves used for the refinement and kthe lattice translation. qn=Pd\ni=1αniqi, where αniare\ninteger numbers for the linear combination of the incommensurate modulation vectors qi.\nSatellite reflections of one order were observed and used for the refinement. Consequently,\none modulation wave for positional and thermal parameters was used for all atoms. Only\nthe symmetry allowed Fourier terms were refined.\nThe ARPES measurements were conducted at the SGM3 beamline at the ASTRID2\nsynchrotron facility [24]. FS data at the SGM3 beamline were collected with an energy\n4resolution of 15-20 meV at photon energies (h ν) of 24 and 28 eV, with an angular resolution\nof 0.2◦(0.008 ˚A−1). These measurements were performed at various temperatures ranging\nfrom 45-340 K, and photon energy-dependent studies were conducted using different photon\nenergies in the range of 16 eV to 30 eV. A linearly polarized photon beam in the horizontal\nplane was incident at an angle of 50◦with respect to the surface normal, which was oriented\nalong the analyzer axis. The analyzer slit was vertically oriented, resulting in a vertical\ndetection plane, the experimental geometry is similar to what has been used in Ref. 1. The\nSTM measurements were conducted under a base pressure of 2 ×10−11mbar employing a\nvariable-temperature STM from Omicron Nanotechnology GmbH in the constant current\nmode. Mechanincally grinded Pt-Ir tips from Unisoku were used and cleaned in-situ using\nvoltage pulse method. LEED was performed using a four grid rear view optics from OCI\nVacuum Microengineering. A third order 2D polynomial background function (with 10\ncoefficients) [25] was subtracted from the LEED image to extract the weak CDW related\nsatellite spots. All the measurements were carried out on freshly peeled surfaces under a\nchamber base pressure of 2 ×10−10mbar.\nB. Density functional theory:\nDFT calculations have been performed using the Vienna Ab-initio Simulation Package\n(VASP) [26, 27] within the framework of the projector augmented wave method (PAW) [26,\n27] to obtain the electronic structure of YTe 3. The exchange-correlation functional is treated\nunder the generalized gradient approximation [28]. The energy cut-off is set to 500 eV\nfor the expansion of the plane waves. The convergence criterion for energy in the self-\nconsistent-field cycle and total force tolerance on each atom are taken to be 10−6eV and\n0.02 eV/ ˚A, respectively. The SOC is employed by a second-variation method as implemented\nin the VASP code [27]. The calculations have been performed for a seven-fold approximate\nstructure with C2cm space group ( SG#40) derived from the experimental atomic positions\nfrom the cif file using the PSEUDO program [29]. This program displaces the atoms to arrive\nat the commensurate seven-fold structure with non-centrosymmetric C2cmspace group (SG\n#40). This is discussed further in section III A.\nEBS has been computed using Pyprocar code and experimental energy and momen-\ntum broadening were convoluted with the unfolded spectral function for comparison with\n5ARPES [1, 30]. All the DFT bands (and consequently the EBS) are rigidly shifted to larger\nbinding energy ( E) by 0.1 eV with respect to the EFfor comparison with the ARPES data.\nVESTA software has been used for crystal structure visualization [31].\nIII. RESULTS AND DISCUSSION\nA. Crystal structure of YTe 3in the CDW state\nX-ray crystallography data show that YTe 3has an orthorhombic structure. The lattice\nconstants and other crystal data are presented in Table S1 of the Supplementary Material\n(SM) [32]. We find that YTe 3hosts a unidirectional incommensurate CDW with the qCDW=\n0.2907c∗and the superspace group is determined to be C2cm(00γ), similar to the other\nmembers of the RTe 3series [17]. C2cmis the basic space group of the C2cm(00γ) superspace\ngroup and γrepresents the zcomponent of qCDW.\nAn incommensurate structure is generally approximated as a commensurate structure\nwith a large unit cell such that its qCDWis close to the incommensurate value. However,\nthe latter becomes an accurate representation of the incommensurate structure if it is based\non the atom positions determined by x-ray crystallography and the qCDWvalue is within\nthe experimental accuracy [33, 34]. By utilizing the continued fraction method [35] to\nfind a rational fraction that could represent qCDW, we arrive at a fiftyfive-fold structure\n(1×1×55 supercell) with C2cm space group with qCDW= 16/55c∗= 0.2909c∗that matches\nthe experimental value of 0.2907(4)c∗within its accuracy. It has 440 atoms in the unit cell\nwith positions almost coinciding with those given by x-ray crystallography. A comparison of\nthe Te2 and Te3 atom positions (green dots) with the experimental positions (open circles)\nshow no perceptible deviation [Fig. S1(a) of SM [32]]. This is also supported by a small\naverage displacement ( u= 0.001 ˚A) of the atoms in the unit cell from the experimentally\ndetermined positions, as obtained from the PSEUDO program [29].\nHowever, because of the large size of the fiftyfive-fold unit cell, DFT calculations turned\nout to be highly resource intensive. So, a smaller seven-fold unit cell with 56 atoms is\nderived, importantly with all symmetries of the fiftyfive-fold structure preserved i.e., with\nthe same space group [Fig. 1(a)]. The corresponding qCDWvalue is 2/7 c∗(= 0.2857 c∗), which\nis an 1.8% deviation from the experimental value, with the Te atoms (orange dots) showing\n6Fourier weightd\n-1.0 -0.5 0.0\nk (c*)0.5 1.0intensity (arb. units)\nPQ\nUVAC\nCEq1\nq2\ne\nmax\nmin\nc\nkx\nkzA\nH\nF GB C\nD\nEq1\nq2\nf\ndomain 1 domain 2\nX\nZ\n0 193\nHeight (pm)\nCDW\n~90°CDW\nq1PQ\nR\nSg\nq2\nTU\nV\nWh\na\nb’\na’\na’b’\nc’\nY\nTe1\nTe2\nTe3\nintensity (arb. units)\n42 41 40 39\nE (eV)0.62 eVhν= 130 eVTe 4d5/2Te2-Te3\nTe1Te 4d3/2b\ni intensity (arb. units)\n120 110 100 90 80 70\n Raman shi� (cm-1)P1 P2 P3 P4 YTe3\n FIG. 1. (a) The primitive unit cell of the seven-fold structure (1 ×1×7) for YTe 3in the CDW state\ncomprising of 56 atoms with q1=2\n7c∗(c∗is the reciprocal lattice vector along kzin the non-CDW\nstate) left: viewed perpendicular and right: parallel to the c′direction. The lattice constants are\na′=b′= 12.781 ˚A,c′= 29.999 ˚A with α′=β′= 90◦, and γ′= 160.76◦. (b) Te 4 dcore level spectrum\nmeasured using h ν= 130 eV. (c) The low energy electron diffraction (LEED) pattern measured\nwith 39 eV primary electron beam energy, the CDW satellite spots are encircled. The color bar\nis in log-scale to enhance the weak satellite spots. (d) The intensity line profiles along AC and\nCE in panel c. The red and blue arrows indicate the CDW wavevectors q1andq2along kzand\nkx, respectively. (e) The intensity line profiles along PQ in panel gand UV in panel h. (f) A\nscanning tunneling microscopy (STM) topography image (12.8 ×12.8 nm2) in the CDW state of\nYTe 3, with bias voltage of 0.2 V and a tunneling current of 0.37 nA. The color scale in the bottom\nis shown in picometers (pm). A zoomed image of the enclosed area inside the yellow box is shown\nin the inset. Fourier transforms (FT) of the (g) top (above the white dashed line that indicates\nthe domain boundary) and (h) bottom (below the white dashed line) part of the STM image. The\nCDW related satellite spots are encircled. Both LEED and STM measurements were carried out\nat room temperature (RT). (i) The Raman spectrum (gray circles) of YTe 3measured at 90 K\nshows four peaks indicated as P1, P2, P3 and P4, where the P2 and P4 peaks (highlighted by the\nlight-blue rectangles) contain the B1symmetry mode.\n7small deviations from the experimental positions (open circles) [Fig. S1(b) of SM [32]].\nFigure 1(a) shows that YTe 3is made up of two main structural units: the Te2-Te3 bilayer\nthat hosts the CDW and the Y-Te1 corrugated slab. The Te bilayer, highlighted by blue\ndouble-sided arrows, is weakly coupled by van der Waals interaction.\nTe 4dcore-level photoemission spectrum of YTe 3in Fig. 1(b) comprises of 4 d5/2and\n4d3/2peaks separated by the spin-orbit splitting of :1.5 eV. Note that each of these peaks\nexhibit two components that are separated by :0.6 eV. These components are the signature\nof different valency of Te in the Y-Te1 slab and the Te2-Te3 layer due to transfer of electronic\ncharge from the former to the latter. A nearly similar splitting was observed in LaTe 34d[1]\nand 3 d[36] spectra. This was attributed to the difference in valencies of Te1 compared to\nTe2 (and Te3), as supported by the DFT calculations.\nB. Twin domains on the YTe 3surface\nIn Fig. 1(c), the LEED pattern of the YTe 3surface in the CDW state at room temperature\n(the CDW transition temperature TCDWbeing 334 K [18]) displays sharp main spots labeled\nas A-H. The satellite spots related to the CDW modulation along the kzdirection are\nhighlighted by red circles. Interesting to note are the relatively weaker satellite spots in the\nkxdirection (blue circles). These are unambiguously visible after performing a background\nsubtraction, as shown in the inset on the right side. The CDW modulation vectors q1along\nkzandq2along kxhave been determined from the LEED intensity profiles by measuring\nthe distance between the satellite and the main peaks. In Fig. 1(d), the intensity line\nprofiles measured along AC and CE demonstrate that both q1andq2have similar value of\n(0.3±0.01)c∗, which is in agreement with that determined by x-ray crystallography. Note\nthat for RTe 3with R= Tb-Tm, coexisting bidirectional CDW has been observed [16, 37, 38],\nand the magnitudes of qCDWin the two mutually perpendicular directions are different (e.g.,\n(2/7)c∗and (1/3) a∗for ErTe 3[15, 39]). On the other hand, the qCDWvalues for the two\ntwin domains are similar here for YTe 3.\nThese twin domains are directly observed in the STM topography image with atomic res-\nolution, where the domain boundary is indicated by a dashed white line [Fig. 1(f)]. Satellite\nspots related to the CDW are also visible in the Fourier transforms (FT) of each domain\nalong kzandkxfor domain 1(top) and domain 2(bottom), respectively [Figs. 1(g,h)].\n8In Fig. 1(e), the qvalues for the two domains from the line profiles measured along PQ\n[Fig. 1(g)] and UV [Fig. 1(h)] are q1= (0.29 ±0.03)c* and q2= (0.3 ±0.05)c*, which are\nequal within the error bar and are in excellent agreement with the values obtained from\nLEED. It was also noted from STM topographies performed at different sample locations\nthat domain 1is more prevalent compared to domain 2, a finding corroborated by the lower\nintensity of the LEED spots corresponding to domain 2[Figs. 1(c)].\nIn Fig. 1(f), atomic resolution enables direct observation of the CDW in the top Te layer.\nA zoomed region in the yellow rectangle shows that the average positions of the neighboring\nTe chains, depicted by the black dashed lines, are not equidistant (orange and green arrows),\nsignifying the breaking of Mxmirror symmetry and, consequently, the inversion symmetry\nas well. The non-centrosymmetry is also demonstrated by the Raman spectrum in Fig. 1(i)\nthrough occurrence of P2 and P4 peaks of B1symmetry mode, which is an irreducible repre-\nsentation of achiral C 2vpoint group. These modes were also observed in LaTe 3[1, 40]. The\nmirror symmetry however remains intact in the non-CDW state that has a centrosymmetric\nCmcm space group [14, 18].\nC. Shadow branches in the twin domain modified Fermi surface\nThe Fermi surface in the CDW state of YTe 3measured by ARPES comprises two\ndiamond-shaped sheets centered around Γ and two smaller oval pockets – γ1andγ2– lo-\ncated near the XandZpoints, respectively; these high symmetry points are indicated by\nred color [Fig. 2(b)]. The high symmetry points are also shown in the Brillouin zone (BZ)\n(black lines) in Fig. 2(a), it is inscribed within the non-CDW BZ (cyan). A signature of\na gapped region in the FS around the Γ Zdirection was observed previously for other RTe 3\ncompounds [1, 15]. Curiously, such a gapped region seems to be absent in Fig. 2(b). The\nFS rather seems to nearly resemble that calculated for the non-CDW state of RTe 3[1, 15],\nexcept for an important difference: occurrence of a narrow gapped region, highlighted by\nthe red arrow at ( kx,kz) = (0.2, 0.2) ˚A−1that splits the inner diamond-shaped sheet ( α).\nWe henceforth refer to the upper (lower) part as α1(α2), as shown in Fig. 2(b). α2resem-\nblesα1rotated by 90◦. A similar replica of the β1(upper part of the outer sheet) branch is\nnamed as β2(lower part). γ2that is observed around Zin the lower right corner, is also a\nreplica of the γ1pocket around X(upper left corner).\n9X2Non-CDW\nCDWΓky\nkzX\nZ’Akx\nT\nZ\nc*YA’\nΓ2\na\n0.8\n0.6\n0.4\nad\n0.2\n0.0kx (Å-1)\n0.8 0.6 0.4 0.2 0.0\nkz (Å-1)X/Z X2\nΓ2 ΓA\nγ1\nγ2α1\nα2β1\nβ2\nZ/Xb\nmax\nmin\nkz(Å-1)kx(Å-1)\n45 K0.8\n0.6\n0.4\n0.2\nheating coolingc\n0.8\n0.6\n0.4\n0.2\n340 K\nd\n0.6 0.4 0.2 0.0 -0.245 K\ne0.8\n0.6\n0.4\n0.2max\nmin\n1.0\n1.00.50.0\n0.50.0\n0.6 0.4 0.2 0.0\n0.6 0.4 0.2X/Z Z/X Γ\nmax\nmin\nE (eV)\nΔ\nmax\nmincut 1\nf\ng\n1domain\n2\nX/Z Z/X Γ\nk|| (Å-1)f\n1.00.50.0\n-0.4 0.0 0.4\n1.00.50.0\nmax\nminX Z Γ\nh\niE (eV)FIG. 2. (a) The Brillouin zone (BZ) of the seven-fold structure of the CDW-state of YTe 3(black\nlines), where the ordering of the conventional lattice is a′\nc< b′\nc[41] within the BZ of the non-CDW\nstate (cyan). The high symmetry points [41] are indicated in the respective colors. Γ, YandX\npoints coincide for both. The Σ line represented by Γ X(and Γ 2X2in the 2ndCDW BZ) is shown\nby green color. (b) The Fermi surface (FS) of YTe 3in the CDW state measured at 45 K with\nhν= 24 eV. In panel b, the horizontal (vertical) black dashed arrow of length q1(q2), connects\na shadow branch with the main branch of the FS. The main branches, α,β, and γ, are denoted\nwith subscripts 1 and 2 to denote the branches of domains 1and2, respectively. The gapped\nregion in the αbranch around kz=kx= 0.2 ˚A−1is highlighted by a red arrow. The yellow arrows\nindicate the shadow branches. The high symmetry points of the BZ are indicated by red and blue\ncolors, denoting domain 1and2, respectively. The color scale is shown on the left. Blue lines on\nthe right axis indicate the directions of the E(kz) cuts in Figs. S2(a-d) of SM [32]. Temperature\ndependent measurement of the FS with h ν= 28 eV at (c) 45 K (d) 340 K and subsequently cooled\nto (e) 45 K. Black dashed ovals highlight the region of interest. The color scale for panels c-eis\nshown on the left of panel e. (f) E(kz) band dispersion along XΓZdirection. Band bending due\nto CDW gap are indicated by the black arrows. (g) The curvature plot of panel f, where the band\nbending is more prominent (black arrows). ∆ is related to the CDW gap. (h) The folded band\nstructure (dashed gray curves) for single domain of YTe 3in the seven-fold structure along XΓZ\nincluding spin orbit coupling (SOC) is overlaid on the effective band structure (EBS) calculated\nfrom it. (i) The EBSs of domain 1(red) and domain 2(blue) overlaid on each other. The black\narrows indicate the band bending related to the CDW gap.\n10The appearance of these replicas in the FS rotated by 90◦can be explained by the presence\nof the twin domains ( 1and2) since the ARPES signal with a photon beam spot size of\n(200×100) µm2probes both the domains simultaneously. The intensity of the FS branches\nrelated to domain 2is less e.g., compare the intensities of the α2sheet and the α1sheet.\nThis indicates that the contribution of domain 1is greater than that of domain 2, which\nis consistent with the LEED and STM results discussed in the previous section. The high\nsymmetry points of domain 1(related to q1) and domain 2(related to q2) are shown by\nred and blue colors, respectively in Fig. 2(b).\nTo examine whether the narrow gapped region (red arrow) mentioned above might be a\nsignature of the CDW gap, we have measured the FS in both CDW and the non-CDW states.\nInterestingly, in contrast to the FS in the CDW state at 45 K [Fig. 2(c)], as the temperature\nis raised above TCDW [Fig. 2(d)], the αsheet is no longer gapped at 340 K (black dashed\noval), indicating CDW melting. Subsequently, as the temperature is lowered back to 45 K,\nthe gap becomes visible again [white arrows in Fig. 2(e)]. Thus temperature dependent FS\nmeasurement establishes that the narrow gapped region in the αsheet corresponds with the\nCDW gap in YTe 3. Additionally, from a comparison of the ARPES bands and the calculated\nEBS considering both domains, we show how the combined effect of CDW gap and twin\ndomains gives rise to the narrow gapped region in the FS in Discussion A of the SM [32].\nThus, unlike the case of RTe 3single crystals (as in Ref. 1), the FS of our YTe 3crystals\nis significantly modified by the presence of twin domains. In spite of this, shadow branches\nthat are related to the CDW modulation are observed, as indicated by dark yellow arrows\nin Fig. 2(b). These are separated from the main branches by qCDW(black dashed arrows).\nThe shadow branches are observed for both the domains; see for example, the vertical and\nhorizontal black dashed arrows of length qCDW.\nD. ARPES and effective band structure along X ΓZ\nThe ARPES intensity plot of YTe 3along XΓZ high-symmetry direction in Fig. 2(f) shows\nthat the bands towards the Γ Z(ΓX) direction of domain 1overlap with those of the Γ X\n(ΓZ) direction of domain 2. Therefore, the intensity plot seems to be essentially identical\nin these two directions, with bands crossing the EFin both directions. Additionally, in both\nthe directions, some bands are bent away from the EF[black arrows] due to the hybridization\n11of the main band and shadow band, resulting in formation of the CDW gap. In Fig. 2(g),\nthese hybridized bands with enhanced clarity are shown by the black and orange arrows\naround E= 0.3 eV and 0.5 eV, respectively. The lower limit of the CDW gap defined from\ntheEFto the band maximum [∆], is estimated to be 0.29 ±0.02 eV. This value is similar in\nboth directions.\nIn Fig. 2(h), the DFT calculated bands of YTe 3along XΓZfor the seven-fold structure\nshow multiple folded bands along both Γ Zand Γ Xdirections (gray dashed curves), which are\ndifficult to compare with the ARPES intensity plots [Figs. 2(f,g)]. So, we have determined\nthe EBS from the DFT bands by band unfolding, as shown by the red markers [1, 42, 43].\nThe EBS shows reasonable resemblance with ARPES: The CDW gap along Γ Zis evident\nin the EBS, as marked by the black arrows in Fig. 2(h). Along Γ X, two parabolic bands\n(related to the γpocket) centered around Xcross the EF, along with two additional bands\n(related to the αsheet) that approach the EF.\nThe agreement between theory [Fig. 2(i)] and experiment [Fig. 2(g)] is excellent when the\ninfluence of the twin domains is accounted for by superimposing the EBSs calculated along\nΓZand Γ X, where the red and blue bands represent the domain 1and2bands, respectively.\nFork||>0, the bands involved in the CDW gap formation are from domain 1, while the bands\nthat cross EFare from domain 2. The roles are reversed for k||<0. Figure S2 of SM [32]\nshows the evolution of the CDW gap across the BZ; a good agreement of ARPES and EBS\nis also apparent here, see Discussion A of SM [32].\nE. Main and Shadow bands towards kznear the BZ boundary\nIn this subsection, we investigate the interaction between the ARPES main bands that\nare present also in the non-CDW state and shadow bands that only appear in the CDW\nstate. Fig. 3(a) shows a stack of kz-kxisosurface plots for different Eranging from E= 0 to\n0.4 eV. The FS i.e., the top plot for E= 0 as well as other isosurface plots at larger Eshow\nthe shadow branches corresponding to the main β1branch, as indicated by yellow arrows.\nThe main and shadow branches appear to cross, and the crossing point shifts to lower kxas\nEincreases (shown by the black arrows). Notably, the crossing point does not shift in kz\nand occurs at kz= 0.209 ˚A−1, (blue dashed arrow). Additionally, the crossings between the\nmain and shadow bands are also identified in the ARPES intensity plots, E(kz), measured\n121.20.80.40.0\n1.20.80.40.0\n1.20.80.40.0\n0.4 0.2 0.0 -0.21.20.80.40.0\nf\ng\nh\n0.4 0.2 0.0 -0.2\n0.4 0.2 0.0 -0.2\nl\nk\nj\ni m2\n1domain\nmax\nmin\nE (eV)\nedcb\nmax\nmin\nkz (Å-1)\n0.4\nkz (Å-1)kx (Å-1)0.2\n0.4\nkz (Å-1)kx (Å-1)0.3\n0.4\nkz (Å-1)kx (Å-1)a\nMBSB 0.1\n0.4\nkz (Å-1)kx (Å-1)\nβ1γ1\nb\neE= 0.0 eV\n0.4\n0.4\nkz (Å-1)kx (Å-1)\nmax\nmin\nkx= 0.61 Å-1kx= 0.67 Å-1\nkx= 0.59 Å-1kx= 0.56 Å-1FIG. 3. (a) A series of kx-kzisosurface plots with Evarying from 0.0 to 0.4 eV measured with\nhν= 28 eV around the Xpoint (red dot), showing the crossings (black arrows) between the main\nand shadow branches. E(kz) ARPES intensity plots at kx= (b) 0.67, (c) 0.61, (d) 0.59, and (e)\n0.56 ˚A−1. The momentum distribution curves measured at the EFare shown by the gray filled\ncircles above the dashed horizontal line in panels b-e. The peaks related to the main and shadow\nbands are indicated by the black and yellow vertical arrows, respectively. The bilayer splitting is\nalso indicated by a pair of black arrows. The color scale for panels a-eis shown on the left of\npanel e. (f-i) 2D curvature plots of panels b-e, respectively. The crossing regions are highlighted\nby black dashed circles. The red and cyan dots at the top of the panels indicate the Γ Xand Γ 2X2\nlines, respectively. The dashed black arrows in panel hdenote q1connecting the main bands and\nshadow bands. The pink arrows in panels c, d, g, h indicate flat bands observed around kz=\n0. The green arrow in panel hindicates the “M” shaped bands, where a dashed curve serves as\nguide to the eye. The color scale is shown within the panel i. (j-m) Calculated EBS at similar kx\nvalues as panels b-e, where bands from both the domains [ 1(red) and 2(blue)] are overlaid. The\npossible crossing region of the shadow and main bands is highlighted by black dashed circles. The\ncolor scale is shown at the top of panel m.\n13atkx= 0.67, 0.61, 0.59, and 0.56 ˚A−1, respectively [Figs. 3(b-e)]. The red horizontal lines\non the right axis of the FS in Fig. 3(a) indicate these directions. In Fig. 3(b), we observe\ntwo main bands (inner and outer) of parabolic shape that are centered around the Xpoint\n(indicated by the red dot at the top horizontal axis). These bands intersect EFatkz≈±0.15\nand±0.21˚A−1. The outer main band related to the β1main branch disperses down to an\nenergy of ∼1.2 eV, while the inner main band related to the γ1main branch has the minimum\nat∼0.8 eV. With decreasing kxin Figs. 3(c-e), the outer main band spreads out in kz, while\nthe inner bands shrink in kzand its bottom moves towards lower E. Additionally, a flat\nband at E= 0.3 eV [pink arrows in Figs. 3(c,d) and (g,h)] along with a “M”-shaped band\ncentered around kz= 0 [green arrow in Fig. 3(h), dashed black curve is guide to the eye]\npartly overlap with the main bands.\nFrom the EBS in Figs. 3(j-m) [calculated at same kxvalues as in Figs. 3(f-i), respectively],\nwhere bands of both the domains are overlaid, we find that these additional bands (blue\ncolor) are related to domain 2[pink and green arrows in Fig. 3(l)]. Furthermore, EBS reveals\na splitting in main band, as highlighted by double black arrows in Fig. 3(k). This splitting\n– referred to as “bilayer splitting”– occurs due to the interaction between the Te bilayers\nthat host the CDW in YTe 3and has been observed in other RTe 3members [1, 15, 44, 45].\nAdditionally, the calculated band structure E(kz) atkx= 0.59 ˚A−1(gray curves in Fig. S3 of\nSM [32]) illustrates that the bilayer splitting for the outer main band increases with energy.\nFor example, at E= 0 eV, the splitting (∆ kz) is 0.2 ˚A−1, while at E= 0.4 eV, it is 0.3 ˚A−1.\nNote that bilayer splitting is not related to the CDW, as has been discussed in detail for\nLaTe 3[1]. For YTe 3also, a comparison of the CDW band structures with the non-CDW\nstate (blue dashed) in Fig. S3 of SM [32] reveals that the bilayer splitting is comparable in\nboth states.\nThe ARPES intensity plots are however complicated by the presence of twin domains,\nbands from domain 2interfere with the bands from domain 1. For example, some of the blue\nbands from domain 2can be seen in between the bilayer split main bands and overlapping\nwith the crossing region within the black dashed circles in Figs. 3(k-m). Nevertheless, bilayer\nsplitting is observed at higher E, where it is pronounced [two black arrows in Figs. 3(c-e)\nand (g-i)].\nThe evidence of shadow bands is obtained in the 2D curvature plots e.g., in Figs. 3(g-i).\nThe shadow bands on both sides of the Xpoint are separated by q1from the corresponding\n140.60.50.40.30.2E (eV)E (eV)0.22 0.20\nkz (Å-1) kz (Å-1) kz (Å-1)\n0.70 0.60 0.50\n1.00.50.0\n0.70 0.60 0.50\nminmax\n1.00.50.0\n0.70 0.60 0.50\n0.050.00-0.05\n0.65 0.64\n0.050.00-0.05\n0.70 0.69 0.68\n0.380.360.34\n0.222 0.22R\n12 meV\n0.380.360.34\n0.198 0.196 0.19412 meVL\nkx (Å-1\n) kx (Å-1\n) kx (Å-1\n)0.2650.2600.255\n0.210 0.2090.4600.4550.450\n0.210 0.209b\nc\nf g\nhi jd e\nR Lmu\nmdsdsuT\nT\nBB0.2 meV\nminmaxminmax\nminmax\nBT\nT\nBBTaFIG. 4. (a) E(kz) bands with SOC at k x= 0.59 ˚A−1in the crossing region of the main and shadow\nbands. The vertical dashed line represents the k zpoint on Γ 2X2i.e., the Σ line in the second BZ\n[see Fig. 2(a)]. The black lines represent the folded bands, whereas the EBS is shown in red color\nscale. Expanded regions around (b) L, (c) R, (d) T, and (e) Bcrossings. The inset in panel e\nshows the expanded region inside the red rectangle and the green arrows in panels d,eindicate\nthe doubly degenerate crossings. (f) EBS along the Γ 2X2line showing the TandBrelated bands\ntraversing the Fermi level. The band E(kx) dispersion from DFT shown in an expanded scale for\nthe (g) Tand (h) Brelated bands. (i) The corresponding E(kx) ARPES intensity plot towards\nΓ2X2[i.e., along the blue dashed arrow in the top plot ( E= 0) in Fig. 3(a)]. (j) 2D curvature plot\nof panel i.\nmain bands, as shown by the black dashed horizontal arrows. In the raw APRES intensity\nplots in Figs. 3(b-e), the momentum distribution curves (MDC) at the EF(gray filled circles)\ndepicts the kzwhere the shadow bands cross EF(yellow vertical arrows). The larger intensity\npeaks in the MDC are related to the main bands (black vertical arrows). As kxdecreases,\n15thekzseparation between the main and the shadow band increases, and their crossing –\nindicated by the black dashed circles – shifts to higher Evalues. However, independent of\nkx, the crossings occur at same kzat the Γ 2X2line [cyan dot at the top of Figs. 3(f-i)].\nIt may be noted that, in good agreement with ARPES, the calculated EBSs in Figs. 3(j-\nm) show that the shadow bands are separated by q1from the main bands, as indicated by\nthe horizontal black dashed arrows. EBS clearly shows that the shadow bands in YTe 3are\nweaker than in LaTe 3[1], which is consistent with an observation from ARPES. So, this is\nnot related to extraneous factors, but is related to the difference in the crystal structure of\nthese two compounds. The most notable difference – although their qCDWare close – is in the\namplitude ( A) of the CDW modulation that is a factor of two smaller in YTe 3(A= 0.07 ˚A\nfrom Fig. S1 of SM [32]) compared to LaTe 3(A= 0.14 ˚A from the supplementary Fig. 2 of\nRef. 1). Additionally, our EBS calculation for LaTe 3showed that if Adecreases to about\none-fifth of the experimental value, the shadow bands are almost completely absent [1].\nThus, it is reasonable to conclude that the weakness of the shadow bands in YTe 3is related\nto the smaller amplitude of the CDW. In spite of this, the crossings of the shadow bands\nwith the main bands are observed [highlighted by the black dashed circles in Figs. 3(j-m)].\nThese show similar trend as ARPES, i.e., the crossings move towards larger Ewith lower\nkx.\nF. Evidence of Kramers nodal line in YTe 3\nFigure 4(a) shows the EBS of the crossing region at kx= 0.59 ˚A−1corresponding to the\nblack dashed circle in Fig. 3(l). EBS shows four crossings ( L,R,TandB) between the\nbilayer split main band ( muandmd) and the shadow band ( suandsd). Since the electronic\nstructure of the structurally similar domains would be same, to avoid the interference from\nthe bands of domain 2, we have shown the EBS for domain 1only and the corresponding\nfolded band structure (black curves) is overlaid on it. Note that the additional parabolic\nband due to band folding is not detected in the unfolded EBS because of its reduced spectral\nweight and is also not observed in ARPES [Figs. 3(d,h)]. Notably, although all the four\ncrossings were observed from ARPES in LaTe 3[1], in case of YTe 3the weakness of shadow\nbands and interference of the twin domains make it difficult to identify them.\nIn Figs. 4(b,c), expanded regions near LandRcrossings of YTe 3show that both are\n16gapped by 12 meV. TandBalso exhibit a mini gap (4-6 meV). To decipher the influ-\nence of SOC, we have calculated the folded band structure near the crossing region without\nSOC (Fig. S4 of SM [32]). Mini-gaps are observed at TandBwithout SOC, signifying\nthe hybridization of bands involved in these crossings. In contrast, LandRexhibit gapless\ncrossings without SOC. Thus nodal lines are formed at LandRthat are gapped out with\nSOC. A similar behavior has been reported for LaTe 3[1]. When SOC is considered, each of\nthe gapped branches of both TandBexhibit splitting that is expected in a noncentrosym-\nmetric system such as YTe 3. However, it is interesting to note that the bands cross at kz=\n0.209 ˚A−1, as indicated by green arrows in Figs. 4(d,e). kz= 0.209 ˚A−1is significant because\nit corresponds to the Γ 2X2direction. The origin of the crossings is discussed later. The\nSOC splitting is more pronounced in case of bands at the Tcrossing [2 meV, Fig. 4(d)]\ncompared to the Bcrossing [0.2 meV, inset of Fig. 4(e)].\nIn Fig. 4(f), the EBS along Γ 2X2direction shows the dispersion of TandBcrossing\nrelated bands, i.e., these bands trace the crossings along the kxdirection. However, due to\nthe broadening used, we do not observe both the branches. Both the mini-gapped branches\nofTandBare however evident in the folded band structure shown zoomed in Figs. 4(g)\nand (h), respectively, where two doubly degenerate branches cross the EF. From the E(kx)\nARPES intensity plot and its 2D curvature plot in Figs. 4(i,j) towards the Γ 2X2direction\natkz= 0.209 ˚A−1, the TandBcrossing related bands are experimentally observed. These\ncross EFatkx= 0.62 and 0.66 ˚A−1, respectively. It is worth mentioning that the EBS\n[Fig. 4(f)] exhibits excellent agreement with the ARPES bands in Figs. 4(i,j).\nAn analysis of the orbital texture of the EBS for kx= 0.59 ˚A−1[Fig. 4(a)] in Figs. S5(a-\nc) of SM [32], shows that both the main and shadow bands are predominantly of Te px–pz\ncharacter near the crossing regions, while Te pycharacter becomes significant for E ≥0.5\neV, indicating the 2D character of both the main and shadow bands. Further, the orbital\ntextures of the EBS along Γ 2X2[Fig. 4(f)] presented in Figs. S5(d-f) of SM [32] reveal that\nboth TandBbands exhibit significant contributions from the in-plane Te orbitals ( pxand\npz), showcasing their 2D nature. However, above E= 0.4 eV, there is some pycontribution\nin the Tband.\nThe double degeneracy of the bands discussed above is in fact true for all the DFT\ncalculated bands along Γ X, as shown in Fig. 5. The zoomed blue rectangular region in\nFig. 5(b) shows this for the different bands crossing the EF. The zoomed yellow and red\n171.00.50.0\n-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6\n0.900.850.80\n0.020 -0.020\n-0.02\n0.00\n0.02\n-0.52 -0.48 -0.44 -0.40 -0.36\n-0.1\n0.0\n0.1\n0.03 -0.03\nk|| (Å-1)k|| (Å-1)E (eV)\nE (eV) E (eV)d cbaX Z Γ\nX Z Γ X Z ΓFIG. 5. (a) DFT calculated folded band structure of seven-fold YTe 3along XΓZwith SOC. (b)\nA zoomed region from the blue rectangle in panel ashows the two fold degeneracy (dashed-green\nand black) of the bands along Γ Xi.e., the KNL. Zoomed regions from (c) the red and (d) yellow\nrectangles in panel ashow the spin-splitting along the Γ Zdirection, where as all bands along the\nKNL are degenerate.\n18rectangle regions in Figs. 5(c,d) show, in contrast, that the degeneracy is lifted by the SOC\nalong other directions i.e., along Γ Z, whereas it remains in tact along Γ Xboth near EFand\nat larger E, respectively. The seven-fold structure of YTe 3with C2cm space group and the\nlittle group that is isomorphic to C2vpoint group, coupled with protection from time reversal\nsymmetry, enforces the double degeneracy along Γ 2X2,i.e.ΓXor the Σ line [1]. This is the\nsignature of the Kramers nodal line (KNL) [11] that always connects two TRIMs. In the\npresent case, we find that both the upper and lower branches of TandBexhibit quadratic\ndispersion in the vicinity of the crossings [Figs. 4(d,e)]. This type of crossing with quadratic\nand cubic dispersion has been described as higher-order Dirac points (two fold degenerate)\nand the Berry phase around a KNL is quantized as m πmod 2 π[11]. The crossings of the\nCDW-induced shadow with the main bands are enforced by the KNL. The crossings of E(kz)\nbands disperse and traverse the EFaskxincreases [Figs. 4(f-j)] and multiple degenerate\nE(kx) bands along the KNL cross EF[Figs. 5(a,b)] establishing that YTe 3is a KNL metal.\nIV. CONCLUSIONS\nWe present a comprehensive investigation of a non-magnetic chalcogenide material YTe 3\nin the CDW state combining results of the electronic band structure using ARPES, DFT\nperformed with an experiment-based structure determined by x-ray crystallography and\nsurface properties using STM and LEED. We find that the crystal structure of YTe 3is\nnoncentrosymmetric with C2cmbasic space group and qCDWof 0.2907c∗. A commensurate\nfiftyfive-fold structure is derived from the atomic positions determined by x-ray crystallog-\nraphy, which has a qCDWindistinguishable from the x-ray crystallography value within the\nexperimental accuracy. However, since this unit cell is very large, the DFT calculations\nhave been performed with a seven-fold structure, albeit with same symmetries.\nEvidence of two mutually orthogonal CDW domains, each exhibiting an unidirectional\nand similar qCDWis obtained from both LEED and STM. The EBS, obtained by unfolding the\nDFT calculated bands, provides excellent agreement with ARPES when the contributions\nfrom both the twin domains are considered. The CDW gap is barely visible in the FS of\nYTe 3because of the twin domains. Nevertheless, the gap could be identified by comparing\nwith the FS measured at high temperature in the non-CDW state. Thus, twin domains\ninfluence the band structure and the Fermi surface of YTe 3.\n19CDW modulation related weak shadow bands near the BZ boundary, which are shifted\nfrom the main bands by qCDW, have been identified in ARPES intensity plots and are further\ncorroborated by the EBS calculation. Additionally, possible existence of multiple crossings\nbetween the bilayer split main and shadow bands is portrayed both by experiment and\ntheory. The bands calculated using DFT with SOC show a nodal line along Σ resulting\nfrom the crossings between the bilayer split main and shadow bands perpendicular to the\nKNL. Doubly degenerate bands are found only along the KNL for all energy values, with\ncertain bands crossing the EF. The KNL exists only in the CDW phase where shadow bands\nare present and the inversion symmetry is broken. The symmetries of the C2cm space group\nand the time reversal symmetry protect the KNL. Thus, our present work establishes that\nYTe 3is a KNL metal. The intricate interplay between CDW, non-trivial topology and twin\ndomains in YTe 3unveils a rich landscape for further exploration and offers a pathway to\ndeeper insights into exotic topological phases in condensed matter systems.\nV. ACKNOWLEDGMENTS\nS.S., P.S. and S.R.B. gratefully acknowledge the financial support from Department of\nScience and Technology, Government of India within the framework of the DST-Synchrotron-\nNeutron Project to perform experiments at ASTRID2 synchrotron facility. A part of this\nwork was supported by VILLUM FONDEN via the Centre of Excellence for Dirac Materials\n(Grant No. 11744). H. S. Kunwar is thanked for help in the Raman measurement. The Com-\nputer division of Raja Ramanna Centre for Advanced Technology is thanked for installing\nthe DFT codes and providing support throughout. At Argonne, this work was supported by\nthe U.S. Department of Energy, Basic Energy Sciences, Office of Science, Materials Sciences\nand Engineering Division (synthesis, crystals growth, property characterization).\n∗shuvamsarkarhere@gmail.com\n†barmansr@gmail.com\n[1] S. Sarkar, J. Bhattacharya, P. Sadhukhan, D. Curcio, R. Dutt, V. K. Singh, M. Bianchi,\nA. Pariari, S. Roy, P. Mandal, T. Das, P. Hofmann, A. Chakrabarti, and S. R. Barman,\nCharge density wave induced nodal lines in LaTe 3, Nat. Commun. 14, 3628 (2023).\n20[2] H. Li, T. T. Zhang, T. Yilmaz, Y. Y. Pai, C. E. Marvinney, A. Said, Q. W. Yin, C. S. Gong,\nZ. J. Tu, E. Vescovo, C. S. Nelson, R. G. Moore, S. Murakami, H. C. Lei, H. N. Lee, B. J.\nLawrie, and H. Miao, Observation of unconventional charge density wave without acoustic\nphonon anomaly in kagome superconductors AV 3Sb5(A= Rb, Cs), Phys. Rev. X 11, 031050\n(2021).\n[3] W. Shi, B. J. Wieder, H. L. Meyerheim, Y. Sun, Y. Zhang, Y. Li, L. Shen, Y. Qi, L. Yang,\nJ. Jena, P. Werner, K. Koepernik, S. Parkin, Y. Chen, C. Felser, B. A. Bernevig, and Z. Wang,\nA charge-density-wave topological semimetal, Nat. Phys. 17, 381 (2021).\n[4] X. Qian, J. Liu, L. Fu, and J. Li, Quantum spin hall effect in two-dimensional transition metal\ndichalcogenides, Science 346, 1344 (2014).\n[5] S. Lei, S. M. L. Teicher, A. Topp, K. Cai, J. Lin, G. Cheng, T. H. Salters, F. Rodolakis, J. L.\nMcChesney, S. Lapidus, N. Yao, M. Krivenkov, D. Marchenko, A. Varykhalov, C. R. Ast,\nR. Car, J. Cano, M. G. Vergniory, N. P. Ong, and L. M. Schoop, Band engineering of dirac\nsemimetals using charge density waves, Adv. Mater. 33, 2101591 (2021).\n[6] B.-J. Yang and H.-Y. Kee, Searching for topological density-wave insulators in multiorbital\nsquare-lattice systems, Phys. Rev. B 82, 195126 (2010).\n[7] S.-M. Huang, S.-Y. Xu, B. Singh, M.-C. Hsu, C.-H. Hsu, C. Su, A. Bansil, and H. Lin, Aspects\nof symmetry and topology in the charge density wave phase of 1t–TiSe2, New J. Phys. 23,\n083037 (2021).\n[8] H. Polshyn, Y. Zhang, M. A. Kumar, T. Soejima, P. Ledwith, K. Watanabe, T. Taniguchi,\nA. Vishwanath, M. P. Zaletel, and A. F. Young, Topological charge density waves at half-\ninteger filling of a moir´ e superlattice, Nat. Phys. 18, 42 (2021).\n[9] X. Zhang, Q. Gu, H. Sun, T. Luo, Y. Liu, Y. Chen, Z. Shao, Z. Zhang, S. Li, Y. Sun, Y. Li,\nX. Li, S. Xue, J. Ge, Y. Xing, R. Comin, Z. Zhu, P. Gao, B. Yan, J. Feng, M. Pan, and\nJ. Wang, Eightfold fermionic excitation in a charge density wave compound, Phys. Rev. B\n102, 035125 (2020).\n[10] N. Mitsuishi, Y. Sugita, M. S. Bahramy, M. Kamitani, T. Sonobe, M. Sakano, T. Shimojima,\nH. Takahashi, H. Sakai, K. Horiba, H. Kumigashira, K. Taguchi, K. Miyamoto, T. Okuda,\nS. Ishiwata, Y. Motome, and K. Ishizaka, Switching of band inversion and topological surface\nstates by charge density wave, Nat. Commun. 11, 2466 (2020).\n[11] Y.-M. Xie, X.-J. Gao, X. Y. Xu, C.-P. Zhang, J.-X. Hu, J. Z. Gao, and K. T. Law, Kramers\n21nodal line metals, Nat. Commun. 12, 3064 (2021).\n[12] T. Shang, J. Zhao, L.-H. Hu, J. Ma, D. J. Gawryluk, X. Zhu, H. Zhang, Z. Zhen, B. Yu,\nY. Xu, Q. Zhan, E. Pomjakushina, M. Shi, and T. Shiroka, Unconventional superconductivity\nin topological Kramers nodal-line semimetals, Science Advances 8, eabq6589 (2022).\n[13] Y. Zhang, Y. Gao, X.-J. Gao, S. Lei, Z. Ni, J. S. Oh, J. Huang, Z. Yue, M. Zonno, S. Gorovikov,\nM. Hashimoto, D. Lu, J. D. Denlinger, R. J. Birgeneau, J. Kono, L. Wu, K. T. Law, E. Mo-\nrosan, and M. Yi, Kramers nodal lines and weyl fermions in SmAlSi, Communications Physics\n6, 10.1038/s42005-023-01257-2 (2023).\n[14] N. Ru and I. R. Fisher, Thermodynamic and transport properties of YTe 3, LaTe 3, and CeTe 3,\nPhys. Rev. B 73, 033101 (2006).\n[15] V. Brouet, W. L. Yang, X. J. Zhou, Z. Hussain, R. G. Moore, R. He, D. H. Lu, Z. X. Shen,\nJ. Laverock, S. B. Dugdale, N. Ru, and I. R. Fisher, Angle-resolved photoemission study of\nthe evolution of band structure and charge density wave properties in RTe 3(R=Y, La, Ce,\nSm, Gd, Tb, and Dy), Phys. Rev. B 77, 235104 (2008).\n[16] K. Yumigeta, Y. Qin, H. Li, M. Blei, Y. Attarde, C. Kopas, and S. Tongay, Advances in\nrare-earth tritelluride quantum materials: Structure, properties, and synthesis, Adv. Sci. 8,\n2004762 (2021).\n[17] C. D. Malliakas and M. G. Kanatzidis, Divergence in the behavior of the charge density wave\nin RETe 3(RE = rare-earth element) with temperature and RE element, J. Am. Chem. Soc.\n128, 12612 (2006).\n[18] N. Ru, Charge density wave formation in rare-earth tritellurides , Ph.D. thesis, Stanford Uni-\nversity, California (2008).\n[19] J. B. He, P. P. Wang, H. X. Yang, Y. J. Long, L. X. Zhao, C. Ma, M. Yang, D. M. Wang,\nX. C. Shangguan, M. Q. Xue, P. Zhang, Z. A. Ren, J. Q. Li, W. M. Liu, and G. F. Chen,\nSuperconductivity in pd-intercalated charge-density-wave rare earth poly-tellurides rete n, Su-\nperconductor Science and Technology 29, 065018 (2016).\n[20] A. Pariari, S. Koley, S. Roy, R. Singha, M. S. Laad, A. Taraphder, and P. Mandal, Inter-\nplay between charge density wave order and magnetic field in the nonmagnetic rare-earth\ntritelluride LaTe 3, Phys. Rev. B 104, 155147 (2021).\n[21] V. Petˇ r´ ıˇ cek, M. Duˇ sek, and L. Palatinus, Crystallographic computing system jana2006: Gen-\neral features, Zeitschrift f¨ ur Kristallographie - Crystalline Materials 229, 345–352 (2014).\n22[22] G. Oszl´ anyi and A. S¨ ut˝ o, ab initio structure solution by charge flipping, Acta Crystallograph-\nica Section A Foundations of Crystallography 60, 134–141 (2004).\n[23] G. Oszl´ anyi and A. S¨ ut˝ o, Ab initio structure solution by charge flipping. ii. use of weak\nreflections, Acta Crystallographica Section A Foundations of Crystallography 61, 147–152\n(2004).\n[24] S. Hoffmann, C. Søndergaard, C. Schultz, Z. Li, and P. Hofmann, An undulator-based spher-\nical grating monochromator beamline for angle-resolved photoemission spectroscopy, Nucl.\nInstrum. Methods Phys. Res. A 523, 441 (2004).\n[25] Igor Pro Manual , version 9 ed. (2021).\n[26] G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab−initio total-energy calcula-\ntions using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).\n[27] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave\nmethod, Phys. Rev. B 59, 1758 (1999).\n[28] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple,\nPhys. Rev. Lett. 77, 3865 (1996).\n[29] C. Capillas, E. S. Tasci, G. de la Flor, D. Orobengoa, J. M. Perez-Mato, and M. I. Aroyo, A new\ncomputer tool at the bilbao crystallographic server to detect and characterize pseudosymmetry,\nZ. Kristallogr. 226, 186 (2011).\n[30] U. Herath, P. Tavadze, X. He, E. Bousquet, S. Singh, F. Mu˜ noz, and A. H. Romero, PyProcar:\nA python library for electronic structure pre/post-processing, Comput. Phys. Commun. 251,\n107080 (2020).\n[31] K. Momma and F. Izumi, Vesta 3 for three-dimensional visualization of crystal, volumetric\nand morphology data, J. Appl. Crystallogr. 44, 1272 (2011).\n[32] See Supplemental Material at xx.xx for Figs. S1-S5, Table S1 and Discussion A.\n[33] T. Janssen, A. Janner, A. Looijenga-Vos, and P. M. de Wolff, Incommensurate and commen-\nsurate modulated structures, in International Tables for Crystallography (International Union\nof Crystallography, 2006) pp. 907–955.\n[34] S. Van Smaalen, Incommensurate crystallography , Vol. 21 (Oxford University Press, 2007).\n[35] M. F. Wyman and B. F. Wyman, An essay on continued fractions, Math. Syst. Theory 18,\n295 (1985).\n[36] S. Sarkar, V. K. Singh, P. Sadhukhan, A. Pariari, S. Roy, P. Mandal, and S. R. Barman, X-ray\n23photoelectron spectroscopy study of a layered tri-chalcogenide system LaTe 3, AIP Conf. Proc.\n2220 , 100005 (2020).\n[37] N. Ru, C. L. Condron, G. Y. Margulis, K. Y. Shin, J. Laverock, S. B. Dugdale, M. F. Toney,\nand I. R. Fisher, Effect of chemical pressure on the charge density wave transition in rare-earth\ntritellurides RTe 3, Phys. Rev. B 77, 035114 (2008).\n[38] A. Fang, N. Ru, I. R. Fisher, and A. Kapitulnik, Stm studies of tbte 3: Evidence for a fully\nincommensurate charge density wave, Phys. Rev. Lett. 99, 046401 (2007).\n[39] R. G. Moore, V. Brouet, R. He, D. H. Lu, N. Ru, J.-H. Chu, I. R. Fisher, and Z.-X. Shen,\nFermi surface evolution across multiple charge density wave transitions in erte 3, Phys. Rev. B\n81, 073102 (2010).\n[40] M. Lavagnini, M. Baldini, A. Sacchetti, D. Di Castro, B. Delley, R. Monnier, J.-H. Chu, N. Ru,\nI. R. Fisher, P. Postorino, and L. Degiorgi, Evidence for coupling between charge density waves\nand phonons in two-dimensional rare-earth tritellurides, Phys. Rev. B 78, 201101 (2008).\n[41] W. Setyawan and S. Curtarolo, High-throughput electronic band structure calculations: Chal-\nlenges and tools, Comput. Mater. Sci. 49, 299 (2010).\n[42] W. Ku, T. Berlijn, and C.-C. Lee, Unfolding first-principles band structures, Phys. Rev. Lett.\n104, 216401 (2010).\n[43] P. B. Allen, T. Berlijn, D. A. Casavant, and J. M. Soler, Recovering hidden bloch character:\nUnfolding electrons, phonons, and slabs, Phys. Rev. B 87, 085322 (2013).\n[44] G.-H. Gweon, J. D. Denlinger, J. A. Clack, J. W. Allen, C. G. Olson, E. DiMasi, M. C.\nAronson, B. Foran, and S. Lee, Direct observation of complete fermi surface, imperfect nesting,\nand gap anisotropy in the high-temperature incommensurate charge-density-wave compound\nSmTe 3, Phys. Rev. Lett. 81, 886 (1998).\n[45] A. Chikina, H. Lund, M. Bianchi, D. Curcio, K. J. Dalgaard, M. Bremholm, S. Lei, R. Singha,\nL. M. Schoop, and P. Hofmann, Charge density wave generated fermi surfaces in ndte 3, Phys.\nRev. B 107, L161103 (2023).\n24"    },    {        "title": "2210.08048v2.Probing_Fermi_sea_topology_by_Andreev_state_transport.pdf",        "content": "Probing Fermi sea topology by Andreev state transport\nPok Man Tam and Charles L. Kane\nDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA\nWe show that the topology of the Fermi sea of a two-dimensional electron gas (2DEG) is reflected in the\nballistic Landauer transport along a long and narrow Josephson \u0019-junction that proximitizes the 2DEG. The\nlow-energy Andreev states bound to the junction are shown to exhibit a dispersion that is sensitive to the Euler\ncharacteristic of the Fermi sea ( \u001fF). We highlight two important relations: one connects the electron/hole\nnature of Andreev states to the convex/concave nature of Fermi surface critical points, and one relates these\ncritical points to \u001fF. We then argue that the transport of Andreev states leads to a quantized conductance that\nprobes\u001fF. An experiment is proposed to measure this effect, from which we predict an I-Vcharacteristic\nthat not only captures the topology of Fermi sea in metals, but also resembles the rectification effect in diodes.\nFinally, we evaluate the feasibility of measuring this quantized response in graphene, InAs and HgTe 2DEGs.\nIntroduction. Topological classification of quantum matter\nhas led to discoveries of quantized responses that remain ro-\nbust under smooth deformation of the physical system [1–4].\nParadigmatic examples include the one-dimensional (1D) Su-\nSchrieffer-Heeger chain characterized by the electric polar-\nization [5, 6], and the two-dimensional (2D) integer quan-\ntum Hall effect (IQHE) characterized by the Hall conduc-\ntance [7, 8]. In these examples, the quantization is associated\nwith the twisting of wavefunction across the Brillouin zone.\nIn metals, there is another type of topology associated with\nthe structure of “holes” in the Fermi sea, as characterized by\nthe Euler characteristic \u001fF. For instance, doping graphene\nabove/below charge-neutrality creates two electron/hole-like\nFermi pockets, giving \u001fF=\u00062. As every metal can be as-\nsigned an Euler number from the topology of its ground state,\na fundamental question arises: does a metal exhibit any quan-\ntized response for its Euler number?\nIn 1D, the above question is answered by the Landauer for-\nmula [9]: for a ballistic conductor with ideal contacts, the lin-\near conductance is 2e2=htimes the number of occupied spin-\ndegenerate bands, which is precisely \u001fFin 1D [10]. While\nthis quantization is not as robust as the Chern number in the\nIQHE, it has nevertheless been observed in quantum point\ncontacts [11], semiconductor nanowires [12, 13], and carbon\nnanotubes [14]. Such successes motivate us to explore higher-\ndimensional generalizations. Recently, frequency-dependent\nnon-linear response [15], equal-time density correlations and\nmultipartite entanglement [16], have been proposed to probe\n\u001fFof a higher-dimensional Fermi sea. The feasibility of mea-\nsuring a quantized non-linear response in an ultracold atomic\ngas has been further investigated in Refs. [17, 18].\nIn this Letter, we introduce a method to probe \u001fFin 2D\nmetals through nonlocal transport along a planar Josephson\n\u0019-junction with normal leads, as depicted in Fig. 1(a). Sim-\nilar setups were previously considered for studying topolog-\nical superconductivity [19–26], while nonlocal transport has\nemerged as a tool to differentiate topological and trivial phases\n[27, 28]. We show that, in the ballistic limit, biasing the volt-\nage in lead 1 ( V1) and measuring the current flowing into lead2 (I2) results in a quantized two-terminal conductance:\nG21\u0011dI2\ndV1=2e2\nh\u0002\nce\u0012(\u0000V1) +ch\u0012(V1)\u0003\n; (1)\nwhere\u0012(x)is the unit step function. Here ceandchare non-\nnegative integers counting the respective number of convex\nand concave critical points on the Fermi surface, with veloc-\nityvk+^x, as illustrated in Fig. 1(b). While ce;hdepend\non the geometry of the Fermi surface, as well as the relative\norientation of the junction, their difference is only sensitive to\nthetopology of the Fermi sea:\n\u001fF=ce\u0000ch: (2)\nEqs. (1) and (2) predict a topological rectification effect: an\nalternating voltage V1(t) =Vsin!tproduces a direct current\nhI2i=\u00002\u001fFe2jVj=\u0019h. Below, we first explain Eq. (2). We\nthen establish a relation between the convex/concave nature of\ncritical points and the particle/hole nature of Andreev states,\nwhich disperse along the junction and contribute to G21.\nEuler characteristic. The Euler characteristic \u001fwas first de-\nvised for classifying polyhedra [29]. For a connected and\norientable 2D manifold, \u001fis related to the genus gand the\nnumber of boundaries b[30, 31]:\u001f= 2\u00002g\u0000b. Here\nFIG. 1. (a) Proposed setup for measuring \u001fFof a 2DEG. Current\nI2, withV2= 0 andjeV1j<j\u0001j, is determined by the Andreev\nstates dispersing along the junction, and encode information about\nFermi surface critical points. (b) An electron-pocket Fermi sea, with\nconvex/concave Fermi surface critical points marked in red/blue.\nFilled/open circles indicate vk\u0006^x.arXiv:2210.08048v2  [cond-mat.mes-hall]  2 Mar 20232\nwe classify metals according to the Euler characteristic \u001fFof\ntheFermi sea [32]. For instance, an electron pocket (being\ntopologically equivalent to a disk) has \u001fF= 1, while a hole\npocket has\u001fF=\u00001(as the Brillouin zone is equivalent to\nag= 1 torus). Generally, for 2D metals, \u001fFis the number\nof electron-like Fermi surfaces minus the number of hole-like\nFermi surfaces.\nMorse theory provides a connection between \u001fFand the\nenergy dispersion E(k), which can be viewed as a Morse\nfunction. We have \u001fF=P\n\r(\u00001)\rC\r, whereC\rcounts\nthe number of critical points inside the Fermi sea with Morse\nindex\r[33, 34]. A critical point where v=rkE(k)=~= 0\ncan be classified as a minimum, saddle or maximum point,\nwith index\r= 0;1;2;respectively. \ris related to the Hes-\nsianHofE(k):(\u00001)\r= sgn[det H]. It follows that\n\u001fF=Z\nd2kf0(k)[@\u0012(vx)\n@kx@\u0012(vy)\n@ky\u0000@\u0012(vy)\n@kx@\u0012(vx)\n@ky];(3)\nwithf0(k)\u0011\u0012(EF\u0000E(k))defining the Fermi sea.\nIt is illuminating to convert the integral in Eq. (3) into\na form that only receives contribution on the Fermi surface.\nThis is achieved by adding zero to the integrand of the form\n\u0012(vx)@kx\u0012(vy)@kyf0[15, 35], and integrate by parts to obtain\n\u001fF=\u0000Z\nd2k@f0(k)\n@kx@\u0012(vy)\n@ky\u0012(vx): (4)\nThe first term @f0=@kxisolates the Fermi surface, while\n\u0012(vx)@\u0012(vy)=@kyisolates the critical points on the Fermi sur-\nface, wherevy= 0 andvx>0. These points are called\nconvex (if@kyvy>0) orconcave (if@kyvy<0), whose local\nneighbourhood resembles an electron -like or a hole-like Fermi\nsurface respectively. Denoting the number of convex/concave\ncritical points as ce=h, we arrive at Eq. (2). Alternatively,\n\u001fF= \u0016ce\u0000\u0016ch, where \u0016ce;hcount critical points with vy= 0\nandvx<0. Time-reversal symmetry requires ce;h= \u0016ce;h.\nWe will first neglect spin-orbit interactions, so ce;h,\u0016ce;h\nand\u001fFare defined for the spin-degenerate Fermi sea. In the\nend, and in the supplementary [36], we address the effect of\nRashba spin-orbit coupling (SOC), and argue that our results\nremain valid.\nCritical points and Andreev states. Let us now relate Fermi\nsurface critical points to properties of Andreev bound states\n(ABS) formed at the SNS junction (S denotes an s-wave su-\nperconductor, N denotes the normal metal of interest). We\nconsider a junction geometry shown in Fig. 1(a), with L\u001d\n\u0018\u001dW. Here,Lis the length of the junction, \u0018the super-\nconducting coherence length and Wthe distance between two\nSN interfaces. The narrow-junction minimizes the number of\nABSs, and the long-junction suppresses crossed Andreev re-\nflections between leads. We also take the adiabatic limit, in\nwhich all potentials vary smoothly on the scale of the Fermi\nwavelength k\u00001\nF. WithkF\u0018\u001d1, the pairing gap does not\nalter the topology of the Fermi sea.\nOur proposal concerns a \u0019-junction, across which the su-perconducting order parameter changes sign. A distinguished\nfeature of\u0019-junctions is the existence of zero energy ABSs\n[37–39]. Intuitively, ABSs are formed by mixing between\nelectron and hole states on the Fermi surface due to Andreev\nreflection [40]. Away from Fermi surface critical points, states\nnear the Fermi surface are governed by a linear dispersion (in\nthey-direction, or perpendicular to the SN-interfaces). Corre-\nspondingly, the Bogoliubov-de Gennes (BdG) equation takes\nthe form of a 1D Dirac equation with a spatially varying mass\nterm [39]. The \u0019phase-change in the pairing potential implies\na “kink” configuration of the mass term, leading to a Jackiw-\nRebbi zero mode localized at the domain wall [41].\nThe above picture breaks down close to a Fermi surface\ncritical point, where the dispersion is quadratic. Aside from\nAndreev reflections, one also needs to consider normal reflec-\ntions that hybridize electrons at +kyand\u0000ky(which coincide\nat the critical point). This gives rise to a dispersion of the ABS\nalong the junction (i.e. the x-direction). As shown in Fig. 2(a,\nb), on one side of a critical point the ABS is dispersionless\n(with almost zero energy), while a significant dispersion be-\ngins near the critical point, such that on the other side the ABS\napproaches the bulk band edge. The dispersion implies that\nthe ABS propagates along the junction, and eventually goes\ninto the lead. The process of entering the lead can be mod-\neled in two ways: fixing Wwhile taking the superconduct-\ning gapj\u0001j!0(as done below), or fixing j\u0001jwhile taking\nW!1 (see the supplementary [36]). In both formulations,\nprovided the transition is adiabatic, the positive-energy ABS\nconverts into a particle/hole if the associated critical point is\nconvex/concave. Combining this with Eq. (2) allows us to\nprobe\u001fFthrough transport. Next, we support these claims by\nexamining the dispersion of the ABS obtained from the BdG\nequation.\nDispersion of ABS. Consider a narrow SNS \u0019-junction in the\nlimitW!0. Expanding around a Fermi surface critical\npoint, we obtain the following BdG Hamiltonian:\nHBdG= (\u0000a@2\ny+vx\u000ekx)\u001cz+sgn(y)j\u0001j\u001cy: (5)\nHere,\u001c’s are the Pauli matrices acting on the particle-hole\nspace, andj\u0001jis the magnitude of the s-wave pairing gap. In\nthe adiabatic limit we assume translation invariance along the\njunction, so the momentum kxis conserved. \u000ekxmeasures\nthe deviation away from the critical point of interest, where\nthe Fermi velocity is (vx;0).ais related to the electron’s\neffective mass, with sgn( a)=\u0006for a convex/concave critical\npoint. Note the W!0limit is taken only for convenience,\nas an analytic solution for the single ABS is available. Finite\nWis studied in the supplementary [36], where a single ABS,\nwith similar features, is found provided W.p\njaj=j\u0001j.\nThe critical-point model has two symmetries useful for\nfinding the ABS. (1) Chiral symmetry \u0005 =\u001cx, with\nfHBdG;\u0005g= 0.\u0005relates states of energy \u0006\"for a fixed\n\u000ekx. (2) Mirror symmetry My=\u001czMy(whereMytakes\ny7!\u0000y), with [HBdG;My] = 0 . Each ABS can be labeled by\nMy=\u00061. Asf\u0005;Myg= 0, particle-hole partners (related3\nby\u0005) acquire opposite mirror eigenvalues. It is sufficient to\nfocus on theMy= 1sector.\nDenote \t(y)as the solution of HBdG\t =\"\t. As \u0001is\nuniform away from y= 0,\t(y)is a linear combination of\nattenuated waves of the form e\u0000\u0014y, with\na\u00142=vx\u000ekx\u0006iq\nj\u0001j2\u0000\"2: (6)\nOnly two of the four solutions, denoted \u0014\u0006(with Re (\u0014\u0006)>0,\nIm(\u0014\u0006)?0), correspond to bound states for y>0, so\n\ty>0=\r+\u00121\niesgn(a)i\u0011\u0013\ne\u0000\u0014+y+\r\u0000\u00121\nie\u0000sgn(a)i\u0011\u0013\ne\u0000\u0014\u0000y:\n(7)\nThe phasee\u0006sgn(a)i\u0011\u0011\"=j\u0001j\u0006isgn(a)p\n1\u0000(\"=j\u0001j)2char-\nacterizes the mixing between particle and hole due to Andreev\nreflections, while \r+and\r\u0000account for normal reflections.\nUsing the mirror symmetry, the ABS wavefunction in the\nlower plane is \t(y < 0) =My\t(y > 0). Matching two\nboundary conditions required by the quadratic BdG equation,\nwe find\n\t(0\u0000) = \t(0+) :\r\u0000=\r+=\u0000e2isgn(a)\u0011; (8a)\n@y\t(0\u0000) =@y\t(0+) :\r\u0000=\r+=\u0000\u0014+=\u0014\u0000: (8b)\nThese, together with Eq. (6), give the dispersion of the ABS:\n\"=\u0006hvx\u000ekx\n2+sgn(a)s\n\u0010vx\u000ekx\n2\u00112\n+j\u0001j2\n2i\n:(9)\nFigure 2 (a) and (b) show a plot of this for a >0anda <0\nrespectively. Right at the Fermi surface critical point ( \u000ekx=\n0), we find\"=\u0006j\u0001j=p\n2. Away from it, and on the inside of\nthe Fermi pocket, we have \"\u0019\u0006j \u0001j2=(2vx\u000ekx). Deep inside\nthe Fermi pocket, the ABS thus becomes an approximate zero\nmode, which is consistent with the familiar result obtained by\nneglecting normal reflections [37–39].\nOn the other side of the critical point ( outside the Fermi\npocket), we obtain \"\u0019\u0006vx\u000ekx. Notice that the band edge of\nthenormal dispersion, which describes physical states in the\nnormal region (without BdG doubling), is precisely located at\nEedge=vx\u000ekx. This leads us to classify two types of ABS:\n(i) Electron-like ABS , which converges to the normal band\nedgeEedgefor\">0and appears near a convex critical point\n(a > 0); (ii) Hole-like ABS , which converges to the normal\nband edgeEedgefor\"<0and appears near a concave critical\npoint (a<0).\nAn electron/hole-like ABS inside the junction would con-\nvert definitively into an electron/hole when it propagates into\nthe normal leads. This is because for a fixed \", asj\u0001j!0,\nthe corresponding ABS is pushed deep into the regime where\nits dispersion coincides with either the electron-like or the\nhole-likeEedge. It thus acquires their electron/hole charac-\nter. The same conclusion is reached when we model the lead\nby widening the junction ( W!1 ) [36]. This is our sec-\nFIG. 2. BdG spectrum of an SNS \u0019-junction, obtained around (a): a\nconvex critical point (i.e. a >0, marked in red) and (b): a concave\ncritical point (i.e. a < 0, marked in blue). Shaded regions repre-\nsent the continuum of bulk states. The ABS dispersion is plotted in\ngreen, using Eq. (9), with the solid curve representing the physical\nstate. The solid ray indicates the normal band edge. Dashed lines\nare the particle-hole partners of the solid ones (related by \u001cx). (c):\nTop panel shows a Fermi sea with \u001fF=\u00001; Lower panel shows the\nschematic BdG spectrum deduced from (a) and (b). There are ce=h\nright-moving (and \u0016ce=hleft-moving) electron/hole-like ABSs.\nond key relation: there is an electron/hole-like ABS for each\nconvex/concave Fermi surface critical point. Moreover, each\nABS (by which we mean specifically the physical state that\ncan adiabatically continue into the normal region) propagates\nalong the junction in the same direction as vx. Figure 2(c)\nillustrates this relation for a Fermi sea with \u001fF=\u00001.\nQuantized conductance and topological rectification. Now\nwe propose an experiment to extract \u001fFfrom a two-terminal\ntransport along an SNS\u0019-junction. The setup is depicted in\nFig. 1(a). Let us fix V2= 0, so that no Andreev reflection hap-\npens between lead 2 and the superconductor. Then I2is solely\ncontributed by normal transmissions between leads. Using the\nLandauer formalism we find [36, 42],\nI2=2e\nhZ1\n\u00001T21(E) [f(E\u0000eV1)\u0000f(E)]dE: (10)\nThe factor of 2 is due to spin (SOC is considered later) and\nf(E) = (e\fE+ 1)\u00001is the Fermi distribution at temperature\n\f\u00001.T21(E)describes transmission from lead 1 to lead 2,\nwhich we denote by Te=h\n21forE?0. ForjeV1j<j\u0001jthe\nonly available channels are the electron/hole-like dispersive\nABSs, which derive from the convex/concave critical points.\nHence\nTe\n21(E) =ce\u0012(E\u0000\u000e); Th\n21(E) =ch\u0012(\u0000E\u0000\u000e);(11)\nwith reflectionless contacts and ballistic transport assumed.\nParameter\u000eis introduced to characterize the splitting of\nABSs. For a \u0019-junction, while \u000e= 0 in the linearized4\n𝐼!/(2𝑒Δ/ℎ)𝑒𝑉!/Δ𝛽Δ=10𝛽Δ=100𝐼!/(2𝑒Δ/ℎ)𝑒𝑉!/Δ\nFIG. 3.I2-V1characteristic curves for transport along the junction,\nrevealing the shape of the Fermi sea. In the ohmic regime, the slope\n(in unit of 2e2=h) is quantized to ce=hforV170. Red/blue curve\nshown here corresponds to an electron/hole-like Fermi pocket with\nce=h= 1andch=e= 0. Here\u000e= 0:1j\u0001j.\nmodel with exact zero modes, there are realistic reasons for\n\u000e > 0. For instance, a parabolic normal dispersion H0=\np2=2m\u0000EFimplies\u000e\u0019\f\f\u00012=(2EF)\f\f. For an ideal graphene\nSNS junction with H0=vp\u0001\u001b\u0000EF, one finds\u000e= 0[43],\nbut an imperfect junction transparency causes \u000e >0[44, 45].\nDifferent ABSs would also have different splittings in gen-\neral. Nonetheless, our working assumption is \u000e\u001cj\u0001j, so we\ninclude one parameter to model these realistic effects.\nFollowing Eqs. (10, 11), we obtain\nI2(V1) =2e\nh\u0002\nceg(V1)\u0000chg(\u0000V1)\u0003\n; (12)\nwith\ng(V) =\f\u00001ln[(1 +e\f(eV\u0000\u000e))=(1 +e\u0000\f\u000e)]\n\f!1\u0000\u0000\u0000\u0000! (eV\u0000\u000e)\u0012(eV\u0000\u000e):(13)\nForkBT=\u000e= 0, the two-terminal conductance G21\u0011\ndI2=dV1is quantized in the form of Eq. (1). This quantization\nremains robust provided j\u0001j>jeV1j\u001d\u000e;kBT.\nEquation (12) implies a unique I-Vcharacteristic for trans-\nport along the SNS \u0019-junction. Figure 3 shows the I2-V1\ncurves for two simplest geometries of the Fermi sea, which\nexhibit distinct unidirectional behaviors. Different from the\nAndreev rectifier proposed in Ref. [23], the rectification pre-\ndicted here is intrinsic to the topology of a metal. For generic\nshapes of the Fermi surface, and for different orientations\nof the junction, integers ce=hcould vary and lead to differ-\nent slopes of the I-Vcurve in the backward/forward-biased\nregime. Nevertheless, \u001fF=ce\u0000chis a robust topological\nquantity, insensitive to either the Fermi surface geometry or\nthe junction orientation.\nDiscussion. While SOC is ignored in the above analysis, its\neffect can be easily incorporated. Let us include the Rashba\nterm\u0015R(k\u0002\u001b)\u0001^zinto the Hamiltonian, with \u001bbeing the spin\nPauli matrix. Turning on \u0015R>0splits the spin-degenerateFermi surfaces by shrinking/enlarging the one with sgn[(k\u0002\n\u001b)\u0001^z] =\u00061. As before, we begin our analysis deep inside\nboth Fermi pockets, so that we can linearize around the Fermi\nsurface. The BdG Hamiltonian for the \u0019-junction is then\nH(kx) = sgn(y)j\u0001j\u001cy+f\u0000i\u0016zvy@y+\n\u0015R[kx\u001by\u0000(\u0016zkF\ny\u0000i@y)\u001bx]g\u001cz;(14)\nwhere\u0016zvy=\u0006vyis the Fermi velocity for the up/down-\nmover around the Fermi point at \u0006kF\ny. Forkx= 0, there are\ntwo pairs of Jackiw-Rebbi zero modes with \u0016z\u001cx= +1 . For\ngenerickx,H(kx)belongs to the symmetry class CII [46, 47],\nwith the chiral symmetry implemented by \u00050=\u0016z\u001cx. Hence\nthe zero modes, which all share the same \u00050-eigenvalue, can-\nnot hybridize for generic kx. Nonetheless, the above argu-\nment breaks down when kxapproaches Fermi surface criti-\ncal points, around which ABSs would begin dispersing due to\nnormal reflections, and eventually merge with the bulk contin-\nuum outside the Fermi pockets. This is confirmed by obtain-\ning the full dispersion of ABSs in the Rashba model [36]. We\nshall thus replace 2\u001fFin previous discussions by \u001fR\nF+\u001fL\nF, i.e.\nthe sum of Euler characteristics for the two Fermi seas with\nright(R)- and left(L)-handed spin-momentum locking. Simi-\nlarly, 2ce;h!cR\ne;h+cL\ne;h. As such, our quantized-transport\nformulas remain valid in the presence of Rashba SOC.\nFinally, we suggest specific avenues for realizing our pro-\nposal. One platform is the InAs or HgTe 2DEG, which has\nbeen fabricated into planar junctions for studying topologi-\ncal superconductivity [24–26]. Measurements of the nonlocal\nconductance ( G21) have been reported recently [27, 28]. An-\nother avenue, where SOC is much weaker, is the graphene-\nbased Josephson junction [43], where tunneling spectroscopy\nhas been performed to measure the spectrum of ABSs [44,\n45]. The ability to tune the chemical potential through charge-\nneutrality, where \u001fF(per spin) changes between \u00062, allows\none to demonstrate an abrupt jump in conductance. To ob-\nserve a robust quantization, a ballistic ,long andnarrow junc-\ntion (`\u001dL\u001d\u0018\u001dW) is required, where `is the mean free\npath. While narrow planar junctions are available [24, 25, 44],\nthe challenge is posed by a short mean free path ( `\u0018\u0018) in\nsuch systems. Away from the ballistic limit, while the quanti-\nzation is no longer exact, G21is still expected to transit dras-\ntically as the topology of Fermi sea changes. In summary, our\nwork calls for near-term experimental effort to look for quan-\ntized transport along a planar SNS \u0019-junction, which probes\nthe intrinsic topology of 2D metals.\nAcknowledgments. We thank Ady Stern for helpful discus-\nsions. This work was supported by a Simons Investigator\nGrant to C.L.K. from the Simons Foundation.\n[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).\n[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).\n[3] C.-K. Chiu, J. C. Y . Teo, A. P. Schnyder, and S. Ryu, Rev. Mod.5\nPhys. 88, 035005 (2016).\n[4] X.-G. Wen, Rev. Mod. Phys. 89, 041004 (2017).\n[5] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. B 22,\n2099 (1980).\n[6] D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48, 4442\n(1993).\n[7] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,\n494 (1980).\n[8] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den\nNijs, Phys. Rev. Lett. 49, 405 (1982).\n[9] R. Landauer, IBM J. Res. Dev. 1, 223 (1957).\n[10] Modern definition of \u001finvolves the alternating sum of Betti\nnumbers:\u001f=b0\u0000b1+b2:::, whereb`counts the topological\nclasses of`-cycles on the manifold. b0counts the number of\nconnected components. For a 1D Fermi sea, b0thus counts the\nnumber of occupied bands. In the main text, we discuss more\ndetailed ways to calculate \u001fin 2D.\n[11] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G.\nWilliamson, L. P. Kouwenhoven, D. van der Marel, and C. T.\nFoxon, Phys. Rev. Lett. 60, 848 (1988).\n[12] T. Honda, S. Tarucha, T. Saku, and Y . Tokura, Japanese Journal\nof Applied Physics 34, L72 (1995).\n[13] I. van Weperen, S. R. Plissard, E. P. Bakkers, S. M. Frolov, and\nL. P. Kouwenhoven, Nano letters 13, 387 (2013).\n[14] S. Frank, P. Poncharal, Z. Wang, and W. A. d. Heer, Science\n280, 1744 (1998).\n[15] C. L. Kane, Phys. Rev. Lett. 128, 076801 (2022).\n[16] P. M. Tam, M. Claassen, and C. L. Kane, Phys. Rev. X 12,\n031022 (2022).\n[17] F. Yang and H. Zhai, arXiv:2206.09845 (2022).\n[18] P. Zhang, arXiv:2207.02382 (2022).\n[19] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).\n[20] B. J. Wieder, F. Zhang, and C. L. Kane, Phys. Rev. B 89,\n075106 (2014).\n[21] M. Hell, M. Leijnse, and K. Flensberg, Phys. Rev. Lett. 118,\n107701 (2017).\n[22] F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and\nB. I. Halperin, Phys. Rev. X 7, 021032 (2017).\n[23] T. O. Rosdahl, A. Vuik, M. Kjaergaard, and A. R. Akhmerov,\nPhys. Rev. B 97, 045421 (2018).\n[24] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portol ´es, A. C. C.\nDrachmann, A. Keselman, S. Gronin, C. Thomas, T. Wang,\nR. Kallaher, G. C. Gardner, E. Berg, M. J. Manfra, A. Stern,\nC. M. Marcus, and F. Nichele, Nature 569, 89 (2019).\n[25] H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky,\nL. Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W.\nMolenkamp, et al. , Nature 569, 93 (2019).\n[26] A. Banerjee, O. Lesser, M. Rahman, H.-R. Wang, M.-R. Li,\nA. Kringhøj, A. Whiticar, A. Drachmann, C. Thomas, T. Wang,\net al. , arXiv:2201.03453 (2022).\n[27] A. Banerjee, O. Lesser, M. Rahman, C. Thomas, T. Wang,\nM. Manfra, E. Berg, Y . Oreg, A. Stern, and C. Marcus,\narXiv:2205.09419 (2022).\n[28] Unlike the measurements reported in Ref. [27], our proposal\nrequires a\u0019-junction without an in-plane magnetic field.\n[29] L. Euler, Novi commentarii academiae scientiarum Petropoli-\ntanae , 109 (1758).\n[30] M. Nakahara, Geometry, topology and physics , Graduate stu-\ndent series in physics (Hilger, Bristol, 1990).\n[31] T. Dieck, Algebraic Topology , EMS textbooks in mathematics\n(European Mathematical Society, 2008).\n[32] Periodicity of the Brillouin zone ensures that the Fermi sea is\na compact manifold, possibly with multiple connected compo-\nnents.[33] J. Milnor, Morse Theory , Annals of Mathematics Studies\n(Princeton University Press, 1963).\n[34] C. Nash and S. Sen, Topology and Geometry for Physicists (El-\nsevier Science, 1988).\n[35] This term is clearly zero, as @f0=@ky=vy@f=@E , while\n@\u0012(vy)=@kxfixesvy= 0.\n[36] Supplementary Materials.\n[37] I. O. Kulik, Soviet Journal of Experimental and Theoretical\nPhysics 30, 944 (1969).\n[38] C. W. J. Beenakker and H. van Houten, Phys. Rev. Lett. 66,\n3056 (1991).\n[39] J. Sauls, Philosophical Transactions of the Royal Society\nA: Mathematical, Physical and Engineering Sciences 376,\n20180140 (2018).\n[40] A. Andreev, Sov. Phys. JETP 19, 1228 (1966); 22, 445 (1966).\n[41] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).\n[42] S. Datta, Electronic Transport in Mesoscopic Systems , Cam-\nbridge Studies in Semiconductor Physics and Microelectronic\nEngineering (Cambridge University Press, 1995).\n[43] M. Titov and C. W. J. Beenakker, Phys. Rev. B 74, 041401(R)\n(2006).\n[44] L. Bretheau, J. I.-J. Wang, R. Pisoni, K. Watanabe,\nT. Taniguchi, and P. Jarillo-Herrero, Nature Physics 13, 756\n(2017).\n[45] S. Park, W. Lee, S. Jang, Y .-B. Choi, J. Park, W. Jung,\nK. Watanabe, T. Taniguchi, G. Y . Cho, and G.-H. Lee, Nature\n603, 421 (2022).\n[46] J. C. Y . Teo and C. L. Kane, Phys. Rev. B 82, 115120 (2010).\n[47]H(kx)has two anti-unitary symmetries, \u00020=\u0016y\u001bz\u001czKand\n\u00040=\u0016x\u001bz\u001cyK, with \u000202= \u000402=\u00001.6\nSupplementary Materials for “Probing Fermi sea topology by Andreev state transport”\nPok Man Tam and Charles Kane\nWe present three supplementary sections to provide further details to our discussion in the main text. In Sec. A, we model\na normal lead as a wide SNS junction, from which we verify our claim about the fate of an Andreev bound state (ABS) as it\nmoves from a narrow junction into the lead. In Sec. B, we apply the Landauer formalism to an SNS junction and derive Eq.\n(10) in the main text, which allows us to relate the transport properties along the junction to the topology of Fermi sea. In Sec.\nC, we solve for the dispersive ABSs in the presence of Rashba spin-orbit coupling (SOC). Our solution confirms that there is\none branch of electron/hole-like ABS for each electron/hole-like Rashba-split Fermi pocket, which disperses drastically around\na Fermi surface critical point.\nA. Modeling the lead: Andreev states in a wide junction\nIn the main text, we found an analytic expression for the dispersion of ABS in an extremely narrow ( W= 0) SNS\u0019-junction,\nand deduced the fate of the ABS excitation as it moves from the junction into the normal lead: the ABS turns into an electron/hole\nexcitation if the normal dispersion is an electron/hole-like band. There the process of propagating from the junction into the lead\nis modeled by adiabatically taking the pairing gap j\u0001j!0. Here we provide an alternative picture by modeling the lead as a\nwide junction with W\u001d\u0018(\u0018is the superconducting coherence length). Propagating from the narrow junction into the normal\nlead then corresponds to following an ABS as Wis increased adiabatically (as represented in Fig. 1(a) of the main text). These\ntwo pictures (j\u0001j!0andW!1 ) are equivalent, as demonstrated below.\nAs in the main text, we focus on the neighbourhood of a Fermi surface critical point, and consider the following Bogoliubov-de\nGennes (BdG) Hamiltonian:\nHBdG= (\u0000a@2\ny+vx\u000ekx)\u001cz+ [\u0012(y\u0000W=2)\u0000\u0012(\u0000y\u0000W=2)]\u0001j\u0001j\u001cy (A.1)\nThis is similar to Eq. (5) in the main text, so we are not explaining again the meaning of parameters introduced before. A major\ndifference is that we now have a normal region of width W > 0, sandwiched between two superconductors with opposite pairing\nphases. Just like Eq. (5), this model has the same chiral symmetry ( \u0005 =\u001cx) and mirror symmetry (My=\u001czMy). Hence, it\nis again sufficient to focus on the My= 1sector, and use \u0005to obtain the full spectrum which is particle-hole symmetric. First\nconsider region (I) with \u0000W=2< y < W= 2. In this normal region, the ABS at energy \"should have a wavefunction of the\nfollowing form,\n\tI(y) = \u0000+\ne\u00121\n0\u0013\neikey+ \u0000\u0000\ne\u00121\n0\u0013\ne\u0000ikey+ \u0000+\nh\u00120\n1\u0013\neikhy+ \u0000\u0000\nh\u00120\n1\u0013\ne\u0000ikhy: (A.2)\nHereak2\ne=h=\u0000vx\u000ekx\u0006\", which is required by the BdG equation HBdG\t =\"\t. AsMy\t = \t , we have \u0000+\ne= \u0000\u0000\neand\n\u0000+\nh=\u0000\u0000\u0000\ne. We can thus express,\n\tI(y) =\u0012\u0000ecoskey\n\u0000hsinkhy\u0013\n: (A.3)\nNext, we consider region (II) with y > W= 2, which is inside a superconductor. A generic bound-state solution takes the exact\nsame form as Eq. (7) in the main text, i.e.\n\tII(y) =\r+\u00121\niesgn(a)i\u0011\u0013\ne\u0000\u0014+(y\u0000W=2)+\r\u0000\u00121\nie\u0000sgn(a)i\u0011\u0013\ne\u0000\u0014\u0000(y\u0000W=2); (A.4)\nwithe\u0006sgn(a)i\u0011\u0011\"=j\u0001j\u0006isgn(a)p\n1\u0000(\"=j\u0001j)2. Again,\u0014\u0006are the two solutions for a\u00142=vx\u000ekx\u0006iq\nj\u0001j2\u0000\"2that satisfy\nRe(\u0014\u0006)>0andIm(\u0014\u0006)?0.\nThe ABS is determined by matching two boundary conditions at the SN interface: \tI(W=2) = \t II(W=2)and@y\tI(W=2) =7\nFIG. A.1. Left: BdG spectrum in an SNS \u0019-junction of width (a) W= 10k\u00001\nFand (b)W= 70k\u00001\nF. Here we have taken ak2\nF=j\u0001j= 100 ,\nwherekFis the Fermi wavevector. Each state in the spectrum is colored according to its Q-value, which reflects its particle-hole nature.\nShaded regions represent the bulk states. Following a particular ABS (of a fixed \") asWincreases, we see that its Q-value approaches\u00061.\nRight: wavefunction \t(y) = (u;v)Tof a representative state marked by a star in the spectrum. The superconducting region, with jyj>W= 2,\nis shaded. The chosen state has \"= 0:5j\u0001j. Comparison between (a) and (b) demonstrates the fate of this ABS as it transits from the narrow\njunction into the normal lead.\n@y\tII(W=2). These can be summarized as B(\u0000e;\u0000h;\r+;\r\u0000)T= 0, with\nB=0\nBB@coskeW\n20\u00001\u00001\n0 sinkhW\n2\u0000iesgn(a)i\u0011\u0000ie\u0000sgn(a)i\u0011\n\u0000kesinkeW\n20 \u0014+\u0014\u0000\n0khcoskhW\n2i\u0014+esgn(a)i\u0011i\u0014\u0000e\u0000sgn(a)i\u00111\nCCA: (A.5)\nA non-trivial solution requires detB= 0. Solving this gives us all the allowed energies at a given \u000ekx, from which we obtain the\ndispersion of all ABSs inside the junction with width W. The corresponding wavefunctions can be read-off from the null-space\nofB. In addition to the spectrum, we are also interested in the particle-hole character of the ABS, as we want to determine the\nfate of an ABS as it propagates into the lead (i.e. as Wincreases). Thus, for the wavefunction \t(y) = (u;v)T, let us assign the\nfollowing quantity to characterize the particle-hole nature of the state:\nQ=R1\n\u00001dy(juj2\u0000jvj2)\nR1\n\u00001dy(juj2+jvj2): (A.6)\nAs such, a definitive particle-state has Q= +1 , while a definitive hole-state has Q=\u00001.\nFigure A.1 (a,b) (left panels) show the BdG spectra for two junction widths W, where each ABS is displayed in a color\nreflecting itsQ-value. For each W, the wavefunction of a representative state (chosen at \"= 0:5j\u0001j) is plotted in the right\npanel. We have illustrated here the case with a>0, so the normal dispersion is electron-like. An ABS with \">0then converts\ndefinitively into an electron as it propagates from a narrow junction ( W\u001c\u0018) into the lead ( W\u001d\u0018). On the other hand, if\na<0, it would be the ABS with \"<0that acquiresQ! +1asW!1 . The physical states thus belong to a hole-like band,\nand hence the ABS excitation would convert definitively into a hole inside the lead. Notice that we have arrived at the same\nconclusion as we did in the main text, while there we considered taking j\u0001j!0instead ofW!1 .\nLast but not least, let us remark on the length scales involved in this problem. In a sense, there are two superconducting coher-\nence lengths. Away from the critical point where the dispersion is linear, i.e. E(ky) =vFky, the level-spacing is roughly vF=W.8\nClose to the critical point where the dispersion is quadratic, i.e. E(ky) =ak2\ny, the spacing becomes jaj=W2. The condition for\nanarrow junction, which hosts one branch of ABS for each Fermi surface critical point, is that min(vF=W;jaj=W2)&j\u0001j.\nDenoting the usual coherence length as \u0018=vF=j\u0001j, and the one associated to the critical point as \u0018c=p\njaj=j\u0001j, we see that\n\u0018c\u001c\u0018forEF\u001dj\u0001j. This feature is reflected in Fig. A.1(b), where we see that the level-spacing is larger for vx\u000ekx=j\u0001j.\u00001,\ncompared to the spacing for vx\u000ekx=j\u0001j\u00190. Our proposal requires a narrow junction that satisfies W.\u0018c.\nB. Landauer formalism for the proposed setup\nHere we derive an expression for the net current Ipflowing into lead pin our proposed setup. In particular, we obtain Eq.\n(10) in the main text. The net current Ipin leadpis contributed by the following scattering processes. For an incident electron-\nexcitation (E > 0) in then-th transport mode moving from lead pinto the junction, it can be either normally-reflected back into\nleadpas an electron (with probability rN\nn;p(E)), or Andreev reflected back into lead pas a hole (with probability rA\nn;p(E)), or\ncrossed Andreev reflected into lead q6=pas a hole (with probability tA\nn;qp(E)). As for an incident hole-excitation ( E < 0),\nthere are three similar scattering processes, and the corresponding scattering probabilities are denoted in the same way, but with\nE < 0. As such, the net current flowing into lead pcan be expressed as\nIp=\u00002e\nhX\nnnZ1\n0dE\u0002\n(1\u0000rN\nn;p+rA\nn;p)fp(E)\u0000X\nq6=p(tN\nn;pq\u0000tA\nn;pq)fq(E)\u0003\n\u0000Z0\n\u00001dE\u0002\n(1\u0000rN\nn;p+rA\nn;p)\u0016fp(E)\u0000X\nq6=p(tN\nn;pq\u0000tA\nn;pq)\u0016fq(E)\u0003o\n:(B.1)\nHerefp(E) =f(E\u0000eVp), withf(E) = (1 +e\fE)\u00001, and \u0016fp(E) = 1\u0000fp(E), which are the respective Fermi distributions\nfor electrons and holes in lead p. The factor of 2 upfront accounts for spin-degeneracy. Unitarity demands\nrN\nn;p+rA\nn;p+X\nq6=p(tN\nn;qp+tA\nn;qp) = 1: (B.2)\nNotice that we assume tA\nn;qp= 0, i.e. crossed Andreev reflection is suppressed. This can be achieved by a long Josephson junc-\ntion, with length Lmuch larger than the coherence length \u0018of the superconductor (see Fig. 1(a) in the main text). Eliminating\nrN\nn;p, we obtain\nIp=\u00002e\nhX\nnnZ1\n0dE\u0002X\nq6=p\u0000\ntN\nn;qpfp(E)\u0000tN\nn;pqfq(E)\u0001\n+ 2rA\nn;pfp(E)\u0003\n\u0000Z0\n\u00001dE\u0002X\nq6=p\u0000\ntN\nn;qp\u0016fp(E)\u0000tN\nn;pq\u0016fq(E)\u0001\n+ 2rA\nn;p\u0016fp(E)\u0003o\n:(B.3)\nLet us define the normal transmission functions, as well as the Andreev reflection functions, for electrons/holes as follows,\nTe=h\nqp(E) =(P\nntN\nn;qp(E); E?0\n0; E 70andRe=h\np(E) =(P\nnrA\nn;p(E); E?0\n0; E 70: (B.4)\nWe then obtain Ip=IN\np+IA\np, with\nIN\np=\u00002e\nhX\nq6=pZ+1\n\u00001[Te\nqp(E)fp(E)\u0000Te\npq(E)fq(E)\u0000Th\nqp(E)\u0016fp(E) +Th\npq(E)\u0016fq(E)]dE; (B.5a)\nIA\np=\u00004e\nhZ1\n\u00001\u0002\nRe\np(E)fp(E)\u0000Rh\np(E)\u0016fp(E)\u0003\ndE: (B.5b)\nNow getting back to the measurement protocol specified in the main text. We are interested in measuring I2as a function of\nV1, by holding V2= 0fixed. As lead 2 is notbiased from the superconductor, IA\n2= 0. We also have I2(V1= 0) = 0 , as there9\nshould be no flow of current at equilibrium. Altogether,\nI2(V1) =IN\n2(V1)\u0000IN\n2(0) =2e\nhZ+1\n\u00001[Te\n21(E) +Th\n21(E)]\u0001[f1(E)\u0000f2(E)]dE\n=2e\nhZ+1\n\u00001[Te\n21(E) +Th\n21(E)]\u0001[f(E\u0000eV1)\u0000f(E)]dE:(B.6)\nThis is Eq. (10) in the main text.\nC. Dispersive Andreev states with Rashba SOC\nHere we solve for the dispersive ABSs in the presence of Rashba SOC. We confirm the argument presented in the main text,\nwhich suggests that there is one branch of electron/hole-like ABS for each electron/hole-like Rashba-split Fermi pocket, see Fig.\nC.1. Similar to the case without SOC, the ABS is an approximate zero-mode ( \"\u0019\f\f\u00012=(2EF)\f\f) inside the Fermi pocket, and\nbecomes strongly dispersive past a Fermi surface critical point. Our proposal for measuring Fermi sea topology thus applies to\ntwo-dimensional electron gas (2DEG) with SOC, leading to a quantized conductance that probes \u001fR\nF+\u001fL\nF, whereR=L refers\nto the chirality of spin-momentum locking: sgn[(k\u0002\u001b)\u0001^z] =\u00061.\nLet us consider the following BdG Hamiltonian:\nHBdG= [a(k2\nx\u0000@2\ny)\u0000EF+\u0015R(kx\u001by+i@y\u001bx)]\u001cz+sgn(y)j\u0001j\u001cy: (C.1)\nAs in the main text, we consider the extreme narrow-junction limit ( W= 0) and model the lead by taking j\u0001j!0. Here,\u001c’s\nare the Pauli matrices acting on the particle-hole space, and \u001b’s are the Pauli matrices acting on the physical spin space. The\nstrength of Rashba SOC is characterized by \u0015R\u00150. The effective mass mof the 2DEG is related to the model parameter a\nbya=~2=(2m), thus an electron/hole-like Fermi pocket has sgn (a) =\u00061. In addition to the chiral symmetry \u0005 =\u001cx, with\nfHBdG;\u0005g= 0, we also have a mirror- ysymmetryMy=My\u001by\u001cz, whereMyflips the real-space coordinate y7!\u0000y. It is\nstraightforward to check that [HBdG;My] = 0 . MoreoverfMy;\u0005g= 0, hence particle-hole partners (at the same kxand with\nenergy\u0006\", as related by \u0005) acquire opposite mirror eigenvalues.\nNext, we construct the bound-state ansatz \ty>0that diagonalizes HBdGin the upper-half plane. Inside the superconductor,\nthe bound-state wavefunction should be a linear combination of decaying modes eikyy, with Im(ky)>0. Each mode eikyyis\nassociated with a spinor d\u001b\ns(ky)that diagonalizes the SOC term:\n(kx\u001by\u0000ky\u001bx)d\u001b\ns(ky) =sq\nk2x+k2yd\u001b\ns(ky) =)d\u001b\ns(ky) =\u0010\n1; s\u0001ikx\u0000kyq\nk2x+k2y\u0011T\n; (C.2)\nwiths=\u00061, andq\nk2x+k2yis defined to have a positive real part. The complex momentum kythen obeys the following\ncharacteristic equation for the BdG eigen-energy \":\nh\na(k2\nx+k2\ny)\u0000EF+s\u0015Rq\nk2x+k2yi2\n=\"2\u0000j\u0001j2\n=)q\nk2x+k2y=1\n2jajh\n\u0000s\u0001sgn(a)\u0015R+r\n\u00152\nR+ 4aEF\u00064aiq\nj\u0001j2\u0000\"2i (C.3)\nThis gives four solutions to kythat satisfy Im(ky)>0. Fors\u0001sgn(a) = +1 , we have two solutions k+;1andk+;2:\nk+;1=2=\u0006s\n\u0000k2x+1\n4a2\u0010\n\u0015R\u0000r\n\u00152\nR+ 4aEF\u00064jajiq\nj\u0001j2\u0000\"2\u00112\n: (C.4)\nHere, the spinor that diagonalizes the physical spin space is d\u001b\nsgn(a), while the corresponding spinor that diagonalizes the particle-\nhole space is (1;ie\u0007sgn(a)i\u0011)T, withei\u0011:=\"=j\u0001j+ip\n1\u0000(\"=j\u0001j)2. Fors\u0001sgn(a) =\u00001, we have two solutions k\u0000;1=2:\nk\u0000;1=2=\u0006s\n\u0000k2x+1\n4a2\u0010\n\u0015R+r\n\u00152\nR+ 4aEF\u00064jajiq\nj\u0001j2\u0000\"2\u00112\n: (C.5)10\nFIG. C.1. Dispersion of Andreev bound states for the two Rashba-split Fermi pockets, labeled as red and blue curves respectively. Here we\nhave chosen ak2\nF=EF= 10j\u0001j, wherekFis the Fermi wavevector for the unsplit Fermi surfaces with \u0015R= 0. Only the positive energy\nspectrum is shown. The shaded region corresponds to the continuum of bulk states. While the inner branch of ABS (shown in red) only exists\nforj\"j<j\u0001j, the outer branch remains a valid bound state for j\"j>j\u0001j. Asj\u0001j!0, both ABSs merge with the bulk continuum and acquire\nthe electron/hole nature as dictated by sgn(a).\nThe spinor that diagonalizes the physical spin subspace is d\u001b\n\u0000sgn(a), and the corresponding spinor that diagonalizes the particle-\nhole subspace is (1;ie\u0007sgn(a)i\u0011)T. Altogether, we can construct a generic ansatz to the uniform BdG equation in the upper-half\nplane:\n\ty>0= (\r+;1eik+;1y\u0001d\u001b\nsgn(a)+\r\u0000;1eik\u0000;1y\u0001d\u001b\n\u0000sgn(a))\n\u00121\nie\u0000sgn(a)i\u0011\u0013\n\u001c\n+(\r+;2eik+;2y\u0001d\u001b\nsgn(a)+\r\u0000;2eik\u0000;2y\u0001d\u001b\n\u0000sgn(a))\n\u00121\niesgn(a)i\u0011\u0013\n\u001c:(C.6)\nThe subscript \u001cindicates spinors in the particle-hole subspace, and \r’s are coefficients to be determined by the boundary\nconditions.\nWe now construct the bound-state wavefunction in the lower-half plane, \ty<0, and match boundary conditions to find \"(kx).\nUsing the mirror symmetry My=My\u001by\u001cz, we shall assume \ty<0=My\ty>0. The chiral symmetry \u0005, which anti-commutes\nwith bothHBdGandMy, allows us to obtain another solution with energy \u0000\"andMy=\u00001, thus forming the full BdG\nspectrum. The quadratic BdG equation requires two boundary conditions, \t(0+) = \t(0\u0000)and@y\t(0+) =@y\t(0\u0000), which\ncan be summarized as B(\r+;1;\r\u0000;1;\r+;2;\r\u0000;2)T= 0, with\nB=0\nBB@f(a;k+;1)f(\u0000a;k\u0000;1) f(a;k+;2) f(\u0000a;k\u0000;2)\nf(\u0000a;k+;1)f(a;k\u0000;1)e2 sgn(a)i\u0011f(\u0000a;k+;2)e2 sgn(a)i\u0011f(a;k\u0000;2)\nf(\u0000a;k+;1)\u0001k+;1f(a;k\u0000;1)\u0001k\u0000;1f(\u0000a;k+;2)\u0001k+;2 f(a;k\u0000;2)\u0001k\u0000;2\nf(a;k+;1)\u0001k+;1f(\u0000a;k\u0000;1)\u0001k\u0000;1e2 sgn(a)i\u0011f(a;k+;2)\u0001k+;2e2 sgn(a)i\u0011f(\u0000a;k\u0000;2)\u0001k\u0000;21\nCCA(C.7)\nandf(a;ky) = 1 +isgn(a)ikx\u0000kyp\nk2x+k2y. From detB= 0, we can self-consistently solve for the dispersion of ABS. Resorting to\nnumerical solutions, we obtain two branches of ABSs, one for each Rashba-split Fermi pocket. The results are shown in Fig.\nC.1."    },    {        "title": "0802.2341v1.Low_frequency__K_π_0______modes_in_deformed_neutron_rich_nuclei__Pairing__and__β__vibrational_modes_of_neutron.pdf",        "content": "arXiv:0802.2341v1  [nucl-th]  16 Feb 2008Low-frequency Kπ= 0+modes in deformed neutron-rich nuclei: Pairing- and\nβ-vibrational modes of neutron\nKenichi Yoshida\nDepartment of Physics, Graduate School of Science, Kyoto Un iversity, Kyoto 606-8502, Japan\nMasayuki Yamagami\nNishina Center for Accelerator Based Science, The Institut e of Physical\nand Chemical Research (RIKEN), Wako, Saitama 351-0198, Jap an\n(Dated: September 15, 2021)\nLow-frequency Kπ= 0+states in deformed neutron-rich nuclei are investigated by means of\nthe quasiparticle-random-phase approximation based on th e Hartree-Fock-Bogoliubov formalism in\nthe coordinate space. We have obtained the very strongly col lectiveKπ= 0+modes not only\nin neutron-rich Mg isotope but also in Cr and Fe isotopes in N= 40 region, where the onset of\nnuclear deformation has been discussed. It is found that the spatially extended structure of neutron\nquasiparticle wave functions around the Fermi level brings about a striking enhancement of the\ntransition strengths. It is also found that the fluctuation o f the pairing field plays an important role\nin generating coherence among two-quasiparticle excitati ons of neutron.\nPACS numbers: 21.10.Re; 21.60.Ev; 21.60.Jz\nI. INTRODUCTION\nPhysics of nuclei far from β-stability has been one of\nthe main current subjects of nuclear physics. Quest for\nnew kinds of collective motion in exotic unstable nuclei\nis one of the most interesting issues in nuclear structure\nphysics and has been actively studied both experimen-\ntally and theoretically [1]. Because low-lying collective\nexcitations are sensitive to the shell structure around the\nFermi level, one can expect unique excitation modes to\nemergeassociatedwith the new spatialstructuressuchas\nneutron skins and novel shell structures which generate\nnew regions of deformation.\nBreaking of the spherical magic number N= 20, 28\nand striking enhancement of B(E2;0+\n1→2+\n1) in Mg iso-\ntopes towards drip line are under lively discussions in\nconnectionwith onsetofthe quadrupoledeformationand\ncontinuum coupling [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Fur-\nthermore in heavier mass region, collectivities and break-\ning of theN= 40 sub-shell have been recently discussed\nfor neutron-rich Cr and Fe isotopes [15, 16, 17, 18, 19].\nAlthough many investigations based on the random\nphase approximation (RPA) or the Quasiparticle RPA\n(QRPA) including the pairing correlations [20, 21, 22,\n23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]\non multipole responses of neutron-rich nuclei have been\ndone, they are largely restricted to spherical systems.\nRecently, low-lying RPA modes in deformed neutron-\nrich nuclei have been investigated by several groups [39,\n40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. These calculations,\nhowever,do not takeintoaccountthe pairingcorrelation,\nor rely on the BCS pairing (except for Ref. [46] and our\nvery recent investigation [49]), which is inappropriate for\ndescribing the pairing correlation in drip line nuclei due\nto the unphysical nucleon gas problem [50].\nIn Ref. [51], we constructed a new computer code\nthat carries out the deformed QRPA calculation basedon the coordinate-spaceHartree-Fock-Bogoliubov(HFB)\nformalism, and investigated quadrupole excitations in\nneutron-rich Mg isotopes close to the drip line around\nN= 28. We obtained low-lying Kπ= 0+and 2+\nquadrupole excitation modes in36,38,40Mg. We found\nthat theKπ= 0+mode in40Mg was generated by co-\nherent superposition of neutron two-quasiparticle (2qp)\nexcitations in loosely bound and resonant states. Among\nthem, the neutron 2qp excitations of up-sloping oblate\nlevels and down-sloping prolate levels play the major\nrole. A deformed gap is formed at N= 28 around\nβ2= 0.3 [12] due to the crossings between the up-sloping\nν[303]7/2level and the down-sloping ν[310]1/2level, and\nthis deformed closed shell approximately corresponds to\nthe (f7/2)−2(p3/2)2configuration in the spherical shell\nmodel representation [9] (see Fig. 1). It was also found\nthat the dynamical pairing correlation, i.e., fluctuation\nof the pairing field played a crucial role in generating\ncoherence among neutron 2qp excitations.\nThe single-particle levels which are (2 j+ 1)−fold de-\ngenerateinthe sphericalpotentialaresplitbythenuclear\ndeformation to 2-fold degenerate levels with the single-\nparticle energy to first order of β2as [13]\n∼β23Ω2−j(j+1)\nj(j+1). (1)\nIn a prolately deformed potential ( β2>0), a level with\nsmall value of Ω ( z−component of the angular momen-\ntumj) has a negative slope. We call this single-particle\nlevel a down-sloping prolate level in this paper, and refer\na positive slope level with large Ω as a up-sloping oblate\nlevel.\nBecausethedeformedshellgapis, inmanycases,made\nby the up-sloping and the down-sloping levels [14], we in-\nvestigatein this paper detailed and systematic properties\nof low-frequency Kπ= 0+modes in deformed neutron-\nrich Mg, Cr and Fe isotopes, and discuss generic features2\nof the soft Kπ= 0+modes uniquely appeared in de-\nformed neutron-rich nuclei.\nThe paper is organized as follows. In the next section,\nwe present our model of the deformed HFB and QRPA.\nNumerical results and discussion are given in §III. Fi-\nnally, we summarize the paper in §IV.\nII. MODEL\nWe briefly summarize our approach (see Ref. [51] for\ndetails). In order to discuss simultaneously effects of nu-\nclear deformation and pairing correlations including the\ncontinuum, we solve the HFB equations [50, 52, 53]\n/parenleftbigg\nhτ(rσ)−λτ˜hτ(rσ)\n˜hτ(rσ)−(hτ(rσ)−λτ)/parenrightbigg/parenleftbiggϕτ\n1,α(rσ)\nϕτ\n2,α(rσ)/parenrightbigg\n=Eα/parenleftbiggϕτ\n1,α(rσ)\nϕτ\n2,α(rσ)/parenrightbigg\n(2)\ndirectly in the cylindrical-coordinatesassumingaxialand\nreflection symmetries. Here τ=ν(neutron) and π(pro-\nton), and r= (ρ,z,φ). For the mean-field Hamiltonian\nh, we employ the deformed Woods-Saxon potential with\nthe parameters used in Ref. [51] for34Mg and those in\nRef. [39] for neutron-rich Cr and Fe isotopes. The pair-\ning field is treated self-consistently by using the density-\ndependent contact interaction [54, 55],\nvpp(r,r′) =V01−Pσ\n2/bracketleftbigg\n1−̺IS(r)\n̺0/bracketrightbigg\nδ(r−r′).(3)\nwith̺0= 0.16 fm−3. Here̺IS(r) denotes the isoscalar\ndensity and Pσthe spin exchange operator. Because the\ntime-reversalsymmetryandthereflectionsymmetrywith\nrespect to the x−yplane are assumed, we have only to\nsolve for positive Ω and positive z. We use the lattice\nmesh size ∆ ρ= ∆z= 0.8 fm and the box boundary\ncondition at ρmax= 10.0 fm andzmax= 12.8 fm. The\nquasiparticle energy is cut off at 50 MeV and the quasi-\nparticle states up to Ωπ= 15/2±are included.\nUsing the quasiparticle basis obtained by solving the\nHFB equation (2), we solve the QRPA equation in the\nmatrix formulation [56]\n/summationdisplay\nγδ/parenleftbigg\nAαβγδBαβγδ\nBαβγδAαβγδ/parenrightbigg/parenleftbiggfλ\nγδ\ngλ\nγδ/parenrightbigg\n=/planckover2pi1ωλ/parenleftbigg\n1 0\n0−1/parenrightbigg/parenleftbiggfλ\nαβ\ngλ\nαβ/parenrightbigg\n.\n(4)\nThe residual interaction in the particle-particle (p-p)\nchannel appearing in the QRPA matrices AandBis the\ndensity-dependent contact interaction (3). On the other\nhand, for the residual interaction in the particle-hole (p-\nh) channel, we use the Skyrme-type interaction\nvph(r,r′) =/bracketleftbigg\nt0(1+x0Pσ)+t3\n6(1+x3Pσ)̺IS(r)/bracketrightbigg\nδ(r−r′),\n(5)-20-15-10-50\n00.2 0.4 0.6 0.8Energy (MeV)\nβ2f7/2\nd5/2122022\n[220]1/2[211]3/2[211]1/2[202]5/2[200]1/2[202]3/2\n[330]1/2[321]3/2[312]5/2[310]1/2[303]7/2[312]3/2\n[440]1/228[301]1/2\ns1/2d3/2p3/2p1/2\np1/2\nFIG. 1: Single-particle energies in the deformed WS po-\ntential for neutrons in40Mg, plotted as functions of the\nquadrupole deformation parameter β2. Solid and dotted\nlines denote positive- and negative-parity levels, respec tively.\nSingle-particle levels are labeled with the asymptotic qua n-\ntum numbers [ Nn3Λ]Ω.\nwitht0=−1100 MeV ·fm3,t3= 16000 MeV ·fm6,x0=\n0.5, andx3= 1.0 [57]. Because the deformed Wood-\nSaxon potential is used for the mean field, we renor-\nmalize the residual interaction in the p-h channel by\nmultiplying a factor fphto get the spurious Kπ= 1+\nmode (representing the rotational mode) at zero energy\n(vph→fph·vph). We cut the 2qp space at Eα+Eβ≤30\nMeV due to the excessively demanding computer mem-\nory for the model space consistent with that adopted in\nthe HFB calculation. Accordingly, we need another fac-\ntorfppfor the p-p channel. We determine this factor\nsuch that the spurious Kπ= 0+mode associated with\nthe particle number fluctuation appears at zero energy\n(vpp→fpp·vpp). The dimension of the QRPA equa-\ntion (4) is 2512 for34Mg and 3804 for72Fe. Within\nthis model space, the energy-weighted sum rule for the\nquadrupole Kπ= 0+excitation is satisfied 92 .9% for\n34Mg and 97.0% for72Fe.\nIII. RESULTS AND DISCUSSION\nLet us now discuss the low-frequency Kπ= 0+modes\nin34Mg in§IIIA and in neutron-rich Cr and Fe isotopes\nwithN≃40 in§IIIB.\nA.34Mg\nFigure 1 shows the single-particle energy diagram for\nthe WS potential as a function of deformation parameter\nβ2. At around β2= 0.5, the deformed shell gap is formed\natN= 22 due to the crossings between the up-sloping\nν[202]3/2 and the down-sloping ν[330]1/2 andν[321]3/2\nlevels. In Fig. 2, we show the low-lying excitation spec-\ntrum below 3 MeV in34Mg. Here excitation energies are3\n0123Energy (MeV)\nExp. Cal.0+2+(4+)\n0+2+4+\n0+2+\nKπ=0+Kπ=2+1−\nKπ=0−\n2+3+\nTV Belyaev34\nMg\nFIG. 2: Excitation energy spectrum in34Mg obtained by the\nQRPA calculation using β2= 0.4 andV0=−400 MeV ·fm3\nand available experimental data [4].\nevaluated by [58]\nE(I,K) =/planckover2pi1ωRPA+/planckover2pi12\n2JTV(I(I+1)−K2),(6)\nwiththevibrationalfrequencies, ωRPA, andtheThouless-\nValatinmomentofinertia, JTV, calculatedbytheQRPA.\nThe pairing strength V0for34Mg is determined in or-\nder to reproduce the experimental excitation energy of\nE(2+\n1) = 660 keV [4]. In this figure, we also show the\nexcitation energies evaluated by using the Belyaev mo-\nment of inertia, JBelyaev, where the residual interaction\nis turned off. We found that JTVbecomes about 11%\nlarger than JBelyaevdue to the time-odd component in\nthe residual interactions of (3) and (5). The Kπ= 2+\nmode at 2.01MeVis dominantlygeneratedby the proton\np-h excitations as that in36,38,40Mg [51]; these are pro-\nton dominated γ−vibrational modes. The negative par-\nityKπ= 0−state at 2.61 MeV is mainly generated by\nthelowestneutron2qpexcitationof ν[202]3/2⊗ν[321]3/2\nwith the excitation energy of 2.74 MeV. Its collectivity\nis thus rather weak.\nThe left panel of Fig. 3 shows the Kπ= 0+isoscalar\nquadrupole transition strengths, and the unperturbed\n2qp transition strengths are shown in the lower panel.\nThe intrinsic transition strength to the lowest Kπ= 0+\nexcited state is about 43 Weisskopf units (1 W.u. ≃6.5\nfm4for34Mg). This state is constructed by the coher-\nent superposition of 2qp excitations of the up-sloping\n(ν[202]3/2)2and the down-sloping ( ν[330]1/2)2and the\n(ν[321]3/2)2levels, together with the high-lying 2 /planckover2pi1ωex-\ncitation of protons.\nGeneration mechanism of the soft Kπ= 0+mode in\ndeformed neutron-richnuclei is understood essentially by\nthe schematic two-levelmodel in Ref. [59]. They consider\nthecasewhereonlytwo λ¯λcomponentsarepresentinthe\nwave functions both of the ground 0+\ngsand of the excited050100150200250300350isoscalar34\nMg\nKπ=0+\n01020304050\n012345Strength (fm4)\n/104ω (MeV)unperturbed00.40.81.21.634\nMg\nP00²\n00.20.40.60.81\n012345Strength\n/104ω (MeV)unperturbed010203040506070 34\nMg\nP20²\n051015\n012345Strength (fm4)\n/104ω (MeV)unperturbed\nFIG. 3: QRPA strength distributions for the Kπ= 0+\nisoscalar quadrupole p-h excitations (left) and the monopo le-\nand quadrupole-pair excitations (middle and right) in34Mg,\ncalculated using fph= 0.74 andfpp= 1.37. For comparison,\nunperturbed 2qp transition strengths are shown in the lower\npanels. The arrow indicates the neutron threshold energy\nEth= 4.55 MeV for 1qp continuum.\n0+\n2states;\n|0+\ngs/an}bracketri}ht=a√\na2+b2|λ1¯λ1/an}bracketri}ht+b√\na2+b2|λ2¯λ2/an}bracketri}ht(7a)\n|0+\n2/an}bracketri}ht=−b√\na2+b2|λ1¯λ1/an}bracketri}ht+a√\na2+b2|λ2¯λ2/an}bracketri}ht.(7b)\nThe transition matrix element for the quadrupole opera-\ntor is then\n/an}bracketle{t0+\ngs|ˆQ20|0+\n2/an}bracketri}ht=2ab\na2+b2[/an}bracketle{tλ1|ˆQ20|λ1/an}bracketri}ht−/an}bracketle{tλ2|ˆQ20|λ2/an}bracketri}ht] (8)\nand it is proportional to the difference in the quadrupole\nmoments of the individual orbitals composing the 0+\nstates. In the case that the quadrupole moments of the\norbitals have opposite signs to each other, this matrix\nelement becomes large. This situation is realized in the\nlevel crossing region, where the up- and down-sloping or-\nbitals exist. As the number of components increases in\nthe QRPA calculations, the wave function becomes more\ncomplicated,andthetransitionmatrixelementreads(see\nAppendix B)\n/an}bracketle{tKπ= 0+|ˆQ20|0/an}bracketri}ht=/summationdisplay\nαβQ(uv)\n20,αβ(fαβ+gαβ)≡/summationdisplay\nαβM(uv)\n20,αβ.\n(9)\nIn the QRPA employing the separable interaction, for\nthe majorcomponent ofthe QRPAeigenmode associated\nwith the pairing fluctuation, the phase of the QRPA am-\nplitudes,fαβandgαβ, are opposite between particle-like\nlevels and hole-like levels (due to the ( u2−v2) factor;\nseeAppendix J of Ref. [60]). In the level crossing re-\ngion, the particle-like level is the up-sloping oblate level,\nand the hole-like level is the down-sloping prolate level.\nBecause the transition matrix elements for the QRPA\neigenmodes are determined by the sum of products of\nthe QRPA amplitudes and the individual 2qp transition4\nTABLE I: QRPA amplitudes for the Kπ= 0+state at 1.37 MeV in34Mg. This mode has the proton strength B(E2) =\n15.9e2fm4, the neutron strength B(Qν2) = 162fm4, and the isoscalar strength B(QIS2) = 279fm4, and the sum of backward-\ngoing amplitudesP|gαβ|2= 0.141. The single-quasiparticle levels are labeled with the a symptotic quantum numbers [ Nn3Λ]Ω.\nOnly components with f2\nαβ−g2\nαβ>0.01 are listed.\nEα+Eβ Q(uv)\n20,αβM(uv)\n20,αβP(uu)\n20,αβM(add)\n20,αβP(uu)\n00,αβM(add)\n00,αβ\nα β (MeV) fαβgαβ(fm2) (fm2) (fm2) (fm2)\n(a)ν[202]3/2ν[202]3/2 2.70 −0.615−0.230−1.814 1.533 −1.128 0.864 0.579 −0.453\n(b)ν[321]3/2ν[321]3/2 2.79 0.688 0.052 4.962 3.673 2.335 1.745 0.443 0.334\n(c)ν[330]1/2ν[330]1/2 4.50 0.371 0.062 6.177 2.676 2.524 1.329 0.278 0.146\n0246810121416\n051015202530020406080100∑M20 (fm2)\n%\nE2qp (MeV)34\nMg\n/104ω=1.37 MeV\ntotal\nneutron\nFIG. 4: Sum of the transition matrix elementP\nαβM(uv)\n20,αβfor\nthe lowest Kπ= 0+state at 1.37 MeV. Solid and dotted lines\ndenote the isoscalar and neutron transitions, respectivel y.\nmatrix elements in Eq. (9), they contribute coherently\nfor the quadrupole transition.\nIn the present case, the main components generating\nthe low-lying Kπ= 0+state are [202]3 /2⊗[202]3/2,\n[321]3/2⊗[321]3/2 and [330]1 /2⊗[330]1/2, which are\nthe up-sloping and down-sloping levels with opposite\nquadrupolemoments and the QRPAamplitudes haveop-\nposite signs as shown in Table I. Therefore, the strength\nfor the quadrupole transition becomes enhanced. These\nthree 2qp excitations contribute about 50 % of the to-\ntal transition strength. Furthermore, many other 2qp\nexcitations coherently participate to generate the lowest\n0+state, which brings about further enhancement of the\ntransition strength.\nFigure 4 showsthe partialsum ofthe transition matrix\nelement for the Kπ= 0+mode/summationtext\nEα+Eβ≦E2qpM(uv)\n20,αβfor\nthe isoscalar and neutron excitations. We see that many\n2qp excitations construct this state with coherence; each\n2qp transition matrix element ( M(uv)\n20,αβ) has a same sign.\nIf the QRPA mode were a destructive state or a single\n2qp excitation, the sum of the transition matrix element\nbecomes zero in total or shows step function.\nFrom this figure, we see another interesting feature:\nSoftKπ= 0+mode in34Mg is generated not only\nby neutron 2qp excitations around the Fermi level with\nthe excitation energy around 5 MeV (three 2qp ex-\ncitations discussed above) but also proton excitationsaround 20 MeV. These proton excitations are 2 /planckover2pi1ωexci-\ntations of giant resonance; π[211]3/2→π[431]3/2 (19.5\nMeV),π[101]3/2→π[321]3/2 (17.3 MeV), π[110]1/2→\nπ[330]1/2 (19.6 MeV) and π[220]1/2→π[440]1/2 (20.1\nMeV). Due to this coupling with giant resonance of pro-\nton, the proton transition strength B(E2) is larger than\nthat in40Mg;B(E2) = 15.9e2fm4in34Mg and 3.4e2fm4\nin40Mg [51]. Hindrance of the proton excitation in drip-\nline nuclei is due to the extreme spatial extension of neu-\ntron wave functions and decoupling with proton wave\nfunctions [51].\nThe discussion for enhancement of the quadrupole\ntransition strength to the low-lying Kπ= 0+mode is\nalso applicable to the pair transition strength. In the\nmiddle and right panels of Fig. 3, we show two kinds of\nstrengthforneutronpairtransitions; oneisthe monopole\npair transition and another the quadrupole pair transi-\ntion,\nˆP†\n00=/integraldisplay\ndrˆψ†(r↑)ˆψ†(r↓), (10a)\nˆP†\n20=/integraldisplay\ndrr2Y20(ˆr)ˆψ†(r↑)ˆψ†(r↓).(10b)\nFor the quadrupole pair transitions, we see a promi-\nnent peak at 1.37 MeV, whereas no peak is seen at\nthis energy for the monopole pair transitions. The ori-\ngin of this contrasting behavior is understood as fol-\nlows: The quadrupole (monopole) pairing matrix ele-\nmentsforindividual2qpexcitationsareinoppositephase\n(in phase) between the down-sloping prolate levels and\nthe up-sloping oblate levels. As the transition matrix el-\nements for the QRPA eigenmodes are determined by the\nsimilar way for the p-h transition matrix elements (See\nEq. (B4)), they contribute coherently and destructively\nforthequadrupoleandmonopolepairtransitions,respec-\ntively. As a consequence, the quadrupole-pair transition\nstrength to the lowest Kπ= 0+collective excited state is\ngreatly enhanced, whereas the monopole-pair transition\nstrength to this state almost vanishes.\nTherefore, in a deformed system where the up- and\ndown-sloping orbitals exist near the Fermi level and the\npairingfluctuationbecomesimportant, wecanexpectthe\nemergence of the low-lying mode which has extremely\nenhanced strengths both for the quadrupole p-h and for5\n0246810\n02468z (fm)\n δρ(π)\n0246810\n02468 \n δρ(ν)\n0246810\n02468z(fm)\nρ(fm)Ppp(ν)\n0246810\n02468 \nρ(fm)Phh(ν)\nFIG. 5: Transition densities to the Kπ= 0+state at 1.37\nMeV. The particle-hole transition densities of protons and\nneutrons (upper panels), the particle-particle and hole-h ole\ntransition densities of neutrons (lower panels). Solid and\ndashed lines indicate positive and negative transition den si-\nties, and the contour lines are plotted at intervals of 3 ×10−4\nfm−3. The thick dotted lines denote the proton and neutron\nhalf densities, 0.041 and 0.049 fm−3, respectively.\nthe quadrupole p-p (pair) transition.\nThe low-lying Kπ= 0+modes possibly emerge both\nin stable and in unstable deformed nuclei, and the unique\nfeature in neutron-richnuclei is that they aremainly gen-\nerated by neutrons, whose wave functions have spatially\nextended structure due to the shallow Fermi level. In or-\nder to see the unique spatial structure, we show in Fig. 5\nthree kinds of transition densities;\nδ̺(τ)(ρ,z) =/an}bracketle{tλ|/summationdisplay\nσψ†\nτ(ρ,z,σ)ψτ(ρ,z,σ)|0/an}bracketri}ht,(11a)\nPpp(ν)(ρ,z) =/an}bracketle{tλ|ψ†\nν(ρ,z,↑)ψ†\nν(ρ,z,↓)|0/an}bracketri}ht,(11b)\nPhh(ν)(ρ,z) =/an}bracketle{tλ|ψν(ρ,z,↓)ψν(ρ,z,↑)|0/an}bracketri}ht,(11c)\nwhere they are called the p-h transition density, the p-p\n(particle-pair) transition density, and the h-h (hole-pair)\ntransition density, respectively [27]. The upper panels\nshow a typical shape vibration of protons and neutrons\nalongthe symmetryaxis( z−axis). Wecanclearlyseethe\nspatial extension of the neutron wave functions; the exci-\ntation takes place in the skin region around 7–8 fm. This\nspatial extension of the neutron wave functions brings\nabout the enhancement of the transition strength. An-\nother unique feature of the low-lying Kπ= 0+mode is\nthat the particle-pair and hole-pair transition densities\nhave an appreciable amplitude and are almost compara-\nble to the p-h transition density.-8-6-4-20\n-0.2 00.20.40.6Energy (MeV)\nβ2g9/2\nf5/2\np1/2\np3/238404446\n[440]1/2[431]3/2[422]5/2[431]1/2[301]3/2[301]1/2[303]5/2[413]7/2[420]1/2\n[312]3/2[404]9/2\nFIG. 6: Same as Fig. 1 but in64Cr.\nB. Cr and Fe isotopes around N= 40\nThe deformation region near the shell closure gives a\nfavorable situation for emergence of the soft Kπ= 0+\nmodesdiscussedinthe previoussubsectionbecausemany\norbitals with opposite quadrupole moments are crossing\naround the Fermi level. Since the low-lying Kπ= 0+\nmode is quite sensitive to the pairing correlation and de-\nformation, we investigate in the present subsection the\nlow-frequency Kπ= 0+excitations in neutron-rich Cr\nand Fe isotopes in N= 40 region, and show generic fea-\ntures of the soft Kπ= 0+modes in deformed neutron-\nrich nuclei.\nIn a prolately deformation region around β2≃0.3,\ndeformed shell gaps associated with different β2values\nappear at various neutron numbers due to the cross-\nings between the down-sloping ν[440]1/2,ν[431]3/2 and\nν[431]1/2 levels and the up-sloping ν[301]3/2,ν[301]1/2\nandν[303]5/2 levels (see Fig. 6).\nAccording to the Skyrme HFB calculations [61, 62],\nthe ground state of the neutron-rich Cr and Fe isotopes\nwithN≃40 has a finite deformation around β2= 0.3.\nThe top panels in Fig. 7 show the isoscalar quadrupole\ntransition strengths for the Kπ= 0+excitations. We\ncan see a prominent peak at around 1 MeV in all nuclei\nunder consideration. All low-lying states obtained here\nhave extremely enhanced transition strengths (note that\n1 W. u. is 14.6–17.1 fm4for62−70Cr and 15.2–17.8 fm4\nfor64−72Fe). These excitation modes are generated by\ncoherent superposition of neutron 2qp excitations among\nthe up- and the down-sloping levels as in34Mg.\nFurthermore, we can see an interesting feature from\nthis systematic calculation; the transition strengths be-\ncome enhanced at N= 38 and 46, and are symmet-\nric atN= 42. This feature can be understood by\nthe role of the Ωπ= 1/2+levels. In62Cr and64Fe,\n2qp excitations of the up-sloping ( ν[301]3/2)2and the\ndown-sloping ( ν[440]1/2)2have largest contributions to\nthe lowest Kπ= 0+state. The transition strength of\nthe (ν[440]1/2)2excitation is 86 fm4(5.9 W.u.) in62Cr\nand 64 fm4(4.2 W.u.) in64Fe, whereas that of the6\n02004006008001000120062\nCr\n050010001500200025003000\n012345Strength (fm4)\n64\nFe64\nCr\n01234566\nFe66\nCr\n012345\n/104ω (MeV)68\nFe68\nCr\n01234570\nFe70\nCr\n01234572\nFe\n00.40.81.21.6262\nCrP00²\n00.40.81.21.62\n012345Strength 64\nFe64\nCr\n01234566\nFe66\nCr\n012345\n/104ω (MeV)68\nFe68\nCr\n01234570\nFe70\nCr\n01234572\nFe\n060120180240300 62\nCr\nP20²\n0120240360480600\n012345Strength (fm4)\n64\nFe64\nCr\n01234566\nFe66\nCr\n012345\n/104ω (MeV)68\nFe68\nCr\n01234570\nFe70\nCr\n01234572\nFe\nFIG. 7: Isoscalar quadrupole (top), the monopole-pair (mid dle), and the quadrupole-pair (bottom) transition strengt hs for the\nKπ= 0+excitations in62−70Cr and64−72Fe obtained by the QRPA calculations using β2= 0.3 andV0=−420 MeV ·fm3. The\nself-consistent factors are fph=0.72–0.74 and fpp= 1.43 for Cr isotopes, and fph=0.74–0.79 and fpp=1.39–1.42 for Fe isotopes.\nThe arrows denote the neutron threshold energies.\n(ν[301]3/2)2is 9.6 fm4and 8.6 fm4, respectively. This\nindicates the ν[440]1/2 level has a spatially extended\nstructure (in62Cr, the root-mean-square radius of the\nν[440]1/2level is 5.1 fm, whereas the total neutron r.m.s.\nradius is 4.2 fm). As the neutron number increases, the\ntransition strength decreases as a consequence of the de-\ncreasing contribution of the ( ν[440]1/2)2excitation.\nAs the neutron number approaches N= 46, the\nν[431]1/2 level becomes close to the Fermi surface (see\nFig. 6). In70Cr and72Fe, 2qp excitations of the up-\nsloping (ν[303]5/2)2and the down-sloping ( ν[431]1/2)2have largest contributions to the lowest Kπ= 0+state.\nThe transition strength of the ( ν[431]1/2)2excitation is\n115fm4(6.7W.u.) in70Crand95fm4(5.4W.u.) in72Fe.\nThis2qptransitionstrengthbecomeslargebecauseofthe\nshallowerFermilevelandthespatiallyextendedstructure\nof the quasiparticle wave function of the ν[431]1/2 level.\nAtN= 42, both of the ν[440]1/2 and theν[431]1/2\nlevels are located far from the Fermi level, and 2qp ex-\ncitations of the ( ν[301]1/2)2and the (ν[422]5/2)2have\nmain contributions to the lowest Kπ= 0+state in66Cr\nand68Fe. These quadrupole transition matrix elements7\nhave opposite signs, but these transition strengths are\nless than 1 W.u.\nIn Fig. 7, we show also the strengths for the monopole-\nand quadrupole-pair transitions. As in34Mg, these low-\nlyingKπ= 0+modes have enhanced strengths for\nthe quadrupole pair transition, whereas disappearing\nstrengths for the monopole pair transition. This system-\natic calculation shows an importance of the dynamical\npairing: Fluctuation of the pairing field, which is de-\nformed as well as the mean field, is crucial for generating\nthe collective Kπ= 0+modes.\nIV. SUMMARY\nWe have studied low-frequency Kπ= 0+modes in\nneutron-rich nuclei, taking account of the effects of nu-\nclear deformation, pairing correlation and continuum\ncoupling simultaneously. New type of this excitation\nmode is generated by coherent superposition of neutron\n2qp excitations near the Fermi level whose wave func-\ntions have spatially extended structure. It is found that\nthe dynamical pairing correlation, i.e., the pairing vibra-\ntion enhances its collectivity.\nIn the spherical neutron-rich nuclei, the effect of the\ndynamical pairing has been investigated in detail in\nRefs. [23, 27, 32]. We have shown in this paper the\nimportance of the dynamical pairing in neutron-rich de-\nformed systems. In a deformed system where the up- and\ndown-sloping orbitals exist near the Fermi level, one ob-\ntains the low-lying mode possessing extremely enhanced\nstrengths both for the quadrupole p-h transition and for\nthe quadrupole p-p (pair) transition induced by the pair-\ning fluctuation.\nWe found that the coupling between the pairing vibra-\ntion and the neutron-skin vibration brings forth the soft\nKπ= 0+mode in deformed Mg region with N= 22.\nFurthermore, it was shown that emergence of low-lying\nKπ= 0+modes is not restricted in the neutron-rich Mg\nisotopes. As an example, in neutron-rich Cr and Fe iso-\ntopes around N= 40, we showed that the coherent cou-\npling between the pairing vibration and the β−vibration\nof the neutron skin brings about the striking enhance-\nment of the strengths for the quadrupole p-h and the\nquadrupole pair transition.\nAcknowledgments\nThe authors thank K. Matsuyanagi for valuable com-\nments and discussions. They also acknowledge N. Van\nGiai for useful discussions. One of the authors (K.Y)\nis supported by Research Fellowships of the Japan So-\nciety for the Promotion of Science for Young Scientists.\nThe numerical calculations were performed on the NEC\nSX-8 supercomputers at Yukawa Institute for Theoreti-\ncal Physics, Kyoto University and at Research Center for\nNuclear Physics, Osaka University.APPENDIX A: MATRIX ELEMENTS FOR\nONE-BODY OPERATORS\nLet us first consider matrix elements for particle-hole\ntype one-body operators\n/an}bracketle{tab|ˆO(uv)\nK|HFB/an}bracketri}ht, (A1)\nwhere\nˆO(uv)\nK=/summationdisplay\nσσ′/integraldisplay\ndrdr′δσ,σ′δ(r−r′)O(uv)\nK(r)ˆψ†(r′σ′)ˆψ(rσ),\n(A2)\nand the HFB ground state and the 2qp excited states\nˆβi|HFB/an}bracketri}ht= 0, (A3a)\n|ab/an}bracketri}ht=ˆβ†\naˆβ†\nb|HFB/an}bracketri}ht (A3b)\nare described by the quasiparticle operators. These op-\nerators are defined by the generalized Bogoliubov trans-\nformation\nˆψ†(rσ) =/summationdisplay\nkϕ1,k(r¯σ)ˆβ†\nk+ϕ∗\n2,k(rσ)ˆβk,(A4a)\nˆψ(rσ) =/summationdisplay\nkϕ∗\n1,k(r¯σ)ˆβk+ϕ2,k(rσ)ˆβ†\nk,(A4b)\nwhere\nϕk(r¯σ) =−2σϕk(r−σ). (A5)\nIn Appendices, we omit the subscript τfor simplicity.\nThe 2qp transition matrix elements are calculated as\n/an}bracketle{tab|ˆO(uv)\nK|HFB/an}bracketri}ht\n=/an}bracketle{tHFB|ˆβbˆβaˆO(uv)\nK|HFB/an}bracketri}ht\n=/summationdisplay\nkk′/summationdisplay\nσσ′/integraldisplay\ndrdr′δσ,σ′δ(r−r′)O(uv)\nK(r)\n×ϕ1,k(r′¯σ′)ϕ2,k′(rσ)/an}bracketle{tHFB|ˆβbˆβaˆβ†\nkˆβ†\nk′|HFB/an}bracketri}ht\n=/integraldisplay\ndrO(uv)\nK(r){−ϕ1,a(r↓)ϕ2,b(r↑)+ϕ1,a(r↑)ϕ2,b(r↓)\n+ϕ1,b(r↓)ϕ2,a(r↑)−ϕ1,b(r↑)ϕ2,a(r↓)}\n≡O(uv)\nK,ab(A6)\nusing the quasiparticle wave functions, and we employed\nthe Wick’s theorem. In the cylindrical coordinate repre-\nsentation, we can rewrite as\n/an}bracketle{tab|ˆO(uv)\nK|HFB/an}bracketri}ht=\n2πδK,Ωa+Ωb/integraldisplay\nρdρdzO(uv)\nK(ρ,z)\n×{ϕ1,a(ρ,z,↑)ϕ2,b(ρ,z,↓)−ϕ1,a(ρ,z,↓)ϕ2,b(ρ,z,↑)\n−ϕ1,b(ρ,z,↑)ϕ2,a(ρ,z,↓)+ϕ1,b(ρ,z,↓)ϕ2,a(ρ,z,↑)},\n(A7)8\nwhere\nO(uv)\nK(ρ,z) =O(uv)\nK(r)eiKφ. (A8)\nNext we considerthe paircreationoperatorsconsisting\nof nucleons with opposite direction of spins\nˆO(uu)\nK=/integraldisplay\ndrdr′δ(r−r′)O(uu)\nK(r)ˆψ†(r′↑)ˆψ†(r↓).(A9)\nThe matrix elements read\n/an}bracketle{tab|ˆO(uu)\nK|HFB/an}bracketri}ht\n=/an}bracketle{tHFB|ˆβbˆβaˆO(uu)\nK|HFB/an}bracketri}ht\n=/summationdisplay\nkk′/integraldisplay\ndrdr′δ(r−r′)O(uu)\nK(r)\n×ϕ1,k(r′¯↑)ϕ1,k′(r¯↓)/an}bracketle{tHFB|ˆβbˆβaˆβ†\nkˆβ†\nk′|HFB/an}bracketri}ht\n=/integraldisplay\ndrO(uu)\nK(r){−ϕ1,a(r↓)ϕ1,b(r↑)+ϕ1,b(r↓)ϕ1,a(r↑)}\n=2πδK,Ωa+Ωb/integraldisplay\nρdρdzO(uu)\nK(ρ,z)\n×{ϕ1,a(ρ,z,↑)ϕ1,b(ρ,z,↓)−ϕ1,a(ρ,z,↓)ϕ1,b(ρ,z,↑)},\n(A10)\nwhere\nO(uu)\nK(ρ,z) =O(uu)\nK(r)eiKφ. (A11)\nAPPENDIX B: QRPA TRANSITION MATRIX\nELEMENTS\nIn terms of the nucleon annihilation and creation op-\nerators in the coordinate representation, the quadrupole\noperator is represented as\nˆQ2K=/summationdisplay\nσ/integraldisplay\ndrr2Y2,−K(ˆr)ˆψ†(rσ)ˆψ(rσ).(B1)\nThe intrinsic matrix elements /an}bracketle{tλ|ˆQ2K|0/an}bracketri}htof the\nquadrupole operator between the excited state |λ/an}bracketri}htand\nthe ground state |0/an}bracketri}htare given by\n/an}bracketle{tλ|ˆQ2K|0/an}bracketri}ht=/summationdisplay\nabQ(uv)\n2K,ab(fλ\nab+gλ\nab) =/summationdisplay\nabM(uv)\n2K,ab,(B2)\nwhere\nQ(uv)\n2K,ab=/an}bracketle{tab|ˆQ2K|HFB/an}bracketri}ht (B3)calculated by using the quasiparticle wave-functions as\n(A7).\nWe can furthermore calculate the intrinsic matrix ele-\nment of the pair creation operator as\n/an}bracketle{tλ|ˆP†\n0|0/an}bracketri}ht=/summationdisplay\nab(fλ\nabP(uu)\n00,ab+gλ\nabP(vv)\n00,ab) =/summationdisplay\nabM(add)\n00,ab,\n(B4a)\n/an}bracketle{tλ|ˆP†\n2K|0/an}bracketri}ht=/summationdisplay\nab(fλ\nabP(uu)\n2K,ab+gλ\nabP(vv)\n2K,ab) =/summationdisplay\nabM(add)\n2K,ab,\n(B4b)\nwhere the monopole- and quadrupole-pair creation oper-\nators are defined by Eq.(10). The matrix element of 2qp\nexcitationP(uu)\n00,ab,P(vv)\n00,ab,P(uu)\n2K,abandP(vv)\n2K,abare given by\n(A10) and by the similar expression for ˆO(vv)\nK.\nAPPENDIX C: CALCULATION OF THE\nMOMENT OF INERTIA\nWe estimate moments of inertia using the spurious so-\nlution of the RPA equation [14]\n/parenleftbigg\nA B\nB∗A∗/parenrightbigg\nαβγδ/parenleftbigg\nJx\n−J∗\nx/parenrightbigg\nγδ= 0, (C1a)\n/parenleftbigg\nA B\nB���A∗/parenrightbigg\nαβγδ/parenleftbigg\nΘ\n−Θ∗/parenrightbigg\nγδ=/planckover2pi1\ni1\nJTV/parenleftbigg\nJx\nJ∗\nx/parenrightbigg\nαβ,(C1b)\nwhereˆJxistheangularmomentumoperatorconsistingof\nthe orbital part ˆlxand the spin part ˆ sx, and its conjugate\noperator ˆΘ.\nThe Thouless-Valatin moment of inertia JTVis deter-\nmined thorough the orthonormal condition\n/parenleftbigJ∗\nxJx/parenrightbig\nαβ/parenleftbigg\nΘ\n−Θ∗/parenrightbigg\nγδ=/planckover2pi1\niδαβ,γδ,(C2)\nas\nJTV= 2/planckover2pi12/summationdisplay\nαβγδ(Jx)∗\nαβ(A−B)−1\nαβγδ(Jx)γδ.(C3)\nTurningofftheresidualinteraction,weobtaintheexpres-\nsion for the Inglis-Belyaev moment of inertia JBelyaev\nJBelyaev= 2/planckover2pi12/summationdisplay\nαβ|Jx|2\nαβ\nEα+Eβ. (C4)\n[1] P.vonNeumann-CoselandT. Aumann(ed.), Proceedings\nof the 2nd International Conference on Collective Mo-\ntions in Nuclei under Extreme Conditions , Nucl. Phys.\nA788, (2007).\n[2] T. Motobayashi, et al., Phys. Lett. B349, 9 (1995).[3] H. Iwasaki, et al., Phys. Lett. B522, 227 (2001).\n[4] K. Yoneda, et al., Phys. Lett. B499, 233 (2001).\n[5] J. A. Church, et al., Phys. Rev. C 72, 054320 (2005).\n[6] Z. Elekes, et al., Phys. Rev. C 73, 044314 (2006).\n[7] T. Baumann, et al., Nature 449, 1022 (2007).9\n[8] T. Otsuka, Y. Utsuno, M. Honma, and T. Mizusaki,\nProg. Part. Nucl. Phys. 46, 155 (2001).\n[9] E. Caurier, F. Nowacki, A. Poves, Nucl. Phys. A742, 14\n(2004).\n[10] J. Terasaki, H. Flocard, P.-H. Heenen, and P. Bonche,\nNucl. Phys. A621, 706 (1997).\n[11] R. Rodor´ ıguez-Guzm´ an, J. L. Egido, L. M. Robledo,\nNucl. Phys. A709, 201 (2002).\n[12] P.-G. Reinhard, D. J. Dean, W. Nazarewicz,\nJ. Dobaczewski, J. A. Maruhn, M. R. Strayer, Phys.\nRev. C60, 014316 (1999) and references therein.\n[13] S. G. Nilsson and I. Ragnarsson, Shapes and Shells in\nNuclear Structure , (Cambridge University Press, 1995).\n[14] P. Ring and P. Schuck, The Nuclear Many-Body Problem\n(Springer, 1980).\n[15] M. Hannawald et al., Phys. Rev. Lett. 82, 1391 (1999).\n[16] O. Sorlin et al., Nucl. Phys. A660, 3 (1999).\n[17] O. Sorlin et al., Eur. Phys. J. A. 16, 55 (2003).\n[18] S. Lunardi et al., Phys. Rev. C 76, 034303 (2007).\n[19] E. Caurier, F. Nowacki, and A. Poves, Eur. Phys. J. A.\n15, 145 (2002).\n[20] I. Hamamoto, H. Sagawa and X. Z. Zhang, Phys. Rev.\nC53, 765 (1996); ibid. 55, 2361 (1997); ibid. 56, 3121\n(1997); ibid. 57, R1064 (1998); ibid. 64, 024313 (2001).\n[21] I. Hamamoto and H. Sagawa, Phys. Rev. C 60, 064314\n(1999); ibid. 62, 024319 (2000); ibid. 66, 044315 (2002).\n[22] S. Shlomo and B. Agrawal, Nucl. Phys. A722, 98c\n(2003).\n[23] M. Matsuo, Nucl. Phys. A696, 371 (2001).\n[24] K. Hagino, H. Sagawa, Nucl. Phys. A695, 82 (2001).\n[25] E. Khan, N. Sandulescu, M. Grasso, N. Van Giai, Phys.\nRev. C66, 024309 (2002).\n[26] M. Yamagami, N. Van Giai, Phys. Rev. C 69, 034301\n(2004).\n[27] M. Matsuo, K. Mizuyama, Y. Serizawa, Phys. Rev. C 71,\n064326 (2005).\n[28] J. Terasaki, J. Engel, M. Bender, J. Dobaczewski,\nW. Nazarewicz, M. Stoitsov, Phys. Rev. C 71, 034310\n(2005) and references therein.\n[29] J. Terasaki, J. Engel, Phys. Rev. C 74, 044301 (2006).\n[30] K. Mizuyama, M. Matsuo, Y. Serizawa, arXiv:0706.1115.\n[31] D. Vretenar, N. Paar, P. Ring, G. A. Lalazissis, Nucl.\nPhys.A692, 496 (2001).\n[32] N. Paar, P. Ring, T. Nikˇ si´ c, D. Vretenar, Phys. Rev. C\n67, 034312 (2003).\n[33] N. Paar, T. Nikˇ si´ c, D. Vretenar, P. Ring, Phys. Rev. C\n69, 054303 (2004).\n[34] N. Paar, T. Nikˇ si´ c, D. Vretenar, P. Ring, Phys. Lett.\nB606, 288 (2005).\n[35] L. G. Cao, Z. Y. Ma, Phys. Rev. C 71, 034305 (2005).\n[36] G. Giambrone, S. Scheit, F. Barranco, P. F. Bortignon,\nG. Col` o, D. Sarchi, E. Vigezzi, Nucl. Phys. A726, 3\n(2003).\n[37] D. Sarchi, P. F. Bortignon, G. Col` o, Phys. Lett. B601,27 (2004).\n[38] S. P´ eru, J. F. Berger, P. F. Bortignon, Eur. Phys. J. A\n26, 25 (2005).\n[39] K. Yoshida, M. Yamagami, K. Matsuyanagi, Prog.\nTheor. Phys. 113, 1251 (2005).\n[40] T. Nakatsukasa and K. Yabana, Phys. Rev. C 71, 024301\n(2005).\n[41] A.Muta, J-I.Iwata, Y.Hashimoto, andK.Yabana, Prog.\nTheor. Phys. 108, 1065 (2002).\n[42] H. Imagawa and Y. Hashimoto, Phys. Rev. C 67, 037302\n(2003).\n[43] T. Inakura, H. Imagawa, Y. Hashimoto, S. Mizutori,\nM. Yamagami and K. Matsuyanagi, Nucl. Phys. A768,\n61 (2006).\n[44] P. Urkedal, X. Z. Zhang and I. Hamamoto, Phys. Rev. C\n64, 054304 (2001).\n[45] K. Hagino, N. Van Giai and H. Sagawa, Nucl. Phys.\nA731, 264 (2004).\n[46] S. P´ eru, H. Goutte, J. F. Berger, Nucl. Phys. A788, 44c\n(2007).\n[47] D. Pe˜ na Arteaga, and P. Ring, Prog. Part. Nucl. Phys.\n59, 314 (2007).\n[48] T. Nakatsukasa, T. Inakura, K. Yabana, Phys. Rev. C\n76, 024318 (2007).\n[49] K. Yoshida, N. Van Giai, arXiv:0802.1687.\n[50] J. Dobaczewski, H. Flocard and J. Treiner, Nucl. Phys.\nA422, 103 (1984).\n[51] K. Yoshida, M. Yamagami, K. Matsuyanagi, Nucl. Phys.\nA779, 99 (2006).\n[52] A. Bulgac, Preprint No. FT-194-1980, In-\nstitute of Atomic Physics, Bucharest, 1980.\n[arXiv:nucl-th/9907088]\n[53] E. Ter´ an, V. E. Oberacker and A. S. Umar, Phys. Rev.\nC67, 064314 (2003).\n[54] G. F. Bertsch, H. Esbensen, Ann. Phys. 209, 327 (1991).\n[55] J. Terasaki, P.-H. Heenen, P. Bonche, J. Dobaczewski,\nH. Flocard, Nucl. Phys. A593, 1 (1995).\n[56] D. J. Rowe, Nuclear Collective Motion , (Methuen and\nCo. Ltd., 1970).\n[57] S. Shlomo and G. F. Bertsch, Nucl. Phys. A243, 507\n(1975).\n[58] L. M. Eisenberg and W. Greiner, Nuclear Models , vol. I\n(North Holland, 1970).\n[59] A. Bohr and B. R. Motteleson, Nuclear Structure , vol. II\n(Benjamin, 1975; World Scientific, 1998).\n[60] D. M. Brink and R. A. Broglia, Nuclear Superfluidity,\nPairing in Finite Systems (Cambridge University Press,\n2005).\n[61] M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, S. Pitte l\nand D. J. Dean, Phys. Rev. C 68, 054312 (2003).\n[62] M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring,\nComp. Phys. Comm. 167, 43 (2005)."    },    {        "title": "1802.00034v1.Oxygen_holes_and_hybridization_in_the_bismuthates.pdf",        "content": "Oxygen holes and hybridization in the bismuthates\nArash Khazraie,\u0003Kateryna Foyevtsova, Ilya El\fmov, and George A. Sawatzky\nDepartment of Physics &Astronomy, University of British Columbia,\nVancouver, British Columbia, Canada V6T 1Z1 and\nStewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4\nMotivated by the recently renewed interest in the superconducting bismuth perovskites, we in-\nvestigate the electronic structure of the parent compounds ABiO 3(A= Sr, Ba) using ab initio\nmethods and tight-binding (TB) modeling. We use the density functional theory (DFT) in the local\ndensity approximation (LDA) to understand the role of various interactions in shaping the ABiO 3\nbandstructure near the Fermi level. It is established that interatomic hybridization involving Bi-\n6sand O-2porbitals plays the most important role. Based on our DFT calculations, we derive a\nminimal TB model and demonstrate that it can describe the properties of the bandstructure as a\nfunction of lattice distortions, such as the opening of a charge gap with the onset of the breathing\ndistortion and the associated condensation of holes onto a1g-symmetric molecular orbitals formed\nby the O-2p\u001borbitals on collapsed octahedra. We also derive a single band model involving the\nhopping of an extended molecular orbital involving both Bi-6 sand a linear combination of six O-2 p\norbitals which provides a very good description of the dispersion and band gaps of the low energy\nscale bands straddling the chemical potential.\nKeywords:\nI. INTRODUCTION\nHole-doped bismuth perovskites ABiO 3(A= Sr, Ba)\nhave recently attracted a lot of attention as one of\nthe few examples of transition-metal-free high-transition-\ntemperature oxide superconductors1{3. The parent com-\npounds are no less interesting, demonstrating a vari-\nety of temperature-driven electronic and structural phase\ntransitions4,5. At low temperature, BaBiO 3and SrBiO 3\nare insulators with some characteristic distortions from\nan ideal cubic perovskite crystal structure. Namely,\noxygen octahedra around the Bi ions exhibit alternat-\ning breathing-in and breathing-out distortions along the\nthree cubic crystallographic directions, resulting in dis-\nproportionated Bi-O bond lengths. Additionally, the O 6\noctahedra are tilted and rotated, following a a\u0000a\u0000c0pat-\ntern in BaBiO 3and aa\u0000a\u0000c+pattern in SrBiO 3in\nGlazer's classi\fcation6,7.\nThe insulating state of ABiO 3is often interpreted in\nterms of a charge-disproportionation model4,5,8,9In this\nmodel, Bi ions with the nominal valency of 4+ dispro-\nportionate into Bi3+and Bi5+as Ba2+\n2Bi3+Bi5+O2\u0000\n6,\nwhich produces shorter Bi5+{O bonds and longer Bi3+{O\nbonds and corresponds to the valence electron occupation\nchanging from 6 s16s1to 6s06s2for the two Bi sites. This\nscenario, however, is not consistent with the on-site re-\npulsion e\u000bects, the high binding energy of the Bi-6 sstates\nas observed in Refs. 10 and 11, or the strongly covalent\nnature of the Bi{O bonding10{12and is not supported\nby spectroscopic measurements \fnding little di\u000berence\nin the Bi valence shell occupations13{15.\nThanks to the weakly correlated nature of the Bi-6 s\nand O-2pelectrons in ABiO 316, an accurate theoretical\ndescription of these systems is already possible in the\nframework of the density functional theory (DFT) and\nlocal density approximation (LDA), althougth a more\nadvanced treatment of exchange and correlation e\u000bects\nhas been shown to result in an enhancement of the gap\nand the electron-phonon coupling. The states bridgingthe Fermi energy are basically unchanged as compared\nto conventional DFT, indicating that the use of LDA+ U\nor hybrid functionals merely increases the gap value.17{22\nPrevious DFT studies12,23{25generally \fnd that the Bi-\n6sstates lie deeper in energy by several electron volts\nthan the O-2 pstates, questioning further the charge-\ndisproportionation model. In Ref. 25, we used DFT\nmethods to validate an alternative microscopic model\nfor the insulating state of ABiO 3, considered initially by\nIgnatov26, in which the hole pairs condense spatially onto\ncollapsed O 6octahedra occupying molecular orbitals of\nthea1gsymmetry while all the Bi ions are close to being\n6s2,i. e., the following process is taking place:\n2Bi3+L!Bi3+L2+ Bi3+;\nwhereLrepresents a ligand hole in an a1g-symmetric\nmolecular orbital on a collapsed O 6octahedron. Note\nthat in this scenario all oxygens remain equivalent\nin terms of charge and the resulting insulating state\nshould be named \\bond-disproportionated\" rather than\n\\charge-disproportionated\".\nIn the present follow up paper, we intend to better\nunderstand the relevance of various interactions in de-\ntermining the electronic structure of ABiO 3and derive\nits minimal tight-binding (TB) model. Such a minimal\nmodel describing the low energy scale states is especially\nuseful in constructing model Hamiltonians including the\nelectron-phonon interactions to discuss aspects such as\nbipolaron formation and superconductivity in hole or\nelectron doped systems. We \frst use DFT calculations\nto study the hybridization strengths between the con-\nstituent elements of ABiO 3, demonstrating the extreme\ne\u000bects of interatomic hybridization involving the Bi-6 s\nand O-2porbitals. Having determined the most rele-\nvant atomic orbitals and interactions, we then study the\nproperties of our derived TB model as a function of struc-\ntural distortions in two and three dimensions. Finally, we\nexplore possible simpli\fcations of the TB model with a\nfocus on describing low-energy electronic excitations.arXiv:1802.00034v1  [cond-mat.mtrl-sci]  31 Jan 20182\n-15-10-5 0 5\n  R ΓX M Γ  Energy (eV)(a) BaBiO 3\nBi-s\n-15-10-5 0 5\n  R ΓX M Γ  (b) BaBi\n-15-10-5 0 5\n  R ΓX M Γ  (c) Bi\n-5 0\n  R ΓX M Γ  Energy (eV)(e) BaO 3\nO-a1g\nTB fit\n-5 0\n  R ΓX M Γ  (f) O 3\n-15-10-5 0 5\n  R ΓX M Γ  (g) BaBiO 3\n-15-10-5 0 5\n  R ΓX M Γ  (h) BaBiO 3(d) O-p σ octahedron\nFIG. 1: The DFT (LDA) bandstructures of (a), (g), (h) BaBiO 3and its (b) BaBi, (c) Bi, (e) BaO 3, and (f) O 3sublattices. In\n(a) - (c), the yellow-colored fat bands represent the contribution of the Bi- sorbital, while in (e) - (h), the red-colored fat bands\nrepresent the contribution of the O- a1gmolecular orbital. The Fermi level is marked with a horizontal dashed black line. Panel\n(d) shows the O- a1gmolecular orbital combination of oxygen- p\u001borbitals in an octahedron. The oxygen sites 1 to 6 are coupled\nvia hopping integrals \u0000tpp= (\u0000tpp\u001b+tpp\u0019)=2. A nearest neighbor TB model \ft of (e) BaO 3(f) O 3and (g)-(h) BaBiO 3is\nshown with dashed lines. In (h), Bi-6 porbitals are added in an extended tight-binding model (ETB) for an improved \ft. The\nparameter values resulting from the \fts are listed in Table II.\nII. RESULTS AND DISCUSSION\nA. Bi, BaBi, O 3, and BaO 3sublattices of BaBiO 3\nIn order to study the hybridization strengths between\nthe constituent elements of ABiO 3, we calculate and com-\npare the electronic band structures of BaBiO 3and of its\nBi, BaBi, O 3, and BaO 3sublattices. It is to be expected\nthat all the conclusions in this section will be equally\napplicable to SrBiO 3because of its close similarity to\nBaBiO 3.\nThe electronic structure calculations are performed\nwith the full-potential linearized-augmented-planewave\ncode WIEN2k27. Exchange and correlation e\u000bects are\ntreated within the generalized gradient density approx-\nimation (GGA)28. For now, we will neglect the e\u000bects\nof lattice distortions and consider an idealized cubic unit\ncell containing one formula unit. A value of 4.34 \u0017A is\nused for the lattice constant, taken as the average over\nthe nearest-neighbor Bi-Bi distances in the experimen-\ntally measured distorted structure4, and a 7\u00027\u00027 k-Octahedron Square plaquette\nO sitea1gt1ut1ut1uegega1geueub1g\n11p\n61p\n20 01p\n301p\n41p\n201p\n4\n21p\n6\u00001p\n20 01p\n301p\n4\u00001p\n201p\n4\n31p\n601p\n20\u00001p\n121p\n41p\n401p\n2\u00001p\n4\n41p\n60\u00001p\n20\u00001p\n121p\n41p\n40\u00001p\n2\u00001p\n4\n51p\n60 01p\n2\u00001p\n12\u00001p\n4\n61p\n60 0\u00001p\n2\u00001p\n12\u00001p\n4\nEnergy \u00004tpp0 0 0 2 tpp2tpp\u00002tpp0 0 2tpp\nTABLE I: The eigenstates and eigenvalues of an octahedron\nand a square plaquette of O- p\u001borbitals coupled via nearest-\nneighbor hopping integrals \u0000tpp= (\u0000tpp\u001b+tpp\u0019)=2. For\noxygen site indexing and relative orbital phases, refer to\nFig. 1 (d). Here, the O- p\u001borbitals' on-site energies are set to\nzero.\npoint grid is used for the Brillouin-zone integration.\nAs discussed earlier12,25, the band structure of BaBiO 3\nnear the Fermi level is featured by a collection of strongly\ndispersive states with predominant Bi-6 sand O-2por-3\nbital characters. Most of the Bi-6 scharacter is concen-\ntrated in the 8 eV broad lowest band centered at \u000010 eV\n[Fig. 1(a)]. In contrast, in either the BaBi [Fig. 1(b)] or\nthe Bi [Fig. 1(c)] sublattices, the Bi-6 sband width is less\nthan 1 eV. This indicates that interatomic hopping inte-\ngrals involving only Ba and Bi atomic orbitals are rather\nsmall.\nLet us now consider the electronic band structures of\nthe BaO 3and O 3sublattices, shown in Figs. 1 (e) and\n(f), respectively. As we demonstrated in Ref. 25, it is\nhelpful to analyze the dispersion of the O-2 pstates in a\nperovskite structure in terms of molecular orbital com-\nbinations of the O-2 p\u001batomic orbitals in an isolated oc-\ntahedron. There are six such combinations listed in Ta-\nble I with the oxygen sites indexed in Fig. 1 (d). For\nfuture reference, Table I also contains molecular orbital\ncombinations of the O-2 p\u001borbitals in an isolated square\nplaquette. The O- a1g-symmetric molecular orbital com-\nbination is of particular interest as it is the only combi-\nnation that is allowed by symmetry to hybridize with the\nBi-6sorbital. In the calculated band structures of BaO 3\nand O 3, its character is concentrated at the bottom of\nthe O-2pband, mirroring the situation in an isolated oc-\ntahedron (see the bottom of Table I). As expected, the\nintensity of the O- a1gcharacter is strongly k-point depen-\ndent in the Brillouin zone of a cubic unit cell, vanishing\nat the \u0000-point and reaching maximum at the R-point.\nWe \fnd little di\u000berence between the widths (1.89 eV ver-\nsus 2.28 eV) and the dispersions of the O-2 pbands in\nthe BaO 3and O 3sublattices, which indicates that the\nhybridization between the Ba and O-2 porbitals is much\nweaker than that between the O-2 porbitals themselves.\nThis is due to a large separation between the O-2 pand\nBa-5patomic energy levels as well as a large distance of\n3.04 \u0017A between the O and Ba atoms.\nWe can now appreciate the enormous e\u000bect that the\nBi-6s| O-2porbital hybridization has on the electronic\nstructure of BaBiO 3, whereby the valence band width\nincreases from 2 eV or less in the isolated sublattices to\n15 eV in the full BaBiO 3structure. After the hybridiza-\ntion, the O- a1gmolecular orbital in an antibonding com-\nbination with the 6 sorbital lands above the Fermi level\n[see Figs. 1 (g) or (h)]. Such behavior of the O- a1gmolec-\nular orbital paves the path for the bipolaronic condensa-\ntion of oxygen holes upon breathing distortion, as will be\ndiscussed later.\nApart from being strongly coupled via sp\u001b-type over-\nlap integrals, the hybridizing Bi-6 satomic orbital and\nthe O-a1gmolecular orbital also take advantage of their\nenergetic proximity. To demonstrate this, the position of\nthe Bi-6sband in, e. g., the Bi sublattice in Fig. 1 (c) has\nbeen aligned with that in BaBiO 3at the \u0000 point where\nthe Bi-6s| O-2phybridization vanishes by symmetry,\nmarking the Bi-6 son-site energy at \u000fs=\u00006.1 eV. Simi-\nlarly, the top of the O-2 pband in the BaO 3sublattice\nin Fig. 1 (e) has been aligned with the top of the O-2 p\nnonbonding states in BaBiO 3. After such alignments,\none clearly sees that the Bi-6 sorbital is only about 2 eV\nbelow the O- a1gmolecular orbital which is much smaller\nthan the hopping integral between the O- a1gand the Bi-O3 BaO 3 BaBiO 3\nTB TB TB WF ETB\n\u000fs \u00004:73\u00006:93\u00006:2\n\u000fp\u001b \u00002:57 \u00002:92\u00005:13\u00004:84\u00003:11\n\u000fp\u0019 \u00002:49 \u00002:82\u00003:06\u00003:73\u00002:78\ntpp\u001b 0:26 0:30 0:63 0:64 0:4\ntpp\u0019 \u00000:05 \u00000:01\u00000:04\u00000:03\u00000:03\ntsp\u001b 2:10 2:31 2:09\n\u000fBa\ns 3:47\ntBa-O\nsp\u001b 0:9\nt0\npp\u001b \u00001:0\n\u000fBi\np 2:1\ntBi-O\npp\u001b 2:34\ntBi-O\npp\u0019 \u00000:53\nTABLE II: On-site energies and hopping integrals in eV for\nBaBiO 3and its O 3and BaO 3sublattices. The values are\nobtained by \ftting either the simplest or the extended tight-\nbinding (TB or ETB) model or by using Wannier functions\n(WF) including Bi-6 sand O-2porbitals. This choice of WF\norbitals results in a large next-nearest-neighbor hopping inte-\ngralt0\npp\u001bbetween the O- p\u001borbitals.\n6sorbitals.\nB. Derivation of tight-binding models for BaO 3,\nO3, and BaBiO 3,\nIn order to quantify the hybridization e\u000bects discussed\nabove, we will now derive minimal tight-binding models\nfor BaBiO 3and its BaO 3and O 3sublattices by \ftting\ntheir DFT bandstructures.\nIn all our nearest-neighbor TB models, there are two\nintersite oxygen hopping integrals tpp\u001bandtpp\u0019. We ad-\nditionally include an sp\u001bhopping integral between the\nBa-6sand O-2porbitals,tBa-O\nsp\u001b, for the BaO 3sublattice\nand ansp\u001bhopping integral between the Bi-6 sand O-2p\norbitals,tsp\u001b, for BaBiO 3. These simple models can nev-\nertheless provide an overall good description of the DFT\nbandstructure, see Figs. 1 (e)-(g). The parameter values\nresulting from the \fts are listed in Table II. The Bi-6 s|\nO-2phybridization parameter tsp\u001b= 2:10 eV is indeed\nfound to be by far the dominant hopping integral in the\nsystem.\nSurprisingly, the ratios jtpp\u001bj=jtpp\u0019j= 5 in O 3, and\njtpp\u001bj=jtpp\u0019j= 10 in BaBiO 3are considerably larger\nthan the empirical ratio of 3 typically assumed in\ncuprates29,30. The origin of such an enhancement of the\njtpp\u001bj=jtpp\u0019jratio is not clear, although we have ruled out\na possible sensitivity of this parameter to the system's\ndimensionality and to the O-O bond length variation by\ncomparing calculations with accordingly modi\fed struc-\ntural parameters.\nFor BaBiO 3, we also \fnd that the TB model parame-\nter values are in good agreement with hopping integrals\ncalculated using Wannier function (WF) projections31,32,\nwhere we only included Bi-6 sand O-2porbitals, since the4\n-10-5 0\nAM ΓXNΓ  Energy (eV)O-a1g\n 0 1 2 3 4\nStates/eV3D: θ = 0, b = 0\nS\na1g\neg\nt1u-10-5 0\nAM ΓXNΓ   0 1 2 3 4\nStates/eV3D: θ = 0, b = 0.1 A\n-10-5 0\nAM ΓXNΓ   0 1 2 3 4\nStates/eV3D: θ = 16o, b = 0\n-10-5 0\nAM ΓXNΓ   0 1 2 3 4\nStates/eV3D: θ = 16o, b = 0.1 A\n-10-5 0\nΓX M Γ  Energy (eV)\n 0 0.5 1 1.5 2\nStates/eV2D: θ = 0, b = 0\nS\na1g\neu\nb1g -10-5 0\nΓX M Γ   0 0.5 1 1.5 2\nStates/eV2D: θ = 0, b = 0.1 A\n-10-5 0\nΓX M Γ   0 0.5 1 1.5 2\nStates/eV2D: θ = 16o, b = 0\n-10-5 0\nΓX M Γ   0 0.5 1 1.5 2\nStates/eV2D: θ = 16o, b = 0.1 A\nFIG. 2: The bandstructures and projected DOS of the 3D (top panels) and 2D (bottom panels) TB models with varying\nstrengths of the breathing and tilting distortions. Here, breathing bis half the di\u000berence between the two disproportionated\nBi-O bond lengths, and \u0012is the tilting angle. Molecular orbital projections are made for the compressed octahedron or square\nplaquette following Table I, while the Bi- sorbital projection is made for the Bi atom located inside the compressed octahedron\nor square plaquette. The red-colored fat bands represent the contribution of the O- a1gmolecular orbital.\nlow energy states spanning the Fermi energy are primar-\nily of O-2pand Bi-6scharacter (see the fourth column\nof Table II). However, this technique gives an unphys-\nically large next-nearest-neighbor hopping between the\nO-p\u001borbitalst0\npp\u001b. This result is a consequence of the\nrather strong hybridization of the O-2 porbitals with the\nempty Bi-6 porbitals, which have not been included in\nthe Wannier basis. It has motivated us to also consider\nan extended tight-binding (ETB) model with added hy-\nbridization between the O-2 pand Bi-6porbitals (see the\n\ffth column of Table II). The ETB model indeed pro-\nvides an improved description of the DFT bandstructure\n[see Fig. 1 (h)], but also gives a more realistic value for\nthe Bi-6sorbital on-site energy \u000fs=\u00006:2 eV, which is\nvery close to the value of \u00006:1 eV corresponding to the\nposition of the strongest Bi-6 scharacter band at \u0000 in\nFig. 1 (a).\nC. Breathing and tilting distortions in two and\nthree dimensions\nIn this section, we will consider lattice distortions\npresent in the real ABiO 3structure. We are interested\nin whether our TB model can capture the changes in theelectronic structure due to the lattice distortions as ob-\nserved in DFT calculations25, such as the opening of the\ncharge gap with the on-set of the breathing distortion.\nBehaviors of the TB model in three and two dimensions\nwill be compared to study the role of dimensionality in\nthe problem.\nSince our focus is mainly on the top valence band cross-\ning the Fermi level, which is of the O- a1gsymmetry and\ndoes not mix with the Bi-6 porbital, we will use here the\nsimpler TB model from the third column of Table II with\nonly Bi-6sand O-2porbitals in the basis. The coupling\nof electrons to lattice distortions is modeled through a\n1=d2dependence of hopping integrals on the interatomic\nseparationd29.\nIn order to study the individual roles of the breath-\ning and tilting distortions, let us consider four model\nstructures of ABiO 3with the following characteristics:\n(i)b= 0\u0017A,\u0012= 0\u000e, (ii)b= 0:1\u0017A,\u0012= 0\u000e, (iii)b= 0\u0017A,\n\u0012= 16:5\u000e, and (iv)b= 0:1\u0017A,\u0012= 16:5\u000e. Here,bis half\nthe di\u000berence between the two disproportionated Bi-O\nbond lengths and \u0012is the tilting angle of the octahe-\ndron in three-dimensions, or the rotation of the square\nplaquette in two-dimensions. The values of b= 0:1\u0017A\nand\u0012= 16:5\u000ein structure (iv) correspond to the respec-\ntive strengths of the breathing and tilting distortions in5\ntsp\u001b(eV)tpp\u001b(eV)tpp\u0019(eV)\nb= 0\u0017A,\u0012= 0\u000e2.1 0.63 -0.04\nb= 0:1\u0017A,\u0012= 0\u000e2.37 1.96 0.71 0.59 -0.045 -0.03\nTABLE III: Variation of the nearest-neighbor hopping inte-\ngrals in response to the Bi-O bond-disproportionation of 0.1\n\u0017A.\nthe experimental SrBiO 3structure. Because the distor-\ntions break translational symmetry, there are four for-\nmula units in the three-dimensional (3D) unit cell and\ntwo formula units in the two-dimensional (2D) unit cell.\nFigure 2 presents the bandstructures and the projected\ndensities of states (DOS) of our 3D (top panels) and\n2D (bottom panels) model structures. Molecular orbital\nprojections are made for the compressed octahedron or\nsquare plaquette following Table I, and the red-colored\nfat bands represent the contribution of the O- a1gmolecu-\nlar orbital. We \fnd that the models' electronic structures\nexhibit similar characteristics irrespective of the dimen-\nsionality. Close to the Fermi level, the tilting distortion\nopens a gap at around -1.5 eV and causes an overall band\nnarrowing while the breathing distortion opens a charge\ngap transforming the system into a semiconductor. This\nmetal-to-semiconductor transition is accompanied by a\nshift of the O- a1gmolecular orbital character into the\nempty states. Its intensity becomes k-point independent\nmeaning that in real space holes spatially condense into\nwell-de\fned molecular orbitals on the collapsed octahe-\ndra. The observed strong tendency towards formation\nof molecular orbitals can be due to the fact that oxy-\ngen hopping integrals are rather sensitive to the Bi-O\nbond-disproportionation. As one can see in Table III,\nbond-disproportionation of 0.1 \u0017A results in 0.4 eV di\u000ber-\nence in the tsp\u001bhopping integrals for the collapsed and\nthe expanded octahedron.\nLet us also have a closer look at the behavior of the\ncharge gap as a function of tandb. It is depicted in\nFigs. 3 (a) and (b) for the 2D and 3D TB models, re-\nspectively. In both cases, the gap is linear in bbut its\n\u0012-dependence is stronger in the 2D case. Qualitatively\nour model calculations can reproduce the DFT results\nfor SrBiO 325[Fig. 3 (c)], but quantitatively the e\u000bects of\nboth the breathing and tilting distortions are rather un-\nderestimated, especially in the 3D case. This is due to the\napproximations we used in the model calculations, such\nas neglecting the variation of on-site energies or limiting\nthe number of orbitals. To exemplify the e\u000bect of the lat-\nter approximation, we compare the DFT gap in SrBiO 3\nwith that in BiO 3[Fig. 3 (d)]. Here, the \u0012-dependence of\nthe gap is noticeably reduced, similarly to what we \fnd\nin the model calculations where the Acation orbitals are\nalso neglected. This suggests that hybridization with the\nAcation orbitals plays a role in determining the size of\nthe charge gap as a function of \u0012.\n 0 0.5 1 1.5Charge gap (eV)(a) 2D TB\nθ=0o\nθ=11.5o\nθ=16.5o\nθ=20o(b) 3D TB\n 0 0.5 1 1.5\n 0  0.05  0.1  0.15Charge gap (eV)\nb (angstoms)(c) SrBiO 3\n 0  0.05  0.1  0.15\nb (angstoms)(d) BiO 3FIG. 3: The charge gap as a function of the breathing\ndistortion at various tilting distortions in (a) the 2D TB\nmodel, (b) the 3D TB model, (c) SrBiO 3from DFT (LDA)\ncalculations25, and (d) the BiO 3sublattice from DFT (LDA)\ncalculations.\nD. Tight-binding models with a reduced number of\norbitals\nFinally, we discuss possible simpli\fcations of the\nABiO 3TB model that would still allow an accurate de-\nscription of low-energy electronic excitations. As was\nshown previously, the bands straddling the Fermi level\nare dominantly of the Bi- sand O-p\u001borbital character.\nTherefore, a natural simpli\fcation of the TB model could\nbe to eliminate the O- p\u0019orbitals from the basis. This\nreduces the basis size from ten to four orbitals per for-\nmula unit. As one can see in Figs. 4 (a) and (b), the\nfour-orbital TB model gives a good agreement with the\nfull ten-orbital model near the Fermi level even without\nadjustment of model parameters. Here, the calculations\nare done for a face-centered cubic unit cell with two Bi\nsites, and the non-distorted lattice [panel (a)] is com-\npared with a lattice featuring an 0.1 \u0017A breathing distor-\ntion and no octahedra tilting [panel (b)]. The compari-\nson illustrates, in particular, that the four-orbital model\nis capable of describing the distortion-induced metal-to-\nsemiconductor transition in ABiO 3.\nDespite its reduced basis size, the four-orbital model,\nhowever, contains redundant degrees of freedom, as far\nas low-energy physics is concerned. They give rise to\nthe bonding Bi- sand O-p\u001bstates at\u000010 eV and the\noxygen nonbonding states at \u00003 eV, i. e. in the energy\nregions deep below the Fermi level. One can take a step\nfurther and write down a single-orbital TB model with an\nA1g-symmetric orbital at each octahedron site. For this\nmodel, which could represent only the low energy scale\nbands, the basis consists of antibonding combinations of6\n-15-10-5 0 5\n  W L Γ X  WEnergy (eV)(a)Full TB\nFour-orbital\n-15-10-5 0 5\n  W L Γ X  W(b)Full TB\nFour-orbital\n-15-10-5 0 5\n  W L Γ X  WEnergy (eV)(c)DFT\nSingle-orbital\n-15-10-5 0 5\n  W L Γ X  W(d)DFT\nSingle-orbital(e)\nFIG. 4: In (a) and (b), the full TB (solid line) and the four-orbital TB (dashed line) models are compared for the b= 0:0\u0017A\nandb= 0:1\u0017A lattices, respectively. In (c) and (d), the DFT bandstructure (solid line) and the single-orbital TB model (dashed\nline) are compared for the b= 0:0\u0017A andb= 0:1\u0017A lattices, respectively. In (e), the single-orbital A1gcoupling to nearest,\nsecond nearest, and fourth nearest neighbors are shown.\nBi-sand O-a1gorbitals:\n\f\f A1g\u000b\n=1p\n\u000b2+\f2\u0010\n\u000b\f\f Bi\u0000s\u000b\n\u0000\f\f\f O\u0000a1g\u000b\u0011\nwhere O\u0000a1gorbital is a symmetric linear combination\nof O-p\u001b[Fig. 1 (d)]. Neglecting spin, the e\u000bective Hamil-\ntonian in this basis can be written as:\nH=X\ni\u000fc\nA1g^cy\ni^ci+X\nj\u000fe\nA1g^dy\nj^dj+Hc\u0000e+Hc\u0000c+He\u0000e:\nHere, indices iandjrun over collapsed and expanded\noctahedron sites, respectively; ^ cy\ni(^ci) create (annihilate)\na hole on site iand ^dy\nj(^dj) create (annihilate) a hole\non sitej;\u000fc\nA1g(\u000fe\nA1g) is the on-site energy of the A1g\norbital on a collapsed (expanded) octahedron site. The\nhybridization terms can be written as:\nHc\u0000e=n:n:X\n<ij>t^cy\ni^dj+h:c:\nHc\u0000c=X\niX\ni02figt0^cy\ni^ci0+X\niX\ni002figt00^cy\ni^ci00+h:c:\nHe\u0000e=X\njX\nj02fjgt0^dy\nj^dj0+X\njX\nj002fjgt00^dy\nj^dj00+h:c:\nwhere<ij > represents sum over nearest-neighbor sites,\ni0andi00(j0andj00) are sites at distancesp\n2aand 2ab\u000fc\nA1g\u000fe\nA1gt t0t00B.D.\n0:00\u0017A\u00000:13\u00000:13\u00000:45\u00000:09 0:10 1:00\n0:05\u0017A 0:35\u00000:51\u00000:47\u00000:10 0:11 1:46\n0:10\u0017A 0:99\u00000:65\u00000:48\u00000:11 0:115 1:68\n0:15\u0017A 1:86\u00000:78\u00000:50\u00000:12 0:125 1:78\nTABLE IV: The single-orbital TB model parameter values in\neV for lattices with a varying degree of the breathing distor-\ntionband no octahedra tilting. t,t0, andt00are the near-\nest, second-nearest, and fourth-nearest neighbor hopping in-\ntegrals, respectively. \u000fc\nA1gand\u000fe\nA1gare the on-site energies of\ntheA1g-like orbitals of the collapsed and expanded octahe-\ndron. Bond disproportionation (B.D.) shows the number of\nAc\n1gorbital holes.\nfrom sitei(j), respectively [Fig. 4(e)]. The model param-\neter values are obtained by \ftting to the DFT states clos-\nest to the Fermi level. The parameter values are given in\nTable IV for lattices with a varying degree of the breath-\ning distortion, and the \fts for the b= 0\u0017A andb= 0:1\u0017A\nlattices are shown in Figs. 4 (c) and (d), respectively. We\n\fnd that within the single-orbital TB model the appear-\nance and growth of the charge gap with an increasing\nbreathing distortion can be well described by a splitting\nof the twoA1gorbital on-site energies, with essentially no\nneed of modifying the hopping integrals [see Table IV].\nThis model can be interpreted as an e\u000bective low energy\nmodel of bismutathes that can well describe the bands\nnear the Fermi level and can now be used for example\nto include electron-phonon coupling keeping in mind the7\norigin of these wave functions and the e\u000bects of electron\nor hole doping looking for possible superconductivity.\nIn order to clarify the physics involved in these ef-\nfective hopping integrals, it is instructive to obtain es-\ntimates for hopping integrals, t,t0, andt00by consider-\ning the composition of the A1gorbitals described above.\nThese hoppings can be directly related to our full ten or-\nbital model parameters, tpp\u001b,tpp\u0019, andtsp\u001b, with takinginto account O-O and Bi-O hoppings only up to nearest-\nneighbor. In the following, we consider for simplicity a\nnon-distorted case, approximate the coe\u000ecients \u000band\f\nto be\u000b=\f= 1, and also neglect the nonorthogonality\nthat occurs in the O- a1gorbitals on nearest-neighbors\nin estimating t, since the overlap is only1\n6. TheA1g\nnearest-neighbor hopping t[see Fig. 4(e)] can be written\nas:\nt=D\n A1g\ni\f\f\fH\f\f\f A1g\njE\n=\n\u001c1p\n2\u0010\n s\u00001p\n6(\u0000p1+p2\u0000p3+p4\u0000p5+p6)\u0011\ni\f\f\f\fH\f\f\f\f1p\n2\u0010\n s\u00001p\n6(\u0000p0\n1+p0\n2\u0000p0\n3+p0\n4\u0000p0\n5+p0\n6)\u0011\nj\u001d\n=\u00001\n2p\n6h s\nijHj(p0\n2)ji+1\n2p\n6hp1ijH\f\f s\nj\u000b\n+1\n12h\u0000p1ijHj(\u0000p0\n3+p0\n4\u0000p0\n5+p0\n6)ji\n+1\n12h(\u0000p3+p4\u0000p5+p6)ijHj(p0\n2)ji=\u00001p\n6tsp\u001b+2\n3tpp=\u00001p\n6tsp\u001b+1\n3(tpp\u001b\u0000tpp\u0019)\u0019\u00000:64 eV;\nThe next-nearest neighbor hopping term can be written:\nt0=D\n A1g\ni\f\f\fH\f\f\f A1g\ni0E\n=\n\u001c1p\n2\u0010\n s\u00001p\n6(\u0000p1+p2\u0000p3+p4\u0000p5+p6)\u0011\ni\f\f\f\fH\f\f\f\f1p\n2\u0010\n s\u00001p\n6(\u0000p00\n1+p00\n2\u0000p00\n3+p00\n4\u0000p00\n5+p00\n6)\u0011\ni0\u001d\n=1\n2\u00021\n6h(p1+p3)ijHj(\u0000p00\n2\u0000p00\n4)i0i=\u00002\n12tpp=1\n12(\u0000tpp\u001b+tpp\u0019)\u0019\u00000:05 eV;\nand the fourth nearest-neighbor term:\nt00=D\n A1g\ni\f\f\fH\f\f\f A1g\ni00E\n=D\n A1g\ni\f\f\f A1g\njED\n A1g\nj\f\f\fH\f\f\f A1g\ni00E\n=D\n A1g\ni\f\f\f A1g\njE\n\u0002t\n=\u001c1p\n2\u0010\n s\u00001p\n6(\u0000p1+p2\u0000p3+p4\u0000p5+p6)\u0011\ni\f\f\f\f1p\n2\u0010\n s\u00001p\n6(\u0000p0\n1+p0\n2\u0000p0\n3+p0\n4\u0000p0\n5+p0\n6)\u0011\nj\u001d\n\u0002t\n=1\n2\u00021\n6h\u0000p1jp0\n2i\u0002t\u0019+0:05 eV\nwhereD\n A1g\ni\f\f\f A1g\njE\n=\u00001\n12is the overlap integral be-\ntween siteiandj. The above estimates are within the\norder of magnitude of the single-orbital parameters in\nTable IV obtained from the \ft. For the on-site energieswe have:\n\u000fA1g=D\n A1g\ni\f\f\fH\f\f\f A1g\niE\n=1\n2D\u0010\n s\u0000 a1g\u0011\ni\f\f\fH\f\f\f\u0010\n s\u0000 a1g\u0011\niE\n=1\n2\u0010\n\u000fs+\u000fa1g\u0011\n\u00006p\n6tsp\u001b8\nwhere on-site energy of O- a1gis\u000fa1g=\u000f\u001b\u00004tppand\n6p\n6tsp\u001bis the coupling energy of Bi- sand O-a1gorbitals.\nThe change in the on-site energies due to breathing can\nnow be written as:\n\u0001\u000f=\u000fc\nA1g\u0000\u000fe\nA1g= (\u00002tc\npp\u00006p\n6tc\nsp\u001b)\u0000(\u00002te\npp\u00006p\n6te\nsp\u001b)\n:\nHere, tc\npp(te\npp) represent O-O hoppings [see Fig.1(d)] of\nthe collapsed (expanded) A1gorbitals, and tc\nsp\u001b(te\nsp\u001b) are\nthe corresponding Bi-O hoppings, and we have assumed\nthat Bi-son-site energies are unchanged. Using hopping\nparameters of a distorted lattice of 0.1 \u0017A from Table III, a\ndirect gap value of \u0001 \u000f\u00191:1 eV is estimated. This value\nof the gap is comparable to \u0001 \u000fin Table IV obtained from\nthe \ft.\nIn Table IV we show an e\u000bective orbital occupation\ncorresponding to the number of holes in the collapsed\nAc\n1gorbital calculated by integrating the projected den-\nsity of states above the chemical potential as a function\nof the breathing distortion. Since we have two Bi and\ntwo holes per unit cell the number of holes in the ex-\npandedAe\n1gorbital is also 1.0 for no breathing. As the\ndistortion increases we see that the number of holes in\nthe collapsed Ac\n1gmolecular orbital gradually moves to-\nwards 2.0 at which point there would be no holes any-\nmore in the expanded Ae\n1gmolecular orbital. This looks\nvery much like charge-disproportionation. However, in\nthe structure all the oxygens are identical which means\nthat this cannot be charge-disproportionation involving\nthe oxygen. Each oxygen indeed participates in both the\ncollapsed and expanded A1gmolecular orbitals however\nit participates more in the collapsed Ac\n1gmolecular or-\nbital than in the expanded one. So again, this has to\ndo with bond-disproportionation. We note however that\nthere is no symmetry change in moving from a bond to\na charge-disproportionation picture and so there is no\nclear boundary but rather a gradual cross-over. A real\ncharge-disproportionation would have to imply an attrac-\ntive coulomb interaction while a bond-disproportionation\nresults from the electron density changes in the bonds\ndriven by an electron-phonon coupling involving the hop-\nping integral changes.\nWe can infer the electron-phonon coupling strength\nfrom our single-orbital model as the A1gmolecular or-\nbital on-site energy lowering in Table IV, which is about\n1 eV in the presence of the experimental bond modulation\nof 0.1 \u0017A or equivalentlyd\u000f\ndx= 10 eV/ \u0017A. Given a Raman\nbreathing mode phonon frequency of !ph\u001970 meV33,34\natq= (\u0019;\u0019;\u0019 ) for BaBiO 3the electron-phonon coupling\ngcan be estimated as:\ng=@\u000f\n@xs\n~\n2Mo!ph\u001910 eV/ \u0017A\u00020:04321 \u0017A\u00190:4321 eV:whereMois the oxygen mass. The electron-phonon cou-\nplinggtranslated into the dimensionless coupling \u0015is:\n\u0015=2g2\n~!phW=2\u00020:43212\n0:07\u00026\u00190:89;\nwhereW\u00196 eV is the single-orbital A1gbandwidth read\nfrom Fig. 4 (d). Following the same argument we can\nestimate a value of \u0015= 1.21 for Ba 0:6K0:4BiO 3given a\nbreathing mode phonon frequency of !ph\u001960 meV35for\nBa0:6K0:4BiO 3and assuming that the bandwidth Wis\nunchanged upon K substitution. These electron-phonon\ncouplings are much higher than what has been obtained\nfrom previous LDA calculations on Ba 0:6K0:4BiO 336{41.\nOf course, to make a real comparison with LDA estimates\none would have to determine the qdependence of this\ncoupling and average this over an assumed Fermi surface.\nIII. CONCLUSIONS\nIn summary, we have studied the electronic structure\nof the bismuth perovskites ABiO 3(A= Sr, Ba) using ab\ninitio calculations and tight-binding modeling. We found\nthat the hopping integrals involving the Bi-6 sand O-2p\norbitals play a leading role in shaping the electronic band-\nstructure of ABiO 3near the Fermi level. A minimal TB\nmodel with ten orbitals per formula unit (one Bi-6 sand\nnine O-2porbitals) was derived and shown to be able\nto describe the changes in the electronic structure due\nto lattice distortions that had been observed in previous\nDFT studies, such as the opening of the charge gap due\nto the breathing distortion and the associated formation\nof molecular orbitals on collapsed octahedra. We also\nshowed that for the purpose of exploring low-energy exci-\ntations inABiO 3this TB model can be further reduced to\na four-orbital one with one Bi-6 sand three O-2 p\u001borbitals\nin the basis and even further down to a single-orbital\none with a single A1g-like orbital at each octahedron\nsite. Within this model, we further estimated electron-\nphonon couplings of \u0015= 0.89 and \u0015= 1.21 for BaBiO 3\nand Ba 0:6K0:4BiO 3respectively. This single band model\nis a good representation of the band structure close to\nthe chemical potential and can be used in a more de-\ntailed study including the in\ruence of electron-phonon\ncoupling and possible mechanisms for superconductivity\nin the doped materials, but one has to keep in mind the\nrather extended molecular orbital character of the basis\nstates in such a model.\nThis work was supported by NSERC, CIFAR, and the\nMax Planck-UBC Stewart Blusson Quantum Matter In-\nstitute.\n\u0003Electronic address: akhazr@phas.ubc.ca1R. J. Cava, B. Batlogg, J. J. Krajewski, R. Farrow, L.9\nW. R. Jr., A. E. White, K. Short, W. F. Peck, and T.\nKometani, Nature (London) 332, 814 (1988).\n2A. Sleight, J. Gillson, and P. Bierstedt, Solid State Com-\nmunications 17, 27 (1975).\n3S. M. Kazakov, C. Chaillout, P. Bordet, J. J. Capponi, M.\nNunez-Regueiro, A. Rysak, J. L. Tholence, P. G. Radaelli,\nS. N. Putilin, and E. V. Antipov, Nature (London) 390,\n148 (1997).\n4D. Cox and A. W. Sleight, Solid State Communications\n19, 969 (1976).\n5D. E. Cox and A. W. Sleight, Acta Crystallographica Sec-\ntion B 35, 1 (1979).\n6A. M. Glazer, Acta Crystallographica Section B 28, 3384\n(1972).\n7A. M. Glazer, Acta Crystallographica Section A 31, 756\n(1975).\n8C. M. Varma, Phys. Rev. Lett. 61, 2713 (1988).\n9I. Hase and T. Yanagisawa, Phys. Rev. B 76, 174103\n(2007).\n10W. A. Harrison, Phys. Rev. B 74, 245128 (2006).\n11G. Vielsack and W. Weber, Phys. Rev. B 54, 6614 (1996).\n12L. F. Mattheiss and D. R. Hamann, Phys. Rev. B 28, 4227\n(1983).\n13J. de Hair and G. Blasse, Solid State Communications 12,\n727 (1973).\n14A. F. Orchard and G. Thornton, J. Chem. Soc. Dalton\nTrans. , 1238 (1977).\n15G. K. Wertheim, J. P. Remeika, and D. N. E. Buchanan,\nPhys. Rev. B 26, 2120 (1982).\n16N. C. Plumb, D. J. Gawryluk, Y. Wang, Z. Ristic, J. Park,\nB. Q. Lv, Z. Wang, C. E. Matt, N. Xu, T. Shang, K.\nConder, J. Mesot, S. Johnston, M. Shi and M. Radovic,\nPhys. Rev. Lett. 117, 037002 (2016).\n17C. Franchini, G. Kresse, and R. Podloucky, Phys. Rev.\nLett. 102, 256402 (2009).\n18C. Franchini, A. Sanna, M. Marsman, and G. Kresse, Phys.\nRev. B 81, 085213 (2010).\n19D. Korotin, V. Kukolev, A. V. Kozhevnikov, D. Novoselov,\nand V. I. Anisimov, J. Phys.: Condens. Matter 24, 415603\n(2012).\n20R. Nourafkan, F. Marsiglio, and G. Kotliar, Phys. Rev.\nLett. 109, 017001 (2012).\n21Z. P. Yin, A. Kutepov, and G. Kotliar, Phys. Rev. X 3,\n021011 (2013).22D. M. Korotin, D. Novoselov, and V. I. Anisimov, J. Phys.:\nCondens. Matter 26, 195602 (2014).\n23L. F. Mattheiss, Phys. Rev. B 28, 6629 (1983).\n24L. F. Mattheiss and D. R. Hamann, Phys. Rev. B 26, 2686\n(1982).\n25K. Foyevtsova, A. Khazraie, I. El\fmov and G. Sawatzky,\nPhys. Rev. B 91, 121114(R) (2015).\n26A. Ignatov, Nuclear Instruments and Methods in Physics\nResearch A 448332(2000).\n27P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,\nand J. Luitz, WIEN2K, An Augmented Plane Wave + Lo-\ncal Orbitals Program for Calculating Crystal Properties\n(2001).\n28J. Perdew, K. Burke and Y.Wang, Phys. Rev. B 54, 16533\n29S. Froyen and W. A. Harrison, Phys. Rev. B 20, 2420\n(1979).\n30A. K. McMahan and R. M. Martin, Phys. Rev. B 38, 6650\n(1988).\n31A. A. Mosto\f, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander-\nbilt, and N. Marzari, Computer Physics Communications\n178, 685 (2008), ISSN 0010-4655.\n32J. Kunes, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, and\nK. Held, Comp. Phys. Commun. 181, 1888 (2010).\n33S. Sugai, S. Uchida, K. Kitazawa, S. Tanaka, and A. Katsui\nPhys. Rev. Lett. 55, 426 (1985).\n34S. Tajima, M. Yoshida, N. Koshizuka, H. Sato, and S.\nUchida Phys. Rev. B 46, 1232 (1992).\n35M. Braden, W. Reichardt, A. S. Ivanov, and A. Yu. Rumi-\nantsev, Anamolous Dispersion of LO Phonon Branches in\nBa0:6K0:4BiO 3, Europhys. Lett. B 34, 531 (1996).\n36N. Hamada, S. Massidda, A. J. Freeman, and J. Redinger,\nPhys. Rev. B 40, 4442 (1989).\n37M. Shirai, N. Suzuki, and K. Motizuki, J. Phys.: Condens.\nMatter 2, 3553 (1990).\n38A. I. Liechtenstein, I. I. Mazin, C. O. Rodriguez, O. Jepsen,\nO. K. Andersen and M. Methfessel, Phys. Rev. B 44, 5388\n(1991).\n39K. Kunc and R. Zeyher, Phys. Rev. B 49, 12216 (1994).\n40V. Meregalli and S. Y. Savrasov, Phys. Rev. B 57, 14453\n(1998).\n41T. Bazhirov, S. Coh, S. G. Louie, and M. L. Cohen, Phys.\nRev. B 88, 224509 (2013)."    },    {        "title": "1904.02522v1.Anomalous_high_magnetic_field_electronic_state_of_the_nematic_superconductors_FeSe___1_x__S__x_.pdf",        "content": "Anomalous high-magnetic field electronic state of the nematic superconductors FeSe 1\u0000xSx\nM. Bristow,1P. Reiss,1A. A. Haghighirad,1, 2Z. Zajicek,1S. J. Singh,1\nT. Wolf,2D. Graf,3W. Knafo,4A. McCollam,5and A. I. Coldea1,\u0003\n1Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK\n2Institut fur Festk ¨orperphysik, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany\n3National High Magnetic Field Laboratory and Department of Physics,\nFlorida State University, Tallahassee, Florida 32306, USA\n4Laboratoire National des Champs Magn ´etiques Intenses (LNCMI-EMFL),\nUPR 3228, CNRS-UJF-UPS-INSA, 143 Avenue de Rangueil, 31400 Toulouse, France\n5High Field Magnet Laboratory (HFML-EMFL), Radboud University, 6525 ED Nijmegen, The Netherlands\n(Dated: April 5, 2019)\nUnderstanding superconductivity requires detailed knowledge of the normal electronic state from which it\nemerges. A nematic electronic state that breaks the rotational symmetry of the lattice can potentially promote\nunique scattering relevant for superconductivity. Here, we investigate the normal transport of superconducting\nFeSe 1\u0000xSxacross a nematic phase transition using high magnetic fields up to 69 T to establish the temperature\nand field-dependencies. We find that the nematic state is an anomalous non-Fermi liquid, dominated by a linear\nresistivity at low temperatures that can transform into a Fermi liquid, depending on the composition xand the\nimpurity level. Near the nematic end point, we find an extended temperature regime with \u0018T1:5resistivity. The\ntransverse magnetoresistance inside the nematic phase has as a \u0018H1:55dependence over a large magnetic field\nrange and it displays an unusual peak at low temperatures inside the nematic phase. Our study reveals anomalous\ntransport inside the nematic phase, driven by the subtle interplay between the changes in the electronic structure\nof a multi-band system and the unusual scattering processes affected by large magnetic fields and disorder.\nMagnetic field is a unique tuning parameter that can sup-\npress superconductivity to reveal the normal low-temperature\nelectronic behavior of many unconventional superconductors\n[1, 2]. High-magnetic fields can also induce new phases of\nmatter, probe Fermi surfaces and determine the quasi-particle\nmasses from quantum oscillations in the proximity of quan-\ntum critical points [1, 3]. In unconventional superconductors,\nclose to antiferromagnetic critical regions, an unusual scaling\nbetween a linear resistivity in temperature and magnetic fields\nwas found [4, 5]. Magnetic fields can also induce metal-to-\ninsulator transitions, as in hole-doped cuprates, where super-\nconductivity emerges from an exotic electronic ground state\n[2].\nFeSe is a unique bulk superconductor with Tc\u00189K which\ndisplays a variety of complex and competing electronic phases\n[6]. FeSe is a bad metal at room temperature and it enters a\nnematic electronic state below Ts\u001887K. This nematic phase\nis characterized by multi-band shifts driven by orbital order-\ning that lead to Fermi surface distortions [6, 7]. Furthermore,\nthe electronic ground state is that of a strongly correlated sys-\ntem and the quasiparticle masses display orbital-dependent\nenhancements [7, 8]. FeSe shows no long-range magnetic or-\nder at ambient pressure, but complex magnetic fluctuations\nare present at high energies over a large temperature range\n[9]. BelowTs, the spin-lattice relaxation rate from NMR ex-\nperiments is enhanced as it captures the low-energy tail of\nthe stripe spin-fluctuations [10, 11]. Furthermore, recent \u0016SR\nstudies invoke the close proximity of FeSe to a magnetic quan-\ntum critical point as the muon relaxation rate shows unusual\ntemperature dependence inside the nematic state [12].\nThe changes in the electronic structure and magnetic fluc-\ntuations of FeSe can have profound implication on its trans-\nport and superconducting properties. STM reveals a highlyanisotropic superconducting gap driven by orbital-selective\nCooper pairing [13]. Due to the the presence of the small inner\nbands, whose Fermi energies are comparable to the supercon-\nducting gap, FeSe was placed inside the BCS-BEC crossover\nregime [14]. In large magnetic fields, when the Zeeman en-\nergy is comparable to the gap and Fermi energies, a peculiar\nhighly-polarized superconducting state may occur [14].\nTo establish the role played by different competing inter-\nactions on nematicity and superconductivity, an ideal route\nis provided by the isoelectronic substitution of selenium by\nsulphur ions in FeSe 1\u0000xSx[15]. This tuning parameter sup-\npresses nematicity and it leads to changes in the electronic\nstructure, similar to the temperature effects, with the Fermi\nsurface becoming isotropic in the tetragonal phase and the\nelectronic correlations becoming weaker [3, 6, 15, 16]. As\nnematicity is suppressed, it creates ideal conditions to explore\na potential nematic critical point [17] in the absence of mag-\nnetism. The superconducting dome extends outside the ne-\nmatic state but anisotropic pairing remains robust [18] and a\ndifferent superconducting state was suggested to be stabilized\nin the tetragonal phase [19].\nIn this paper we study the normal electronic state across\nthe nematic transition in FeSe 1\u0000xSxusing magnetotransport\nstudies in high-magnetic fields up to 69 T. We find that the\nnematic state has a non-Fermi-liquid behaviour with an un-\nusual transverse magnetoresistance ( \u0018H1:55), reflecting an\nunconventional scattering mechanism. Just outside the ne-\nmatic phase, resistivity is dominated by a \u0018T1:5dependence,\nsimilar to studies under pressure [20]. The transverse mag-\nnetoresistance is significant inside the nematic phase and it\nshows an unusual change in slope at low temperatures. Inside\nthe nematic phase at low temperatures, we find linear resis-\ntivity followed by Fermi-liquid behaviour for certain xandarXiv:1904.02522v1  [cond-mat.supr-con]  4 Apr 20192\n0 20 40 60\n 0H  (T)050100150 xx  ( cm)\nx = 0.251.5 K\n4.2 K\n10.0 K14.0 K\n20.0 K\n40.0 K\n0 20 40 60 80 100 120\n T  (K)020406080100 xx  ( cm)x = 0.25\n0 20 40 60 80 100 120\n T  (K)050100150200 xx  ( cm)x = 0.17\nTsT*\n0 20 40 60 80 100 120\n T  (K)050100150200 xx  ( cm)x = 0.11\nTsT*\n0 20 40 60 80 100 120\n T  (K)050100150 xx  ( cm)x = 0.07\nT*\nTs\n0 20 40 60 80 100 120\n T  (K)0100200300400 xx  ( cm)x = 0T* 0 T\n10 T\n15 T\n20 T25 T\n30 T\n34 T\nTs0 10 20 30\n 0H  (T)050100150 xx  ( cm)\nx = 0.172.2 K\n4.2 K\n8.1 K13.5 K\n24.9 K\n52.9 K\n0 10 20 30\n 0H  (T)050100150200 xx  ( cm)\nx = 0.111.9 K\n4.3 K\n10.3 K21.1 K\n45.3 K\n65.4 K\n0 10 20 30\n 0H  (T)020406080100120 xx  ( cm)\nx = 0.071.9 K\n4.3 K\n10.3 K21.1 K\n45.3 K\n65.4 K\n0 10 20 30\n 0H  (T)0100200300400 xx  ( cm)\nx = 04.2 K\n13.5 K\n24.9 K36.9 K\n52.9 K\n77.2 K\nk l m n o x = 0 x = 0 x = 0.1 1 x = 0.18 x = 0.18-40\n-20\n0\n20\n40E (meV)\nk (Å)-0.4 -0.2 0 0.2 0.4\nk (Å)-0.4 -0.2 0 0.2 0.4\nk (Å)-0.4 -0.2 0 0.2 0.4\nk (Å)-0.4 -0.2 0 0.2 0.4\nk (Å)-0.4 -0.2 0 0.2 0.4A Z A Z Za b c d e\nf g h i j\nNematic A Nematic B Tetragonal\nFIG. 1. Transverse magnetoresistance of the nematic and tetragonal FeSe 1\u0000xSx.(a-e) Field-dependent in-plane resistivity at different\nconstant temperatures for different compositions, x, inside and outside the nematic phase. The magnetic field is applied along the c-axis,\nperpendicular to the in-plane electrical current. A strong magnetoresistance develops inside the nematic phase. (f-j) Resistivity against\ntemperature in zero field (solid line) and at fixed magnetic fields (symbols), as extracted from the top panel for different x. The peak in\nmagnetoresistance is indicated by T\u0003and the nematic phase emerges at Ts. (k-o) Schematic band dispersion at low temperatures at two\nhigh symmetry points at the top of the Brillouin zone, ZandAfor different x(based on ARPES data reported in Refs.6, 7, 15, and 16).\nThe horizontal lines represent the location of distinct regions in the magnetotransport behaviour called nematic A ( x= 0;0:07), nematic B\n(x= 0:11;0:17) and the tetragonal phase for x&0:18. In the tetragonal phase, the compensated semi-metal is formed of two electron and\ntwo-hole like bands. Deep inside the nematic phase the inner hole band and inner electron bands are brought in the vicinity of the Fermi level.\nimpurity levels. Our study reveals anomalous transport in the\nnematic state due to the subtle changes in the electronic struc-\nture and/or scattering, which are also influenced by impurity\nlevels.\nRESULTS AND DISCUSSION\nFigs. 1a-e show the transverse magnetoresistance, \u001axx, of\ndifferent single crystals of Fe(Se 1\u0000xSx) up to 35 T at various\nfixed temperatures inside the nematic phase and up to 69 T for\nx\u00180:25in the tetragonal phase. From these constant tem-\nperature runs, we can extract the magnetoresistance at fixed\nfields for each composition x, as shown in Fig. 1f-j, which re-\nveals several striking features. Firstly, the magnetoresistance\nincreases significantly once a system enters the nematic state\natTs, and its magnitude dependents on the concentration x,\nbeing largest for FeSe, just above Tc. Secondly, in the vicin-\nity ofTcin magnetic fields much larger than the upper critical\nfield, the magnetoresistance shows an unusual temperature de-\npendence that varies strongly with xacross the phase diagram,\nas shown in Fig. 1(f-g). The resistivity slope d\u001axx=dT in 34 T\nof FeSe changes sign around a crossover temperature, T\u0003\u0018\n14 K, as shown in Fig.1f (also in the colour plot of the slope in\nFig. 3d). With increasing sulphur substitution from FeSe to-\nwardsx\u00180:07(defined as the nematic A region), the positionofT\u0003shifts to a slightly higher temperature of \u001820K, and the\npeak in magnetoresistance is much smaller than for FeSe. For\nhigher concentrations, approaching the nematic phase bound-\nary, (x\u00180:11\u00000:17defined as the nematic B region), there\nis a small peak at T\u0003but the negative slope d\u001axx=dT in 34 T\nis strongly enhanced at low temperatures, different from the\nnematic A phase (see Fig. 1(h,i) and Fig. 3(d)). Lastly, in\nthe tetragonal phase, the magnetoresistance shows a conven-\ntional behaviour and increases quadratically in magnetic fields\n(Fig. 1(e) and (j)).\nThe unusual downturn in resistivity in high-field fields be-\nlowT\u0003inside the nematic A phase was previously assigned to\nlarge superconducting fluctuations in FeSe in magnetic fields\nup to 16 T [10, 11]. We find that this behaviour remains ro-\nbust in magnetic fields at least a factor of 2 higher than the\nupper critical field of \u001816 T forHjjc[10]. Furthermore, it\nalso manifests in x\u00180:07inside the nematic A phase but it\ndisappears for higher x&0:1. AsTcand the upper critical\nfield inside the nematic phase for different xremain close to\nthat of FeSe [3, 21], the changes in the resistivity slope in high\nmagnetic fields are likely driven by field-induced effects that\ninfluence scattering and/or the electronic structure.\nThe Hall coefficient, RH=\u001axy=\u00160H, extrapolated in the3\n0 50 100 150 200\n (0H)1.55  (T1.55)050100150200250 xx  ( cm)\nx = 04.2 K\n13.5 K\n24.9 K36.9 K\n62.7 K0 20 40 60 80 100 120\n T  (K)-30-20-10010 xy /0H  (10-9m3C-1)\n0H = 1 T\nx = 0 (S1)\n0 (S2)\n0.07\n0.11\n0.17\n0.17 (S2)\n0.250 500 1000\n T 1.5  (K1.5)00.20.40.6 /300K  x = 0.17\n0.18\n0.19\n0.25x = 0\nH ||(ab)\nTFL~5 K~T 2~T\n0 5 10 15\n T  (K)0510152025   ( cm)\n(T)\nH0\nx = 0.25\n H ||(ab)\nTFL~9 K\n~T 2\n0 5 10 15\n T  (K)024681012   ( cm)\n(T)\nH0x = 0.04\nH ||(ab)\nTFL~1.5 K~T 2~T\n0 5 10 15\n T  (K)0102030   ( cm)\n(T)\nH0a d\nb e\nc f\nFIG. 2. Normal electronic state of FeSe 1\u0000xSx.(a) Temperature de-\npendence of resistivity versus T1:5over a large temperature region\njust outside the nematic phase. (b) Hall effect coefficient in low mag-\nnetic fields, indicating the change in sign and the dominance of dif-\nferent highly mobile carriers across the nematic phase. (c) Resistivity\nversusH1:55for FeSe inside the nematic phase at constant temper-\natures. (d-f) The low-temperature linear resistivity. The solid lines\nare the zero-field resistivity data. Solid circles represent the zero-\nfield extrapolated values of \u001axxwhenHjj(ab)plane. The dashed\nlines represent fits to a Fermi-liquid behaviour found below TFL, as\nindicated by arrows.\nlow-field limit (below 1 T) for FeSe 1\u0000xSxhas an unusual tem-\nperature dependence, as shown in Fig.2b. For a compensated\nmetal, the sign of the Hall coefficient depends on the differ-\nence between the hole and electron mobilities [22]. In the\ntetragonal phase above Tsand forx&0:18,RHis close to\nzero (Fig.2b), as expected for a two-band compensated metal.\nOn the other hand, in the low-temperature nematic A phase\nthe sign ofRHis negative suggesting that transport is domi-\nnated by a highly mobile electron band [15, 23]. It becomes\npositive inside the nematic B phase, dominated by a hole-like\nband (Fig. 2(a)). It is worth mentioning that inside the ne-\nmatic B phase the quantum oscillations are dominated by a\nlow-frequency pocket with light-mass that disappears at the\nnematic end point [3]. Thus, the behaviour of RHis linked\nto the disappearance of a small 3D hole pocket center at the\nZ-point in FeSe below Tsand its re-emergence in the nematic\nB phase with xsubstitution around x\u00180:11, as found in\nARPES studies [15] and sketched in Fig.1(m). Interestingly,\nthe subtle changes in the electronic structure in FeSe 1\u0000xSx\nseem to correlate with the different features observed both inmagnetoresistance (Fig.1(f-i)) and in the Hall coefficient jRHj\nthat shows a maximum near T\u0003(Fig.2(b)). In a high magnetic\nfield, the Hall component of FeSe is complex, changing sign\nand being non-linear [15, 21]. A magnetic field can induce\nchanges in scattering and/or field-induced Fermi-surface ef-\nfects in the limit when the cyclotron energy is close to the\nZeeman energy. The smallest inner bands of FeSe 1\u0000xSxshift\nin energy as a function of composition x(and temperature\n[3]), as shown in Figs.1(k-o). Furthermore, Hall effect in iron-\nbased superconductors can be affected by the spin fluctuations\nthat induce mixing of the electron and hole currents [24].\nNext, we attempt to quantify the magnetoresistance across\nthe phase diagram and in the vicinity of the nematic end point\nin FeSe 1\u0000xSx, as shown in Fig.1(a-e). At the lowest temper-\nature, inside the nematic phase, the transverse magnetoresis-\ntance of most samples is dominated by quantum oscillations\n[3] making difficult to quantify its dependence. A near-linear\nmagnetoresistance is detected for x\u00180:07in Fig. 1b and\nfor a dirty sample (with low residual resistivity ratio \u00188:5)\nin Fig. S9. The quasi-linear field magnetoresistance at low\ntemperature can arise from squeezed trajectories of carriers in\nsemiclassically large magnetic fields in case of small Fermi\nsurfaces (!c\u001c\u001d1) [25, 26]. Another explanation for an\nalmost linear magnetoresistance is the presence of mobility\nfluctuations caused by spatial inhomogeneities, as found in\nlow carrier density systems [26–28].\nClassical magnetoresistance in systems with a single domi-\nnant scattering time is expected to follow a H2dependence\n[25]. This results in Kohler’s rule, which is violated in\nFeSe 1\u0000xSxsuggesting that the magnetoresistance is not dom-\ninated by a single scattering time, as shown in Fig. S2(a-c).\nMagnetoresistance is quadratic in magnetic fields up to 69 T\nin the tetragonal phase ( x\u00150:19) (see Fig.1e and Fig.S4(e-f))\nbut not inside the nematic phase. FeSe 1\u0000xSxare compensated\nmulti-band systems [6] where the high-field magnetoresis-\ntance is expected to be very large and dependent on scattering\ntimes of electron and hole bands [22]. Magnetoresistance has\na complex form and instead simpler scaling have been sought\nto reveal its importance, in particular in the vicinity of critical\npoints [4, 5]. For example, in BaFe 2(As1\u0000xPx) forx\u00180:33\nat the antiferromagnetic critical point, a universal H\u0000Tscal-\ning was empirically found between the linear resistivity in\ntemperature and magnetic field [4]. For FeSe 1\u0000xSxnear the\nnematic end point at x\u00180:17we find that a H\u0000Tdepen-\ndence collapses onto a single curve, as shown in Fig. S2(e).\nDespite this, the energy scaling of magnetoresistance used to\ndescribed the antiferromagnetic critical point in Ref. [4] is not\nobeyed in the vicinity of the nematic end point in FeSe 1\u0000xSx,\nas detailed in Fig. S2(g-i). This could be due to additional\nconstrains to be included either to account for the nematoelas-\ntic coupling [29] and/or the effect of impurities. For example,\na very dirty sample of FeSe 1\u0000xSxclose toxnom\u00180:18was\nrecently suggested to obey H\u0000Tscaling [30].\nFor reasons described above, we propose a different ap-\nproach to model the magnetoresistance data in the nematic\nstate of FeSe 1\u0000xSx, using a power law in magnetic fields4\ngiven by\u001axx(H) =\u001a0(H) +bH\u000e. Strikingly, we find that\nall the magnetoresistance data inside the nematic phase can be\ndescribed by a unique exponent \u000e\u00181:55(5) over a large field\nwindow, as shown by the colour plot in Fig.3(c) as well as in\nFigs.2(c) and S4(a-d). A detailed method of the extraction of\n\u000eand its stability over a large temperature and field window is\nshown in Fig.S3. Furthermore, this gives \u000e\u00182for samples in\nthe tetragonal phase (see Fig.3(c)). Inside the nematic phase,\nthe Fermi surface of FeSe 1\u0000xSxdistorts anisotropically [6, 7]\nand an unusual type of scattering could become operational\ndue to presence of hot and cold spots along certain directions\n[31].\nIn the absence of magnetic field the transport behaviour can\nalso be described by a power law, \u001a(T) =\u001a0+AT\r. Fig. 3a\nshows a colour plot of the exponent \r, which is close to unity\nat low temperatures inside the nematic phase and becomes\nsublinear close to the nematic phase boundary, indicating a\nsignificant deviation from Fermi-liquid behaviour (a value of\n\r=1.1(2) was previously reported for FeSe [32]). Outside the\nnematic phase a T1:5dependence of resistivity describes the\ndata well over a large temperature range up to 120 K (see\nFig. 2(a) and Fig.3(a)), in agreement with previous studies\nof FeSe 1\u0000xSxunder pressure [20]. Using the high-magnetic\nfield data below Tc, we extract the low-temperature resistiv-\nity in the absence of superconductivity, \u001aH!0(T). Fig. 2(d-\nf) shows resistivity against temperature for different values\nofx, together with the extrapolated high-field points, using\nlongitudinal magnetoresistance when Hjj(ab)plane, shown in\nFig.S5. We also use transverse magnetoresistance data to ex-\ntract the zero-field resistivity, using the established power law\nH1:55, as shown in Fig. S7. From both measurements, we find\nstrong evidence for a linear resistivity in the low temperature\nregime, below T\u0003, inside the nematic phase. Linear resistiv-\nity was also detected from the 35 T temperature dependence\nof the longitudinal magnetoresistance in Ref.[33], however, it\nwas assumed to occur near the nematic critical point defined\nasxnom\u00180:16, which corresponds to x\u00180:13in our phase\ndiagrams in Fig.3 and Fig.S1(b) (as the resistivity derivative in\nRef.[33] show a Ts\u001851 K). At low temperatures, we observe\nthat Fermi-liquid behaviour recovers in the tetragonal phase\n(see also Refs. [33, 34]) and inside the nematic phase, below\nTFL(see Figs. 2(d-f) and 3(b)). This is strongly dependent\non composition and impurity level, even in the vicinity of the\nnematic end point (see Figs. S8 and S9). We find that TFLis\nhighest for the samples with the largest residual resistivity ra-\ntio (above\u001816) (see Figs.S1(c) and S6). Theoretical models\nsuggest that the temperature exponent, \r, in vicinity of criti-\ncal points is highly dependent on the presence of cold spots on\ndifferent Fermi surfaces, due to the symmetry of the nematic\norder parameter [31, 35]. On the other hand, near a antifer-\nromagnetic critical point in the presence of spin fluctuations\nthe impurity level also affects the temperature exponent [36].\nFurthermore, the scale at which the crossover to Fermi liq-\nuid behavior occurs at TFLin nematic critical systems could\ndepend on the strength of the coupling to the lattice [29].\nAn overall representation of the resistivity sloped\u001axx(34 T)=dT in 34 T for FeSe 1\u0000xSxas a function of\ntemperature is shown in the phase diagram in Fig. 3d. The\nlow-temperature manifestation of the nematic A and B phases\nis clearly different below T\u0003. In order to identify possible\nsources of scattering responsible for these changes, we\nconsider the role of spin fluctuations. Recent NMR data\nfound that anti-ferromagnetic spin fluctuations are present\ninside the nematic phase of FeSe 1\u0000xSx, being strongest\naroundx\u00180:1[37]. In FeSe, spin fluctuations are rather\nanisotropic [37, 38] and strongly field-dependent below 15 K\n[11]. Interestingly, the spin-fluctuations relaxation rate is\nenhanced below T\u0003(Fig. 3(d)), suggesting a correlation\nbetween spin-dependent scattering, the high-field magnetore-\nsistance and the low-temperature transport inside the nematic\nstate. High-magnetic fields are expected to align magnetic\nspins and could affect the energy dispersion of low-energy\nspin excitations and spin-dependent scattering in magnetic\nfields. In FeSe, the spin-relaxation rate in different magnetic\nfields up to 19 T deviates at T\u0003[11] but it remains relatively\nconstant in 19 T at the lowest temperatures. This may\nsuggest the variation in magnetoresistance in high magnetic\nfields at low temperatures in FeSe 1\u0000xSxis more sensitive\nto the changes in the electronic behaviour rather to the spin\nfluctuations across the nematic phase.\nThe low-temperature regime below T\u0003displays linear re-\nsistivity, which is a potential manifestation of scattering in-\nduced by critical spin-fluctuations in clean systems [36]. \u0016SR\nstudies place FeSe near an itinerant antiferromagnetic quan-\ntum critical point at very low temperatures [12] and spin-\nfluctuations are only found inside the nematic state [11,\n37]. On the other hand, close to the nematic end point in\nFeSe 1\u0000xSxwe find that resistivity is not linear in tempera-\nture but is dominated by a T1:5dependence. This is contrast\nto the linear resistivity found near a antiferromagnetic criti-\ncal point in BaFe 2(As1\u0000xPx) [32]. Theoretically, \r= 3=2\ncould describe the resistivity caused by strong antiferromag-\nnetic critical fluctuations in the dirty limit [36, 39]. However,\nin FeSe 1\u0000xSxthe spin fluctuations are suppressed and a Lif-\nshitz transition was detected at the nematic end point [3]. At\na nematic critical point the divergent fluctuations for different\nFermi surfaces could display unusual power laws in resistiv-\nity, as discussed in Refs. [31, 35, 40]. To asses the critical\nbehaviour, it is worth emphasizing that the effective masses\nassociated to the outer hole bands do not show any divergence\nclose to the nematic end point x\u00180:18[3]. This agrees\nwith the variation of the A1=2coefficient (see Fig. S11) and\nprevious studies under pressure [20], suggesting the critical\nnematic fluctuations could be quenched by the coupling to the\nlattice along certain directions in FeSe 1\u0000xSx.\nThe striking difference in magnetotransport behaviour be-\ntween the nematic and tetragonal phase in FeSe 1\u0000xSxcan\nhave significant implications on what kind of superconductiv-\nity is stabilized inside and outside the nematic phase as differ-\nent pairing channels may be dominant in different regions, as\nfound experimentally [18, 19]. Linear resistivity found at low\ntemperatures inside the nematic state is present in the region5\nTcTsFeSe1-xSx ~T \n~T 2~T1.5\n0 0.1 0.2\n x  020406080100120 T  (K)\n0.60.811.21.41.61.82\nNematicTetragonalTs\nT *\nA B0H = 34 T\n0 0.1 0.2\n x  010203040506070 T  (K)\n-1-0.500.51\n dxx(34T)/ dT  (arb. units)\nFeSe1-xSx xx~H \nTs\nH 1.55(5)H 2\n0 0.1 0.2\n x  020406080100 T  (K)\n11.21.41.61.82\n  a b\nc d\nFIG. 3. Phase diagrams of the resistivity exponents and high-field transport in FeSe 1\u0000xSx.The colour plot of the temperature exponent,\n\r, extracted from (a) zero-resistivity data, as shown in Fig. S10. (b) The low-temperature resistivity exponent below T\u0003, extrapolated from\nhigh magnetic fields as shown in Figs. S5 and S7, indicating the non-Fermi liquid behaviour of the nematic phase. Fermi liquid recovers below\nTFLfor the compositions xwith lowest disorder both inside the nematic phase and in the tetragonal phase. (c) The temperature dependence\nof field exponent \u000eshowing a dominant \u0018H1:55power law inside the nematic phase (based on Fig.S3). (d) The colour plot of the slope of\nresistivity in 34 T between the nematic A and B phases. Solid squares represent Tsand solid triangles Tc.T\u0003indicated by stars represents the\npeak in magnetoresistance and the maximum in jRHj. Solid lines indicate the nematic and superconducting phase boundaries and the dashed\nlines are guides to the eye. The hashed region at low temperatures in (b) has not yet been accessed experimentally.\nwhere spin-fluctuations are likely to be present. Furthermore,\nthe absence of superconductivity enhancement at the nematic\nend point in FeSe 1\u0000xSxis supported by the lack of diver-\ngent critical fluctuations, found both with chemical pressure\n[3] and applied pressure [20]. It is expected that the coupling\nto the relevant lattice strain restricts criticality in nematic sys-\ntems only to certain high symmetry directions [29, 41].\nIn conclusion, we have studied the evolution of the low-\ntemperature magnetotransport behaviour in FeSe 1\u0000xSxin\nhigh-magnetic fields up to 69 T. We find that the nematic\nstate has non-Fermi liquid behaviour and displays unconven-\ntional power laws in magnetic field, reflecting the dominant\nanomalous scattering inside the nematic phase. In high mag-\nnetic fields, well-above the upper critical fields, the transverse\nmagnetoresistance shows a change in slope that reflects the\nchanges in the spin-fluctuations and/or the electronic struc-\nture. In the low-temperature limit, high magnetic field sup-\npresses superconductivity and it reveals an extended linear re-\nsistivity in temperature followed by a Fermi-liquid like de-\npendence, highly dependent on the composition and impuritylevel. Our study reveals the anomalous transport behaviour\nof the nematic state, strikingly different from the tetragonal\nphase, that influences how superconductivity is stabilized in\ndifferent phases.\nMATERIALS AND METHODS\nSingle crystals of FeSe 1\u0000xSxwere grown by the KCl/AlCl 3\nchemical vapor transport method [42]. The composition for\nsamples from the same batch were checked using EDX as re-\nported previously in Ref. [3]. Note that in Refs.[30, 33] the\nnominal,xnom were can be at least 80% less than the real\nx(see also Ref. [3, 17, 37]). The structural transition at Ts\nalso provides useful information about the expected xvalue,\nas shown in Fig.S1. More than 30 samples were screened for\nhigh magnetic field studies to test their physical properties.\nResidual resistivity ratio varies between 15-44, as shown in\nFig.S1c. We observed the variation within the same batch due\nto the inhomogeneous distribution of sulfur with increasing x\n(see Figs.S1 and S8). We estimate that the nematic end point\nis located close to x\u00180:180(5) (see Figs.S1) and S11).\nIn-plane transport measurements ( Ijj(ab)) were performed6\nin a variable temperature cryostat in dc fields up to 38 T at\nHFML, Nijmegen and up to 70 T at LNCMI, Toulouse with\nthe magnetic field applied mainly along the c-axis (transverse\nmagnetoresistance) but also in the ( ab) conducting plane (lon-\ngitudinal magnetoresistance) at constant temperatures. Low-\nfield measurements were performed in a 16 T Quantum De-\nsign PPMS. The resistivity \u001axxand Hall\u001axycomponents\nwere measured using a low-frequency five-probe technique\nand were separated by (anti)symmetrizing data measured in\npositive and negative magnetic fields. Good electrical con-\ntacts were achieved by In soldering along the long edge of the\nsingle crystals and electrical currents up to 3 mA were used\nto avoid heating. Magnetic fields along the c-axis suppress\nsuperconductivity in fields higher than 20 T for all xvalues\n[3].\nACKNOWLEDGMENTS\nWe thank Lara Befatto, Dmitrii Maslov, Rafael Fer-\nnandes, Erez Berg, Shigeru Kasahara, Steve Simon, Sid-\ndharth Parameswaran and Stephen Blundell for useful com-\nments and discussions. We thank and acknowledge pre-\nvious contributions from Matthew Watson, Mara Bruma,\nSamuel Blake, Abhinav Naga and Nathaniel Davies. This\nwork was mainly supported by EPSRC (EP/L001772/1,\nEP/I004475/1, EP/I017836/1). A.A.H. acknowledges the fi-\nnancial support of the Oxford Quantum Materials Platform\nGrant (EP/M020517/1). A portion of this work was performed\nat the National High Magnetic Field Laboratory, which is sup-\nported by National Science Foundation Cooperative Agree-\nment No. DMR-1157490 and the State of Florida. Part of\nthis work was supported supported by HFML-RU/FOM and\nLNCMI-CNRS, members of the European Magnetic Field\nLaboratory (EMFL) and by EPSRC (UK) via its membership\nto the EMFL (grant no. EP/N01085X/1). Part of this work\nwas supported by Programme Investissements d Avenir under\nthe programme ANR-11-IDEX-0002-02, reference ANR-10-\nLABX-0037-NEXT We also acknowledge the Oxford Centre\nfor Applied Superconductivity and the Oxford John Fell Fund.\nA.I.C. acknowledges an EPSRC Career Acceleration Fellow-\nship (EP/I004475/1).\nFOOTNOTES\nTo whom correspondence may be addressed:\namalia.coldea@physics.ox.ac.uk\nAuthor contributions: A.I.C designed, planned and super-\nvised the research. M.B., P.R., Z.Z. and A.I.C. performed\nexperiments in Nijmegen with support from A.M.; P.R. and\nA.I.C. performed experiments in Tallahassee with support\nfrom D.G.; M.B., P.R., and A.I.C. performed experiments\nin Toulouse with support from W.K.; A.A.H., T.W. and S.S\ngrew single crystals. M.B. and A.I.C. performed data analy-\nsis. A.I.C and M.B. wrote the paper with contributions and\ncomments from all the authors.\u0003corresponding author:amalia.coldea@physics.ox.ac.uk\n[1] B. J. Ramshaw, S. E. Sebastian, R. D. McDonald, James\nDay, B. S. Tan, Z. Zhu, J. B. Betts, Ruixing Liang,\nD. A. Bonn, W. N. Hardy, and N. Harrison, “Quasi-\nparticle mass enhancement approaching optimal doping in\na high-tc superconductor,” Science 348, 317–320 (2015),\nhttp://science.sciencemag.org/content/348/6232/317.full.pdf.\n[2] G. S. Boebinger, Yoichi Ando, A. Passner, T. Kimura,\nM. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku,\nN. Ichikawa, and S. Uchida, “Insulator-to-Metal Crossover in\nthe Normal State of La2\u0000xSrxCuO 4Near Optimum Doping,”\nPhys. Rev. Lett. 77, 5417–5420 (1996).\n[3] A. I. Coldea, S. F. Blake, S. Kasahara, A. A. Haghighi-\nrad, M. D. Watson, W. Knafo, E. S. Choi, A. McCollam,\nP. Reiss, T. Yamashita, M. Bruma, S. Speller, Y . Matsuda,\nT. Wolf, T. Shibauchi, and A. J. Schofield, “Evolution of the\nlow-temperature Fermi surface of superconducting FeSe 1\u0000xSx\nacross a nematic phase transition,” npj Quantum Materials 4, 2\n(2019).\n[4] Ian M. Hayes, Ross D. McDonald, Nicholas P. Breznay,\nToni Helm, Philip J. W. Moll, Mark Wartenbe, Arkady\nShekhter, and James G. Analytis, “Scaling between magnetic\nfield and temperature in the high-temperature superconductor\nBaFe 2(As1\u0000xPx)2,” Nature Physics 12, 916 (2016).\n[5] P. Giraldo-Gallo, J. A. Galvis, Z. Stegen, K. A. Modic, F. F.\nBalakirev, J. B. Betts, X. Lian, C. Moir, S. C. Riggs, J. Wu,\nA. T. Bollinger, X. He, I. Bo ˇzovi´c, B. J. Ramshaw, R. D. Mc-\nDonald, G. S. Boebinger, and A. Shekhter, “Scale-invariant\nmagnetoresistance in a cuprate superconductor,” Science 361,\n479–481 (2018).\n[6] A. I. Coldea and M. D. Watson, “The key ingredients of the\nelectronic structure of FeSe,” Annu. Rev. Cond. Matt. Phys. 9\n(2018), 10.1146/annurev-conmatphys-033117-054137.\n[7] M. D. Watson, T. K. Kim, A. A. Haghighirad, N. R. Davies,\nA. McCollam, A. Narayanan, S. F. Blake, Y . L. Chen, S. Ghan-\nnadzadeh, A. J. Schofield, M. Hoesch, C. Meingast, T. Wolf,\nand A. I. Coldea, “Emergence of the nematic electronic state in\nFeSe,” Phys. Rev. B 91, 155106 (2015).\n[8] Matthew D. Watson, Steffen Backes, Amir A. Haghighirad,\nMoritz Hoesch, Timur K. Kim, Amalia I. Coldea, and Roser\nValent ´ı, “Formation of Hubbard-like bands as a fingerprint of\nstrong electron-electron interactions in FeSe,” Phys. Rev. B 95,\n081106 (2017).\n[9] Q. Wang, Y . Shen, B. Pan, X. Zhang, K Ikeuchi, K. Iida,\nA D Christianson, H C Walker, D T Adroja, M. Abdel-Hafiez,\nX Chen, D A Chareev, A. N. Vasiliev, and J Zhao, “Magnetic\nground state of FeSe,” Nat. Commun. 7, 12182 (2016).\n[10] S. Kasahara, T. Yamashita, A. Shi, R. Kobayashi, Y . Shi-\nmoyama, T. Watashige, K. Ishida, T. Terashima, T. Wolf,\nF. Hardy, C. Meingast, H. v. L ¨ohneysen, A. Levchenko,\nT. Shibauchi, and Y . Matsuda, “Giant superconducting fluc-\ntuations in the compensated semimetal FeSe at the BCS-BEC\ncrossover,” Nat. Commun. 7, 12843 (2016).\n[11] Anlu Shi, Takeshi Arai, Shunsaku Kitagawa, Takayoshi Ya-\nmanaka, Kenji Ishida, Anna E. B ¨ohmer, Christoph Mein-\ngast, Thomas Wolf, Michihiro Hirata, and Takahiko\nSasaki, “Pseudogap Behavior of the Nuclear SpinLattice\nRelaxation Rate in FeSe Probed by 77Se-NMR,” Jour-\nnal of the Physical Society of Japan 87, 013704 (2018),\nhttps://doi.org/10.7566/JPSJ.87.013704.\n[12] V . Grinenko, R. Sarkar, P. Materne, S. Kamusella, A. Ya-7\nmamshita, Y . Takano, Y . Sun, T. Tamegai, D. V . Efremov, S.-L.\nDrechsler, J.-C. Orain, T. Goko, R. Scheuermann, H. Luetkens,\nand H.-H. Klauss, “Low-temperature breakdown of antiferro-\nmagnetic quantum critical behavior in FeSe,” Phys. Rev. B 97,\n201102 (2018).\n[13] P. O. Sprau, A. Kostin, A. Kreisel, A. E. B ¨ohmer, V . Taufour,\nP. C. Canfield, S. Mukherjee, P. J. Hirschfeld, B. M. Ander-\nsen, and J. C. S ´eamus Davis, “Discovery of Orbital-Selective\nCooper Pairing in FeSe,” Science 357, 75 (2016).\n[14] Shigeru Kasahara, Tatsuya Watashige, Tetsuo Hanaguri,\nYuhki Kohsaka, Takuya Yamashita, Yusuke Shimoyama, Yuta\nMizukami, Ryota Endo, Hiroaki Ikeda, Kazushi Aoyama,\nTaichi Terashima, Shinya Uji, Thomas Wolf, Hilbert von\nL¨ohneysen, Takasada Shibauchi, and Yuji Matsuda, “Field-\ninduced superconducting phase of FeSe in the BCS-BEC cross-\nover.” Proc. Natl. Acad. Sci. U. S. A. 111, 16309 (2014).\n[15] M. D. Watson, T. K. Kim, A. A. Haghighirad, S. F. Blake, N. R.\nDavies, M. Hoesch, T. Wolf, and A. I. Coldea, “Suppression\nof orbital ordering by chemical pressure in FeSe 1\u0000xSx,” Phys.\nRev. B 92, 121108 (2015).\n[16] P. Reiss, M. D. Watson, T. K. Kim, A. A. Haghighirad, D. N.\nWoodruff, M. Bruma, S. J. Clarke, and A. I. Coldea, “Suppres-\nsion of electronic correlations by chemical pressure from FeSe\nto FeS,” Phys. Rev. B 96, 121103 (2017).\n[17] S. Hosoi, K. Matsuura, K. Ishida, Hao Wang, Y . Mizukami,\nT. Watashige, S. Kasahara, Y . Matsuda, and T. Shibauchi, “Ne-\nmatic quantum critical point without magnetism in FeSe 1\u0000xSx\nsuperconductors,” PNAS 113, 8139 (2016).\n[18] Yuki Sato, Shigeru Kasahara, Tomoya Taniguchi, Xiangzhuo\nXing, Yuichi Kasahara, Yoshifumi Tokiwa, Youichi Yamakawa,\nHiroshi Kontani, Takasada Shibauchi, and Yuji Matsuda,\n“Abrupt change of the superconducting gap structure at the ne-\nmatic critical point in FeSe 1\u0000xSx,” Proceedings of the National\nAcademy of Sciences (2018), 10.1073/pnas.1717331115.\n[19] T. Hanaguri, V . Iwaya, Y . Kohsaka, T. Machida, T. Watashige,\nS. Kasahara, T. Shibauchi, and Y . Matsuda, “Two distinct su-\nperconducting pairing states divided by the nematic end point\nin FeSe 1\u0000xSx,” Sci. Adv. 4, eaar6419 (2018).\n[20] P. Reiss, D. Graf, A. A. Haghighirad, W. Knafo, L. Drigo,\nM. Bristow, A. J. Schofield, and A. I. Coldea, “Quenched ne-\nmatic criticality separating two superconducting domes in an\niron-based superconductor under pressure,” arXiv:1902.11276\n(2019).\n[21] M Bristow, P. Reiss, A. A. Haghighirad, and A. I. Coldea, in\npreparation (2019).\n[22] M. D. Watson, T. Yamashita, S. Kasahara, W. Knafo, M. Nar-\ndone, J. B ´eard, F. Hardy, A. McCollam, A. Narayanan,\nS. F. Blake, T. Wolf, A. A. Haghighirad, C. Meingast, A. J.\nSchofield, H. v. L ¨ohneysen, Y . Matsuda, A. I. Coldea, and\nT. Shibauchi, “Dichotomy between the Hole and Electron Be-\nhavior in Multiband Superconductor FeSe Probed by Ultrahigh\nMagnetic Fields,” Phys. Rev. Lett. 115, 027006 (2015).\n[23] Yue Sun, Sunseng Pyon, and Tsuyoshi Tamegai, “Electron car-\nriers with possible Dirac-cone-like dispersion in FeSe 1\u0000xSx\n(x=0 and 0.14) single crystals triggered by structural transi-\ntion,” Phys. Rev. B 93, 104502 (2016).\n[24] L. Fanfarillo, E. Cappelluti, C. Castellani, and L. Benfatto,\n“Unconventional hall effect in pnictides from interband inter-\nactions,” Phys. Rev. Lett. 109, 096402 (2012).\n[25] A.B. Pippard, Magnetoresistance in Metals , Cambridge Stud-\nies in Low Temperature Physics (Cambridge University Press,\n1989).\n[26] Xu Du, Shan-Wen Tsai, Dmitrii L. Maslov, and Arthur F.Hebard, “Metal-insulator-like behavior in semimetallic bismuth\nand graphite,” Phys. Rev. Lett. 94, 166601 (2005).\n[27] A. Narayanan, M. D. Watson, S. F. Blake, N. Bruyant, L. Drigo,\nY . L. Chen, D. Prabhakaran, B. Yan, C. Felser, T. Kong, P. C.\nCanfield, and A. I. Coldea, “Linear Magnetoresistance Caused\nby Mobility Fluctuations in n-Doped Cd3As2,” Phys. Rev.\nLett. 114, 117201 (2015).\n[28] John Singleton, “A simple transport model for the temperature-\ndependent linear magnetoresistance of high-temperature super-\nconductors,” arXiv:1810.01998 (2018).\n[29] I. Paul and M. Garst, “Lattice Effects on Nematic Quantum\nCriticality in Metals,” Physical Review Letters 118, 1–5 (2017).\n[30] S. Licciardello, N. Maksimovic, J. Ayres, J. Buhot, M. Culo,\nB. Bryant, S. Kasahara, Y . Matsuda, T. Shibauchi, V . Nagara-\njan, J. G. Analytis, and N. E. Hussey, “Coexistence of or-\nbital and quantum critical magnetoresistance in FeSe 1\u0000xSx,”\narXiv:1903.05679 (2019).\n[31] Xiaoyu Wang and Erez Berg, “Scattering mechanisms and elec-\ntrical transport near an Ising nematic quantum critical point,”\narXiv:1902.04590 (2019).\n[32] S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tone-\ngawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hi-\nrata, T. Terashima, and Y . Matsuda, “Evolution from non-\nFermi- to Fermi-liquid transport via isovalent doping in\nBaFe 2(As1\u0000xPx)2superconductors,” Phys. Rev. B 81, 184519\n(2010).\n[33] S. Licciardello, J. Buhot, J. Lu, J. Ayres, S. Kasahara, Y . Mat-\nsuda, T. Shibauchi, and N. E. Hussey, “Electrical resistiv-\nity across a nematic quantum critical point,” Nature (2019),\n10.1038/s41586-019-0923-y.\n[34] Takahiro Urata, Yoichi Tanabe, Khuong Kim Huynh, Hidetoshi\nOguro, Kazuo Watanabe, and Katsumi Tanigaki, “Non-\nFermi liquid behavior of electrical resistivity close to the ne-\nmatic critical point in Fe 1\u0000xCoxSe and FeSe 1\u0000ySy,” (2016),\narXiv:1608.01044.\n[35] Dmitrii L. Maslov, Vladimir I. Yudson, and Andrey V .\nChubukov, “Resistivity of a Non-Galilean–Invariant Fermi Liq-\nuid near Pomeranchuk Quantum Criticality,” Phys. Rev. Lett.\n106, 106403 (2011).\n[36] A. Rosch, “Interplay of disorder and spin fluctuations in the\nresistivity near a quantum critical point,” Phys. Rev. Lett. 82,\n4280–4283 (1999).\n[37] P. Wiecki, K. Rana, A. E. B ¨ohmer, Y . Lee, S. L. Bud’ko, P. C.\nCanfield, and Y . Furukawa, “Persistent correlation between su-\nperconductivity and antiferromagnetic fluctuations near a ne-\nmatic quantum critical point in FeSe 1\u0000xSx,” Phys. Rev. B 98,\n020507 (2018).\n[38] R. X. Cao, Jun Dong, Q. L. Wang, X. S. Ye, J. B. Zhang, Y . F.\nXu, D. A. Chareev, A. N. Vasiliev, Bing Wu, Guoqing Wu, and\nX. H. Zeng, “Observation of orbital ordering and origin of ne-\nmatic order in FeSe,” arxiv.1810.09639 (2018).\n[39] T. Moriya, Spin fluctuations in itinerant electron magnetism ,\nSpringer series in solid-state sciences (Springer-Verlag, 1985).\n[40] Luca Dell’Anna and Walter Metzner, “Electrical resistivity near\npomeranchuk instability in two dimensions,” Phys. Rev. Lett.\n98, 136402 (2007).\n[41] D. Labat and I. Paul, “Pairing instability near a lattice-\ninfluenced nematic quantum critical point,” Phys. Rev. B 96,\n195146 (2017).\n[42] A. E. B ¨ohmer, F. Hardy, F. Eilers, D. Ernst, P. Adelmann,\nP. Schweiss, T. Wolf, and C. Meingast, “Lack of coupling\nbetween superconductivity and orthorhombic distortion in sto-\nichiometric single-crystalline FeSe,” Phys. Rev. B 87, 180505\n(2013).8\nSupplemental Materials for\nAnomalous high-magnetic field electronic state of the nematic superconductors Fe(Se 1\u0000xSx)\n0 100 200 300\n T  (K)00.20.40.60.811.21.41.61.8 /300K  x = 0\n0.04\n0.07\n0.11\n0.17\n0.17 (S2)\n0.18\n0.19\n0.25\n0 50 100\n T  (K)012345678910 d/dT  (arb. units)x = 0\n0.04\n0.07\n0.11\n0.17\n0.17 (S2)\n0.18\n0.19\n0.25Ts\n0 0.05 0.1 0.15 0.2 0.25\n x  1015202530354045 RRR  a b\nc\nFIG. S1. Temperature dependence of the resistivity of FeSe 1\u0000xSx.(a) Resistivity, normalised to the 300K value, against temperature for\ndifferent sulphur concentrations. (b) The first derivative of the resistivity with respect to temperature for the same data. The curves for different\nsulphur concentrations have been offset for clarity. The location of the structural transition, Tsis defined by the the intercept of the linear fits\non either side of the transition, as indicated by arrows. (c) The residual resistivity ratio (defined as the ratio between the room temperature\nand the onset of superconductivity resistivity), RRR , as a function of x. The complete suppression of the structural transition occurs at\nxc\u00180:180(5) , which agrees with previous reports [3, 17]. This value however differs from that reported in Ref. [33], where the nominal\nconcentrations have been used. For example, in Ref [33] xnom= 0:16hasTs\u001851K, which would correspond to x\u00180:13, based on our\nphase diagram and previous reports [3, 17]. The two x\u00180:17and thex\u00180:18samples come from the same batch and their differences\nreflect the sulphur variation and the degree of disorder ( x\u00180:18is cleaner with an RRR of\u001824compared with\u001816for the twox\u00180:17\nsamples). For x\u00180:18the derivative in (b) evolves more gradually, without a well-defined structural transition as compared to the others,\nand we believe that this sample is the closest to the nematic end point (just inside the nematic state).9\n0 5 10 15\n 0H / T  (TK-1)0102030405060 xx / T  ( cmK-1)\nx = 0.17\nxx= xx-02.2 K\n2.7 K\n4.2 K\n8.1 K13.5 K\n18.8 K\n24.9 K0 0.2 0.4 0.6 0.8\n(0H / H=0)2  (T( cm)-1)200.511.5 xx / H=0 x = 0.17\n13.5 K\n18.8 K\n24.9 K\n30.8 K\n36.9 K52.9 K\n62.7 K\n77.2 K\n86.4 K\n109.4 K\n0123\n(0H / H=0)2  (T( cm)-1)2051015 xx / H=0 x = 0\n13.5 K\n18.8 K\n24.9 K\n30.8 K\n36.9 K52.9 K\n62.7 K\n77.2 K\n86.4 K\n109.4 K\n0 10 20 30 40 50\n(0H / H=0)2  (T( cm)-1)2024681012 xx / H=0 x = 0.25\n10.0 K\n14.0 K\n20.0 K40.0 K\n70.0 K\n0 5 10 15\n 0H / T  (TK-1)020406080100 xx / T  ( cmK-1)\nx = 0\nxx= xx-0\n0=3.5 cm2.2 K\n2.7 K\n4.2 K\n8.1 K\n13.5 K\n18.8 K\n24.9 K30.8 K\n36.9 K\n52.9 K\n62.7 K\n77.2 K\n86.4 K\n109.4 K\n0 10 20 30 40\n 0H / T  (TK-1)020406080 xx / T  ( cmK-1)\nx = 0.25\nxx= xx-0\n0=7.5 cm1.5 K\n4.2 K\n10.0 K\n14.0 K20.0 K\n40.0 K\n70.0 K\n0 20 40 60 80 100 120\n /kB  (K)050100150200250 xx  ( cm)x = 0\n/ = 1\n0 20 40 60 80 100 120\n /kB  (K)020406080100120140 xx  ( cm)x = 0.17\n/ = 1\n0 20 40 60 80 100 120\n /kB  (K)010203040 xx  ( cm)x = 0.19\n/ = 1a b c\nd e f\ng h i\nFIG. S2. The magnetoresistance scaling in FeSe 1\u0000xSx.(a-c) Kohler’s plots, \u0001\u001axx(\u00160H)=\u001axx(H= 0)\u0018(\u00160H=\u001axx(H= 0))2, where\n\u0001\u001axx=\u001axx\u0000\u001axx(H= 0) forx= 0;0:17and0:25, respectively. Kohler’s rule is clearly violated for all samples over a large temperature\nrange, providing evidence that electrical transport in these systems is not governed by a single scattering time. (d-f) H\u0000Tscaling of\n\u0001\u001axx=\u001a0\u0018\u00160H=T, where \u0001\u001axx=\u001axx\u0000\u001a0and\u001a0is the zero-temperature zero-field resistivity. There is clearly no scaling for x= 0and\n0:25; for thex= 0:17sample in the vicinity of the nematic end point, data are dominated by a low-frequency quantum oscillation and collapse\nonto a single curve at low temperature, as shown in (e), but we have not identified yet an appropriate scaling law for it. However, in Ref. [30]\nfor a particularly dirty sample of xnom= 0:18withRRR\u00185and weak magnetoresistance (a factor \u0018100smaller than our x\u00180:17)\nH=T scaling was proposed suggesting that disorder may play an important role in this type of scaling in FeSe 1\u0000xSx. (g-i) Energy scaling\nof resistivity as \u0000, where \u0000 =\u000bkBTp\n1+ (\f=\u000b)2(\u0016B\u00160H=(kBT))2using\u000b= 1 and\f= 1, forx= 0;0:17and0:19, respectively.\nOur data do not follow the proposed energy scaling for any reasonable value of \u000b=\f and for any sulphur concentration in FeSe 1\u0000xSx. This\nmagnetoresistance scaling was used to describe the antiferromagnetic critical region in BaFe 2(As1\u0000xPx)2[4].10\nTsx = 0\n~H \n0 20 40 60 80 100 120\n T  (K)05101520253035 0H  (T)\n11.11.21.31.41.51.61.71.81.92\n0 5 10 15 20 25 30 35\n 0H  (T)05101520253035 dxx/dH  ( cmT -1) x = 04.2 K\n8.1 K\n13.5 K24.9 K\n36.9 K\n77.2 K\n- 1 0123\nln(0H)  -101234ln(dxx/dH)  x = 0~1.52(3)\n~1.59(5)4.2 K\n8.1 K\n13.5 K\n24.9 K\n36.9 K\n77.2 K\nTsx = 0.11\n~H \n0 20 40 60 80 100 120\n T  (K)05101520253035 0H  (T)\n11.11.21.31.41.51.61.71.81.92\n0 5 10 15 20 25 30 35\n 0H  (T)0123456 dxx/dH  ( cmT -1) x = 0.1110.3 K\n21.1 K\n33.4 K45.3 K\n65.4 K\n93.2 K\n0123\nln(0H)  -101234ln(dxx/dH)  x = 0.11\n~1.52(5)10.3 K\n21.1 K\n33.4 K45.3 K\n65.4 K\n93.2 K\n0 5 10 15 20 25 30 35\n 0H  (T)012345 dxx/dH  ( cmT -1) x = 0.178.1 K\n13.5 K\n18.8 K24.9 K\n36.9 K\n62.7 K\n0123\nln(0H)  -101234ln(dxx/dH)  x = 0.17\n~1.60(6)\n~1.88.1 K\n13.5 K\n18.8 K\n24.9 K\n36.9 K\n62.7 K\nTsx = 0.17\n~H \n0 20 40 60 80 100 120\n T  (K)05101520253035 0H  (T)\n11.11.21.31.41.51.61.71.81.92a b c\nd e f\ng h i\nFIG. S3. Extraction of the magnetic field exponent \u000e, from the relationship \u001axx=\u001aH!0+bH\u000e, for FeSe 1\u0000xSx.(a, d, g) Derivative of the\nsymmetrized transverse resistivity with respect to the applied magnetic field for x= 0;0:11and0:17, respectively. (b, e, h) Natural logarithm\nof the derivative against the natural logarithm of magnetic field for the same data in (a, d, g). Here linear fits to the high-field region allow the\nextraction of the magnetic field exponent from the gradient \u000e\u00001. (c, f, i) Colour plots of \u000ein the temperature-magnetic field plane, extracted\nabove the superconducting transition temperature using the relationship dln(\u001axx\u0000\u001axx(H= 0))=d(ln(\u00160H)) =\u000e. Consistent values of \u000e\nwere obtained from linear fits over small regions using ln(\u001axx\u0000\u001axx(H= 0)) = ln( b) +\u000eln(\u00160H). Our findings clearly show the evolution\nof the magnetic field exponent from \u000e\u00181:55(5) inside the nematic state towards \u000e\u00182outside of the nematic state.11\n0 50 100 150 200\n (0H)1.55  (T1.55)050100150200250 xx  ( cm)\nx = 04.2 K\n13.5 K\n24.9 K\n36.9 K\n62.7 K\n0 50 100 150 200\n (0H)1.55  (T1.55)020406080100 xx  ( cm)\nx = 0.074.3 K\n10.3 K\n21.1 K33.4 K\n55.7 K\n0 50 100 150 200\n (0H)1.55  (T1.55)050100150200 xx  ( cm)\nx = 0.114.3 K\n10.3 K\n21.1 K33.4 K\n45.3 K\n0 50 100 150 200\n (0H)1.55  (T1.55)020406080100120140 xx  ( cm)\nx = 0.174.2 K\n8.1 K\n13.5 K\n18.8 K\n24.9 K\n0 200 400 600 800 1000 1200\n (0H)2  (T2)01020304050 xx  ( cm)\nx = 0.192.6 K\n4.2 K11.4 K\n29.7 K\n0 1000 2000 3000 4000\n (0H)2  (T2)050100150 xx  ( cm)\nx = 0.251.5 K\n4.2 K\n10.0 K\n14.0 K\n20.0 K\n40.0 Ka b c\nd e f\nFIG. S4. Field dependence of the transverse magnetoresistance of FeSe 1\u0000xSxover a large magnetic field window. (a-d) Resistivity as\nfunction of (\u00160H)1:55showing linear dependence for samples inside the nematic phase at constant temperatures below Tsup to a magnetic\nfield of 35T. At the lowest temperatures, below \u00184:2K, the magnetoresistance is dominated by quantum oscillations. (e-f) Resistivity versus\n(\u00160H)2at constant temperatures for samples in the tetragonal phase. Data for x= 0:25were measured up to \u001869T.12\nx = 0.18\nH ||(ab)\nTFL~2.0 K\n0 5 10 15\n T  (K)05101520   ( cm)\n(T)\n(16 T)\n(35 T)\nx = 0\nH ||(ab)\n0 5 10 15\n T  (K)0510152025   ( cm)\n(T)\nH0TFL~5 K\n0 10 20 30\n 0H  (T)02468   ( cm)x = 0   H ||( ab)\n0.36 K\nx = 0.11\nH ||(ab)\n0 5 10 15\n T  (K)05101520   ( cm)\n(T)\nH0\n0 10 20 30\n 0H  (T)051015   ( cm)x = 0.11   H ||( ab)\n0.36 K\n4.27 K\nx = 0.17 (S2)\nH ||(ab)\n~T\n0 5 10 15\n T  (K)05101520   ( cm)\n(T)\nH0\n0 10 20 30\n 0H  (T)051015   ( cm)x = 0.17 (S2)\nH||(ab)\n0.36 K\n4.27 K\n0 5 10 15 20 25 30 35\n 0H  (T)024681012   ( cm)x = 0.18   H ||( ab)\n0.38 K\nx = 0.25\n H ||( ab)\n0 5 10 15\n T  (K)024681012   ( cm)\n(T)\nH0TFL~9 K\n0 5 10 15 20 25 30 35\n 0H  (T)02468   ( cm)x = 0.25   H ||( ab)\n0.38 K\nx = 0.04\nH ||(ab)\n0 5 10 15\n T  (K)05101520253035   ( cm)\n(T)\nH0TFL~1.3 K\n0 10 20 30 40\n 0H  (T)051015   ( cm)x = 0.04   H ||( ab)\n0.44 Ka b c\nd e f\ng h i\nj k l\nFIG. S5. The low-temperature resistivity extracted from longitudinal magnetoresistance data of FeSe 1\u0000xSx.(a-f) Temperature depen-\ndence of resistivity at low temperatures for different compositions used to build the low-temperature phase diagram in Fig. 3(b). Solid squares\nshow the extrapolated normal state resistivity, \u001aH!0from (g-l), and the solid triangles in (e) are the resistivity data at 35 T from (k), when\nthe magnetic field is along the conducting ( ab) plane. Solid lines are the zero-field resistivity for each sample. Fermi-liquid like behaviour is\nobserved in certain samples (with the largest resistivity ratio in Fig.S1(c)) below TFL, as indicated by arrows. (g-l) Magnetic field-dependence\nof the resistivity at the lowest temperature when Hjj(ab). Dashed lines are the linear extrapolation towards H!0. Forx= 0:11and0:17,\nthe upper critical field is too large to reach the normal state in magnetic fields up to 38T in this orientation.13\n0123456\n T 2  (K 2)12.212.312.412.512.6   ( cm)x = 0.18\nH ||(ab)\n0 = 12.21(2)\nA = 0.067(7)(35 T)\n=0+AT 2\nx = 0.18\nH ||(ab)\nTFL~2.0 K\n0 5 10 15\n T  (K)05101520   ( cm)\n(T)\n(16 T)\n(35 T)\nx = 0.04\nH ||(ab)\n0 5 10 15\n T  (K)05101520253035   ( cm)\n(T)\nH0TFL~1.3 K\n0 0.5 1 1.5 2\n T 2  (K 2)11.411.611.81212.212.412.6   ( cm)x = 0.04\nH ||(ab)\n0 = 11.4(1)\nA = 0.70(8)\n(T)\nH0\n=0+AT 2\nx = 0.25\n H ||( ab)\n0 5 10 15\n T  (K)024681012   ( cm)\n(T)\nH0TFL~9 K\n0 20 40 60 80\n T 2  (K 2)7.588.599.5   ( cm)x = 0.25\nH ||(ab)\n0 = 7.50(1)\nA = 0.029(2)(T)\nH0\n=0+AT 2\nx = 0\nH ||(ab)\n0 5 10 15\n T  (K)0510152025   ( cm)\n(T)\nH0TFL~5 K\n0 5 10 15 20 25 30\n T 2  (K 2)4567891011   ( cm)x = 0\nH ||(ab)\n0 = 3.9(1)\nA = 0.31(2)(T)\nH0\n=0+AT 2a b\nc d\ne f\ng h\nFIG. S6. Evidence of Fermi liquid behaviour in certain samples of FeSe 1\u0000xSx.(a, c, e, g) Temperature dependence of resistivity at low\ntemperatures for different samples which show Fermi-liquid behaviour, extracted as detailed previously in Fig. S5. The red dashed lines are\nfits of resistivity to a quadratic temperature dependence below TFL, and the blue dashed lines show a linear dependence between \u0018TFLand\nT\u0003. Resistivity data taken at 16T forx\u00180:18is also shown in (e) and follows the zero field curve at high temperatures as expected for\nlongitudinal magnetoresistance. (b, d, f, h) The temperature dependence of resistivity against T2illustrating the Fermi-liquid behaviour, given\nby\u001a=\u001a0+AT2. Here the dashed red lines are linear fits in T2and the zero-temperature resistivity values, \u001a0, andAparameters are listed\nin each panel. We find that the samples with the larger RRR also display larger TFL.14\n0 5 10 15 20 25 30\n T  (K)010203040   ( cm)x = 0(T)\n~T fit\nH0 H||c\n0 5 10 15 20 25 30\n T  (K)01020304050   ( cm)x = 0.07(T)\n~T fit\nH 0 H||c\n0 5 10 15 20 25 30\n T  (K)01020304050   ( cm)x = 0.11(T)\n~T fit\nH 0 H||c\n0 5 10 15 20 25 30\n T  (K)01020304050   ( cm)x = 0.17\n~T\n(T)\nH0 H||c\nH0 H||(ab)\n0 5 10 15 20 25 30\n T  (K)0510152025   ( cm)x = 0.19(T)\n~T 2 fit\nH 0 H||c\n0 5 10 15 20 25 30\n T  (K)05101520   ( cm)x = 0.25(T)\n~T 2 fit\nH 0 H||ca b c\nd e f\nFIG. S7. The low-temperature transport behaviour based on transverse magnetoresistance for FeSe 1\u0000xSx.(a-f) Low temperature\nresistivity against temperature for different compositions. The open circles in each panel correspond to the extrapolated resistivity (from\nsymmetrized magnetic field data) up to 35T, using the magnetic field exponents of \u000e= 1:55inside the nematic state ( x.0:18) and\u000e= 2in\nthe tetragonal state ( x= 0:19\u00000:25), as shown in Fig. S4. Inside the nematic phase the presence of quantum oscillations at low temperatures,\nmakes the extrapolation of \u001aH!0more difficult in this orientation when compared with longitudinal magnetoresistance studies, shown in\nFig. S5. In the tetragonal phase, Fermi-liquid behaviour is confirmed, similar to the longitudinal magnetoresistance studies in Fig. S5(f).\nDashed lines are either linear fits to the resistivity inside the nematic state or quadratic fits in the tetragonal state. Linear resistivity is found in\nthe vicinity of the nematic end point for x\u00180:17using both transverse and longitudinal magnetoresistance studies.15\n0 5 10 15 20 25 30\n T  (K)01020304050   ( cm)x = 0.17\n~T(T)\nH0 H||c\nH0 H||(ab)\nx = 0.17 (S2)\nH ||(ab)~T\n0 5 10 15\n T  (K)05101520   ( cm)\n(T)\nH0\n0 10 20 30\n 0H  (T)051015   ( cm)x = 0.17 (S2)\nH||(ab)\n0.36 K\n4.27 K\n0 5 10 15 20 25 30 35\n 0H  (T)024681012   ( cm)x = 0.18   H ||( ab)\n0.38 K\n0 5 10 15\n T  (K)05101520   ( cm)x = 0.18\nH||(ab)~T\nT 2\n(T)\n(16 T)\n(35 T)0 10 20 30 40\n 0H  (T)05101520   ( cm)\nx = 0.17\nH||(ab)\n0.38 K0 = 19(1) cma b\nc d\ne f\nFIG. S8. The low-temperature transport behaviour in the proximity of the nematic end point in FeSe 1\u0000xSx.The nematic end point\nis defined as the complete suppression of the structural transition at xc\u00180:180(5) , which agrees with previous reports [3, 17]. (a, c, e)\nResistivity against magnetic field up to 45 T at the lowest temperatures ( \u00180:38K) for three different samples from the same batch with\nHjj(ab). For sample x\u00180:17(S2) extracting \u001aH!0at the lowest temperatures is not possible up to 38 T and measurements at \u00184:2K, are\nalso shown in (c). (b, d, f) Resistivity against temperature for the same samples. Here the solid curves show the zero-field resistivity data and\nsolid black squares are the extrapolated \u001aH!0from Hjj(ab)field sweeps in (a, c). Solid triangles in (f) are the resistivity at 35T shown in\n(e). Dashed blue lines in (b) and (d) show a linear dependence, and the red dashed line in (f) show a Fermi-liquid behaviour for x\u00180:18.\nThe sample in which we find Fermi-liquid behaviour has a larger RRR of\u001824compared to\u001816of the two other samples, suggesting that\nFermi-liquid behaviour is observed only in cleaner samples.16\nFIG. S9. Magnetoransport data of a dirtier sample of FeSe 1\u0000xSx, withx\u00180:10. (a) Resistivity as a function of magnetic field at constant\nlow temperatures with Hjjc. (b) Resistivity against magnetic field for two different orientations at base temperature, HjjcandHjj(ab). (c)\nZero-resistivity data (solid curve) together with zero-field extrapolation using both linear extrapolation for Hjj(ab)(solid square) from (b)\nand zero-field extrapolation using a H1:55dependence for Hjjc(open circles) from (a). (d) Resistivity versus temperature for x\u00180:10with\nRRR\u00188:5. The strong suppression of quantum oscillations and the low RRR in this sample are evidence that it is a dirtier system which\nalso display linear resistivity down to the lowest temperatures measured ( \u00180:44K).17\n0 20 40 60 80 100 120\n T  (K)1/214/33/25/32   \n~T x = 0\n0.04\n0.070.11\n0.17\n0.180.19\n0.25\nFIG. S10. The temperature dependence of the exponent \rfrom\u001a=\u001a0+AT\rfor FeSe 1\u0000xSx.The\rexponent is estimated a \r=\ndln(\u001a\u0000\u001a0)=dln(T)for various sulphur concentrations. The \u001a0values are the zero-field zero-temperature resistivity extracted from Figs. S5,\nS6 and S7. The curves shown here were used to generate the colour plots in Fig. 3(a) and (b) in the main body of the paper. Dashed black lines\nshow possible extrapolations of the exponent towards the zero temperature limit. The horizontal lines indicated different possible exponents\npredicted by theory in the vicinity of a critical point [31, 35, 40]. The \rexponent depends on the value of \u001a0, which was extracted at the lowest\ntemperature from longitudinal magnetoresistance in Fig. S5.18\nFIG. S11. Comparison between the A1=2temperature coefficient and the effective masses for different compositions of FeSe 1\u0000xSx.\nThe band masses of the \u000eorbit (from Ref. [3]) are compared to the Fermi liquid coefficient, A, extracted at low temperatures (solid triangles)\nfor compositions that show Fermi-liquid behaviour, as shown in Fig.S6. Data reported in Ref.[33] are also included for comparison as open\ntriangles. Please note that Ref.[33] uses nominal xnom values which are shifted to smaller values (as indicated by horizontal arrows) to match\nthe realxvalues based on EDX studies reported previously [3, 17]. The nematic end point (NEP) is indicated by an arrow and circle around\nx\u00180:180(5) . Solid thick lines are guides to the eye."    },    {        "title": "1804.00665v1.Electronic_hydrodynamics_and_the_breakdown_of_the_Wiedemann_Franz_and_Mott_laws_in_interacting_metals.pdf",        "content": "easter egg\nElectronic hydrodynamics and the breakdown of the\nWiedemann-Franz and Mott laws in interacting metals\nAndrew Lucasaand Sankar Das Sarmab\naDepartment of Physics, Stanford University, Stanford, CA 94305, USA\nbCondensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland,\nCollege Park, MD 20742 USA\najlucas@stanford.edu April 4, 2018\nAbstract: We present the theory of thermoelectric transport in metals with long-lived quasiparticles,\ncarefully addressing the interplay of electron-electron scattering as well as electron-impurity\nscattering, but neglecting electron-phonon scattering. In Fermi liquids with a large Fermi sur-\nface and weak electron-impurity scattering, we provide universal and simple formulas for the\nbehavior of the thermoelectric conductivities across the ballistic-to-hydrodynamic crossover.\nIn this regime, the electrical conductivity is relatively unchanged by hydrodynamic e\u000bects.\nIn contrast, the thermal conductivity can be parametrically smaller than predicted by the\nWiedemann-Franz law. A less severe violation of the Mott law arises. We quantitatively\ncompare the violations of the Wiedemann-Franz law arising from ( i) momentum-conserving\nelectron-electron scattering in the collision integral, ( ii) hydrodynamic modi\fcations of the\nelectron-impurity scattering rate, and ( iii) thermal broadening of the Fermi surface, and\nshow that ( i) is generally the largest e\u000bect. We present simple formulas for electrical and\nthermal magnetoconductivity across the ballistic-to-hydrodynamic limit, along with a more\ncomplicated formula for the thermoelectric magnetoconductivity. In a \fnite magnetic \feld,\nthe Lorenz number may be smaller or larger than predicted by the Wiedemann-Franz law,\nand the crossover between these behaviors is a clear prediction for experiments. The arbi-\ntrarily strong violation of the Wiedemann-Franz law found in our work arises entirely from\nelectron-electron interaction e\u000bects within the Fermi liquid paradigm, and does not imply\nany non-Fermi liquid behavior. We predict clear experimental signatures of bulk hydrody-\nnamics in high-mobility 2D GaAs semiconductor structures, where a spectacular failure of\nthe Wiedemann-Franz law should persist down to very low temperatures in high-quality and\nlow-density samples.\n1 Introduction 2\n1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2 Main Results 4\n1arXiv:1804.00665v1  [cond-mat.str-el]  2 Apr 20183 Kinetic Theory Formalism 6\n3.1 Linearized Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n3.2 Perturbative Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3.3 Rotational Invariance and a Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n4 Large Fermi Surfaces 10\n4.1 An E\u000ecient Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n4.2 Approximating the Collision Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n4.3 The Strongly Interacting Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n4.4 The Weakly Interacting Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n4.5 Hydrodynamic Fluctuations and Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n5 Magnetotransport 17\n5.1 The Strongly Interacting Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n6 Discussion and Conclusions 19\nAcknowledgements 21\nA Low Temperature Expansion 21\nB Thermodynamic Identities 22\nC The First Three Basis Vectors 23\nD Thermal Conductivity in the Hydrodynamic Limit 24\nReferences 25\nIntroduction1\nOne of the most important open problems in condensed matter physics is a non-perturbative understand-\ning of interaction e\u000bects on the electronic transport properties of metals. The subject is vast and an\nenormous literature exists, spanning almost a century. Due to unresolved experimental puzzles such as\nthe high temperature linear-in-temperature resistivity of the normal phase of high-temperature supercon-\nductors at optimal doping [1], the study of interaction-limited transport has taken on much signi\fcance\nover the last two decades. Typically, one assumes that strong electron-electron interactions lead to a\nbreakdown of Fermi liquid physics and of a quasiparticle description, such that the resulting metal is a\nnon-Fermi liquid (NFL) [2, 3, 4, 5]. Despite much e\u000bort, the understanding of NFLs is very challenging\nand remains incomplete.\nA common heuristic used in experiments to probe the breakdown of Fermi liquid phenomenology is\nthe Wiedemann-Franz (WF) law [6, 7], which states that the ratio of thermal and electrical conductivity\nis\n\u0014\n\u001bT\u0019\u00192\n3k2\nB\ne2: (1)\nThe ratio on the left hand side above is called the Lorenz number, and the constant on the right hand side is\nthe ideal Lorenz number experimentally observed in most normal elemental metals at room temperatures.\n2The WF law arises in a conventional metal at temperatures T\u001cTF, the Fermi temperature, whenever\nelectron-electron collisions are negligible, and elastic scattering processes dominate [8]. Transport prop-\nerties of normal metals are dominated by electron-phonon scattering, which is essentially quasi-elastic at\nroom temperatures. The theoretical prediction (1) therefore holds very well for most normal metals [9].\nFurthermore, NFLs will usually (but not always) manifest a Lorenz number much smaller than predicted\nby (1) [3]. In some instances, the implication that bad metals (apparent NFLs with large resistivity\n[2, 3, 4]) violate the WF law has been \\inverted\" to conclude that a failure of the WF law by itself is\nevidence for the existence of a NFL [10]. Of course, this conclusion is not logically necessary: it is possible\nfor Fermi liquids to violate (1) even in the very low temperature limit [11]. In particular, (1) is violated\nwhenever the scattering processes which dominate transport are inelastic.\nThe purpose of this paper is to { within the Fermi liquid paradigm { describe the interplay of electron-\nelectron interactions and impurities. A careful understanding of this interplay is essential to develop a NFL\ntransport theory. We will see that some of the thermoelectric transport phenomena that occur in NFLs\nalso occur in FLs. Our description of `unconventional' FL transport phenomena both provides a \\less\nstrange\" setting in which to understand interaction-limited transport physics, and sheds light on which\ntransport phenomena rely on the breakdown of FL theory, and which do not. Both the hydrodynamic\nFLs which we consider, and the NFLs outlined previously, are strongly interacting electron systems.\nOne main di\u000berence is that NFL behavior is commonly believed to emerge due to a putative nearby\nquantum critical point. The WF violation and related `exotic' transport properties being studied in our\ncurrent work arise not from any hidden quantum criticality, but from strong interaction e\u000bects within the\nmetallic Fermi liquid itself. The hidden quantum criticality which may be responsible for NFL physics\nwill most strongly manifest itself in speci\fc temperature-dependent corrections to the e\u000bective electron-\nimpurity scattering rate.1It is an interesting question, beyond the scope of the current work, whether\nan interplay of quantum criticality and hydrodynamic e\u000bects coexist in strongly correlated materials; see\nrecent discussion in [13, 14].\nExpanding upon earlier works such as [3, 11], we provide a comprehensive and quantitative theory\nof the \fnite temperature breakdown of the WF law in metals described by Fermi liquid theory. In\nparticular, we con\frm the results of [11] that the Lorenz ratio becomes parametrically smaller than (1)\nin the hydrodynamic limit [15], where non-umklapp electron-electron scattering processes occur much\nfaster than electron-imputiy or electron-phonon processes. This e\u000bect is universal and does not rely on\nany details of the interactions or band structure. Any metal, if driven to a hydrodynamic regime,2will\nnecessarily violate (1). This violation does not necessarily imply NFL behavior.\nWe will also describe the interaction-driven breakdown of the related Mott law for thermoelectric\nconductivity:\n\u000b=\u0000\u00192\n3k2\nBT\ne@\u001b\n@\u0016; (2)\nwith\u0016=kBTFthe chemical potential, or Fermi energy. We will see that the Mott law holds in the same\nparameter regimes as the WF law; however, its breakdown is not as dramatic.\nThe reason that hydrodynamic breakdowns of the WF and Mott laws are not observed in typical\nmetals, even at very low temperatures, is simply a re\rection of the fact that most metals are rather\ndirty, and exhibit electron-electron scattering rates which are negligibly small compared to electron-\nimpurity scattering rates. Elemental 3D metals are never in the hydrodynamic regime, even when they\nare relatively pure { at low (high) temperatures, electron-impurity (-phonon) interactions are stronger\n1At \fnite temperatures there is generally simply a crossover (not a phase transition) between the quantum critical fan\nand conventional portions of the phase diagram [12]. The impact of quantum critical \ructuations is most clearly seen in the\nunconventional temperature dependence of scattering rates.\n2In the absence of umklapp, this can be done by making the crystal exceptionally pure so as to drastically reduce the\nelectron-impurity scattering rate.\n3than the corresponding electron-electron interaction. However, e\u000bectively 2D metals (e.g. graphene, high-\nmobility 2D semiconductor layers) can often be clean enough for the electron-electron scattering rate to\nsurpasss the electron-impurity scattering rate at low to intermediate temperatures (where electron-phonon\nscattering is relatively weak). These systems can be in the hydrodynamic regime. We urge experiments in\nhigh-mobility low-density modulation-doped 2D GaAs electron and hole systems [16] in order to verify our\npredictions of hydrodynamics-induced breakdowns of the WF and Mott relations arising from electron-\nelectron interactions. These high-quality GaAs systems, which are routinely used for studies of the\nfractional quantum Hall e\u000bect, are exceptionally pure and easily enter the hydrodynamic regime in a\n0.1{10 K temperature range.\nIn fact, recent experiments on graphene [17, 18], PdCoO 2[19], WP 2[20] and GaAs [21] have all\nobserved evidence for the collective hydrodynamic \row of electrons. Interaction-limited transport phe-\nnomena in low density oxides such as SrTiO 3[22, 23] may also have a hydrodynamic character. Our results\ncomplement the existing theoretical literature on hydrodynamic transport [24, 25, 26, 27, 28, 29, 30] and\ncontain new hydrodynamic predictions for experiments in the materials listed above. In graphene, break-\ndowns of the WF [31] and Mott [32] laws have been reported and attributed to hydrodynamic e\u000bects. A\ndemonstration of quantitative consistency between bulk transport phenomena and more direct probes of\nviscous e\u000bects such as nonlocal resistance will be strong evidence for the hydrodynamic nature of electron\n\row in these systems.\nConsistent with our goal of understanding the role of electron-electron interaction in metallic transport\n(and particularly, the hydrodynamic regime), we ignore electron-phonon interaction in our work. This\nis not because electron-phonon interaction is generically unimportant { as we noted previously, room\ntemperature transport in normal metals is dominated by phonons { but because the role of electron-\nphonon interaction in metallic transport is well-understood [8]. Neglecting phonons keeps our theory\ntransparent and tractable. It is straightforward to add electron-phonon scattering e\u000bects to our theory.\n1.1 Outline\nThe rest of the paper is organized as follows. In Section 2 we summarize some key results and experimental\npredictions. Section 3 outlines the computation of transport coe\u000ecients from kinetic theory. Detailed\ncalculations of transport coe\u000ecients at low temperature in any Fermi liquid with an isotropic dispersion\nrelation are provided in Section 4 in the absence of a background magnetic \feld, and in Section 5 in\na background magnetic \feld. We conclude in Section 6 emphasizing our key results and pointing out\nexperimental implications. Appendices contain a few technical details of our calculations.\nMain Results2\nIn this paper, we compute the dc thermoelectric transport coe\u000ecients of a metal:\n\u0012Ji\nQi\u0013\n=\u0012\u001bijT\u000bij\nT\u000bijT\u0016\u0014ij\u0013\u0012Ej\n\u00001\nT@jT\u0013\n; (3)\nwhereJiis the charge current, Qiis the heat current, Ejis an externally applied electric \feld, and @jT\nis an \\externally applied temperature gradient\".3In experiments, one often measures the open circuit\nthermal conductivity, where no electrical current \rows in the system:\n\u0014ij= \u0016\u0014ij\u0000T\u000bik\u001b\u00001\nkl\u000blj: (4)\nFor much of this paper, we will assume that the conductivity tensors are isotropic: e.g. \u001bij=\u001b\u000eij.\n3A formal discussion of how this can be done may be found in [5].\n4A number of systems ranging from graphene [31, 32] to more exotic \\strange metals\" [10, 33, 34]\nviolate the relations (1) and (2). Some of these violations could arise from the dominance of inelastic\nscattering processes, while others appear to have a hydrodynamic origin [31]. The purpose of this paper\nwill be to elucidate and expand upon the breakdown of (1) and (2) due to hydrodynamic e\u000bects. Indeed,\nit is well-understood why in the hydrodynamic regime (1)) and (2) will fail. For example, a new universal\ntransport relation arises deep in the hydrodynamic regime: [35, 36, 37]\n\u001b=e2n2\nTs2\u0016\u0014=\u0000en\ns\u000b: (5)\nHeresis the entropy density, nis the number density of electrons, and \u0000eis the charge of the elec-\ntron. These ratios emerge because all thermoelectric transport phenomena become linked to momentum-\nrelaxing scattering processes. Also observe that in this hydrodynamic regime, \u0014\u001c\u001bbecause \u0016\u0014\u001b\u0019T\u000b2\n[3]; furthermore, \u0016 \u0014\u001d\u0014.\nThe \frst main result of this paper is the quantitatively accurate description of the transition between\nthe conventional \\collisionless\" regime (1) and (2), and the hydrodynamic regime (5), carefully accounting\nfor all hydrodynamic e\u000bects. In metals with large Fermi surfaces and weak disorder, we will show that\n\u0014\n\u001bT\u0019\u00192\n3\u0000\n\u0000+\r; (6)\nwhere\rdenotes the electron-electron scattering rate, and \u0000denotes the electron-impurity scattering rate.\nWhile (6) was derived previously in [11], our derivation will naturally generalize to the Mott law, and\nto more complicated settings. We will describe the computation of \u0000in some detail below, including\na qualitative discussion of hydrodynamic e\u000bects on \u0000[38, 39] in Section 4.5. We brie\ry discuss \fnite-\ntemperature corrections to (6) in T=T Fin Section 4.4. Eq. (6) makes clear that as observed prior, \u0014=\u001bT\nis generally smaller than the Wiedemann-Franz prediction in an interacting metal at \fnite density. Eq.\n(6) also implies the expected FL behavior at T= 0, where \r= 0. This is in contrast to a NFL, where it\nis possible for the WF relation to be violated even at T= 0. It would be interesting to experimentally\nstudy the behavior of the Lorenz number at very low temperatures in strange metals (although in some\ncases, the putative critical point is unstable to superconductivity, e.g. cuprates, making a low temperature\nmeasurement impractical). The recovery of the WF law at low temperatures, or lack thereof, is a simple\ncheck on the FL paradigm.\nThe second main result of this paper is the generalization of (6) to magnetotransport. Including a\nbackground magnetic \feld which leads to cyclotron frequency !c, we \fnd that when T\u001cTF:\n\u0014xx\n\u001bxxT\u0019\u00192\n3e2\u0000+\r\n\u0000\u00002+!2\nc\n(\u0000+\r)2+!2c; (7a)\n\u0014xy\n\u001bxyT\u0019\u00192\n3e2\u00002+!2\nc\n(\u0000+\r)2+!2c: (7b)\nLike (6), the ratio of the Hall thermal to Hall electrical conductivity strictly decreases with increasing\nelectron-electron interactions. However, the dissipative conductivities exhibit a richer structure. When\n\u0000+\r < ! c, enhancing electron-electron interactions increases the Lorenz number above the ballistic\nprediction; only when \u0000+\r >! cdoes the Lorenz number become smaller than (1). This implies that a\nnon-monotonic temperature dependence of the Lorenz number will occur whenever \u0000 <! c. It is possible,\nthough di\u000ecult, to accurately measure electronic thermal conductivity directly [31, 40, 41]. Eq. (7a) is\na key prediction for future experiments and will provide a quantitative test of both the hydrodynamic\norigin of Wiedemann-Franz violations, and the validity of the Fermi liquid paradigm.\n5Kinetic Theory Formalism3\nIn this section, we introduce the general formalism to solve the transport problem. We will start out with\nrather minimal assumptions beyond the existence of quasiparticles with lifetime \u001d~=kBT, and as the\nsection continues we will make more and more speci\fc assumptions. It is (at least in principle) simple to\nrelax many of these assumptions. In practice, going beyond some of these assumptions may necessitate\nextensive numerical work focusing on speci\fc materials, which is beyond the scope of our current work.4\n3.1 Linearized Boltzmann Equation\nIn order to compute the thermoelectric conductivity matrix, we solve the Boltzmann equation for a Fermi\nliquid of dispersion relation \u000f(p), linearized about thermal equilibrium at temperature T. We assume that\nthe band structure is inversion-symmetric and time-reversal invariant. We also assume the presence of an\ninhomogeneous single-particle potential energy Vimp(x) caused by random impurities. We ignore the spin\nof the electronic quasiparticles, as we are explicitly interested in spin-independent transport properties,\nthough it is easy to keep track of spin if necessary.\nWe denote by f(x;p) the distribution function of (quantum) kinetic theory. Letting f=feq+\u000ef,\nwhere\nfeq(x;p) =1\n1 + e(\u000f(p)+Vimp(x)\u0000\u0016)=kBT; (8)\ndenote the distribution function in the absence of any external electric \felds or temperature gradients,\nand\u000efdenote the in\fnitesimal correction arising in in\fnitesimal electric \felds/temperature gradients, we\n\fnd that the Boltzmann equation reduces to\n@t\u000ef+v\u0001@\u000ef\n@x+Fimp\u0001@\u000ef\n@p+\u000eFext\u0001@feq\n@p=\u0000\u000eC[\u000ef] (9)\nwithin linear response. As we are only interested in the dc transport problem, we may drop the time\nderivative from now on. Here\nv=@\u000f\n@p(10)\nFimp=\u0000rVimpare the forces arising from the impurity potential, and\n\u000eFext=\u0000e\u0012\n\u000eE\u0000\u000f\u0000\u0016\nTr\u000eT\u0013\n: (11)\n\u000eC[\u000ef] denotes the collision integral, written only to leading (\frst) order in \u000ef. Explicit discussions of\nthe linearized collision integral may be found in [42]. A solution of the linearized Boltzmann equation (9)\nwill provide \u000ef. As there are no charge and heat currents in equilibrium, the charge and heat currents\nare given entirely by this perturbation:\nJi(x) =\u0000eZddp\n(2\u0019~)dvi\u000ef(x;p); (12a)\nQi(x) =Zddp\n(2\u0019~)d(\u000f+Vimp\u0000\u0016)vi\u000ef(x;p); (12b)\nFor simplicity in this paper, and without loss of generality, we will shift \u000f(p) by a constant such that the\nFermi surface is located at \u0016= 0, which simpli\fes a few of the formulas below.\n4Also, the detailed system parameters for carrying out such numerical calculations for speci\fc strongly correlated materials\nmay not be known. For such systems it would not yet be useful to perform serious numerical computations.\n6Mathematically, the Boltzmann equation is simply a linear algebra problem in a high-dimensional\nvector space. In order to make analytic progress, we cannot simply invert the linearized Boltzmann\noperator { it is far too complicated. We instead proceed in a number of steps, which may (for the\nmoment) appear rather cumbersome, but will in the end reduce the calculation of the conductivities to\nthe inversion of few-dimensional matrices, in certain limits. The notation and formulations below follow\nclosely [38, 39].\nThe \frst step is to de\fne a suitable inner product for the p-indices on our vector space:\nhfjgi=Zddp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013\nf(p)g(p); (13)\nWriting the distribution function as\nf(x;p) =feq(x;p) +Zddk\n(2\u0019)d\u0012\n\u0000@feq\n@\u000f\u0013\n\b(x;p); (14)\nwith\btreated as in\fnitesimally small, and de\fning\njJii=Z\nddpvi(p)jpi; (15a)\njQii=Z\nddp\u000f(p)vi(p)jpi; (15b)\n(15c)\nwe \fnd that\nh\b(x)jJii=Ji(x); (16a)\nh\b(x)jQii=Qi(x)\u0000Vimp(x)Ji(x): (16b)\nThe advantage of introducing the inner product (13) is now clear: the highly singular structure of the\nFermi function has been absorbed into the integration measure in the inner product. The vectors which\nwe are studying are now smooth functions of momentum across the Fermi surface. We also emphasize\nthat theVimp-dependent correction to the heat current will be negligible in the limits studied in this\npaper. We will think of jJiias the vector that encodes the charge current, and jQiias the vector that\nencodes the heat current.\nLetWdenote the linearized collision integral. It is a symmetric, positive-semide\fnite matrix with the\nassumptions made above; its null vectors are associated with local conservation laws of electron-electron\ncollisions. We will assume that one of these conservation laws is associated with momentum, because\ntranslation invariance is only broken by Vimp. The linearized Boltzmann equation (9) can then be written\nin our vector notation as\nhpj\u0014\nv\u0001@\n@x+Fimp\u0001@\n@p\u0015\nj\bi=\u0000hpjWj\bi+EihpjJii\u0000@iT\nT(hpjQii+VimphpjJii): (17)\nThe ( x;p)-dependence in the inner product has been properly taken into account { the derivatives above\nonly act on \b. Clearly, we may undo the inner product with jpi, since the equation is valid for all jpi.\n3.2 Perturbative Limit\nWe now focus on a perturbative limit where the amplitude of the smooth impurity potential Vimpis\nperturbatively weak. This is a natural limit of interest for the study of hydrodynamic interaction e\u000bects,\n7and helps ensure that electron-electron interaction e\u000bects are stronger than electron-impurity interactions.\nNote that although Vimpis small, it cannot vanish because then strict translational invariance is restored\nand the electrical conductivity is in\fnite, in the absence of umklapp (which is ignored throughout this\nwork). This can be seen by writing down Newton's second law:\ndPtot\ndt=QtotE (18)\nwhere Ptotis the total momentum in the theory, and Qtotis the total charge. In the absence of translation\nsymmetry breaking, no further terms may be written down on the right hand side, and transport properties\nin the strict dc limit are ill-posed. In a quantum theory, (18) continues to hold: it is the Ward identity\nassociated with translation symmetry. Therefore, we work consistently in the limit of small, but \fnite,\nVimp.\nAs is well-known, and we will see explicitly below, all thermoelectric conductivities now scale like V\u00002\nimp\nas a consequence of the slowness of momentum relaxation [5, 15, 36, 37]. This means that most { but not\nall {Vimp-dependence in the problem can be ignored. To see where we can neglect Vimp, recall that our\ngoal is to compute\nJavg\ni=Zddx\nVdJi(x)\u0019Zddx\nVdh\b(x)jJii (19a)\nQavg\ni=Zddx\nVdQi(x)\u0019Zddx\nVdh\b(x)jQii: (19b)\nHereVddenotes the spatial volume { we are simply taking a spatial average. The latter approximations\nabove come from the fact that both \u001band\u0014will scale as V\u00002\nimpat leading order. It will be useful to Fourier\ntransform\b:\nj\b(k)i\u0011Zddx\nVdj\b(xie\u0000ik\u0001x: (20)\n(9) can be written as\n \nW L0\ndis\n\u0000L0\ndisyW+ ik\u0001v+Ldis(k;k0)!\u0012j\b(k=0)i\nj\b(k6=0)i\u0013\n=\u0012\nEijJii\u0000@iT\nTjQii\n0\u0013\n; (21)\nwhere\nLdis(k;k0) =\u0000i(k\u0000k0)iVimp(k\u0000k0)\u0000\u0000!@\n@pi; (22)\nand we have denoted L0\ndis=Ldis(0;k). SinceVimpis perturbatively small, and i k\u0001v+Whas no null\nvectors in general, we do not need to worry about the Ldisexcept in the o\u000b-diagonal blocks in (21).\nCombining all of the above together, we conclude that the thermoelectric conductivity matrix is given\nby\u0012\u001bijT\u000bij\nT\u000bijT\u0016\u0014ij\u0013\n\u0019 \nhJijeW\u00001jJji h JijeW\u00001jQji\nhQijeW\u00001jJji h QijeW\u00001jQji!\n; (23)\nwhere we have de\fned the matrix\neW\u0019W+Zddk\n(2\u0019)dkikjjVimp(k)j2 \u0000\u0000@\n@pi(W+ ik\u0001v)\u00001\u0000\u0000!@\n@pj+ O(V3\nimp): (24)\nThe \frst term in eWcorresponds to electron-electron interactions. The second term corresponds to\nelectron-impurity collisions, which will not conserve momentum. Formally speaking, we observe that the\n8presence of Win the electron-impurity term means that the electron-electron interactions can substan-\ntially modify the nature of electron-impurity scattering. This interplay of electron-electron and electron-\nimpurity interactions is described in some detail in [38, 39], and in Section 4.5. However, for most of this\npaper, we will focus on a limit where Wisalsotaken to be perturbatively small: W\u0018V2\nimp. In this limit,\nimpurity scattering is well described by the single-particle theory: we may approximate\neW\u0019W+Wimp; (25)\nwhere\nWimp=Zddk\n(2\u0019)dkikjjVimp(k)j2 \u0000\u0000@\n@pi\u000e(kivi)\u0000\u0000!@\n@pj: (26)\nEq. (25) is reminiscent of Mattheisen's rule { the impurity-scattering and electron-scattering diagrams\nmay be added independently to the (disorder-averaged) collision integral of kinetic theory. However, we\nemphasize that in the \fnal transport coe\u000ecients, Mattheisen's rule for resistivities can be violated.\n3.3 Rotational Invariance and a Basis\nThe \fnal assumption that we will make in this paper is that the dispersion relation (and disorder dis-\ntribution) are rotationally invariant. This is a reasonable assumption for many correlated Fermi liquids\nwhere the hydrodynamic regime is likely experimentally accessible, including graphene and GaAs. We\nfurther assume that Wis rotationally invariant; by construction in (26), it is clear that (after performing\nthek-integral) Wimpis rotationally invariant. Thus, eWis rotationally invariant. What this means is that\nthe natural basis of vectors jpiare:\nj\u000b;m 1\u0001\u0001\u0001md\u00001i=Z\nddpf\u000b(p)Ym1\u0001\u0001\u0001md\u00001(\u00121;:::;\u0012d\u00001)jpi (27)\nIn the above equation, and henceforth, p=jpj,\u00121;:::;\u0012d\u00001denote the angular coordinates of p, andf\u000b\nis a set of polynomial functions that we will detail below. In this basis,\nh\u000b;m 1\u0001\u0001\u0001md\u00001jeWj\f;m0\n1\u0001\u0001\u0001m0\nd\u00001i=Wm1\u0001\u0001\u0001md\u00001\n\u000b\f\u000em1m0\n1\u0002\u0001\u0001\u0001\u0002\u000emd\u00001m0\nd\u00001: (28)\nOur goal is to compute the thermoelectric conductivity matrix. Both jJiiandjQiiare vectors under\nspatial rotation, and without loss of generality, we may compute \u001b=hJxjeW\u00001jJxi, etc. So a natural\nchoice of basis vectors to study corresponds to\nj\u000bi\u0011px\npf\u000b(p); (29)\nfor a basis of polynomials ff\u000bg. For these basis vectors, we observe that\nh\u000bj\fi=Zddp\n(2\u0019~)df\u000b(p)f\f(p)p2\nx\np2=1\ndZddp\n(2\u0019~)df\u000b(p)f\f(p): (30)\nUp to the overall prefactor of 1 =d, we may simply compute the radial integrals over the Fermi surface.\nAnother useful simpli\fcation in this basis arises when computing the matrix elements h\u000bjWimpj\fi.\nUsing\nki@\n@pi\u0012px\npf\u000b\u0013\n=kx\npf\u000b+\u0000\nf0\n\u000b\u00001\u0001kipipx\np2; (31)\nand observing that the \u000e(kivi) in (26) constrains kipi= 0 in an isotropic theory, we conclude that\nh\u000bjWimpj\fi=\ndZpd\u00001dp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013Zddk\n(2\u0019)dk2\nxjVimp(k)j2\u000e(kivi)f\u000bf\f\np2\n9=\ndZpd\u00001dp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013f\u000bf\f\np2v\u0014Zddk\n(2\u0019)dk2\ndjVimp(k)j2\u000e(kx)\u0015\n=Aimp\u0002\ndZpd\u00001dp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013f\u000bf\f\np2v; (32)\nwhere\ndis the area of the unit sphere. Hence, all of the physics of impurity scattering becomes captured\nby a single constant Aimp, which is de\fned as the object in square brackets in the second line of (32). We\ncaution that there can be non-trivial temperature dependence in Aimp, in the presence of temperature-\ndependent screening of a long-range Coulomb impurity potential Vimp(k) [43, 44], an e\u000bect that we ignore\nin the present work.\nLarge Fermi Surfaces4\nTo make further progress, we will now work in the limit where the Fermi surface is large. To be precise,\nwe will assume that T\u001cpFvF. In this limit, we can further simplify the evaluation of the conductivity\nby neglecting all but a few low order polynomials f\u000bin the basis (29).\n4.1 An E\u000ecient Basis\nLet the dispersion relation be\n\u000f(p) =vFq+a\n2q2+b\n3q3+\u0001\u0001\u0001; (33)\nA natural set of basis functions can be constructed as follows. Letting q\u0011p\u0000pF, we de\fne the states\njqni, corresponding to the polynomials f\u000b=qn. Of great importance to us is that in this basis,\njJxi=\u0000e\u0000\nvFjq0i+ajq1i+bjq2i+\u0001\u0001\u0001\u0001\n; (34a)\njQxi=v2\nFjq1i+3\n2vFajq2i+\u0001\u0001\u0001: (34b)\nUnfortunately, the basis functions jqniare not normalized. It would be more convenient if we could\nwork with an orthonormal basis. This can be constructed straightforwardly by the Gram-Schmidt pro-\ncedure, and we will denote the result of this procedure with jni. The calculation of jnifromjqniis\nstraightforward but tedious: details are provided in Appendix A. A useful qualitative fact for the discus-\nsion that follows is that\nhqnjqmi\u0018\u001aTn+mn+meven\nTn+m+1n+modd: (35)\nIn what follows, subleading numerical coe\u000ecients in Tare speci\fc to d= 2 spatial dimensions , but the\ngeneral structure is unchanged for all dimensions. The \frst few basis functions are\nj0i=c00r\nd\n\u0017jq0i; (36a)\nj1i=r\nd\n\u0017hc11\nTjq1i+Tc10jq0ii\n; (36b)\nwith\u0017the density of states (we remind the reader of useful thermodynamic formulas for a Fermi liquid\nin the low temperature limit in Appendix B), the factor of darising from (30), and\nc00= 1\u0000\u00192T2\n2\u0012a2\n2v4\nF\u0000a\n2pFv3\nF\u0000b\n3v3\nF\u0013\n+ O(T4); (37a)\n10c11=p\n3vF\n\u0019+\u0019T2\u0000\n2v2\nF+ 14bvFp2\nF+ 15apFvF\u000027a2p2\nF\u0001\n4p\n3v3\nFp2\nF+ O(T4); (37b)\nc10=\u0019(3apF\u00002vF)\n2p\n3pFv2\nF+ O(T2): (37c)\nA general basis function is\njni=r\nd\n\u0017\u0010vF\nT\u0011n\ncnnjqni+\u0010vF\nT\u0011n\u00002\u0000\ncn;n\u00001jqn\u00001i+cn;n\u00002jqn\u00002i\u0001\n+\u0001\u0001\u0001: (38)\nA key observation is that { up to O( T2=T2\nF) corrections { the orthonormal basis is equivalent to jqni, up\nto a prefactor which is propotional to T\u0000n. Also observe that jqnicanexactly be expressed as a function\nofjmiwithm\u0014n.\n4.2 Approximating the Collision Integrals\nFrom the form of (34) and (36), it is clear that as long as components of Wimpare \\well-behaved\" in the\nlowTlimit, then we can e\u000eciently compute the thermoelectric conductivity matrix by focusing only on\nthe \frst few basis vectors: j0i;j1i;:::. Indeed, this is the case:\nhmjWimpjni.Tjm\u0000nj: (39)\nTo show this, we \frst write\nAimp\np2v(p)=\u0000+\u00001q+\u00002q2+\u0001\u0001\u0001: (40)\nBy de\fnition, qkjqni=jqn+ki. (We are denoting the linear operation which multiplies by qasq, in the\nobvious way.) Now, assuming that m>n , without loss of generality, observe that\nhmjqm\u0000njni\u0018hmj\u00141\nTnjqmi+T2\nTnjqm\u00001i+T2\nTnjqm\u00002i+\u0001\u0001\u0001\u0015\n\u0018hmjqmi\nTn\u0018Tm\u0000n: (41)\nIn fact, staring at the above expression, we also \fnd that hm\u0000`jqm\u0000njqni.Tm\u0000nfor any`\u00150.\nTherefore,\nhmjWimpjni\u0018\u0000m\u0000nTm\u0000n+\u0001\u0001\u0001: (42)\nThe\u0001\u0001\u0001in (42) includes terms proportional to \u0000m\u0000n+2Tm\u0000n+2, etc., which are all subleading in the low\ntemperature limit. At T= 0, there is a residual resistivity:\nhmjWimpjni=\u0000\u000emn: (43)\nWe have not dropped any constant prefactor in the above formula (43).\nWe next observe that\nhmjW\u00001\nimpjni.Tjm\u0000nj: (44)\nOne way to show this result is recursively. From the form of the T= 0 result (43), we show the base case\njn\u0000mj= 0. For \fnitejn\u0000mj, we may use block matrix inversion identities. Without loss of generality,\nwe again take m>n . Let us write\nW=\u0000\nhk<mj hk\u0015mj\u0001\u0012W\u0000\u0000 W\u0000+\nW+\u0000W++\u0013\u0012jk<mi\njk\u0015mi\u0013\n(45)\n11Since there are no singular eigenvalues at low temperatures,\nhmjW\u00001\nimpjni\u0018X\nk<m;`\u0015m\u0000\nW++\u0000W+\u0000W\u00001\n\u0000\u0000W\u0000+\u0001\u00001\nm`(W+\u0000)`k(W\u00001\n\u0000\u0000)kn.X\nk<m;`\u0015mTjk\u0000`j+jk\u0000nj.Tjm\u0000nj:\n(46)\nSince physical observables such as jJxiandjQxiare naturally expressed in terms of the jqnibasis, as\nin (34), we are only interested in evaluating W\u00001\nimpon vectors which take the form\njJxi=X\nnanTnjni;jQxi=X\nnbnTnjni;etc: (47)\nObserving that\nX\nmnamanTm+nhmjW\u00001\nimpjni\u0018X\nmnamanTm+n+jm\u0000nj=X\nmnamanT2 max(m;n); (48)\nwe conclude that to compute a transport coe\u000ecient up to O( T2k), we only need to keep basis vectors\nj0i;:::;jki. To leading order, we know from the Wiedemann-Franz law and Mott law that T\u0016\u0014andT\u000b\nwill both scale as T2at low temperatures. As we show in Section 4.3, keeping only the j0iandj1ibasis\nvectors is su\u000ecient to recover these relations in the non-interacting limit.\nNow that we understand the form of the \\collisionless\" electron-impurity scattering, let us discuss the\nform of the electronic collision integral. Since we are interested in the hydrodynamic regime of electron\n\row, we focus on theories where momentum is conserved in all electron-electron collisions (there is no\numklapp). This means that the momentum vector\njPxi=pFjq0i+jq1i\u0019r\u0017\nd\u0014\npFj0i+\u0019Tp\n3j1i\u0015\n(49)\nis a null vector of W. One simple choice of Wthat is consistent with this requirement is\nW\u0019\r\u0012\nj1i\u0000\u0019Tp\n3pFvFj0i\u0013\u0012\nh1j\u0000\u0019Tp\n3pFvFh0j\u0013\n+1X\nn=2\rjnihnj: (50)\nThis choice, which is used extensively in transport theories, is often called the relaxation time approxima-\ntion [45]. Note that the relative decay of longitudinal \ructuations above necessitates inelastic scattering\nprocesses. The coe\u000ecient \r\u0018\u001c\u00001\neeis associated with the rate at which electron-electron collisions relax\nlongitudinal \ructuations of the distribution function, proportional to px:\n\r\u0011~\u000b2T2\nTF: (51)\nwhere ~\u000b2, an e\u000bective coupling constant in the problem which we do not specify, is (roughly) proportional\nto the interaction constant. In lower dimensions, it is possible for this scattering time to be enhanced by\nfactors of log( TF=T) [46, 47, 48, 49, 50, 51], and we will not worry about these logarithms in this paper.\nWe note that the Fermi liquid prediction (51) for the inelastic scattering rate { with suitable logarithms\n{ has been veri\fed experimentally (see e.g. [52] for a direct veri\fcation in a 2D GaAs system). The\nT2dependence in (51) is completely consistent with the classic Fermi liquid paradigm of well-de\fned\nquasiparticles. Indeed, our kinetic picture is perturbative in interactions and so a NFL does not emerge\nin our model in 2D or 3D electron liquids. This statement is independent of ~ \u000b2, although for large enough\n~\u000b2our kinetic theory is not valid. Normal 3D metals are strongly interacting in the sense that ~ \u000b(also\ncommonly called ` rs' in interacting Coulomb systems) \u00186. However, because TF\u0018104K,\ris still small\nrelative tokBT=~(the scattering rate below which a quasiparticle picture does not make sense [5]). In\n2D GaAs hole systems, the interaction coupling constant can be \u001840 in dilute experimental systems,\nputting the system deep in the hydrodynamic regime [53].\n124.3 The Strongly Interacting Limit\nWe are now \fnally ready to study transport phenomena across the ballistic-to-hydrodynamic crossover.\nTo start o\u000b, let us work in a simple limit where\n\u0000\u001c~\u000b2TF: (52)\nWe will relax this assumption in Section 4.4. This inequality can be satis\fed in 2D systems experimentally,\nbut not in the usual metals. (52) implies that \r=\u0000at temperatures T\u001cTF, where an expansion in\nlow order basis vectors ought to be sensible, even in the hydrodynamic regime. Thus, we will work only\nto the lowest non-trivial order in T=T FinWimp, to capture the e\u000bects of interactions. From the form of\nthe Wiedemann-Franz law (1), Mott relation (2), and the de\fnition (23), we conclude that it will su\u000ece\nto keep track of only the vectors j0iandj1i. This is notequivalent to a Taylor series expansion in T, as\n\ritself depends on Tvia (51). We will return to this point in Section 4.4.\nConverting thejqnibasis to thejnibasis, the charge and heat current vectors are\njJxi\u0019\u0000er\u0017\nd\u0014\nvFj0i+a\u0019Tp\n3vFj1i\u0015\n; (53a)\njQxi\u0019r\u0017\nd\u0014\u00192T2\n3pFj0i+\u0019TvFp\n3j1i\u0015\n: (53b)\nThe net collision integral ~Wis given by\neW\u0019\u0012\u00192T2\n3p2\nFv2\nF\r+\u0000\u0013\nj0ih0j+\u0019Tp\n3vF\u0012\n\u00001\u0000\r\npF\u0013\n(j0ih1j+j1ih0j) + (\u0000+\r)j1ih1j (54)\nComputing the thermoelectric conductivity matrices has reduced to simply inverting a 2 \u00022 matrix:\neW\u00001\u00191\n\u0000j0ih0j\u0000\u0019Tp\n3vF\u00001\u0000\r\npF\n\u0000(\u0000+\r)(j0ih1j+j1ih0j) +\u00192T2\n3p2\nFv2\nF\r+\u0000\n\u0000(\u0000+\r)j1ih1j (55)\nIn this step, we have made approximations consistent with the assumptions stated at the start of this\nsubsection.\nFirst, let us study the limit \r= 0, which corresponds to the T!0 limit. After some algebra we\nobtain\n\u001b\u0019\u0017e2v2\nF\nd\u0000; (56a)\n\u0014\u0019\u0016\u0014\u0019\u00192T\n3\u0017v2\nF\nd\u0000\u0011\u0014non\u0000int; (56b)\n\u000b\u0019\u0000e\u00192T\u0017\n3d\u0014\n\u0000vF\u00001\n\u00002+a\n\u0000+vF\npF\u0000\u0015\n\u0011\u000bnon\u0000int: (56c)\nIn each case, we have only written down the leading order terms in the T!0 limit (where, by de\fnition,\nwe should recover the standard Fermi liquid WF law since interaction-induced scattering vanishes at the\nFermi surface). It is clear that we recover the Wiedemann-Franz law (1). The Mott relation (2) is also\nobeyed, because\n@\u001b\n@\u0016=1\ndvF@\u001b\n@pF=\nd\n(2\u0019~)ddvF@\n@pFpFvF\n\u0000=\nd\n(2\u0019~)ddvF\u0014vF\n\u0000\u0000pFvF\u00001\n\u00002+pFa\n\u0000\u0015\n13=\u0000\u0017vF\u00001\nd\u00002+\u0017a\nd\u0000+\u0017vF\ndpF\u0000: (57)\nHaving con\frmed that our theory correctly reproduces the non-interacting \\collisionless\" limit of\nelectron-impurity scattering, let us crank up the electron-electron collision rate \r. The \frst thing we\nobserve is that\n\u001b\u0019\u0017e2v2\nF\nd\u0000+ O\u0000\nT2;T2\r\u0001\n: (58)\nThis is an important result: interactions a\u000bect the electrical conductivity only weakly (i.e. at higher-\norder terms) through hydrodynamic corrections. By contrast, as we show below, interaction does a\u000bect\nthe thermal conductivity and the thermoelectric conductivity. The electrical conductivity is, in this\napproximation, essentially independent of electron-electron scattering.\nNext, let us discuss the thermal conductivity \u0016 \u0014. In this case, we obtain\n\u0016\u0014=\u0017\u00192v2\nFT\n3d\u00002\n4\u00192T2\n3p2\nFv2\nF\r+\u0000\u00002\u00192T2\n3pFv2\nF(\u00001\u0000\r\npF)\n\u0000+\r+\u00192T2\n3p2\nFv2\nF3\n5\u0019\u0014non\u0000int\u0000\n\u0000+\r+ \u0016\u0014int\r\n\u0000+\r; (59)\nwith\n\u0016\u0014int=4\u00194T3\u0017\n9dp2\nF\u0000: (60)\nThe thermoelectric conductivity is given by\n\u000b\u0019\u0000e\u0017\u00192Tv2\nF\n3d\u0000+\r\npFvF\u00001\nvF(\u00001\u0000\r\npF) +a\nv2\nF(\u00192T2\n3p2\nFv2\nF\r+\u0000)\n\u0000(\u0000+\r)\u0019\u000bnon\u0000int\u0000\n\u0000+\r+\u000bint\r\n\u0000+\r; (61)\nwhere\n\u000bint=\u0000e2\u00192T\u0017v F\n3dpF\u0000: (62)\nBoth \u0016\u0014and\u000badmit an elegant interpretation in this limit: they are a weighted average of the \\col-\nlisionless\" transport coe\u000ecient and an \\interaction-limited\" transport coe\u000ecient. The weights in front\nof each term correspond to the fraction of the total scattering rate that is in the electron-impurity vs.\nelectron-electron channel.\nWe can go further. In the interaction-dominated limit, we expect that the thermoelectric conductivity\nmatrix is limited entirely by momentum relaxation [5, 15, 35, 36, 37]. Writing down the momentum\nconservation equation, we obtain\n\u0000enE\u0000srT=\u0000\u0003v; (63)\nwithvthe velocity of the interacting \ruid, and \u0000\u0003a coe\u000ecient related to weak momentum relaxation.\nThe left hand side corresponds to the momentum added to the system by the external drives; the right\nhand side corresponds to the system's internal momentum dissipation. Approximating the charge current\nasJ\u0019nvand the heat current as Q\u0019Tsv, we arrive at the formulas\n\u0012\u001b T\u000b\nT\u000b T \u0016\u0014\u0013\n\u00191\n\u0000\u0003\u0012e2n2\u0000eTsn\n\u0000eTsn (Ts)2\u0013\n: (64)\nUsing the thermodynamic relations of a low temperature Fermi liquid given in Appendix B, we \fnd that\nin our toy model above,\n\u0000\u0003=M\u0000; (65)\nwhereMis the momentum-momentum susceptibility (analogous to a mass density). Since \u0000is the\nmomentum relaxation rate, this agrees with the predictions of the memory matrix formalism [36, 37].\n14Using (60) and (62) with these thermodynamic identities, we can con\frm that \u000bintand \u0016\u0014inttake the form\ndemanded by (64), relative to \u001b. Note also the identity\n\u001b\u0016\u0014int=T\u000b2\nint; (66)\nwhich is tied to the fact that one process { momentum relaxation { limits all transport coe\u000ecients in the\nhydrodynamic regime.\nNow, let us return to the fate of the Wiedemann-Franz law. Conventionally one compares not \u0016 \u0014to\u001b,\nbut\u0014to\u001b. As a consequence, using the identity (66), we obtain\n\u0014\u0019\u0000\n\u0000+\r\u0014non\u0000int+\r\u0000\n(\u0000+\r)2\u0012\n\u0016\u0014int\u00002T\u000bint\u000bnon\u0000int\n\u001b\u0013\n\u0019\u0000\n\u0000+\r\u0014non\u0000int\u0019\u00192T\u0017v2\nF\n3d(\u0000+\r)(67)\nAs before, we are neglecting contributions that are subleading in T, at each order in \r. The Wiedemann-\nFranz law will be violated, \\strongly interacting\" limit, in the particularly elegant manner given in (6).\nIn this simple model, the amount of violation of the Wiedemann-Franz law corresponds directly to the\nfraction of scattering processes which are momentum-conserving. This interpretation fails to hold if T=T F\nis not so small, and/or if the disorder strength is comparable to the interaction strength (Section 4.4).\nEarlier discussions emphasizing that the Wiedemann-Franz law will be violated in the hydrodynamic limit\nof a \fnite density metal, because \u0014=\u001bT is very small, can be found in [3, 11, 54]. In the hydrodynamic\nlimit, the value of \u0014matches the dissipative hydrodynamic coe\u000ecient \u0014q, at leading order in T=T F: see\nAppendix D.\nThe fate of the Mott relation is less dramatic. In the limit T\u001cTF, both\u000bnon\u0000intand\u000bintare linear\ninT. The collisionless-to-hydrodynamic crossover will simply correspond to a gradual change in the slope\nof\u000b(T). We note that this crossover can be more dramatic near charge neutrality ( T&TF). Some\nexperimental evidence for this e\u000bect in graphene, and further discussion, can be found in [32, 55].\n4.4 The Weakly Interacting Limit\nIn this section, we will consider the limit where (52) is not obeyed. This corresponds to metals with\nrelatively low Fermi temperature (where the WF law is a\u000bected by thermal smearing of the Fermi function\neven without any interaction e\u000bects), and/or relatively small interaction coupling ~ \u000b. Indeed, smearing of\nthe Fermi surface leads to comparable O\u0000\nT2\u0001\ncorrections, similar to (but distinct from) electron-electron\ninteractions.\nTo correctly compute the thermal conductivity, we must compute T\u0016\u0014to O(T4), and so we must keep\nthe basis vectorj2iin our expansion. For simplicity, we will take the dispersion relation \u000f(p) to be exactly\nquadratic. This means that in (33), b= 0,apF=vF, andpFvF= 2TF. We will also treat electron-electron\ncollisions within a relaxation time approximation (50), but at next-to-leading order this does not change\nthe form of the answer. The forms of jJxi,jQxiandWimpare written explicitly in Appendix C. We then\nexplicitly compute\n\u001b=\u0017\n2v2\nF\n\u0000\u0012\n1 +\u00192T2\n3T2\nF\u0013\n+ O\u0000\nT4\u0001\n; (68a)\n\u0014=\u0017\n2\u00192T\n3e2v2\nF\n\u0000\u0012\n1\u000017\u00192T2\n30T2\nF\u0000~\u000b2T2\n\u0000TF\u0013\n+ O\u0000\nT5\u0001\n: (68b)\nThe Lorenz number\n\u0014\n\u001bT=\u00192\n3e2\u0014\n1\u00009\u00192T2\n10T2\nF\u0000~\u000b2T2\n\u0000TF\u0015\n+\u0001\u0001\u0001 (69)\nis thus decreased both by thermal smearing of the Fermi surface and by electron-electron interactions.\nWhen disorder is very weak, the dominant e\u000bect will be that of electron-electron interactions. Also observe\n15that even the \frst subleading temperature-dependent correction to \u001barises only from thermal smearing\nof the Fermi surface, and not from electron-electron scattering.\nThus,\u0000,\r, andTFare all relevant parameters determining the e\u000bective Lorenz number as a function of\nT. Remembering that TF\u0018n2=d, where n is the electron density of the d-dimensional electron liquid,5and\nthat\r\u0018T2=TF, we conclude that lowering electron density may enhance both the e\u000bect of interaction\nand the e\u000bect of Fermi surface thermal smearing depending on the temperature of the system. If the\ne\u000bective impurity disorder Gamma itself has a temperature dependence through electronic screening (as\nit does in 2D systems [56]), the situation gets much more complicated. Disentangling all of these e\u000bects\nquantitatively will require materials-speci\fc computations beyond the scope of the present paper.\n4.5 Hydrodynamic Fluctuations and Impurities\nIn this section, we will qualitatively describe the interplay of electron-electron interactions with the\nimpurity potential. Mathematically, this is done by keeping track of the full Wmatrix when performing\nthe matrix inverse in (24). For the purposes of this section, we will keep the discussion qualitative, since\nthe quantitative details depend on far too many unknown material-speci\fc parameters and the precise\nnature of the disorder [38, 39]. After a microscopic calculation, what one \fnds is that Wimpis no longer\napproximately a single scattering rate \u0000, times the identity matrix. Instead, the eigenvalues of Wimpwill\nsensitively depend on the nature of disorder, and on the modes j\u000biwhich are being sourced.\nWe assume that the inhomogeneous potential Vimp(x) varies on the length scale \u0018. Let\u0012\u001c1 denote\nthe typical de\rection angle of a quasiparticle after it moves a distance \u0018\u0018through the potential landscape.\nA typical degree of freedom j\u000biwill obey\n\u0000\u0018h\u000bjWimpj\u000bi\u0018\u00122vF`ee\n\u0018(\u0018+`ee); (70)\nwhere\n`ee\u0011vF\n\r(71)\nThis will hold, at least qualitatively, for all modes j0i,j1i,j2i, etc., in the isotropic Fermi liquid with\npotential disorder [38]. In these theories, \u0000decreases as electron-electron interactions increase [38]:\n\u0000\u0019\u00122vF\n\u0018\u0014\n1\u0000\u0018\r\nvF+\u0001\u0001\u0001\u0015\n\u0019\u00122vF\n\u0018\u0000\u00122\r+ O\u0000\n\r2\u0001\n; (72)\nWe conclude from (6) that the Wiedemann-Franz law is violated even more strongly when hydrodynamic\ncorrections to impurity scattering are accounted for, since the e\u000bective disorder strength \u0000is suppressed\nby interactions. From (72), we also estimate that the hydrodynamic corrections to impurity scattering\nare relatively weak compared to intrinsic electron-electron scattering in the collision integral by a factor\n\u00122. This arises from the inherent di\u000berence between elastic (impurity) and inelastic (electron-electron)\nscattering processes.\nHowever, in some cases it is possible for hydrodynamic corrections to impurity scattering to behave\nrather di\u000berently. In particular, in the presence of \\magnetic\" disorder, one obtains [39]\nh0jWimpj0i\u0019\u00122vF\u00121\n\u0018+1\n`ee\u0013\n=\u00122vF\n\u0018+\u00122\r\u0011\u0000\u0003: (73)\nj1i,j2i, etc. will have the same decay rate as before. The origin of the peculiar behavior of the decay\nrate of thej0imode is related to the fact that the \\magnetic\" disorder exerts large shear stresses which\n5This relation changes for systems with non-parabolic dispersion relations, such as monolayer graphene.\n16in turn create large local velocity \felds in the electronic \ruid. We must now recompute \u001band\u0014in a \ruid\nwhere\nWimp=\u0000\u0003j0ih0j+\u0000j1ih1j: (74)\nFollowing the procedure of Section 4.3, we obtain\n\u001b\u0019\u0017\nde2v2\nF\n\u0000\u0003; (75a)\n\u0014\u0019\u0017\nd\u00192T\n3v2\nF\n\u0000+\r: (75b)\nWe therefore \fnd that for this \ruid,\n\u00122.3e2\n\u00192\u0014\n\u001bT\u00141: (76)\nThe Wiedemann-Franz law is still violated, but not as strongly as before. Furthermore, the Lorenz ratio\nwill no longer become arbitrarily small as the sample becomes more pure. This scenario is analogous to\nthe presence of electron-electron umklapp (momentum-relaxing) scattering at rate \u00122\r.\nFinally, another well-known [56] example of an interaction-suppressing e\u000bect of disorder is the screening\nof long range Coulomb impurities by the electrons themselves, which leads to a strongly temperature-\ndependent e\u000bective disorder in 2D systems, manifesting a \frst-order positive thermal correction to the\nresistivity which is linear in T=T Fin 2D (but not in 3D). This e\u000bect, which is beyond the scope of the\ncurrent hydrodynamic theory as it arises from the singular nature of the 2D polarizability function at\n2kF[57, 58], tends to increase \u0000, and therefore the Lorenz number, with increasing temperature. As this\ne\u000bect will increase the Lorenz number it does not provide an alternate explanation for reduced Lorenz\nnumbers in an interaction-limited regime.\nMagnetotransport5\nIn this section, we will discuss magnetotransport phenomena in the presence of a weak magnetic \feld,\nfocusing speci\fcally on the experimentally relevant case of a two-dimensional electron \ruid. When the\ncyclotron radius is very large compared to \u0015F, the magnetic \feld can be treated within our classical\nkinetic description. If the cyclotron radius is large compared to the size of the impurity potential, then\nwe can neglect magnetic \feld corrections to Wimp. We will make both of these assumptions to simplify\nthe calculations below.\nAccounting for the magnetic \feld amounts to modifying eWtoeW=W+Wimp+Wmag, with\nWmag=\u0000eB\u000fijvi@\n@pj(77)\nand\u000fxy=\u0000\u000fyx= 1 the antisymmetric Levi-Civita tensor. In a rotationally invariant Fermi liquid, we\ncan further simplify this. Generalizing (29) to\nj\u000bii\u0011pi\npf\u000b(p); (78)\nand generalizing the bases discussed in Section 4.1, we obtain that\nWmagj\u000bii=e\u000fijpj\npBv(p)\npf\u000b(p); (79)\nhence, Wmagwill mixxandyvectors (under spatial rotation), but will not couple vectors to higher/lower\nrank tensors.\n175.1 The Strongly Interacting Limit\nFor simplicity, let us now focus on the strongly interacting limit described in Section 4.3. As before, we\nwill (up to the T-dependence of \r) only be interested in the leading order temperature dependence of all\nconductivities, which will allow us to neglect all basis vectors but j0x;yiandj1x;yi. We may write\neW= (W+Wimp)\n\u000eij+eWmag\n\u000fij; (80)\nwhere\neWmag=eBv F\npF(j0ih0j+j1ih1j) +\u0019TeB(apF\u0000vF)p\n3vFp2\nF(j0ih1j+j1ih0j): (81)\nUsing (23), we can invert the 4 \u00024 matrixeWto compute all conductivities of interest. Taking the same\nlimit as Section 4.3, we \fnd that the electrical conductivity is given by\n\u001bxx=\u001byy=\u0017\nde2v2\nF\u0000\n\u00002+!2c; (82a)\n\u001bxy=\u0000\u001byx=\u0017\nde2v2\nF!c\n\u00002+!2c; (82b)\nwhere we have de\fned the cyclotron frequency\n!c\u0011eBv F\npF: (83)\nAs before, we observe that \u001bijis independent of electron-electron interactions, within this simple model.\nFurthermore, the form of \u001bijis completely consistent with the Drude model of magnetotransport: see e.g.\n[15]. The resistivity tensor \u001aij=\u001b\u00001\nijexhibits no classical magnetoresistance, following the conventional\nlore. When \u0000= 0, we obtain the Hall conductivity\n\u001bxy=\u0017\n2e2v2\nF\n!c=en\nB: (84)\nSince\u0000enis the charge density, we \fnd the classical Hall conductivity. In fact, using quantum Ward\nidentities, this relation can be derived for any translation invariant quantum system (in the absence of\nBerry curvature): see e.g. [5]. The open-circuit thermal conductivities are given by\n\u0014xx=\u0014yy=\u0017\nd\u00192T\n3v2\nF(\u0000+\r)\n(\u0000+\r)2+!2c; (85a)\n\u0014xy=\u0000\u0014yx=\u0000\u0017\nd\u00192T\n3v2\nF!c\n(\u0000+\r)2+!2c; (85b)\nwhile the closed-circuit thermal conductivity is\n\u0016\u0014xx= \u0016\u0014yy\u0019\u0014xx+\u0017\nd4\u00194T3\u0000\r2\np2\nF(\u00002+!2c)((\u0000+\r)2+!2c); (86a)\n\u0016\u0014xy=\u0000\u0016\u0014yx\u0019\u0014xy\u0000\u0017\nd4\u00194T3!c\r2\np2\nF(\u00002+!2c)((\u0000+\r)2+!2c): (86b)\nThe thermoelectric conductivity is given by\n\u000bxx=\u000byy=\u0000e\u0017\nd\u00192T\n3\u0000(\u0000+\r)(apF\u0000+vF(\u0000\u0000\u00001pF+ 2\r)) +!2\nc(3\u0000vF+\u00001pFvF\u0000a\u0000p F)\npF(\u00002+!2c)((\u0000+\r)2+!2c);(87a)\n18\u000bxy=\u000byx=e\u0017\nd\u00192T\n3pF(a\u0000\u0000\u00001vF)(2\u0000+\r) +vF(2\r2+ 2!2\nc+ 3\u0000\r)\npF(\u00002+!2c)((\u0000+\r)2+!2c): (87b)\nWe now unpack the thermal and thermoelectric conductivities. We \frst focus on the non-interacting\nlimit\r= 0. The Wiedemann-Franz law is obeyed component-wise:\n\u0016\u0014ij\u0019\u0014ij\u0019\u00192T\n3e2\u001bij; (88)\nas is the Mott relation:\n\u000bij=\u0000\u00192T\n3e@\u001bij\n@\u0016: (89)\nThis relation can be checked using methods analogous to (57). In the limit of large \r, we \fnd that\n\u0012\u001bxxT\u000bxx\nT\u000bxxT\u0016\u0014xx\u0013\n=1\nM\u0012e2n2\u0000eTsn\n\u0000eTsn (Ts)2\u0013\u0000\n\u00002+!2c=\u0012e2n2\u0000eTsn\n\u0000eTsn (Ts)2\u0013M\u0000\n(M\u0000)2+ (enB)2\n(90a)\n\u0012\u001bxyT\u000bxy\nT\u000bxyT\u0016\u0014xy\u0013\n=1\nM\u0012e2n2\u0000eTsn\n\u0000eTsn (Ts)2\u0013\u0000!c\n\u00002+!2c=\u0012e2n2\u0000eTsn\n\u0000eTsn (Ts)2\u0013\u0000enB\n(M\u0000)2+ (enB)2:\n(90b)\nThese latter equations are consistent with the predictions of the memory matrix formalism at weak\ndisorder and weak magnetic \feld strength [37].\nWhen\u0000,\rand!care all comparable, we conclude from (85) and (87) that the Wiedemann-Franz and\nMott relations are no longer valid. The thermal conductivity depends rather simply on interactions: to\nleading order in T=T F, we may simply replace \u0000!\u0000+\r, as in Section 4.3. However, the thermoelectric\nconductivity \u000bijis far more complicated, and we do not see a simple way to disentangle the e\u000bects of\n\fnite magnetic \feld, disorder and interaction strength.\nDiscussion and Conclusions6\nThe WF and Mott laws are strongly a\u000bected by electron-electron interactions in the hydrodynamic regime\nentirely within the Fermi liquid paradigm. Even an arbitrary suppression of the WF ratio well below the\nideal Lorenz number does not necessarily signify any NFL, but may only indicate strong interaction e\u000bects\nin the hydrodynamic regime [11]. The fact that most normal 3D metals manifest the ideal WF behavior is\na consequence of the fact that \u0000\u001d\rby virtue of the large TF\u0018104{105K, and also the relative strength\nof electron-phonon scattering. This is in spite of 3D normal metals being 'strongly interacting' in the sense\nof having a large dimensionless coupling constant rs\u00186. At low temperatures, where electron-phonon\ninteraction is negligible compared with electron-electron interaction, the electron-impurity scattering is\nstronger than electron-electron interaction. We do not see any obvious theoretical obstruction to a 3D\nmetal where electronic hydrodynamics occurs so long as the condition \r >\u0000 exists; a recent experiment\nsuggests such 3D systems exist [20].\nWe expect that at least some of the experimentally observed strange metals are in a hydrodynamic\nregime. Earlier discussion of this point may also be found in [13, 14]. Is it also possible that this hydro-\ndynamic regime is reasonably described by the Fermi liquid approach detailed in this paper? Answering\nthis question requires careful material-speci\fc considerations of each system to investigate ( i) whether\n\r\u001d\u0000is satis\fed, and ( ii) if the failure of the WF law is happening in a temperature range consistent\nwith our Eq. (6). Such an investigation is well beyond the scope of the current work, but is worth future\nconsideration. Perhaps (at least in some portions of an often complicated phase diagram) strange metals\n19are 'strange' only in the sense of being hydrodynamic metals. Indeed, most strange metals have rather\nlowTF\u0018103K, so strong interaction e\u000bects could easily drive the system into the hydrodynamic regime.\nIn many strange metals, the large T-linear resistivity, violating the Mott-Io\u000be-Regel `bound' [59], is\nused as additional evidence for NFL physics [1, 4]. The validity of our kinetic approach is suspect in\nthis limit. Nevertheless, building upon [14], we propose a more careful study of thermal and electrical\ntransport in the Fermi liquid regime of strange metals, where \\conventional\" T2resistivity is observed just\noutside of the \\critical fan\" of T-linear resistivity, and where the kinetic theory of transport should apply.\nIt is reasonable to assume that disorder has the same origin in both strange and conventional portions\nof the phase diagram [14], so we expect that a clear understanding of the roles of umklapp, phonon\nscattering, and hydrodynamic e\u000bects in this regime will shed light on the origin of T-linear resistivity.\nAll solid state materials have (at least) three distinct scattering mechanisms a\u000becting transport:\nelectron-impurity scattering (controlling low-temperature transport), electron-phonon scattering (con-\ntrolling high-temperature transport), and electron-electron scattering (controlling the ballistic to hydro-\ndynamic crossover, in the absence of umklapp). Hydrodynamics can only be observed if \r\u0018T2=TFis\nthe dominant of these three scattering mechanisms.6This rules out hydrodynamic observations at high\ntemperatures, since electron-phonon scattering rates (not considered in our work) scaling as \u0018T=T D(for\nT >T D), whereTD(\u0018102{ 103K) is the typical phonon energy scale in most materials, is the dominant\nresistive scattering mechanism in all electronic materials at higher temperatures. At low temperatures,\nthe requirement \r\u001d\u0000necessitates both a very pure system so that the impurity disorder strength is\nweak and also a Fermi temperature low enough for \rnot to be too small. For a Coulomb-interacting\nFermi liquid with parabolic band dispersion with an e\u000bective mass m, the requirement for hydrodynamics\nbecomesm=n2=dbeing very large, where nis the carrier density and dthe spatial dimension. In addition,\nthe \"low-temperature\" condition T\u001cTDmust be satis\fed in order for phonon scattering to be unim-\nportant. It turns out, as emphasized already, this quantity is rather small in 3D metals by virtue of n\nbeing too large. However, in 2D GaAs systems, the hydrodynamic condition can easily be satis\fed, both\nbecause the carrier density can be made very low and the impurity disorder can be made very small by\nutilizing modulation doping.\nIndeed, the systems where we believe that our hydrodynamic predictions are most likely going to be\nveri\fed are low-disorder, high-mobility 2D systems such as modulation-doped 2D GaAs systems and high-\nquality graphene layers. Converting the available experimental 2D mobility values and theoretical (but\nexperimentally veri\fed) electron-electron scattering strengths to e\u000bective scattering times, we \fnd that\nthe condition \r\u001d\u0000is obeyed in 2D GaAs structures (both n-doped and p-doped) down to 1 K or below in\ndilute systems of density \u00181011cm\u00002[60, 61]. At these low temperatures, phonon e\u000bects are completely\nnegligible, and therefore, our approximations should apply uncritically with the dilute 2D GaAs system\nbeing deep in the hydrodynamic regime. We predict a strong failure of the WF law in these systems,\nand our prediction of (6) can be directly veri\fed by varying the temperature. To be speci\fc, in the 2D\nn-GaAs system (assuming a mobility of 107cm2=V\u0001s), the electron-electron and the electron-impurity\nscattering times at T= 3 K andn= 3\u00021011cm\u00002are respectively \u001cee= 1=\r= 20 ps and \u001cei= 1=\u0000= 400\nps, making \r\u001820\u0000, thus placing the system deep into the hydrodynamic regime. Lowering Tto 100\nmK in the same sample hardly modi\fes the mobility (i.e. \u0000), but\u001ceeincreases to 25000 ps, driving the\nsystem to\r\u001c\u0000. SinceTF= 125 K for n= 3\u00021011cm\u00002, the low-temperature condition necessary for\nthe applicability of our Eq. (6) remains valid throughout. Thus, changing temperature from 3 K to 100\nmK in a high mobility 2D n-GaAs system would be an ideal experiment for the veri\fcation of our theory\nof the failure of the WF law. The same is even more true for 2D p-GaAs holes also, since the hole mass\nis larger than the electron mass in GaAs, thus making \re\u000bectively larger at the same temperature since\n6We cannot formally rule out momentum-conserving electron-phonon scattering as playing an important role, but this\nseems unlikely in most materials.\n20TF1=m. In both cases, electron-phonon interaction is at least an order of magnitude weaker than even\nthe electron-impurity interaction at these cryogenic temperatures, making the high-mobility 2D GaAs\nstructures the ideal systems for studying hydrodynamic transport e\u000bects.\nUnfortunately, we \fnd that the other well-known 2D semiconductor system, namely Si MOSFETs [62],\ndoes not satisfy the hydrodynamic condition (it comes close, but falls below the minimal requirement of\n\r&\u0000) in any accessible density or temperature range. Even the purest 2D Si system is simply not pure\nenough, at present.\nIn graphene, one has to go rather close to the Dirac point ( n\u0018109cm\u00002) to achieve the hydrodynamic\nconstraint at T\u001850 K. Here phonon e\u000bects are weak, but not negligible. One therefore requires extremely\nclean samples so that impurity-induced puddle e\u000bects do not overwhelm the hydrodynamic e\u000bects at such\nlow densities. Such experiments have been performed and interpreted based on hydrodynamic theories\nrecently [27, 31], but more work should be done to directly verify our predictions. Recently, the relevance\nof clean bilayer graphene in the context of 2D hydrodynamic transport has been pointed out [63].\nIn this paper, we have focused on the limit T\u001cTF. Assuming reasonable forms for the electron-\nelectron interactions, in this limit we rigorously reduced the calculation of thermoelectric conductivities to\na \fnite-dimensional linear algebra problem. We emphasize, however, that the techniques described in this\npaper immediately and straightforwardly generalize to T\u0018TF(so long as long-lived quasiparticles exist),\nand to systems without rotational invariance. In these more complicated settings, it is likely that the\nevaluation of the thermoelectric conductivity matrix must be done numerically, following the theoretical\nformalism described in this work.\nAcknowledgements\nAL was supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4302.\nSDS was supported by Laboratory for Physical Sciences.\nLow Temperature ExpansionA\nIn this appendix, we describe more explicit details about the low temperature expansion of inner products\nin thejqniandjnibases, in Section 4. We employ the integral\n1Z\n\u00001dx\n(1 + ex)ne(m+1)x=\u0000(1 +m)\u0000(n\u0000m\u00001)\n\u0000(n)\u0011F(m;n); (91)\nIgnoring the singularity at the origin of momentum space only leads to exponentially small corrections\n(e\u0000TF=T) in our theory. We then de\fne\nG(m;n;p )\u0011@pF(m;n)\n@mp=1Z\n\u00001dx\n(1 + ex)ne(m+1)xxp: (92)\nHaving de\fned G(m;n;p ), all integrals over momentum space in the inner products can be analytically\nevaluated. But, the process is a little bit subtle. In particular, observe that the derivative of the Fermi\nfunction must be carefully expanded:\ne\u000f(q)=T\nT(1 + e\u000f(q)=T)2=evFq=T\n(1 + evFq=T)2\"\n1\u0000(evFq=T\u00001)\n(1 + evFq=T)\u0012ap2\n2+bp3\n3\u0013\n+\u0000(e2vFq=T\u00004evFq=T+ 1)\n2(1 + evFq=T)2\u0012ap2\n2+bp3\n3\u00132#\n+\u0001\u0001\u0001\n(93)\n21This means that\nhqnjqmi\u0019\u0017\nd1Z\n\u00001dq\u0012\n1 +q\npF\u0013d\u00001e\u000f(q)=T\nT(1 + e\u000f(q)=T)2qm+n: (94)\nis not straightforwardly given by something proportional to G(0;2;m+n). Ind= 2, for example, we \fnd\nthe complicated Taylor series\nhqnjqmi\u0019\u0017\nd\u0012T\nvF\u0013m+n\u0014\nG(0;2;m+n)\u0000aT\n2v2\nF(G(1;3;m+n+ 2)\u0000G(0;3;m+n+ 2))\n\u0000bT2\n3v3\nF(G(1;3;m+n+ 3)\u0000G(0;3;m+n+ 3))\n+a2T2\n8v4\nF(G(2;4;m+n+ 4)\u00004G(1;4;m+n+ 4) +G(0;4;m+n+ 4))\n+T\nvFpF\u0012\nG(0;2;m+n+ 1)\u0000aT\n2v2\nF(G(1;3;m+n+ 3)\u0000G(0;3;m+n+ 3))\u0013\u0015\n+ O\u0000\nT3\u0001\n:(95)\nThe expansion of the inner product to O( T2), as above, will be adequate for the purposes of this paper.\nThermodynamic IdentitiesB\nIn this appendix, we describe the relationship between entropy, charge and the inner products in our\nkientic theory.\nFirst we re-derive an identity observed in [38]:\nhJxjPxi=Zddp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013\nvxpx=Zddp\n(2\u0019~)d\u0012\n\u0000@feq\n@px\u0013\npx=Zddp\n(2\u0019~)dfeq=n: (96)\nObserve that the zero temperature charge density is given by\nn0=pd\nF\nd(2\u0019~)d\nd; (97)\nwhere\nddenotes the volume of the unit sphere in dspatial dimensions. Secondly, the density of states\nis given by\n\u0017=@n0\n@\u0016=1\nvF@n0\n@pF=\ndpd\u00001\nF\nvF(2\u0019~)d; (98)\nIn the large Fermi surface limit, in d= 2, we have\nn\u0019\u0017vFpF\n2: (99)\nNext, we \fnd\nhQxjPxi=Zddp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013\nvx\u000fpx=Zddp\n(2\u0019~)d\u0012\n\u0000@feq\n@px\u0013\npx\u000f=Zddp\n(2\u0019~)dfeq(\u000f+pxvx): (100)\nThe pressure is given by\nP=Zddp\n(2\u0019~)dTlog\u0010\n1 + e\u0000(\u000f\u0000\u0016)=T\u0011\n(101)\n22and we \fnd (using the same integration by parts tricks as before)\nZddp\n(2\u0019~)dfeqpxvx=Zddp\n(2\u0019~)dTlog\u0010\n1 + e\u0000\u000f=T\u0011\n: (102)\nWe conclude that\nhQxjPxi=h\u000fi+P=Ts; (103)\nwheresis the entropy density. In the last step, we have used the Gibbs-Duhem identity together with\nthe fact that we have \fxed \u0016= 0 by our conventions. In a Fermi liquid, we also have\ns=\u0012@P\n@T\u0013\n\u0016\u0019\u00192T\n3\u0017: (104)\nFinally, the momentum-momentum susceptibility is given by\nM= lim\nvx!01\nvxZddp\n(2\u0019~)dpxfeq(\u000f\u0000pxvx) =Zddp\n(2\u0019~)dp2\nd\u0012\n\u0000@feq\n@\u000f\u0013\n\u0019\u0017p2\nF\nd: (105)\nNote that (in d= 2, at least)n\nM=vF\npF: (106)\nThis identity allows one to show (90).\nThe First Three Basis VectorsC\nWe must keep track of j0i,j1iandj2i, as well as keep track of O\u0000\nT2\u0001\ncorrections to Wimp, in order to\ncorrectly compute (68). The vectors\njJxi\u0019\u0012\nvF\u0000\u00192T2\n6p2\nFv3\nF\u0013\nj0i+\u0019Tp\n3pFj1i; (107a)\njQxi\u0019\u0012\u00192T2\n3pF+21\u00194T4\n8p3\nFv2\nF\u0013\nj0i+ \n\u0019TvFp\n3\u000031\u00193T3\n15p\n3p2\nFvF!\nj1i+ \n2\u00192T2\np\n5pF+149\u00194T4\n15p\n5p3\nFv2\nF!\nj2i: (107b)\nWe have included the \frst non-trivial subleading corrections in T, which will be necessary to compute\nthe \frst non-trivial subleading corrections to conductivity. As noted in Section 4.4, we have assumed a\nquadratic dispersion relation; hence 2 TF=pFvF, andapF=vF. Using (40) and the fact from (32) that\nAimpis a constant, we obtain\n\u00001\n\u0000=\u00002\npF; (108a)\n\u00002\n\u0000=3\np2\nF: (108b)\nFinally,\nWimp\u0019\u0000\u0012\n1 +5\u00192T2\n8T2\nF\u0013\nj0ih0j\u0000p\n3\u0019\u0000T\n2TF(j1ih0j+j0ih1j) +\u0000\u0012\n1 +137\u00192T2\n40T2\nF\u0013\nj1ih1j+2\u00192\u0000T2\np\n5T2\nF(j2ih0j+j0ih2j)\n\u0000r\n12\n5\u0019\u0000T\nTF(j2ih1j+j1ih2j) +\u0000\u0012\n1 +2523\u00192T2\n280T2\nF\u0013\nj2ih2j (109)\n23Employing (23), we \fnd (68).\nFinally, let us justify that the relaxation time approximation for electron-electron collisions does not\nchange the form of (68). Using (107b), the electron-electron contribution to T\u0016\u0014is\nT\u0016\u0014\u0019\u00192T2v2\nF\n3h1jeW\u00001j1i=\u00192T2v2\nF\n3\u0000\u0014\n1\u00001\n\u0000h1j(eW\u0000\u0000)j1i+\u0001\u0001\u0001\u0015\n=\u00192T2v2\nF\n3\u0000\u0014\n1\u0000h1jWj1i\n\u0000+\u0001\u0001\u0001\u0015\n:(110)\nAsh1jWj1iis O(T2), we conclude that so long as \u000b2T2=TF\u0011h1jWj1i, (68) is unchanged for arbitrary\ncollision integrals.\nThermal Conductivity in the Hydrodynamic Limit D\nIn this appendix we explicitly compute the hydrodynamic dissipative coe\u000ecient \u0014q. In a Galilean-invariant\n\ruid, the heat current is given by\nQ=Tsvhydro\u0000\u0014qrT: (111)\nTo see how this arises within kinetic theory, consider the following component of the time-independent\nBoltzmann in a translation-invariant \ruid:\nvj@jjQii\u0019\u0000\r\u0012\njQii\u0000hQijPji\nhPjPijPji\u0013\n(112)\nIn a hydrodynamic limit, vj@jjQiiis dominated by any vector associated with a conserved quantity {\nsince these gradients are long lived modes. Assume for simplicity that charge, energy and momentum are\nthe only conserved quantities: the \frst two are given by\njni=Z\nddpjpi; (113a)\njei=Z\nddp\u000f(p)jpi: (113b)\nNote the identities\nhnjni=Zddp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013\n=@n\n@\u0016\u0011\u001f; (114a)\nhejei=Zddp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013\n\u000f2\u0019\u00192T2\u0017\n3; (114b)\nhnjei\u0018T2(114c)\nwith\u001fthe charge compressibility; note \u001f\u0018T0. Since\nhnjvjjQii=\u000eij\ndZddp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013\nv2\u000f\u0018T2; (115a)\nhejvjjQii=\u000eij\ndZddp\n(2\u0019~)d\u0012\n\u0000@feq\n@\u000f\u0013\nv2\u000f2\u0019hejeiv2\nF\nd\u000eij (115b)\nusing (103), we conclude that\njQii\u0000TsjPii\nhPjPi=\u0000v2\nF\nd\r@ijei+ O\u0000\nT2@ijni\u0001\n(116)\n24In order to relate \ructuations in temperature \u000eTtojei, observe that\nfeq(T+\u000eT)\u0000feq(T) =\u0010\u000f\nT\u000eT\u0011\u0012\n\u0000@feq\n@\u000f\u0013\n; (117)\nwhich implies that\nhej\bi=\u000eT\u0002hejei\nT=\u000eT\u0002\u00192T\u0017\n3: (118)\nTaking the inner product of (116) with a generic distribution function j\bi, and noting that the approximate\nlow-temperature identity J=nvhydro implies thath\bjPii=hPjPivhydro;i, we conclude that (111) holds\nwith\n\u0014q=\u00192\u0017Tv2\nF\n3d\r: (119)\nThis expression precisely agrees with the open-circuit thermal conductivity (67) in the hydrodynamic\nlimit\r\u001d\u0000.\nReferences\n[1] J. A. N. Bruin, H. Sakai, R. S. Perry, and A. P. Mackenzie. \\Similarity of scattering rates in metals\nshowingT-linear resistivity\", Science 339804 (2013).\n[2] V. J. Emery and S. A. Kivelson. \\Superconductivity in bad metals\", Physical Review Letters 74\n3253 (1995).\n[3] R. Mahajan, M. Barkeshli, and S. A. Hartnoll. \\Non-Fermi liquids and the Wiedemann-Franz law\",\nPhysical Review B88 125107 (2013), arXiv:1304.4249 .\n[4] S. A. Hartnoll. \\Theory of universal incoherent metallic transport\", Nature Physics 1154 (2015),\narXiv:1405.3651 .\n[5] S. A. Hartnoll, A. Lucas, and S. Sachdev. \\Holographic quantum matter\", arXiv:1612.07324 .\n[6] R. Franz and G. Wiedemann. \\Ueber die W arme-Leitungsf ahigkeit der Metalle\", Annalen der Physik\n165497 (1853).\n[7] R. Lorenz. \\Ueber das Leitungsverm ogen der Metalle f ur W arme und Electricit at\", Annalen der\nPhysik 249582 (1881).\n[8] J. Ziman. Electrons and Phonons (Oxford University Press, 1960).\n[9] G. S. Kumar, G. Prasad, and R. O. Pohl. \\Experimental determinations of the Lorenz number\",\nJournal of Materials Science 284261 (1993).\n[10] R. W. Hill, C. Proust, L. Taillefer, P. Fournier, and R. L. Greene. \\Breakdown of Fermi-liquid theory\nin a copper-oxide superconductor\", Nature 414711 (2001).\n[11] A. Principi and G. Vignale. \\Violation of the Wiedemann-Franz law in hydrodynamic electron\nliquids\", Physical Review Letters 115 056603 (2015).\n[12] S. Sachdev. Quantum Phase Transitions (Cambridge University Press, 2nded., 2011).\n25[13] R. A. Davison, K. Schalm, and J. Zaanen. \\Holographic duality and the resistivity of strange metals\",\nPhysical Review B89 245116 (2014), arXiv:1311.2451 .\n[14] A. Lucas and S. A. Hartnoll. \\Resistivity bound for hydrodynamic bad metals\", Proceedings of the\nNational Academy of Sciences 11411344 (2017), arXiv:1704.07384 .\n[15] A. Lucas and K. C. Fong. \\Hydrodynamics of electrons in graphene\", Journal of Physics: Condensed\nMatter 30053001 (2018), arXiv:1710.08425 .\n[16] E. H. Hwang and S. Das Sarma. \\Limit to two-dimensional mobility in modulation-doped GaAs\nquantum structures: how to achieve a mobility of 100 million\", Physical Review B77 235437 (2008),\narXiv:0804.4684 .\n[17] D. A. Bandurin et al. \\Negative local resistance due to viscous electron back\row in graphene\",\nScience 3511055 (2016), arXiv:1509.04165 .\n[18] R. Krishna Kumar et al. \\Super-ballistic \row of viscous electron \ruid through graphene constric-\ntions\", Nature Physics 131182 (2017), arXiv:1703.06672 .\n[19] P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and A. P. Mackenzie. \\Evidence for hydrodynamic\nelectron \row in PdCoO 2\",Science 3511061 (2016), arXiv:1509.05691 .\n[20] J. Gooth, F. Menges, C. Shekhar, V. S uss, N. Kumar, Y. Sun, U. Drechsler, R. Zierold, C. Felser,\nand B. Gotsmann. \\Electrical and thermal transport at the Planckian bound of dissipation in the\nhydrodynamic electron \ruid of WP 2\",arXiv:1706.05925 .\n[21] E. V. Levinson, G. M. Gusev, A. D. Levin, E. V. Levinson, and A. K. Bakarov. \\Viscous electron \row\nin mesoscopic two-dimensional electron gas\", AIP Advances 8025318 (2018), arXiv:1802.09619 .\n[22] X. Lin, B. Fauque, and K. Behnia. \\Scalable T2resistivity in a small single-component Fermi\nsurface\", Science 349945 (2015), arXiv:1508.07812 .\n[23] E. Mikheev, S. Raghavan, J. Y. Zhang, P. B. Marshall, A. P. Kajdos, L. Balents, and S. Stemmer.\n\\Carrier density independent scattering rate in SrTiO 3-based electron liquids\", Scienti\fc Reports 6\n20865 (2016), arXiv:1512.02294 .\n[24] A. V. Andreev, S. A. Kivelson, and B. Spivak. \\Hydrodynamic description of transport in strongly\ncorrelated electron systems\", Physical Review Letters 106 256804 (2011), arXiv:1011.3068 .\n[25] I. Torre, A. Tomadin, A. K. Geim, and M. Polini. \\Non-local transport and the hydrodynamic shear\nviscosity in graphene\", Physical Review B92 165433 (2016), arXiv:1508.00363 .\n[26] L. Levitov and G. Falkovich. \\Electron viscosity, current vortices and negative nonlocal resistance\nin graphene\", Nature Physics 12672 (2016), arXiv:1508.00836 .\n[27] A. Lucas, J. Crossno, K. C. Fong, P. Kim, and S. Sachdev. \\Transport in inhomogeneous quan-\ntum critical \ruids and in the Dirac \ruid in graphene\", Physical Review B93 075426 (2016),\narXiv:1510.01738 .\n[28] H. Guo, E. Ilseven, G. Falkovich, and L. Levitov. \\Higher-than-ballistic conduction of viscous electron\n\rows\", Proceedings of the National Academy of Sciences 1143068 (2017), arXiv:1607.07269 .\n[29] F. M. D. Pellegrino, I. Torre, A. K. Geim, and M. Polini. \\Electron hydrodynamics dilemma:\nwhirlpools or no whirlpools\", Physical Review B94 155414 (2016), arXiv:1607.03726 .\n26[30] T. Sca\u000edi, N. Nandi, B. Schmidt, A. P. Mackenzie, and J. E. Moore. \\Hydrodynamic electron \row\nand Hall viscosity\", Physical Review Letters 118 226601 (2017), arXiv:1703.07325 .\n[31] J. Crossno et al. \\Observation of the Dirac \ruid and the breakdown of the Wiedemann-Franz law in\ngraphene\", Science 3511058 (2016), arXiv:1509.04713 .\n[32] F. Ghahari, H-Y. Xie, T. Taniguchi, K. Watanabe, M. S. Foster, and P. Kim. \\Enhanced thermo-\nelectric power in graphene: violation of the Mott relation by inelastic scattering\", Physical Review\nLetters 116 136802 (2016), arXiv:1601.05859 .\n[33] R. P. Smith, M. Sutherland, G. G. Lonzarich, S. S. Saxena, N. Kimura, S. Takashima, M. Nohara, and\nH. Takagi. \\Marginal breakdown of the Fermi-liquid state on the border of metallic ferromagnetism\",\nNature 4551220 (2008).\n[34] H. Pfau et al. \\Thermal and electrical transport across a magnetic quantum critical point\", Nature\n484493 (2013), arXiv:1307.1066 .\n[35] S. A. Hartnoll, P. K. Kovtun, M. M uller, and S. Sachdev. \\Theory of the Nernst e\u000bect near quantum\nphase transitions in condensed matter, and in dyonic black holes\", Physical Review B76 144502\n(2007), arXiv:0706.3215 .\n[36] S. A. Hartnoll and D. M. Hofman. \\Locally critical umklapp scattering and holography\", Physical\nReview Letters 108 241601 (2012), arXiv:1201.3917 .\n[37] A. Lucas and S. Sachdev. \\Memory matrix theory of magnetotransport in strange metals\", Physical\nReview B91 195122 (2015), arXiv:1502.04704 .\n[38] A. Lucas and S. A. Hartnoll. \\Kinetic theory of transport for inhomogeneous electron \ruids\", Physical\nReview B97 045105 (2018), arXiv:1706.04621 .\n[39] A. Lucas. \\Kinetic theory of electronic transport in random magnetic \felds\", Physical Review Letters\n120 116603 (2018), arXiv:1710.11141 .\n[40] K. C. Fong and K. C. Schwab. \\Ultrasensitive and wide-bandwidth thermal measurements of\ngraphene at low temperatures\", Physical Review X2031006 (2012), arXiv:1202.5737 .\n[41] J. Crossno, X. Liu, T. A. Ohki, P. Kim, and K. C. Fong. \\Development of high frequency\nand wide bandwidth Johnson noise thermometry\", Applied Physics Letters 106 023121 (2015),\narXiv:1411.4596 .\n[42] P. Ledwith, H. Guo, and L. Levitov. \\Fermion collisions in two dimensions\", arXiv:1708.01915 .\n[43] S. Das Sarma and E. H. Hwang. \\Charged impurity scattering limited low temperature\nresistivity of low density silicon inversion layers\", Physical Review Letters 83164 (1999),\narXiv:cond-mat/9812216 .\n[44] S. Das Sarma and E. H. Hwang. \\Metallicity and its low-temperature behavior in dilute two-\ndimensional carrier systems\", Physical Review B69 195305 (2004), arXiv:cond-mat/0302047 .\n[45] P. L. Bhatnagar, E. P. Gross, and M. Krook. \\A model for collision processes in gases. I. Small\namplitude processes in charged and neutral one-component systems\", Physical Review 94511 (1954).\n[46] C. Hodges, H. Smith, and J. W. Wilkins. \\E\u000bect of Fermi surface geometry on electron-electron\nscattering\", Physical Review B4302 (1971).\n27[47] G. F. Giuliani and J. J. Quinn. \\Lifetime of a quasiparticle in a two-dimensional electron gas\",\nPhysical Review B26 4421 (1982).\n[48] T. Jungwirth and A. H. MacDonald. \\Electron-electron interactions and two-dimensional-two-\ndimensional tunneling\", Physical Review B53 7403 (1996), arXiv:cond-mat/9603001 .\n[49] L. Zheng and S. Das Sarma. \\Coulomb scattering lifetime of a two-dimensional electron gas\", Physical\nReview B53 9964 (1996), arXiv:cond-mat/9602066 .\n[50] B. N. Narozhny, G. Zala, and I. L. Aleiner. \\Interaction corrections at intermediate temperatures:\ndephasing time\", Physical Review B65 180202 (2002), arXiv:cond-mat/0201379 .\n[51] Q. Li and S. Das Sarma. \\Finite temperature inelastic mean free path and quasiparticle lifetime in\ngraphene\", Physical Review B87 085406 (2013), arXiv:1211.6430 .\n[52] S. Q. Murphy, J. P. Eisenstein, L. N. Pfei\u000ber, and K. W. West. \\Lifetime of two-dimensional electrons\nmeasured by tunneling spectroscopy\", Physical Review B52 14825 (1995).\n[53] H. Noh, M. P. Lilly, D. C. Tsui, J. A. Simmons, E. H. Hwang, S. Das Sarma, L. N. Pfei\u000ber, and\nK. W. West. \\Interaction corrections to two-dimensional hole transport in the large- rslimit\", Physical\nReview B68 165308 (2003), arXiv:cond-mat/0206519 .\n[54] M. M uller, L. Fritz, and S. Sachdev. \\Quantum-critical relativistic magnetotransport in graphene\",\nPhysical Review B78 115406 (2008), arXiv:0805.1413 .\n[55] H-Y. Xie and M. S. Foster. \\Transport coe\u000ecients of graphene: Interplay of impurity\nscattering, Coulomb interaction, and optical phonons\", Physical Review B93 195103 (2016),\narXiv:1601.05862 .\n[56] S. Das Sarma and E. H. Hwang. \\Screening and transport in 2D semiconductor systems at low\ntemperatures\", Scienti\fc Reports 516655 (2015).\n[57] F. Stern. \\Calculated temperature dependence of mobility in silicon inversion layers\", Physical\nReview Letters 441469 (1986).\n[58] S. Das Sarma. \\Theory of \fnite-temperature screening in a disordered two-dimensional electron gas\",\nPhysical Review B33 5401 (1986).\n[59] N. E. Hussey, K. Takenaka, and H. Takagi. \\Universality of the Mott-Io\u000be-Regel limit in metals\",\nPhilosophical Magazine 842847 (2004), arXiv:cond-mat/0404263 .\n[60] M. P. Lilly, J. L. Reno, J. A. Simmons, I. B. Spielman, J. P. Eisenstein, L. N. Pfei\u000ber, K. W.\nWest, E. H. Hwang, and S. Das Sarma. \\Resistivity of dilute 2D electrons in an undoped GaAs\nheterostructure\", Physical Review Letters 90056806 (2003).\n[61] M. J. Manfra, E. H. Hwang, S. Das Sarma, L. N. Pfei\u000ber, K. W. West, and A. M. Sergent. \\Transport\nand percolation in a low-density high-mobility two-dimensional hole system\", Physical Review Letters\n99236402 (2007), arXiv:0710.3542 .\n[62] B. Hu, M. M. Yazdanpanah, B. E. Kane, E. H. Hwang, and S. Das Sarma. \\Strongly metallic electron\nand hole 2D transport in an ambipolar Si-vacuum \feld e\u000bect transistor\", Physical Review Letters\n115 036801 (2015), arXiv:1502.02956 .\n28[63] D. Y. H. Ho, I. Yudhistra, N. Chakraborty, and S. Adam. \\Theoretical determination of hydro-\ndynamic window in monolayer and bilayer graphene from scattering rates\", Physical Review B97\n121404 (2018), arXiv:1710.10272 .\n29"    },    {        "title": "0909.3552v1.Phenomenology_of_ESR_in_heavy_fermion_systems__Fermi_liquid_and_non_Fermi_liquid_regime.pdf",        "content": "arXiv:0909.3552v1  [cond-mat.str-el]  19 Sep 2009Phenomenology of ESR in heavy fermion systems: Fermi liquid and non-Fermi liquid\nregime\nPeter W¨ olfle1and Elihu Abrahams2\n1Institut f¨ ur Theorie der Kondensierten Materie,\nUniversit¨ at Karlsruhe, D-76128 Karlsruhe, Germany\n2Center for Materials Theory, Serin Physics Laboratory, Rut gers University, Piscataway, NJ 08854-8019\n(Dated: November 5, 2018)\nWe extendandapply arecent theoryofthe dynamical spinresp onse of Andersonlattice systems to\ninterpret ESR data on YbRh 2Si2. Starting within a semiphenomenological Fermi liquid desc ription\nat low temperatures T < T x(a crossover temperature) and low magnetic fields B≪Bx,we extend\nthedescription tothenon-Fermiliquid regime byadoptinga quasiparticle picturewith effective mass\nand spin susceptibility varying logarithmically with ener gy/temperature, as observed in experiment.\nWe find a sharpESR resonance line slightly shifted from the local f-level resonance and broadened\nby quasiparticle scattering (taking unequal g-factors of conduction and felectrons) and by spin-\nlattice relaxation, both significantly reduced by the effect of ferromagnetic fluctuations. A detailed\ncomparison of our theory with the data shows excellent agree ment in the Fermi liquid regime. In\nthe non-Fermi liquid regime we find a close relation of the T-dependence of the specific heat/spin\nsusceptibility with the observed T-dependence of line shift and linewidth.\nI. INTRODUCTION\nIn several recent experiments [1, 2], low-temperature\nESR has been observed in some heavy-fermion metals, in\nparticular YRh 2Si2(YRS) [1, 3]. The phase diagram of\nYRS has a magnetic-field induced quantum critical point\nand is a model system for the study of quantum critical-\nity in the Kondo lattice. Consequently, the observation\nof a narrow ESR resonance in this compound aroused\ngreat interest, especially since it was commonly believed\nthat heavy-fermion ESR would be unobservable due to\nan enormous intrinsic linewidth ∆ Bof orderkBTK/gµB\n[1]. Here TKis the lattice coherence (“Kondo”) temper-\nature for the onset of heavy-fermion behavior and gµB\nis the gyromagnetic ratio for the resonance. These were\nthe first observations of ESR in Kondo lattice systems at\nT < T K.\nA common feature of the compounds in which ESR\nhas been observed appears to be the existence of ferro-\nmagnetic fluctuations [2, 4] These findings challenge our\nunderstanding of heavy fermion compounds: How does\na sharp electron spin resonance emerge despite Kondo\nscreening and spin lattice relaxation, and why is this pro-\ncessinfluenced byferromagneticfluctuations? Inarecent\npaper (“AW”) [5], we discussed the background of these\nquestions and answered them in the framework of Fermi-\nliquid theory. An alternative explanation based on lo-\ncalized spins was subsequently proposed by Schlottmann\n[6]. The general derivation of Fermi liquid theory from\nthe microscopic theory for a two-band Anderson lattice\nmodel has been given by Yip [7]\nIn YRS, the observed narrow dysonian [8] ESR line\nshape was originally interpreted [1] as indicating that\nthe resonance was due to local spins at the Yb sites.\nTherefore, initially the authors speculated that the nar-\nrowESRlinemightindicatethesuppressionoftheKondo\neffect near the quantum critical point, since, as ex-plained above, carrying over Kondo impurity physics to\nthe Kondo lattice, one might expect the local spins to be\nscreened by the Kondo effect, giving rise only to a broad\nspin excitation peak, too wide to be observed in ESR\nexperiments. However, a closer look [9] revealed that\nitinerant (heavy) electron ESR could give rise to a simi-\nlar line shape since the carrier diffusion in YRS is quite\nslow. Thus, whether the resonance is that of localized or\nitinerant spins remained an open question.\nIn this paper, we extend our previous work [5] to the\nnon-Fermiliquid (NFL) regionoftheYRS phasediagram\nand make a detailed comparisonwith the data. Excellent\nagreement is obtained for the Fermi-liquid (FL) regime.\nIn particular, the ratio of the contributions ∝T2and\n∝B2to the linewidth in the FL region is very well re-\nproduced. In addition, we account for the anomalous be-\nhavior observed in the NFL region for the resonance line\nshift and the linewidth, One absolutely essential aspect\nof our theory is the lattice coherence of the quasiparticles\nin the Anderson or Kondo lattice model: it is this lattice\ncoherencethatisresponsiblefortheabsencein thelattice\ncaseof the stronglocal spin relaxationthat is observedin\nsingle Kondo impurity physics. Attempts to account for\nthe observed logarithmic temperature dependence of the\nlineshift as arising from single Kondo ion physics above\nthe Kondo temperature are therefore problematic, since\nlattice coherence is lost in that case\nII. ESR IN THE KONDO-SCREENED\nANDERSON LATTICE MODEL: FERMI LIQUID\nREGIME.\nThis was analyzed in Sec. III of AW. The Hamilto-\nnian of the simplest Anderson lattice model, assuming2\nmomentum independent hybridization is given by\nH=/summationdisplay\nk,σǫkσc+\nkσckσ+/summationdisplay\ni,σǫfσnfiσ+U/summationdisplay\ninfi↑nfi↓\n+V/summationdisplay\ni,k,σ(eik·Rif+\niσckσ+h.c.), (1)\nwhereǫkσ=ǫk−ωcσ/2, σ=±1,is the conduction-\nelectron energy spectrum and ωc=gcµBBis its Zeeman\nsplitting; c+\nkσ,f+\niσarecreationoperatorsoftheconduction\nelectrons in momentum and spin eigenstates ( kσ), and of\nelectrons in the local flevel at site Ri, respectively. The\noperator nfiσ=f+\niσfiσcounts the number of electrons\non the local level, and ǫfσ=ǫf−ωfσ/2.VandUare\nthehybridizationamplitudeandtheCoulombinteraction\nmatrixelement. WetaketheZeemansplittings ωcandωf\nto be unequal, as they are in real materials. We consider\nthe limit ωf−ωc→0 in the Appendix.\nWe now review the results for the dynamical spin sus-\nceptibility χ+−(Ω) obtained in AW [5]. We find a single\nresonance peak\nχ+−(Ω) =χ+−(0)−ωr+iΓ\nΩ−ωr+iΓ, (2)\nwhere the resonance frequency is given by [10]\nωr=ωf−m\nm∗(ωf−ωc), (3)\nHerem∗/mis the quasiparticle effective mass ratio. We\nnote that for equal g-factors the line position is not\nshifted. In the Appendix, we discuss the complete re-\nsult, showing that even in the case of unequal g-factors\nthere is only a single resonance peak. We also show that\nthe residual Fermi liquid interaction effects drop out of\nthe resonance frequency.\nThe linewidth Γ has contributions from quasiparticle\nscattering and from the conduction electron spin lattice\nrelaxation γ\nΓ =A[α(πT)2+1\n4(Rωf)2]1\nR+2γm\nm∗1\nR.(4)\nHereR= [1+/tildewideUχ+−(0)] is identified as the Wilson ratio,\n/tildewideUis the Fermi-liquid spin-exchange interaction [11] and\nχ+−(0) =M/Bisthestatictransversespinsusceptibility\n(Mis the spin polarization). The numerical coefficient α\ndepends on the band structure and is of order unity. In\nthe case of a sizeable ferromagnetic interaction, ( /tildewideU >0),\nR >>1,the linewidth getsnarrowedbya factor1 /R.We\nsuggest that this effect is responsible for the fact that so\nfar an ESR line has only been observed in compounds\nthat exhibit signatures of ferromagnetic fluctuations.A. Magnetic anisotropy of ESR line\nThe magnetic response of YRS is strongly anisotropic,\nlargely because of the single ion anisotropy. The Zee-\nman Hamiltonian of the f-electron ground state dou-\nblet of Yb in tetragonal symmetry has the form HZ=\n−µBgf⊥(SxBx+SyBy)−µBgf/bardblSzBz, wherethe z-axisis\nalong the crystallographic c-axis. The anisotropy of the\ng-factor is about a factor of 20, gf⊥= 3.6 andgf/bardbl= 0.17\n. We assumeforthe presentthat the anisotropyofthe in-\nteraction is negligible. The Hamiltonian is then diagonal\nin the coordinate system in the spin space that diagonal-\nizesHZ. The eigenvalues of HZare found as\nωf(φ) =∓µBB/radicalBig\ng2\nf/bardblcos2φ+g2\nf⊥sin2φ ,(5)\nwhereφis the angle between the magnetic field Band\nthec-axis.\nThe above results, Eq. (2), for the dynamical spin sus-\nceptibility χ+−(Ω) obtained in [5] for the isotropic model\nmay then be generalized to the anisotropic model by re-\nplacingωfbyωf(φ) and taking the tensor of spin suscep-\ntibilityprojectedontothedirectionofthestaticmagnetic\nfield. According to Ref. [12], the angle dependence of the\nresonance frequency is well-represented by Eq. (5). This\nindicates that the anisotropic part of the residual Fermi-\nliquid spin-exchange interaction is small. As we shall\nsee later, the temperature dependence of the line shift\nin the non-Fermi liquid regime suggests that there may\nbe a small anisotropic interaction component. We shall\nexplore the consequences of such a non-spin rotation in-\nvariant term in Sec. IIC, below.\nB. ESR line shift and linewidth in the Fermi liquid\nregime\nIn Fig. 1, we show a sketch of the phase diagram of\nYRS, including the B, Tranges in which ESR experi-\nments have been carried out. Here, the crossover to the\nFermi liquid (FL) regime is determined by the onset of\nFL behavior in thermodynamic quantities [13]. The T∗\ncrossover is primarily determined by Hall effect results\n[14, 15] that can be interpreted as a transition (from left\nto right) to a large Fermi surface.\nThe high magnetic field ESR data reported in [3] show\nacrossoverfromalow-temperatureFermi-liquid(FL)like\nregime to a higher temperature non-Fermi liquid behav-\nior at a temperature Tx≃5 Kelvin. In the FL regime\nthe line shift appears to be temperature independent.\nRelative to the ionic g-factor of 3 .86 [1], the resonance\nis shifted to lower values g⊥≃3.42 independent of mag-\nnetic field in the range 5 .15 to 7.45 Tesla. Estimating the\neffective mass ratio, which is also temperature indepen-\ndent below Txin the magnetic field range considered, as\nm∗/m∼40, we obtain from Eq. (3) a g-shift ∆g≃0.04,\nwhich is an order of magnitude too small. This discrep-\nancy may point to an additional small anisotropic spin3\n.02 T \n.69 T 1.1 T \nT\nBAFM T* \nLFL \nAFM NFL \nLFL T* 1.9 T \n5.0-7.5 T \n1.1 T \n.69 T .19 T \nFIG. 1: Phase diagram of YRSshowing field and temperature\nranges of ESR experiments\ninteraction, which we consider in the following subsec-\ntion.\nAs for the linewidth, in the FL regime Schaufuß, et al\n[3] find a linewidth that follows the law Γ( T,B)∼T2,\nextrapolating to Γ(0 ,B)∼B2asT→0. The experi-\nmental ratio of the prefactors of the T2andB2terms,\nrexp=B2[Γ(T,B)−Γ(0,B)]/[Γ(0,B)T2] turns out to be\nrexp∼2. Estimating the Wilson ratio from the available\nspecific heat data [16, 17] at Tx= 5 K and in mag-\nnetic fields B≈6 Tesla, ∆ C/T= 0.032 K−1Yb−1and\nspin susceptibility data [17, 18] ∆ χ=M/B= 0.224µ2\nB\nK−1Yb−1asR= [χ/(gfµB/2)2](π2T/3∆C)≃7.5, we\ncalculate from Eq. (4) the theoretical ratio rth∼1.2α,\nin good agreement with the experimental value. Note\nthat the large enhancement of the single particle Zeeman\nsplitting by the Fermi liquid interaction (a factor R) is\nessential in obtaining this agreement.\nC. Effect of non spin-rotation invariant Fermi\nliquid interaction.\nThe spin-orbit interaction in conjunction with the\ntetragonal lattice anisotropy may be expected to lead to\na small admixture of a non-spin-symmetric component\nto the Fermi liquid interaction of the form −4I(− →S·ˆc)2,\nwhere ˆcis the unit vector along the c-axis of the tetrag-\nonal lattice. Taking the magnetic field along the b-axis,\nwe employ a coordinate system in spin space, in which\nthez-axis is oriented along the magnetic field and the\nx-axis along the c-axis (see Fig. 2).\nThe ESR oscillating transverse magnetic field is cir-\ncularly polarized in the x−yplane, which is the a−c\nplane of the crystal, perpendicular to the static magnetic\nfield. The screening of the static magnetic field is ef-\nfected in linear order in Ionly when the component of\nthe magnetic field along ˆ cis nonvanishing. In turn, the\ndynamic screening is changed at linear order in I, for\nany component of Bstaticperpendicular to ˆ c. Thus, the\nstatic and dynamic screening are effected differently by\nFIG. 2: Specification of axes for magnetic fields in ESR ex-\nperiment\nI, which givesriseto a resonanceline shift, which we now\ncalculate. As we show in the Appendix, the dynamical\nscreening of the ffcomponent of the dynamical suscep-\ntibility is modified in the presence of Ito\nχ+−\nff(iΩm) =χ+−\nff,H(iΩm)[1+/tildewideUdχ+−\nff(iΩm)]\nThus, in the transverse spin response, the Fermi-liquid\ninteraction is changed to /tildewideUd=/tildewideU+Isin2φ. The notation\nis as in AW [5], Eqs. (16-18): χff,His the susceptibility\nof Fermi-liquid quasiparticles in the absence of vertex\ncorrections (bubble diagram only) and the subscript H\nindicates that the bare Zeeman energy ωfis replaced\neverywhere by\n/tildewideωf=ωf[1+/tildewideUsχ+−\nff(0)] =ωf[1−/tildewideUsχ+−\nff,H(0)]−1.\nHere/tildewideUs=/tildewideU+4Icos2φand/tildewideUis the renormalized on-\nsiteffrepulsion that appears in the effective low-energy\nHamiltonian (see AW for further details). The resonance\nposition is therefore shifted as\nωr=/tildewideω−\nkF[1−/tildewideUdχ+−\nff,H(0)] =ω−\nkF1−/tildewideUdχ+−\nff,H(0)\n1−/tildewideUsχ+−\nff,H(0)\nSubstituting ω−\nkFas obtained in the Appendix and using\nthe definition of R, we find that the resonance frequency\nis modified from Eq. (3) to\nωr≃ω−\nkF[1−Iχ+−\nff,H(0)R]\n≃ωf−m\nm∗(ωf−ωc)−ωfI(1−5cos2φ)χ+−\nff(0),\nor\ngr≃gf−m\nm∗(gf−gc)−gfI(1−5cos2φ)χ+−\nff(0).(6)\nSince most of the ESR data have been taken in the\nconfiguration of magnetic field perpendicular to the c-\naxis,i.e.φ=π/2 , we concentrate on this case from4\nnow on. We may try to determine Iby fitting the low-\ntemperature line shift [3]. In the following, we assume\nIto be independent of temperature and magnetic field.\nIn the NFL regime, experimentally χ+−\nff(0;T) is a de-\ncreasing function of temperature, such that the g-shift\nincreases as observed in experiment, provided I >0. We\nmay relate Itoχ+−\nff(0;T1) at a reference point T1= 4K,\nB= 0.2T, where χ+−\nff(0;T1)≈1.3×10−6m3/mol [27] as\nI= (gion\nf−gr)/(gfχ+−\nff) [the reference g-factorgion\nfis\nactually reduced by the factor (1 −m/m∗)]. The data at\nlow magnetic fields, B= 0.18T and B= 0.68T show a\ng-factor of g≃3.5 at the lowest temperature, T= 2K,\nwhereas the high-field data show g= 3.42 in the Fermi\nliquid regime. It follows that I≈0.075×106m−3mol.\nFrom a comparison with the temperature dependence of\nthe resonance frequency we determine in the next Sec-\ntion a value of I= 0.063×106m−3mol, which agrees\nvery well with the independently-obtained value above.\nWe observe in passing that the magnetic susceptibility\ndata indicate that in the non-Fermi liquid regime /tildewideUap-\npears to depend on both temperature and magnetic field.\nIII. ESR IN THE NON-FERMI LIQUID\nREGIME.\nWe now attempt to phenomenologically relate the\nframework we have set up to the ESR data in the NFL\nregime by using the observed specific heat and suscepti-\nbilityTandBdependences. The non-Fermi liquid be-\nhavior in the temperature range T > T xappears in the\nESR data as a nearly logarithmic increase of the g-factor\nwith temperature and a change in the temperature de-\npendence of the linewidth from T2toT. This change\ninto the NFL regime occurs at about the same tempera-\nture as the observed changes in the specific heat and spin\nsusceptibility.\nA. Resonance shift\nIn the theoretical resonance shift, Eq. (6), the T- and\nB- dependences enter in two ways, if we continue to as-\nsume that the temperaturedependence ofthe anisotropic\nFermi-liquid interaction parameter Imay be neglected:\n1) through the susceptibility χ+−\nff(0), which we get from\nexperiment, and 2) through the effective mass ratio,\nwhich we extract from the measured specific heat γ-\ncoefficient (∆ C=γT), by taking γ∝m∗/m. Using\nthese experimentally determined quantities, we shall use\nEq. (6) to evaluate the theoretical resonance shift and\ncompare it to the observed one.\nWe shall use Eq. (6) to calculate the g-shiftδg=\ng−gion\nfat two reference temperatures T= 4K and\nT= 10K. The inputs are the values of m/m��from the\nobserved specific heat [16], taking m∗/m= 40 atT= 5\nK andB= 6 T as a reference point, the observed sus-ceptibility [18, 27] and the value of the anisotropic FL\ninteraction I. These data and the calculated g-shifts are\ncollectedinTableI.Sincewehaveassumedthat Iisinde-\npendent of TandB, we can evaluate it from Eq. (6) us-\ning experimental data at T= 4K,B= 0.2T as discussed\nin the previous Section. The result is I= 0.063×106\nmol/m3The data on the ESR line shift given in Fig. 2\nof [3] show an approximately linear ln Tdependence in\ntheT-range 4K < T < 10K. Therefore, to check the\naccuracy of our theoretical result, Eq. (6), we fit the two\ncalculated δgvalues to a linear ln Tfunction and give the\nresulting theoretical slope in the last column of Table I.\nThe comparison of the calculated and observed values of\nthe slope ∆ g/∆lnTis shown in Fig. 3. It is seen that the\nagreement is quite good, supporting our assumption of\na constant interaction I. The theory explains the rather\nstrong dependence of the ln Tterm on magnetic field, de-\ncreasing by approximately a factor of 8 as the magnetic\nfield is stepped up from 0 .19 T to 7 .45 T. The present\ntheory predicts that the slope of the ln Tdependence of\ntheg-shift depends sensitively on the magnetic field ori-\nentation (the angle φ) and it reverses sign when Bstatic\nis oriented along the ˆ c-axis of the crystal.\nTABLE I: Experimental values of susceptibility and specific\nheat coefficient and calculated g-shift [from Eq. (6)] at differ-\nentBand two temperatures T1= 4K,T2= 10K. We chose\nI= 0.063×106mol/m3. Units: χin10−6m3/mol,γinJ/(mol\nK2)\nB(T)χ(T1)χ(T2)γ(T1)γ(T2)δg(T1)δg(T2)(∆g/∆lnT)th\n7.5 0.82 0.65 0.24 0.185 -0.222 -0.197 0.027\n6.0 0.91 0.70 0.26 0.18 -0.239 -0.210 0.032\n5.0 0.98 0.74 0.27 0.17 -0.254 -0.222 0.035\n1.85 1.13 0.78 0.29 0.15 -0.283 -0.238 0.050\n1.0 1.2 0.80 0.29 0.15 -0.298 -0.243 0.060\n0.68 1.23 0.80 0.29 0.15 -0.305 -0.243 0.068\n0.5 1.25 0.80 0.29 0.15 -0.309 -0.243 0.072\n0.2 1.3 0.80 0.29 0.15 -0.320 -0.243 0.085\nB. Linewidth\nTurning now to the linewidth, we use the analyticity\nproperties of the self energy Σ( ω) to infer the linewidth\nfrom the temperature dependence of the specific-heat co-\nefficientγ(T). Here one has to observe that only part of\nthe specific heat enhancement is coming from the non-\nanalytic contribution of the self energy Σ. An addi-\ntional part is coming from the regular (analytic) contri-\nbution to Σ. Therefore one may split the effective mass\ninto two components, m∗/m= (m∗/m)reg+(m∗/m)sing.\nThe specific heat data show a ln Tvariation over a wide\nrange on top of a background. If we identify the back-5\ntheory \nexperiment\nFIG. 3: Comparison of g-shift slopes for different B. The the-\noretical and experimental values are identical at B= 0.68T\nground with ( m∗/m)reg, the singular part at the ref-\nerence point T= 5 K and B= 6 T is about 60%\nof the total, i.e. ( m∗/m)s= 24, taking the FL value\nm∗/m= 40. The singular part can now be associated\nwith the non-Fermi-liquid logarithmic temperature de-\npendence of γ. Thus, γsing∝(m∗/m)sing. To account\nfor the crossover from NFL to FL behavior at T=Tx,\nwe adopt an interpolation formula γsing=−cln[(T2+\nT2\nx)/T2\n0]. Since ( m∗/m) = [1−Re{∂Σ/∂ω}|0], we may\nwrite (m∗/m)sing=−Re{∂Σsing/∂ω}|0. The tempera-\nture dependence ( m∗/m)sing=−aln[(T2+T2\nx)/T2\n0] may\nbe approximately converted into a frequency dependence\nRe{∂Σsing/∂ω}=aln[(ω2+T2\nx)/T2\n0] of the nonanalytic\nreal part of the self energy. The self energy in the com-\nplex plane may be inferred as\nΣsing(ω) = 2aωln[(−iω+Tx)/T0]\nFrom this approximate model the imaginary part of the\nself energy and hence the quasiparticle contribution [the\nfirst term in Eq. (4)] to the ESR line width follows as\nΓqp= 2m\nm∗1\nRImΣs(ω=T) =pTtan−1(T/Tx)\nIn the limit of T≪Txthe above expression recovers the\nFermi liquid resultΓ qp∝T2, asdiscussed above, while at\nT≫Txthe non-Fermi liquid result Γ qp∝Tis obtained.\nThis is in qualitative agreement with experiment. It is\nworth pointing out that a similar structure of the self\nenergy has been proposed for the “strange metal” phase\nof the cuprates under the name “marginal Fermi liquid\ntheory” [28]. By comparison with the effective mass ra-\ntio we find ath≈6 and using R≈7 we get pth≈0.1.\nThe experimental value is pex1≈0.02, which is quite a\nbit smaller. Similarly, from the experimentally observed\ncoefficient of the T2term of the linewidth, Γ /T2≈0.004K−1intheFermiliquidregime,oneextractsagainavalue\npex2≈0.02 . The discrepancy may come from our as-\nsumption that Σ singis entirely due to spin-flip scattering\nand from our very approximate determination of Σ sing.\nVertex corrections will for example remove any non-spin\ninteraction contribution to Σ singfrom the linewidth Γ.\nIf this is correct, it would imply that the fluctuations\ncontributing most to the ln Tterm in the specific heat\nare nonmagnetic in origin. Finally we comment on the\npossible contribution to Γ caused by the regular part of\nΣreg(ω). In this case the prefactors cr,ciof the low\nenergy limiting forms ReΣ reg(ω) = Σ reg(0) +crωand\nImΣreg(ω) =ciω2arenotdirectlyrelated. TheKramers-\nKronig relations imply in this case that, e.g., the higher\nfrequency parts of ReΣ reg(ω) will predominantly deter-\nmine the coefficient ci, while the coefficient crhas little\ninfluence in this. In the present case, the resulting imag-\ninary part and coefficient ciis apparently small.\nIV. CONCLUSION\nWe extended and applied our recent theory [5] of\nthe dynamical spin response of Anderson lattice sys-\ntems to interpret ESR data on YbRh 2Si2. Starting with\na semiphenomenological Fermi-liquid description at low\ntemperatures T < T x(a crossover temperature) and low\nmagnetic fields B≪Bx,we extended the description to\nthe non-Fermi liquid regime by adopting a quasiparticle\npicturewitheffectivemassandspinsusceptibilityvarying\nlogarithmically with energy/temperature, as observed in\nexperiments. We find a sharpESR resonance line that\nis broadened by quasiparticle scattering and spin lattice\nrelaxation, both significantly reduced by the effect of fer-\nromagnetic fluctuations. A more complete evaluation of\nthe results presented in our first paper shows that the\nESR-line position is shifted by an amount ∝(gf−gc)\n, thus reducing to zero for equal g-factors. In the case\nof different g-factors there is only one sharp resonance\nline atω≃ωf, the local f-resonance frequency. The ob-\nserved strong anisotropy of the ESR response is shown to\nfollow from the single-ion spin anisotropy, assuming an\napproximately spin-conserving exchange interaction.\nA detailed comparison of our theory with the data\nshows excellent agreement in the Fermi-liquid regime,\nwhen the model is amended by a small anisotropic part\nof the spin exchange interaction, induced by spin-orbit\ncoupling. We assumed the strength of the latter to be\nindependent of temperature and magnetic field through-\nout the regime considered in the experiments. In partic-\nular, the ratio of the contributions ∝T2and∝B2to\nthe linewidth in the FL region is very well reproduced by\ntheory.\nIn the non-Fermi liquid regime we find a close relation\nof theT-dependences of the specific heat and spin sus-\nceptibility with the observed T-dependences of the line\nshift and linewidth. There are two terms contributing\nto the temperature dependence of the lineshift [Eq. (6)]6\nin opposite ways. The first and dominant one is propor-\ntionaltothespinsusceptibility χandleadstoaresonance\nfrequency increasing with temperature, while the second\nand smaller one is proportional to the inverse specific\nheat coefficient 1 /γ, leading to a decreasing behavior.\nThe observed approximately linear ln Tdependence of\ntheg-shift is determined by χand 1/γ, (in the restricted\ntemperature regime, where the remaining curvature in\nbothχand 1/γtends to compensate), while the mag-\nnitude of the shift is fitted by adjusting the anisotropic\nexchange interaction constant I. The observed rather\nstrong magnetic field dependence of the prefactor of ln T\nis very well accounted for by χ. Finally, we attempted\nto relate the linewidth to the singular part of the self\nenergy Σ sing, by identifying the ln Tcontribution to the\nspecfic heat coefficient with the effective mass deduced\nfromΣ sing. Onaqualitativelevel, the observedcrossover\nfromT2to linear Tbehaviorof the linewidth upon enter-\ning the non-Fermi liquid regime is reproduced. However,\nthe line width is found to be approximately a factor of\n5 too large compared to experiment. The most likely\nexplanation of this discrepancy is our neglect of vertex\ncorrections, which would remove any non-magnetic con-\ntribution to Σ singfrom the linewidth.\nOverall the extended quasiparticle picture used to ac-\ncount for the ESR-properties in the non-Fermi liquid\nregimeappearstoworkquitewell. Itwouldbeinteresting\nto compare with data taken at much lower temperatures,when the present theory would predict, e.g. nonlinear\nvariationofthe g-shift with ln T, as exhibited by χ. Also,\na cleaner identification of the linear Tdependence of the\nlinewidth would be essential to corroborate the extended\nquasiparticle picture. Finally, our theory makes definite\npredictions for the anisotropy of the lineshift.\nAcknowledgments\nWe thank the Aspen Center for Physics, where part of\nthisworkwascompleted. P.W.acknowledgespartialsup-\nportfromtheDFG-CenterforFunctionalNanostructures\nand the DFG-Forschergruppe “Quantum Phase Transi-\ntions”.\nAPPENDIX\n1. Derivation of dynamical susceptibility:\nspin-rotation invariant Fermi liquid interaction\nIn this Appendix wederivethe quasiparticleproperties\nand the dynamical spin susceptibility in a more detailed\nand complete way than was done in AW [5]. We start\nfrom the Dyson equation for the single particle Green’s\nfunctions,\nG−1G=/parenleftBigg\niωn−ǫfσ−Σfσ(iωn,k) −V\n−V iω n−ǫkσ−Σcσ(iωn,k)/parenrightBigg/parenleftBigg\nGff\nkσGcf\nkσ\nGfc\nkσGcc\nkσ/parenrightBigg\n=1. (A.1)\nWe approximate the conduction electron retarded self-\nenergy by Σ cσ(ω+i0,k) =−iγ, where γis the\nconduction-electron spin-lattice relaxation rate. To care-\nfully derive the quasiparticle Zeeman energies, we make\nuse of the conservation of total spin in the model con-\nsidered here, to remove the f-electron Zeeman term of\nthe Hamiltonian by performing a gauge transformation\nthat shifts the zero of energy of ↑-spins and ↓-spins by\n∓ωf/2 respectively. As a consequence iωnis replaced by\niωn+σωf/2 ,ǫfσbyǫfand the conduction electron Zee-manenergyischangedto −σ(ωf−ωc)/2. Foranisotropic\nband structure ( ǫk=ǫk), the magnetic field dependence\nof the self energy (neglecting small band edge terms) is\nthen of the form\nΣfσ(ω+i0,k)≃Σf(ω+σωf/2+i0,ǫk−σ(ωf−ωc)/2).\nInG−1\n11, we expand this f-self energy about the Fermi\nenergy:\nω+σωf/2−ǫf−Σfσ(ω+i0,k) = (ω+σωf/2)(1−∂Σf/∂ω|0)−ǫf−Σf(i0,ǫkF)\n−(∂Σf/∂ǫk|kF)(ǫk−ǫkF−σ(ωf−ωc)/2)+iImΣf(ω+i0,ǫk)\n=z−1[ω−/tildewideǫfkσ+iγfk], (A.2)\nwherez−1= 1−∂Σf/∂ω|0,/tildewideǫfkσ=/tildewideǫfk−σωf/2 +σ(∂Σf/∂ǫk|kF)(ωf−ωc)/2,/tildewideǫfk=z[ǫf+ Σf(i0,ǫkF) +\n(∂Σf/∂ǫk|kF)(ǫk−ǫkF)], andγfk=zImΣf(ω+i0,ǫk)7\nThen for low energies one has a quasiparticle description, with Gff\nkσ(ω) =zσ/tildewideGff\nkσ,Gcf\nkσ=√zσ/tildewideGcf\nkσand the\nrenormalized hybridization amplitude /tildewideV2=zσV2. The complex energy eigenvalues are given by\nζ±\nkσ=1\n2(/tildewideǫfσ−iγfk+ǫkσ−iγ)±/radicalbigg\n1\n4(/tildewideǫfσ−iγfk−ǫkσ+iγ)2+/tildewideV2=ǫ±\nk−1\n2σω±\nk−iγ±\nk, (A.3)\nwhere, assuming |ǫkF|>>|/tildewideǫfk|,/tildewideV,\nǫ±\nk=ǫ±\nkF+1\n2(ǫk−ǫkF)(1+z(∂Σf/∂ǫk|kF))[1±ǫkF−/tildewideǫf/radicalBig\n(/tildewideǫf−ǫkF)2+4/tildewideV2],\nω±\nk=ωf+1\n2[(ωc−ωf)(1+z(∂Σf/∂ǫk|kF))[1±ǫkF−/tildewideǫf/radicalBig\n(/tildewideǫf−ǫkF)2+4/tildewideV2]\nγ±\nkσ=1\n2(γfk+γ)∓1\n2(γfk−γ)ǫkσ−/tildewideǫfσ/radicalBig\n(/tildewideǫf−ǫkF)2+4/tildewideV2(A.4)\nFrom now on we can safely neglect the term involving\n(∂Σf/∂ǫk|kF), since it is small - of order z <<1.\nUsing partial fraction decomposition, we construct the\nretarded Green function\n/tildewideGff\nkσ(ω+i0) =aff,+\nkσ\nω−ζ+\nkσ+aff,−\nkσ\nω−ζ−\nkσ(A.5)\nand similar expressions for /tildewideGcf\nkσandGcc\nkσ, where, with\nukσ=ζ+\nkσ−ζ−\nkσ,\naff,±\nkσ=±(ζ±\nkσ−/tildewideǫkσ)/ukσ,\nacc,±\nkσ=±(ζ±\nkσ−ǫfσ)/ukσ\nacf,±\nkσ=±/tildewideV/ukσ.\nFor sufficiently small imaginary parts, γ≪(/tildewideV,/tildewideǫfσ), we\nmay neglect them in the weight factors aff,±\nkσ,...and re-\nplaceζ±\nkσbyǫ±\nkσ.\nThe quasiparticles interact via the residual Fermi liq-\nuid interaction. For ESR, the relevant component of the\nFermi liquid interaction is the spatially isotropic spin-\nantisymmetric part described by the Landau parameter\nFa\n0=−2N0/tildewideU. Here/tildewideUis the coupling constant of a\nspin isotropic exchange interaction Hex=−/tildewideU− →S·− →S,\nwhich leads to a quasiparticle energy shift δωk=/tildewideUM,\nwhere the spin polarization Mis given by M=χH/tildewideωkF,\nwithχHthe unscreened static spin susceptibility (in the\nabsence of the Fermi liquid interaction) and /tildewideωkFthe fully\nrenormalized quasiparticle Zeeman splitting at the Fermi\nenergy.\nFor definiteness in the following we assume a band fill-\ning of somewhat less than two electrons per site, suchthat the Fermi level lies in the lower quasiparticle band\n(energyζ±\nkσ), and/tildewideǫfσis close to the Fermi energy. The\nscreened Zeeman splitting is then obtained by solving\nthe self-consistent equation /tildewideωkF=ω−\nkF+/tildewideUχH/tildewideωkF. The\nsolution is\n/tildewideωkF=ω−\nkF\n1−/tildewideUχH=ω−\nkFR, (A.6)\nwhereR= 1/(1−/tildewideUχH) is the Wilson ratio. One\nobserves that in the case of ferromagnetic correlations,\nwhenR >>1, the single particle Zeeman splitting is\nenhanced by a large factor. We conjecture that this ef-\nfect should be observable in tunneling experiments. In\nthe two-particle spectrum probed by electron spin res-\nonance, the enhancement is completely removed by dy-\nnamical screening (see AW).\nAs derived in AW, the dynamical transverse suscepti-\nbilityχ+−(Ω), whereΩ isthe frequencyofana.c.electro-\nmagnetic field polarizedtransverseto the static magnetic\nfield, is given by\nχ+−(Ω) =µ2\nB[g2\ncχ+−\ncc(Ω)+g2\nfχ+−\nff(Ω)+2gcgfχ+−\ncf(Ω)].\nThe partial susceptibilities are obtained by evaluating\nFeynman bubble diagrams dressed by vertex corrections\noftheladdertypereferringtotheFermiliquidinteraction\n(local electrons) and the spin-orbit interaction (impurity\ncorrelation lines for the conduction electrons). The final\nresult obtained in AW may be reexpressed as8\nχ+−(Ω+i0) =µ2\nB{[(gfχ+−\nff(0)+gcχ+−\ncf(0))2/χ+−\nff(0)]−ωr+iΓ\nΩ−ωr+iΓ\n+g2\nc−ω−\nkF+2iγ−\nkF\nΩ−ω−\nkF+2iγ−\nkF[χ+−\ncc,H(0)−(χ+−\ncf,H(0))2/χ+−\nff,H(0)]}\nThere appear to be two different resonance denomina-\ntors in the above expression. Only the resonance at ωr\nhas been observed in ESR experiments The resonance\natω−\nkF, is shifted to much higher frequencies (the fac-\ntorR!). However, a closer look reveals that the weight\nof the resonance at ω−\nkFis zero in the regime consid-\nered. Indeed, by inserting the quasiparticle weight fac-\ntors,χ+−\nab,H(0)≃2N0(aab,−\nkFσ)2, wherea,biscorf, one\nfinds that the prefactor of the second resonance is zero.\nIn the limit of equal g-factors, the prefactor of the first\nresonancesimplifiesto2 N0Randitspositionisunshifted,\nωr=ωf. In the absence of spin lattice relaxation, the\nlinewidth shrinks to zero in this case. This is not repro-\nduced in the above calculation, since we did not take into\naccountthevertexcorrectionsbelongingtotheimaginary\npart of the self energy (the scattering-in term in Boltz-\nmann equation language).\n2. Non-spin rotation invariant Fermi liquid\ninteraction\nTherelativelylarge g-shiftobservedinexperimentsug-\ngests the presence of a small additional spin-symmetry\nbreaking Fermi liquid interaction I. Since this inter-\naction can only be mediated by the spin-orbit interac-\ntion it should have preferred direction given by the lat-\ntice symmetry. We therefore assume the form Iαβ;γδ=\n−I(c·ταβ)(c·τγδ), where τis the vector of Pauli ma-\ntrices and cis a unit vector in the direction of the\ncrystallographic c-axis of the tetragonal lattice. The\nscreened and unscreened tensor susceptibilities X,XH,whereXij=∝an}bracketle{t∝an}bracketle{tSi;Sj∝an}bracketri}ht∝an}bracketri}ht,i=x,y,z, are connected by the\nBethe-Salpeter equation\nX=XH+/tildewideU(XHX)+4I(XH·c)(c·X).\nThe solution is given by\nX= (1−/tildewideUXp)−1Xp\nwhereXpis the projected unscreened susceptibility\nXp=XH+4I\n1−4Iχcc\nH(XH·c)(c·XH)., χcc\nH= (c·XH·c)\nTo linear order in Ithe general expression simplifies to\nX= [1−/tildewideUXH−4IX−1\nH(XH·c)(c·XH)]−1XH\nIn the main configuration of the experiments, the static\nmagnetic field is oriented parallel to the ab-plane, say\nalong the a-axis. We take this to be the z-axis in spin\nspace and identify the c-axis with the x-axis. Then we\nsee that the screening of the static field is not changed to\nlinear orderin I,asX+z=Xz+= 0,etc. The dynamical\nresponsewith the time-dependent magneticfield oriented\nperpendicular to the z-axis is, however modified. Using\nχ+xχx−=1\n4(χ+−)2we find\nχ+−=χ+−\nH/[1−(/tildewideU+I)χ+−\nH]\n[1] J. Sichelschmidt, V.A. Ivanshin, J. Ferstl, C. Geibel an d\nF. Steglich, Phys. Rev. Lett. 91, 156401-1 (2003).\n[2] C. Krellner, T. Forster, H. Jeevan, C. Geibel and J.\nSichelschmidt, Phys. Rev. Lett. 100, 066401 (2008).\n[3] U. Schaufuß, V. Kataev, A.A. Zvyagin, B. B¨ uchner, J.\nSichelschmidt, J. Wykhoff, C. Krellner, C. Geibel, and\nF. Steglich, arXiv0804.4105v1.\n[4] P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, and F.\nSteglich, Phys. Rev. Lett. 94, 076402 (2005)\n[5] Elihu Abrahams and Peter W¨ olfle, Phys. Rev, B 78,\n104423 (2008).\n[6] P. Schlottmann, Phys. Rev. B 79, 045104 (2009).\n[7] S.-K. Yip, Phys. Rev. B 38, 8785 (1988)\n[8] George Feher and A.F. Kip, Phys. Rev. 98, 337 (1955).[9] E. Abrahams, unpublished.\n[10] The term proportional to m/m∗was inadvertently mis-\nevaluated in Eq. (20) of AW.\n[11] In the familiar Fermi liquid notation, Fa\n0=−2N0eU.\n[12] J. Sichelschmidt, J. Wykhoff, H.-A. Krug von Nidda, J.\nFerstl, C. Geibel, and F. Steglich J. Phys. Condens. Mat-\nter19116204 (2007).\n[13] See for example, Fig. 1 of Ref. [3].\n[14] S. Paschen, T. L¨ uhmann, S. Wirth, P. Gegenwart, O.\nTrovarelli, C. Geibel, F. Steglich, P. Coleman, Q. Si, Na-\nture413, 804 (2004)\n[15] P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa,\nS. Paschen, C. Geibel, F. Steglich, E. Abrahams, Q. Si,\nScience315, 969 (2007).9\n[16] O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer,\nF.M. Grosche, P. Gegenwart, M. Lang, G. Sparn, and F.\nSteglich, Phys. Rev. Lett. 85, 626 (2000).\n[17] P. Gegenwart, Y. Tokiwa, T. Westerkamp, F. Weickert,\nJ. Custers, J. Ferstl, C. Krellner, C. Geibel, P. Kerschl,\nK-H.M¨ uller, and F. Steglich, New J. Phys. 8, 171 (2006).\n[18] P. Gegenwart, Y. Tokiwa, J. Custers, C. Geibel, and F.\nSteglich, J. Phys. Soc. Jpn. 75(Suppl.) 155 (2006)\n[19] A.C. Hewson, The Kondo Problem to Heavy Fermions .\n(Cambridge University Press, 1993)\n[20] J. Kondo, Prog.Theor. Phys. 32, 37 (1961).\n[21] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n[22] For references to the early experimental and theoretic al\npapers, see W. Brenig, W. G¨ otze and P. W¨ olfle, Phys.Rev. B2, 4533 (1970); J. Sweer, D.C. Langreth and J.W.\nWilkins, Phys. Rev B 13, 192 (1976).\n[23] U. Walter, E. Holland-Moritz and Z. Fisk, Phys Rev B\n43, 320 (1991)\n[24] P. Nozi` eres, J. Low Temp. Phys. 17, 31 (1974).\n[25] A.C. Hewson, J. Bauer and W. Koller, Phys. Rev. B 73,\n045117 (2006).\n[26] S.E. Barnes and J. Zitkova-Wilcox, Phys. Rev. B 7, 2163\n(1973).\n[27] O. Trovarelli et al., Physica B 284-288, 1507 (2000)\n[28] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E.\nAbrahams, and A. E. Ruckenstein, Phys. Rev. Lett. 63,\n1996 (1989)."    },    {        "title": "2104.10409v1.Fermiology_of_two_dimensional_titanium_carbide_and_nitride_MXenes.pdf",        "content": "Fermiology of two-dimensional titanium carbide and nitride MXenes\nMohammad Bagheri1, Rina Ibragimova2, and Hannu-Pekka Komsa1,2\n1Microelectronics Research Unit, University of Oulu, Oulu, Finland\n2Department of Applied Physics, Aalto University, Aalto, Finland.\n(Dated: April 22, 2021)\nMXenes are a family two-dimensional transition metal carbide and nitride materials, which often exhibit very\ngood metallic conductivity and are thus of great interest for applications in, e.g., flexible electronics, electro-\ncatalysis, and electromagnetic interference shielding. However, surprisingly little is known about the fermiology\nof MXenes, i.e, the shape and size of their Fermi-surfaces, and its effect on the material properties. One reason\nfor this may be that MXene surfaces are almost always covered by a mixture of functional groups, and studying\nFermi-surfaces of disordered systems is cumbersome. Here, we study fermiology of four common Ti-based\nMXenes as a function of the surface functional group composition. We first calculate the effective band struc-\ntures of systems with explicit mixed surfaces and observe gradual evolution in the filling of the Ti-d band and\nresulting shift of Fermi-level. We then demonstrate that these band structures can be closely approximated by\nusing pseudohydrogenated surfaces, and also compare favorably to the experimental angle-resolved photoemis-\nsion spectroscopy results. By modifying the pseudohydrogen charge we then proceed to plot Fermi-surfaces for\nall systems and extract their properties, such as the Fermi-surface area and average Fermi-velocity. These are\nin turn used to evaluate the electrical conductivity with the relaxation time fitted to experimentally measured\nconductivities.\nI. INTRODUCTION\nMXenes are a large class of two-dimensional (2D) materi-\nals of transition metal (M) carbides and nitrides (X) [1, 2].\nDistinct from other 2D materials, MXenes are synthesized\nby etching out layers of atoms from a layered bulk precur-\nsor phase. As a result of the etching process, the dangling\nbonds on the surface are passivated by a mixture of organic\ngroups from the etching solution, such as -O, -OH, and -F [3].\nMXenes are metallic and exhibit very high conductivity in ad-\ndition to good mechanical properties and easy solution pro-\ncessing. As a result, they have shown great promise for appli-\ncations in conductive inks, battery electrodes, electromagnetic\ninterference shielding, and various types of sensors [4–6, 31].\nGiven the large interest on these materials, surprisingly lit-\ntle effort has been devoted to understanding the fundamen-\ntal properties of their Fermi-surfaces, i.e., fermiology, and its\neffect on the material properties [7]. Experimentally, Fermi-\nsurfaces are commonly investigated using angle-resolved pho-\ntoemission spectroscopy (ARPES) [7, 8]. ARPES results for\ndelaminated Ti 3C2, the prototypical MXene, were reported\nby Schultz et al. [9], but since the measured film consists of\nrandomly oriented flakes, the ARPES data was azimuthally\naveraged, which prevented the extraction of the Fermi-surface\nshape. Quasiparticle interference within scanning tunneling\nmicroscope (STM-QPI) is also often used to gain insight on\nthe Fermi-surface, but requires high quality surface with small\ndensity of scattering centers [10] — a luxury not afforded in\nthe case of MXenes, where the surfaces are covered by a ran-\ndom mixture of organic groups and other residues from the\nsynthesis. In fact, to the best of our knowledge, clean STM\nimages of MXene surfaces have not been reported in the lit-\nerature. Finally, Fermi-surfaces could be probed via de Haas\nvan Alphen effect [11], but again no reports exist for MXenes.\nAlso computational studies of Fermi-surfaces are very rare,\neven when the band structures have been reported in numer-\nous papers and are relatively well understood [12–14]. Huet al. calculated the Fermi-surface of monolayer and bulk\nTi2C2(OH) 2for two different layer stackings [15]. In addi-\ntion, some insight can be gained from the Fermi-surfaces of\nthe precursor MAX phases, which were presented in Ref. 11.\nWhile the Fermi-surfaces for pure O or OH terminated struc-\ntures are straightforward to evaluate, since the surfaces are\npassivated by a mixture of groups (stabilized by strong near-\nest neighbor interactions between the groups [16, 17]) and the\nelectronic structure of O and OH-terminated surfaces differ\nmarkedly, it is not clear how the Fermi-surface evolves with\nchanging functional group composition. Calculations for the\nmixed group surfaces are cumbersome due to requiring the\nuse of supercells and the resulting band folding. The band\nstructures can be unfolded with the effective band structure\nmethod, where each state from the supercell Brillouin zone is\nprojected to each of the folded planewaves from the primitive\ncell Brillouin zone [18, 19], but smearing of the bands still\ncomplicates the analysis.\nIn this paper, we show how to overcome these problems and\nreport results for the Fermi-surfaces of Ti 3C2, Ti2C, Ti 4N3,\nand Ti 2N monolayers with varying O/OH composition. We\ncalculate the effective band structures for mixed group sur-\nfaces and show that they can be well approximated by pseu-\ndohydrogenated surfaces. We analyze the Fermi-surface prop-\nerties as a function of the surface group composition and com-\npare the evaluated electrical conductivities to the experiments.\nWe also compare the Ti 3C2band structure to those obtained\nfrom recent ARPES experiments.\nII. METHODS\nAll density-functional theory calculations are carried out\nusing V ASP software [20, 21], together with projector aug-\nmented plane wave method [22]. We adopt the Perdew-\nBurke-Ernzerhof exchange-correlation functional for solids\n(PBEsol)[23]. The plane wave cutoff is 550 eV and the Bril-arXiv:2104.10409v1  [cond-mat.mtrl-sci]  21 Apr 20212\nlouin zone of the primitive cell is sampled with a 16 ×16 k-\npoint mesh.\nThe effective band structures (EBS) are calculated using\nBandUP software [24]. The atomic structures for the mixed\nsurfaces were determined via combination of cluster expan-\nsion and Monte Carlo simulations in our previous publication\n[17], and we use the same 4 ×4 supercell special quasi-ordered\nstructures as constructed therein.\nElectronic band structures are interpolated using Boltz-\nTrap2 software, which is also used for the integration of the\ntransport properties [25]. We use k-point mesh interpolation\nfactor 200 and temperature 300 K. Fermi-surfaces are ana-\nlyzed and plotted using the ifermi software [26], with interpo-\nlation factor 20.\nIII. RESULTS\nA. Band structures of mixed surface MXenes\nMore realistic structural models for the distribution of func-\ntional groups were developed by us in Refs. 16, 17. As an\nexample, the structure of Ti 3C2with O 0.5OH0.5surface com-\nposition is shown in Fig. 1(a). In the left panels of Fig. 1(b-e)\nwe show the effective band structures for the four Ti-based\nMXenes with O 0.5OH0.5surface composition. From these\nresults, and from comparison to the band structure of pure\nsurfaces (see Figs. 2–3 below), it is clear that the band struc-\ntures undergo a gradual evolution and there are no extrane-\nous bands arising from the surface groups. The band(s) at\nthe Fermi-level arise from the non-bonding t 2gstates of Ti-d\ncharacter [27]. Consequently, to a very good approximation,\nthe surface groups only control the electron concentration and\nconsequently the Fermi-level position in the metal d-bands, as\nhas been previously assumed [13, 14]. This also suggests that\nother charge acceptor or donor species could be used just as\nwell for controlling the Fermi-level position.\nGiven that effective band structures are unwieldy for de-\ntailed studies of these materials’ fermiology, we propose to\nuse pseudohydrogenated surfaces to control the Fermi-level\nposition. In practice, we use the primitive cell of a fully OH-\nterminated structures, but with the H atom replaced by a pseu-\ndohydrogen atom with atomic number Z= 0.25,0.50, or\n0.75(i.e., proton charge is Zeand total electron charge −Ze).\nThe pseudohydrogen band structures for Z= 0.5are shown\non the right panels of Fig. 1(b-e), overlaid on top of the effec-\ntive band structures. In all cases, the band structures of pseu-\ndohydrogenated MXenes agree extremely well with the EBS,\nthereby further corroborating the physical picture described\nabove.\nIt is worth noting that we also calculated the band structures\nfor systems with one side fully terminated by O and the other\nby OH, but the resulting band structures did not agree favor-\nably with the EBS due to broken symmetry. Moreover, such\napproach would obviously be limited to a O 0.5OH0.5compo-\nsition.\nThe band structure evolution with O/OH composition, for\nall materials and five different O/OH compositions, are col-lected in Fig. 2 and 3. From these plots, the gradual change in\nthe band energies is easily resolved.\nBased on the analysis by Hu et al. [27], the states be-\nlow Fermi-level (red colored band and below in Figs. 2 and\n3) arise from bonding state between Ti-d and C-p/N-p ( eg),\nand similar to bulk carbides and nitrides [28]. The states at\nand above Fermi-level (orange colored band and above) arise\nfrom non-bonding t2gstates of Ti-d. In most cases there is\na gap between these two band manifolds, but the Fermi-level\nresides in the Ti-d manifold in all cases except for the fully\nO-terminated Ti 2C. Due to the larger electronegativity of N,\nas compared to C, the Ti-N bonding states are clearly lower in\nenergy and lead to a larger gap below the nonbonding states.\nThe Ti-C bands cross the Fermi-level around Γ-point only for\nO-rich Ti 3C2, but thereby prevent it from becoming semicon-\nductor as happens with Ti 2CO2.\nDue to the additional electron(s) in N, as compared to C,\nthere is one additional electron per X atom to fill the M-d\nbands and consequently Fermi-level resides higher within this\nband. For instance, the lowest energy band (orange) is exactly\nhalf-filled by the one additional electron in the case of Ti 2N,\nwhereas the band is empty in the case of Ti 2C. Similar half-\nfilling of the band can be obtained in the case of O 0.5OH0.5\nsurface composition of Ti 3C2or Ti 2C.\nThe band fillings can also be readily obtained via electron\ncounting. In the case of Ti 3C2O2and Ti 2CO2, Ti, C, and O\nare in the oxidation states +4, -4, and -2, respectively, as in TiC\nand TiO 2. Upon addition of H atoms (oxidation state +1) and\nthe subsequent occupation of the non-bonding Ti-d band, the\ncharge balance is achieved by decreasing the oxidation state\nof Ti. The nitrogen atoms (oxidation state -3) have a similar\neffect.\nO/OH composition controls the occupation of the t 2gband,\nbut since it is nonbonding, it should have relatively minor\neffect on the energetics. Among the purely terminated sur-\nfaces, O-terminated surfaces have been predicted to be the\nmost stable, but the mixed surfaces arise in practice due to the\nstrong attractive interactions between the opposite type sur-\nface groups [16, 17]. Consequently, since the surface com-\nposition is dominated by the interactions between surface\ngroups, it is rather independent on the M and X species, as\nwe observed in Ref. 17 (we also note, that the energy gain\nupon mixing is not captured in the pseudohydrogen models).\nEven if the O/OH mixing cannot be easily avoided, we expect\nthat the Fermi-level position can still be tuned by introducing\nalloying in the M and X sublattices.\nB. Fermi-surface properties\nThe Fermi-surfaces are plotted below the band structures in\nFigs. 2 and 3, and colored by the Fermi-velocity. When Fermi-\nlevel crosses the lowest energy Ti-d band, it leads to hexago-\nnally symmetric ellipsoids around the M-points. These have\nfairly large Fermi-velocities, reaching values above 0.5·106\nm/s, which is only about factor of two lower than in the highly\nconducting elemental metals. At large OH concentrations (or\nany concentration in the case of Ti 4N3) when the Fermi-level3\nFIG. 1: (a) Atomic structure for Ti 3C2with a surface functionalized by a mixture of O and OH groups (top) and the corresponding pseudohy-\ndrogenated structure (bottom). (b-e) Effective band structures for all materials (left panel) and band structures of pseudohydrogenated surfaces\noverlaid on the EBS (right panel). Fermi-level is set to zero.\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nTi3C2(O0.75OH0.25)2 Ti3C2O2 Ti3C2(O0.5OH0.5)2 Ti3C2(O0.25OH0.75)2 Ti3C2(OH)2\nTi2C(O0.75OH0.25)2 Ti2CO2 Ti2C(O0.5OH0.5)2 Ti2C(O0.25OH0.75)2 Ti2C(OH)2\nFIG. 2: Evolution of the band structures of Ti 3C2and Ti 2C with surface group composition. The Fermi-surfaces are plotted under each panel,\nwith the color denoting Fermi-velocity.\nrises to cross the set of states around Γ-point, the Fermi- surfaces develop more complex features.4\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nK M-2-1012Energy (eV)\nTi4N3(O0.75OH0.25)2 Ti4N3O2 Ti4N3(O0.5OH0.5)2 Ti4N3(O0.25OH0.75)2 Ti4N3(OH)2\nTi2N(O0.75OH0.25)2 Ti2NO2 Ti2N(O0.5OH0.5)2 Ti2N(O0.25OH0.75)2 Ti2N(OH)2\nFIG. 3: Evolution of the band structures of Ti 4N3and Ti 2N with surface group composition. The Fermi-surfaces are plotted under each panel,\nwith the color denoting Fermi-velocity.\nTi3C2 Ti2C Ti4N3 Ti2N\n0 0.5 1\nO1-xOHx012345672D/0(1/ s)1011\n0 0.5 1\nO1-xOHx0123456(1/ m)105\n0 0.5 1\nO1-xOHx0510152025Fermi surface length (1/Å)\n0 0.5 1\nO1-xOHx234567Avg. Fermi velocity (m/s)105\n0 0.5 1\nO1-xOHx012345DOS (states/eV/u.c.)(a) (b) (c) (d) (e)□\nFIG. 4: Fermi-surface properties: (a) sheet conductivity, (b) conductivity, (c) Fermi-surface length, (d) Fermi-velocity, and (e) density of\nstates.\nWith access to Fermi-surfaces of (effectively) mixed sur-\nface MXenes, we can next estimate their properties. Since\nMXenes are known to exhibit excellent electrical conductiv-\nity, we evaluated the sheet conductivity within the constantrelaxation time approximation (in x-direction) as\nσ2D=e2\n2π2/planckover2pi1τ0/integraldisplay\nE=EFv2\nx(k)\n|v(k)|dkF (1a)\n=e2\n2π2/planckover2pi1τ0/angbracketleftvF/angbracketright\n2·lF (1b)5\nsee Supplemental Material for more detailed description.\nThe sheet conductivities are shown in Fig. 4(a). The solid\nlines show the results from the full integral (Eq. 1a) and\nthe dashed lines from average Fermi-velocity approximation\n(Eq. 1b). All four considered MXenes show similar σ2D/τ0,\nmostly falling within 2–4 1/Ω/squares. The approximation (Eq.\n1b) works very well and thus by inspecting the /angbracketleftvF/angbracketrightandlF\nshown in Fig. 4(c,d), we can study their roles in governing\nthe conductivity. Fermi-surface lengths and average Fermi-\nvelocity are similar in Ti 3C2, Ti2C, and Ti 2N. Ti 4N3shows\nlarger total length due to large number of states crossing the\nFermi-level, but the average Fermi-velocities in these states\nare low. Overall, our results indicate that by simply tuning the\nsurface composition the sheet conductivity cannot be changed\nby more than about a factor of two.\nTrends in electrical conductivity are often evaluated from\nthe density of states (DOS) at Fermi-level, since lF∼\nD2D(EF)/angbracketleftvF/angbracketright(cf. Supplemental Material). The DOS at\nFermi-level are presented in 4(e) and shows that the trends\nin the conductivity changes with composition or between dif-\nferent materials cannot be extracted solely from the DOS due\nto missing the effect of /angbracketleftvF/angbracketright.\nExperimentally measuring the sheet conductivity of MX-\nene monolayers is challenging, but the conductivity of MXene\nfilms has been reported in several publications. In the case of\nTi3C2films, conductivities above 105S/m have been reported\noften [29–32]. Recently a value as high as 1.1·106S/m was\nachieved for a high crystalline quality Ti 3C2monolayer [33].\nFor other considered materials, the reports are more scarce,\nbut nevertheless fairly similar values can be found: 5·105\nS/m for Ti 2N [34], and 2.3·105S/m for Ti 2C [35]. It is worth\nnoting, that the conductivity may be governed by interflake\ncontacts and thus also sensitive to the presence of any interca-\nlated species [36].\nIn order to estimate the film conductivity, we calculate\nσ= (σ2D/τ0)·τ0/t, whereτ0is the relaxation time and tis\nthe interlayer separation taken from experiments: 10.3, 7.5,[1]\n14,[37] and 8.7 Å[34] for Ti 3C2, Ti2C, Ti 4N3, and Ti 2N, re-\nspectively. Note, that the interlayer separation depends on the\npreparation method, as, e.g., intercalation of Li can increase\nthe spacing by a few Å. Here we do not attempt to evaluate τ0\nfrom first-principles calculations, but instead from the com-\nparison to experimentally measured conductivities. In Fig.\n4(b) we show the results for τ0= 1 fs, which yields σval-\nues comparable to majority of the experimental reports. The\nresult of Ref. 33 can be reproduced by taking slightly larger\nτ≈2−3fs, depending on the surface composition. The lower\nτ0in earlier reports may then be ascribed to larger number of\ndefects caused by the strong etching, although the interflake\ncontacts may also contribute.\nC. Angle-resolved photoemission spectrum (ARPES)\nTo verify that our calculations properly capture the main\nfeatures in the electronic structures of MXenes, we compare\nthem to the only available experimental piece of information:\nthe ARPES results for Ti 3C2reported in Ref. 9. The exper-imental data is reproduced in Fig. ??(d) and shows: (i) flat\nband at the Fermi-level, (ii) band with negative dispersion\nstarting at -1.5 eV , (iii) band with negative dispersion starting\nat -2 eV , (iv) flat band at -3 eV , (v) flat band segment at -4 eV ,\n(vi) broad flat band between -5 and -6 eV , and (vii) broad flat\nband between -8 and -9 eV . The composition was determined\nto be Ti 3C2F0.8O0.8.\nIn Fig. 5(a) we replot the effective band structure of\nTi3C2(O0.5OH0.5)2overlaid with the pseudohydrogenated\nband structure from Fig. 1(a), but in a wider energy range. As\ndiscussed, the non-bonding Ti bands at and above 0 eV and the\nTi-C bonding bands between -4 and 0 eV are well reproduced\nin the pseudohydrogenated systems. The Ti-O bands between\n-7 and -4 eV are not reproduced very well, and the O-H states\nat -9 eV are understandably poorly described by pseudohydro-\ngens. Also some parts of the effective band structure are more\nstrongly smeared by the surface group disorder than others,\nwhereas the pseudohydrogen band structures contain no dis-\norder. These factors limit the use of the pseudohydrogenated\nstructures for reproducing ARPES results of deeper states.\nSince the features close to Fermi-level [features (i)–(iii)\nabove] are properly described by pseudohydrogen models, we\ncan start by comparing them to the band structures in Fig.\n??. Feature (i) indicates that the non-bonding band at Γ-\npoint must be occupied, and therefore O content must be less\nthan about 0.25. The gap of about 1.5 eV between features\n(i) and (ii) would match to the pure OH-terminated surface.\nWe then calculated effective band structures for few different\nmixtures with small O content and found overall best agree-\nment with Ti 3C2(O0.25OH0.25F0.5)2, as shown in Fig. 5(b)\nand processed such that it is more comparable to the experi-\nmental spectrum in Fig. 5(c). In this case, features (i)–(iv) are\nall reproduced very well. The EBS also shows flat band away\nfrom the Γ-point at about -4 eV , matching well with feature\n(v) in the experimental spectrum. We also reproduce fairly\nwell the broad band between -5 and -6 eV , feature (vi). The\nlowest energy feature (vii) is not reproduced in our calcula-\ntions. The band at -7 eV in EBS originates from Ti-F bonds\nand the band at below -9 eV from O-H bonds. Since semilocal\nfunctionals, such as PBEsol one used here, are known to un-\nderestimate band widths and experimentally measured sample\nshould contain significant amount of F, we ascribe feature (vii)\nto Ti-F bonds, in agreement with the discussion in Ref. 9.\nD. Conclusions\nIn conclusion, we presented a study of the fermiology of\ntitanium carbide and nitride MXenes via density-functional\ntheory calculations. We first calculated the effective band\nstructures for surfaces with mixtures of O and OH groups.\nThe electronic structure near the Fermi-level was found to\nevolve gradually with the composition, and we further showed\nthey are very well reproduced by surfaces fully terminated\nwith pseudohydrogenated OH groups. With the pseudohydro-\ngenated surfaces we could readily access Fermi-surface prop-\nerties, such as Fermi-surface area (or length in the case of 2D\nmaterials) and Fermi-velocity. We calculated the electrical6\nK M-10-8-6-4-202Energy (eV)\n0 0.3 0.6 0.9 1.2\nk|| (Å-1)-10-8-6-4-20Energy (eV)\n(i)\n(ii)\n(iii)\n(iv)\n(v)\n(vi)\n(vii)(a) (b) (c) (d)\nK M-10-8-6-4-20Energy (eV)Ti3C2(O0.5OH0.5)2 Ti3C2(O0.25OH0.25F0.5)2 Expt. Ti3C2 ARPES Ti3C2(O0.25OH0.25F0.5)2\nFIG. 5: (a) Calculated effective band structure of Ti 3C2(O0.5OH0.5)2overlaid with the band structure from pseudohydrogenated system. (b)\nEffective band structure of Ti 3C2(O0.25OH0.25F0.5)2. (c) The same as panel (b), but showing only the Γ-K line, broadened, and colormap\nchosen to match that in the experiments. (d) Experimentally measured ARPES spectrum of Ti 3C2Tx, adapted from Ref. 9.\nconductivity within the constant relaxation time approxima-\ntion and by comparing to the experimentally determined con-\nductivities we obtained relaxation time of about 1 fs. For the\nsame relaxation time, the conductivities for all studied MX-\nenes were rather similar and also fairly independent of the sur-\nface composition, changing at most by a factor of two. Finally,\nwe compared our effective band structures to those obtained\nfrom recent ARPES experiments [9] for Ti 3C2, where the\nbest agreement was obtained using Ti 3C2(O0.25OH0.25F0.5)2\nmodel.\nThe approaches demonstrated here can be used also for\nother MXenes, including double MXenes [38] and even solid\nsolutions in the M or X sublattice if their positions and those\nof the surface functional groups are uncorrelated. With the\npseudohydrogenation approach it should be possible to calcu-\nlate many other material properties of MXenes with an effec-tive mixed surface composition, such as optical properties and\nelectron-electron and electron-phonon scattering rates. It will\nalso allow investigating how instabilities in electronic struc-\nture, leading to e.g. magnetism or charge density waves, de-\npend on the surface composition.\nAcknowledgments\nWe thank Prof. Norbert Koch and Dr. Thorsten Schultz for\nproviding us the original ARPES data, previously published\nin Ref. 9. We are grateful to the Academy of Finland for sup-\nport under Academy Research Fellow funding No. 311058.\nWe also thank CSC–IT Center for Science Ltd. for generous\ngrants of computer time.\n[1] M. Naguib, O. Mashtalir, J. Carle, V . Presser, J. Lu, L. Hult-\nman, Y . Gogotsi, and M. W. Barsoum, ACS Nano 6, 1322\n(2012), http://dx.doi.org/10.1021/nn204153h, URL http://\ndx.doi.org/10.1021/nn204153h .\n[2] Y . Gogotsi and B. Anasori, ACS Nano 13, 8491 (2019), pMID:\n31454866, https://doi.org/10.1021/acsnano.9b06394, URL\nhttps://doi.org/10.1021/acsnano.9b06394 .\n[3] M. Naguib, M. Kurtoglu, V . Presser, J. Lu, J. Niu, M. Heon,\nL. Hultman, Y . Gogotsi, and M. W. Barsoum, Advanced Ma-\nterials 23, 4248 (2011), ISSN 1521-4095, URL http://dx.\ndoi.org/10.1002/adma.201102306 .\n[4] B. Anasori, M. R. Lukatskaya, and Y . Gogotsi, Nature Reviews\nMaterials 2, 16098 (2017), URL https://www.nature.\ncom/articles/natrevmats201698 .\n[5] Z. Fu, N. Wang, D. Legut, C. Si, Q. Zhang, S. Du,\nT. C. Germann, J. S. Francisco, and R. Zhang, Chem-\nical Reviews 119, 11980 (2019), pMID: 31710485,\nhttps://doi.org/10.1021/acs.chemrev.9b00348, URL https:\n//doi.org/10.1021/acs.chemrev.9b00348 .\n[6] E. Lee and D.-J. Kim, Journal of The Electrochemical Society167, 037515 (2020), URL https://doi.org/10.1149%\n2F2.0152003jes .\n[7] I. Matsuda and S. Hasegawa, Journal of Physics: Condensed\nMatter 19, 355007 (2007), URL https://doi.org/10.\n1088/0953-8984/19/35/355007 .\n[8] K. Lasek, J. Li, S. Kolekar, P. M. Coelho, L. Guo,\nM. Zhang, Z. Wang, and M. Batzill, Surface Sci-\nence Reports p. 100523 (2021), ISSN 0167-5729, URL\nhttps://www.sciencedirect.com/science/\narticle/pii/S016757292100008X .\n[9] T. Schultz, N. C. Frey, K. Hantanasirisakul,\nS. Park, S. J. May, V . B. Shenoy, Y . Gogotsi, and\nN. Koch, Chemistry of Materials 31, 6590 (2019),\nhttps://doi.org/10.1021/acs.chemmater.9b00414, URL\nhttps://doi.org/10.1021/acs.chemmater.\n9b00414 .\n[10] L. Chen, P. Cheng, and K. Wu, Journal of Physics: Condensed\nMatter 29, 103001 (2017), URL https://doi.org/10.\n1088/1361-648x/aa54da .\n[11] T. Ouisse and M. W. Barsoum, Materials Research Letters 5,7\n365 (2017), https://doi.org/10.1080/21663831.2017.1333537,\nURL https://doi.org/10.1080/21663831.2017.\n1333537 .\n[12] Y . Xie and P. R. C. Kent, Phys. Rev. B 87, 235441\n(2013), URL http://link.aps.org/doi/10.1103/\nPhysRevB.87.235441 .\n[13] M. Khazaei, M. Arai, T. Sasaki, C.-Y . Chung, N. S. Venkatara-\nmanan, M. Estili, Y . Sakka, and Y . Kawazoe, Advanced Func-\ntional Materials 23, 2185 (2013), ISSN 1616-3028, URL\nhttp://dx.doi.org/10.1002/adfm.201202502 .\n[14] M. Khazaei, A. Ranjbar, M. Arai, T. Sasaki, and S. Yunoki,\nJ. Mater. Chem. C 5, 2488 (2017), URL http://dx.doi.\norg/10.1039/C7TC00140A .\n[15] T. Hu, H. Zhang, J. Wang, Z. Li, M. Hu, J. Tan, P. Hou, F. Li,\nand X. Wang, Scientific Reports 5, 16329 (2015).\n[16] R. Ibragimova, M. J. Puska, and H.-P. Komsa, ACS Nano 13,\n9171 (2019), https://doi.org/10.1021/acsnano.9b03511, URL\nhttps://doi.org/10.1021/acsnano.9b03511 .\n[17] R. Ibragimova, P. Erhart, P. Rinke, and H.-P. Komsa, The\nJournal of Physical Chemistry Letters 12, 2377 (2021),\nhttps://doi.org/10.1021/acs.jpclett.0c03710, URL https://\ndoi.org/10.1021/acs.jpclett.0c03710 .\n[18] W. Ku, T. Berlijn, and C.-C. Lee, Phys. Rev. Lett. 104, 216401\n(2010), URL https://link.aps.org/doi/10.1103/\nPhysRevLett.104.216401 .\n[19] V . Popescu and A. Zunger, Phys. Rev. Lett. 104, 236403\n(2010), URL https://link.aps.org/doi/10.1103/\nPhysRevLett.104.236403 .\n[20] G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996).\n[21] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).\n[22] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994), URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.\n50.17953 .\n[23] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E.\nScuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev.\nLett. 100, 136406 (2008), URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.100.136406 .\n[24] P. V . C. Medeiros, S. Stafström, and J. Björk, Phys. Rev. B\n89, 041407 (2014), URL https://link.aps.org/doi/\n10.1103/PhysRevB.89.041407 .\n[25] G. K. Madsen, J. Carrete, and M. J. Verstraete, Computer\nPhysics Communications 231, 140 (2018), ISSN 0010-4655,\nURL https://www.sciencedirect.com/science/\narticle/pii/S0010465518301632 .\n[26] A. M. Ganose, A. Searle, A. Jain, and S. M. Griffin, Journal of\nOpen Source Software 6, 3089 (2021), URL https://doi.\norg/10.21105/joss.03089 .\n[27] T. Hu, Z. Li, M. Hu, J. Wang, Q. Hu, Q. Li, and\nX. Wang, The Journal of Physical Chemistry C 121,\n19254 (2017), https://doi.org/10.1021/acs.jpcc.7b05675, URL\nhttps://doi.org/10.1021/acs.jpcc.7b05675 .[28] J. Häglund, A. Fernández Guillermet, G. Grimvall, and M. Kör-\nling, Phys. Rev. B 48, 11685 (1993), URL https://link.\naps.org/doi/10.1103/PhysRevB.48.11685 .\n[29] J. Halim, M. R. Lukatskaya, K. M. Cook, J. Lu, C. R. Smith,\nL.-Å. Näslund, S. J. May, L. Hultman, Y . Gogotsi, P. Ek-\nlund, et al., Chemistry of Materials 26, 2374 (2014), pMID:\n24741204, https://doi.org/10.1021/cm500641a, URL https:\n//doi.org/10.1021/cm500641a .\n[30] Z. Ling, C. E. Ren, M.-Q. Zhao, J. Yang, J. M. Giammarco,\nJ. Qiu, M. W. Barsoum, and Y . Gogotsi, Proceedings of the\nNational Academy of Sciences 111, 16676 (2014), ISSN 0027-\n8424, https://www.pnas.org/content/111/47/16676.full.pdf,\nURL https://www.pnas.org/content/111/47/\n16676 .\n[31] F. Shahzad, M. Alhabeb, C. B. Hatter, B. Ana-\nsori, S. Man Hong, C. M. Koo, and Y . Gogotsi,\nScience 353, 1137 (2016), ISSN 0036-8075,\nhttps://science.sciencemag.org/content/353/6304/1137.full.pdf,\nURL https://science.sciencemag.org/\ncontent/353/6304/1137 .\n[32] J. Liu, Z. Liu, H.-B. Zhang, W. Chen, Z. Zhao, Q.-W. Wang, and\nZ.-Z. Yu, Advanced Electronic Materials 6, 1901094 (2020),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1002/aelm.201901094,\nURL https://onlinelibrary.wiley.com/doi/\nabs/10.1002/aelm.201901094 .\n[33] A. Lipatov, A. Goad, M. J. Loes, N. S. V orobeva, J. Abourahma,\nY . Gogotsi, and A. Sinitskii, Matter 4, 1413 (2021), ISSN\n2590-2385, URL https://www.sciencedirect.com/\nscience/article/pii/S2590238521000540 .\n[34] B. Soundiraraju and B. K. George, ACS Nano 11, 8892 (2017),\npMID: 28846394, http://dx.doi.org/10.1021/acsnano.7b03129,\nURL http://dx.doi.org/10.1021/acsnano.\n7b03129 .\n[35] J. Halim, I. Persson, E. J. Moon, P. Kühne, V . Darakchieva,\nP. O. Å. Persson, P. Eklund, J. Rosen, and M. W. Barsoum,\nJournal of Physics: Condensed Matter 31, 165301 (2019),\nURL https://doi.org/10.1088%2F1361-648x%\n2Fab00a2 .\n[36] J. L. Hart, K. Hantanasirisakul, A. C. Lang, B. Anasori,\nD. Pinto, Y . Pivak, J. T. van Omme, S. J. May, Y . Gogotsi, and\nM. L. Taheri, Nature Communications 10, 522 (2019).\n[37] P. Urbankowski, B. Anasori, T. Makaryan, D. Er, S. Kota, P. L.\nWalsh, M. Zhao, V . B. Shenoy, M. W. Barsoum, and Y . Gogotsi,\nNanoscale 8, 11385 (2016), URL http://dx.doi.org/\n10.1039/C6NR02253G .\n[38] B. Anasori, Y . Xie, M. Beidaghi, J. Lu, B. C. Hosler,\nL. Hultman, P. R. C. Kent, Y . Gogotsi, and M. W.\nBarsoum, ACS Nano 9, 9507 (2015), pMID: 26208121,\nhttps://doi.org/10.1021/acsnano.5b03591, URL https://\ndoi.org/10.1021/acsnano.5b03591 .Supplemental Material to “Fermiology of two-dimensional titanium carbide and nitride MXenes”\nMohammad Bagheri1, Rina Ibragimova2, and Hannu-Pekka Komsa1,2\n1Microelectronics Research Unit, University of Oulu, Oulu, Finland\n2Department of Applied Physics, Aalto University, Aalto, Finland.\n(Dated: April 22, 2021)\n3D DOS at the Fermi-level is shown in Fig. S1(b) in the\nunits of 1/eV/unitcell and including a factor of two for spin.\nWithin the free-electron model, it could also be calculated\nfrom\nD(EF) =2\n(2π)3/planckover2pi1·aF\n/angbracketleftvF/angbracketright·Vu.c. (S1)\nwhereaFis the Fermi-surface area, /angbracketleftvF/angbracketrightthe average Fermi-\nvelocity, and Vu.c.the unit cell volume in real space. This\napproximation is shown by dashed lines in Fig. S1(b) and\nshows that it holds fairly well when the Fermi-surface con-\nsists of only one set of ellipsoids from a single band. Note,\nthat 3D DOS is equal to 2D DOS (given here for the free-\nelectron model)\nD2D(EF) =2\n(2π)2/planckover2pi1·lF\n/angbracketleftvF/angbracketright·Au.c. (S2)\nwherelFis the Fermi-surface length and Au.c.the unit cell\narea. This can be immediately verified by plugging in lF=\naF/bzandAu.c.=Vu.c./Lz, whereLzthe unit cell length\nin z-direction and bz= 2π/Lzis the reciprocal space basis\nvector in the out-of-plane direction.\nThe electrical conductivity is calculated from\nσij=e2\n4π3/planckover2pi1/integraldisplay\nE=EFvi(k)vj(k)\n|v(k)|τ(k)dkF (S3a)\n≈e2\n4π3/planckover2pi1τ0/integraldisplay\nE=EFvi(k)vj(k)\n|v(k)|dkF (S3b)\n≈e2\n4π3/planckover2pi1τ0/angbracketleftvF/angbracketright\n2·aF (S3c)We have assumed τ(k)is constantτ0and/angbracketleftvF/angbracketrightaf=/integraltext\n|v|=/integraltext\nv2\nx/|v|+/integraltext\nv2\ny/|v|≈2/integraltext\nv2\nx/|v|.\nThe sheet conductivity is written similarly:\nσ2D=e2\n2π2/planckover2pi1τ0/integraldisplay\nE=EFvi(k)vj(k)\n|v(k)|dkF (S4a)\n=e2\n2π2/planckover2pi1τ0/angbracketleftvF/angbracketright\n2·lF (S4b)\n=σ·2π\nbz=σ·Lz (S4c)\nWe first use Boltztrap2 to calculate σand then use Eq. S4c\nto getσ2D, whereLzis the length of the lattice constant cof\nthe computational cell. In order to evaluate σof MXene films,\nwe again use Eq. S4c, but this time Lzis the layer separation.\nThe “3D quantities” obtained for the monolayer are shown\nin Fig. S1, but depend on the size of the vacuum region. The\ncell volume and lattice constant c, which are required to trans-\nlate the 2D and 3D conductivities, are listed in Table S1.arXiv:2104.10409v1  [cond-mat.mtrl-sci]  21 Apr 20212\n0 0.5 1\nO1-xOHx0123456Fermi surface area (Å2)\n0 0.5 1\nO1-xOHx012345DOS (states/eV/u.c.)\n0 0.5 1\nO1-xOHx01234/0(1/( m s)1020\nTi3C2\nTi2C\nTi4N3\nTi2NTi3C2\nTi2C\nTi4N3\nTi2N(a) (b) (c)\nFIG. S1: Fermi-surface properties as obtained using the bulk (3D) equations: (a) Fermi-surface area, (b) density of states, and (c) electrical\nconcductivity divided by relaxation time σ/τ 0.\nTABLE S1: Unit cell volumes V(in Å3) and the lattice constants in z-direction c(in Å) for all considered systems.\nO O 0.75OH0.25 O0.5OH0.5O0.25OH0.75 OH\nsystem V c V c V c V c V c\nTi3C2232.41 29.53 242.71 30.97 242.71 30.70 242.71 30.30 242.71 29.98\nTi2C 154.70 19.75 154.70 19.96 154.70 19.78 154.70 19.54 154.70 19.34\nTi4N3258.45 33.58 258.45 33.61 258.45 33.46 258.45 33.33 258.45 33.20\nTi2N 154.70 20.20 154.70 20.21 154.70 19.99 154.70 19.71 154.70 19.52"    },    {        "title": "1108.5539v2.Anderson_Orthogonality_in_the_Dynamics_After_a_Local_Quantum_Quench.pdf",        "content": "Anderson Orthogonality in the Dynamics After a Local Quantum Quench\nWolfgang M under,1Andreas Weichselbaum,1Moshe Goldstein,2Yuval Gefen,3and Jan von Delft1\n1Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience,\nLudwig-Maximilians-Universit at, Theresienstrasse 37, 80333 Munich, Germany\n2Department of Physics, Yale University, 217 Prospect Street, New Haven, Connecticut 06520, USA\n3Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel\n(Dated: November 26, 2021)\nWe present a systematic study of the role of Anderson orthogonality for the dynamics after a\nquantum quench in quantum impurity models, using the numerical renormalization group. As shown\nby Anderson in 1967, the scattering phase shifts of the single-particle wave functions constituting the\nFermi sea have to adjust in response to the sudden change in the local parameters of the Hamiltonian,\ncausing the initial and \fnal ground states to be orthogonal. This so-called Anderson orthogonality\ncatastrophe also in\ruences dynamical properties, such as spectral functions. Their low-frequency\nbehaviour shows nontrivial power laws, with exponents that can be understood using a generalization\nof simple arguments introduced by Hop\feld and others for the X-ray edge singularity problem. The\ngoal of this work is to formulate these generalized rules, as well as to numerically illustrate them\nfor quantum quenches in impurity models involving local interactions. As a simple yet instructive\nexample, we use the interacting resonant level model as testing ground for our generalized Hop\feld\nrule. We then analyse a model exhibiting population switching between two dot levels as a function\nof gate voltage, probed by a local Coulomb interaction with an additional lead serving as charge\nsensor. We con\frm a recent prediction that charge sensing can induce a quantum phase transition\nfor this system, causing the population switch to become abrupt. We elucidate the role of Anderson\northogonality for this e\u000bect by explicitly calculating the relevant orthogonality exponents.\nPACS numbers: 02.70.-c, 05.10.Cc, 71.27.+a, 72.10.Fk, 73.21.La, 75.20.Hr, 78.20.Bh\nI. INTRODUCTION\nThe Anderson orthogonality (AO) catastrophe1refers\nto the response of a Fermi sea to a change in a local scat-\ntering potential, described, say, by a change in Hamil-\ntonian from ^Hito^Hf. Such a change induces changes\nin the scattering phase shifts of all single-particle wave\nfunctions. This causes the initial ground state jGiiof^Hi\nand the \fnal ground state jGfiof^Hf, both describing a\n\flled Fermi sea but w.r.t. di\u000berent single-particle wave\nfunctions, to be orthogonal in the thermodynamic limit,\neven if the changes in the single-particle wave functions\nare minute. The overlap of the respective ground states\nscales as1,2\njhGijGfij\u0018N\u00001\n2\u00012\nAO; (1)\nwhereNis the number of particles in the system, and the\nexponent \u0001 AOcharacterizes the degree of orthogonality.\nAO underlies the physics of numerous dynamical\nphenomena such as the Fermi edge singularity,2{5the\nAltshuler-Aronov zero bias anomaly6in disordered con-\nductors, tunnelling into strongly interacting Luttinger\nliquids,7{11slow relaxation in electron glasses,12,13and\noptical absorption involving a Kondo exciton,14{16where\nphoton absorption induces a local quantum quench, to\nname but a few. Recently, AO has also been evoked17,18\nin an analysis of population switching (PS) in quantum\ndots (the fact that the population of individual levels of a\nquantum dot may vary non-monotonically with the gate\nvoltage), and was argued to lead, under certain condi-\ntions involving a local Coulomb interaction with a nearbycharge sensor, to a quantum phase transition.\nOne of the goals of the present work is to analyse the\nlatter prediction in quantitative detail. Another is to\ngeneralize arguments that were given in Refs. 14{16, for\nthe role of AO for spectral functions of the excitonic An-\nderson model, to related models with a similar struc-\nture. Thus, we present a systematic study of the role of\nAnderson orthogonality for the dynamics after a quan-\ntum quench in quantum impurity models involving local\ninteractions, using the numerical renormalization group\n(NRG).19,20We thereby extend a recent study,21which\nshowed how \u0001 AOcan be calculated very accurately (with\nerrors below 1%) by using NRG to directly evaluate over-\nlaps such ashGijGfi, to the domain of dynamical quan-\ntities.\nThe spectral functions that characterize a local\nquantum quench typically show power-law behaviour,\n\u0018!\u00001+2\u0011, in the limit of small frequencies, where \u0011typ-\nically depends on \u0001 AO.2{5For the case of the X-ray edge\nsingularity, Hop\feld4gave a simple argument to explain\nthe relation between \u0001 AOand\u0011. We frame Hop\feld's\nargument in a more general setting and numerically il-\nlustrate the validity of the resulting generalized Hop-\n\feld rule (Eq. (25) below) for several nontrivial models.\nIn particular, we also analyse how this power-law be-\nhaviour is modi\fed at low frequencies when one adds to\nthe Hamiltonian an extra tunnelling term, that describes\ntransitions between the Hilbert spaces characterizing the\n\\initial\" and \\\fnal\" con\fgurations. This e\u000bect plays a\ncrucial role in understanding the abovementioned quan-\ntum phase transition for population switching.\nThe paper is organized as follows. In Sec. II we reviewarXiv:1108.5539v2  [cond-mat.str-el]  3 Sep 20112\nvarious consequences of AO in di\u000berent but related set-\ntings, and formulate the abovementioned generalization\nof Hop\feld's rule. In Sec. III we illustrate this rule for the\nspinless interacting resonant level model (IRLM), involv-\ning a single localized level interacting with the Fermi sea\nof a single lead. We consider this model without and with\ntunnelling, and study a quantum quench of the energy\nof its local level, focussing on signatures of AO in each\ncase. Finally, in Sec. IV and Sec. V we discuss population\nswitching without and with a charge sensor, respectively,\ncon\frming that if the sensor is su\u000eciently strongly cou-\npled, AO indeed does cause population switching to be-\ncome a sharp quantum phase transition. Section VI o\u000bers\nconcluding remarks and outlines prospective applications\nof the present analysis.\nII. VARIOUS CONSEQUENCES OF\nANDERSON ORTHOGONALITY\nIn this section we review various consequences of AO,\nin di\u000berent but related settings. We begin by recalling\ntwo well-known facts: \frst, the relation between the ex-\nponent \u0001 AOand the charge that is displaced due to the\nquantum quench, \u0001 ch; and second, the role of \u0001 AOin\ndetermining the asymptotic long-time power-law decay\nof correlation functions GX(t) involving an operator ^Xy\nthat connects the initial and \fnal ground state.\nThen we consider the spectral function AX(!) asso-\nciated withGX(t), which correspondingly shows asymp-\ntotic power-law behaviour, \u0018!\u00001+2\u0011, for small frequen-\ncies, where the exponent \u0011depends on \u0001 AO. We recall\nand generalize an argument due to Hop\feld, that extends\nthe relation between \u0011and \u0001 AOto composite local op-\nerators. Finally, we recapitulate how all these quantities\ncan be calculated using NRG.\nFor simplicity, we assume in most of this section that\nthe Fermi sea consists only of a single species of (spinless)\nelectrons.The generalization to several channels needed\nin subsequent sections (in particular for discussing PS),\nis straightforward and will be introduced later as needed.\nAlthough the concepts summarized in subsections II B\nto II E below apply quite generically to a wide range of\nimpurity models, for de\fniteness we will illustrate them\nby referring to a particularly simple example, to be called\nthe \\local charge model\" (LCM), which we de\fne next.\nA. Local charge model\nThe LCM describes a single spinless localized level,\nto be called dot level (alluding to a localized level in\na quantum dot), interacting with a single Fermi sea of\nspinless electrons [see Fig. 1(a)]:\n^HLCM(^nd) =U^nd^cy^c+X\n\"\"^cy\n\"^c\": (2)\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\n(g)Figure 1: (a) Cartoon of the Hamiltonian (2) for the LCM.\n(b) to (g) Cartoons of the occupation of the dot and a half-\n\flled lead, for U > 0, for several states discussed in the text.\n(b) and (c) give two equivalent depictions of the ground state\njG0iof^H0. (c) depicts the fact that jg0ican be written as\na superposition of the form j0icjQirest+j1icjQ\u00001irest, in-\ndicating complementary occupations of the \frst site and the\nrest of a half-\flled Wilson chain (de\fned in Sec. II E below).\nHerej0ic(which obeys ^ cj0ic= 0) andj1ic= ^cyj0icdescribe\nthe \frst site of the Wilson chain being empty or \flled, re-\nspectively; the charge in the rest of the Wilson chain is corre-\nspondingly distributed in such a way that both components of\nthe superposition have the same total charge, Q. (d) depicts\nthe ground statejG1iof^H1, indicating that charge on the dot\npushes charge in the lead away from the dot site. (e) shows\nthe e\u000bect of applying ^dytojG0i, the latter depicted according\nto (b). Similarly, (f) and (g) show the e\u000bect of applying ^ cy^dy\nor ^c^dytojG0i, the latter depicted according to (c). The dis-\nplaced charge \rowing inwards from in\fnity towards the dot\nas each of the states (e) to (g) evolves to the \fnal ground\nstatejG1iof (d) is \u0001 d<0, \u0001d\u00001<0 or \u0001d+ 1>0, re-\nspectively. Comparison of (f) and (g) with (e) shows average\ncharge di\u000berences of +1 and \u00001, respectively, in accord with\nthe Hop\feld-type argument summarized by Eq. (15).\nHere ^c\"and ^dare annihilation operators for Fermi sea\nstates and the dot state, respectively, ^ nd=^dy^dcounts the\nnumber of dot electrons, and ^ c\u0011^ (0)\u0011P\n\"^c\"destroys\na Fermi sea electron at the position of the dot. The\ninteraction is taken to be repulsive, U > 0. There is no\ntunnelling between dot and sea. Therefore, the Hilbert\nspace separates into two distinct sectors, in which the\nlocal charge operator ^ ndhas eigenvalues nd= 0 andnd=\n1, respectively. The Hamiltonians describing the Fermi3\nsea in the two distinct sectors are\n^H0=^HLCM(nd= 0) =X\n\"\"^cy\n\"^c\"; (3a)\n^H1(U) = ^HLCM(nd= 1) =X\n\"\"^cy\n\"^c\"+U^cy^c: (3b)\nWe will denote their respective ground states [illustrated\nin Figs. 1(b,c) and 1(d), respectively] by\njG0i=j0ijg0i;jG1i=j1ijg1i; (4)\nwherej0iandj1i=^dyj0idescribe the dot state with\ncharge 0 or 1, respectively, and jg0iandjg1ithe corre-\nsponding Fermi sea ground states.\nThe LCM contains all ingredients needed for AO, hence\nwe will repeatedly refer to it below as an explicit example\nof the general arguments to be presented. [Corresponding\nLCM passages will sometimes appear in square brackets,\nso as not to disrupt the general \row of the discussion.]\nExplicit numerical results for the LCM will be presented\nin Sec. III A below.\nB. AO and the displaced charge\nFor the ensuing discussions, it will be useful to distin-\nguish between two types of quenches, to be called type 1\nand 2, which we now discuss in turn.\nType 1 quench : For a type 1 quench, some parameter\nof the Hamiltonian is changed abruptly (e.g. by a sudden\nchange of gate voltage for one of the gates de\fning a\nquantum dot). Taking the LCM as an example, suppose\nthat the value of the interaction in the LCM is changed\nsuddenly from UtoU0for a \fxed local charge of nd= 1.\nThis corresponds to a type 1 quench with\n^Hi=^H1(U); ^Hf=^H1(U0); (5a)\njGii=j1ijg1;ii;jGfi=j1ijg1;fi: (5b)\nThe overlap of initial and \fnal ground states,\njhGijGfij=jhg1;ijg1;fij\u0018N\u00001\n2\u00012\nAO; (6)\nwill vanish in the thermodynamic limit due to AO, since\nthe two Fermi sea states jg1;iiandjg1;fifeel scattering\npotentials of di\u000berent strengths.\nIn his classic 1967 paper, Anderson showed that for this\ntype of situation the exponent \u0001 AOin Eq. (6) is equal to\nthe change in scattering phase shifts at the Fermi surface\ndivided by \u0019, in reaction to the change in the strength\nof the scattering potential. According to the Friedel sum\nrule,22{25the change in phase shifts divided by \u0019, in turn,\nis equal to the displaced charge \u0001ch(in units of e) that\n\rows inward from in\fnity into a large but \fnite volume\n(sayVlarge) surrounding the scattering site, in reaction to\nthe change in scattering potential, so that \u0001 AO= \u0001 ch.\nTo be explicit,\n\u0001ch\u0011hGfj^ntotjGfi\u0000hGij^ntotjGii; (7)where ^ntot\u0011^nsea+ndotcounts the total number of elec-\ntrons within Vlarge, with ^nseacounting the Fermi sea elec-\ntrons and ^ndotcounting the electrons on the dot. [For the\nLCM, ^ndot= ^nd.]\nThe relative sign between \u0001 AOand \u0001 ch(+ not\u0000) is\na matter of convention, which does not a\u000bect the orthog-\nonality exponent \u00012\nAO. Our convention,21which agrees\nwith standard usage,26is such that \u0001 AO>0 (or<0)\nif the change in local potential induces electrons to \row\ninward toward (outward away from) the scattering site.\nFor the LCM quench of Eq. (5) above, the initial and\n\fnal states have the same dot charge, nd= 1, hence\nthe displaced charge reduces to \u0001 ch\u0011hg1;fj^nseajg1;fi\u0000\nhg1;ij^nseajg1;ii. However, such a simpli\fcation will not\noccur for more complex impurity models involving tun-\nneling between dot and lead [of the form ( ^dy^c+ ^cy^d)], so\nthat the local charge is not conserved. Examples are the\ninteracting resonant level model [Eq. (36) below], or the\nsingle-impurity Anderson model [Eq. (47) below].\nFor such a model, consider a type 1 quench from ^Hi\nto^Hf, implemented by a sudden change in one or several\nmodel parameters, in analogy to Eq. (5a). Although the\ncorresponding ground states jGiiandjGfiwill no longer\nhave the simple factorized form of Eq. (5b), they will still\nexhibit AO as in Eq. (1). Moreover, the decay exponent is\nstill equal to the displaced charge, \u0001 AO= \u0001 ch, given by\nEq. (7). (For a NRG veri\fcation of this fact, see Ref. 21.)\nType 2 quench : For a type 2 quench, all model param-\neters are kept constant, but the system is switched sud-\ndenly between two dynamically disconnected sectors of\nHilbert space characterized by di\u000berent conserved quan-\ntum numbers. Taking again the LCM of Eq. (2) as an ex-\nample, suppose that the local charge is suddenly changed,\nsay fromnd= 0 to 1, while all model parameters are kept\nconstant. This corresponds to a type 2 quench with\n^Hi=^H0^Hf=^H1; (8a)\njGii=j0ijg0i;jGfi=j1ijg1i: (8b)\nA physical example of such a quench would be core level\nX-ray photoemission spectroscopy (XPS), where an inci-\ndent X-ray photon is absorbed by an atom in a crystal,\naccompanied by the ejection of a core electron from the\nmaterial.27This amounts to the sudden creation of a core\nhole, which subsequently interacts with the Fermi sea of\nmobile conduction electrons (but does not hybridize with\nthem). Thus, in this example ^ ndwould represent the hole\nnumber operator ^ nh=^hy^h.\nMore generally, a type 2 quench presupposes a Hamil-\ntonian ^H(^nx) that depends on a conserved charge, say ^ nx\n[such as ^ndfor the LCM], with eigenvalues nx[such as\nnd= 0 or 1]. The Hilbert space can then be decom-\nposed into distinct, dynamically disconnected sectors,\nlabelled by nxand governed by e\u000bective Hamiltonians\n^H(nx), whose ground states have the form jG(nx)i=\njnxijg(nx)i. A type 2 quench is induced by an operator,\nsay^Xy[such as ^dyfor the LCM], whose action changes\nthe conserved charge, thereby connecting two distinct4\nsectors, sayhn0\nxj^Xyjnxi= 1, with n0\nx6=nx. For such\na quench we make the identi\fcations\n^Hi=^H(nx) ^Hf=^H(n0\nx); (9a)\njGii=jnxijg(nx)i;jGfi=jn0\nxijg(n0\nx)i: (9b)\nThe overlaphGijGfi= 0 vanishes trivially, because\nhnxjn0\nxi= 0. However, de\fne\nj ii\u0011^XyjGii (10)\nto be the \\initial post-quench state\" obtained by the ac-\ntion of the charge switching operator ^Xyon the initial\nground state. [Fig. 1(e) illustrates this state for the LCM\nwith ^Xy=^dy.] Then the overlap\nOX\u0011jh ijGfij=jhg(nx)jg(n0\nx)ij\u0018N\u00001\n2\u00012\nX (11)\nagain shows AO, since it is equal to the overlap of two\nFermi sea ground states corresponding to di\u000berent local\ncharges. The corresponding exponent in Eq. (11) can\nagain be related to a displaced charge, \u0001 X= \u0001ch\nX, but\nnow the latter should compare the total charge within\nVlargedescribed by the states jGfiandj ii:\n\u0001ch\nX\u0011hGfj^ntotjGfi\u0000h ij^ntotj ii: (12)\n\u0001ch\nXcan be interpreted as the charge (in units of e) that\n\rows intoVlarge during the post-quench time evolution\nfromj iitojGfisubsequent to the action of ^Xy. To\nsimplify notation, we will often omit the superscript ch\ndistinguishing the displaced charge \u0001ch\nXfrom the AO ex-\nponent \u0001 X, since the two are equal in any case.\nComposite type 2 quench : Let us now consider a more\ncomplicated version of a type 2 quench, induced by a\ncomposite operator of the form ^Yy=^Cy^Xy. Here ^Xy\nswitches between disconnected sectors of Hilbert space\nas above, while ^Cydoes not; instead, ^Cyis assumed to\nbe a local operator which acts on the dot or in the Fermi\nsea at the location of the dot, but commutes with ^ nx.\nFor the LCM, an example would be ^Cy= ^cy, so that ^Yy\ncreates two electrons, one on the dot, one in the Fermi\nsea at the site of the dot.\nA physical realization hereof is furnished by the edge-\nray edge e\u000bect occurring in X-ray absorption spec-\ntroscopy (XAS), where an incident X-ray photon is ab-\nsorbed by an atom in a crystal, accompanied by the\ncreation of a core hole ( ^Xy=^hy) and the transfer of\na core electron into the conduction band of the metal\n(^Cy= ^cy).27Another example is the Kondo exciton dis-\ncussed in Refs. 15,16, where the absorption of a photon\nby a quantum dot is accompanied by the creation of an\nelectron-hole pair on the dot, described by ^Cy= ^eyand\n^Xy=^hy, respectively. In this example, the hole num-\nber ^nh=^hy^his conserved, but the dot electron number\n^ne= ^ey^eis not, since the Hamiltonian contains dot-lead\nhybridization terms of the form (^ ey^c+^cy^e) (see Refs. 15,16\nfor details).For a composite type 2 quench, the initial and \f-\nnal Hamiltonians and ground states are de\fned as in\nEqs. (9), but the post-quench initial state is given by\nj 0\nii\u0011^YyjGii=^Cyj ii: (13)\nIts overlap with the \fnal ground state jG0\nfito which it\nevolves in the long time limit has the form\nOY\u0011jh 0\nijG0\nfij=jhg(nx)j^Cjg0(n0\nx)ij\u0018N\u00001\n2\u00012\nY:(14)\nThe exponent \u0001 Yarising here is related to \u0001 Xand can\nbe found using the following argument, due to Hop\feld.4\nDue to the action of ^Cy, the statesj 0\niiandj iidescribe\ndi\u000berent amounts of initial post-quench charge within the\nvolumeVlarge. We will denote the di\u000berence by\n\u0001C\u0011h 0\nij^ntotj 0\nii\u0000h ij^ntotj ii: (15)\nFor example, if ^Cyis a local electron creation or an-\nnihilation operator, then \u0001 C= 1 or\u00001, respectively [as\nillustrated in Figs. 1(f) and (g)]. However, since an initial\ncharge surplus or de\fcit at the scattering site is compen-\nsated, in the long-time limit, by charges \rowing to or\nfrom in\fnity, the ground states jG0\nfiandjGfitowards\nwhichj 0\niiandj iievolve, respectively, will di\u000ber only\nby one Fermi sea electron at in\fnity, and hence for prac-\ntical purposes describe the same local physics. In par-\nticular, the charge within Vlarge is the same for both,\nhG0\nfj^ntotjG0\nfi=hGfj^ntotjGfi. Therefore, the total dis-\nplaced charge associated with the action of ^Yyis\n\u0001Y\u0011hG0\nfj^ntotjG0\nfi\u0000h 0\nij^ntotj 0\nii= \u0001X\u0000\u0001C;(16)\nwhere the second equality follows from Eqs. (15) and\n(12). The exponent governing the AO decay in Eq. (14)\nis thus given by Eq. (16). Since \u0001 Cis a trivally known\ninteger, knowledge of \u0001 Xfor a type 2 quench su\u000eces to\ndetermine the AO exponents \u0001 Yfor an entire family of\nrelated composite quenches.\nTo conclude this section, we note that a type 1 quench\ncan always be formulated as a type 2 quench, by intro-\nducing an auxiliary conserved degree of freedom (say ^ nh),\nwhose only purpose is to divide the Hilbert space into two\nsectors (labelled by nh= 0 or 1), within which some pa-\nrameters of the Hamiltonian take two di\u000berent values.\nFor example, if the quench involves changing UtoU0,\nthis can be modelled by replacing UbyU+ ^nh(U0\u0000U)\nin the Hamiltonian. For an example, see Sec. III C.\nC. AO and post-quench time evolution\nAfter a sudden change in the local Hamiltonian, AO\nalso a\u000bects the long-time limit of the subsequent time\nevolution, and hence the low-frequency behaviour of cor-\nresponding spectral functions. A prominent example is\noptical absorption,2{5,14{16for which AO leaves its im-\nprint in the shape of the absorption spectrum, by re-\nducing the probability for absorption. This is familiar5\nfrom the x-ray edge problem.3In particular, in the limit\nof absorption frequency !very close to (but above) the\nthreshold for absorption, the zero-temperature absorp-\ntion spectrum has a power-law form, with an exponent\nthat is in\ruenced by AO. Recent demonstrations of this\nfact can be found in studies, both theoretical14,15and\nexperimental,16of exciton creation in quantum dots via\noptical absorption, whereby an electron is excited from a\nvalence-band level to a conduction band level.\nIn this subsection, we will analyse the role of AO for\nthe time evolution after a type 2 quench of the form (8).\nWe consider the following generic situation: For t <0,\na system is in the ground state jGiiof the initial Hamil-\ntonian ^Hi(with ground state energy Ei), describing a\nFermi sea under the in\ruence of a local scattering poten-\ntial. Att= 0, a sudden change in the local potential\noccurs, described by the action the local operator ^Xy. It\nswitches sector nxton0\nx, yielding the post-quench initial\nstatej ii=^XyjGiiat timet= 0+, and switches the\nHamiltonian from ^Hito^Hf.\nThe subsequent dynamics can be characterized by the\ncorrelator\nGX(t)\u0011\u0000iei!0t\u0012(t)hGij^X(t)^XyjGii; (17)\nwhere ^X(t) =ei^Hit^Xe\u0000i^Hft, re\recting the fact that ^X\nswitches ^Hfto^Hi. The phase factor ei!0tis included\nfor later convenience, with !0to be speci\fed below [after\nEq. (22)].\nSince the Fermi sea adjusts in reaction to the sudden\nchange in local potential at t= 0, AO builds up and the\noverlap functionGX(t) decreases with time. It is known\nsince 1969 that in the long-time limit it decays in power-\nlaw fashion as2,4\nGX(t)\u0018t\u0000\u00012\nX; (18)\nwhere \u0001Xis the exponent governing the AO decay of\nOXin Eq. (11). This can be understood heuristically by\nexpanding Eq. (17) as\nie\u0000i(Ei+!0)tGX(t) =\u0012(t)h ije\u0000i^Hftj ii (19a)\n=\u0012(t)h ij i(t)i (19b)\n=\u0012(t)X\nne\u0000iEntjh ijnij2;(19c)\nwherej i(t)i=e\u0000i^Hftj iidescribes the time-evolution\nfort >0, andjniandEnrepresent a complete set of\neigenstates and eigenenergies of ^Hf. In the long-time\nlimit Eq. (19c) will be dominated by the ground state\njGfiof^Hf(with eigenenergy Ef), yielding a contribution\njh ijGfij2that scales as N\u0000\u00012\nX[by Eq. (11)]. Now, as\ntime increases, the e\u000bect of the local change in scattering\npotential is felt at increasing length scales L(t)\u0018vft,\nwithvfthe Fermi velocity; regarding jGfias the lowest\neigenstate of ^Hfin a box of size N\u0018L(t), the AO of\njh ijGfij2\u0018L(t)\u0000\u00012\nXimplies Eq. (18).For a composite type 2 quench induced by ^Yy=^Cy^Xy,\nwe can conclude by analogous arguments that\nGY(t)\u0018t\u0000\u00012\nY; (20)\nwhere \u0001Yis the displaced charge of Eq. (16).\nFor future reference, we also introduce the correlator\nGeq\nC(t)\u0011\u0000i\u0012(t)hGjei^Ht^Ce\u0000i^Ht^CyjGi\u0018it\u00002\u0011eq\nC(21)\nof an operator ^Cythat does not switch between dynam-\nically disconnected sectors, i.e. that commutes with ^ nx\n[examples of such operators are given in the discussion\nbefore Eq. (13) above]. Then Eq. (21) is a standard\nequilibrium correlator, with ^Hi=^Hf, in contrast to the\nquench correlatorGX(t) of Eq. (21), where ^Hi6=^Hf. For\nsuch an equilibrium correlator the decay exponent \u0011eq\nCis\ncalled the scaling dimension of ^Cy. A local operator ^Cy\nis relevant, marginal or irrelevant under renormalization\nif\u0011eq\nC<1, = 1 or>1, respectively.28\nD. AO and spectral functions\nNext we consider the spectral function corresponding\ntoGX(t),\nAX(!)\u0011 \u00001\n\u0019=0\n@1Z\n0dtei(!+i0+)tGX(t)1\nA (22a)\n=X\nnjhnj^XyjGiij2\u000e(!\u0000En+Ei+!0):(22b)\nIt evidently has the form of a golden-rule transition rate\nfor^Xy-induced transitions with excitation energy !+!0\nand is nonzero only for !above the threshold frequency\n!th= (Ef\u0000Ei)\u0000!0. For simplicity, we will here and\nhenceforth set !th= 0 by choosing !0=Ef\u0000Ei. Note\nthe sum ruleR\nd!A(!) =hGij^X^XyjGii, which can be\nused as consistency check for numerical calculations.\nEquation (18) implies that in the limit !!!th= 0,\nthe spectral function behaves as\nAX(!)\u0018!\u00001+2\u0011X; \u0011X=1\n2\u00012\nX: (23)\nThe de\fnition of \u0011Xis deliberately chosen such that\nEq. (23) parallels the form of the equilibrium spectral\nfunction corresponding to Geq\nC(t) of Eq. (21), namely\nAeq\nC(!)\u0018!\u00001+2\u0011eq\nC: (24)\nNow consider the spectral function AY(!) involving\nthe composite type 2 quench operator ^Yy=^Cy^Xy.\nEquations (20) and (16) immediately lead to the pre-\ndiction\nAY(!)\u0018!\u00001+2\u0011Y; \u0011Y=1\n2(\u0001X\u0000\u0001C)2; (25)6\nto be called the generalized Hop\feld rule , since the\nessence of the argument by which we have obtained it\nwas \frst formulated by Hop\feld.4\nA physical situation for which Eq. (25) is relevant is the\nedge-ray edge e\u000bect occurring in X-ray absorption spec-\ntroscopy (XAS). There we have ^Yy= ^cy^hy(as explained\nabove), and \u0001 C= 1. Thus Eq. (25) yields\nAhc(!)\u0018!\u00001+(\u0001h\u00001)2=!\u00002\u0001h+\u00012\nh (26)\nreproducing a well-established result for the X-ray edge\nabsorption spectrum [Ref. 4, p. 48; Ref. 5, Eq. (66)]. In\nthe literature,\u00002\u0001his often called the \\Mahan contri-\nbution\" to the exponent, and \u00012\nhthe AO contribution.\nSince \u0001h\u00141, one has 2\u0001 h>\u00012\nh, i.e. \\Mahan wins\", and\nAhc(!) diverges at small frequencies. For present pur-\nposes, though, it is perhaps somewhat more enlightening\nto adopt Hop\feld's point of view, stated in Eq. (25), ac-\ncording to which both terms, \u00002\u0001hand \u00012\nharise from\nthe AO exponent (\u0001 h\u00001)2.\nEquations (11), (23) and (25) will play a central role\nin this work. Their message is that the near-threshold\nbehaviour of spectral functions of the type de\fned in\nEq. (22) is governed by an AO exponent that can be ex-\ntracted from the overlap h ijGfibetween the initial post-\nquench statej iiand the ground state jGfito which it\nevolves in the long-time limit.\nTo conclude this section, we remark that the above\nanalysis generalizes straightforwardly to models involving\nseveral species or channels of electrons, say with index \u0016,\nprovided that the channel index is a conserved quantum\nnumber (i.e. no tunnelling between channels occurs).21\nThen the initial and \fnal ground states will be products\nof the ground states for each separate channel, so that\nEq. (1) generalizes to\njhGijGfij\u0018Y\n\u0016N\u00001\n2\u00012\nAO;\u0016\u0016: (27)\nAll power laws discussed above that involve \u00012\nAO(or\nquantities derived therefrom) in the exponent can be sim-\nilarly generalized by including appropriate products over\nchannels.\nE. AO exponents and NRG\nResults of the above type have been established analyt-\nically, in the pioneering papers from 1969, Refs. 2{5, only\nfor the simple yet paradigmatic case of the X-ray edge ef-\nfect. Nevertheless, Eq. (25) can be expected to hold for\na larger class of models, as long as the setting outlined\nabove applies. Indeed, it has recently been found to hold\nalso in the context of the Kondo exciton.14{16The pur-\npose of this work, therefore, is to establish the validity\nof the connections between Eqs. (11), (23) and (25) for a\nseries of models of increasing complexity. We shall do so\nnumerically using NRG, since for most of these modelsan analytical treatment along the lines of Refs. 2 and 5\nwould be exceedingly tedious, if not impossible. How-\never, the requisite numerical tools are available within\nNRG,29,30and have become very accurate quantitatively\ndue to recent methodological re\fnements.15,31,32\nNRG, developed in the context of quantum impurity\nmodels, o\u000bers a very direct way of evaluating the overlap,\nsince it allows both ground states jGiiandjGfito be\ncalculated explicitly. Models treatable by NRG have the\ngeneric form ^H=^HB+^Hd. Here\n^HB=ncX\n\u0016=1X\n\"\"^cy\n\"\u0016^c\"\u0016; (28)\ndescribes a free Fermi sea involving ncchannels of\nfermions, with constant density of states \u001aper channel\nand half-bandwidth D= 1=(2\u001a). (When representing\nnumerical results, energies will be measured in units of\nhalf-bandwidth by setting D= 1.) ^Hd, which may in-\nvolve interactions, describes local degrees of freedom and\ntheir coupling to the Fermi sea.\nWilson discretized the spectrum of ^H0on a logarithmic\ngrid of energies\u0006D\u0003\u0000k(with \u0003>1,k= 0;1;2;:::),\nthereby obtaining exponentially high resolution of low-\nenergy excitations. He then mapped the impurity model\nonto a semi-in\fnite \\Wilson tight-binding chain\" of sites\nk= 0 to1, with the impurity degrees of freedom coupled\nonly to site 0. To this end, he made a basis transforma-\ntion from the set of Fermi sea operators f^c\"\u0016gto a new\nsetf^fk\u0016g, with ^f0\u0016/^c\u0016\u0011 \u0016(0)\u0011P\n\"^c\"\u0016, chosen such\nthat they bring ^H0into the tridiagonal form\n^HB'ncX\n\u0016=11X\nk=1tk(^fy\nk\u0016^fk\u00001;\u0016+ h:c:); (29)\nwith hopping matrix elements tk/D\u0003\u0000k=2that decrease\nexponentially with site index kalong the chain. Because\nof this separation of energy scales, the Hamiltonian can\nbe diagonalized iteratively by solving a Wilson chain of\nlengthk(restricting the sum in Eq. (29) to the \frst k\nterms) and increasing kone site at a time. The number\nof kept states at each iteration will be denoted by Nk.\nFor a Wilson chain of length k, the e\u000bective level spac-\ning of its lowest-lying energy levels is set by the small-\nest hopping matrix element of the chain, namely \u0003\u0000k=2;\nsuch a Wilson chain thus represents a real space system\nof volumeVlarge\u0018\u0003k=2. Thus, the overlap between the\ntwo ground states of a Wilson chain of length kcan be\nexpressed as21\njkhGijGfikj\u0018\u0003\u0000k\n4\u00012\nAO\u0011e\u0000\u000bk; (30)\nwhere\u000b\u0011(log \u0003=4)\u00012\nAO. Explicit calculations show21\nthat an exponential decay of the form Eq. (30) applies for\nthe overlap between any two states jEiikandjEfikrep-\nresenting low-lying excitations w.r.t. jGiikandjGfikat\niterationk, respectively. More technically, khEijEfik\u00187\ne\u0000\u000bkholds wheneverjEiikandjEfikrepresent NRG\neigenstates with matching quantum numbers from the\nk-th NRG shell for ^Hiand ^Hf, respectively, and their\noverlap is calculated for increasing k. For multi-chain\nmodels, we note that channel-speci\fc exponents such as\n\u0001AO;\u0016[see Eq. (27)] can be calculated, if needed, by con-\nsidering Wilson chains with channel-dependent lengths.21\nWithin the framework of NRG, a consistency check is\navailable for the value of \u0001 AOextracted from Eq. (30):\n\u0001AOshould be equal to the displaced charge \u0001 chof\nEq. (7), which can also be calculated directly from NRG\nby calculating the expectation value of ^ ntotforjGii\nandjGfiindividually.21This check was successfully per-\nformed, for example, in Refs. 14 and 15, within the con-\ntext of the single impurity Anderson model; for a recent\nsystematic study, see Ref. 21. We have also performed\nthis check in the present work wherever it was feasible.\nWithin NRG, it is also possible to directly calculate\nspectral functions such as AX(!) of Eq. (22). To this\nend, one uses two separate NRG runs to calculate the\nground statejGiiof^Hiand an approximate but com-\nplete set of eigenstates jniof^Hf.31,32The Lehmann sum\nin Eq. (22) can then be evaluated explicitly,33,34while\nrepresenting the \u000e-functions occurring therein using a log-\nGaussian broadening scheme. To this end, we follow the\napproach of Ref. 34, which involves a broadening param-\neter\u001b. (The speci\fc choice of NRG parameters \u0003, Nk\nand\u001bused for spectral data shown below will be spec-\ni\fed in the legends of the corresponding \fgures.) That\nthis approach is capable of yielding spectral functions\nwhose asymptotic behaviour shows power-law behaviour\ncharacteristic of AO has been demonstrated recently in\nthe context of the Kondo exciton problem.14{16In the ex-\namples to be discussed below, we will compare the power-\nlaw exponents extracted from the asymptotic behaviour\nof such spectral functions to the values expected from\nAO, thus checking relations such as Eq. (23) for AX(!)\nand Eq. (25) forAY(!).\nIII. INTERACTING RESONANT LEVEL\nMODEL\nIn this section we consider the e\u000bect of AO on dy-\nnamical quantities in the context of the spinless inter-\nacting resonant level model (IRLM).11,35(The e\u000bects of\nAO for some static properties of this model were studied\nin Ref. 36.) The purpose of this exercise is to illustrate\nseveral e\u000bects that will be found to arise also for more\ncomplex models considered in subsequent sections. The\nIRLM involves a single localized level, to be called dot\nlevel (alluding to localized levels in a quantum dot), in-\nteracting with and tunnel-coupled to a single Fermi sea.\nWe consider \frst the case without tunnelling, in which\ncase the IRLM reduces to the LCM introduced in Sec. II\nabove, where adding an electron to the dot at time t= 0\nconstitutes a type 2 quench. This leads to AO between\nthe initial and \fnal ground states, and corresponding\n−0.5−0.250∆d,∆chd∆d\n∆ch\nd\n−tan−1(πρU )/π(a)\n0 1 2 3 4 501\nU /Dηexp spec\nηd\nηdc\nηdc†(b)\n10−1210−810−410010−1010−51001051010\nω /UU· A(ω)\nAd\nAdc\nAdc†\n(Λ=2 , Nk=1024 , σ=0.6)U= 1(c)Figure 2: (Color online) Numerical results for the LCM of\nEq. (2), for the type 2 quench of Eq. (8), whose initial, \fnal\nand post-quench initial states jGii,jGfiandj iiare depicted\nin Figs. 1(b,c), 1(d) and 1(e-g), respectively. (a) Comparison\nof the decay exponent \u0001 dobtained from Eq. (31) (crosses)\nwith the displaced charge \u0001ch\ndfrom Eq. (32) (pluses), for a\nnumber of di\u000berent values of U. The two values agree very\nwell (they di\u000ber by less than 0 :1%), also with the analytic\nprediction Eq. (33) (solid line). As expected, \u0001 d!\u0000 1=2\nforU! 1 . (b) Comparison of two ways of determining\nthe AO exponents \u0011that govern the low-energy asymptotic\nbehaviourA\u0018!\u00001+2\u0011of the spectral functions of Eqs. (35),\nrelated to Figs. 1(e-g): exponents obtained by \ftting a power\nlaw to the corresponding spectra [shown in (c)] are shown\nas crosses (marked \\spec\", for \\spectra\"); the corresponding\nexponents expected from Eq. (35), using the results of (a) for\n\u0001d, are shown as dots (marked \\exp\" for \\expected\"). We\n\fnd a maximal deviation of less than 1%. Here and in all\nsimilar \fgures below, the dashed lines are only guides to the\neye. (c) Asymptotic low-frequency dependence of the spectra\nEqs. (35), for U= 1, on a double logarithmic plot, allowing\nthe corresponding exponents \u0011to be extracted.\nnontrivial AO power laws, !\u00001+2\u0011, in spectral functions.\nWe then turn on tunnelling, which connects the sectors\nof Hilbert space for which the dot is empty or \flled, and\nhence counteracts AO. Correspondingly, the power-laws\nget modi\fed at frequencies smaller than the renormal-\nized level width, !.\u0000ren, where the AO behaviour is\nreplaced by simple Fermi liquid behaviour; the e\u000bects of\nAO do survive, however, in a regime of intermediate fre-\nquencies, \u0000 ren<!<D . Finally, we consider quenches of\nthe position of the dot level, in which case AO reemerges.8\nA. Without tunnelling: LCM\nIn this subsection we present numerical results for\nthe IRLM without tunnelling, corresponding to the lo-\ncal charge model of Eq. (2), depicted in Fig. 1(a). We\nconsider the type 2 quench of Eq. (8), with ^Xy=^dy.\nThe initial and \fnal ground states jGiiandjGfiare il-\nlustrated in Figs. 1(b,c) and 1(d), respectively, and the\npost-quench initial state j ii=^dyjGiiin Fig. 1(e). With\nthese choices the overlap jh ijGfijof Eq. (11) becomes\nOd\u0011jhG0j^djG1ij=jhg0jg1ij\u0018N\u00001\n2\u00012\nd: (31)\nThe corresponding displaced charge obtained from\nEq. (12) is\n\u0001ch\nd=hg1j^nseajg1i\u0000hg0j^nseajg0i; (32)\nsincejGfiandj iidescribe the same dot charge, nd= 1.\nWe used NRG to calculate the overlap Odof Eq. (31)\nand extract the exponent \u0001 dfrom its exponential decay\nwith Wilson chain length [Eq. (30)], for several values of\nU. As consistency check, we also calculated the displaced\ncharge \u0001ch\nd[Eq. (32)]. As shown in Fig. 2(a), the results\nfor \u0001d(crosses) and \u0001ch\nd(pluses) agree very well. The\ndisplaced charge \u0001ch\ndis<0, since the repulsive interac-\ntion pushes charge away from the local site. Its magni-\ntudej\u0001ch\ndjdepends on the interaction strength: as Uis\nincreased from 0 to 1, the displaced charge goes from 0\nto\u00001\n2, re\recting the complete depletion of the initially\nhalf-\flled Wilson chain site directly adjacent to the dot\nsite [compare Figs. 1(b) and 1(d)]. Figure 2(a) shows\nthat the numerical results for \u0001 dand \u0001ch\nd(symbols) also\nagree with the analytical result (solid line) obtained for\nthe phase shift obtained from elementary scattering the-\nory [see e.g. Ref. 11, Eq. (25.29)],\n\u0001d=\u00001\n\u0019tan\u00001(\u0019\u001aU); (33)\nwith\u001athe density of states in the Fermi sea (cf. Sec. II E).\nTo study the in\ruence of AO on dynamical quantities,\nwe consider simple and composite type 2 quenches in-\nduced by acting on the initial ground state jGii=jG0i\nwith the operators\n^Xy=^dy; ^Yy\n1= ^cy^dy; ^Yy\n2= ^c^dy: (34)\nAll three operators describe transitions between the nd=\n0 and 1 sectors. The analysis of Sec. II D applies directly,\nwith the identi\fcations ^Hi=^H0and ^Hf=^H1, while\n\u0001C=\u00061 for ^Yy\n1or^Yy\n2, respectively [see Figs. 1(e-g)]. In\nparticular, Eqs. (23) and (25) imply:\nAd(!)\u0018!\u00001+2\u0011d\u0011d=1\n2\u00012\nd; (35a)\nAdc(!)\u0018!\u00001+2\u0011dc\u0011dc=1\n2(\u0001d\u00001)2;(35b)\nAdcy(!)\u0018!\u00001+2\u0011dcy\u0011dcy=1\n2(\u0001d+ 1)2:(35c)\nUsing NRG, we calculated these three spectra for several\nvalues ofU(cf. Fig. 2(c)). In the limit of small !, the\n(a)\n10−810−4100104100102104\nU /DΓren/ΓΓ 1/πχ cEq.(37)\n10−8\n10−6\n10−4\n10−2(b)Figure 3: (Color online) (a) Cartoon of the Hamiltonian\n(36) for the IRLM. (b) The renormalized level width \u0000 ren,\ncalculated via the dot's charge susceptibility, 1 =\u0019\u001fc,36(dots)\nor via Eq. (37) (solid line), shown as a function of Ufor\"d= 0\nand several values of \u0000. As Uincreases from 0, \u0000 ren=\u0000 begins\nto di\u000ber signi\fcantly from its initial value, namely 1, only\nonceUbecomes comparable to the band-width, reaching its\nmaximal value (\u0000 =D)\u00001=2forU\u001dD.\nspectra show clear power law behaviour, !\u00001+2\u0011. The\nexponents\u0011d,\u0011dc,\u0011dcyextracted from these spectra are\nshown in Fig. 2(b) (crosses, marked \\spec\", for \\spec-\ntra\"). They agree well with the values expected (dots,\nmarked \\exp\", for \\expected\") from Eqs. (35), based on\nthe value for \u0001 dextracted from Eq. (31). Thus, all ways\nof determining \u0001 dare completely consistent, con\frming\nthe validity of the above analysis.\nB. With tunnelling: IRLM\nThe previous subsection focused on a switch between\ntwo sectors of the Hilbert space, with nd= 0 andnd= 1,\nthat were not coupled dynamically, but governed instead\nby two distinct Hamiltonians, ^Hiand ^Hf. In the present\nsubsection, we consider the case that the sectors with\nnd= 0 andnd= 1 are coupled by tunnelling between dot\nand lead, so that the notion of an initial and \fnal Hamil-\ntonian, acting in decoupled sectors of Hilbert space, does\nnot apply. The dynamics is governed instead by the single\nHamiltonian ^Hi=^Hf=^HIRLM , given by [see Fig. 3(a)]\n^HIRLM =\"d^dy^d+U(^dy^d\u00001=2)(^cy^c\u00001=2) (36)\n+X\n\"\"^cy\n\"^c\"+s\n\u0000\n\u0019\u001aX\n\"(^dy^c\"+ ^cy\n\"^d):\nWe assume, here and in all later settings, that the hy-\nbridization of the dot level with the Fermi sea states is \"-\nindependent, with \u0000 being the bare width of the dot level.\nHere, in contrast to the local charge model of Eq. (2), the\ninteraction term is taken to be particle-hole symmetric,\nso that the model is particle-hole symmetric for \"d= 0.9\nThe presence of the interaction, U, is known to e\u000bec-\ntively modify the level width,35,36both by reducing the\ndensity of states of the leads near the dot, and by induc-\ning AO in the leads when the dot occupancy changes.\nThe precise interplay between these e\u000bects can be quite\nintricate and was studied in Ref. 36. A quantitative anal-\nysis can be performed by de\fning a renormalized level\nwidth in terms of the charge susceptibility, \u0000 ren\u00111=\u0019\u001fc.\nAt the point of particle-hole (ph) symmetry ( \"d= 0), an\nanalytic formula for the latter is available,35\n\u0000ren=D= (\u0000=D)1=(2\u0000(1+\u0001ph\nd)2); (37)\nwhere \u0001ph\ndis given by\n\u0001ph\nd=\u00002\n\u0019tan\u00001(\u0019\u001aU= 2): (38)\n\u0001ph\ndcan be interpreted as the change in scattering phase\nshift that a system with \u0000 = 0, \"d= 0 experiences if the\nlocal occupancy is changed abruptly from nd= 0 to 1.\nThe form of Eq. (38) is analogous to Eq. (33) for \u0001 d,\nwith two di\u000berences: since the \fnal scattering potentials\nbeing compared have amplitude \u0000U=2 andU=2 (instead\nof 0 andU), the argument of tan\u00001has an extra factor\nof 1=2, and there is an extra prefactor of 2.\nThe dependence of \u0000 renonUis illustrated in Fig. 3(b),\nwhich shows good agreement between the NRG results\nfor 1=\u0019\u001fc(dots) and the analytic formula (37) (lines).\nForUmuch smaller than the bandwidth D, \u0000ren=\u0000 is es-\nsentially equal to 1; it strongly increases once Ubecomes\nof the order D, and saturates to (\u0000 =D)\u00001=2forU\u001dD.\nLet us now consider the equilibrium spectral functions\nfor the operators of Eq. (34), Aeq\nd,Aeq\ndcandAeq\ndcy. They\nare de\fned as in Eq. (22) but with ^Hf=^Hi, because for\nthe IRLM, where ndis not conserved, none of these oper-\nators induces a quench. Therefore, the !!0 behaviour\nof their correlators is expected (and indeed found) to be\nindependent of AO. However, quite remarkably, traces\nof AO do show up in an intermediate frequency regime,\n!\u0003< ! < D , where 1=!\u0003corresponds to the time scale\nwithin which charge equilibration takes place. Below the\nenergy scale !\u0003the quantum impurity becomes strongly\ncorrelated with the Fermi sea and for the present model\nwe have!\u0003'\u0000ren. Let us therefore discuss the two\nregimes,!below or above !\u0003, separately.\nIn the regime !\u001c!\u0003, the spectral functions are found\nto have the following asymptotic form Aeq\u0018!\u00001+2\u0011eq\n[cf. Figs. 4(a-c)]:\nAeq\nd(!)\u0018!0; \u0011eq\nd= 1=2; (39a)\nAeq\ndc(!)\u0018!3; \u0011eq\ndc= 2; (39b)\nAeq\ndcy(!)\u0018!3; \u0011eq\ndcy= 2: (39c)\nThe exponents arising here can be understood analyti-\ncally using elementary, though not entirely trivial argu-\nments, based on the fact that the lowest-lying excitations\nof this model have Fermi liquid properties. We refer the\nreader to the Appendix for a detailed analysis.\n10−2010−1510−1010−5100\nΓ = 10−8Aeq\nd(ω)/Aeq\nd(Γren)(a)\n10−1510−1010−51001051010Aeq\ndc(ω)/Aeq\ndc(Γren)(b)\n10−210010210410610810−2010−1510−1010−5100\nω /ΓrenAeq\ndc†(ω)/Aeq\ndc†(Γren)\n(Λ=2 , Nk=4096 , σ=0.6)(c)U Γren\n0 10−8\n10−810−8\n10−610−8\n10−410−8\n10−310−8\n10−210−8\n10−15·10−8\n1 10−5\n37·10−5\n59·10−5\n10−810−610−410−2100−0.5012\nηeq\ndηeq\ndcηeq\ndc† η0\ndη/prime\nd η0\ndcη0\ndc†\nexp\nspec\nU /Dη(d)Figure 4: (Color online) (a)-(c) The equilibrium spectral\nfunctionsAeq\nd(!),Aeq\ndc(!) andAeq\ndcy(!) for the IRLM, showing\na crossover from trivial power laws, !\u00001+2\u0011eq, for!<!\u0003, to\nAO power laws, !\u00001+2\u00110, for!\u0003<!<D , with the crossover\nfrequency!\u0003given by \u0000 ren. (d) Comparison of the exponents\n\u0011eq(triangles) and \u00110or\u00110\nd(crosses) extracted from the spec-\ntra shown in (a-c), with the values expected from Eqs. (39) for\n\u0011eq(solid lines), and from Eqs. (41) for \u00110or from Eq. (40)\nfor\u00110\nd(dashed lines with dots), for several values of U. In\n(a), arrows indicate the scale \u0016 !\u0003that separates the regimes\n!\u0003<!< \u0016!\u0003and \u0016!\u0003<!<D , whereAeq\ndscales according to\nEqs. (40) or (41a), respectively.\nNow consider the regime !\u0003<!<D . As shown in the\ncorresponding regime of !=\u0000ren>1 in Figs. 4(a-c), each\nof the equilibrium spectral functions Aeq\nd,Aeq\ndcandAeq\ndcy,\nexhibits another, di\u000berent power-law there. For Aeq\ndwe\nactually \fnd that within this regime two di\u000berent power-\nlaws can be discerned: First, in a regime !\u0003< ! < \u0016!\u0003\nwe \fnd,\nAeq\nd(!)\u0018!\u00001+2\u00110\nd; \u00110\nd=\u00001\n2\u0000\u0001ph\nd+ (\u0001ph\nd)2;(40)10\nwhere \u0001ph\ndis given by Eq. (38). The exponent \u00110\ndcor-\nresponds to the leading correction for weak interactions\n(U=D\u001c1) to the pure Lorentzian decay of the spectral\nfunction of the d-level (see Eq. (A.2) of the Appendix),\nas can be shown using methods discussed in Refs. 37 and\n38. The scale \u0016 !\u0003that sets the upper limit for this be-\nhaviour is marked by arrows in Fig. 4(a) and decreases\nwith increasing U=D . ForU=D su\u000eciently small that \u0016 !\u0003\nlies far below the bandwidth D, we \fnd a second power\nlaw within the window \u0016 !\u0003<!<D , namely\nAeq\nd(!)\u0018!\u00001+2\u00110\nd; \u00110\nd=1\n2(\u0001ph\nd)2: (41a)\nFor the other two spectral functions we \fnd throughout\nthe regime !\u0003<!<D :\nAeq\ndc(!)\u0018!\u00001+2\u00110\ndc; \u00110\ndc=1\n2(\u0001ph\nd\u00001)2;(41b)\nAeq\ndcy(!)\u0018!\u00001+2\u00110\ndcy; \u00110\ndcy=1\n2(\u0001ph\nd+ 1)2:(41c)\nRemarkably, Eqs. (41) have the same form as Eqs. (35),\nexcept that \u0001 dis replaced by \u0001ph\ndof Eq. (38), i.e. by\nthe AO exponent involved in abruptly changing the lo-\ncal occupancy from 0 to 1 (in the absence of tunnelling).\nThat this exponent should emerge is natural, since the\ncorresponding correlators Gd,GdcandGdcyall involve an\noperatordythat places an electron on the dot at time\nt= 0. Although the dot occupancy nd(t) will relax\nback to its initial value ni\ndin the long time limit, this\nrequires times t\u001d1=!\u0003. In contrast, the lead electrons\nreact to the change in local charge on the much shorter\ntime scale 1 =D. Thus, in the window of intermediate\ntimes, 1=D\u001ct\u001c1=!\u0003, corresponding to frequencies\n!\u0003\u001c!\u001cD, the situation is similar to that of the pre-\nvious subsection, where we had \u0000 = 0 and a change in\ndot occupancy from 0 to 1 induced changes in the lead\nphase shifts, accompanied by AO. Thus, the exponents\n\u00110arising in Eq. (41) can be identi\fed as the (equilib-\nrium) scaling dimensions of the corresponding operators\ncalculated in the absence of tunnelling (which is why we\nuse a superscript 0 on such exponents, here and below).\nThis explains the similarity between the behaviour de-\nscribed by Eqs. (41) and Eqs. (35). Note that the scaling\ndimension\u00110\ndcy[Eq. (41c)] of the tunnelling operators ^d^cy\nand ^cy^dsatisfy 0\u0014\u00110\ndcy\u00141=2 [since for U > 0, we have\n\u00001\u0014\u0001ph\nd\u00140, by Eq. (38)], thus tunnelling is always\nrelevant for this model.\nWe conclude this subsection with a comment on the\nfact thatAeq\nd(!) crosses over from non-AO behaviour\n[Eq. (40)] to AO behaviour [Eq. (41a)] as U=D is in-\ncreased past 1. AO behaviour is absent for U=D\u001c1\nbecause this situation corresponds essentially to a non-\ninteracting resonant-level model, for which Geq\nd(t) does\nnotshow power-law behaviour of the type assumed in\nEq. (18); instead it decays exponentially ( \u0018e\u0000\u0000t), caus-\ning the spectral function Aeq\nd(!) to have an essentially\nLorentzian form. Equation (40) is the large-frequency\nlimit of the latter, but including the leading corrections\ninU=D , calculated using methods discussed in Refs. 37\n(a)\n00.250.50.75\n∆h\n∆ch\nh∆h,∆chhΓ = 10−6\nU= 0.1\nΓren/Γ = 3 .1(b)\n0 1 2 3012\nW /Γrenηexp spec\nηhd\nηhdc\nηhdc†(c)Figure 5: (Color online) (a) Cartoon of the quench which\noccurs when an electron-hole pair is created at time t= 0,\nsee Eq. (43). (The cartoon depicts the situation relevant for\nexciton creation by absorption of a photon, which excited an\nelectron from a valence-band to a conduction band level of a\nsemiconducting quantum dot.) (b) The exponent \u0001 h[from\nEq. (11)] and the displaced charge \u0001ch\nh[from Eq. (12)], for\nthe quench of Eq. (42), as function of the quench range W.\n(c) Corresponding values of the AO exponents \u0011hd,\u0011hdcand\n\u0011hdcy, extracted from the asymptotic behaviour !\u00001+2\u0011of\nspectral functions (crosses), or as expected from Eqs. (46)\n(dots). Typically, relative errors are less than 1%.\nand 38. However, once U=D becomes &1, AO does be-\ngin to matter, implying a regime of power-law decay for\nGeq\nd(t) on intermediate time scales, leading to Eq. (41a)\nforAeq\nd(!).\nC. Quantum quench of level position\nIn the previous subsection we emphasized the impor-\ntance of the scale !\u0003, which separates the low- and\nintermediate-frequency regimes, showing trivial expo-\nnents or AO exponents, respectively. It is instructive\nto study the role of the scale !\u0003in a slightly di\u000berent\nbut related context, namely quench spectral functions\ninvolving a quantum quench of the level position. This\nwill shed further light on the AO between states with\ndi\u000berent local level occupancies.\nConcretely, we consider initial and \fnal Hamiltonians\nthat both are of the form Eq. (36), but with initial and\n\fnal level positions that are symmetrically spaced on op-\nposite sides of the model's symmetry point at \"d= 0:\n\"i\nd=W=2quench\u0000!\"f\nd=\u0000W=2: (42)11\n10−310−210−1100∆hU Γren/Γ\n0 1\n10−2 1\n1 2·102\n3 7·102\n5 9·102Γ = 10−6(a)\n10−210−110010110210300.51\nW /Γrenndnf\nd\nni\nd(b)\n10−610−410−210010100.51\nU/D∆max\nhexp calc(c)\nFigure 6: (Color online) (a) The AO exponent \u0001 h[extracted\naccording to Eq. (11)] as function of W=\u0000ren, for several\nvalues ofU. ForW=\u0000ren\u001d1, \u0001happroaches its maxi-\nmal value \u0001max\nh. AsWis reduced below \u0000 ren, \u0001hdrops\nbelow its maximal value and decreases with W[linearly so\nforW=\u0000ren\u001c1]. (b) The \fnal and initial occupancies nf\nd\n(squares, upper curves) and ni\nd(circles, lower curves) as func-\ntions ofW=\u0000ren, for the same values of U[same color code\nas in (a)]. (c) The maximal value \u0001max\nhof the AO exponent\n\u0001h, extracted from the W=\u0000ren\u001d1 regime of (a) (crosses,\n\\calc\"), or expected from Eq. (44) (solid line, \\exp\"); the\nrelative deviations are well below 1%.\nAlthough this is an example of a type 1 quench, it will be\nconvenient (mainly for notational reasons) to reformulate\nthis situation as a type 2 quench. To this end we use the\nHamiltonian\n^H=W(1=2\u0000^nh)^nd+U(^nd\u00001=2)(^cy^c\u00001=2)\n+X\n\"\"^cy\n\"^c\"+s\n\u0000\n\u0019\u001aX\n\"(^dy^c\"+ ^cy\n\"^d); (43)\nwhere we have introduced an auxiliary degree of freedom,\ncalled \\hole\" (in analogy to the role of holes in exciton\ncreation by optical absorption14{16), with hole counting\noperator ^nh=^hy^h. The hole has no dynamics; its only\nrole is to distinguish two distinct sectors of Hilbert space,\nin which the dynamics is described by ^Hior^Hf, with hole\nnumbernh= 0 or 1, respectively [see Fig. 5(a)]. The type\n2 quench that switches between these sectors is induced\nby^Xy=^hy. The overlapOh\u0018N\u00001\n2\u00012\nhbetween the\ninitial and \fnal ground states is characterized by an AO\nexponent \u0001 h[Eq. (11)] that is equal to the charge \u0001ch\nh\ndisplaced by the quench [Eq. (12)].\nThe magnitude of \u0001 hincreases with the range Wof the\nquench, as shown in Fig. 5(b) (linear scale) and Fig. 6(a)\n(log-log scale). Note, in particular, that the scale onwhich the quenching range, W, needs to change in order\nfor the AO exponents to change signi\fcantly, is given by\n\u0000ren. This is natural: when W\u001d\u0000ren, the two states\njGiiandjGficonnected by the quench describe dots with\nstrongly di\u000berent occupancies, ni\nd'0 vs.nf\nd'1, see\nFig. 6(b). Hence the AO [Eq. (11)] of the corresponding\nFermi seas will be strong. The maximum possible value\nof the exponent \u0001 his\n\u0001max\nh= 1 + \u0001ph\nd; (44)\nwith \u0001ph\nd(U)<0 given by Eq. (38). The \frst term simply\ngives theU!1 value of the change in dot occupancy\ninduced by the quench, namely 1; the second term re\rects\nthe reaction of the Fermi sea to this change, cf. Sec. III B.\nFollowing the arguments of Sec. II D, the nonequilib-\nrium spectral functions AY(!), de\fned for\n^Yy\n1=^dy^hy; ^Yy\n2= ^cy^dy^hy; ^Yy\n3= ^c^dy^hy;(45)\nare expected to show the following AO behaviour for\n!!0:\nAhd(!)\u0018!\u00001+2\u0011hd\u0011hd=1\n2(\u0001h\u00001)2;(46a)\nAhdc(!)\u0018!\u00001+2\u0011hdc\u0011hdc=1\n2(\u0001h\u00002)2;(46b)\nAhdcy(!)\u0018!\u00001+2\u0011hdcy\u0011hdcy=1\n2\u00012\nh: (46c)\nThe reason for the speci\fc form of the exponents is that\nfor the correlators Ghd,GhdcorGhdcy, att= 0 the local\ncharge (on the d-level or in the Fermi sea) is increased\nby one, two or zero, respectively [i.e. \u0001 C= 1, 2 or 0 in\nEq. (16)]. Figure 5(c) shows that the exponents (crosses)\nextracted from the asymptotic behaviour AY(!) are in-\ndeed in good agreement with values expected (dots) from\nEqs. (46).\nIV. POPULATION SWITCHING WITHOUT\nSENSOR\nThe models investigated so far served as testing ground\nfor the in\ruence of AO on various types of spectral func-\ntions. The following two sections have the concrete mo-\ntivation to clarify the role of AO in the context of quan-\ntum dot models that display the phenomenon of popula-\ntion switching (PS).17,18,39{44In such models, a quantum\ndot, tunnel-coupled to leads, contains levels of di\u000berent\nwidths and is capacitively coupled to a gate voltage that\nshifts the levels energy relative to the Fermi level of the\nleads. Under suitable conditions, an (adiabatic) sweep of\nthe gate voltage induces an inversion in the population\nof these levels (a so-called population switch), implying\na change in the local potential seen by the Fermi seas in\nthe leads. Goldstein, Berkovits and Gefen (GBG) have\nargued in Ref. 17,18 that in this context AO can play an\nimportant role. In particular, they pointed out that for\na model involving a third lead acting as a charge sensor,\nthe e\u000bects of AO can be enhanced to such an extent that12\npopulation switching becomes abrupt, i.e. turns into a\nphase transition. Our goal is to elucidate the in\ruence\nof AO by using the tools developed above in the context\nof the IRLM.\nIn the present section, we will study population switch-\ning in a two-lead model (without charge sensor), which is\nequivalent to an anisotropic Kondo model.17,18,45{48The\ncorresponding Kondo temperature, TK, sets the width of\nthe population switch as function of gate voltage. We\ncalculate the spectral function Aeq\nY(!) of the pseudospin-\n\rip operator and show that TKalso acts as the crossover\nscale!\u0003that separates a low-frequency regime showing\nFermi-liquid power laws from an intermediate-frequency\nregime revealing AO exponents. We investigate the ori-\ngin of the latter by a quantum quench analysis similar\nto that of Sec. III C above. In the following section, we\nwill generalize the model by adding a charge sensor and\nanalyse how this enhances the e\u000bects of AO.\nA. Width of switching regime\nWe consider a model involving two single-level dots\n(\u0016= L;R) and for convenience choose their level ener-\ngies\"dto be equal, so that the PS always occurs at the\nparticle-hole symmetric point, \"d=\u0000U=2. (Note that PS\noccurs also for nondegenerate levels, as long as their level\nspacing is smaller than the di\u000berence of their level widths\n\u0000\u0016.) The levels have an electrostatic coupling U > 0 and\nare each tunnel-coupled to its own lead [see Fig. 7(a)]:\n^HSIAM =X\n\u0016\"d^dy\n\u0016^d\u0016+U^dy\nL^dL^dy\nR^dR (47)\n+X\n\"\u0016\"^cy\n\"\u0016^c\"\u0016+X\n\u0016s\n\u0000\u0016\n\u0019\u001aX\n\"(^dy\n\u0016^c\"\u0016+ ^cy\n\"\u0016^d\u0016);\n(We use notation analogous to that of Sec. III B.) We\nchoose the level widths to be strongly asymmetric and\nwill use a \fxed value of their ratio, \u0000 R=\u0000L= 20, through-\nout. The model thus has the form of a spin-asymmetric\nsingle-impurity Anderson model (SIAM), where \u0016acts as\npseudospin index.\nAs illustrated in Fig. 7(b), this model shows PS when\n\"dis decreased past \"d=\u0000U=2 (the particle-hole sym-\nmetric point): as this \\switching point\" is crossed, the\noccupancy of the broad level (dashed lines) changes from\nnear 1 to near 0, and vice versa for the narrow level (solid\nlines). We de\fne the width of the switching regime, WPS,\nas the di\u000berence,\nWPS\u0011\"d(nR+)\u0000\"d(nR\u0000); (48)\nbetween those two values of \"d, located symmetrically on\neither side of the switching point, at which the occupation\nof the right level is nR+\u00113\n4nmax\nR+(>1\n2) ornR\u0000= 1\u0000\nnR+(<1\n2), respectively, where nmax\nR+is the largest value\nreached by nRfor\"d>\u0000U=2, to the right of the PS.\n(a)\n−2 −1 0 101\nεd/UnΓR/U nLnR\n0.2\n0.15\n0.1\nΓR/ΓL= 20\nU= 0.2(b)\n−0.501 −0.49901\n0.05 0.1 0.15 0.210−1210−910−610−3\nΓR/Ub0WPS, TK\n0.6b0WPS/U\nTK/U\n(1/χs)/UU= 0.2\nΓR/ΓL= 20(c)Figure 7: (Color online) (a) Cartoon of the Hamiltonian (47)\nfor the asymmetric SIAM. (b) The occupations nL(solid lines)\nandnR(dashed lines) of the left and right level, respectively,\nas functions of \"d, for several values of \u0000 R, at a \fxed ratio of\n\u0000R=\u0000L= 20. As\"dis lowered past the particle-hole symmet-\nric point at \"d=\u0000U=2, population switching occurs, with nR\nchanging from near 1 to near 0, and vice versa for nL. Inset:\nzoom into the switching region around \"d=\u0000U=2, showing\nthat the switch is continuous (as function of \"d) even though\nthe switching region becomes narrower for decreasing \u0000 R. (c)\nComparison of b0WPS[from Eqs. (48) and (51), crosses], TK\n[from Eq. (52), solid], and the inverse pseudospin suscepti-\nbility 1=\u001fs(pluses). All three quantities evidently decrease\nsimilarly with decreasing \u0000 R=U.\nFigure 7(b) and its inset show that the width of the\nswitching regime decreases with decreasing \u0000 \u0016, without,\nhowever, dropping to zero as long as the level widths\nare nonzero. This behaviour can be understood as\nfollows.17,18,45{48In the vicinity of the particle-hole sym-\nmetric point, only two local charge con\fgurations are rel-\nevant, namely those with occupancies ( nL;nR) equal to\n(0;1) or (1;0). The spin-asymmetric SIAM can thus be\nmapped onto an anisotropic Kondo model by a Schrie\u000ber-\nWol\u000b transformation. This leads to an anisotropic pseu-\ndospin exchange interaction of the form\n^Hexch = (UL+UR)(^dy\nL^dL\u0000^dy\nR^dR)(^cy\nL^cL\u0000^cy\nR^cR)\n+2p\nULUR(^cL^cy\nR^dy\nL^dR+ h:c:)\n+Be\u000b(^dy\nL^dL\u0000^dy\nR^dR)=2; (49)13\nrespectively, with coupling constants given by\n\u001aU\u0016=\u0000\u0016\n\u0019\u00121\n\"d+U+1\nj\"dj\u0013\n; (50)\nand e\u000bective magnetic \feld\nBe\u000b=b0(\"d+U=2); b 0=4(\u0000R\u0000\u0000L)\n\u0019U:(51)\nThe corresponding Kondo temperature is given by the\nfollowing expression:45\nTK=p\nU(\u0000L+ \u0000R)\n\u0019exp\u0014\u0019\"d(U+\"d)\n2U(\u0000L\u0000\u0000R)ln\u0000L\n\u0000R\u0015\n:(52)\nNote thatTKdecreases exponentially if \u0000 \u0016is decreased\nwith a \fxed ratio of \u0000 R=\u0000Land actually becomes zero\nfor \u0000\u0016= 0 (the argument of the exponent in Eq. (52) is\nnegative, since \"d<0).\nNow,TKcan be associated with the energy gained\nby forming a ground state involving a screened local\npseudospin, which in the present setting translates to a\nground state involving a coherent superposition of con\fg-\nurations with local occupancies (0 ;1) and (1;0). Screen-\ning will cease when \"ddeviates su\u000eciently from the sym-\nmetry point\u0000U=2 that the e\u000bective magnetic \feld jBe\u000bj\nexceedsTK, in which case the ground state will be dom-\ninated solely by the (0 ;1) or (1;0) con\fguration, instead\nof involving a coherent superposition of both. Thus the\nswitching width will be set by b0WPS'TK, up to a nu-\nmerical constant of order unity.\nFigure 7(c) con\frms this expectation. It shows that b0\ntimes the switching width WPS[from Eq. (48)] (crosses)\nand the Kondo temperature TKat\"d=\u0000U=2 [from\nEq. (52)] (solid line), when plotted as functions of \u0000 R=U\nat \fxed \u0000 R=\u0000L, are indeed almost perfectly proportional\nto each other. As a numerical consistency check, Fig. 7(c)\nalso shows the inverse of the zero-temperature pseu-\ndospin susceptibility of the dot levels, 1 =\u001fs(pluses), con-\n\frming that TK= 1=\u001fs. (This is analogous to the rela-\ntion \u0000 ren= 1=\u0019\u001fcof Sec. III B.)\nB. AO in dynamics of pseudospin-\rip operator\nLet us now explore the role of AO in population switch-\ning. To this end, we note that the e\u000bective exchange\ninteraction ^Hexchof Eq. (49) is similar in structure to\nthe IRLM of Eq. (36): both involve two charge con\fgu-\nrations (the former (0,1) and (1,0), the latter 0 and 1),\nwhich induce di\u000berent phase shifts in the leads due to a\ndot-lead interaction term, and which are connected by a\ntunnelling term. More formally, the relation between the\nIRLM and PS is revealed by the equivalence of both mod-\nels to the Kondo model (for the IRLM, this equivalence\nis discussed, e.g., in Refs. 11,35,36). Thus, we may ex-\npect AO to play a similar role for both models, and hence\nperform an analysis similar to that in Sections III B and\nIII C.\n10−3100103106109101210−610−3100103106109\nω /T KAeqY(ω)/AeqY(TK)ΓR/U\n0.200\n0.175\n0.150\n0.125\n0.100\n0.075\n0.050AO: Γ R/U= 0.20\nAO: Γ R/U= 0.05(Λ=2 , Nk=4096 , σ=0.6)U= 0.2\nΓR/ΓL= 20 ω3(a)\n0.05 0.1 0.15 0.20.71\nexp\nspec\nΓR/Uη0Y(b)Figure 8: (Color online) (a) The pseudospin-\rip spectral\nfunctionAeq\nY(!) [cf. Eq. (53)] for the PS model without charge\nsensor, for several values of \u0000 R=Uwith \fxed ratio \u0000 R=\u0000L,\ncalculated at \"d=\u0000U=2: plottingAeq\nY(!)=Aeq\nY(TK) versus\n!=T Kyields a scaling collapse. The frequency dependence of\nthe curves qualitatively changes at TK: for! < T Kwe \fnd\nFermi liquid behaviour, \u0018!3(dotted line), while for !>T K\neach curve shows a nontrivial AO power-law, \u0018!\u00001+2\u00110\nY\n[Eq. (54)], exempli\fed by the dashed and dash-dotted lines\nfor \u0000 R=U= 0:2 and 0:05, respectively. (b) Comparison of\nthe values for \u00110\nYexpected from Eq. (54b) (solid line), or ex-\ntracted from the spectral function Aeq\nY(!) by using Eq. (54a)\nin the intermediate-frequency regime between TKand the high\nfrequency maximum (crosses). The accuracy of this extrac-\ntion deteriorates with increasing \u0000 R=U, since this reduces the\nwidth of the latter regime [see (a)], causing the relative er-\nror between crosses and solid line to increases from 1% for\n\u0000R=U= 0:05 to 10% for \u0000 R=U= 0:2.\nSpeci\fcally, let us study the spectral function Aeq\nY(!)\nof the pseudospin-\rip operators occurring in ^Hexch,\n^Yy= ^cL^cy\nR^dy\nL^dR; ^Y=^dy\nR^dL^cR^cy\nL: (53)\nThese induce transitions between the local charge con-\n\fgurations (0,1) and (1,0) and simultaneously add an\nelectron to one lead while removing an electron from\nthe other. (Such a transition does not constitute a\nquench, since for the present model ndis not conserved.)\nAeq\nYshould, in some respects, be analogous to Adcyof\nSec. III B. We have thus calculated Aeq\nYnumerically, us-\ning the Hamiltonian ^HSIAM of Eq. (47). Indeed, Fig. 8(a),\nwhich showsAeq\nY(!) for several values of \u0000 R=U, exhibits\nseveral features reminiscent of Fig. 4(c) for Adcy(!): (i)\nA crossover scale !\u0003, separating a regime of very low fre-\nquencies from one of intermediate frequencies, is clearly\ndiscernible; it is given by !\u0003'TK. (ii) When prop-\nerly rescaled by plotting Aeq\nY(!)=Aeq\nY(!\u0003) versus!=!\u0003,\nall curves collapse onto each other in the regime !.D.14\n(iii) In the low-frequency regime !\u001c!\u0003we \fnd the\nsame Fermi-liquid power law, Aeq\nY(!)\u0018!3(dotted line),\nas forAdcy(!) [cf. Eq. (39c)]. (An analytical explana-\ntion for this fact is given in at the end of the Appendix.)\n(iv) In an intermediate-frequency regime !\u0003.!.!he,\nwhose upper limit !heis a high-energy scale set by the\nminimum of the bandwidth or the cost of local charge\n\ructuations, we \fnd an AO-dominated power law,\nAeq\nY(!)\u0018!\u00001+2\u00110\nY; !\u0003.!.!he:(54a)\nThough the numerical calculation of Aeq\nY(!) was per-\nformed using the full Hamiltonian ^HSIAM of Eq. (47),\ntunnelling is not important on the short time-scales that\ngovern the frequency regime ! > !\u0003. Hence, we expect\nthe exponent \u00110\nYfound from Eq. (54a) to be equal in\nvalue to that which one would obtain in the !!0 limit\nof a calculation performed in the absence of pseudospin-\n\rips, i.e. using ^Hexchwithout the pseudospin-\rip terms\nin the second line of Eq. (49) (but retaining its \frst and\nthird lines).\nWithout pseudospin-\rips, the correlator involving ^Y\nwould actually constitute a type 2 quench correlator,\nbecause ^Yychanges (^nL\u0000^nR), which is a conserved\nquantum number for Hamiltonians without pseudospin-\n\rips. Therefore, the expected value of \u00110\nYcan be pre-\ndicted using the generalized Hop\feld rule [Eq. (25)]. For\nthe present case of two channels that are not intercon-\nnected by tunnelling, so that the total charge within each\nchannel is conserved, it can be applied to each chan-\nnel separately, adding the corresponding exponents21[cf.\nEq. (27)]:\n\u00110\nY=1\n2(\u0001L+ 1)2+1\n2(\u0000\u0001R\u00001)2: (54b)\nHere \u0001\u0016describes the change in phase shift, divided by\n\u0019, induced in lead \u0016by a pseudospin-\rip; it is given by\nEq. (38), with Ureplaced by U\u0016[from Eq. (50)]. The ap-\nplicability of these arguments is con\frmed by Fig. 8(b),\nwhich shows that the exponents extracted from the nu-\nmerical spectra (crosses) agree quite well with the values\nexpected from Eq. (54b) (solid line). The quality of the\nagreement deteriorates with increasing \u0000 R=U, because\nthis reduces the width of the intermediate-frequency\nregime, making an accurate extraction of \u00110\nYfromAeq\nY(!)\nincreasingly di\u000ecult.\nEquation (54b) allows us to understand why PS is al-\nways continuous in this model: Since \u00001\u0014\u0001\u0016\u00140,\nthe scaling dimension of ^Ysatis\fes\u0011Y\u00141, implying\nthat this operator always remains a relevant perturba-\ntion that does not \row to zero at low energy scales. This\nmeans that AO, although present, is not strong enough\nto completely suppress the amplitude for pseudospin-\rip\ntransitions. Hence, the two sectors (0,1) and (1,0) are\nalways coupled by the e\u000bective low-energy Hamiltonian,\nso that population switching is continuous.17,18\n10−1210−810−410010210−410−2100\nW /U∆AOΓR/U\n0.200\n0.175\n0.150\n0.125\n0.100\n0.075\n0.050U= 0.2\nΓR/ΓL= 20Figure 9: (Color online) AO for the PS model without charge\nsensing. The exponent \u0001 AO[extracted from Eq. (1)] (solid\nlines with dots) is shown as function of quench size W=U\n[Eq. (55)], for several values of \u0000 R=Uwith \fxed ratio \u0000 R=\u0000L,\nshowing that AO becomes signi\fcant once Wincreases past\nTK=b0(indicated by dashed vertical lines). The exponent\n\u0001AOincreases linearly with WforW\u001cTK=b0, and satu-\nrates to a maximal value ofp\n2 [Eq. (56)] for W\u001dTK=b0.\nThe corresponding values of \u0001 ch[from Eq. (7)] are not shown\nbut agree with \u0001 AOwith relative errors of a few percent.\nC. AO induced by quench of level positions\nAs mentioned above, the operators ^Yyand ^Yconnect\ntwo con\fgurations with di\u000berent local occupancies, (0,1)\nand (1,0). To shed further light on the AO between such\ncon\fgurations, we now perform a quantum quench anal-\nysis similar to that of Sec. III C. We consider a type 1\nquench, ^Hi!^Hf, induced by changing the level posi-\ntion\"dfrom a value above the symmetry point, favouring\n(0,1), to one below, favouring (1,0):\n\"i\nd=\u0000U=2 +W=2quench\u0000!\"f\nd=\u0000U=2\u0000W=2:(55)\nThe corresponding ground states, jGiiandjGfi, will dis-\nplay AO as in Eq. (1). Based on the lessons learnt from\nSec. III C, the corresponding exponent \u0001 AOwill increase\nwith the width Wof the quench. Indeed, Fig. 9 [to be\ncompared with Fig. 6(a)] shows that \u0001 AOincreases from\nclose to 0 for Wmuch below TK=b0(indicated by vertical\ndashed lines) to a maximal value of\n\u0001max\nAO=p\n(1)2+ (1)2=p\n2 (56)\nforW\u001dTK=b0. This maximal value re\rects the dis-\nplaced charge \u0001 ch[cf. Eq. (7)] induced by a very strong\nquench: both nLandnRare'0 (or'1) if the\nlevel position is far above (or below) the Fermi energy,\n\"i\nd=\u0000U=2 +W=2\u001d0 (or\u001c0), cf. Fig. 7(b), thus the\ndisplaced charge associated with both nLandnRis 1.\n(The contribution to \u0001 chfrom the leads turns out to be\nnegligible here,21since for su\u000eciently large Wthe Fermi\nsea is essentially decoupled from the dot.)\nD. Summary for PS without sensor\nThe results of this section can be summarized as fol-\nlows: (i) The energy scale setting the width of PS is pro-15\nportional to TK. (ii) This can directly be attributed to\nAO: as shown in Fig. 9, the ground states of two con\fgu-\nrations on opposite sides of the switching points exhibit\nstrong AO when their level positions di\u000ber by more than\nTK=b0. Thus, quantum \ructuations between them, in-\nduced by operators such as ^Yand ^Yy, are strongly sup-\npressed. (iii) For the present model PS will always be\ncontinuous as a function of \"d, because (for given U)TK\nis nonzero for any \fxed choice of \u0000 Land \u0000 R(although\nexponentially small), and AO ceases to be important\n(\u0001AO'0) once\"dcomes within TK=b0of the switch-\ning point. Conversely, however, it should now also be\nplausible that an essentially abrupt PS will be achiev-\nable if, by a suitable modi\fcation of the model, the de-\ngree of AO between the con\fgurations (0,1) and (1,0)\ncan be enhanced su\u000eciently to push TKto zero even for\n\fnite \u0000 Land \u0000 R. As pointed out by GBG,17,18this can\nbe achieved by adding a charge sensor, to which we turn\nnext.\nV. POPULATION SWITCHING WITH SENSOR\nFigure 10: Cartoon of the Hamiltonian (57), describing an\nasymmetric SIAM with an additional sensor lead coupled elec-\ntrostatically to the left dot.\nIn this section we study the e\u000bects of adding an electro-\nstatically coupled charge sensor to the model of the pre-\nvious section, as proposed by GBG,17,18and analyse how\nthis enhances the e\u000bects of AO. In particular, we show\nthat by increasing the sensor coupling strength ( US), the\ne\u000bective Kondo temperature ( TS\nK) can be driven to zero,\nimplying that population switching becomes abrupt. (A\nstudy of how additional leads increase the e\u000bects of AO\nfor static quantities has recently been performed in sim-\nilar context, involving a multi-lead IRLM.36)\nA. Width of switching regime\nGBG proposed to extend the asymmetric SIAM stud-\nied above by introducing a third lead as \\charge sensor\"\nfor the left dot (see Fig. 10). For simplicity, it is taken\nto have the same density of states as the other two leads,\nbut in contrast to the latter, it couples to the left dot\nonly electrostatically ( notby tunnelling), with interac-\n10−5100105nR\nnL\n−105−100−10−500.20.40.60.81\n(εd+U /2)/WPSnUS/U\n0\n1\n2\n3\n4\n5(a) U= 0.2ΓR/U= 0.1 ΓR/ΓL= 20\n2 4 611.251.51.75\n(U∗\nS−US)/Ul nTS\nK/l nTK\nl nTS\nK/l nTK\nF i t :\nU∗\nS/U = 6.78±0.32\nγ=−0.54±0.06(b)\n024600.51\nUS/ UTS\nK/ TKFigure 11: (Color online) Population switching for the charge\nsensor model of Eq. (57). (a) nR(\"d) (solid lines) and nL(\"d)\n(dashed lines) for several values of US=U, plotted versus\n(\"d+U=2)=WPSin a pseudo-logarithmic fashion (\\pseudo\"\nin that the x-axis is plotted logarithmic with positive and\nnegative values to the left and right of the switching point,\nrespectively, represented by the vertical solid line). The hor-\nizontal light solid lines indicate the values of nRwhich de\fne\nthe widths WS\nPSof the PS regimes. The noisy behaviour of\nthe curves for US= 5Uat small values of \"dindicates that\nour analysis cannot resolve smaller values for \"das we are\nreaching the limits of double precision numerical accuracy.\n(b) Inset:TS\nK=TKas function of US=U, showing the rapid de-\ncrease of the Kondo temperature with increasing coupling.\nMain panel: ln TS\nK=lnTKversus (U\u0003\nS\u0000US)=U, plotted on a\nlog-log scale (dashed line with dots), together with a linear\n\ft using Eq. (59) (solid line).\ntion strength US(with ^cS\u0011P\n\"^c\"S):\n^H=^HSIAM (57)\n+X\n\"\"^cy\n\"S^c\"S+US(^dy\nL^dL\u00001\n2)(^cy\nS^cS\u00001\n2):\nA plot ofnLandnRas functions of \"dfor this model\nlooks essentially similar to Fig. 7(b), showing population\nswitching at \"d=\u0000U=2. However, when the strength of\nthe coupling USis increased, the width of the PS, say\nWS\nPS, is strongly reduced below the value WPSit had\nforUS= 0, as predicted by GBG. This is illustrated in\nFig. 11(a), which shows nR(solid lines) and nL(dashed\nlines) as functions of ( \"d+U=2)=WPS, using a logarithmic\nscale to zoom in on the immediate vicinity of the PS. In\nfact, asUSapproaches a critical value U\u0003\nS, the width WS\nPS\ndrops exponentially towards zero, until it becomes too16\nsmall to be resolved within double precision numerical\naccuracy.\nThe behaviour of WS\nPSis mimicked by that of the\nKondo temperature, calculated via the pseudospin sus-\nceptibility, TS\nK\u00111=\u001fs. We \fnd that it decreases relative\nto itsUS= 0 valueTK, precisely in proportion to WS\nPS,\nsuch that\nTS\nK\nTK=WS\nPS\nWPS(58)\nholds within our numerical accuracy.\nThe transition from a continuous to an abrupt PS as\nUScrossesU\u0003\nShas been predicted to be of the Kosterlitz-\nThouless type.17,18This implies that TS\nKis expected to\napproach zero according to\n\u0000lnTS\nK\u0018(U\u0003\nS\u0000US)\r; (59)\nwhere\r=\u00001=2. To test whether our data is con-\nform to this expectation, Fig. 11(b) shows ln( TS\nK)=ln(TK)\nvs. (U\u0003\nS\u0000U) on a log-log plot. Indeed, we \fnd a\nstraight line for USnot too close to U\u0003\nS, consistent with\nEq. (59). We extract the values \r=\u00000:54\u00060:06 and\nU\u0003\nS=U= 6:78\u00060:32, by making linear \fts over several\nsomewhat di\u000berent \ftting ranges and taking the average\nand standard deviation of the \ft parameters as \fnal \ft-\nting results. The relatively large errors of about 10% are\na consequence of the fact that it is not possible to obtain\ndata forUScloser toU\u0003\nS, since this would drive TS\nKbelow\nthe level of numerical noise.\nWe note that analytical calculations based on Refs. 17\nand 18 [using the more accurate criterion, Jz(U\u0003\nS) =\nJxy(U\u0003\nS) in the notation of these papers] predict the crit-\nical interaction to be U\u0003\nS=U\u00187:6. The agreement of\nthis prediction with the numerical result of 6 :8 is quite\nrespectable, given the inaccuracies in both the numerical\nand analytical calculations [for the latter, inaccuracies\narise since the cuto\u000b scheme employed in the analyti-\ncal calculation is di\u000berent from the one realized numeri-\ncally. The cuto\u000b appears explicitly in the arguments of\nthe functions Qin Eqs. (6) to (10) of Ref. 17].\nThough the above results unambiguously show that\nthe width of PS decreases exponentially as USapproaches\na critical value U\u0003\nS, an analysis based purely on WS\nPScan\nnot access the critical point itself or the regime beyond.\nWe therefore proceed now with a numerical calculation of\nthe dynamics of the pseudospin-\rip operator, for which\nwe are not constrained to US<U\u0003\nS.\nB. AO in dynamics of pseudospin-\rip operator\nThe reason for the US-dependence of WPSandTS\nKis\nthat the introduction of the sensor ( US6= 0) increases the\nin\ruence of AO in the leads. As pointed out by GBG,\nthe scaling dimension of ^Yacquires an extra contribution\n1\n2\u00012\nSdue to the sensor lead:\n\u0011S\nY=1\n2(\u0001L+ 1)2+1\n2(\u0000\u0001R\u00001)2+1\n2\u00012\nS; (60)\n10−3100103106109101210−610−21021061010\nω /TS\nKAeqY(ω)/AeqY(TS\nK) US/U\n0\n1\n2\n3\n4\n5AO: US/U= 0\nAO: US/U= 5(Λ=2 .5, Nk=16384 , σ=0.6) U= 0.2\nΓR/U= 0.1\nΓR/ΓL= 20ω3(a)\n0123456780.80.911.1\nUS/UηS\nY\nexp\nspec(b)\n10−610−410010410810−610−4100104108\nUS/U= 8\nω /T KAeqY(ω)/AeqY(TK)TK/U= 3.8·10−7(c)Figure 12: (Color online) (a) The pseudospin-\rip spectral\nfunctionAeq\nY(!) [cf. Eq. (53)] for the PS model with charge\nsensor, for several values of US=U, calculated at \"d=\u0000U=2:\nplottingAeq\nY(!)=Aeq\nY(TS\nK) versus!=TS\nKyields a scaling col-\nlapse. The general shape of the curves is similar to those\nshown in Fig. 8: for ! <TS\nKwe \fnd Fermi liquid behaviour,\n\u0018!3(dotted line), while for ! > TS\nKeach curve shows a\nnontrivial AO power-law, \u0018!\u00001+2\u0011S\nY[cf. Eq. (54)], exempli-\n\fed by the dashed and dash-dotted lines for US=U= 0 and\n5, respectively. (b) Comparison of the values for \u0011S\nYexpected\nfrom Eq. (60) (solid line), or extracted from the spectral func-\ntionAeq\nY(!) in the intermediate-frequency regime between TK\nand the high frequency maximum (crosses). The relative er-\nrors are below 5%, where the errors decrease with increasing\nUSfor similar reasons as in Fig. 8. The light horizontal line\nindicates\u0011S\nY= 1. (We were unable to obtain reliable data\nforUSaround 7U, presumably because this is too close to\nU\u0003\nS.) (c)Aeq\nY(!)=Aeq\nY(TK) versus!=T KforUS= 8U. The AO\npower-law behaviour !\u00001+2\u0011S\nYextends down to the smallest\nfrequencies accessible, illustrating that the crossover scale TS\nK\nhas become undetectably small.\nwhere \u0001 Sis given by Eq. (38), with USreplacingU.\nBy increasing USand thereby \u00012\nS, it is thus possible to\ndrive\u0011S\nYbeyond 1. This will render the pseudospin-\rip\noperators ^Yand^Yyirrelevant , thus suppressing quantum\n\ructuations between the (0,1) and (1,0) con\fgurations,\nand, hence, pushing TS\nKdown to zero.\nTo check this scenario explicitly, we have studied the\nUS-dependence of \u0011S\nYby extracting it from the spectral\nfunctionAeq\nY(!), calculated at the particle-hole symmet-17\n10−1210−1010−810−610−410−210010−310−210−1100\nW /U∆AOUS/U\n876543210\nU= 0.2\nΓR/U= 0.1\nΓR/ΓL= 20\nFigure 13: (Color online) AO for the PS model with charge\nsensing. The exponent \u0001 AO[extracted from Eq. (1)] (solid\nlines with dots) is shown as function of quench size W=U\n[Eq. (55)], for several values of US=U, with \fxed values of \u0000 R\nand \u0000 L. We see that TS\nK=b0(indicated by dashed vertical lines)\nis pushed to zero as USincreases past U\u0003\nS=U'6:78. Already\nforUS=U\u00156 the curves are essentially indistinguishable, in\nthat they do not deviate from their constant value for all\nW=U -values accessible to our analysis. For W\u001dTS\nK=b0the\nexponent \u0001 AOsaturates to a maximal value given by Eq. (61).\nThe corresponding values of \u0001 ch[from Eq. (7)] are not shown\nbut agree with \u0001 AOwith relative errors of a few percent.\nric point for several values of US. The general shape of\nAeq\nY, shown in Fig. 12(a), is similar to that of Fig. 8(a)\nforUS= 0: For frequencies well below TS\nK,Aeq\nY(!) scales\nas!3, while in the regime of intermediate frequencies,\nTS\nK.!.!he(cf. Sec. IV B), the spectrum shows AO\npower-law behaviour, \u0018!\u00001+2\u0011S\nY. Indeed, Fig. 12(b)\nshows that the values for \u0011S\nYextracted from the spec-\ntra (crosses) agree fairly well with those expected from\nEq. (60). Moreover, for su\u000eciently large US=U, the expo-\nnents\u0011S\nYincrease past 1, con\frming that the pseudospin-\n\rip operators become irrelevant.\nC. AO induced by quench of level positions\nTo further highlight the e\u000bect of AO on TS\nK, let us con-\nsider again the quench of level position [Eq. (55)] studied\nin Sec. IV C, and repeat the analysis presented there, but\nnow for several di\u000berent values of US=U. Figure 13 shows\nthe results for the exponent \u0001 AO. For large values of W\nthe AO factor reaches its maximal value\n\u0001max\nAO=q\n(1)2+ (1)2+ \u00012\nS: (61)\nThis is similar to Eq. (56) for the model without sen-\nsor, but includes the additional contribution \u00012\nS[given\nby Eq. (38), with USreplacingU] from the displaced\ncharge induced in the sensor lead by the change in local\noccupancy of the left dot from nL= 0 to 1.\nThe most important feature of Fig. 13 is the fact that\nthe crossover scale TS\nK=b0(indicated by vertical dashed\nlines) is rapidly pushed to extremely small values as\nUS=Uis increased. Indeed, for US= 8U, which lies be-yond the critical value of U\u0003\nS=U'6:78 discussed above,\n\u0001AOis essentially pinned to its maximal value down to\nthe smallest values of quench range Wthat we can access\nnumerically. This is consistent with the fact that the cor-\nresponding spectral function Aeq\nY(!) atUS= 8U, shown\nin Fig. 12(c), shows nontrivial AO power laws down to the\nlowest frequencies accessible, with no trace of a Fermi-\nliquid!3. This demonstrates very clearly, if somewhat\nindirectly, that the PS will be abrupt for US>U\u0003\nS.\nD. Summary for PS with sensor\nLet us summarize the results of this section, by way of\nextending the list of salient points collected in Sec. IV D.\n(iv) The presence of a charge sensor reduces the crossover\nscaleTS\nK, which reaches zero at a critical coupling U\u0003\nS\n[Fig. 11]. (v) This reduction is due to the increased\ne\u000bect of AO in the leads, which increases the scaling\ndimension\u0011S\nY[Fig. 12]; when the latter passes 1 (cor-\nresponding to US=U\u0003\nS), the pseudospin-\rip operators\nbecome irrelevant and TS\nKequals zero, rendering the PS\nabrupt. (vi) Correspondingly, for US> U\u0003\nS, the spec-\ntrumAeq\nY(!) shows nontrivial AO power-law behaviour,\n!\u00001+2\u0011S\nY, all the way down to the smallest frequencies ac-\ncessible [Fig. 12(c)], and a low-frequency regime showing\nFermi-liquid exponents does not exist.\nVI. CONCLUDING REMARKS\nThe goal of this paper was to elucidate the role of\nthe Anderson orthogonality catastrophe in giving rise to\nanomalous scaling dimensions in dynamical correlation\nfunctions for quantum impurity models. To this end, we\nhave studied several setups involving (interacting) quan-\ntum dots and (non-interacting) leads. The quantum dots\nand leads may be interconnected electrostatically, or also\nthrough tunnel-coupling. In our analysis we focussed on\nthe asymptotic behaviour of various correlation functions\nG(t) and the corresponding spectral functions A(!) in\nthe limit of long times or low frequencies, respectively.\nTheir asymptotic behaviour could be understood via a\ngeneralized version of Hop\feld's rule, whose validity was\nchecked and con\frmed through an extensive NRG analy-\nsis. As a particular application, we performed a detailed\nstudy of population switching, both without and with a\nthird lead that acts as a charge sensor. We con\frmed\na previous prediction17,18that when the charge sensor\nis su\u000eciently strongly coupled, population switching can\nturn into an abrupt quantum phase transition.\nAside from presenting a systematic discussion of the\ngeneralized Hop\feld rule, which, hopefully, will be use-\nful for practitioners in the \felds, several general features\nhave emerged from our analysis:\n(1) In the context of a local quantum quench of type\n1, where a change of parameters switches the Hamilto-\nnian from ^Hito^Hf, each lead-dot electrostatic coupling18\nFermi-liquid\nexponents\nAO exponents(a)\n(b)\nAO exponents\nFigure 14: Schematic depiction of an equilibrium spectral\nfunctionAeq\nY(!) for the cases that the local charge relaxation\nrate!\u0003is (a) larger or (b) smaller than the lead level spacing\n\u000eE.\ngives rise to an AO factor in the ground state overlap\njhGijGfij, re\recting a change in the many-body con\fgu-\nration of the lead when the charging state of the dot is\nmodi\fed. This AO factor scales as N\u00001\n2\u00012\n\u0016\u0016 , whereN\u0016\nis the number of electrons in lead \u0016and \u0001\u0016the change\nin the scattering phase, divided by \u0019, in that lead. (AO\nfactors from leads that are not interconnected by tun-\nnelling, so that the total charge within each channel is\nconserved, are multiplicative [Eq. (27)].21)\n(2) AO also arises for a type 2 quench, induced by an\noperator ^Xythat connects initial and \fnal ground states\njGiiandjGfilying in dynamically disconnected sectors\nof Hilbert space. In particular, AO in\ruences the corre-\nsponding quench spectral function AX(!) which scales as\nAX(!)\u0018!\u00001+\u00012\nX[Eq. (23)]. For a Hamiltonian without\ntunnelling terms such as the LCM of Eq. (2), the spectral\nfunction for ^Xy=^dythus scales asAd(!)\u0018!\u00001+\u00012\nd.\n(3) When a type 2 quench has the form of a tunnelling\noperator, ^Yy= ^c^dy, the asymptotic power law is modi\fed\nto becomeAdcy\u0018!\u00001+(\u0001d+1)2[Eq. (35c)], implying a\nscaling dimension \u00110\ndcy=1\n2(\u0001d+ 1)2. For a particle-\nhole symmetric interaction term [as in Eq. (36)], we have\n\u00001\u0014\u0001d\u00140 [Eq. (38)], implying that 0 \u0014\u00110\ndcy\u00141=2,\nthus tunnelling between a dot and a single lead is always\na relevant perturbation.\n(4) The scaling exponent can be increased, and AO\nstrengthened, by coupling the dot(s) to further leads. In\nparticular, leads that couple to the dot only electrostat-\nically (not via tunnelling) contribute AO exponents of\nthe form1\n2\u00012\n\u0016, and thus enhance AO more strongly than\nleads that are tunnel-coupled [cf. point (3)]. In this way,\nthe scaling dimension of the tunnelling operator can be\nincreased past 1 [cf. Eq. (60)], and tunnelling rendered\nirrelevant. In such a situation, population switching be-\ncomes a quantum phase transition, tuned by gate voltage.\n(5) A particularly revealing way of demonstrating thee\u000bect of AO for population switching is to calculate the\nexponent \u0001 AOfor a type 1 quench in which the level\nposition is abruptly changed from lying above to below\nthe PS point (see Figs. 9 and 13, which are analogous to\nFig. 6(a) for the IRLM).\n(6) In the presence of tunnelling terms of the form\n(^cy^d+^dy^c), operators such as ^Yy=^dy, ^c^dyand ^cy^dydo\nnot induce a quench, since they do not cause a switch be-\ntween disconnected sectors of Hilbert space. Thus, when\nsuch an operator acts on the ground state, the result-\ning state will relax back to the ground state over long\ntime scales, say t\u001d1=!\u0003, where!\u0003represents the local\ncharge relaxation rate.\n(7) The corresponding equilibrium spectral function\nAeq\nY(!) thus typically shows trivial Fermi-liquid expo-\nnents [e.g. Eq. (39)] in the regime of very small frequen-\ncies,\u000eE.!\u001c!\u0003, where\u000eErepresents an infrared\ncuto\u000b such as the level spacing in the lead. (Throughout\nthis paper we took \u000eE'0, since in NRG calculations \u000eE\ncan be made arbitrarily small by using su\u000eciently long\nWilson chains.)\n(8) In an intermediate frequency regime ! > !\u0003, the\nequilibrium spectral function Aeq\nY(!) may nevertheless\ncontain signatures of anomalous AO exponents, scaling as\n!\u00001+2\u00110\nY[e.g. Eq. (41)], where \u00110\nYrepresents the scaling\ndimension of ^Ycalculated in the absence of tunnelling.\nThus, such exponents may be extracted by focussing on\nthis regime of intermediate frequencies [as done in Figs. 4,\n8 and 12]. This is schematically indicated in Fig. 14(a).\n(9) If AO can be made so strong that the scaling di-\nmension\u00110\nYof the operator ^Yyis larger than 1, the scale\n!\u0003is pushed below \u000eE(or, in the context of NRG, below\nthe level of numerical noise). In this case, the regime of\nanomalous AO scaling !\u00001+2\u00110\nYwill extend all the way\ndown to the smallest frequencies accessible [e.g. 12(c)],\nas schematically indicated in Fig. 14(b).\nTo conclude, we note that cases where AO dominates\nin the low frequency limit such that !\u0003'0, [as in point\n(9)], quantum \ructuations of the charge on the dot(s)\nare essentially completely frozen out. At zero tempera-\nture and in the absence of any extraneous decay mech-\nanism, the system will remain localized in a particular\nlocal charge con\fguration. Thus, varying the gate volt-\nage in such a situation may lead to hysteretic behaviour.\nIt would be very interesting to experimentally search for\nsuch signatures of the freezing out of charge \ructuations\nby performing linear response measurements at the PS\npoint.\nAcknowledgments\nWe thank R. Berkovits, L. Borda, Y. Imry, Y. Oreg, A.\nSchiller, G. Zar\u0013 and, and A. Zawadowski for helpful dis-\ncussions. This work received support from the DFG (SFB\n631, De-730/3-2, De-730/4-2, SFB-TR12, WE4819/1-1),\nin part from the NSF under Grant No. PHY05-51164,19\nfrom the Israel-Russia MOST grant, the Israel Science\nFoundation, and the EU grant under the STREP pro-\ngram GEOMDISS. Financial support by the Excellence\nCluster \\Nanosystems Initiative Munich (NIM)\" is grate-\nfully acknowledged. M.G. is supported by the Adams\nFoundation of the Israel Academy of Sciences and Hu-\nmanities, the Simons Foundation, the Fulbright Founda-\ntion, and the BIKURA (FIRST) program of the Israel\nScience Foundation.\nAppendix: Fermi-liquid spectral functions\nIn this appendix we study analytically the low energy\n(!<!\u0003) behaviour of the spectral functions of the IRLM\n(Sec. III B) and the PS setup (Sections IV B and V B).\nLet us start from the noninteracting resonant level\n(Eq. (36) with U= 0). In that case an elementary cal-\nculation gives for the retarded dot Green function,49\nGR\nd(!) =1\n!\u0000\"d\u0000\u0000\n\u0019\u001aGR;0\nc(!)=1\n!\u0000\"d+i\u0000;(A.1)\nwhereGR;0\ncis the retarded cGreen function for \u0000 = 0,\nand we assumed the wide band limit (used just to sim-\nplify expressions, but actually not essential for any of the\nfollowing arguments) GR;0\nc(!) =\u0000i\u0019\u001a=\u0000i\u0019=2 in units\nwhereD= 1. The imaginary part of the retarded Green\nfunction gives (up to a factor of \u00001=\u0019) the well-known\nLorentzian spectral function\nAeq\nd(!) =1\n\u0019\u0000\n(!\u0000\"d)2+ \u00002: (A.2)\nThus, at low energies ( !\u001c\u0000)Aeq\nd(!) becomes a con-\nstant, corresponding to \u0011eq\nd= 1=2 [which reproduces\nEq. (39a)]. This behaviour is easy to understand: In the\nabsence of tunnelling Aeq;0\nc(!) =\u001ais constant, re\recting\nthe constant local density of states of the lead electrons\nnear the end of the lead. In the presence of tunnelling,\nat low energy the dot level is well-hybridized with the\nlead, and assumes the role of the end of the lead, thus\nfeaturing the slowly-varying low-energy spectral function\nAeq\nd(!).\nBased on similar arguments, one would expect that,\nin the presence of tunnelling, Aeq\nc(!) is still constant at\nlow-energies, since in that limit the small spatial separa-\ntion between the dot and the end of the lead should be\nunimportant. However, commensurability at half \flling\n(particle-hole symmetry) makes things bit more compli-\ncated. An explicit calculation gives:\nGR\nc(!) =GR;0\nc(!) +GR;0\nc(!)s\n\u0000\n\u0019\u001aGR\nd(!)s\n\u0000\n\u0019\u001aGR;0\nc(!)\n=\u0000i\u0019\u001a!\u0000\"d\n!\u0000\"d+i\u0000: (A.3)\nThus, when \"dis nonzero, we indeed get a constant low\nenergy limit, i.e. \u0011eq\nc= 1=2. However, when \"d= 0(the value used throughout this paper for the IRLM),\nGR\nc(!)\u0018!whileAeq\nc(!)\u0018!2, corresponding to \u0011eq\nc=\n3=2. To understand this behaviour, let us examine a\nhalf in\fnite tight-binding chain with lattice spacing aand\nHamiltonian ^HTB=P1\nn=1(^\ty\nn+1^\tn+ h:c:). Taking the\ncontinuum limit in the standard way, we can expand the\nfast-varying annihilation operators \t nin terms of slowly-\nvarying (on the scale of the Fermi wavelength) right/left\nmoving \felds  R=L(x), withx=na:\n\tn=eikFna R(na) +e\u0000ikFna L(na); (A.4)\nwherekFis the Fermi wavevector. From the boundary\ncondition \t 0= 0 one gets  L(0) =\u0000 R(0), so we can de-\n\fne the single slowly-varying \feld  (x) by (x) = R(x)\nifx>0 and (x) =\u0000 L(\u0000x) ifx<0. Then:\n\tn=eikFna (na)\u0000e\u0000ikFna (\u0000na): (A.5)\nAt half \flling, kFa=\u0019=2, we get at the site next to the\nboundary\n\tn=2=\u0000 (2a) + (\u00002a)\u0018\u00004a@x (0);(A.6)\nThe same thing happens at the \frst site (n= 1) when\nwe attach a dot, since at low energies the dot behaves\nas the new \frst site . The spatial derivative is equiva-\nlent to a time derivative, up to the Fermi velocity vF.\nThis extra time derivative is responsible for the vanish-\ning of the spectral function Aeq\nc(!) for!!0. Since\nwe have derivative for both ^ cand ^cyin the Green func-\ntion, and each gives an extra factor of !, we end up with\nAeq\nc(!)\u0018!2. This behaviour depends on being at half\n\flling (particle-hole symmetry), hence is modi\fed when\n\"dis not zero.\nNow we can discuss the higher spectral functions,\nAeq\ndcy(!), andAeq\ndc(!). These are the imaginary parts of\nthe corresponding retarded Green functions, up to a fac-\ntor of\u00001=\u0019. The retarded Green functions are in turn the\nanalytical continuation of the temperature Green func-\ntions to the real frequency axis. And the temperature\nGreen functions can be found in the noninteracting case\nusing Wick's theorem.49\nPerforming these calculations for Aeq\ndcy(!), one gets:\nAeq\ndcy(!) =\u001a\n\u0019=\u0014\nln!\u0000\"d+i\u0000\n\u0000\"d+i\u0000(A.7)\n\u0000\u00002\n!(!+ 2i\u0000)ln\"2\nd\u0000(!+i\u0000)2\n\"2\nd+ \u00002\u0015\n:\nConcentrating on !\u001c\u0000 one \fnds \u0011eq\ndcy=\u0011eq\nc+\u0011eq\nd= 1\nfor\"d6= 0 and\u0011eq\ndcy=\u0011eq\nc+\u0011eq\nd= 2 for\"d= 0 [the data in\nFig. 4(c) corresponds to the latter case, which reproduces\nEq. (39c)]. This simple summation of scaling dimensions\nis natural here, since there is only one possible di\u000berent-\ntime Wick-pairing, of each single-particle operator with\nits conjugate.\nForAeq\ndc(!), however, there are two di\u000berent-time\nWick-pairings, causing cancellations, and resulting in:\nAeq\ndc(!) =\u001a\n\u0019=\u0014!\u00002\"d+ 2i\u0000\n!\u00002\"dln!\u0000\"d+i\u0000\n\u0000\"d+i\u0000\u0015\n:(A.8)20\nConcentrating again on !\u001c\u0000 one \fnds now that \u0011eq\ndc= 2\nforallvalues of\"d[which reproduces Eq. (39c)].38The\nreason is that in the low-energy continuum limit the\nproduct ^d^cbecomes the product of annihilation opera-\ntors at almost the same point. Hence, one should expand\nin the distance between ^dand ^c(lattice spacing). The\nleading term (with no spatial derivatives) vanishes by\nPauli's principle; the next term involves a spatial deriva-\ntive, leading to a factor of !, similarly to the arguments\nabove. Another factor of !comes from the operator ^ cy^dy\nappearing in the de\fnition of Geq\ndc(!). Thus, at low ener-\ngies we end up with Aeq\ndc(!)\u0018!3even for\"d6= 0.\nAlthough the above calculations were performed for\nthe noninteracting case, the qualitative arguments ex-\nplaining the low-energy behaviour are valid even when\nU > 0. Moreover, since the system \rows to the same\n\fxed point for all values of U > 0, the low energy power-\nlaws are in any case independent of U. Our numerical\nresults (Fig. 4) are in agreement with this picture.Let us now turn to the low-energy behaviour of the PS\nsetup in the case where PS is continuous. At low-energy\nthe system is governed by Kondo physics, Eq. (49),45{48\nwhere the L-R degree of freedom plays the role of a\npseudo-spin. The equivalence to Kondo continues to\nhold even in the presence of a charge sensor, as shown\nby GBG.17,18The operator ^Y=^dy\nR^dL^cR^cy\nL(which is\nrelevant in the continuous-PS phase) is the pseudospin-\n\rip local exchange term between the dot and the lead.\nSimilarly, the IRLM is also equivalent to the Kondo\nmodel,11,35,36with the role of spin replaced by the charg-\ning state of the dot. The pseudo-spin local exchange term\nis simply ^d^cy. Hence, when the parameters are properly\nmapped, the spectral functions Aeq\nY(!) andAeq\ndcy(!) are\nequivalent when the Kondo description applies (i.e. for\n!\u001cDfor the IRLM, and !\u001c!hefor the PS setup).\nIn particular,Aeq\nY(!) should exhibit an !3behaviour at\nlow energy, similarly to Aeq\ndcy(!) for\"d= 0, as the NRG\ndata shows [dotted line in Fig. 8(a) and Fig. 12(a)].\n1P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967).\n2K. D. Schotte and U. Schotte, Phys. Rev. 185, 509 (1969).\n3G. D. Mahan, Phys. Rev. 163, 612 (1967).\n4J. J. Hop\feld, Comments Solid State Phys. 2, 40 (1969).\n5B. Roulet, J. Gavoret, and P. Nozi\u0012 eres, Phys. Rev. 178,\n1072 (1969); P. Nozi\u0012 eres, J. Gavoret, and B. Roulet, Phys.\nRev.178, 1084 (1969); P. Nozi\u0012 eres, and C. T. De Dominics,\nPhys. Rev. 178, 1097 (1969).\n6B. L. Altshuler and A. G. Aronov, Pis'ma Zh. Eksp. Teor.\nFiz.30, 514 (1979).\n7A. O. Gogolin, Phys. Rev. Lett. 71, 2995 (1993).\n8N. V. Prokof'ev, Phys. Rev. B49, 2148 (1994).\n9I. A\u000feck and A. W. W. Ludwig, J. Phys. A 27, 5375\n(1994).\n10A. Furusaki, Phys. Rev. B 56, 9352 (1997).\n11A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik,\nBosonization and Strongly Correlated Systems (Cambridge\nUniversity Press, Cambridge, 2004).\n12A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.\nFisher, A. Garg, W. Zwerger, Rev. Mod. Phys. 59, 1\n(1987).\n13Z. Ovadyahu, Phys. Rev. Lett. 99, 226603 (2007).\n14R. W. Helmes, M. Sindel, L. Borda, and J. von Delft, Phys.\nRev. B72, 125301 (2005).\n15H. E. T ureci, M. Hanl, M. Claassen, A. Weichselbaum,\nT. Hecht, B. Braunecker, A. Govorov, L. Glazman, A.\nImamoglu, and J. von Delft, Phys. Rev. Lett. 106, 107402\n(2011).\n16C. Latta, F. Haupt, M. Hanl, A. Weichselbaum, M.\nClaassen, W. Wuester, P. Fallahi, S. Faelt, L. Glazman,\nJ. von Delft, H. E. T ureci, and A. Imamoglu, Nature 474,\n627 (2011).\n17M. Goldstein, R. Berkovits, and Y. Gefen, Phys. Rev. Lett.\n104, 226805 (2010).\n18M. Goldstein, Y. Gefen, and R. Berkovits, Phys. Rev. B\n83, 245112 (2011).\n19K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n20R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys.80, 395 (2008).\n21A. Weichselbaum, W. M under, and J. von Delft, Phys.\nRev. B84, 075137 (2011).\n22J. Friedel, Phil. Mag. 43, 153 (1952).\n23J. Friedel, Can. J. Phys. 34, 1190 (1956).\n24D. C. Langreth, Phys. Rev. 150, 516 (1966).\n25A. C. Hewson, The Kondo problem to heavy Fermions\n(Cambridge University Press, Cambridge, 1993).\n26Compare Ref. 22, p. 159; Ref. 23, Eq. (7); Ref. 24, Eq. (21);\nRef. 25, Sec. 1.2.\n27O. Gunnarsson and K. Sch onhammer, Phys. Rev. B28,\n4315 (1983).\n28J. Cardy, Scaling and Renormalization in Statistical\nPhysics (Cambridge University Press, Cambridge, 1996).\n29L. N. Oliveira and J. W. Wilkins, Phys. Rev. B24, 4863\n(1981); D. L. Cox, H. O. Frota, L. N. Oliveira, and J. W.\nWilkins, Phys. Rev. B32, 555 (1985).\n30T. A. Costi, P. Schmitteckert, J. Kroha, and P. W ol\re,\nPhys. Rev. Lett. 73, 1275 (1994).\n31F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801\n(2005).\n32F. B. Anders and A. Schiller, Phys. Rev. B74, 245113\n(2006).\n33R. Peters, T. Pruschke and F. B. Anders, Phys. Rev. B74,\n245114 (2006).\n34A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99,\n076402 (2007).\n35P. Schlottmann, Phys. Rev. B22, 622 (1980).\n36L. Borda, A. Schiller, and A. Zawadowski, Phys. Rev. B\n78, 201301 (2008).\n37M. Goldstein, Y. Weiss, and R. Berkovits, Europhys. Lett.\n86, 67012 (2009); Physica E 42, 610 (2010).\n38M. Goldstein and R. Berkovits, Phys. Rev. B82, 235315\n(2010).\n39G. Hackenbroich, W. D. Heiss, and H. A. Weidenm uller,\nPhys. Rev. Lett. 79, 127 (1997).\n40R. Baltin, Y. Gefen, G. Hackenbroich and H. A. Wei-\ndenm uller, Eur. Phys. J. B10, 119 (1999).21\n41P. G. Silvestrov and Y. Imry, Phys. Rev. Lett. 85, 2565\n(2000).\n42P. G. Silvestrov and Y. Imry, Phys. Rev. B65, 035309\n(2001).\n43D. I. Golosov and Y. Gefen, Phys. Rev. B74, 205316\n(2006).\n44C. Karrasch, T. Hecht, A. Weichselbaum, Y. Oreg, J. von\nDelft, and V. Meden, Phys. Rev. Lett. 98, 186802 (2007).\n45V. Kashcheyevs, A. Schiller, A. Aharony, and O. Entin-\nWohlman, Phys. Rev. B75, 115313 (2007).46H. W. Lee and S. Kim, Phys. Rev. Lett. 98, 186805 (2007).\n47P. G. Silvestrov and Y. Imry, Phys. Rev. B75, 115335\n(2007).\n48V. Kashcheyevs, C. Karrasch, T. Hecht, A. Weichselbaum,\nV. Meden, and A. Schiller, Phys. Rev. Lett. 102, 136805\n(2009).\n49G. D. Mahan, Many-Particle Physics (Kluwer, New York,\n2000)."    },    {        "title": "1010.4016v1.Fermi_liquid_theory_of_Fermi_Bose_mixtures.pdf",        "content": "arXiv:1010.4016v1  [cond-mat.other]  19 Oct 2010Fermi liquid theory of Fermi-Bose mixtures\nE. V. Thuneberg and T. H. Virtanen\nDepartment of Physics, P.O.Box 3000, FI-90014 University o f Oulu, Finland\n(Dated: November 23, 2018)\nWe write down the basic equations of Fermi-liquid theory for mixtures of fermions and bosons,\nan example being3He-4He mixtures at low temperatures. Basically the theory is ide ntical to the\none derived by Khalatnikov, but it is derived in a different wa y, and includes more discussion. A\nsimplifying transformation of the equations is found where the coupling of the normal and superfluid\ncomponents appears in a simple form. The boundary condition s are discussed.\nI. INTRODUCTION\nThe Fermi liquid theory formulated by Landau has become a paradigm of what can be the effect of interactions\nin a Fermi system1. Landau formulated the theory originally for liquid3He. Khalatnikov generalized this theory to\nmixtures of fermions and bosons, and applied it to mixtures of3He and4He [2]. The purpose of this article is to\nreformulate Khalatnikov’s theory. Although the theory is basically t he same, our approach is different. We avoid\nseveral complications by concentrating on the laboratory frame, by directly starting with the linearized theory, and by\nusing the osmotic energy. We construct a transformation that elim inates the coupling between the superfluid velocity\nand the quasiparticle momentum. We discuss boundary conditions. W e also try to interpret the theory by discussing\nhow the momentum of a quasiparticle can be divided into three contrib utions. The purpose is to formulate general\nFermi-Bose liquid equations that are needed to calculate the respon se on a vibrating wire3,4.\nA general introduction to Fermi liquid theory can be found in Refs. 5 –8. Although our emphasis is on subjects not\nfound in these reviews, we try to be self-contained. Note that Kha latnikov’s theory2as well as the present theory\nmake no assumption about the diluteness of the fermion component in the mixture, which is assumed in several other\ntreatments8–10. Thus the Fermi-liquid theory of a pure fermion system can be obtain ed as a limiting case of the\ntheory.\nWe start in Sec. II with some basic definitions and introduce the nonin teracting system. The basic assumptions of\nthe Fermi-Bose liquid theory are given in Sec. III. In Sec. IV we defin e the phenomenological parameters that enter\nthe theory. The equations of motion are formulated in Sec. V. The c onserved currents are identified in Sec. VI and\nsome discussion is given. In Sec. VII we make a change to new variable s where the coupling between the normal\nand superfluid components appear only through their densities. Th e scattering of quasiparticles in the bulk and at\nsurfaces is discussed in Secs. VIII and IX. The hydrodynamic limit of the theory is discussed in Sec. X.\nII. BASIC DEFINITIONS\nAn important thing to realize is that the mass current density Jis the same as the momentum density P. The\nequivalence J≡Pfollows because the momentum of a particle is mass times the velocity, p=mv, i.e. the mass mis\ntransported at velocity v. This relation is valid in condensed matter under standard (nonrelat ivistic) conditions since\nthe changes of momentum and mass associated with the interaction field are negligible. This is valid also in crystalline\nmaterial, where even in the idealized limit of infinitely rigid lattice, one sho uld allow part of the momentum or mass\ncurrent carried by the lattice. The equivalence J≡Pis essential in connection of equations (19)-(21) below.\nWe consider a system of particles having one type of fermions and on e type of bosons, with masses mFandmB. In\nthis section, we describe the system in the absence of interactions . The free fermions have momenta pand energies\nǫp=p2/2mF. The state of the fermion system is described by distribution funct ionnpthat takes values 0 and 1 for\neach momentum state. The number density of fermions nFis given by\nnF=/integraldisplay\nnpdτ, (1)\nwheredτ= 2d3p/(2π/planckover2pi1)3. The fermions have a spin but since we are not considering spin-depe dent phenomena, it\nonly appears as a factor of 2 in dτ. The ground state consists of a Fermi sphere, which has filled stat es,np= 1, for\nmomentap < pFand empty states, np= 0, for momenta p > pF. The Fermi momentum pFis determined by the\nnumber density nF=p3\nF/3π2/planckover2pi13. The momentum density of fermions is JF=/integraltext\npnpdτ. The bosons are assumed to be\ncondensed to a state with velocity vBand all excited states of the boson system are neglected. The num ber density of\nbosons is denoted by nBand the momentum density JB=mBnBvB. The total momentum density of both fermions2\nand bosons J=JF+JBis thus given by\nJ=mBnBvB+/integraldisplay\npnpdτ. (2)\nIII. BASIC ASSUMPTIONS\nWe now turn to the discussion of the interacting system. Landau’s F ermi-liquid theory applies to low energy\nexcitations of the system. The basic assumption is that when intera ctions are turned on, the low-energy part of the\nexcitation spectrum remains qualitatively the same as it is in the non-in teracting system. More precisely, one assumes\nthat the momenta of the excitations remain the same when the interactions are turned on at constant densities of both\nfermions and bosons. The energies of the excitations can be shifte d, butthe equilibrium Fermi surface is assumed to\nremain unchanged . Alsothe quasiparticle energy is assumed to be linear in pclose to the Fermi surface. Thus the\nexcited states can still be specified by a quasiparticle distribution fu nctionnp, and the ground state corresponds to\nthe filled Fermi sphere.\nA particular consequence of the assumptions is that Eq. (1) remain s valid in the interacting system. For the total\nmomentum of an arbitrary interacting state we write, following Khala tnikov2\nJ=mBnBvs+/integraldisplay\npnpdτ. (3)\nThis is the same as Eq. (2) except that instead of the velocity of bos ons it defines the superfluid velocity vs.\nIn order to stress the non-triviality of Eq. (3) we mention that it is incorrect to deduce that the first term would\nbe the momentum density of bosons and the latter that of fermions . The correct decomposition will be given later in\nequations (22) and (24). Note that an interacting system does no t have a single boson velocity vB, and therefore it\ncannot appear in Eq. (3).\nIn addition we use in the following the spherical symmetry of the syst em, the conservation laws of particle number,\nmomentum and energy, and some estimates of orders of magnitude .\nIV. PARAMETERIZATION\nAbove we used variables np,nBandvsto specify the state of the system. In particular, the energy den sity could\nbe written ˜E({np},nB,vs). Here the curly brackets indicate that ˜Eis a functional of np, i.e. it depends on npat all\nvalues of p. With respect to the variable nB, it is more convenient to change to the “osmotic energy” defined by\nE({np},µB,vs) =˜E({np},nB,vs)−nBµB−nFµ(0)\nF. (4)\nHere the term −nBµBeffects the standard Legendre transformation from the density nBto the chemical potential\nµB=∂˜E({np},nB,vs)/∂nB. The term −nFµ(0)\nFlooks formally similar for the fermions, but is different because the\nchemical potential of fermions in a non-equilibrium state is not define d. Instead, we define a reference state. It has the\nequilibrium distribution n(0)\np= Θ(pF−p), where Θ( x) is the step function. This equilibrium is taken to correspond\nchemical potentials µ(0)\nBandµ(0)\nF, superfluid velocity vs= 0 and temperature T= 0. Thus the osmotic energy (4)\nstill depends on the distribution function np, and the effect of the −nFµ(0)\nFterm is merely a shift of energy so that\nquasiparticle energies are counted from µ(0)\nF.\nThe quasiparticle energy is defined as\nǫp=δE({np},µB,vs)\nδnp, (5)\nwhere the functional derivative is interpreted as δE=/integraltext\nǫpδnpdτ. The linearity of the quasiparticle energy near the\nFermi surface is satisfied by the choice\nǫp=vF(p−pF)+δǫp. (6)\nThe coefficient of p−pFdefines the Fermi velocity vF. WritingvF=pF/m∗it defines the effective mass m∗. The\nsecond term δǫpin (6) can be written as\nδǫp= (1+α)δµB+Dp·vs+/integraldisplay\nf(p,p1)(np1−n(0)\np1)dτ1, (7)3\nwhereδµB=µB−µ(0)\nBis the deviation of the boson chemical potential from its equilibrium va lue. We see that δǫp\nis linear in the deviation from equilibrium, and thus appears less importa nt than the first term in (6). It was an\nimportant observation of Landau that this term still has to be kept in order to make a consistent theory. Being linear\nin the deviations of µB,vsandnpfrom equilibrium, we see that δǫp(7) has the most general form that is allowed by\nsymmetry. For example the most general function of pthat is linear in vsand does not depend on any other direction,\nhas the form D(p)p·vswith some function D(p). For the same reason f(p,p1) cannot depend on the directions of ˆp\nandˆp1separately but only on their scalar product ˆp·ˆp1.\nWecanfurthersimplify δǫp(7). SinceweareconsideringonlyexcitationsneartheFermisurfa ce,wecanapproximate\np≈pFˆpin theDp·vsterm. Also, we estimate that f(p,p1) changes essentially only on the momentum scale of pF.\nSincenp−n(0)\npis nonzero only in a thin shell around the Fermi surface, we can negle ct the dependence of f(p,p1)\non the magnitudes pandp1. Similarly, we can assume αandDas constants not depending on p. Thusδǫpdepends\nonly on the momentum direction ˆp, not on the magnitude p. We can now define an energy integrated distribution\nfunction\nφˆp=/integraldisplay\n(npˆp−n(0)\np)vFdp, (8)\nwhere the argument of npisp=pˆp. We get that fp,p1depends only on the scalar product ˆp·ˆp1. Thus we can\nexpandfp,p1using Legendre polynomials Pl(x)\nF(ˆp·ˆp′)≡m∗pF\nπ2/planckover2pi13f(ˆp·ˆp′) =∞/summationdisplay\nl=0FlPl(ˆp·ˆp′), (9)\nwhereP0(x) = 1,P1(x) =x, etc andm∗pF/π2/planckover2pi13is the quasiparticle density of states at the Fermi surface. As a\nresult we can write (7) into the form\nδǫˆp= (1+α)δµB+DpFˆp·vs+∞/summationdisplay\nl=0Fl/an}b∇acketle{tPl(ˆp·ˆp′)φˆp′/an}b∇acket∇i}htˆp′, (10)\nwhere/an}b∇acketle{t.../an}b∇acket∇i}htˆpdenotes the average over the unit sphere of ˆp.\nWe have defined parameters m���,α,DandFl’s in equations (6) and (10). There is one constraint between these\nrequired by translational invariance. The relation is conveniently st ated by expressing Din terms of other parameters,\nD= 1−mF\nm∗/parenleftbigg\n1+1\n3F1/parenrightbigg\n. (11)\nThis relation will be justified below in connection of equation (23). Not e that because of using the osmotic energy (4)\nright from the start, we need to consider only one set of Flparameters, in contrast to the three sets used in Ref. 2.\nWe also need to consider changes in the boson density. We paramete rize\nδnB≡nB−n(0)\nB=−(1+α)δnF+nB\nmBs2δµB, (12)\nwhereδnF=nF−n(0)\nF. The equality of the coefficient 1+ αwith the one in (7) and (10) follows because they are\nsecond partial derivatives of E(4),\n∂2E\n∂nF∂µB=∂µF\n∂µB=−∂nB\n∂nF= 1+α. (13)\nIn the limit of vanishing concentration of fermions, αreduces to the parameter αused in Refs. 8,9. In the same limit,\nthe parameter sin (12) reduces to the velocity of sound [as will be evident from equat ions (18) and (36) below].\nV. EQUATIONS OF MOTION\nIn general the distribution function npdepends on location and time, np(r,t), and similarly for other variables\nµB(r,t) andvs(r,t). The quasiparticle distribution obeys the kinetic equation\n∂np\n∂t+∇np·∂ǫp\n∂p−∂np\n∂p·∇ǫp=Ip, (14)4\nwhereIpis the collision term. The fermion number and momentum conservation in collisions requires that\n/integraldisplay\nIpdτ= 0,/integraldisplay\npIpdτ= 0. (15)\nAssuming small deviation from equilibrium, we linearize the kinetic equat ion (14) and get\n∂np\n∂t+vFˆp·∇/parenleftBigg\nnp−∂n(0)\np\n∂ǫpδǫˆp/parenrightBigg\n=Ip, (16)\nwhere the derivative ∂n(0)\np/∂ǫpshould be evaluated at the unperturbed energy ǫ(0)\np=vF(p−pF).\nThe superfluid velocity vsappearing in (3) is assumed to be curl free11\n∇×vs= 0. (17)\nIt is assumed to obey the ideal-fluid equation of motion,\n∂vs\n∂t+1\nmB∇µB= 0. (18)\nVI. CURRENTS\nBased on the equations above one should be able to derive conserva tion laws. In particular, one should be able to\nderive separate continuity equations for the two components\nmF∂nF\n∂t+∇·JF= 0, (19)\nmB∂nB\n∂t+∇·JB= 0, (20)\nand the momentum conservation\n∂J\n∂t+∇·Π↔= 0. (21)\nwhereJ=JF+JB. In the following we derive linearized expressions for the fermion mas s current JF, for the boson\nmass current JBand for the momentum flux tensor Π↔.\nWe integrate the kinetic equation (16) over all momenta. Using (1) a nd (10) and comparing to (19) we get\nJF=DmFnFvs+mF\nm∗(1+1\n3F1)/integraldisplay\np(np−n(0)\np)dτ. (22)\nFrom equations (3) and (22) we can eliminate the integral term and g et\nJ=/parenleftbigg\nmBnB−Dm∗nF\n1+1\n3F1/parenrightbigg\nvs+m∗\nmF(1+1\n3F1)JF. (23)\nThis can be used to justify relation (11). Consider equilibrium but the whole system moving at constant velocity v.\nBecause it has to be that J= (mFnF+mBnB)v,vs=vandJF=mFnFv, we get (11).\nUsing (3) and (22) we calculate JB=J−JFand get\nJB= (mBnB−DmFnF)vs+D/integraldisplay\np(np−n(0)\np)dτ. (24)\nLet us consider Eqs. (22) and (24) in the frame where vs= 0. From Eq. (24) we see that a single quasiparticle of\nmomentum phas fraction Dof the momentum carried by bosons. As the principal fermion formin g the quasiparticle\ntravels with velocity v=p/m∗, it carries the fraction mF/m∗of the quasiparticle momentum. The rest fraction of\nthe momentum mFF1/3m∗is carried by other fermions. The total fermion contribution is visible in the second term\nofJF(22).5\nIn order to clarify the nature of the quasiparticle, we consider a sim ple model. Assume that the principal fermion\nforms a sphere of radius aand the bosons and other fermions are modeled as an incompressible ideal fluid of density\nρ. In this model one can calculate m∗=mF+2πa3ρ/3, i.e. the effective mass is the fermion mass plus the fluid mass\ncorresponding to half of the sphere. This result is obtained by calcu lating the momentum p=m∗vfrom Newton’s\nsecond law when a force is applied to accelerate the sphere, or, equ ivalently, by calculating the kinetic energy Ek\nand writing Ek=1\n2m∗v2. This model gives a simple picture how the effective mass is increased d ue to the medium\ndragged along by the principal fermion. This is sometimes called “back flow” but this may be misleading since the\nmomentum is in the “forward” direction (as m∗>mF).\nWenotethatwhenconsideringasingleexcitedquasiparticleinafinitec ontainer,onemustalsoallowacompensating\nflow. Using the simple quasiparticle model as discussed above, the to tal momentum of an incompressible fluid in finite\nstationary container must vanish. Thus the momentum p=m∗vof a quasiparticle must be compensated by an\nopposite flow of the medium. This compensating momentum arises fro m force applied by the walls of the container,\nand thus is separate from the momentum of the quasiparticle.\nFor further insight, consider the case that the fermions are in equ ilibrium but moving with velocity v, i.e.JF=\nmFnFv. We assume vs= 0. The mass density dragged in such a normal current is called normal fluid density ρn,\ni.e.J=ρnv. The complementary fraction of the total density ρ=mFnF+mBnBthat remains at rest, is called\nsuperfluid density ρs=ρ−ρn. From (23) we get\nρs=mBnB−Dm∗nF\n1+F1/3, ρn=m∗nF\n1+F1/3. (25)\nThese expressionshavea straightforwardinterpretationin term s of the momentum division introduced above. Namely\nJF=mF(1+1\n3F1)J/m∗andJB=DJ. Solving Jfrom the former and using JF=mFnFvone gets the normal fluid\ndensity (25). One also sees that the mass density Dm∗nF/(1+F1/3) subtracted from the boson density in ρs(25)\njust corresponds to the boson fraction that is bound to the quas iparticles.\nNext we check the momentum conservation (21). We take the time d erivative of J(3), and use the kinetic equation\nof quasiparticles (16) and the equation of motion of the Bose compo nent (18). Using (15) one finds the momentum\nconservation with the momentum flux tensor\nΠ↔=P(0)1↔+nBδµB1↔+1\nm∗/integraldisplay\npp/parenleftBigg\nnp−n(0)\np−∂n(0)\np\n∂ǫpδǫˆp/parenrightBigg\ndτ. (26)\nHere the constant term arising from the equilibrium pressure P(0)is added.\nVII. ELIMINATION OF SUPERFLUID-VELOCITY COUPLING\nOur purpose in this section is to rewrite the theory so that the coup ling between the superfluid and normal fluid\ncomponents appears in a minimal way. For that purpose we define ne w quantities δ¯npandδ¯ǫp, and write the kinetic\nequation in the form\n∂\n∂t/parenleftbigg\nδ¯np+∂n(0)\n∂ǫpδ¯ǫˆp/parenrightbigg\n+vFˆp·∇δ¯np=Ip, (27)\nso thatδ¯ǫpwill only depend on δµBand onδ¯npbut not on vs. In order to achieve this, we scalar multiply the\nsuperfluid equation (18) by AvFˆp∂n(0)/∂ǫpand add it to the kinetic equation (16). Ais a constant that is fixed\nbelow. Comparing this to (27) we find\nδ¯np=np−n(0)\np−∂n(0)\n∂ǫp/parenleftbigg\nδǫˆp+A\nmBδµB/parenrightbigg\n, (28)\nδ¯ǫˆp=δǫˆp+A\nmBδµB−AvFˆp·vs. (29)\nSimilar toφˆp(8), it is convenient to define\nψˆp=/integraldisplay\nδ¯npˆpvFdp. (30)6\nThe next task is to express δ¯ǫˆpin terms of ψˆp. Doing this one notices that the dependence on vsdrops out by\nchoosingA=Dm∗/(1+F1/3) and we get\nδ¯ǫˆp=K\n1+F0δµB+∞/summationdisplay\nl=0Fl\n1+1\n2l+1Fl/an}b∇acketle{tPl(ˆp·ˆp′)ψˆp′/an}b∇acket∇i}htˆp′, (31)\nwhere we have defined\nK=m∗D\nmB(1+1\n3F1)+1+α=m∗\nmB(1+1\n3F1)−mF\nmB+1+α. (32)\nIt is useful to express the densities and currents in terms of δ¯np(28) orψˆp(30). Here we give the fermion density\n(1)\nδnF≡nF−n(0)\nF=1\n1+F0/parenleftbigg/integraldisplay\nδ¯npdτ−m∗pFK\nπ2/planckover2pi13δµB/parenrightbigg\n=m∗pF\nπ2/planckover2pi13(1+F0)(/an}b∇acketle{tψˆp/an}b∇acket∇i}htˆp−KδµB), (33)\nthe fermion momentum (22)\nJF=mF\nm∗/integraldisplay\npδ¯npdτ=mFp2\nF\nπ2/planckover2pi13/an}b∇acketle{tˆpψˆp/an}b∇acket∇i}htˆp, (34)\nand the momentum flux tensor (26)\nΠ↔=P(0)1↔+ρs\nmBδµB1↔+1\nm∗/integraldisplay\nppδ¯npdτ=P(0)1↔+ρs\nmBδµB1↔+3nF/an}b∇acketle{tˆpˆpψˆp/an}b∇acket∇i}htˆp. (35)\nIn the momentum flux (35) the second and third terms arise from th e superfluid and the normal components, respec-\ntively. We also rewrite the boson conservation (20) using (12) as\nnB\ns2∂δµB\n∂t+ρs∇·vs−mBK∂nF\n∂t= 0. (36)\nThe equations (27), (36), and (18) constitute equations of motio n in variables δ¯np,δµB, andvs.\nWe see that in the system of equations (18), (27), (31), and (36) , the coupling of the superfluid and normal\ncomponents takes place throughtheir densities via the terms prop ortionaltoK. Let us estimate this effect considering\na problem of length scale a, frequency ω, and relatively long mean free path ℓ/greaterorsimilara. We first consider fermion flow with\nvelocity scale u, for which we estimate δnF∼nFu/vF. Assuming ωa/s≪1, the first term in (36) can be neglected,\nand it is important to balance the last two terms. We get that a fermio n flow with velocity ugenerates a superfluid\nvelocity of order K(nF/nB)(ωa/vF)u. The opposite effect of vs∼uon the quasiparticles can be estimated from the\nfermion conservation (19) and applying (33) and (34). We get that forωa/vF/lessorsimilar1 the induced fermion velocity is\nof orderK(mB/mF)(ωa/vF)2u. For small frequencies ωa/vF≪1 the velocities induced in the other component are\nsmall, and the superfluid and quasiparticle components can be treat ed independently of each other.\nVIII. COLLISION TERM\nFordetailed discussionofthe collision term Ipwe refer toSykes and Brooker12. In the collisionoftwo quasiparticles,\ntheir total energyis conserved. It followsthat the collision integra lcan be written as a function δ¯np(28). The collision\nterm can in a few cases be analyzed exactly12. A simpler approach is to use relaxation-time approximation. In its\nsimplest version one assumes13\nIp=−1\nτ/bracketleftBig\nnp−n(0)(ǫ(0)\np+δǫp−c−b·p)/bracketrightBig\n, (37)\nwhereτis the relaxation time and coefficients candbare chosen so that conditions (15) are satisfied. The relaxation\ntime can also be specified by the mean free path ℓ=vFτ. The relaxation-time approximation leads to essential\nsimplification because instead of using δnporδ¯np, one can construct equations for ψˆp(30), which does not depend\non the magnitude of the momentum. In the relaxation-time approxim ation the kinetic equation (27) takes the form\n∂\n∂t(ψˆp−δ¯ǫˆp)+vFˆp·∇ψˆp=−1\nτ(ψˆp−/an}b∇acketle{tψˆp′/an}b∇acket∇i}htˆp′−3/an}b∇acketle{tˆp·ˆp′ψˆp′/an}b∇acket∇i}htˆp′). (38)\nIt is also possible to introduce different relaxation times τl(withl= 2,3,...) corresponding to spherical harmonic\ndecomposition of ψˆp(54).7\nIX. BOUNDARY CONDITIONS\nThe boundary conditions at walls depend on the scattering propert ies of the wall. Two common models for surface\nscattering are specular and diffuse scattering. We generalize the a nalysis of Ref. 13 to Fermi-Bose liquids and include\ngeneralinteractions (9). We first consider the boundary conditio ns in the rest frame of a wall, and after that generalize\nto the case of a moving wall.\nWe assume a wall with normal ˆnthat is inpenetrable to both bosons and fermions. From equation (2 3) we see\nthat the vanishing of the particle fluxes JandJFnormal to the wall imply vanishing of the normal component of the\nsuperfluid velocity,\nˆn·vs= 0. (39)\nSpecular scattering means that the quasiparticle momentum chang es fromptopR=p−2ˆn(ˆn·p), and thus\nnp=np−2ˆn(ˆn·p). (40)\nBefore accepting this one should check that it conserves the quas iparticle energy (6), i.e. we calculate\nδǫˆp−δǫˆpR=DpF(ˆp−ˆpR)·vs+/an}b∇acketle{tF(ˆp·ˆp′)φˆp′−F(ˆpR·ˆp′)φˆp′/an}b∇acket∇i}htˆp′\n= 2DpF(ˆn·ˆp)(ˆn·vs)+/an}b∇acketle{tF(ˆp·ˆp′)φˆp′−F(ˆpR·ˆp′\nR)φˆp′\nR/an}b∇acket∇i}htˆp′\n= 2DpF(ˆn·ˆp)(ˆn·vs)+/an}b∇acketle{tF(ˆp·ˆp′)[φˆp′−φˆp′\nR]/an}b∇acket∇i}htˆp′= 0 (41)\nwhere we have used (10), ˆpR·ˆp′\nR=ˆp·ˆp′, (39) andφˆp′−φˆp′\nR= 0, which follows from (40).\nDiffuse scattering means that the quasiparticles reflected from th e wall are in equilibrium evaluated at the exact\nquasiparticle energy (6). Thus the distribution of the reflected qu asiparticles is\nnp=n(0)(ǫ(0)\np+δǫp−c) =n(0)\np+dn(0)\ndǫp(δǫp−c) forˆn·p>0, (42)\nwhere the constant cis determined by the condition that the particle flux (34) to the wall v anishes, ˆn·JF= 0. More\nconveniently, the boundary is expressed as\nδ¯np=−dn(0)\ndǫp¯cforˆn·p>0 (43)\nwith a new constant ¯ c. The constant can be determined by evaluating the condition ˆn·JF= 0 for reflected quasipar-\nticles, which allows to write the diffuse boundary condition as\nδ¯np=4π2/planckover2pi13\nm∗pFdn(0)\ndǫp/integraldisplay\nˆn·ˆp′<0ˆn·ˆp′δ¯np′dτ′forˆn·p>0. (44)\nThe boundary conditions in the case of a moving wall can be found by c hanging the reference frame. In the\nlaboratory frame the quasiparticle momentum p=p′+mFu, wherep′is the momentum in the rest frame of the\nwall, and uis the velocity of the wall. (This relation follows because the momenta a re assumed to remain unchanged\nwhen the interactions are switched on.) The quasiparticle distributio ns in the two frames are the same, np=n′\np′. A\nchange arises in δnp,\nδnp=np−n(0)\np=n′\np′−n(0)\np′+mFu=δn′\np′−dn(0)\ndǫpmFvFˆp·u (45)\nBased on this we find φˆp=φ′\nˆp+mFvFˆp·u, and using vs=v′\ns+uwe getδǫˆp=δǫ′\nˆp+ (m∗−mF)vFˆp·u. The\ntransformation for δ¯npis then\nδ¯np=δ¯n′\np′−dn(0)\ndǫppFˆp·u (46)\nwhich implies ψˆp=ψ′\nˆp+pFˆp·u.\nApplying the transformation rules to (39), (40) and (44) gives the superfluid condition\nˆn·vs=ˆn·u, (47)8\nthe specular boundary condition\nδ¯np=δ¯np−2ˆn(ˆn·p)+dn(0)\ndǫp2pFˆn·ˆpˆn·u, (48)\nand the diffusive boundary condition\nδ¯np= 4dn(0)\ndǫpdǫp\ndτ/integraldisplay\nˆn·ˆp′<0ˆn·ˆp′δ¯np′dτ′−dn(0)\ndǫppF(ˆp+2\n3ˆn)·uforˆn·p>0. (49)\nIn terms of ψˆpthe specular and diffuse conditions can be written\nψˆp=ψˆp−2ˆn(ˆn·ˆp)+2pF(ˆn·ˆp)(ˆn·u). (50)\nψˆpout=−2/an}b∇acketle{tˆn·ˆpinψˆpin/an}b∇acket∇i}htˆpin+pF(ˆpout+2\n3ˆn)·u (51)\nwith an average over half of the unit sphere ( ˆn·ˆpin<0).\nA consistency check for the boundary conditions is that the fermio n particle current (34) behaves as expected\nˆn·JF=mFnFˆn·u. (52)\nFrom the specular boundary condition (50) and from (35) we deduc e the vanishing of the transverse momentum flux,\nˆn·Π↔׈n= 0. (53)\nX. HYDRODYNAMIC LIMIT\nThe kinetic theory reduces to hydrodynamic theory in the limit of sma ll mean free path of quasiparticles. We sketch\nthe derivation here starting from the relaxation-time approximatio n. The quasiparticle distribution function ψˆpcan\nbe expanded in spherical harmonics\nψˆp=∞/summationdisplay\nl=0l/summationdisplay\nm=−lcm\nlYm\nl(ˆp). (54)\nThe lowest order coefficients c0\n0andcm\n1can be related to fermion number (33) and current (34). Due to ra pid\nscattering, all higher order coefficients are assumed to be small. We definenormal fluid velocity vnby writing the\ncurrentJF=nFvn. We multiply the kinetic equation (38) by ˆpˆp−1↔/3 and average over ˆp. We get\n∂\n∂t/an}b∇acketle{t(ˆpˆp−1↔/3)[ψˆp−δ¯ǫˆp]/an}b∇acket∇i}ht+vF/an}b∇acketle{t(ˆpˆp−1↔/3)ˆp·∇ψˆp/an}b∇acket∇i}ht=−1\nτ/an}b∇acketle{t(ˆpˆp−1↔/3)ψˆp/an}b∇acket∇i}ht. (55)\nIn the limit of small gradients and time derivatives, the time-derivativ e term and the coupling to third order term of\n(54) in the gradient term of (55) can be neglected. This allows to writ e the momentum flux tensor (35) into the form\nΠij=P0δij+/parenleftbiggρs\nmBδµB+δP∗/parenrightbigg\nδij−η/parenleftbigg∂vni\n∂xj+∂vnj\n∂xi−2\n3∇·vnδij/parenrightbigg\n, (56)\nwhere effective pressure of the normal component\nδP∗=KnFδµB+1\n3m∗v2\nF(1+F0)δnF (57)\nand the coefficient of viscosity\nη=1\n5nFvFpFτ. (58)9\nThe momentum conservation (21) combined with the flux (56) gives a n equation of motion. Combined with the\nsuperfluid equation of motion (18), the fermion conservation law (1 9) and the boson conservation law (36) gives a\ncomplete set of linearized hydrodynamic equations. Using variables ( δnF,δµB,vn,vs) the set can be written as\n∂δnF\n∂t+nF∇·vn= 0, (59)\nnB\ns2∂δµB\n∂t+ρs∇·vs+mBKnF∇·vn= 0, (60)\nρn∂vn\n∂t+∇δP∗=η∇2vn+1\n3η∇∇·vn, (61)\nmB∂vs\n∂t+∇δµB= 0. (62)\nHere (61) is the Navier-Stokes equation describing the viscous nor mal component.\nOne application of the equations (59)-(62) is to determine the veloc ities of first and second sound modes. The\nresults are identical to those found by Khalatnikov2.\nXI. DISCUSSION\nForstationary phenomena, the Fermi-Bose liquid theory does not differ from the no ninteracting system, when\nwritten in terms of proper variables. This follows because dropping t he time derivative term in the kinetic equation\n(27), theonlydifferencetononinteractingcaseisthatthedistribu tionδnpisreplacedby δ¯np. Theboundaryconditions\n(48)-(49) and the observables (33)-(34) also are functions of δ¯npand they depend on the interaction parameters m∗,\nα,DandFl’s in a simple scaling manner, if at all.\nThe force applied to slowly moving objects in a Fermi liquid in the ballistic lim it was calculated in Ref. 14. All the\nresults presented in Ref. 14 [Equations (13)-(19)] concern the t ime-indepent case. Therefore, their independence of the\ninteractions parameters m∗,α,DandFlfollows most simply from the general argument of the preceding par agraph.\nIn time dependent problems, the interactions haveimportant effec t through the δ¯ǫ-term in the kinetic equation (27).\nFor small frequencies, however, the coupling to the superfluid mot ion can be neglected, as argued in Sec. VII. Thus\nat low frequencies the response of Fermi-Bose liquid to external pe rturbation is the sum of independent superfluid\nand Fermi-liquid responses.\nThe theory formulated here is applied to calculate the force on a vibr ating cylinder in Refs. 3, 4, 14, and 15.\n1L. D. Landau, Sov. Phys. JETP 3, 920 (1957).\n2I. M. Khalatnikov, Sov. Phys. JETP 28, 1014 (1969).\n3T. H. Virtanen and E. V. Thuneberg, ”Pendulum in Fermi liquid ”, arXiv:1010.xxxx.\n4T. H. Virtanen and E. V. Thuneberg, to be submitted to arXiv:c ond-mat.\n5A. A. Abrikosov and I. M. Khalatnikov, Rep. Prog. Phys. 22, 329 (1959).\n6P. Nozieres, Theory of interacting Fermi systems (Bejamin, New York 1964).\n7E. M. Lifshitz and L. P. Pitaevski˘ ı, Statistical Physics, Part 2 (Pergamon, Oxford 1980).\n8G. Baym and C. Pethick, Landau Fermi-liquid theory (Wiley, Ney York 1991).\n9J. Bardeen, G. Baym, and D. Pines, Phys. Rev. 156, 207 (1967).\n10C. Ebner and D. O. Edwards, Phys. Rep. 2, 77 (1971).\n11I. M. Khalatnikov, An introduction to the theory of superfluidity (Benjamin, New York 1965).\n12J. Sykes and G. A. Brooker, Ann. Phys. (N.Y.) 56, 1 (1970).\n13I. L. Bekarevich and I. M. Khalatnikov, Sov. Phys. JETP 12, 1187 (1961).\n14T. H. Virtanen and E. V. Thuneberg, AIP conference proceedin gs850, 113-114 (2006). The diffusive boundary condition,\nEq. (8) of this reference, should be replaced by equation (51 ) of the present paper.\n15T. H. Virtanen and E. V. Thuneberg, J. Phys. Conference Serie s150, 032115 (2009)."    },    {        "title": "1810.00102v1.Influence_of_host_matrices_on_krypton_electron_binding_energies_and_KLL_Auger_transition_energies.pdf",        "content": "1 \n Influence of host matrices  on krypton electron binding energies and KLL \nAuger transition energies  \nА.Kh. Inoyatova,b, L.L. Perevoshchikov a, A. Kovalíka,c, D.V. Filosofova, Yu.V. Yushkevicha,  \nM. Ryšavýc, B.Q. Leed, T. Kibédid, A.E. Stuchberyd, V.S Zhdanove,  \n \na Laboratory of Nuclear Problems, JINR, Dubna, Moscow Region, Russian Federation  \nb Institute of Applied Physics, National University, Tashkent, Republic of Uzbekistan  \nc Nuclear Physics Institute of the ASCR, CZ -25068 Řež near Prague, Czech Republic  \nd Department of Nuclear Physics, RSPE, The Australian National University,  \n  Canberra,  ACT 0200, Australia  \ne Nuclear Physics Institute, Almaty, Kazakhstan  \n \nKeywords:  Electron spectroscopy; Ion implantation; Vacuum evaporation;  83Kr; 83Rb; 83Sr; \nAtomic environment ; Electron binding energy ; Chemical shift; Conversion electrons;  Auger \neffect; KLL transitions;  Transition energy ; Atomic structure effect; Natural atomic level  width ; \nMulti configuration Dirac -Fock calculations;   \nAbstract  \n The low -energy electron spectra  emitted in the radioactive decay of the 83Rb and  83Sr \nisotopes  were  measured with a combined electrostatic electron spectrometer. Radioactive sources \nused were prepared by ion implantation of 83Sr into a  high purity polycrystalline  platinum foil at \n30 keV and by vacuum -evaporation deposition of 83Rb on the same type of foil. From the \nmeasured conversion electron spectra , the electron binding energies  (referenced to the Fermi \nlevel ) for the K, L 1, L2, L3, M 1, M 2, and M 3 shell/subshells of  krypton  in the  platinum host  were \ndetermined to be 14  316.4(12), 1  914.3(9), 1  720.3(9), 1  667.6(9), 281.5(9), 209.6(13), and \n201.2(15) eV , respectively , and those for the evaporated layer were observed  to be lower by \n0.7(1) eV . For bot h host matrices , values of 2.3(2), 4.6(2), 1.7(2), 1.3(2), and 3.2(3) eV were \nobtained for the krypton K, L 1, L2, L3, and M1 natural atomic level widths , respectively . The \nabsolute energies  of 10  838.5( 9) and 10  839.5(1 0) eV were measured for the KL 2L3(1D2) Auger \ntransition in krypton implanted in Pt and generated in the evaporated rub idium layer, \nrespectively . A value of 601.0( 8) eV was measured for the energy difference of the KL 2L3(1D2) \ntransitions in Rb and Kr in the Pt host. Multiconfiguration Dirac -Fock calculations of the krypton \nKLL transition energies and intensities were  also performed.  \n1. Introduction  \nExperimental investigation s of the influence of the atomic environment on absolute \nbinding energies of core electron s in medium  heavy  elements by means of “classical” electron \nspectroscopy methods (e.g. such as X -ray photoelectron spectroscopy  (XPS) ) are very rare also \ndue to  their technical limitations. This is especially true for the study of solid -host effects on \nabsolute core level binding energies in noble gases , whic h are chemically inert and are the only \nfree atoms . Thus, e.g., in  the pioneer works  [1,2] the binding energies of core electrons in Ne, \nAr, Kr, and Xe implanted in Cu, Ag, and Au were investigated by  X-ray photoemission and were \nfound to be 2 -4 eV smaller in magnitude than the corresponding binding energies obtained from \ngas-phase measurements. The aim of the se measurements  was to study contributions from both \nthe change in the self-consistent potent ial experienced by the core levels upon implantation (the \ninitial state) and the polarization of metal -host electrons upon photoionization (the final state) \nwhich are thought  to be responsible for the observed shifts.  The most studied core levels for \nwhich ex perimental data are published in Ref. [ 1,2] were K (Ne), L 3 (Ar), M 3 (Kr), and M 5 (Xe), 2 \n i.e. atomic levels with relatively low bi nding energies (several hundred  eV) easily accessible  by \nstandard XPS -apparatus.  \nThe “low energy nuclear electron spectroscopy” used, e.g., in the present work exhibits  \nconsiderable  advantages in this field  such as: i)  a high signal -to-background ratio;  ii) a wider \nspectrometer energy range  (from “0” to 50  000 eV in the case of our combined electrostatic \nelectron spectrometer [ 3]) enabling one to study much deeper core levels ; iii) various methods \nfor incorporation of parent  isotopes of daughter noble gases into different solid hosts ; iv) a \npossibility to study trace  amounts of elements ; v) electrons are emitted only by radioactive \natoms , etc.  \nIt should be noted that e xperimental information on the influence of the atomic \nenvironment on absolute core level binding energies in noble gases plays an essential role, for \nexample , in the preparation of stable electron energy calibration standards for some significant \nphysical experiments  also in the field of high energy ph ysics  and in electron spectroscopy  in \ngeneral . \nThere is also a lack of experimental data on the influence of atomic environments on the \nKLL Auger spectra especially for medium and heavy elements. T his information is important for \nboth basic research in this field and interpretation of weak effects in complex experimental \nAuger ele ctron spectra . \nIn the present paper we report results of our experimental investigation of environmental \neffects on th e electron binding energies, the absolute KLL Auger transition energies , and natural \natomic level  widths in krypton  generated in two different solid hosts. For this study , radioactive \nsources of both 83Sr (T1/2=32.4 h , Fig.  1) prepared by ion implantation at 30 keV into a high \npurity polycrystalline  platinum foil and 83Rb ( T1/2=86.2 d , Fig.  1) produced by vacuum \nevaporation on the same type of Pt foil were used.  \nOur investigations were performed in the frame of the development of a new technique \nfor the preparation of super stable cali bration 83Rb/83mKr electron sources [ 5,6] for the KATRIN \nneutrino mass experiment [ 7]. \n2. Experimental  \n2.1. Source preparation  \n Radioactive isotopes of strontium and rubidium were obtained by spallation of metallic \nyttrium by 300 MeV protons from  the internal beam of the phasotron particle accelerator at the \nJINR, Dubna, Russia.  After “cooling ” for three days, the irradiated target (1 g weight) was \ndissolved in the concentrated nitric acid.  Strontium isotopes were chemically separated from the \ntarget m aterial and other elements using  “Sr resin” (TrisKem International ). An additional \npurification of strontium was then carried out on a cation -exchange chromatography column (70 \nmm length, 2 mm diameter, A6 resin) also in а nitric acid medium.  Afterwards , the strontium \nfraction (82Sr (T1/2=25.3 d ), 83Sr (T1/2=32.4 h ), 85Sr (T1/2=64.9 d )) obtained was used for the \nelectron source preparation .  \n Chemical separation of the rubidium fraction (83Rb (T1/2=85.2 d ), 84Rb (T1/2=32.8 d) ) \nfrom the target material and other elements was also performed on the above “Sr resin” followed \nby twofold purification on the cation -exchange chromatography column  in a nitric acid medium . \n The m ass separation of the strontium isotopes was performed  on a mass separator at  the \nJINR, Dubna . At the same time , the strontium ions were embedded  at the energy of 30 keV  into \nthe high purity polycrystalline  platinum foil . Prior to use, the surface of the Pt foil s was  cleaned \nonly by alcohol. A part of the foil containing t he Sr isotope  with the atomic mass number A  = 83 \nwas cut out and used for the electron spectrum measurements. The size of the “active” spot  was \nabout 2x 2 cm2. The a ctivity of the 83Sr isotope in the source upon  the preparation  was 11.7 MBq.  \nAccording to the simulations [ 5] performed for the implantation of 83Rb ions  at 30 keV  \ninto the same type of platinum foil as used in our experiment, the mean projected range and 3 \n standard deviation of the distribution of the implanted atoms along the depth of the foil \namounted to about 9 and 4 nm , respectively . Real circumstances of the implantations [ 5] such as \nthe zero ion incident angle (relative to the source foil normal), p olycrystalline structure of the Pt \nfoil, and an adsorbed surface contamination layer of water and hydrocarbons  (so called “rest gas \nlayer ”) were  represented  in the simulations  by an additional 3 nm thick pure carbon layer on the \nfoil surface . These experimental conditions were  very close to our ones.  It was , moreover,  found \nin Ref. [ 5] that a portion of 5-10% of the incident 83Rb ions gets stopped in the  “rest gas layer ”, \nwhich represents a non -metal lic environment. As surface  of the Pt foil was  not cleaned (by ion \nsputtering or any other means ) after the 83Rb ion  collections, a certain portion of 83Rb thus \nremain s in this non -metal lic environment . This feature also pertains to  our 83Sr source . After the \ncollection  of the 83Sr ions, the source was  exposed to air during its transfer to the electron \nspectrometer. The 83Sr source  surface  was also not cleaned before measurements of electron \nspectra. Thus a portion of 83Sr ions stopped on the Pt foil surface  in the non -metallic \nenvironme nt were  bound with atoms of oxygen in all possible forms (oxides, hydroxides, \ncarbonates, hydrocarbonates, etc.) a nd had the oxidation number +1  (see below).   \n Thermal evaporation deposition of rubidium  on the Pt foil  took place at 900 °C. Prior to \nthe use, the surface of the foil was cleaned  by alcohol . An aqueous solution of a carrier free \nrubidium nitrate was transferred to an annealed Ta evaporation boat and dried up. To remove \npossible volatile organic compounds, the Ta evaporation boat with deposited activity was first \nheated at lower te mperature for about 30 s. The source backing was shielded throughout this \nprocedure . During the source evaporation, the source backing rotated around its ax is at a speed \nof 3000 turns/min  at a d istance of 10  mm from the Ta evaporation boat to improve homogeneity \nof the evaporated layer. Activities of the 83Rb and 84Rb isotopes in the prepared source  of 8 mm \ndiamet er were  2.2 and 0.3 MBq, respectively.  \n The exact “chemical state ” of the deposited trace amount s of 83Rb on the surface of the Pt \nsource backing in vacuum  was unknown. However, the source was  transferred to the electron \nspectrometer in air after the preparation and thus the evaporated layer was  exposed to  it. Due to \nextreme rubidium reactivity  to air , ions of 83Rb were bound with atoms of oxygen in all possible \nforms ( oxides, hydroxides, carbonates, hydrocarbon ates, etc. of  different proportions ) and  had \nthe oxidation number  +1. This statement is based on : (i) specif ic chemical properties of \nrubidium , (ii) its known macro -chemistry , (iii)  the inner self -consistency of the physicochemical \nmethods used for the preparation of the source , and (iv) conditions of its treatment . After the  EC \ndecay  of 83Rb, the daughter 83Kr atoms were  stabilized in the above rubidium matrix and \napparently they were not charged.  \n 2.2. Measurement s and energy calibration  \nElectron spectra  were measured in sweeps using the above mentio ned combined electrostatic \nelectron spectrometer [ 3] consisting of a retarding sphere followed by a double -pass cylindrical \nmirror energy analyzer. The choice of the a bsolute spectrometer energy resolution and the \nspectrum scanning step depended on the intensity  of the  radioactive source being measured  and \nthe complexity of the i nspected spectrum.  Examples of the measured spectra are shown in Figs. \n2-5. \nAltogether 14 low energy conversion electron lines (listed in parentheses) of nuclear \ntransitions in 169Tm with energies Eγ = 8.41008 (21) [9] (M 1,2, N1,3), 20.74378 (10) [10] (L 1-3, M 1-\n3, N 1-3), and 63.12081 (5) keV [ 11] (K) along with five lines (K, L 1-3, M 1) of the 14.41300 (15) \nkeV [ 11] nuclear transition in 57Fe were used for calibration of the spectrometer energy scale . \nEnergies of the calibration lines related to the Fermi level EF(i) (i is the atomic subshell index) \nwere evaluated as EF(i) = Eγ - Eb,F(i) making  use of the experimental Fe and Tm electron binding \nenergies Eb,F(i) [12] related to the Fermi level . Experimental uncertainties from 0.4 to 1.6 eV and \nfrom 0.4 to 0.9 eV are quoted in Ref. [ 12] for thulium and iron electron binding energies, \nrespectively. Thus in all cases the Eb,F(i) uncertainties fully dominate uncertainties of the above 4 \n nuclear transition energies. Therefore uncertainties in the calibration line energies were almost \nidentical with those of the relevant electron binding energies.  \nIn the case of Fe, the electron binding energies [ 12] are given for an oxide but the chemical \nform f or Tm is not specified . Chemical techniques applied in the preparation of our calibration \nsources by vacuum evaporation on polycrystal line carbon substrate should guarantee the oxide \nstate for both elements. For the case that it is not true , we took into account maximum measured \nchemical shifts of the outer  subshell electron binding energies of about 2 and 4 eV [ 13] among \ndifferent chemical states of Tm and Fe, respectively. We found that the influence of these shifts \non the energy calibration is well below the standard deviations of the measured absolute energies \nof the studied electron lines (quoted in Table s 1, 7 and in the text).  \n2.3. Spectra  evaluation  \nThe measured electron spectra were evaluated using the approach and the computer code  \ndescribed in Ref. [14]. The individual spectrum line shape was expressed by a convolution of a \nGaussian (“spectrometer response function  to monoenergetic electrons ”) and an “artificial” \nfunction which describes the natural energy distribution of the investigated electrons leaving  \natoms (Lorentzian) distorted by manif old inelastic scattering of the electrons in the source \nmaterial (surface and volume plasmon excitations, shake -up/-off, atomic and lattice excitations, \netc.). In order to  find the proper form of the  complex discrete energy loss peak  and its long low \nenerg y tail (going down to the “zero” energy), multiple fitting of th e measured electron spectru m \nwith random variations ( Monte Carlo method) of the shape of the  fitted lines in  this “energy loss  \nregion ” within the pre -set shape limits was applied. It should be  noted  that t he position and width \nof the discrete energy loss peak  (DEL – see, e.g., Figs. 3-5) depend on the source material , while \nits intensity relies on a ratio of the energy -dependent mean free path for inelastic electron \nscattering and the effective  source thickness . (Contrary to the inelastically scattered electrons , \nthe electrons which left the electron source without any energy loss create a so -called zero loss \n(or no -loss) peak . This peak can be described by a  simple convolution of a Gaussian  and a \nLorentzian  resulting in a Voigt function. ) \nIn the evaluation , the fitted parameters were the position, the height, and the width  (of the \nno-loss peak)  of each spectrum line, the constant background and the width  of the spectrometer \nresponse function. Results of the evalu ations are shown in Figs. 3 and 4 (continuous lines), in \nTables  1,6,7 and in the text.  The quoted uncertainties are our estimates of standard deviations \n(σ). \n3. Results and discussion  \nAs can be seen from Figs. 3 -5, the discrete energy loss peaks  (DEL)  of the conversion \nelectron lines  for the 83Rb source prepared by vacuum evaporation on the Pt foil are higher than \nthose for the 83Sr source prepared by ion implantation into the same foil . On the other hand , the \nlow energy tails of the spectrum lines converge faster to the background level for the evaporated \nsource. This featur e is caused by manifold scattering of the electrons in the Pt foil.  \n Energies (re ferenced  to the Fermi level) of the K -32, L1-3-9.4, and M 1-3-9.4 conversion \nelectron lines from the deexcitation of  83mKr (generated in the EC decay chain of 83Sr, see Fig.1 ) \nmeasured with the  83Sr source prepared by  ion implantation into the Pt foil are shown in Table 1 . \nIn Table 2, the derived  (with the use of these conversion line energies)  electr on binding energies \nin Kr incorporated  into the Pt foil are compared with the experimental values for gaseous \nkrypton  (referenced to the vacuum level) . Electron binding energies measured for the M 3-\nsubshell of Kr implanted in Cu and Ag metals (previous works), and Pt (this  work) are displayed \nin Table 3 . In Tables 4 and 5, e lectron binding energies for some subshells  of Kr in different \nenvironments related  to those of  the L 1 and K ones , respectively, are given. Experimental natural \nwidths of the K, L 1,2,3, and M 1 atomic levels of krypton in different surroundings are compared  in \nTable 6 . Measured and calculated (also in this work) energies and relative intensities of the KLL 5 \n Auger transitions in Kr loc ated in different enviro nments are displayed in Table 7 . In Table 8, \nthe energy difference of the KL 2L3(1D2) transitions in Rb and Kr measured in the present work is \ncompared with theoretical predictions.  \n 3.1. The krypton K -, L-, and M -subshell electron binding energies  \n From the electron spectra  (see Figs. 3-5) measured with the  83Sr source prepared by ion \nimplantation into the Pt  foil, we determined energies  EF(i) (referenced to  the Fermi level) of the \nK and L, M conversion electron lines of the 32  keV E3  and 9.4 keV M1+E2 nuclear transitions \nin 83mKr (T1/2=1.83 h) , respectively  (see Fig.1) . They are pres ented in Table 1 . Using the latest  \nenergy values  of the above nuclear transitions in 83mKr, namely  Eγ,9.4 = 9 405.8 (3) eV [15] and  \nEγ,32 = 32 151.7 (5) eV [16], the electron binding energies  Eb,F(i) (referenced  to the Fermi level ) \nin krypton  in the matrix of the platinum host  were obtained using the following equation:  \n   Eb,F(i) = Eγ,j + Eγ,j,rec – EF(i) – Erec(i)   (1) \nHere, i is the krypton atomic shell/subshell index, j = 9.4 or 32.1  keV, Eγ,j,rec is the energy of the \nrecoil atom after γ-ray emission ( Eγ,9.4,rec  = 0.002 eV, Eγ,32.1,rec  = 0.007 eV)  and Erec(i) is the \nenergy of the recoil atom after conversion electron emission  (Erec(K) = 0.12 eV, Erec(L1) = 0.05 \neV). They are compared in Table 2  with the expe rimental values for gaseous krypton of the \ncompilation [ 12] and results of the latest evaluation [ 17] for the K shell and L subshells of gas-\nphase  krypton . These values  are, however, referenced to the vacuum level which differ s from the \nFermi level by the work function of  our spectrometer . It is seen from the table  that (as expected ) \nthe electron binding energies in implanted krypton are considerably lower  in magnitude  (also due \nto the different reference level ) and, generally , the observed difference  increase s with increasing \natomic shell/subshell . In the case of th e gas -phase values [ 12], the M -subshell differences are \nhigher than the L -subshell ones by about 20% a nd for the data [ 17], the L 2,3 difference exceed  \nthe L 1 difference by a factor of 1.5. This finding  contradicts the conclusion made from results of \nthe above me ntioned  measurements [1,2] based on the X -ray photoelectron spectroscopy  that the \nbinding energy shifts are “essentially identical” for the core levels of the investigated rare gases  \nNe, Ar, Kr, and Xe  implanted at 1 keV in the noble metals Cu, Ag, and Au  (the determine d \naverage range s of the implanted  atoms at this energy were comparable to the mean escape depth \nfor the investigated photoelectrons and span from 1 to 2 nm ). Moreover, a drastic decrease by \n6.4(15) eV (compared with 0.8 and 0.9 eV determine d in Ref.  [2] for Ne and Ar, respectively) of \nthe M 3 binding energy  in Kr  is seen from Table 3  between the Ag  [2] and Pt  (this work)  host \nmatrices  (no data are given in Ref. [ 2] for Au host  due to troubles with the Kr implantation ). We \nhave no explanation for this observation.  \n Because of the different reference levels used and their approx imations , in Table 4  we \ncompared  the relative elect ron binding energies  in Kr for different environments , namely the \nhigh purity Pt polycrystalline  host matrix  of the present work , the evaporated layer of the parent \n83Rb on a naturally  oxidized aluminum foil  of Ref. [18], and 83mKr frozen at 4.2 K onto a highly \noriented pyrolytic graphite  in Ref. [ 19]. Values for gaseous krypton of both the compilation  of \nRef. [12] and the latest evaluation [ 17] are provided in Table 4  too. Our values were obtained \nfrom relative positions of the L - and M -9.4 keV conversion electron lines on the scale of the \nelectron retarding voltage  applied  [3]. Thus a higher precision was reached.  The L1-3 and M 1-3 \nelectron binding energies were related to  the L 1-subshell. It is seen from the table that \nexperimenta l values for the L -subshells in  different surroundings agree very well with each other \nand also with the compil ed data [ 12] for gas -phase Kr while the evaluation [ 17] gives differenc es \nlower  in ma gnitude  by almost the same value of 4.5 eV  (i.e. as if the L 2,3 levels are located closer \nto the L 1 level  in Ref. [ 17]). Nevertheless, a very good agreement is seen for the M 1 subshell  \namong the experimental differences  for the M 1-3 subshells in solids . The M 2,3 subshells are, \nhowever, moving away from  the L1 subshell  starting from the frozen Kr to implanted Kr into the \nPt foil (but the agreement within 2σ is kep t). The M 1-3 subshells as a whole in gaseous Kr  [12] 6 \n are closer to the L 1 level  in comparison with our data  for implanted Kr  (but again within 2σ) . \nThus i t generally seems that the M 2,3 shells are shifted  more and more away from the L 1 as one \ngoes from the gas -phase Kr through the frozen and “evaporated ” forms to Kr in the platinum \nhost matrix.  This can be explained by the changes in initial -state potential at the core produced \nby the compression of the valence wave functions in going from the atom s to solids.  \nIn Table 5, the L and M 1 electron binding energies  related to the K shell one determined \nfor Kr in corporated into  the Pt host matrix and  83mKr froze n onto a backing at 4.2 K [ 19] are \ncompared with v alues  obtained from data for gaseous krypton of both the compilation [ 12] and \nthe latest evaluation [ 17]. An agreement within 1σ is seen for corresponding shells , not only \nbetween the different solids , but also between the solids and gas -phase Kr , with the exception of \nthe L 1 shell value [ 17] which is higher by 3.7(8) eV  (4.6 σ) than ours and by 3.3(11) eV tha n the \ngas-phase difference [ 12]. As mentioned above, the L 2,3 subshells are closer to the L 1 one in Ref. \n[17] by 4.5(4) eV and 4.1(10) eV than in the present  work  and compilation [12], respectively  \n(see Table 4 ). This finding indicates that the L 1 electron binding energy [ 17] in gaseous Kr may \nbe incorrect.  From the abov e agreement observed in Table 5  for the L and M 1 electron binding \nenergies related to the K shell , a conclusion can be made that different surroundings have no \ninfluence on the relative K, L 1-3 and M 1 binding energies within the uncertainties quoted.  \n By the above approach, we also determined the K-, L1-, and M 1-electron binding energies \nin krypton generated in the EC decay of 83Rb in a n evaporated rubidium layer on the  same type \nplatinum foil.  The measured spectrum of the corresponding  conversion electron lines of the 9.4 \nkeV nuclear transition  is shown in Fig. 4 . The obtained  values  were lower by 0.7(1) eV than \nthose measured  for the implanted Kr.   \n3.2. Natural widths of the krypton K -, L1,2,3, and M 1 atomic leve ls \nFrom the well resolved and sufficiently intense conversion electro n lines , we also \nextracted  their natural widths  by two different ways . First, t he computer code [ 14] used for the \nspectrum evaluation give s them as fitted parameters. In addition, we performed a dedicated \nevaluation  by the computer code  [25] using only a part of the measured conversion lines , namely \nits high -energy slope and a s few points below the peak as possible, usually two (to set the li ne at \nits proper position but still eliminate the effect of the line asymmetry).  These spectrum line \nfragment s were  fitted by the Voig t function . In the fitting, the Gaussian half -widths (the absolute \ninstrumental resolution of the spectrometer) was kept fixed. The determined values of the natural \nwidths  of the K, L 1, L2, L3, and M 1 levels  of Kr in the  host of the  platinum  are compared in  Table \n6 with those obtained for: i) Kr arising from  the EC decay of 83Rb evaporated on a naturally \noxidized aluminum  [18]; ii) frozen Kr on  highly oriented pyrolytic graphite [ 19], and iii) gas-\nphase Kr  [20] (a compilation  of experimental data ). As can be seen from the table, our values \ngenerally agree well with those of Ref. [ 18] for Kr arising from  a rubidium layer evaporated on a \nnaturally oxidized aluminum . They are, however, substantially lower than the values measured in \nRef. [ 19] for frozen  Kr for the K, L 1, and M 1 levels . A considerable difference  between the \nnatural widths measured for the gas -phase krypton and Kr in solids  is observed for the L 1 \nsubshell . However,  “a considerable line broadening compared with widths of corresponding lines \nobserved in spectra of free gaseous atoms” as stated in Ref.  [2] for Ne, Ar, Kr, and Xe implanted \nin Cu, Ag, and Au was  not observed in our work.  Within the quoted errors, we did not find any \nnatural width difference for the given  atomic level between  the source s prepared by the 83Sr \nimplantation in the Pt foil and by the vacuum evaporation of 83Rb on the Pt foil.  \n 3.3. The krypton KLL Auger spect rum 7 \n  In the measured electron spectra  from  both the decay chain of 83Sr impl anted i nto the \nplatinum foil and  the EC decay of 83Rb in the evaporated layer on the same type  of platinum foil, \nthe K LL Auger lines of krypton  were also observed (see Fig. 2 ). We concentrated only on the \ndetermination of the absolute transition energies as the KLL Auger spectrum of Kr was \nthoroughly studied in Ref. [ 23]. From the measured spectra we obtained energies of 10  838.5(9) \nand 10  839.5(10) eV (referenced to the Fermi level) for the KL 2L3(1D2) transition in Kr \nimplanted into Pt and generated in the evaporated layer, respectively, i.e., higher by 1.0(2) eV in \nthe latter case (see also Table 7). The observed increase of the krypton KL 2L3(1D2) transition \nenergy for the eva porated 83Rb source agrees very well with the difference of +0.8(2) eV \ndetermined in our work [24] for the rubidium KL 2L3(1D2) transition energy between the 85Sr \nsources prepared by vacuum evaporation and implantation into the same type of platinum foil.   \nThe measured KL 2L3(1D2) transition energy of 10  839.5(10) eV for Kr  arisen  in the \nevaporated layer on the Pt backing is higher by 2.1(14) eV than the energy of 10  837.4(10) eV \n(related to the Fermi level) measured in Ref. [23] for the KL 2L3(1D2) transition  in Kr generated \nin the EC decay of 83Rb evaporated on a naturally oxidized aluminum foil.  This fact indicates \nthat in the case of evaporated sources, the source backing material also plays an import role as \nregards the “chemical state” of the radioactive atoms emitting the studied electron s. \nThe semi -empirical energy of 10  824.9 eV [ 21] for the KL 2L3(1D2) transition in gaseous \nkrypton is lower by 13.6(9) and 14.6(10) eV than our above experimental values for Kr \nimplanted in platinum and generat ed in the evaporated layer on the Pt foil, respectively. The \nKLL transition energies [21] were calculated for recombination of the initial K vacancies created \nby, e.g.,  the internal conversion in daughter 83Kr (our case) and are referenced  to the vacuum \nlevel  while the primary K vacancies in our solid -phase sources are produced by the electron \ncapture decay and the measured transition energies are referenced to the Fermi level. The \nobserved differences between the theoretical [ 21] and exper imental transition energies cannot be \ncompensated by corrections of the semi -empirical value for both the solid state effect (increasing \nthe Auger transition energies by 6.2 and 7.5 eV [ 21] for rubidium and platinum, respectively) \nand a change of the vacuu m reference level to the Fermi one (decreasing the transition energies \n[21] roughly by the work function of our spectrometer equal to about 5.7 eV) as these corrections \nnearly cancel each other as a result.  \nHaving the electron binding energies determined i n the present work for Kr implanted in \nthe platinum host (Table 2), we modified the semi -empirical KL 2L3(1D2) transition energy [ 21] \nfollowing the recommendation given in Ref. [ 21] (with the exception of accounting for a solid -\nstate correction  term not provided in Ref. [ 21] for our case ). The obtained value of 10  829.9 (17) \neV is higher by 5.0(17) eV than the original prediction [ 21] but still lower by 8.6(19) eV than the \nmeasured transition energy for implanted Kr (see Table 7 ). The omitted sol id-state correction  \nterm increases  (as mentioned above)  the calculated Auger transition energies, e.g., by  6.2 and 7.5 \neV for Rb and Pt  solids  [21], respectively. Thus i ts consideration  as well as the below  “atomic \nstructure effect” (increasing transition energies  by about 6.5 eV for Kr)  shoul d bring the \nmodified value  [21] to a better  agreement with the experiment.   \n \n We also performed calculations of energies  and intensities of the KLL Auger transition in \nkrypton (Z=36) following the creation of initial vacancies by both the electron capture decay of \nthe parent 83Rb isotope and internal conversion of gamma rays in the daughter 83Kr. The aim was \nto obtain also an estimati on of a possible contribution from  the “atomic structure effect” to the \nabove discrepancy between the semi -empirical  [21] and experimental KL 2L3(1D2) transition \nenergy  values . This effect  was revealed for the first time in X -rays (see, e.g., Ref. [26]). Higher \nobserved energies of the K Auger transitions following the EC decay  (see, e.g., Ref. [27] and \nreferences therein ) are explained as a result of additional screening of the daughter nucleus by a \n“spectator” electron , as the 10-16-10-17 s lifetime of the 1 s atomic hole produced in the EC decay \nis so short that the intermediate state has the outer -electron configuration close to the parent \natom.  Thus i n the calculations, we assumed the atomic configuration of neutral krypton for the 8 \n IC de cay while for the EC decay, a 5s electron was added to a neutral Kr atom. These \" ab initio \" \nmulticonfiguration Dirac -Fock (MCDF) calculations used the actual electronic configurations of \nboth the initial and final states in the KLL Auger transitions, inclu ding the vacancies. Both \nrelativistic effects and quantum -electrodynamic  (QED ) corrections were taken into account. \nEach of the Auger transition energies were evaluated as the difference of the total energies of the \ninitial and final states and are referen ced to the vacuum level. The calculations were performed \nfor free atoms , i.e., without accounting for the solid state effects. A more detailed description of \nthe calculations is given in  the appendix of Ref.  [24]. Results of the calculations are presented in \nTable  7. It is seen from the table  that the “atomic structure effect” increases the KL 2L3(1D2) \ntransition  energy i n Kr by 6.5 eV. O ur transition energy calculated for the IC processes is , \nhowever,  higher by 5.4 eV than the semi -empirical value [ 21] though both values are referenced \nto the vacuum level  and concern gas -phase Kr . No explanation is available for this finding  \nIt should be  also noted th at, as in the case of rubidium [24], relative energies of the basic \nkrypton spectrum components calculated in the present work are substantially greater than  both \nthe measured [ 23] and semi -empirical [ 21] values with the exception of the lines  lying close to \nthe KL 2L3(1D2) one. In other words, our calculated spectrum occupies an energy interval wider  \nby 13.1 (7) and 18.9 eV than the measured [ 23] and calculated [ 21] ones, respectively . As stated \nin Ref. [ 24], no explanation is  also available for this finding at present time.  \nIn regard to  the relative transition intensities , our calculated values agree well with both \nthe relativistic calculations [22] in intermediate coupling with the configuration interaction and \nthe experimental data [ 23] (see Table 7 ). \n \n3.4. The energy difference of the KLL transitions in Kr and Rb  \n \nAs me ntioned above, in the EC decay chain of the 83Sr isotope  (see Fig. 1) the KLL \nAuger electrons of both rubidium and krypton are also emitted. After about four 83Sr half-lives \nfrom the source preparation , the KLL Auger lines of the two  elements were clearly observed and \nwell resolved  in our spectra (see  Fig. 2)  measured with the 83Sr source prepared by the ion \nimplantation into the Pt foil . From the relative position  of the measured KLL spectra on the \nelectron retarding voltage scale [ 3] we determined with higher accuracy the energy difference of \nthe KL 2L3(1D2) transitions in rubidium and krypton to be 601.0 (8) eV. It is seen from Table 8 \nthat this valu e agrees very well with the  one determined from the absolute energies of the \nKL 2L3(1D2) transition s in krypton and rubidium measured with the same 83Sr source in the \npresent work  (Table 7)  and in Ref. [ 24], respectively.  \nThe transition energy difference of 605.0  eV obtained from  the MCDF  calculations  \nperformed in the present work for Kr and  in Ref. [ 24] for Rb is higher by 4 .0(8)  eV (i.e. by 5σ)  \nthan the measured value  (see Table 8) . For both elements, the MCDF calculations  were \naccomplished for the gas -phase system and for K-vacancy produc ed by the EC -decay .  \nThe semi -empirical calculations  [21] give the corresponding  transition energy difference \nto be  610.5 eV, i.e. higher by 9.5(8) eV than the experimental value  (Table 8) . However, in this \ncase the comparison is not correct because the KL 2L3(1D2) transition  energy [ 21] for Kr is \nreferenced  to the vacuum level and is valid for gas -phase system while that one  [21] for Rb was \ncalculated for a solid and is related to the Fermi level. But (as mentioned above)  a correction of \nthe krypton KL 2L3(1D2) transition  energy [ 21] for the referenced level and the solid -state effect  \nhas almost no influence on the energy value because the particular contrib utions nearly cancel \neach other . \nA value of 605.5(17) eV for the KL 2L3(1D2) transition energy  difference is obtained from \nthe calculations [ 21] when the modified krypton KL 2L3(1D2) transition energy  is used (see Table \n8). In this case, however,  the krypton  KL 2L3(1D2) transition energy  is lowered because no \ncorrection was performed for the solid -state effect  (increasing the transition energies [ 21]).Thus  \nits consideration should decrease the above energy difference by several eV but the exact value \nis unknown for our case (e.g., values of 6.2 and 7.5 eV are given in Ref. [ 21] for the Rb and Pt 9 \n solid -state correction terms,  respectively) . In addit ion, in our experiment the primary K \nvacancies in both elements are produced by the  EC decay. According to the results of the \" ab \ninitio \" MCDF transition energy calculations performed in the present work and in Ref. [ 24], the \n“atomic structure effect” connected with the EC decay increases the KLL transition energies by \n6.5 and 7.0 eV for Kr and Rb, respectively. As a result, the “atomic structure effect” increases \nthe KL 2L3(1D2) transition energy difference  of the two el ements  by 0.5 eV. Thus after the both \ncorrections the resulting transition energy difference can well approximates the exper imental \nvalue . This fact also indicates the meaningfulness  of the adaptation of the Auger transition \nenergies calculated for free atoms for the solid state hosts using the corresponding experimental \nelectron binding energies.  \n \n 4. Conclusion  \nUsing the high-resolution -low-energy “nuclear” electron spectroscopy developed in our \nlaboratory, we  determined electron binding energies, natur al atomic level widths, and the \nKL 2L3(1D2) transition energies in krypton in two different solid hosts , namely the  high purity \npolycrystalline platinum and the vacuum evaporated rubidium layer on the same type  of \nplatinum foil.  The results obtained clearly indicate sensitivity of the studied quantities to the \nenvironment of the Kr atoms. This fact has substantial importance  for, e.g., the development of \nelectron energy calibration standards for the KATRIN neutrino mass experiment [ 7]. Results of \nthe performed \" ab initio \" multiconfiguration Dirac -Fock calculations of the rubidium KLL \ntransition energies revealed a significant role of the “atomic structure effect” even for krypton.  \nAcknowledgement  \nThe work was partly supported b y grants G ACR P 203/12/1896, RFFI 13 -02-00756 , and the \nAustralian Research Council  (grant no. DP140103317).  \n \n \n  10 \n Table 1 \nEnergies EF(i) (in eV) related to the Fermi level of the K and L, M conversion electron lines of the 32 .1 and 9.4 keV \nnuclear transitions in 83mKr, respectively, measured  with a source  prepared by implantation of the 83Sr ions in Pt  foil. \n \nConversion line  K-32 L1-9.4 L2-9.4 L3-9.4 M1-9.4 M2-9.4 M3-9.4 \nEF(i) 17 835.3(11)a) 7 491.5(8)  7 685.5(8)  7 738.2(8)  9 124.3(8)  9 196.2(13)  9 204.6(15)  \na) Numbers in parentheses represent uncertainties in last digit(s).  \n \n \n \nTable 2 \nKrypton electron binding energies Eb,F(i) (in eV) related to the Fermi level determined from the measured K, L, and M \nconversion line energies listed in Table 1 and the corresponding nuclear transition energies [ 15,16] for Kr atoms im planted \ninto the  high purity polycrystalline platinum foil . Electron binding energies Eb,V(i) for gaseous Kr [ 12,17 ] are referenced to the \nvacuum level.  \n \nKr shell/subshell  (i) K L1 L2 L3 M1 M2 M3 \nEb,F(i) (this work ) 14 316.4(12)a) 1 914.3(9)  1 720.3(9)  1 667.6(9)  281.5(9)  209.6(13)  201.2(15)  \nEb,V(i) (gaseous Kr [ 12]) 14 327.2 (8) 1 924.6 (8) 1 730.9 (5) 1 678.4 (5) 292.8(3 ) 222.2(2 ) 214.4(2 ) \nImplanted – Gaseous [ 12]     - 10.8(14)   - 10.3(12 )    - 10.6(10)    - 10.8(10) - 11.3(10 ) - 12.6(13 ) - 13.2(15 ) \nEb,V(i) (gaseous Kr [ 17]) 14 327.26(4 ) 1 921.4(3 ) 1 731.91(3)  1 679.21(3)     \nImplanted – Gaseous [ 17]      - 10.9(12)      - 7.1(9)    - 11.6(9)    - 11.6(9)    \na) Numbers in parentheses represent uncertainties in last digit(s).  \n 11 \n Table 3 \nElectron binding energies Eb,F(M3) (in eV) referenced to  the Fermi \nlevel as measured for the M 3-subshell of Kr implanted in Cu, Ag [ 2] \nand Pt (this work).  \n \nHost matrix  Cu [2] Ag [2] Pt (this work)  \nEb,F(M3) 207.27(6) a) 207.60(5)  201.2(15)  \na) Numbers in parentheses represent uncertainties in last digit(s).  \n \n \n \nTable 4 \nThe L 2,3 and M 1-3 subshell electron binding energies  (in eV)  of Kr  referenced to the L 1 \nsubshell for  different solid s: a high purity Pt host matrix (this work);  an evaporated layer \nof the parent  83Rb on a naturally oxidized aluminum foil  (Ref [ 18]); 83mKr frozen onto a \nhighly oriented pyrolytic graphite  at 4.2 K (Ref. [ 19]). Values for gaseous krypton were \ntaken from  the compilation [ 12] and the latest evaluation [ 17]. \n \nSubshell  Experiment  Compilation/Evaluation  \nThis work  Ref. [ 18] Ref. [ 19] Ref. [ 12] Ref. [ 17] \nL1        0.0        0.0        0.0        0.0     0.0 \nL2    194.0(2)  a)    193.8(2)     193.7(6)     193.7(9 ) 189.5(3)  \nL3    246.6(2)     246.4(2)     246.4(6)     246.2(9 ) 242.2(3)  \nM1 1 632.8(2)  1 633.1(8)  1 632.8(6)  1 631.8(9 ) --- \nM2 1 704.7(8)  1 703.5(8 ) 1 702.7(6)  1 702.4(8)  --- \nM3 1 713.2(11)  1 711.1(9 ) 1 710.7(6)  1 710.2(8)  --- \na) Numbers in parentheses represent uncertainties in last digit(s).  \n  12 \n Table 5 \nThe L and M 1 electron binding energies (in eV) referenced to the K shell one measured for  Kr in the \nhigh pu rity Pt host matrix (this work) and  83mKr frozen onto a carbon backing at 4.2 K [ 19]. Data \nfor g aseous Kr  were obtained from  the compilation [ 12] and the latest evaluation [ 17]. \n \nShell  Experiment  Gaseous Kr  \nThis work  Ref. [ 19] Ref. [ 12] Ref. [ 17] \nK          0.0          0.0          0.0          0.0 \nL1 12 402.2(7) a) 12 401.3(9)  12 402.6(11)  12 405.9(3)  \nL2 12 595.9( 7) 12 595.0(9)  12 596.3(9)  12 595.4(3)  \nL3 12 648.8(7)  12 647.7(9)  12 648.8(9)  12 648.1(1)  \nM1 14 035.0(7)  14 034.1(9)  14 034.4(9)       ------        \na) Numbers in parentheses represent uncertainties in last digit(s).  \n \n \n \n \nTable 6 \nNatural widths  (in eV) of some atomic levels of  krypton in different surroundings: this work – \nhigh purity Pt  foil; Ref. [18] - naturally oxidized aluminum foil; Ref. [19] - gaseous Kr frozen \nat 4.2  K on a  highly oriented pyrolytic graphite ; Ref. [ 20] (a compilation)  – gaseous Kr.  \n \nAtomic level  K L1 L2 L3 M1 \nThis work  2.3(2) a) 4.6(2)  1.7(2)  1.3(2)  3.2(3)  \nRef. [ 18] 2.5(2)/2.4(1)c) 4.5(3) /4.5(1)c) 1.0(4) /0.8(2)c) 1.2(2) /1.2(1)c) 2.7(2) /3.0(3)c) \nRef. [ 19] 2.83(12)  5.30(4)  1.84(5)  1.40(2)  4.27(5)  \nRef. [ 20]b) 2.7(3)  3.8(9)  1.3(4 ) 1.2(4 ) 3.5(4) \na) Numbers in parentheses represent uncertainties in last digit(s).  \nb) Interpolated values from experimental data.  \nc) A value  obtained from our re -evaluation of the conversion electron spectra measured in  \n   Ref. [18]. \n  13 \n Table 7 \nMeasured and calculated energies (in eV) and relative intensities (KL iLj/ΣKLL , %) of the KLL Auger transitions in Kr located in \ndifferent environments: gaseous Kr [ 21,22 ,24]; Kr generated in the EC decay  of the parent  83Rb evaporated on a naturally oxidized \naluminum foil  [23]. EC - the initial vacancies  produced by the electron capture  decay; IC - the initial vacancies  produced by internal \nconversion.  \n \nTransition  Energy (eV)  Intensities (%) \nExperiment  Theory  Experiment  Theory  \nRef. [ 23] This work  Ref. [ 21] This work  [24] Ref. [ 23] This work  [24] Ref. [ 22] \nKL 1L1(1S0) - 430.4(6)a)  - 426.7  - 443.7  7.6(3)  8.0 7.7 \nKL 1L2(1P1) - 241.7(4)   - 240.0  - 248.9  15.2(3)  16.1 16.1 \nKL 1L2(3P0) - 214.1(7)   - 208.5  - 217.8  1.7(2)  1.7 1.7 \nKL 1L3(3P1) - 181.6(4)   - 178.7  - 186.6  6.8(2)  6.6 6.3 \nKL 1L3(3P2) - 159.6(5)   - 156.2  - 163.5  3.4(2)  3.6 3.4 \nKL 2L2(1S0)   - 58.2(5)     - 55.4   - 58.4 3.5(2)  3.5 3.5 \nKL 2L3(1D2) 10 837.4(10)b) EC 10 838.5(9)b,c) EC \n10 839.5(10)b,d) EC 10 824.9e)IC \n10 829.9(17 )b,f)IC 10836.8e) EC \n10830.3e) IC 47.4(8)  46.3 47.0 \nKL 3L3(3P0) + 42.7(4)   + 42.6  + 43.3 2.9(2)  2.8 2.8 \nKL 3L3(3P2) + 62.5(3)   + 60.4  + 62.3  11.5(2)  11.4 11.5 \na) Numbers in parentheses represent uncertainties in last digit(s).  \nb) Referenced to the Fermi level.  \nc) Kr implanted into the  high pu rity Pt foil.  \nd) Kr generated in the EC decay of the parent 83Rb evaporated on the  high pu rity Pt foil.  \ne) Referenced to the vacuum level (gas -phase Kr).  \nf) A value obtained in the present work by the use of the relevant electron binding  energies from Table 2  for Kr in the platinum host.  \n  14 \n Table 8  \nA comparison of t he energy difference s (in eV) of the KL 2L3(1D2) transitions in \nRb and Kr directly measured in the present work for the 83Sr source prepared by \nion implantation into the platinum foil with values obtained from results of the \nsemi -empirical calculations [ 21] (for the IC processes)  and the \" ab initio \" MCDF \ncalculations of the present work and Ref. [ 24] (for the EC decay) . \n \nExperiment  Theory  \nThis work  This work a) Ref. [ 21] Ref. [ 21] b) This work and  Ref. [ 24] \n601.0(8) c) 601.8(14)  610.5  605.5(17)  605.0  \na) The value obtained from the absolute KL 2L3(1D2) transition energies of Kr  (this \nwork)  and Rb (Ref. [ 24]) measured for the same 83Sr source.  \nb) The krypton KL 2L3(1D2) transition energy [ 21] modified by the use of the \nelectron binding energies determined in the present work (Table 2)  without \naccounting for the solid state effect .  \nc) Numbers in parentheses represent uncertainties in last digit(s).  \n 15 \n References  \n[1]   P.H. Citrin, D.R. Hamann, Chem. Phys. Letters, 22(1973)301.  \n[2]   P.H. Citrin, D.R. Hamann, Phys. Rev. B, 10(1974)4948.  \n[3]   Ch. Briançon, B. Legrand, R.J. Walen, Ts. Vylov, A. Minkova, A. Inoyatov,  \n Nucl. Instrum. Methods, 221(1984)547.  \n       (DOI: 10.1016/0167 -5087(84)90062 -0) \n[4]   R.B. Firestone, V.S. Shirley, Tables of Isotopes, 8th ed., Wiley, New York, 1996 , \n pp. 2119, 2152.  \n[5]   M. Zbořil, S. Bauer, M. Beck, J. Bonn, O. Dragoun, J. Jakůbek, K. Johnston, A. Kovalík,  \nE.W. Otten, K. Schlösser, M. Slezák, A. Špalek, T. Thümmler, D. Vénos, J. Žemlička,  Ch. \nWeinheimer, Journal of Instrumentation 8, P03009  \n (DOI: 10.1088/1748 -0221/ 8/03/P03009)  \n[6]   D. Vénos, M. Zbořil, J. Kašpar, O. Dragoun, J. Bonn, A. Kovalík, O. Lebeda,  \n N.A. Lebedev, M. Ryšavý, K. Schlösser, A. Špalek, Ch. Weinheimer, Measurement \nTechniques, 53(2010)573.  \n       (DOI: 10.1007/s11018 -010-9545 -3) \n[7]   http://www.katrin.kit.edu/  \n[8]   V.M. Gorozhankin, V.G. Kalinnikov, A. Kovalík, A.A. Solnyshkin,  A.F. Novgorodov,  \n N.A. Lebedev, N.Yu. Kotovskij, E.A. Yakushev, M.A. Mahmoud and M. Ryšavý,  \n J. Phys. G: Nucl. Part. Phys., 22(1996)377.  \n (DOI: 10.1088/0954 -3899/22/3/011  ) \n[9]   G.L. Borchert, W. Scheck, K.P. Wieder, Z. Naturf. A, 30(1975)274.  \n[10]  M. Lederer, V.S. Shirley, Tables of Isotopes,  7th ed., Wiley, New York, 1978,  \nAppendix 3.  \n[11]  R.B. Firestone, V.S. Shirley, Tables of Isotopes, 8th ed., Wiley, New York, 1996,  \n        Appendix C -3. \n[12]  K.D. Sevier, Atom. Data Nucl. Data Tables, 24(1979)323.  \n        (DOI: 10.1016/0092 -640X(79)90012 -3) \n[13]  A. V. Naumkin, A. Kraut -Vass, S.W. Gaarenstroom, and C.J. Powell,  \n         NIST Standard Reference Database 20, Version 4.1, (“X -ray Photoelectron  \n        Spectroscopy Database”), http://srdata.nist.gov/xps/selEnergyType.aspx  \n[14]  A. Inoyatov, D.V. Filosofov, V.M. Gorozhankin, A. Kovalík, L.L. Perevoshchikov,  \n  and Ts. Vylov,  J. Electron Spectrosc. Relat. Phenom., 160(2007)54.  \n        (DOI: 10.1016/j.elspec.2007.06.005 ) \n[15]  M. Slezák, D. Vénos, O. Lebeda, and T. Trojek, Eur. Phys. J A, 48(2012)12  \n  (DOI 10.1140/epja/i2012 -12012 -y) \n[16]  D. Vénos, O. Dragoun, A. Špalek, and M. Vobecký,  \n         Nucl. Instrum. Meth. A, 560(2006)352.  16 \n          (DOI: 10.1016/j.nima.2005.12.213 ) \n[17]  O. Dragoun, A. Špalek, F.J.Wuilleumier, Czech. J. Phys, 54(2004)833.  \n         (DOI: 10.1023/B:CJOP.0000038591.13369.e1)  \n[18]  A. Kovalík and V.M. Gorozhankin, J. Phys. G: Nucl. Part. Phys., 19(1993)1921.  \n         (DOI: 10.1088/0954 -3899/19/11/018 ) \n[19]  A. Picard, H. Backe, J. Bonn, B. Degen, R. Haid, A. Hermanni, P. Leiderer,  \n         A. Osipowicz, E.W. Otten, M. Przyrembel, M. Schrader, M. Steininger, and  \n         Ch. Weinheimer, Z. Phys. A – Hadrons and Nuclei, 342(1992)71.  \n         (DOI: 10.10 07/BF01294491 ) \n[20]  J.L. Campbell, T. Papp, Atom. Data Nucl. Data Tables 77, 1(2001).  \n         (DOI: 10.1006/adnd.2000.0848 ) \n[21]   F.P. Larkins, At. Data Nucl. Data Tables, 20(1977)311.  \n       (DOI: 10.1016/0092 -640X(77)90024 -9) \n[22]   M.N. Chen, B. Crasemann, H. Mark, Phys. Rev. A, 21(1980)442.  \n        (DOI: 10.1103/PhysRevA.21.442 ) \n[23]  A. Kovalík, V.M. Gorozhankin, A.F. Novgorodov, A. Minkova, M.A. Mahmoud,  \n  J. Electron Spectrosc. Relat. Phenom., 58(1992)49.  \n        (DOI: 10.1016/0368 -2048(92)80006 -T) \n[24]  А.Kh. Inoyatov, L.L. Perevoshchikov, A. Kovalík, D.V. Filosofov, Yu.V. Yushkevich,  \n  M. Ryšavý, T. Kibédi, B.Q. Lee, A.E. Stuchbery, V.S Zhdanov, 2014 (to be published).  \n[25]  M. Ryšavý, M. Fišer, Comput. Phys. Commun., 29(1983)171.  \n         (DOI: 10.1016 /0010 -4655(83)90072 -3) \n[26]  G.L. Borchert, P.G. Hansen, B. Jonson, I. Lindgren, H.L. Ravn, O.W.B. Schult and  \n   P. Tidemand -Petersson, Phys. Lett. A, 66(1978)374.  \n         (DOI  : 10.1016/0375 -9601(78)90063 -4) \n[27]  А.Kh. Inoyatov, L.L. Perevoshchikov, A. Kovalík, D.V. Filosofov, V.S. Zhdanov,  \n  A.V. Lubashevskii, Z. Hons, J. Electron Spectrosc. Relat. Phenom., 187(2013)61.  \n        (DOI: 10.1016/j.elspec.2013.04.009 ) \n  17 \n Figure captions  \n \nFig. 1 The incomplet e decay schemes of the 83Sr and 83Rb isotopes  [4]. \nFig. 2  An overview low -energy electron spectrum emitted in  the 83Sr decay measured with 21 eV \ninstrument al resolution and 7 eV step  and with 30 s exposition time per spectrum point  after four \n83Sr half-lives from the source preparation . The  spectrum was not corrected for the 83Sr decay \nand for the spectrometer trans mission drop  [3,8] with increasing electron retarding voltage. In \nthe spect rum, electrons following the EC decay of 83Sr to 83Rb and 83Rb to 83Kr are seen . In the \ninsert, a spectrum region including a part  of the KLL Auger spectrum of Kr  and the full KLL \nAuger spectrum of Rb  is shown on  an enlarge d scale.   \n \nFig. 3  The L and M subshell conversion electron lines of the 9.4 keV M1+E2 nuclear transition \nin 83Kr emitted in the EC decay chain of of 83Sr (see Fig. 1) . The source used was prepared by \n83Sr ion implantation into a high purity polycrystalline Pt foil at 30 keV. The spectrum was \nmeasured after twelve 83Sr half-lives from the source preparation with 7 eV instrumental \nresolution and 2  eV step size in 8 sweeps . The exposition time per spectrum point was 60 s . A \ndecomposition of the measured spectrum into individuals are displayed by continuous lines. The \n“discrete energy loss” pea ks (see section 2.3.  for explanation) are denoted by DEL.  The M1-3-9.4 \nconversion lines are shown in the insert in the enlarge scale.  \n \nFig. 4 The L and M subshell conversion electron lines of the 9.4 keV M1+E2 nuclear transition \nin 83Kr emitted in the EC decay of 83Rb. The 83Rb source was prepared by  vacuum evaporation \non a high purity polycrystalline  Pt foil. The spectrum was  measured with  instrumental resolution \nof 7 eV and 2  eV step  in 4 sweeps . The exposition per spectrum point  in each sweep  was 60 s. \nResults of spectrum decomposition are shown by continuous lines.  DEL means “discrete energy \nloss” pea k (see section 2.3. ). The insert  shows , on an enlarged scale, the M 1-3-9.4 conversion line \ngroup.   \nFig. 5  Examples of the K conversion electron line of the 32 keV E3  nuclear transition in 83mKr \n(generated in the  EC decay of 83Rb, see Fig. 1 ) taken at instrumental resolution of 7 eV with 2 \neV step  size. The 83Rb source s used were  prepared by vacuum evaporation on the Pt foil (full \ncircles) and  by implantation  of 83Sr ions at 30 keV into the same foil (open circles).  The “discrete \nenergy loss” pea k (see section 2.3. ) is denoted by DEL (upper spectrum).  18 \n Figure 1  \n \n \n19 \n Figure 2  \n \n \n  \n20 \n Figure 3  \n \n \nFigure 4  \n \n \n21 \n Figure 5  \n \n \n"    },    {        "title": "2309.10026v1.Origin_of_magic_angles_in_twisted_bilayer_graphene__The_magic_ring.pdf",        "content": "Origin of magic angles in twisted bilayer graphene: The magic ring\nWei-Chen Wang,1, 2,∗Feng-Wu Chen,1,∗Kuan-Sen Lin,1, 2Justin T. Hou,1, 2Ho-Chun Lin,1, 2and Mei-Yin Chou1, 2,†\n1Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106319, Taiwan\n2Department of Physics, National Taiwan University, Taipei 106319, Taiwan\nThe unexpected discovery of superconductivity and strong electron correlation in twisted bilayer\ngraphene (TBG), a system containing only spelectrons, is considered as one of the most intriguing\ndevelopments in two-dimensional materials in recent years. The key feature is the emergent flat\nenergy bands near the Fermi level, a favorable condition for novel many-body phases, at the so-\ncalled “magic angles”. The physical origin of these interesting flat bands has been elusive to date,\nhindering the construction of an effective theory for the unconventional electron correlation. In\nthis work, we have identified the importance of charge accumulation in the AA region of the moir´ e\nsupercell and the most critical role of the Fermi ring in AA-stacked bilayer graphene. We show that\nthe magic angles can be predicted by the moir´ e periodicity determined by the size of this Fermi\nring. The resonant criterion in momentum space makes it possible to coherently combine states on\nthe Fermi ring through scattering by the moir´ e potential, leading to flat bands near the Fermi level.\nWe thus establish the physical origin of the magic angles in TBG and identify the characteristics of\none-particle states associated with the flat bands for further many-body investigations.\nI. INTRODUCTION\nTwo graphene sheets with a small rotation angle θbe-\ntween them creates a special system of twisted bilayer\ngraphene (TBG), in which a moir´ e pattern emerges with\na spatial periodicity inversely proportional to the twist\nangle [1, 2]. It has been shown that the variation of\nthe twist angle in moir´ e two-dimensional (2D) materi-\nals could modify the electronic properties, giving rise to\nan interesting research area of twistronics [3]. Previous\ncalculations on TBG using different theoretical methods\nincluding the low-energy continuum model [2, 4, 5], the\ntight-binding method [1, 6–8], and density functional the-\nory (DFT) [9] all had similar findings: as θdecreases, the\nFermi velocity of the linear bands also becomes progres-\nsively smaller. This results from an enhanced interlayer\nhybridization between the two rotated Dirac cones of dif-\nferent layers as they get closer in momentum space when\nθgets smaller. These calculations also revealed that as\nthe bandwidth of the low-lying bands is reduced at small\nθangles, the corresponding electron charge becomes lo-\ncalized to the AA-stacked region in the moir´ e supercell\n[2, 7, 9]. One would anticipate that the Fermi velocity\nshould decrease smoothly as θapproaches zero; however,\nit was found in these calculations that at certain so-called\n“magic angles” the velocity deviates away from the ex-\npected monotonic curve and drops to almost zero [2, 5, 8],\nwith the first of these magic angles being around 1 .1◦.\nThe resulting unusually flat bands thus have a large ratio\nbetween the Coulomb and kinetic energies and favor the\nformation of possible many-body phases.\nExperimentally, with the “tear-and-stack” technique\n[10–14], researchers have been able to control the rotation\nangle of TBG precisely. It was first discovered by Y. Cao\n∗These authors contributed equally to this work.\n†Corresponding author. Email: mychou6@gate.sinica.edu.twet al. [15] that around the first magic angle there exist\ninsulating gaps when the doping level is tuned to integer\nfillings. Moreover, superconducting states appear [16] in\nthe intermediate filling regions. These surprising reports\nstimulated further transport measurements for detailed\nstudies on correlated insulating states at several integer\nfillings [17], superconductivity domes in a wide, contin-\nuous range of doping [17–21], ferromagnetism [17, 22],\nand the integer quantum anomalous Hall effect [23] at\ncertain odd fillings. The existence of flat bands in TBG\nwas confirmed by scanning tunneling spectroscopy (STS)\nstudies [24–26] that revealed two pronounced van Hove\nsingularities very close to each other in the density of\nstates (DOS) near the first magic angle . Using angle-\nresolved photoemission spectroscopy with nanoscale reso-\nlution (nano-ARPES), energy bands with little dispersion\nwere directly observed in momentum space at θ≈0.96◦\n[27] and 1 .34◦[28]. The charge localization in the AA\nregion at small θwas also confirmed using scanning tun-\nneling microscope (STM) [12, 24–26]. Furthermore, both\ntransport and STS measurements in the presence of mag-\nnetic field have recently identified correlated gaps that\naccount for both integer [29–35] and fractional [31, 36]\nChern insulating states, as a result of strong correlation\nin combination with the flat band topology.\nIt is unprecedented that a carbon material with only\nspelectrons can exhibit such rich phenomena of elec-\ntron correlation driven by interaction. Thus, magic-angle\ntwisted bilayer graphene (MATBG) and related systems\nhave presented special challenges in our understanding\nof two-dimensional physics. In order to explain these\nexotic properties observed, various theoretical investiga-\ntions have been reported. Most of these studies [37–40]\nrelied on building a minimum model/basis set consisting\nof, for example, Wannier orbitals, which reproduces the\nsingle-particle flat band dispersion while satisfying the\noriginal symmetry and topology properties as much as\npossible. Many-body interactions were then included in\nthe framework of (generalized) Hubbard model to shedarXiv:2309.10026v1  [cond-mat.mes-hall]  18 Sep 20232\nlight on the nature of many-body states observed exper-\nimentally. Some other studies [41–43] pursued alterna-\ntive approaches to avoid possible bias toward the basis\nset. Nevertheless, no convincing theory that has been\nproposed to date can fully explain the interesting and\nintriguing experimental findings in MATBG yet.\nThe existence of multiple magic angles with extremely\nflat bands was found in many one-particle calculations.\nYet their origin and why a series of flat bands occur at\nthese specific magic angles are still not understood until\nnow. Building an effective many-body theory for electron\ncorrelation and superconductivity requires, as the start-\ning point, a good knowledge of the characteristics of one-\nparticle electronic states at the magic angles. Therefore,\nin this theoretical work we aim to uncover the unique\nfeatures in the electronic properties of TBG. Since the\nsystem is too large for full-scale DFT calculations, we\nmostly used the tight-binding method with newly devel-\noped parameters for accurate angle-dependent interlayer\ninteractions [44, 45]. First, we provide an explanation us-\ning DFT calculations for the reason why electrons near\nthe Fermi level become localized in the AA region of TBG\nwhen the twist angle θgets small. As a result, the lo-\ncal electronic structure of AA-stacked bilayer graphene\n(AABLG) with a characteristic Fermi ring becomes the\nkey feature to be considered for generating a special ef-\nfect at certain twist angles. Second, we demonstrate that\nwhen the size of this Fermi ring matches the reciprocal\nlattice vectors of the moir´ e superlattice at a set of angles,\nthe Dirac points of TBG fall on the Fermi ring. Multiple\nstates on the ring can then be coupled coherently by the\nmoir´ e potential, leading to energy bands with extremely\nsmall dispersions. This matching condition in momen-\ntum space generates a series of discrete magic angles in\nTBG. Computational results on TBG with different in-\nterlayer coupling strengths when it is under uniaxial pres-\nsure are presented to illustrate our theory. This work\nprovides the physical explanation on the origin of magic\nangles. The result also indicates that magic angles are\nspecial features in twisted graphene systems arising from\ntheir particular electronic structure.\nII. RESULTS\nElectron accumulation in the AA region\nIn our tight-binding calculation, we have considered\ncommensurate TBG configurations with moir´ e lattice\nvectors of L1=ma1+ (m+ 1)a2andL2=−(m+\n1)a1+ (2m+ 1)a2, where a1anda2are lattice vectors\nfor monolayer graphene, and mis an integer [2]. The ro-\ntation starts from AABLG with the axis passing through\na vertical pair of carbon atoms. We neglect the pos-\nsible interlayer corrugation and intralayer relaxation in\nthis work. Figure 1a shows the atomic arrangement of\na typical moir´ e pattern in TBG with a rotation angle of\nθ= 3.15◦(m= 10). Because of the relative rotationbetween two graphene layers, the local atomic arrange-\nments exhibit different stacking patterns at different lo-\ncations. Some noticeable ones, such as the AA-stacked\nand Bernal-stacked (AB or BA) regions, are labeled in\nFig. 1a, as well as six other representative stacking pat-\nterns (T1-T6) in between. Thus, the Bloch electrons will\nexperience different lattice potentials in different stacking\nregions within a single moir´ e supercell, which is an impor-\ntant feature for this system. To understand the stacking\neffect, we have calculated the electronic DOS for these\ndifferent lattice potentials with DFT, and the results are\nshown in Fig. 1b. It turns out that the AA stacking\ngives significantly more low-energy states near the Fermi\nlevel, while the AB(BA) stacking has the fewest states in\nthe same energy range. This can be understood by ex-\namining the band dispersion. Compared with monolayer\ngraphene with one Dirac cone (Fig. 1c), AABLG has two\ncones shifted vertically in energy (Fig. 1d), giving rise to\na sizable constant DOS between the two original band\ncrossings. In contrast, the Bernal stacking gives roughly\ntwo low-energy parabolic bands joined at the Dirac point\nwith a smaller DOS. A tight-binding model with only the\nnearest-neighbor interlayer interaction will give a ratio of\n4:1 for the low-energy DOS in these two systems. As the\nsize of the moir´ e superlattice increases with decreasing\nθ, so is the size of a region with a particular stacking\npattern. Therefore, this explains the observation that,\nat small θvalues, the electrons near the Fermi level tend\nto accumulate around the AA region in the moir´ e cell,\nbecause more states are available to them there.\nTo illustrate the major change in the electronic struc-\nture as θvaries, we present in the upper panels of Fig.\n2a-e the band structure of TBG by tight-binding calcu-\nlations for five different twist angles (9.43◦, 3.89◦, 2.13◦,\n1.12◦and 0.86◦). Note the change of the energy scale in\nthese plots. The size of the moir´ e supercell is 1.5 nm, 3.6\nnm, 6.6 nm, 13 nm, and 16 nm, respectively. The four\nbands in red near the Fermi energy gradually flatten as\nthe twist angle decreases in Fig. 2a-c, which is expected\nfor an increased coupling between the Dirac cones of two\nlayers. Then there is a sudden change near the first magic\nangle of 1.1◦where the bands become extremely flat (Fig.\n2d). After passing through the first magic angle, the\nbands regain their slopes (Fig. 2e). This trend can be\nseen clearly from Fig. 2f in which the red curve shows the\nnormalized Fermi velocity of TBG (with respect to the\nFermi velocity v0\nFof monolayer graphene) as a function\nof angle θand exhibits a clear sharp dip at the first magic\nangle. The fact that the extremely flat bands only occur\nat certain discrete magic angles indicates that there ex-\nists a characteristic condition that breaks the expected\nmonotonic behavior.\nIt is interesting to examine the developed electron ac-\ncumulation in the AA region. We present the probability\nof finding electrons at each atomic site in the lower pan-\nels of Fig. 2a-e. The dark and light dots mark the atoms\nin the first and second layer, respectively, and the dashed\ndark hexagon marks the Wigner-Seitz cell of the moir´ e3\na b\nc d\nFIG. 1. |Moir´ e effect on electronic density of states. a , Moir´ e pattern of TBG at θ= 3.15◦with six representative local\natomic configurations T1 - T6 in addition to AA and AB(BA). The black (gray) atoms are in the first (second) layer. The arrow\nin the insets shows how the second layer of AA-stacked graphene is shifted to generate a particular local stacking configuration.\nThe thick hexagon marks the Wigner-Seitz cell of the moir´ e supercell. b, Density of states (DOS) of representative bilayer\nstacking configurations in a, obtained from density-functional-theory (DFT) calculations. The uppermost panel covers a larger\nenergy range, while the lower two panels compare the DOS variations near the Fermi level along two paths (AA →T1→T2→\nT3→AB and AA →T4→T5→T6). It is noted that the AA region has significantly more low-energy states around the\nFermi level. c, Dirac cone in momentum space for monolayer graphene. d, Two Dirac cones shifted in energy for AA-stacked\nbilayer graphene with the Fermi ring marked by a dashed circle.\nsuperlattice. The electron distribution of the zero-energy\nstates at Kis found to be similar to that integrated over\nthe four red bands in the upper panels. Therefore, we\nshow only the distribution of the four degenerate states\natK, represented by the size of red dots in the lower\npanels of Fig. 2a-e. As the twist angle is reduced and\nthe AA region clearly develops, the electrons gradually\nbecome accumulated in the AA region at the center of\nthe Wigner-Seitz cell. We measure the degree of local-\nization by the integrated fraction of electrons within the\ndashed blue circle of a radius 0.2 L, where Lis the lat-\ntice constant of the moir´ e supercell at each angle. This\nlocalization measure is found to increase smoothly as the\ntwist angle decreases, as shown by the blue curve in Fig.\n2f. As discussed above, this behavior can be explained by\nthe fact that the AA region provides significantly more\nlow-energy states than other stacking patterns. Surpris-\ningly, this smooth behavior is not affected by the presence\nof the magic angle at all, in contrast to the variation of\nthe Fermi velocity in Fig. 2f. It was proposed previously[2, 8] that the series of magic angles are associated with\nquantization conditions of the confined states in the AA\nregion. However, results in Fig. 2f indicate that the flat\nband dispersion at the magic angle may not reflect the\nmost localized electronic distribution and that the emer-\ngence of the magic angle requires a mechanism other than\nelectron confinement alone.\nConnecting AABLG Fermi ring and magic angles in\nTBG\nAs the low-energy electrons are accumulated in the\nAA region at small angles, apart from the moir´ e poten-\ntial they mostly experience the local lattice potential of\nAABLG, which has a ring-shaped Fermi surface. This\nFermi ring (Fig. 1d) has a particular radius ( QF) in\nmomentum space that is determined by the interlayer\ninteraction. Next we propose a physical explanation for\nthe development of flat bands at the magic angles that4\n0 1 2 3 4 5 610-310-210-1100vF /v0\nF\nTwist angle (degree )00.20.40.60.8\nLocalizationa\nfb c\nd e\nFIG. 2. |Fermi velocity and localized electrons. a-e , Band structure (upper panel) of twisted bilayer graphene with\nangles of 9 .43◦, 3.89◦, 2.13◦, 1.12◦and 0 .86◦, respectively, calculated by the tight-binding method. The four bands near the\nFermi level (energy zero) are marked in red. Note that the bands become flatter when the angle decreases, with a minimal\nFermi velocity near 1 .12◦, but the slope at 0 .86◦bounces back after passing through this magic angle. The lower panel shows\nthe electronic probability distribution of the zero-energy states at K. The dark and light dots mark the atoms in layer 1 and\n2, respectively, and the dashed dark hexagon represents the Wigner-Seitz cell with the real-space dimension labels in units of\n˚A. The size of the red dots represents the probability of finding the electrons at each atom. f, Calculated Fermi velocity (red\ncurve) and electron localization (blue curve) as a function of the twist angle θ. The Fermi velocity is normalized with respect\nto that in monolayer graphene v0\nF. The electron localization is measured by the total fraction of Kelectrons within the the\nblue dashed circle centered at the AA region that has a radius of 20% of the moir´ e lattice constant. The Fermi velocity has a\nsharp dip at 1 .12◦near the magic angle. In contrast, the electron localization measure increases smoothly and monotonically\nas the twist angle decreases.\nis connected to the Fermi ring of AABLG. In particular,\nwe will show that a special situation exists if the TBG\nDirac points (in the extended Brillouin zones) fall on the\nFermi ring of AABLG. We will then demonstrate that\nthe magic angles occur under these matching conditions\nand discuss the characteristics of the states associated\nwith the resulting flat bands.\nThe interaction between the layers can be changed by\napplying uniaxial pressure, namely, by imposing a com-\npressive vertical stress. With different reduced interlayer\ndistances, both the magic angle and the size of the Fermi\nring are separately modified. This provides a collection\nof sample systems that allow us to examine our proposed\nmechanism systematically by tight-binding calculations.\nThe band dispersion of TBG with different compressedTABLE I. First magic angle θand corresponding moir´ e lat-\ntice constant Lof TBG obtained from tight-binding calcu-\nlations for different vertical compression levels . These are\ndetermined by a commensurate supercell of mthat has the\nsmallest Fermi velocity.\ncompression 0% 3% 5% 10% 15% 20%\nm 29 26 23 17 13 10\nθ(◦) 1.12 1.25 1.41 1.89 2.45 3.15\nL(nm) 12.6 11.3 10.0 7.46 5.75 4.45\nvertical strain values (0%, 3%, 5%, 10%, 15%, and 20%)\nwas evaluated as a function of the twist angle in order to\ndetermine their corresponding magic angles. The com-5\n0 10 20 30 40 50 600306090120150\n20 30 40 50 60 70 80 901006090120150180210Moiré size L (Å)\nπ / QF (Å)Slope = 4\n3\n ε = 0\n ε = 3%  \n ε = 5%\n ε = 10%\n ε = 15%\n ε = 20%(S = 1)\nS = 3)(Moiré size L (Å)\nπS / QF (Å)Slope = 4\n3\nFirstSecond Third\n(S = 1)(S = 2)S = 7)(Fortha c\nb de\nf\nFIG. 3. |Connection between magic-angle moir´ e periodicity and AABLG Fermi ring. a-d , Brillouin zone (dashed\ngreen hexagon) and Fermi ring (red circle) of AABLG superimposed on the Brillouin zones (small black hexagons) of TBG at\ndifferent angles. For a clear illustration, the Fermi ring for AABLG at 20% compression of the interlayer separation is shown.\nBoth the dashed green hexagon and the red ring are of the same size in a-d, but the Brillouin zones of TBG shrink as the moir´ e\nsize increases with a decreasing twist angle. The matching condition (see text) is reached at θ= 3.15◦, which is the magic\nangle for TBG at 20% compression. e, Correlation between the moir´ e size Lat the first magic angle ( S= 1) and the inverse of\nthe radius QFof the Fermi ring in AABLG with different interlayer compression. The linear behavior follows the relationship\nL= 4/√\n3×(π/Q F) (dashed line). The horizontal error bars on the data points mark the variation in the size of the Fermi\nring due to trigonal warping, while the vertical error bars represent the uncertainty in determining the magic angle of TBG\nsince our calculations used a particular collection of commensurate moir´ e supercells. f, Similar linear connection is found for\nthe second ( S=√\n3), third ( S= 2), and the fourth ( S=√\n7) magic angles for the TBG system with 10% compression.\nplete results are listed in Table 1. When the interlayer\nseparation is compressed, the magic angle occurs at a\nlarger angle [45]. For example, the first magic angle shifts\nfrom θ= 1.12◦at 0% compression to 3.15◦at 20% com-\npression. At the same time, the radius of the Fermi ring\nin AABLG also enlarges as the bilayer is compressed.\nFor large compression values, a trigonal warping may be\nfound, and an approximated Fermi ring is determined\nby a circle with the same area. This deviation will be\nconsidered in later discussions.\nSince the compressed bilayer graphene has a smaller\nmoir´ e supercell at the first magic angle, it is easier\nto graphically visualize this effect in a compressed bi-\nlayer system. We plot in Fig. 3a-d the 2D Brillouin\nzone (dashed green hexagon) and Fermi ring (red cir-\ncle) of AABLG with 20% vertical compression. Super-\nimposed in the figures are the Brillouin zones (small black\nhexagons) of TBG at different angles, where the Kand\nK′Dirac points are marked by solid and open circles,\nrespectively. Figure 3a shows the configuration for θ=\n7.34◦, with an enlarged view in Fig. 3b. At this angle,the Dirac points of TBG do not seem to be influenced by\nthe AABLG Fermi ring. Note that the Brillouin zone of\nTBG shrinks as the moir´ e size increases with a decreas-\ning twist angle. A special matching condition is found\nin Fig. 3d, where the Fermi ring of AABLG centered\natKreaches the first star of Dirac Kpoints in the ex-\ntended Brillouin zones of TBG. We will show below that\nthis particular matching condition yields the first magic\nangle in systems we have investigated.\nWe plot in Fig. 3e the moir´ e supercell size Lat the\nfirst magic angle found in the calculation (determined by\nthe dip in the calculated Fermi velocity) versus the in-\nverse of QF(the radius of the AABLG Fermi ring) for\nbilayer systems with various compression levels. These\ntwo quantities were independently obtained for two dif-\nferernt systems by our tight-binding calculations. The\nhorizontal error bars on the data points mark the vari-\nation in the size of the Fermi ring due to the trigonal\nwarping: the smallest (largest) radius in the KΓ(KM)\ndirection gives the upper (lower) bound of π/Q F. The\nvertical error bars represent the uncertainty in determin-6\n1 2 310-210-1\nTwist angle (degree)vF / v0\nFGa b c\nbc aFQ G 3FQ G 2FQ G\nd e\nFIG. 4. |Higher-order magic angles. a-c , Matching conditions between the AABLG Fermi ring (red circle) and the first,\nsecond, and third stars of the Kpoints (black filled circles) of TBG in the extended zones (black hexagons). These give rise to\nthe first, second, and third magic angles, respectively. The magnitude of G, the smallest reciprocal lattice vector of the moir´ e\nsupercell decreases in a-c.d, Calculated (normalized) Fermi velocity as a function of the twist angle for TBG with 10% vertical\ncompression, showing four dips corresponding to the first four magic angles. e, Enlarged matching condition in ashowing the\nsixKpoints on the Fermi ring and possible scattering between them via reciprocal lattice vectors of the moir´ e supercell.\ning the exact magic angle, because our calculations used\na particular collection of commensurate moir´ e supercells.\nIf the largest Fermi velocity dip was found in the calcula-\ntion for a moir´ e supercell with a particular mvalue, then\nthe moir´ e size for the exact magic angle is bounded by\nthose corresponding to m−1 and m+1 moir´ e supercells.\nAs one can see from Fig. 3e, an almost perfect linear\nbehavior is found with a slope of 4 /√\n3. In other words,\nthe size of the AABLG Fermi ring determines the moir´ e\nperiodicity where the magic angle occurs . And the first\nmagic angle appears when the radius of the Fermi ring\nmatches G, the smallest magnitude of nonzero reciprocal\nlattice vectors of the moir´ e supercell, as shown in Fig. 3d\nand Fig. 4a. As will be shown below, the critical match-\ning condition is QF=SG, where S= 1 corresponds to\nthe first magic angle.\nWhen the radius of the AABLG Fermi ring matches\nthe second and third stars of the Dirac Kpoints, as\nshown in Fig. 4b and c, we expect to see the second\n(S=√\n3) and third ( S= 2) magic angles, respectively.\nWe show in Fig. 4d a more complete plot of the calcu-\nlated Fermi velocity versus θfor TBG with a 10% com-pression, and data up to the fourth magic angle (0.71◦,\nS=√\n7) was obtained. The corresponding moir´ e size at\nthe first four magic angles versus πS/Q Fis plotted in\nFig. 3f. Again, a consistent linear behavior is found that\nmatches the relationship\nL=4√\n3π\nQFS, (1)\nwhere S= 1,√\n3,2,√\n7. Given that the minimal length of\nthe reciprocal lattice vector is G= 4π/(√\n3L),QF=S G\nis then the criterion in momentum space that deter-\nmines the occurrence of the series of magic angles. Pos-\nsible Svalues are expected to reflect the radius of the\nstars of Kpoints: S= 1,√\n3,2,√\n7,3,2√\n3, .... Under\nthese matching conditions, the zero-energy states within\nAABLG (Fermi ring) and the zero-energy states within\nTBG (Dirac points) are aligned.\nAt small angles, we have L≈a/θ, where ais the lattice\nconstant of monolayer graphene. Therefore, the magic\nangles in TBG can be expressed as7\nθmagic≈QF√\n3lK1\nS, (2)\nwhere lK= 4π/(3a) is the distance from ΓtoKfor\nmonolayer graphene, and Sis not necessarily an inte-\nger. Our DFT value for QFis about 0.055 ˚A−1. There-\nfore, the predicted magic angles are 1.07◦, 0.62◦, 0.53◦,\n0.40◦,. . .for TBG. If a tight-binding model considers\nonly the nearest neighbor coupling t⊥for the interlayer\ninteraction, the AABLG Fermi ring will have a radius of\nQF=t⊥/(ℏv0\nF).\nWave functions of the Dirac point at magic angles\nAway from the magic angle, the energy bands in Fig.\n2 can be explained by the following picture: As the two\nlayers are rotated by an angle in real space, the Dirac\ncones of the two layers are also rotated with respect to\neach other in momentum space; the interlayer interaction\ninduces coupling between the two sets of linear bands,\ngiving rise to a reduced slope (Fermi velocity) that is sup-\nposed to vary monotonically with the angle. This simple\npicture is understandable, since each moir´ e supercell has\nregions of different stacking patterns (see Fig. 1a) that\ncannot influence the energy bands in an abrupt way. On\nthe other hand, when the angle gets small, the relevant\nlow-energy electrons are accumulated in the AA region,\nso they should be highly influenced by the lattice poten-\ntial of AABLG. When the matching condition QF=S G\nis satisfied at the magic angles, it would be reasonable to\nconstruct the wave functions for the Dirac point of TBG\nfrom the Bloch states on the AABLG Fermi ring.\nWe use the first magic angle ( QF=G) as an exam-\nple. Figure 4e shows the detailed geometry with six Dirac\npoints on the AABLG Fermi ring; they could be folded in\nTBG to the Kpoint at the ring center (marked by a red\npoint). The Bloch states on the AABLG Fermi ring are\ndoubly degenerate and half occupied in a neutral system.\nUsing the nearest-neighbor tight-binding model with an\northogonal basis of pzorbitals on the AandBsublattices\nin layer 1 and 2: {A1,A2,B1,B2}, these two degener-\nate states can be chosen as |k⟩ ∝(1 0 0 −eiϕ)Tand\n(0 1−eiϕ0)Twith ϕ= tan−1(ky/kx). These two de-\ngenerate states will not be coupled with each other by a\nlocal moir´ e potential, so in the following analysis we make\nthis assumption and deal with one set of them at a time.\nEach pair of the six kpoints in Fig. 4e are connected by\na reciprocal lattice vector of the moir´ e supercell, so they\nwill be coupled with each other via the scattering by a\n2D moir´ e potential U(r) = Σ G′UG′eiG′·r, where G′is a\nreciprocal lattice vector of the moir´ e superlattice. With\nthe combination of C3(vertical axis) and C2(horizontal\naxes) rotation symmetries and the fact that the moir´ e po-\ntential is real ( UG′=U∗\n−G′), the number of independent\nFourier coefficients is greatly reduced. We denote that\nUG′=0=U0,Uk1−k2=Uk2−k3=. . .=U1, and Uk1−k3=U∗\nk2−k4=Uk3−k5=. . .=U2.U1is real, and U2can\nbe made real by choosing the 2D origin at an inversion\ncenter.\nAfter including the symmetry of the system and as-\nsuming that the long-scale moir´ e potential varies little\nover a single graphene unit cell (see Methods), the per-\nturbation Hamiltonian of the moir´ e potential for one set\nof the wave functions at the six kpoints ( k1,k2,. . .,k6\nin Fig. 4e) has this form:\nH=\nu11u12u13 0 u∗\n13 u∗\n12\nu11u12ei2π/3u∗\n130e−i2π/3u13\nu11 u12 u13 0\nu11 u12ei2π/3u∗\n13\nc.c. u 11 u12\nu11\n,\n(3)\nwhere uij=⟨ki|U|kj⟩with the following values: u11\n=U0,u12= (√\n3/2)U1eiπ/6,u13= (1/2)U2eiπ/3, and\nu14= 0. This perturbation Hamiltonian turns out to\nhave three doubly degenerate eigenstates. After a con-\nstant shift of U0−Re[U2] to align the energy of the middle\nstate with zero, we have the three eigenvalues:\nE0= 0 ; E±=3\n2\"\nRe[U2]±r\nU2\n1+\u0010\nIm[U2]/√\n3\u00112#\n.\n(4)\nIn the space of the six kstates on the Fermi ring of\nAABLG, the two degenerate eigenvectors v1andv2for\nE0are:\nv1=\n0\n1\n0\nei2π/3\n0\nei4π/3\nand v2=\n1\n0\nei2π/3\n0\nei4π/3\n0\n, (5)\nwhere v1is a linear combination of k1,k3, and k5states,\nwhile v2is made of k2,k4, and k6states. Both v1andv2\nare invariant under the C3rotation and with the same\neigenvalue ω. (The value of ωcould be e−i2π/3or 1 with\nthe rotation axis chosen at a C atom or the center of a\nsix-atom ring.) The results for the other set of degenerate\nBloch states at the six kpoints are the same. Therefore,\nwe obtain four zero-energy degenerate states at the K\npoint of TBG that all transform in the same way under\ntheC3rotation. For higher magic angles, the matrix\nelements in Eq. (3) will involve higher Fourier coefficients\nof the moir´ e potential. Nevertheless, the features of the\nfinal eigenvectors will be similar to those for S= 1.\nNext we show that the bands turn out to be flat going\naway from the Kpoint by the k·pperturbation theory.\nThe main O(k) correction in energy away from the K\npoint comes from the k·pintraband coupling among the8\nfourfold degenerate states at E=0. These intraband cou-\nplings, however, turn out to vanish because all the four\neigenstates ( vl) atE=0 have the same C3rotation sym-\nmetry with the same eigenvalue of ω. This can be seen\nas follows. First, we rewrite k·ˆp=k+ˆp−+k−ˆp+,where\nk±= (kx±iky)/√\n2 and ˆ p±= (ˆpx±iˆpy)/√\n2. The op-\nerator ˆ p±transforms as Cnˆp±C†\nn=e∓i2π/nˆp±under the\nCnrotation. Then ⟨vl|ˆp+|vm⟩=⟨vl|C†\n3C3ˆp+C†\n3C3|vm⟩\n=⟨vl|ω∗(e−i2π/3ˆp+)ω|vm⟩=e−i2π/3⟨vl|ˆp+|vm⟩. There-\nfore,⟨vl|ˆp+|vm⟩= 0, and similarly ⟨vl|ˆp−|vm⟩= 0. As\nsuch, the resonant scattering by the moir´ e potential reg-\nulates the C3rotation symmetry of the degenerate states\natE=0, kills the intraband k·pcoupling, and thus sig-\nnificantly flattens the bands away from K.\nIn addition, the net O(k2) correction from the inter-\nband coupling also tends to be small. If we neglect the\nlong-range interaction and set U2=0, we have E+=−E−\nandσHσ =−Hwith σ= Diag[1,-1,1,-1,1,-1]. Conse-\nquently, contributions from the interband coupling can-\ncel each other: the E+bands tend to bend downward the\nE0band, while the E−bands tend to bend it upward,\nrendering a net zero correction in energy to the order of\nO(k2).\nIII. DISCUSSION\nIn this work we have identified the following key factors\nfor the occurrence of magic angles in TBG, which may\nsuggest the directions for searching for this interesting\nphenomenon in other 2D systems.\nFirst, there exists a particular stacking pattern within\nthe the moir´ e supercell that attracts low-energy electrons\nto accumulate. This charge concentration becomes signif-\nicant at small twist angles, because the region associated\nwith the particular stacking pattern becomes substantial\nover many bond lengths. This happens in the AA re-\ngion of TBG that has more low-energy states available\ncompared with regions with other stacking patterns. If\nthis condition does not occur and the electrons spread\nout throughout the moir´ e supercell, the averaging effect\nwill only decrease the bandwidth monotonically as the\nsupercell size increases.\nSecond, this particular region has its characteristic\nband structure near the Fermi level that will respond to\nthe perturbing moir´ e potential in a particular way only at\ncertain twist angles, where the resulting electronic states\ncan have special features. In the case of TBG, the AA\nregion has a characteristic Fermi ring, and the states on\nthe ring can be coherently coupled when the size of the\nring matches the reciprocal lattice vectors of the moir´ e\nsupercell ( QF=SG). Therefore, the size of this Fermi\nring determines the moir´ e periodicity where the series of\nmagic angles occur, and the corresponding TBG wave\nfunctions at the Dirac Kpoint are special linear combi-\nnations of wave functions on the AABLG Fermi ring.\nThird, the characteristic wave functions at the magic\nangles have certain symmetry properties that are able toinduce a flat band dispersion away from the reference k\npoint. This is possible because with a proper symmetry\ntransformation property for ˆ p±, the first-order correction\nwithin the k·pperturbation theory can vanish. Possible\nsymmetries include inversion, Cnrotations, and reflec-\ntions. At the magic angles in TBG, the fourfold degen-\nerate E=0 states at Khave the same eigenvalue under\ntheC3rotation. The transformation property C3ˆp±C†\n3=\ne∓i2π/3ˆp±sets the intraband k·pcoupling to zero, giving\nrise to flat bands out of Kto the linear order.\nThe calculations presented in this work did not include\nthe lattice relaxation effect in TBG. Previous theoretical\nstudies have shown that the relaxation pattern in the\nAA region can simply be described as adding an extra\nsmall local rotation around the center of the AA region\n[46, 47]. For example, for the case of θ= 1.05◦, the\nlocal rotation angle between the two layers is increased\nfrom 1 .05◦to 1.63◦upon relaxation [46], and for very\nsmall θvalues, the relaxed local twist angle at the AA\nstacking converges to 1 .7◦[47]. As a consequence, the\nAA stacking remains intact after relaxation, albeit with\na reduced extent. Nevertheless, the remaining size is still\nof the order of a few nanometers for small twist angles\n(see Fig. 5 in [9] for θ= 0.99◦and Fig. 7 in [46] for\nθ= 1.05◦). This is consistent with the STM topographic\nimages at around θ= 1.1◦that also showed bright spots\nof AA stacking regions with a size of a few nanometers\n[26, 48]. Since there still exist sizable AA regions after\none takes into account the lattice relaxation, our theory\nbased on the Fermi ring of the AA stacking should remain\nvalid.\nA previous study focusing on an idealized continuum\nmodel by switching off the interlayer coupling parameter\nwAAfor the AA stacking yielded perfectly flat bands at\nmultiple magic angles [49]. Intriguingly, the wave func-\ntions of these flat bands at the magic angles in this par-\nticular model were found to be reminiscent of quantum\nHall wave functions on the torus. Because the condition\nwAA= 0 gives rise to perfectly flat bands, it was fur-\nther speculated that the fundamental features of TBG\nare mainly connected with the interlayer coupling wABin\nthe AB stacking. Since this idealized model does not cor-\nrespond to any realistic graphene systems, it was hoped\nthat one can extend the results to cases with nonzero\nwAAby treating wAA/wABas a perturbation. However,\nthis may be difficult because wAAandwABshould be of\nthe same order of magnitude in realistic TBG systems. In\ncontrast, the current work focuses on the important effect\nof the wAAinteraction based on the fact that low-energy\nelectrons in TBG are concentrated in the AA region. Our\nproposed theory is therefore able to fully explain the oc-\ncurrence and positions of magic angles in terms of the\nlocal electronic properties of the AA region.\nIn conclusion, this work resolves the long-standing\nquestion why magic angles happen in twisted bilayer\ngraphene. The origin of the magic angles turns out to be\nthe “magic” Fermi ring in AA-stacked bilayer graphene.\nWe have theoretically derived the series of magic angle9\nvalues. In addition, we have identified the composition of\nthe one-particle wave functions associated with the flat\nbands for further many-body investigations.\nMETHODS\nTight-binding calculations\nTo calculate the electronic properties of TBG, we used\nan extended tight-binding model proposed by S. Fang\nand E. Kaxiras [44] with parameters determined from\ncalculations with density functional theory (DFT). This\nmodel includes intralayer couplings up to eighth near-\nest neighbors, and a strong angular dependence of in-\nterlayer couplings between atoms in two different layers.\nThe spin-orbit interaction is neglected. The interlayer\ncouplings under uniaxial pressure were also obtained us-\ning similar procedures [45]. The Hamiltonian was then\ndiagonalized by the FEAST eigenvalue solver [50], and\nthe eigenvalues and eigenvectors were verified by Math-\nematica in some cases. The density of states were cal-\nculated by the tetrahedral method [51], and the slope\nat the Kpoint (Fermi velocity) was evaluated based on\nthe Hellmann-Feynman theorem [52] as follows. The ve-\nlocity operator is given by ⃗ v(⃗k) =∂⃗kH(⃗k), where ⃗kis\nthe crystal momentum, and H(⃗k) is the tight-binding\nHamiltonian of the moir´ e structure. Due to the fourfold\ndegeneracy at the Dirac point, we formed a velocity ma-\ntrixvij=⟨i|v(K)|j⟩, where iandjare labels for the\ndegenerate states at the Dirac point. The eigenvalues of\nthe velocity matrix give the velocities; the one with the\nsmallest absolute value was taken as the velocity result.\nSince the velocity is almost isotropic, we show the result\nfor the K→Mdirection in most cases. When there was\nuncertainty, the average between K→MandK→Γ\nwas taken.\nDensity functional calculations\nWe also performed DFT calculations for bilayer\ngraphene with different stacking patterns using the Vi-\nenna Ab initio Simulation Package (VASP) [53, 54].\nWe used the projector augmented wave (PAW) method\n[55, 56] to treat core electrons and the optB86b functional\nto include the van der Waals correction [57, 58], with a\nplane-wave cutoff energy of 400 eV. The Monkhorst-Pack\nk-point mesh of 21 ×21×1 was used in the self-consistent\ncalculations for the 1 ×1 unit cell. The calculation of the\ndensity of states used a more denser 91 ×91×1 grid.\nMatrix elements of the moir´ e potential\nThe Bloch-state components constructed from the lo-\ncal orbitals at any of the four sites {A1, A2, B1, B2}inthe unit cell of AABLG have the form:\n⟨r|αi⟩=1p\nNα,iX\nRα,jeik·Rα,jζpz(r−Rα,j−δi,2cˆz),\nwhere Rα,jis the in-plane position vector for sublattice\nαin the jth unit cell; cis the interlayer separation, i\nrefers to the layer index {1,2}, and δi,2= 1 if i=2.ζpz(r)\ndescribes the orthonormal pzatomic orbital, and Nα,i=\nNis the total number of atoms pertaining to sublattice α\nand layer iin AABLG. For the six kvectors on the Fermi\nring in Fig. 4e, there are two degenerate states, and the\nwave functions can be chosen as |k⟩ ∝(1 0 0 −eiϕ)Tand\n(0 1−eiϕ0)Twith ϕ= tan−1(ky/kx). These two states\nwill not be coupled with each other by the local moir´ e\npotential, so we only have to deal with one of them at a\ntime. Both of them are eigenstates of the C3rotation.\nTheC3rotation symmetry exists for both TBG [and\nthus the moir´ e potential U(r)] and AABLG. It con-\nnects the matrix elements uij=⟨ki|U|kj⟩, for exam-\nple,⟨k1|U|k2⟩=⟨k1|C†\n3(C3UC†\n3)C3|k2⟩=⟨k3|U|k4⟩.\nTherefore, we have (as shown in Fig. 4e) u12=u34=u56\n(navy dashed lines); u23=u45=u61(blue lines); u13=\nu35=u51(orange dashed line); u24=u46=u62; and\nu14=u36=u52(brown dashed line). As a consequence,\ntheC3symmetry reduces the number of variables from\u0000\n62−6\u0001\n/2 = 15 to 5, which are u12, u23, u13, u24,andu14.\nSince the moir´ e supercell is much larger than the unit\ncell of graphene, we can assume that the effective moir´ e\npotential varies little within one unit cell of AABLG.\nTherefore, we have the following approximation:\nZ\nd3rζ∗\npz(r−Rα,j−δi,2cˆz)U(r)ζpz(r−Rα′,j′−δi′,2cˆz)\n≈U(Rα,j)δi,i′δj,j′δα,α′.\nTherefore, the matrix elements of the 2D moir´ e potential\nfor six |k⟩ ∝(1 0 0 −eiϕ)Tare:\nulm=⟨kl|U|km⟩\n≈1\n2N\nX\nRA,jei(km−kl)·RA,jU(RA,j)\n+ei(ϕm−ϕl)X\nRB,jei(km−kl)·RB,jU(RB,j)\n\n≈1\n2A\u0010\n1 +ei(ϕm−ϕl)\u0011Z\nei(km−kl)·rU(r)d2r\n=1\n2\u0010\n1 +ei(ϕm−ϕl)\u0011\nUkl−km,\nwhere Adenotes the whole area of the TBG. With the\ncombination of C3andC2rotation symmetries and the\nfact that the moir´ e potential is real ( UG′=U∗\n−G′), we\nhave Uk1−k2=Uk2−k3=. . .=U1, and Uk1−k3=U∗\nk2−k410\n=Uk3−k5=. . .=U2.U1is real, and U2is generally\ncomplex but can be made real by choosing the 2D origin\nat an inversion center. Given that ϕ2-ϕ1=π/3,ϕ3-ϕ1\n= 2π/3,ϕ4-ϕ1=π, (1+ eiθ)/2 = cos( θ/2)eiθ/2, and ulm\n=u∗\nml, we then have u11=UG=0,u12= (√\n3/2)U1eiπ/6\n=u23,u13= (1/2)U2eiπ/3,u24=ei2π/3u∗\n13andu14= 0.ACKNOWLEDGMENTS\nWe gratefully acknowledge Li-Feng Yen and Po-Tung\nFang for sharing the tight-binding codes and for help-\ning with technical matters. Helpful discussions with\nWen-Ying Ruan, Chi-Ruei Pan, Jie-Cheng Chen, Wei-\nEn Tseng, and Martin Callsen are acknowledged. This\nwork is supported by Academia Sinica, Taiwan.\n[1] S. Shallcross, S. Sharma, E. Kandelaki, and O. A.\nPankratov, Electronic structure of turbostratic graphene,\nPhys. Rev. B 81, 165105 (2010).\n[2] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H.\nCastro Neto, Continuum model of the twisted graphene\nbilayer, Phys. Rev. B 86, 155449 (2012).\n[3] S. Carr, D. Massatt, S. Fang, P. Cazeaux, M. Luskin,\nand E. Kaxiras, Twistronics: Manipulating the electronic\nproperties of two-dimensional layered structures through\ntheir twist angle, Phys. Rev. B 95, 075420 (2017).\n[4] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H.\nCastro Neto, Graphene bilayer with a twist: Electronic\nstructure, Phys. Rev. Lett. 99, 256802 (2007).\n[5] R. Bistritzer and A. H. MacDonald, Moir´ e bands in\ntwisted double-layer graphene, Proceedings of the Na-\ntional Academy of Sciences 108, 12233 (2011).\n[6] E. Su´ arez Morell, J. D. Correa, P. Vargas, M. Pacheco,\nand Z. Barticevic, Flat bands in slightly twisted bilayer\ngraphene: Tight-binding calculations, Phys. Rev. B 82,\n121407 (2010).\n[7] G. Trambly de Laissardi` ere, D. Mayou, and L. Magaud,\nLocalization of Dirac Electrons in Rotated Graphene Bi-\nlayers, Nano Lett. 10, 804 (2010).\n[8] G. Trambly de Laissardi` ere, D. Mayou, and L. Magaud,\nNumerical studies of confined states in rotated bilayers\nof graphene, Phys. Rev. B 86, 125413 (2012).\n[9] K. Uchida, S. Furuya, J.-I. Iwata, and A. Oshiyama,\nAtomic corrugation and electron localization due to\nmoir´ e patterns in twisted bilayer graphenes, Phys. Rev.\nB90, 155451 (2014).\n[10] Y. Cao, J. Y. Luo, V. Fatemi, S. Fang, J. D. Sanchez-\nYamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras,\nand P. Jarillo-Herrero, Superlattice-Induced Insulating\nStates and Valley-Protected Orbits in Twisted Bilayer\nGraphene, Phys. Rev. Lett. 117, 116804 (2016).\n[11] K. Kim, M. Yankowitz, B. Fallahazad, S. Kang,\nH. C. P. Movva, S. Huang, S. Larentis, C. M. Corbet,\nT. Taniguchi, K. Watanabe, S. K. Banerjee, B. J. LeRoy,\nand E. Tutuc, Van der Waals Heterostructures with High\nAccuracy Rotational Alignment, Nano Lett. 16, 1989\n(2016).\n[12] K. Kim, A. DaSilva, S. Huang, B. Fallahazad, S. Laren-\ntis, T. Taniguchi, K. Watanabe, B. J. LeRoy, A. H. Mac-\nDonald, and E. Tutuc, Tunable moir´ e bands and strong\ncorrelations in small-twist-angle bilayer graphene, PNAS\n114, 3364 (2017).\n[13] R. Frisenda, E. Navarro-Moratalla, P. Gant, D. P. D.\nLara, P. Jarillo-Herrero, R. V. Gorbachev, and\nA. Castellanos-Gomez, Recent progress in the assembly\nof nanodevices and van der Waals heterostructures by de-\nterministic placement of 2D materials, Chem. Soc. Rev.47, 53 (2018).\n[14] R. Ribeiro-Palau, C. Zhang, K. Watanabe, T. Taniguchi,\nJ. Hone, and C. R. Dean, Twistable electronics with dy-\nnamically rotatable heterostructures, Science 361, 690\n(2018).\n[15] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,\nJ. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,\nT. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-\nHerrero, Correlated insulator behaviour at half-filling\nin magic-angle graphene superlattices, Nature 556, 80\n(2018).\n[16] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,\nE. Kaxiras, and P. Jarillo-Herrero, Unconventional super-\nconductivity in magic-angle graphene superlattices, Na-\nture556, 43 (2018).\n[17] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir,\nI. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang,\nA. Bachtold, A. H. MacDonald, and D. K. Efetov, Su-\nperconductors, orbital magnets and correlated states in\nmagic-angle bilayer graphene, Nature 574, 653 (2019).\n[18] Y. Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F.\nYoung, Independent superconductors and correlated in-\nsulators in twisted bilayer graphene, Nat. Phys. 16, 926\n(2020).\n[19] P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe,\nT. Taniguchi, F. H. L. Koppens, J. Lischner, L. Levitov,\nand D. K. Efetov, Untying the insulating and supercon-\nducting orders in magic-angle graphene, Nature 583, 375\n(2020).\n[20] Y. Cao, D. Rodan-Legrain, J. M. Park, N. F. Q. Yuan,\nK. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu,\nand P. Jarillo-Herrero, Nematicity and competing orders\nin superconducting magic-angle graphene, Science 372,\n264 (2021).\n[21] X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek,\nand J. I. A. Li, Tuning electron correlation in magic-\nangle twisted bilayer graphene using Coulomb screening,\nScience 371, 1261 (2021).\n[22] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney,\nK. Watanabe, T. Taniguchi, M. A. Kastner, and\nD. Goldhaber-Gordon, Emergent ferromagnetism near\nthree-quarters filling in twisted bilayer graphene, Science\n365, 605 (2019).\n[23] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,\nK. Watanabe, T. Taniguchi, L. Balents, and A. F. Young,\nIntrinsic quantized anomalous Hall effect in a moir´ e het-\nerostructure, Science 367, 900 (2020).\n[24] G. Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Cas-\ntro Neto, A. Reina, J. Kong, and E. Y. Andrei, Observa-\ntion of van hove singularities in twisted graphene layers,\nNature Physics 6, 109 (2010).11\n[25] I. Brihuega, P. Mallet, H. Gonz´ alez-Herrero, G. Tram-\nbly de Laissardi` ere, M. M. Ugeda, L. Magaud, J. M.\nG´ omez-Rodr´ ıguez, F. Yndur´ ain, and J.-Y. Veuillen, Un-\nraveling the intrinsic and robust nature of van hove sin-\ngularities in twisted bilayer graphene by scanning tunnel-\ning microscopy and theoretical analysis, Phys. Rev. Lett.\n109, 196802 (2012).\n[26] A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian,\nM. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi,\nJ. Hone, C. Dean, A. Rubio, and A. N. Pasupathy, Maxi-\nmized electron interactions at the magic angle in twisted\nbilayer graphene, Nature 572, 95 (2019).\n[27] M. I. B. Utama, R. J. Koch, K. Lee, N. Leconte, H. Li,\nS. Zhao, L. Jiang, J. Zhu, K. Watanabe, T. Taniguchi,\nP. D. Ashby, A. Weber-Bargioni, A. Zettl, C. Jozwiak,\nJ. Jung, E. Rotenberg, A. Bostwick, and F. Wang, Vi-\nsualization of the flat electronic band in twisted bilayer\ngraphene near the magic angle twist, Nat. Phys. 17, 184\n(2021).\n[28] S. Lisi, X. Lu, T. Benschop, T. A. de Jong, P. Stepanov,\nJ. R. Duran, F. Margot, I. Cucchi, E. Cappelli,\nA. Hunter, A. Tamai, V. Kandyba, A. Giampietri,\nA. Barinov, J. Jobst, V. Stalman, M. Leeuwenhoek,\nK. Watanabe, T. Taniguchi, L. Rademaker, S. J. van der\nMolen, M. P. Allan, D. K. Efetov, and F. Baumberger,\nObservation of flat bands in twisted bilayer graphene,\nNat. Phys. 17, 189 (2021).\n[29] K. P. Nuckolls, M. Oh, D. Wong, B. Lian, K. Watanabe,\nT. Taniguchi, B. A. Bernevig, and A. Yazdani, Strongly\ncorrelated Chern insulators in magic-angle twisted bi-\nlayer graphene, Nature , 1 (2020).\n[30] Y. Choi, H. Kim, Y. Peng, A. Thomson, C. Lewandowski,\nR. Polski, Y. Zhang, H. S. Arora, K. Watanabe,\nT. Taniguchi, J. Alicea, and S. Nadj-Perge, Correlation-\ndriven topological phases in magic-angle twisted bilayer\ngraphene, Nature 589, 536 (2021).\n[31] S. Wu, Z. Zhang, K. Watanabe, T. Taniguchi, and\nE. Y. Andrei, Chern insulators, van Hove singularities\nand topological flat bands in magic-angle twisted bilayer\ngraphene, Nat. Mater. 20, 488 (2021).\n[32] Y. Saito, J. Ge, L. Rademaker, K. Watanabe,\nT. Taniguchi, D. A. Abanin, and A. F. Young, Hof-\nstadter subband ferromagnetism and symmetry-broken\nChern insulators in twisted bilayer graphene, Nat. Phys.\n17, 478 (2021).\n[33] I. Das, X. Lu, J. Herzog-Arbeitman, Z.-D. Song,\nK. Watanabe, T. Taniguchi, B. A. Bernevig, and\nD. K. Efetov, Symmetry-broken Chern insulators and\nRashba-like Landau-level crossings in magic-angle bilayer\ngraphene, Nat. Phys. 17, 710 (2021).\n[34] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and\nP. Jarillo-Herrero, Flavour Hund’s coupling, Chern gaps\nand charge diffusivity in moir´ e graphene, Nature 592, 43\n(2021).\n[35] A. T. Pierce, Y. Xie, J. M. Park, E. Khalaf, S. H. Lee,\nY. Cao, D. E. Parker, P. R. Forrester, S. Chen, K. Watan-\nabe, T. Taniguchi, A. Vishwanath, P. Jarillo-Herrero, and\nA. Yacoby, Unconventional sequence of correlated Chern\ninsulators in magic-angle twisted bilayer graphene, Nat.\nPhys. 17, 1210 (2021).\n[36] Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf,\nP. Ledwith, Y. Cao, S. H. Lee, S. Chen, P. R. Forrester,\nK. Watanabe, T. Taniguchi, A. Vishwanath, P. Jarillo-\nHerrero, and A. Yacoby, Fractional Chern insulators inmagic-angle twisted bilayer graphene, Nature 600, 439\n(2021).\n[37] N. F. Q. Yuan and L. Fu, Model for the metal-insulator\ntransition in graphene superlattices and beyond, Phys.\nRev. B 98, 045103 (2018).\n[38] J. Kang and O. Vafek, Symmetry, Maximally Localized\nWannier States, and a Low-Energy Model for Twisted\nBilayer Graphene Narrow Bands, Phys. Rev. X 8, 031088\n(2018).\n[39] M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi,\nK. Kuroki, and L. Fu, Maximally Localized Wannier Or-\nbitals and the Extended Hubbard Model for Twisted Bi-\nlayer Graphene, Phys. Rev. X 8, 031087 (2018).\n[40] H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Ori-\ngin of Mott Insulating Behavior and Superconductivity\nin Twisted Bilayer Graphene, Phys. Rev. X 8, 031089\n(2018).\n[41] L. Rademaker and P. Mellado, Charge-transfer insulation\nin twisted bilayer graphene, Phys. Rev. B 98, 235158\n(2018).\n[42] M. Xie and A. H. MacDonald, Nature of the Correlated\nInsulator States in Twisted Bilayer Graphene, Phys. Rev.\nLett. 124, 097601 (2020).\n[43] Z.-D. Song and B. A. Bernevig, Magic-angle twisted bi-\nlayer graphene as a topological heavy fermion problem,\nPhys. Rev. Lett. 129, 047601 (2022).\n[44] S. Fang and E. Kaxiras, Electronic structure theory of\nweakly interacting bilayers, Phys. Rev. B 93, 235153\n(2016).\n[45] S. Carr, S. Fang, P. Jarillo-Herrero, and E. Kaxiras, Pres-\nsure dependence of the magic twist angle in graphene\nsuperlattices, Phys. Rev. B 98, 085144 (2018).\n[46] N. N. T. Nam and M. Koshino, Lattice relaxation and en-\nergy band modulation in twisted bilayer graphene, Phys.\nRev. B 96, 075311 (2017).\n[47] S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin,\nand E. Kaxiras, Relaxation and domain formation in in-\ncommensurate two-dimensional heterostructures, Phys.\nRev. B 98, 224102 (2018).\n[48] N. Tilak, X. Lai, S. Wu, Z. Zhang, M. Xu, R. d. A.\nRibeiro, P. C. Canfield, and E. Y. Andrei, Flat band car-\nrier confinement in magic-angle twisted bilayer graphene,\nNature Communications 12, 4180 (2021).\n[49] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Ori-\ngin of magic angles in twisted bilayer graphene, Phys.\nRev. Lett. 122, 106405 (2019).\n[50] E. Polizzi, Density-matrix-based algorithm for solving\neigenvalue problems, Phys. Rev. B 79, 115112 (2009).\n[51] G. Lehmann and M. Taut, On the numerical calculation\nof the density of states and related properties, physica\nstatus solidi (b) 54, 469 (1972).\n[52] O. E. Alon and L. S. Cederbaum, Hellmann-Feynman\ntheorem at degeneracies, Phys. Rev. B 68, 033105 (2003).\n[53] G. Kresse and J. Furthm¨ uller, Efficient iterative schemes\nfor ab initio total-energy calculations using a plane-wave\nbasis set, Phys. Rev. B 54, 11169 (1996).\n[54] G. Kresse and J. Furthm¨ uller, Efficiency of ab-initio total\nenergy calculations for metals and semiconductors using\na plane-wave basis set, Computational Materials Science\n6, 15 (1996).\n[55] P. E. Bl¨ ochl, Projector augmented-wave method, Phys.\nRev. B 50, 17953 (1994).\n[56] G. Kresse and D. Joubert, From ultrasoft pseudopoten-\ntials to the projector augmented-wave method, Phys.12\nRev. B 59, 1758 (1999).\n[57] J. Klimeˇ s, D. R. Bowler, and A. Michaelides, Van der\nWaals density functionals applied to solids, Phys. Rev. B\n83, 195131 (2011).[58] G. Rom´ an-P´ erez and J. M. Soler, Efficient Implementa-\ntion of a van der Waals Density Functional: Application\nto Double-Wall Carbon Nanotubes, Phys. Rev. Lett. 103,\n096102 (2009)."    },    {        "title": "2304.14269v1.Performance_Analysis_of_DNA_Crossbar_Arrays_for_High_Density_Memory_Storage_Applications.pdf",        "content": "1 \n Performance Analysis of DNA Crossbar Arrays for High -Density Memory \nStorage Applications  \n \nArpan De1, Hashem Mohammad1,2, Yiren Wang1, Rajkumar Kubendran3, Arindam K. Das4, \nM.P.Anantram1,** \n \n1Department of Electrical and Computer Engineering, University of Washington, 98195 Seattle, WA, USA  \n2Department of Electrical Engineering, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait  \n3Department of Electrical and Computer Engineering, University of Pittsburgh, 15261 Pittsburgh, PA, USA  \n4Department of Computer Science and Electrical Engineering, Eastern Washington University, 99004 Cheney, WA, USA.  \n \n** Corresponding author: M. P. Anantram (e -mail: anant@uw.edu ) \n \nABSTRACT  \n \nDeoxyribonucleic acid (DNA) has emerged as a promising building block for next -generation ultra -high density \nstorage devices. Although DNA has high durability and extremely high density in nature, its potential as the basis \nof storage devices is currently hindered by limitations such as expensive and complex fabrication processes and \ntime-consuming read -write operations. In this article, we propose the use of a DNA crossbar array architecture \nfor an electrically readable Read -Only Memory (DNA -ROM). While information can be ‘written’ error -free to a \nDNA -ROM array using appropriate sequence encoding, its read accuracy can be affected by several factors such \nas array size, interconnect resistance, and Fermi energy deviations from HO MO levels of DNA strands employed \nin the crossbar. We study the impact of array size and interconnect resistance on the bit error rate of a DNA -ROM \narray through extensive Monte Carlo simulations. We have also analyzed the performance of our proposed DNA \ncrossbar array for an image storage application, as a function of array size and interconnect resistance. While we \nexpect that future advances in bioengineering and materials science will address some of the fabrication challenges \nassociated with DNA crossb ar arrays, we believe that the comprehensive body of results we present in this paper \nestablishes the technical viability of DNA crossbar arrays as low power, high -density storage devices. Finally, our \nanalysis of array performance vis -à-vis interconnect r esistance should provide valuable insights into aspects of \nthe fabrication process such as proper choice of interconnects necessary for ensuring high read accuracies.  \n \n1. INTRODUCTION  \n \nProduction of digital data is increasing exponentially, driven predominant ly by the adoption of artificial \nintelligence (AI) and deep learning. According to current projections, approximately 175 zettabytes (109 \nterabytes) of new data will be generated annually by 2025. Archival storage of even a fraction of such immense \ndata re quires a fundamental rethink in durable, low power, accurate, and extreme high -density storage technology \nsince traditional storage devices (e.g., magnetic tapes, hard disks, Blu -ray, and solid -state devices like flash) are \nfast approaching their scaling a nd performance limits. The key figures -of-merit of a storage technology are: (i) \nthe maximum density of any storage device in terms of bits per unit volume, (ii) durability (total time for which \nthe data can be stored), and (iii) energy cost. With the proj ected explosion of data storage requirements, it is widely \naccepted that existing technologies are inadequate with respect to the figures -of-merit. Silicon -based devices are \nfast approaching their density limits and further downscaling of such devices will  result in performance \ndegradation. This motivates the demand for development of alternate ultra -high-density storage technologies \nwhich are durable and energy -efficient.  \n \n      Deoxyribonucleic acid (DNA) is one of the most promising candidates for high -density storage devices. The \nconcept of using DNA for data storage dates to the mid -1960s when the idea of ‘genetic memory’ was proposed \nby Neiman and Wiener  [1], [2] . However, technological limitations at the time regarding  DNA sequencing and \nsynthesis limited the practical viability of such memory devices. Around 1986, Davis et al [3] and  Clelland et al \n[4] made the first experimental demonstrations of DNA storage. A breakthrough was achieved in 2012 -13 by \nsuccessfully storing hundreds of kilobytes of data [5]. \n \n      DNA -based molecular -level data storage is an excellent choice for future high -density storage applications \nsince only a few layers of atoms are required to store one bit of information. DNA storage can be significantly \ndenser than the densest storage device a vailable at present and is also more durable. When insulated from heat, \nhumidity, and light, DNA can last for centuries to millennia, compared to decades for traditional media devices. \nTable 1 provides a comparison of conventional silicon -based storage dev ices, resistive random -access memory \n(RRAM), and DNA -based devices with respect to (w.r.t) density, random access, read latency, and durability.  \n \n      DNA based data storage involves four steps: (i) encoding (mapping of digital information to appropriate DNA \nsequences), (ii) synthesis (writing into DNA sequences), (iii) reading (accessing the data), and (iv) decoding 2 \n (converting to digital format). Current DNA -in-solution storage systems are engineered to store data in chunks of \nDNA  [6], [7] . Traditionally, information in DNA -in-solution systems is encoded into appropriate DNA sequences \nusing a dedicated algorithm that maps digital bits to DNA bases [8]. Different methodologies for DNA sequence \nsynthesis exist, for instance, phosphoramidite -based synthesis or enzymatic synthesis [9]. Polymerase chain \nreaction (PCR) is a proven high -throughput method that can facilitate la rge-scale DNA replication. The read \nprocess for DNA -in-solution systems has significant time complexity since it requires an indexing system for \nproper reconstruction of the data.  \n \n      A major drawback of DNA -in-solution  storage systems concerns random a ccess. Even if a small segment of \nthe stored data  needs to be accessed and retrieved, the entire DNA pool must  be read and sequenced, which is \ntime-consuming and inefficient. However, some recent work exists that aims to address/improve random access, \ne.g., the magnetic bead extraction approach or PCR with primers discussed in [10]. We note that the read accuracy \nof DNA -in-solution storage systems depends largely on the synthesis process, which, as mentioned previously, is \nsusceptible to errors.  \n \n      Even though DNA -based storage systems have a transformative potential for next -generation storage \napplications, the technology is still at a nascent stage . In this paper, we address the random access and read latency \nissues by adopting a crossbar ar ray architecture for an electrically readable Read -Only Memory (ROM) device \n[11]. Our proposed device is composed of interconnects and two different  DNA sequences serving as memory \ncells, as illustrated in Figure 1. In contrast to existing DNA -in-solution storage systems where bit strings are \nencoded into a DNA sequence, we choose an ‘appropriate’ pair of DNA sequences and exploit the difference \nbetwe en their conductance to reliably store binary information. Motivated by recent advances in the field of \nsynthetic biology, we propose self -assembly of crossbar nanostructures using DNA origami -inspired approaches \n[12]. The assembly of DNA crossbars will consist of separately forming the memory elements and the DNA \nnanowires, followed by their integration. Finally, after the self -assembly phase, the DNA crossbar will be \nconnected to electrodes at the edges of the array to enable readout [11]. From a fabrication perspective, we propose \nleveraging synthetic biology approaches for fabricat ing and programming DNA memory cells/interconnects and \nconventional semiconductor and lithography processes for contact formation. Challenges related to DNA crossbar \narray fabrication include: (i) High precision contact formation between DNA memory cells a nd DNA nanowires \n(interconnects). Since DNA crossbar formation is a self -assembly process, a suitable fabrication recipe is needed \nwhich will yield good contacts between the memory cells and the nanowires. (ii) Fabrication of highly conductive \nDNA nanowire s, an area which is mostly unexplored. The work by Ke and Hihath is a pioneering effort that has \njust stared to experimentally address this issue [13]. As seen from our computational analysis, interconnects \nshould be conductive to achieve good read -out acc uracy. Since DNA nanowires are typically very resistive, \nmaking them more conductive is a challenging part of the process. However, the relative contributions of the \ncontact and intrinsic resistance of the DNA are unclear at this stage. Finally, as noted i n [14], initialization of large \nDNA crossbar arrays can incur a large spatiotemporal overhead. Exceptional spatial control of self -assembly has \nbeen demonstrated for DNA origami [15], [16]. From a temporal perspective, self -assembly process can take up \nto 1-3 days [17], [ 18]. We estimate that self -origami of a DNA array can take up to 2 -4 days, but the process is \nextremely parallelized such that billions of copies can be made in parallel. ` \n \n     Exponential growth in data generation and the need for durable and long -term data storage have exposed the \nshortcomings of current memory technologies such as flash, RRAM, phase change memory (PCM), and magnetic \ntunnel junctions (MTJ). According to the Int ernational Roadmap for Devices and Systems (IRDS 2020), DNA \nstorage technologies are a promising candidate for long -term, large -scale data storage applications. The \ndevelopment of commercially viable DNA -based storage devices is now widely recognized as a critical emergent \nresearch area and has rightfully attracted a proliferation of research in recent times. Nevertheless, the state -of-the-\nart in this area is still quite nascent and the path from ideation to commercialization is riddled with significant \nengineering and scientific challenges. Overcoming these challenges will require potentially ground -breaking \ninnovations from researchers in diverse disciplines such as molecular biology, materials science, and electrical \nengineering. Some notable research cha llenges include making conducting DNA origami and the development of \nrobust biotic -abiotic interfaces between DNA and the electrode array. Recent research trends in the domain of \nCMOS -DNA systems [19], [20]  may provide a solution for integrating silicon -based devices/circuits with DNA. \nEqually important are the design of peripheral circuitry and the choice of interconnect material for na noscale \ndevices.  \n \n     Despite the advantages of a crossbar topology over DNA -in-solution systems, two factors can adversely affect \nits read accuracy. First, Fermi energy variations  [21] can alter the conductance state of DNA, leading to errors \nduring the read -out process. Second, sneak paths, which distort the read current dist ributions in memory cells, \naffect all crossbar topologies and DNA -based arrays are no exception. Existing approaches for alleviating sneak \npath issues include the use of a selector switch coupled with an appropriate biasing scheme. While the ideal \nmechanis m for mitigating sneak path effects in a DNA crossbar array is an open question, the feasibility of ‘DNA 3 \n diodes’ [22] is an interesting proposition and merits further investigation. Additionally, the development of the \nread-out circuitry for DNA crossbar arrays offers significa nt engineering challenges and is subject to further \nresearch.  \n \n       The rest of the paper is structured as follows. The calculation of current -voltage characteristics is addressed \nin Section 2, which is followed by a discussion of architectural concepts of a DNA -ROM array in Section 3. In \nSection 4, we study the performance of the array for different DNA strands and interconnect resistances. The \nimpact of Fermi energy variation on array performance is addressed in Section 5. In Section 6, we present a \ndetailed scalability analysis in terms of two key metrics – bit error rate and power consumption. We conclude this \npaper with a brief discussion of ongoing and future work in Section 7. \n \n2. DNA CURRENT SIMULATION FRAMEWORK & RESULTS  \n \nWe adopted two B -form DNA sequences 3’ -CCCTCCC -5’ and 3’ -TTTCTTT -5, referred to as B -CT 1C and B -\nTC 1T respectively in this paper, which are studied in [23]. The simulation methodology involves four main steps: \nstructure building, energy minimization, density functional theory calculations, and charge transport calculations. \nFirst, the Nucleic Acid Builder (NAB) software package [24] is used to obtain the atomic coordinates of double -\nstranded B -DNA structures. Next, we used PACMOL [25] to obtain energy minimized structures with explicit \nsolvent (water) environment within a 60 Å x 60 Å x 60  Å cubic solvation box. We added Na counterions to \nneutralize the phosphate groups in the DNA backbone. We also added Na and Cl ions randomly into the solution \nto represent a homogenous mixture of 150 mM [26], [27] . We used General Amber Force Field [28]. For the water \nsolvent, a truncated octahedron box of SPC/E water model was used to solvate the  DNA. Amber force field with \nparmbsc1 correction [29] was used to energy minimize the structures. The structure minimization was done in \ntwo stages. The first stage consists of restraining the DNA with a  force of 25 kcal/mol -Å2 for the first 1000 cycles, \nwhile the solvent environment is minimized. The second stage involves minimizing the whole system for \nadditional 1000 cycles. Both stages of minimization have 500 steps of the steepest descent method, fol lowed by \n500 steps of the conjugate gradient method.  \n        Next, we delete the solvent atoms and only keep the DNA atoms along with the closest Na counterions \nnecessary to neutralize the DNA molecule. We then performed density functional theory calculati ons and obtain \noverlap and Fock matrices, denoted by S and F respectively. In this step, we used the B3LYP functional and 6 -\n31G(d,p) basis set. We used the polarizable continuum model (PCM) to include the impact of water solvent \nenvironment. The Fock and o verlap matrices resulting from this step are used as inputs for the charge transport \ncalculations. The system Hamiltonian is obtained by transforming the F and S matrices to an orthogonal basis by \napplying the L ὅewdin transformation [30]:  \n \n𝐻𝑎= 𝑆−1\n2 𝐹 𝑆−1\n2 (1) \n \nwhere 𝐻𝑎 is the system Hamiltonian. The diagonal elements of 𝐻𝑎 represent the atomic orbitals and the off -\ndiagonal elements represent the electronic coupling between these atomic orbitals. We chose to partition the DNA \ninto nucleotides (backbone + base). We achi eved this by transforming the Hamiltonian into a block -diagonal \nmatrix, through the unitary transformation:  \n \n𝐻𝑏= 𝑈†𝐻𝑎𝑈 (2) \nwhere U consists of the eigenvectors of each nucleotide. The eigenvectors in U are arranged accordi ng to the order \nof the DNA nucleotides. As a result, the diagonal elements of each diagonal block in 𝐻𝑏  represent the molecular \norbital energies of the nucleotide and the off -diagonal blocks in 𝐻𝑏 represent the electronic coupling between the \ndifferen t nucleotides. Both dimensions of a diagonal block for 𝐻𝑎  and 𝐻𝑏  are equal to the number of orbitals \nused to represent the DNA nucleotide. We also note here that in obtaining 𝐻𝑏, we omitted the atomic orbital \ncontributions of the counterions and focused the charge transport calculations on the DNA molecule itself. T his \nis in accordance with the finding in  [23] that the counterions do not impact the DNA conductance, especially in \nenergy ranges within the vicinity of the highest occupied molecular orbital (HOMO) of the DNA.  \n      After the 𝐻𝑏 matrix is obtained, the transmission versus ene rgy of a DNA molecule is computed using the \nGreen’s function approach to account for decoherence [30]. We use this m ethod to model the system shown in \nFigure 2(a). A DNA strand is placed between two metallic contacts and a voltage bias is applied. The governing \nGreen’s function equation is defined as:  \n \n[𝐸−(𝐻𝑏+𝛴𝐿+𝛴𝑅+𝛴𝐵)]𝐺𝑟= I (3) \n 4 \n where 𝐸 is the energy, 𝛴𝐿(𝑅) is the left (right) contact self -energy, 𝛴𝐵 is the decoherence self -energy, 𝐺𝑟 is the \nretarded Green’s function and I is the identity matrix. In this formalism, we use the contact’s self -energy matrices \n(𝛴𝐿(𝑅)) to account for the two electrodes shown in Figure 2(a). The self -energy of the contacts is defined as 𝛴𝐿(𝑅)=\n−𝑖𝛤𝐿(𝑅)/2 , 𝑖=√−1  and 𝛤𝐿(𝑅)=1000  meV  is the coupling value . The contact self-energ ies are added to the 3’ \nand 5’ nucleotide ends of the strand , respectively. The decoherence probe’s self -energy is defined in a similar \nmanner, 𝛴𝐵= ∑(−𝑖𝛤𝑘\n2) 𝑘 , where the subscript 𝑘 represents the 𝑘th decoherence  probe, and Γ𝑘 is the coupling \nstrength between the decoherence probe and the system, with 𝛤𝑘=10 meV . The decoherence probes ( 𝛴𝐵) are \nincluded at each nucleotide (the diagonal elements of 𝐻𝑏), except at the two contact locations. These are fictitious \nprobes which mimic extracting electrons from the molecule and re -injecting them after phase breaking. The \ncurrent at the 𝑘th probe is:  \n \n𝐼𝑘↑=2𝑞\nℎ∑∫𝑇𝑘𝑙(𝐸)[𝑓𝑘(𝐸)−𝑓𝑙(𝐸)]𝑑𝐸+∞\n−∞𝑁\n𝑙=1=∫𝐽𝑙(𝐸)𝑑𝐸+∞\n−∞ (4) \nwhere 𝑘=1,2,3,...,𝑁, 𝑇𝑘𝑙=𝛤𝑘𝐺𝑟𝛤𝑙𝐺𝑎 is the transmission function between probes 𝑘  and 𝑙, 𝐺𝑎=(𝐺𝑟)†  is the \nadvanced Green’s function, 𝑓𝑘(𝐸)=(1+exp (𝐸−𝐸𝑓𝑘\n𝑘𝑇))−1\n is the Fermi distribution and 𝐽𝑘(𝐸) is the current with \nrespect to energy at probe 𝑘. The condition for the fictious decoherence probes is that the current 𝐽𝑘(𝐸)=0 at \nevery energy.  These yields 𝑁𝑏 independent equations, where 𝑁𝑏 is the number of dec oherence probes in the \nsystem. The following relation can then be derived  [31]: \n \n𝑓𝑘(𝐸)−𝑓𝐿(𝐸)=(∑𝑊𝑘𝑙−1𝑇𝑙𝑅𝑁𝑏\n𝑙=1)(𝑓𝑅(𝐸)−𝑓𝐿(𝐸)) (5) \nwhere  𝑘=1,2,3,…,𝑁, 𝑊𝑘𝑙−1 is the inverse of  𝑊𝑖𝑗=(1−𝑅𝑘𝑘)𝛿𝑘𝑙−𝑇𝑘𝑙(1−𝛿𝑘𝑙), where 𝑅𝑘𝑘 is the reflection \nprobability at probe 𝑘, and is given by  𝑅𝑘𝑘=1−∑𝑇𝑘𝑙𝑁\n𝑘≠𝑙. Since the currents at the left (𝐼𝐿↑) and right (𝐼𝑅↑) are \ngoverned by the conservation of electron number,  𝐼𝐿↑+𝐼𝑅↑=0, this yields the equation for the current at the lef t \ncontact  [31], which is the current flowing through the DNA:  \n \n𝐼𝐷𝑁𝐴= 𝐼𝐿↑=2𝑒\nℎ∫𝑇𝑒𝑓𝑓(𝐸)[𝑓𝐿(𝐸)−𝑓𝑅(𝐸)]𝑑𝐸+∞\n−∞ (6) \n \nThe effective transmission term is:  \n \n𝑇𝑒𝑓𝑓=𝑇𝐿𝑅+∑∑𝑇𝐿𝑘 𝑊𝑘𝑙−1 𝑇𝑙𝑅𝑁𝑏\n𝑙=1𝑁𝑏\n𝑘=1 (7) \n  \nIn equation (7), 𝑇𝐿𝑅 is the coherent transmission from the left electrode to the right electrode. The second term is \nthe decoherence contribution to transmission via the decoherence probes.  \n      The final step involves the calculation of current under different biases and Fermi energies. The DNA structure \nis static throughout the calculations. We apply a potential shift in the DNA Hamiltonian to represent voltage drop \nacross the DNA. In a me tal-molecule -metal setup, obtaining the electrostatic potential profile across the  molecule \nand finding the exact location of the Fermi energy requires using the density matrix and solving Poisson’s equation \nin a self -consistent manner [32]. However, this is very difficult to achieve for large mol ecules such as the studied \nDNA (which consists of several hundred atoms). In addition, the details of the contacts such as the orientation and \ngeometry at the metal -molecule interface are unknown, which further increases the complexity of the problem. \nTher efore, the general approach is to treat the Fermi energy as a variable and calculate the current or conductance \nat different Fermi energies. Further, at nonzero bias, a common approach is to use a linear ramp potential profile \nacross the molecule by shifti ng the molecular orbitals in the DNA Hamiltonian ( 𝐻𝑏) prior to calculating the \ntransmission and the current. Following prior work [32], the potential drop is taken to be : \n 5 \n 𝑉𝑛=\n{  0,𝑛=1\n𝑞𝑉𝑏𝑖𝑎𝑠×(0.4+0.2(𝑛−2)\n𝑁−3),2≤𝑛≤𝑁−1\n𝑞𝑉𝑏𝑖𝑎𝑠,𝑛=𝑁 \n (8) \nwhere 𝑉𝑛 is the voltage at base pair 𝑛. The electrostatic potential energy at base pair 𝑛, 𝑞𝑉𝑛, is added to the  diagonal \nelements of the Hamiltonian 𝐻𝑏 of block 𝑛. Note that block 𝑛 consists of a base pair, 𝑞 is the magnitude of electron \ncharge (1.6×10−19 C), and 𝑉𝑏𝑖𝑎𝑠 is the bias voltage applied across the strand. Locations 1 and 𝑁 represent the \ncontact coupling sites. We set 40% of the voltage to drop on each end of the strand, while the remaining 20% of \nthe bias drops linearly along the DNA.  \n      The next step involves the calculation of current under different voltage biases and  Fermi energies. The DNA \nstructure is static in the calculations. After shifting the molecular orbitals in 𝐻𝑏, we calculate the effective \ntransmission and use equation (6) to calculate the current at a certain bias with  \n \n 𝐸𝑓𝑅=𝐸𝑓𝐿+𝑞𝑉𝑏𝑖𝑎𝑠 (9) \n  \nwhere 𝐸𝑓𝐿 and 𝐸𝑓𝑅 are the Fermi energies of the left and right contacts, respectively.  \n       Figure 2(a) shows the schematic for the DNA conductance calculations along with the non -linear potential \ndrop, illustrating how the molecular orbitals shift along the strand. Figure 2(b,c) shows the transmission profiles \nof the two DNA strands (B -CT 1C and B -TC 1T) under different biases -1V (yellow), 0V (green), and 1V (blue). \nThe transmission profile carries information regarding the density of states (DOS) of the  system. Since DOS is \ndirectly related to the energy distribution of the molecular orbit als, the effect of shifting of the orbitals’ energy \ndue to the applied bias is observable in the transmission. The peaks in transmission suggest high DOS at those \nenergies, which in turn yields high current conduction. From Figure 2(b,c), we can observe th at the peaks shift to \nthe right (higher energy) when a positive bias is applied and vice versa. Numerically, for B -CT 1C DNA, the \nHOMO peak is at -5.8 eV, -5.2 eV, and -4.78 eV corresponding to bias values of -1 V, 0 V, and 1  V respectively. \nThus, we have a  10.38% and 8.47% shift in HOMO when  the bias changes from 0 to -1 V and from 0 to 1  V \nrespectively. Similarly, we get a shift of 8.6% and 9.4% in the case of B -TC 1T DNA when the bias changes from \n0 to -1 V and from 0 to 1  V respectively. The transmission profiles at different energies and biases are plugged \ninto equations (6) and (9), to get the transfer characteristics shown in Figure 2(d,e). The current -voltage curves \nfor both DNA strands are shown at different Fermi energy values.  \n \n      We have chosen  to operate near the HOMO since the comparison between the gold work function and the \nionization potentials of the DNA base pairs estimates the Fermi energy of gold to be close to the HOMO of the \nDNA. From Figure 2(d,e), we observe that there is a signific ant current difference between the two strands in the \nvoltage range of 0  to 1 V. Therefore, we use these two strands to represent logic 0 and 1 memory states in our \nstudy of the DNA -ROM crossbar.  \n \n3. ARCHITECURE OF DNA CROSSBAR ARRAY  \n \nTraditional memory stora ge devices are fast reaching their scaling limits. Additionally, these devices suffer from \nlow packing density of cells. To mitigate these problems, crossbar architectures are preferred for Resistive \nRandom -Access Memory (RRAM) applications  such as neuromorphic computing, in -memory computing, and \nimage processing. Such an architecture is advantageous because of fast  read and write operations which are \nessential for many digital and analog applications. Random access is enabled by selective ap plication of a bias to \nappropriate wordlines and grounding of corresponding bitlines. For example, to access the memory element at the \n(1,3) th location in the array shown in Figure 3(a), a bias 𝑉1 = 1V has to be applied to wordline 1 with bitline 3 \ngroun ded (all other bitlines are kept floating). A high current measured in bitline 3 under this biasing scheme \nrepresents logic state ‘1’ and a low current represents logic state ‘ 0.’ \n \n      In this article, we propose and analyze the performance of a DNA cros sbar array as a storage device. Figure \n1(c) shows our proposed DNA ROM architecture. In this  arrangement,  a double -stranded (ds) DNA is assembled \nat every junction  (alternately referred to as grid points or memory cell locations in this article) of a cro ssbar array. \nA circuit schematic of a DNA crossbar array with interconnect resistances and two different DNA strands, \ndepicted as different resistors, is shown in Figure 3(a). Since the performance of the array depends significantly \non the difference in co nductance of the two dsDNA sequences, a proper choice of the DNA pair is critical.  \n \n      Unlike an RRAM crossbar technology, DNA crossbars are not associated with a WRITE process which \nrequires that resistances be updated by application of an external bias. Hence, we study the applicability of DNA \nin a ROM system instead of a RAM. In DNA -based storage devices, information can be easily loaded onto the 6 \n device using sequence encoding. We use a readout scheme proposed by Liao et al. in [33]. In their scheme, to \nread the 𝑖th row, an external bias voltage 𝑉𝑖𝑛 is applied to that row (wordline) and the currents through the \ngrounded columns (bitlines) are recorded. Extensive computational experiments have demonstrated that the \nreadout approach outlined in [33] maintains high accuracy, while reducing the computational complexity from \nlog(m2n2) to log( m2+n2+mn) for an m×n array. In performing our simulations, we have grounded all bitlines under \nthe assumption that the information is stored in the whole crossbar.   \n \n     The interconnect resistance 𝑅𝑖𝑛𝑡 of an RRAM crossbar between adjacent memory cells is about 7 -10 Ω wh en \nthe half -pitch is 14 nm.  If the cross -sectional area of the interconnects is reduced in the interest of scalability, \n𝑅𝑖𝑛𝑡 increases. However, large interconnect resistances (> 10 Ω) in RRAM arrays are problematic for READ and \nWRITE operations, espe cially the WRITE process, since the resistive state of the array cannot be set properly. \nOur proposed  DNA crossbar  is intended for data storage and retrieval  application s. Therefore, it  involves only a \nREAD proces s. The resistance of DNA strands is typically on the order of mega -Ohms, which permits the use of \nhigher interconnect resistances before sneak path issues start to adversely affect the performance of DNA crossbar \narrays.  \n \n     In this paper, we consider 𝑅𝑖𝑛𝑡 as purely a simulation parame ter which models the interconnect resistance \nbetween two adjacent DNA memory cells and affects the accuracy and power consumption of the array. For DNA \ncrossbar -based storage technology, potential candidates for interconnects  include DNA nanowires  (resista nce of \nwhich is not yet known) , carbon nanotubes  (1-20 KΩ) [34], [35] , and graphene nanoribbons  (10-1000 KΩ) [36], \n[37]. Carbon nanotube interconnects can have a resistance of less than 10 kΩ depending on the number of shells. \nThe resistance of DNA nanowires currently being pursued experimentally is expected to be larger than that of \nnanotubes.  Conventional metal inter connects like tungsten and copper have a resistivity of 5.6×10−8 Ω.m and \n1.68×10−8 Ω.m respectively.  Accordingly, we have considered a wide range of 𝑅𝑖𝑛𝑡 values from 10 kΩ to 1 \nMΩ, without explicitly making a choice regarding the interconnect material or its dimensional aspects such as the \ncross -sectional area or aspect ratio. Our simulation results should provide valuable guidelines regarding the \nphysical  design parameters of a DNA crossbar array vis -à-vis its key performance parameters such as bit error \nrate and power dissipation.  \n \n     In the subsequent sections, we analyze the performance of our proposed DNA crossbar array w.r.t bit error rate \nand power  consumption due to READ operations only. We also demonstrate in detail how parameters such as the \nvalue of interconnect resistance and Fermi energy variations of the DNA strands from their HOMO levels can \naffect the performance of such an array.  \n \n4. ARRAY  SIMULATION RESULTS  \n \nThe detailed simulation methodology is shown in Figure 3(b). From DNA transport simulations at various voltage \nbiases, a look -up table is constructed to calculate the DNA currents. Since the current -voltage characteristic of \nDNA strands is non -linear (see Figure 2), we have adopted an iterative approach for computing the voltage drop \n𝑉𝑖𝑗 across the DNA at the ( i, j)th location of the memory a rray, after the interconnect resistances and the resistances \nof the other DNA memory elements in the array have been accounted for. In this paper, we compute 𝑉𝑖𝑗 when a \nvoltage of 𝑉𝑖𝑛=1 𝑉 is applied to row i while other rows are left floating, and the bottom of each column in Figure \n3 is grounded . The iterative process commences with a small, applied volt age of 0.05 V to obtain the initial \nconductance matrix by interpolating from the look -up table. Based on the conductance values of DNA memory \ncells, the voltage distribution in the array is calculated using the following equation:  \n \n𝑉𝑖𝑗= 1+∑(𝐺𝑖𝑘∗∑1\n𝑔𝑙𝑤𝑘\n𝑤=𝑗+1) 𝑛\n𝑘=𝑗+1\n1+∑(𝐺𝑖𝑘∗∑1\n𝑔𝑙𝑤𝑘\n𝑤=𝑗+1) 𝑛\n𝑘=1     (𝑉𝑖𝑛𝛼𝑖) (10) \n \nwhere i and j are the row and column indices, 𝐺𝑖𝑗 is the conductance of the DNA sequence at the (𝑖,𝑗)th grid point, \n𝑔𝑙𝑤 is the interconnect conductance between adjacent memory cells along wordlines, and 𝛼𝑖 is a row -specific \nsneak path parameter (see eqn. (8) in  [33]) which is independent of the bias voltage. After the  first iteration, the \nresistance states of DNA memory cells are updated according to the node voltages and the current -voltage \nrelationship. The updated resistance states are then used to refine the voltage distribution. This process continues \nuntil the voltage difference between two successive iterations at each node is within an acceptable tolerance. The \nvoltage distribution obtained after convergence is then used to compute the final read -out current distribution. We \nrefer to the set of voltages, {𝑉𝑖𝑗\n𝑉𝑖𝑛 𝛼𝑖:1  𝑗  𝑛}, as normalized vo ltages . 7 \n         All simulations in this section were performed considering homogeneous arrays, which we define as arrays \ncomposed of identical DNA sequences at all junction points. We have modelled the crossbar array as a resistive \nnetwork and each memory cell is modelled as a non -linear resistor. Capacitive effects, which are important in \nunderstanding the transient response, are not accounted for in our current study. While such effects can contribute \nto latency and play a critical  role in matrix -vector multiplication applications , its impact on storage applications \nisn’t major. To compute the latency, we need to calculate the quantum capacitance of DNA memory cells and \ninterconnects. Carbon nanotubes offer a quantum capacitance of 100-400 aF/μm [38], [39] . Similar estimates for \nDNA origami nanowires have not been derived t ill date. Figure 4 shows the normalized voltages (applicable for \nall rows) for a homogeneous 64×64 array, co mposed with either B -CT 1C or B -TC 1T dsDNA sequences and 𝑅𝑖𝑛𝑡 \n= 10 kΩ or 1 MΩ. We chose a value of 10 kΩ on the low end to better demonstrate the sensitivities of array \nvoltages (Figure 4) and currents (Figure 5) to the interconnect resistance.   \n \n      Figure 4 also shows the normalized voltages for two different Fe rmi energies, at HOMO and (HOMO+0.2) \neV. First, we observe that the voltage degradation is worse the farther a node is from the input, irrespective of the \ntype of sequence and the value of 𝑅𝑖𝑛𝑡. Since the read current of a memory cell is directly propo rtional to the \nvoltage across it, we expect this voltage degradation to be reflected in the spatial distribution of read currents, \nwhich in turn should affect the read accuracy. We address this issue in the subsequent paragraph. Second, we \nobserve that the  voltage degradation is worse for the B -CT 1C sequence compared to the B -TC 1T dsDNA sequence, \nfor all values of interconnect resistance and Fermi energy. This is due to the presence of more low resistance cells \nin B-CT 1C crossbar configurations compared to B-TC 1T crossbar configurations, which leads to more sneak paths. \nFrom Figure 2(d,e), we see that the conductance of B -CT 1C dsDNA is higher than that of B -TC 1T strand, \nparticularly when the applied bias is in the range of 0  to 1 V. At a particular interconn ect resistance, voltage \ndegradation is less at a higher value of Fermi energy for both the homogenous arrays. The reason for this can be \nattributed to the reduced conductance of the DNA strands at a higher Fermi energy (see Figure 2(d,e)).  \n \n      From the preceding discussion, we conclude that the B -CT 1C dsDNA sequence is more susceptible to changes \nin the interconnect resistance due to its high conductivity. This is manifested in Figure 5 which shows the current \ndistribution heatmaps for a 6464 homogeneous array composed of either  the B -CT 1C dsDNA sequence or the \nB-TC 1T dsDNA sequence, with 𝑅int = 10 kΩ and 1 MΩ and two diff erent Fermi energies. The current 𝐼𝑖𝑗 used to \ngenerate the 2 -D heatmaps is the current that can be measured experimentally, i.e., it represents the current flowing \nout at the bottom of column j when a voltage of 𝑉𝑖𝑛=1 𝑉 is applied to row i while other rows are left floating. \nThe expression for this current is:  \n𝐼𝑖𝑗=  𝛽𝑗∗ 𝐼𝐷𝑁𝐴(𝑉𝑖𝑗) (11) \nwhere 𝐼𝐷𝑁𝐴(𝑉𝑖𝑗) is the current flowing through the DNA at the ( i, j)-th location of the memory array calculated \nusing eqn. (6), 𝑉𝑖𝑗 is the voltage drop across this  DNA strand calculated using eqn.10 , and j is a column -specific \nsneak path parameter (see eqn. (9) in [33] and Figure 3). Like  𝛼𝑖, j is also independent of the bias voltage. Figure \n6 shows the dependence of 𝛼𝑖 and j on row and column indices respectively. Sequential application of the bias \nvoltage to the rows of a crossbar array yields an output current matrix which is  depicted as a heatmap in Figure 5. \n      Comparing the rows in Figure 5, we first observe that for both values of interconnect resistance, the spatial \ndistortion of the read current distribution is lower for the B -TC 1T array compared to the B -CT 1C array. Second, \nwe note that the distortion is greatest in the northeast corner of the array. This is due to the combined effect of the \nsneak path parameters, {𝛼𝑖,𝛽𝑗:1≤𝑖≤𝑚,1≤𝑗≤𝑛}, which are plotted in Figure 6. It follows from Figure 6 that \nthe largest distortion should occur when 𝑖 = 1 (row) and j = n (column). As shown in  [33], [40] , these parameters \nare estimated from an approximation of Kirchhoff’s laws (KCL and KVL) and is based on an  average conductance \nof the memory cells. The results obtained from this parametric model with 𝛼 and 𝛽 are in good agreement with \nthe solution of  Kirchhoff’s equations, particularly for array sizes in the range 64×64 to 128×128 (see Figure 5(c) \nin [33]). By estimating the output cu rrent with 𝛼 and 𝛽 parameters, this method provides remarkable improvement \nover a Kirchhoff’s laws -based solution procedure in terms of time complexity, while maintaining almost similar \naccuracy. Finally, for both sequence types, we notice that the disto rtion in Figure 5 is significantly higher when \n𝑅int = 1 MΩ. This can be attributed to an increase in the number of sneak paths and a higher voltage drop across \nthe crossbar when the interconnect resistance is within one or two orders of magnitude of the resistance of the \nmemory cell. Consequently, if binar y information is stored in a crossbar that uses a B -CT 1C sequence to represent \nlogic 0 and a B -TC 1T sequence to represent logic 1, we can expect some overlap between the current values \nrecorded at the logic 1 locations and the logic 0 locations, leading to  bit errors during the read process. Finally, \nwe observe that the distortion is small when the Fermi energy of DNA strands is deep inside the bandgap. The \nreduced conductance at higher Fermi energy causes a smaller voltage to drop, which results in a small er distortion.  \n 8 \n 4. IMPACT OF FERMI ENERGY VARIABILITY  \n \nConductance dispersion in single -molecule systems can be attributed to several factors, including the geometry of \natoms, the orientation of atoms at the interface, the bonding energy between molecule and  contact, and stochastic \nrupture  [21]. When a molecule touches a contact, partial charge transfer occurs due to hybridization whi ch partially \ncharges the molecule  [41]. This in turn induces a change in the potential  of the molecule. As a result, the molecular \norbitals shift with respect to the Fermi energy of the contact, which we assume is fixed. We relate this orbital shift \nto the energy separation between the contact’s Fermi energy and the HOMO level of the DNA. U ltimately, this \ntranslates to a Fermi energy variation of the DNA sequence. Since each dsDNA memory cell can be viewed as a \nsingle molecule system, the performance of the crossbar array depends on the Fermi energy variation of each DNA \nsequence as well as the spatial voltage/current distortion across the crossbar, as discussed in the previous section. \nIn this section, we explore the extent to which Fermi energy variations can affect the performance of a DNA \ncrossbar array. As before, we will use B -CT 1C and B-TC 1T dsDNA sequences to represent logic 0 and logic 1 \nrespectively.  \n      For a better understanding of the sensitivity of DNA currents to Fermi energy variations and their impact on \ncrossbar performance, we have conducted 1000 simulations with a bias of  1 V, choosing a small random deviation \nof Fermi energy from the HOMO level in the range [0, 𝛿] for each simulation, where 𝛿 = 0.1 eV and 0.2 eV. The \narray size is 64 64 and the interconnect resistance, 𝑅int, is chosen to be 100 kΩ or 1 MΩ.  \n     The first row of Figure 7 shows the average voltage distribution for low ( 𝛿 = 0.1 eV) and high ( 𝛿 = 0.2 eV) \nFermi energy variability. We observe that the mean voltage of the crossbar increases with an increase in Fermi \nenergy variability 𝛿. Specifically, for 𝑅int = 100 kΩ and 1 MΩ, the mean crossbar voltage increases by 13.33% \nand 30.76%  respectively when 𝛿 changes from 0.1 eV to 0.2 eV. From Figure 2(d, e), we observe that the \nconductance of both DNA strands decreases with an increase in 𝛿. This leads to a reduction in sneak paths in the \ncrossbar architecture which translates to higher node voltages. For a fixed 𝛿, the mean crossbar voltage is higher \nfor 𝑅int = 100 kΩ  compared to 1 MΩ. This can be attributed to fewer sneak paths in the cr ossbar caused by lower \ninterconnect resistance w.r.t. DNA resistance. Higher mean voltages are desirable since they translate to improved \nreadout current distributions (reduced overlap) and enhanced read accuracy.  \n     The second row in Figure 7 shows the distributions of read currents for a 64 64 array with different values of \n𝛿 and 𝑅int. For clarity, logic ‘1’ currents are shown in blue, and logic ‘0’ currents are shown in red (the probability \nof ‘1 cell’ in the array is equal to the probability of ‘0 cells’). With 𝑅int= 100 kΩ, the overlap between bit 1 and \nbit 0 currents is very small for both low and high 𝛿. However, as the interconnect resista nce increases to 1 MΩ, \nthe current distributions overlap due to spatial distortion of current and sneak path effects, which implies that error -\nfree reconstruction is not possible. Lower conductance of DNA strands with an increase in 𝛿 (see Figure 2(d,e)) \nand larger sneak path currents which are associated with higher interconnect resistance are the two main reasons \nfor the overlap between the two class distributions.  \n     In terms of bit -error rates (see the third row in Figure 7), we can see a small mean BER of 0.23% for 𝛿 = 0.1 \neV and 1.48% for 𝛿 = 0.2 eV, when 𝑅int = 100 kΩ. The mean errors increase to 20.3% for 𝛿 = 0.1 eV and 25.8% \nfor 𝛿 = 0.2 eV when 𝑅int= 1 MΩ. Thus, we see that Fermi energy variation plays a crucial role in determining \nthe accuracy of the DNA cro ssbar as it causes an increase in mean BER by 1.25% and 5.2% at 𝑅int = 100 k Ω and \n1 MΩ respectively. Even though we have not shown results for 𝑅int = 10 k Ω, we note that the distributions for ‘bit \n1’ and ‘bit 0��� currents do not overlap, implying 0% BER ( perfect readout), for both 𝛿 = 0.1 eV and 0.2 eV.  \n      The preceding discussion leads to important observations. Since the optimal threshold level changes with 𝛿, \nideally, one should determine it after fabrication of the array, based on the distributio n of read currents. However, \nsince the optimal threshold levels are in the order of nano -Amperes, the threshold detector must be accurate and \nprecise enough to read and distinguish between current values of a similar or smaller order. This is the practical  \nlimitation which can affect the accuracies that may be achievable in practice.  \n \n5. EFFECT OF SCALABILITY ON IMAGE RETRIEVAL ACCURACY  \n \nIn this section, we evaluate the performance of a DNA crossbar by loading 1000 randomly chosen images from \nthe Linnaeus  [42] dataset. Our evaluation criteria are BER  and power consumption as a function of the bit load \n(in %), defined as the percentage of bits in an image that are represented by B -CT 1C DNA (logic ‘1’). Figure 8(a) \nshows the histogram of the bit load of all images in our dataset.  \n      Figure 8 (b-g) show the bit error rate distributions as a function of bit load for different array sizes and \ninterconnect resistances. The spatial bit distribution in a crossbar can be different for  a particular bit load, leading \nto different bit errors. Therefore, we have presented box plots of BER for each bit load. As discussed in the \nprevious sections, the bit error rate increases with an increase in array size and interconnect resistance. We obs erve 9 \n that the BER is higher when the bit load is in the range of 50 -70% (the critical range), and highest when it is \napproximately 60%. For reference, the BER is nearly 0% (in other words, the distributions of bit 1 and bit 0 \ncurrents do not overlap) for a ll interconnect resistances below 100 kΩ and 50 kΩ in cases of 64 ×64 and 128 ×128 \narrays respectively while at 10kΩ the BER is exactly 0% for both the array sizes (see Figure 8(b,e)) resulting in \nperfect readout. Note that in Figures 4 and 5, we chose the interconnect resistance to be 10 kΩ and 1 MΩ in order \nto demonstrate the sensitivity of voltage and current distribution across the crossbar to extreme values of Rint. \nHowever, in Figures 7 and 8, we have chosen Rint to be 100 kΩ and 1 MΩ since there are no bit errors when Rint \n= 10 kΩ . \n      Figure 8 (h,i) show the average power consumed in the crossbar (memory cells and interconnects), as a \nfunction of 𝑅int and array size. With an increase in interconnect resistance, the mean voltage of  the crossbar \ndecreases, as seen from the trends in Figure 4 and Figure 7(a). This leads to lower read currents of the DNA \nstrands. Both these factors collectively lead to lower power consumption, as can be observed in Figure 8 (h, i). \nFor an array size of 64 ×64, the power consumed is 42.8 µW for  𝑅int = 10 k Ω, which reduces to 2.84 µW for 𝑅int \n= 1 M Ω. For an array size of 128 ×128, the power consumption is 60 µ W for  𝑅int = 10 k Ω and 11.3 µW for 𝑅int \n= 1 M Ω. It follows therefore that a DNA array formed with a large interconnect resistance will scale better in \nterms of power consumption, but at the cost of an enhanced bit error rate.  \n \n6. CONCLUSION  \n \n \nTo the best o f our knowledge, this is the first paper of its kind to propose an electrically readable DNA storage \ndevice based on crossbar architecture. Our study leverages an idea that blends molecules created by synthetic \nbiology with traditional semiconductor proces ses for contact formation, to yield a novel DNA storage technology. \nWe have characterized the performance of DNA crossbar arrays as read -only memory devices in terms of readout \naccuracy and array power consumption. Since these performance metrics depend on  the interconnect resistance, \narray size, and Fermi energy deviations of DNA strands, each of these parameters has been studied in detail. Our \nquantitative results provide a comprehensive benchmark for evaluating such (and competing) devices and shed \nimpor tant insights into how array size affects the trade -off between accuracy and power consumption. One \nimportant result from our analysis pertains to the role of interconnect resistance. We find that if the interconnect \nresistance is less than 10 kΩ on averag e, read errors are minimal for  a 128× 128 array . Second, the performance \nof a DNA -based crossbar array may be impacted by Fermi energy variations, which may be impossible to avoid \ncompletely in a single molecule system like DNA. Our study includes a comprehensive analysis of the impact of \nsuch variatio ns on array performance. In our current analysis, we have studied the impact of different interconnect \nresistances and typical non -idealities in nanodevices which result in Fermi energy variability. Other parameters \nwhich should be considered in the future  include contact engineering, non -uniformity of interconnects , and \nvariability due to self -assembly process.  \n \n7. REFERENCES  \n \n[1] M. S. Neiman, “On the molecular memory systems and the directed mutations,” Radiotekh nika, vol. 1, p. 8, 1965.  \n[2] “Machines Smarter Than Men? Interview with Dr. Norbert Wiener, Noted Scientist - Joshua Lederberg - Profiles in \nScience.” https://profiles.nlm.nih.gov/spotlight/bb/catalog/nlm:nlmuid -101584906X7699 -doc (accessed Sep. 04, \n2021) . \n[3] J. Davis, “Microvenus,” Art J. , vol. 55, no. 1, pp. 70 –74, 1996, doi: 10.1080/00043249.1996.10791743.  \n[4] C. T. Clelland, V. Risca, and C. Bancroft, “Hiding messages in DNA microdots,” Nat. 1999 3996736 , vol. 399, no. \n6736, pp. 533 –534, Jun. 1999, do i: 10.1038/21092.  \n[5] G. M. Church, Y. Gao, and S. Kosuri, “Next -generation digital information storage in DNA,” Science (80 -. )., vol. \n337, no. 6102, p. 1628, Sep. 2012, doi: 10.1126/SCIENCE.1226355.  \n[6] N. Goldman et al. , “Towards practical, high -capacity, low -maintenance information storage in synthesized DNA,” \nNat. 2013 4947435 , vol. 494, no. 7435, pp. 77 –80, Jan. 2013, doi: 10.1038/nature11875.  \n[7] K. N. Lin, K. Volkel, J. M. Tuck, and A. J. Keung, “Dynamic and scalabl e DNA -based information storage,” Nat. \nCommun. 2020 111 , vol. 11, no. 1, pp. 1 –12, Jun. 2020, doi: 10.1038/s41467 -020-16797 -2. \n[8] B. JL et al. , “Random access DNA memory using Boolean search in an archival file storage system,” Nat. Mater. , \nvol. 20, no. 9 , 2021, doi: 10.1038/S41563 -021-01021 -3. \n[9] H. H. Lee, R. Kalhor, N. Goela, J. Bolot, and G. M. Church, “Terminator -free template -independent enzymatic DNA \nsynthesis for digital information storage,” Nat. Commun. 2019 101 , vol. 10, no. 1, pp. 1 –12, Jun. 2 019, doi: \n10.1038/s41467 -019-10258 -1. \n[10] J. Bornholt, R. Lopez, D. M. Carmean, L. Ceze, G. Seelig, and K. Strauss, “A DNA -based archival storage system,” \nACM SIGPLAN Not. , vol. 51, no. 4, pp. 637 –649, Apr. 2016, doi: 10.1145/2872362.2872397.  \n[11] J. Hihath, M. P. Anantram, Yonggang Ke, “NUCLEIC ACID -BASED ELECTRICALLY READABLE, READ -\nONLY MEMORY , Patent application # US17/253,008, filed on 07/11/2019.”  \n[12] D. Wang, L. Yu, C. M. Huang, G. Arya, S. Chang, and Y. Ke, “Programmable Transformations of DNA  Origami 10 \n Made of Small Modular Dynamic Units,” J. Am. Chem. Soc. , vol. 143, no. 5, pp. 2256 –2263, Feb. 2021, doi: \n10.1021/JACS.0C10576/SUPPL_FILE/JA0C10576_SI_002.PDF.  \n[13] J. Marrs, Q. Lu, V. Pan, Y. Ke, and J. Hihath, “Structure -Dependent Electrical Cond uctance of DNA Origami \nNanowires,” ChemBioChem , vol. 24, no. 2, p. e202200454, Jan. 2023, doi: 10.1002/CBIC.202200454.  \n[14] F. M. Gartner, I. R. Graf, and E. Frey, “The time complexity of self -assembly,” Proc. Natl. Acad. Sci. U. S. A. , vol. \n119, no. 4, p.  e2116373119, Jan. 2022, doi: \n10.1073/PNAS.2116373119/SUPPL_FILE/PNAS.2116373119.SAPP.PDF.  \n[15] L. N. Green, H. K. K. Subramanian, V. Mardanlou, J. Kim, R. F. Hariadi, and E. Franco, “Autonomous dynamic \ncontrol of DNA nanostructure self -assembly,” Nat. Che m. 2019 116 , vol. 11, no. 6, pp. 510 –520, Apr. 2019, doi: \n10.1038/s41557 -019-0251 -8. \n[16] J. A. Paulson, A. Mesbah, X. Zhu, M. C. Molaro, and R. D. Braatz, “Control of self -assembly in micro - and nano -\nscale systems,” J. Process Control , vol. 27, pp. 38 –49, Mar. 2015, doi: 10.1016/J.JPROCONT.2014.10.005.  \n[17] F. Zhang, J. Nangreave, Y. Liu, and H. Yan, “Structural DNA nanotechnology: State of the art and future perspective,” \nJ. Am. Chem. Soc. , vol. 136, no. 32, pp. 11198 –11211, Aug. 2014, doi: \n10.1021/JA5051 01A/ASSET/IMAGES/LARGE/JA -2014 -05101A_0009.JPEG.  \n[18] D. Mathur and I. L. Medintz, “The Growing Development of DNA Nanostructures for Potential Healthcare -Related \nApplications,” Adv. Healthc. Mater. , vol. 8, no. 9, p. 1801546, May 2019, doi: 10.1002/ADHM.2 01801546.  \n[19] C. L. Hsu, H. Jiang, A. G. Venkatesh, and D. A. Hall, “A Hybrid Semi -Digital Transimpedance Amplifier with Noise \nCancellation Technique for Nanopore -Based DNA Sequencing,” IEEE Trans. Biomed. Circuits Syst. , vol. 9, no. 5, \npp. 652 –661, Oct. 2015, doi: 10.1109/TBCAS.2015.2496232.  \n[20] O. Ghadami et al. , “Helix: An Electrochemical CMOS DNA Synthesizer,” Dig. Tech. Pap. - Symp. VLSI Technol. , \nvol. 2022 -June, pp. 66 –67, 2022, doi: 10.1109/VLSITECHNOLOGYANDCIR46769.2022.9830446.  \n[21] Z. Li, L. Mej ía, J. Marrs, H. Jeong, J. Hihath, and I. Franco, “Understanding the Conductance Dispersion of Single -\nMolecule Junctions,” J. Phys. Chem. C , vol. 125, no. 6, pp. 3406 –3414, Feb. 2020, doi: 10.1021/ACS.JPCC.0C08428.  \n[22] V. Periasamy, N. Rizan, H. M. J. Al -Ta’ii, Y. S. Tan, H. A. Tajuddin, and M. Iwamoto, “Measuring the Electronic \nProperties of DNA -Specific Schottky Diodes Towards Detecting and Identifying Basidiomycetes DNA,” Sci. Reports \n2016 61 , vol. 6, no. 1, pp. 1 –9, Jul. 2016, doi: 10.1038/srep29879.  \n[23] S. R. Patil et al. , “Quantum Transport in DNA Heterostructures: Implications for Nanoelectronics,�� ACS Appl. Nano \nMater. , vol. 4, no. 10, pp. 10029 –10037, Oct. 2021, doi: \n10.1021/ACSANM.1C01087/SUPPL_FILE/AN1C01087_S I_001.PDF.  \n[24] K. Kim, M. Yoon, J. Koo, and Y. Roh, “Electrical characteristics of oxygen doped DNA molecules,” Thin Solid \nFilms , vol. 519, no. 20, pp. 7057 –7059, Aug. 2011, doi: 10.1016/J.TSF.2011.04.085.  \n[25] L. Martínez, R. Andrade, E. G. Birgin, and J . M. Martínez, “PACKMOL: A package for building initial configurations \nfor molecular dynamics simulations,” J. Comput. Chem. , vol. 30, no. 13, pp. 2157 –2164, Oct. 2009, doi: \n10.1002/JCC.21224.  \n[26] Richard  Owczarzy et al. , “Effects of Sodium Ions on DNA D uplex Oligomers: Improved Predictions of Melting \nTemperatures,” Biochemistry , vol. 43, no. 12, pp. 3537 –3554, Mar. 2004, doi: 10.1021/BI034621R.  \n[27] R. Galindo -Murillo, D. R. Roe, and T. E. Cheatham, “Convergence and reproducibility in molecular dynamics  \nsimulations of the DNA duplex d(GCACGAACGAACGAACGC),” Biochim. Biophys. Acta - Gen. Subj. , vol. 1850, \nno. 5, pp. 1041 –1058, May 2015, doi: 10.1016/J.BBAGEN.2014.09.007.  \n[28] J. Wang, R. M. Wolf, J. W. Caldwell, P. A. Kollman, and D. A. Case, “Development and testing of a general amber \nforce field,” J. Comput. Chem. , vol. 25, no. 9, pp. 1157 –1174, Jul. 2004, doi: 10.1002/JCC.20035.  \n[29] I. Ivani et al. , “Parmbsc1: a refined force field for DNA simulations,” Nat. Methods 2016 131 , vol. 13, no. 1, pp. 55 –\n58, Nov. 2015, doi: 10.1038/nmeth.3658.  \n[30] J. Qi, N. Edirisinghe, M. G. Rabbani, and M. P. Anantram, “Unified model for conductance through DNA with the \nLandauer -Büttiker formalism,” Phys. Rev. B - Condens. Matter Mater. Phys. , vol. 87, no. 8, pp. 1 –10, 2013 , doi: \n10.1103/PhysRevB.87.085404.  \n[31] M. Buttiker, “COHERENT AND SEQUENTIAL TUNNELING IN SERIES BARRIERS.,” IBM J. Res. Dev. , vol. \n32, no. 1, pp. 63 –75, 1988, doi: 10.1147/RD.321.0063.  \n[32] G. C. Liang, A. W. Ghosh, M. Paulsson, and S. Datta, “Electrosta tic potential profiles of molecular conductors,” \nPhys. Rev. B , vol. 69, no. 11, p. 115302, Mar. 2004, doi: 10.1103/PhysRevB.69.115302.  \n[33] Y. Liao et al. , “A Compact Model of Analog RRAM with Device and Array Nonideal Effects for Neuromorphic \nSystems,” IEEE Trans. Electron Devices , vol. 67, no. 4, pp. 1593 –1599, Apr. 2020, doi: 10.1109/TED.2020.2975314.  \n[34] A. Nieuwoudt and Y. Massoud, “Evaluating the impact of resistance in carbon nanotube bundles for VLSI \ninterconnect using diameter -dependent modeling t echniques,” IEEE Trans. Electron Devices , vol. 53, no. 10, pp. \n2460 –2466, 2006, doi: 10.1109/TED.2006.882035.  \n[35] A. Naeemi and J. D. Meindl, “Physical modeling of temperature coefficient of resistance for single - and multi -wall \ncarbon nanotube interconne cts,” IEEE Electron Device Lett. , vol. 28, no. 2, pp. 135 –138, Feb. 2007, doi: \n10.1109/LED.2006.889240.  \n[36] T. Ragheb and Y. Massoud, “On the modeling of resistance in Graphene Nanoribbon (GNR) for future interconnect \napplications,” IEEE/ACM Int. Conf. Co mput. Des. Dig. Tech. Pap. ICCAD , pp. 593 –598, 2008, doi: \n10.1109/ICCAD.2008.4681637.  \n[37] R. Murali, K. Brenner, Y. Yang, T. Beck, and J. D. Meindl, “Resistivity of graphene nanoribbon interconnects,” IEEE \nElectron Device Lett. , vol. 30, no. 6, pp. 611 –613, 2009, doi: 10.1109/LED.2009.2020182.  \n[38] J. Li and P. J. Burke, “Measurement of the com bined quantum and electrochemical capacitance of a carbon nanotube,” \nNat. Commun. 2019 101 , vol. 10, no. 1, pp. 1 –9, Aug. 2019, doi: 10.1038/s41467 -019-11589 -9. \n[39] M. P. Anantram and F. Léonard, “Physics of carbon nanotube electronic devices,” Reports Pr og. Phys. , vol. 69, no. 11 \n 3, p. 507, Feb. 2006, doi: 10.1088/0034 -4885/69/3/R01.  \n[40] Y. Liao et al. , “Diagonal Matrix Regression Layer: Training Neural Networks on Resistive Crossbars with \nInterconnect Resistance Effect,” IEEE Trans. Comput. Des. Integr. Ci rcuits Syst. , vol. 40, no. 8, pp. 1662 –1671, Aug. \n2021, doi: 10.1109/TCAD.2020.3021309.  \n[41] H. Mohammad et al. , “Role of intercalation in the electrical properties of nucleic acids for use in molecular \nelectronics,” Nanoscale Horizons , vol. 6, no. 8, pp. 651 –660, Jul. 2021, doi: 10.1039/D1NH00211B.  \n[42] “Linnaeus 5 dataset.” http://chaladze.com/l5/ (accessed Aug. 18, 2022).  \n[43] M. Mallary, A. Torabi, and M. Benakli, “One terabit per square inch perpendicular recording conceptual design,” \nIEEE Trans. Magn. , vol. 38, no. 4 I, pp. 1719 –1724, Jul. 2002, doi: 10.1109/TMAG.2002.1017762.  \n[44] G. RN, H. R, P. M, P. D, and  S. WJ, “Robust chemical preservation of digital information on DNA in silica with \nerror -correcting codes,” Angew. Chem. Int. Ed. Engl. , vol. 54, no. 8, pp. 2552 –2555, Feb. 2015, doi: \n10.1002/ANIE.201411378.  \n[45] F. Zahoor, T. Z. Azni Zulkifli, and F. A. K handay, “Resistive Random Access Memory (RRAM): an Overview of \nMaterials, Switching Mechanism, Performance, Multilevel Cell (mlc) Storage, Modeling, and Applications,” \nNanoscale Res. Lett. 2020 151 , vol. 15, no. 1, pp. 1 –26, Apr. 2020, doi: 10.1186/S11671 -020-03299 -9. \n[46] M. G. T. A. Rutten, F. W. Vaandrager, J. A. A. W. Elemans, and R. J. M. Nolte, “Encoding information into \npolymers,” Nat. Rev. Chem. 2018 211 , vol. 2, no. 11, pp. 365 –381, Oct. 2018, doi: 10.1038/s41570 -018-0051 -5. \n[47] S. Pi et al. , “Mem ristor crossbar arrays with 6 -nm half -pitch and 2 -nm critical dimension,” Nat. Nanotechnol. , vol. \n14, no. 1, pp. 35 –39, 2019, doi: 10.1038/s41565 -018-0302 -0. \n[48] P. O. Vontobel, W. Robinett, P. J. Kuekes, D. R. Stewart, J. Straznicky, and R. S. Williams, “Writing to and reading \nfrom a nano -scale crossbar memory based on memristors,” Nanotechnology , vol. 20, no. 42, p. 425204, Sep. 2009, \ndoi: 10.1088/0957 -4484/20/42/425204.  \n[49] M. A. Zidan et al. , “Single -Readout High -Density Memristor Crossbar,” Sci. Repo rts 2016 61 , vol. 6, no. 1, pp. 1 –9, \nJan. 2016, doi: 10.1038/srep18863.  \n \n \n \n \n8. ACKNOWLEDGMENT  \n \nWe acknowledge NSF Grant Numbers 1807391 (SemiSynBio Program) and 2036865 (Future of Manufacturing) \nfor support. HM acknowledges Kuwait University Fellowship.  \n \n9. AUTHORS’ CONTRIBUTION  \n \nMPA who works on DNA Nanotechnology defined the DNA memory framework. The transport calculation code \nwas written by HM who also performed the calculations. The circuit modeling and image analysis were performed \nby AD. RK advised on circuit modeling and rea d technology. YW contributed to circuit -level data collection. The \nfirst draft of the manuscript was written by AD and supervised by AKD . All authors contributed to the subsequent \ndraft of the paper and edited the manuscript.  \n \n10. DATA AVAILABILITY  \n \nAll data a re provided in this manuscript.  \n \n11. ADDITIONAL INFORMATION  \n \nAll the authors declare no competing interests.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 12 \n 12. FIGURES & FIGURE LEGENDS  \n \nFigure no. 1 : \n \nFigure no. 2 : \n \n \nFigure 2: (a) A DNA strand is placed between two electrodes and bias is applied across it. For our work, B conformation \nof two double stranded DNA sequences is considered, CT 1C and TC 1T. (b, c) Transmission versus energy for CT 1C \nand TC 1T strands as a function of Fermi energy for diffe rent bias voltages. (d, e) Current -voltage transfer \ncharacteristics for CT 1C and TC 1T B-DNA double stranded sequences. The collection of plots in these panels is for \ndifferent Fermi energies ( 𝜹). \n(a)(b) (c)\n(d) (e)BCT1C DNA BTC1T DNA\nBCT1C DNA BTC1T DNATransmission Profile\nTransfer CharacteristicsTransfer CharacteristicsDNA Schematic & Potential Drop \nFigure 1: Illustration of a DNA based nano system for high -density data storage. (a) DNA crossbar components including \nmemory cells and interconnects. (b) Process flow for the proposed technology. DNA origami principles can be invoked for \nnano -structure generation to form long DNA nanowires as interconnects and short DNA seque nces as memory cells.  A \nlarge number of these elements can be built through methods like PCR amplification.  These components can  then  be self -\nassembled to create a crossbar architecture . An electrical read -out mechanism can be adopted for reading the infor mation \nin the DNA crossbar array. (c) 3D conceptual schematic of a DNA crossbar array, where TE and BE denote the top and \nbottom electrodes respectively.  \n \n \n \n \n \n \n \n \n \n13 \n Figure no. 3 : \nFigure 3: (a) Circuit schematic of a 4 4 array, where αi and βj are array size dependent scaling factors which account \nfor the interplay between interconnect resistance and sneak paths. (b) Flow chart for crossbar simulation. ‘Voltage \nBlock’ comprises of the set of equations governing the voltage distribution in the c rossbar (see eqn. 10) while ‘Current \nBlock’ is a look -up-table created with the DNA currents ( 𝑰𝑫𝑵𝑨) obtained from DNA current simulations (see eqn. 6).  \nFigure no. 4: \n \n \n \nFigure 4: Normalized voltage distribution for homogenous arrays composed of B -CT 1C DNA (a, c) and B -TC 1T DNA \n(b, d). Array size is 64 64 for (a, b) and 128 128 for (c, d). The pink lines correspond to 𝑹𝒊𝒏𝒕 = 10 kΩ and the purple \nlines correspond to 𝑹𝒊𝒏𝒕 = 1 MΩ. Moreover, the solid lines repres ent Fermi energy at corresponding HOMO level while \nthe dotted lines represent Fermi energy at (HOMO+0.2) eV.  \nFigure no. 5: \n \n \nFigure 5: Read -out current distributions (see eqn. (11) for homogenous arrays composed of B -CT 1C DNA (a -d) and B -\nTC 1T DNA (e -h). The interconnect resistance is 10 k Ω for (a, b, e, f) and 1 M Ω for (c, d, g, h ). Array size is kept fixed \nat 6464. For panels (a, c, e, g), the Fermi energy is at corresponding HOMO level while for panels (b, d, f, h), Fermi \nenergy is at (HOMO+0.2) eV.  \n(a) (b) (c) (d)\n(a) (b) (c) (d)\n(e) (f) (g) (h)                    \n                                                                \n                                                      (a)                                                                                          (b) \n \n \n14 \n Figure no. 6: \n \n \nFigure 6: (a) Plot of sneak path parameter α as a function of row index. (b) Plot of sneak path parameter   as a function \nof column index.  \n \n \n \n \n \nFigure no. 7: \n \n \nFigure 7 Monte Carlo simulation results for a 64 64 array: (a) voltage distributions, (b) current distributions, (c) bit \nerror rate (BER) distributions for low ( δ = 0.1 eV) and high ( δ = 0.2 eV) Fermi energy variations. All results (including \nthe means) are comput ed from 1000 Monte Carlo simulations.  \n \n \n \n \n \n \n \n \n \n \nδ= 0.1 eV δ= 0.2 eV\na.\nVoltage\nDistribution\nb.\nCurrent\nDistribution\nc.\nBER\nDistribution‘0’ Current ‘1’ Current\n ‘0’ Current ‘1’ Current\n15 \n Figure no. 8: \n \n \nFigure 8: Storage application results for a simulation setup where 1000 images from the Linnaeus dataset are loaded \nonto 64×64 and 128×128 crossbar arrays. (a) distribution of bit load (%) in the dataset, (b -g) distributions of bit error \nrates for each bit load (% ) vs. array size and interconnect resistance, (h, i) average power consumed as a function of \n𝑹𝒊𝒏𝒕 for 64×64 and 128×128 arrays. Note that the read -out accuracy is ~100% for interconnect resistances below 100 \nkΩ and 50 kΩ for 64×64 and 128×128 arrays re spectively.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nLinnaeus Dataset\n Loading onto \nDNA Crossbar \nSimulation\nSetup\nBit Error Rate\nDistribution\nPower\nConsumption(a)\n(c) (d)\n(e) (f) (g)\n(h) (i)(b)16 \n TABLE 1  \nComparison of various storage technologies  (existing and proposed) . \nNature of \nTechnology  2D Density  \n(bits/inch2) 3D Density  \n(bits/inch3) Random** \nAccess  Read$$ \nTime  Lifetime  \n(Years)  \nConventional  ~ 1012 [43] ~ 109 [44] Yes Low > 10 [45] \nDNA -in-solution  − ##6.04×1022 [6] \n \n##1.07×1023 [46]  \n \n##1.18×1024 [7]  No High  > 100  \n(@ T = 25oC) \n \n> 1000  \n(@ T < 10oC) \n  [44] \nRRAM crossbar  4.5×1012 [47] \n \n4.1×1012 [48] \n \n4.2×1012 [49] − Yes Low > 10 [45] \nDNA&& crossbar  \n(This work) 3.47×1012 (12 nm ) 7.35×1018 (12 nm ) \nYes Low > 100  \n(@ T = 25oC) \n \n> 1000  \n(@ T < 10oC) \n   [44] 4.72×1012 (10 nm ) 1×1019 (10 nm ) \n6.78×1012  (8 nm ) 1.44×1019 (8 nm ) \n \n(##) For  3D density of DNA -in-solution technology, bulk density is that of pure water. ($$) Read time is classified to be ‘High’ \nif it is > 1 hour and ‘Low’ if it is < 10 ns . (&&) 2D and 3D density metrics for DNA crossbar storage are based on  topological \nand DNA parameters. Italicized numbers ( 8 nm , 10 nm , 12 nm ) refer to the half -pitch of the crossbar array.  \n "    },    {        "title": "1709.00549v1.C__4_N__3_H_monolayer__A_novel_two_dimensional_organic_Dirac_material_with_high_Fermi_velocity.pdf",        "content": "C4N3H monolayer: A novel two-dimensional organic Dirac material with high Fermi\nvelocity\nHongzhe Pan,1, 2Hongyu Zhang,3Yuanyuan Sun,1, 2Jianfu Li,2Youwei Du,1and Nujiang Tang1,\u0003\n1National Laboratory of Solid State Microstructures,\nCollaborative Innovation Center of Advanced Microstructures and Jiangsu Provincial\nKey Laboratory for Nanotechnology, Nanjing University, Nanjing 210093, China\n2School of Physics and Electronic Engineering, Linyi University, Linyi 276005, China\n3Department of Physics, East China University of Science and Technology, Shanghai 200237, China\n(Dated: September 5, 2017)\nSearching for two-dimensional (2D) organic Dirac materials, which have more adaptable practi-\ncal applications in comparing with inorganic ones, is of great signi\fcance and has been ongoing.\nHowever, only two kinds of these materials with low Fermi velocity have been discovered so far.\nHerein, we report the design of an organic monolayer with C 4N3H stoichiometry which possesses\nfascinating structure and good stability in its free-standing state. More importantly, we demonstrate\nthat this monolayer is a semimetal with anisotropic Dirac cones and very high Fermi velocity. This\nFermi velocity is roughly one order of magnitude larger than that in 2D organic Dirac materials\never reported, and is comparable to that in graphene. The Dirac states in this monolayer arise from\nthe extended \u0019-electron conjugation system formed by the overlapping 2 pzorbitals of carbon and\nnitrogen atoms. Our \fnding opens a door for searching more 2D organic Dirac materials with high\nFermi velocity.\nI. INTRODUCTION\nFollowing with the great development of graphene [1]\nand topological insulators [2], an emerging \feld of Dirac\nphysics is being established for investigating the quantum\nrelativistic properties of a class of special materials with\nDirac cones, namely Dirac materials [3]. Such materi-\nals exhibit linear electronic band dispersion at the Fermi\nlevel, that is Dirac band, and thus have charge carriers\nbehaved like massless Dirac fermions. The unique Dirac\nbands endow these materials with many novel phenom-\nena in electronic transport, such as ballistic charge trans-\nport and high carrier mobility [4], Klein tunneling [5],\nvarious quantum Hall e\u000bects [1, 6, 7], etc. These speci\fc\ntransport properties o\u000ber Dirac materials a wide range of\npromising applications in high-speed low-dissipation de-\nvices. In addition, the continuous reduction in the size\nof devices highly needs to develop low-dimensional ma-\nterials. Thus, two-dimensional (2D) Dirac materials are\nmuch more desirable for the applications in nanoscale in-\ntegrated circuits.\nTheoretically, based on the generalized von Neumann-\nWigner theorem [8], at least three conditions are re-\nquired to achieve Dirac bands in 2D materials: (i) speci\fc\nsymmetries, (ii) proper parameters and (iii) appropriate\nFermi level and band overlap [9]. The hexagonal sym-\nmetry was widely believed to the most favorable for the\nexistence of Dirac bands due to the fact that many 2D\nDirac materials are observed in hexagonal lattice. How-\never, it is not a necessary precondition for the presence of\n2D Dirac materials. For example, 6,6,12-graphyne [10],\nsquare-MoS 2[11] and Pmmn -boron [12] have rectangular\n\u0003Corresponding author: tangnujiang@nju.edu.cnlattices while also possess Dirac cones. Besides symme-\ntries, proper structure parameters such as bond lengths\nand angles, which are related to the crystal lattice, are\nalso required. Furthermore, the Fermi level should pre-\ncisely lie at the Dirac points, while there should be not\nany other bands overlap at the Fermi level. This con-\ndition is required to ensure the experimental observa-\ntion and practical applications of the novel properties\nparticular to the Dirac cones. Unfortunately, 2D Dirac\nmaterials are found to be very rare due to these rigor-\nous conditions [9]. For instance, among hundreds of 2D\nmaterials, only graphene [1, 4], phagraphene [13], gra-\nphynes [10], silicene and germanen [14], borophenes [12],\nFeB 2monolaye [15], etc., have been con\frmed to be 2D\nDirac materials. Moreover, most of the existing 2D Dirac\nmaterials are inorganic compounds. In general, organic\nmaterials have the additional advantages of mechanical\n\rexibility and tunable properties in comparison with in-\norganic ones. Therefore, it is of great practical signi\f-\ncance to search for more 2D organic Dirac materials with\nthe development of Dirac physics.\nActually, an extensive search for 2D organic Dirac\nmaterials has been ongoing. However, to the best of\nour knowledge, only two kinds of 2D organic systems\nhave been discovered to be Dirac materials so far. The\n\frst successfully realized example is the 2D layered\norganic conductor \u000b-(BEDT-TTF) 2I3(BEDT-TTF =\nbis(ethylenedithio)-tetrathiafulvalene) which turns into a\nDirac material with a pair of titled Dirac cones when ap-\nplying a high hydrostatic pressure above 1.2 GPa [16, 17].\nThe other one is a theoretical design about some 2D\nconjugated polymers. These polymers can be theoreti-\ncally constructed by replacing the insulating connectors\n(1,3,5-triazine, etc.) of conventional semiconducting 2D\ncovalent organic frameworks (COFs) with the conductive\nones (trivalent carbon atoms, etc.) [18]. Unfortunately,arXiv:1709.00549v1  [cond-mat.mtrl-sci]  2 Sep 20172\nboth of the two kinds of 2D organic materials have very\nlow Fermi velocities, which are roughly one order of mag-\nnitude lower than that in 2D inorganic Dirac materials\n[4, 10, 12{15]. It is known that a high Fermi velocity is\nfavorable to the transport property of a Dirac material\n[19]. Thus, designing stable 2D organic Dirac materials\nwith high Fermi velocities is urgent and of great practical\nsigni\fcance.\nInspired by the experimental synthesis of several 2D\ncarbon nitride sheets, such as graphitic carbon nitride (g-\nC3N4) [20, 21], C 2N holey 2D crystal (C 2N-h2D) [22] and\nC3N sheets [23] based on condensation reactions, we de-\nsign a novel 2D organic material named as C 4N3H mono-\nlayer by its stoichiometry. This organic monolayer has a\nintriguing structure with evenly distributed heart-shaped\nangstrom-scale pores and rather good dynamic, thermal\nand mechanical stabilities in its freestanding state. More\ninterestingly, we demonstrate that this organic monolayer\nis a 2D Dirac material with anisotropic Dirac cones and\na very high Fermi velocity of 1 :1\u0002106m s\u00001. Remark-\nably, this Fermi velocity is roughly one order of magni-\ntude larger than that in 2D organic Dirac materials ever\nreported [17, 18], and is even comparable to that in 2D\ninorganic Dirac materials [4, 10, 12{15]. The Dirac points\nin this monolayer locate at o\u000b-symmetry points between\nthe \u0000 and K points, and arise predominantly from the\noverlapping 2 pzorbitals of carbon (C) and nitrogen (N)\natoms. In addition, we also comment on the experimental\nfeasibility of producing the predicted C 4N3H monolayer\nand propose two hypothetical synthesis routes.\nII. COMPUTATIONAL DETAILS\nStructural optimization, energy, density of states\n(DOS), band structure, electron localization function\n(ELF) [24] and deformation charge density calculations\nbased on density functional theory were performed in\nthe framework of the generalized gradient approximation\nwith the PBE functional [25] using Vienna ab initio sim-\nulation package (VASP) [26, 27]. Projector-augmented\nplane wave approach [28] was used to represent the elec-\ntron interaction. A 500 eV energy cuto\u000b of the plane-\nwave basis sets and the \frst Brillouin zone sampled with\na 45\u000245\u00021 Monkhorst-Pack k-points grid were adopted\nin the structural relaxations and self-consistent calcula-\ntions. In all the calculations, a vacuum distance of 15 \u0017A\nwas applied along the perpendicular direction to ensure\nnegligible interaction between adjacent layers. All atoms\nwere allowed to relax without any constraint in the ge-\nometric optimizations until the total energy change was\nless than 1:0\u000210\u00005eV and the force on each atom was\nsmaller than 0.001 eV \u0017A\u00001. According to the crystal\nsymmetry, the band structure of the C 4N3H monolayer\nwas calculated along the special lines connecting the fol-\nlowing high-symmetry points: \u0000 (0, 0, 0), K (0.4, 0.4, 0),\nM (0.5, 0, 0), R (0.6, -0.4, 0), S (0.5, -0.5, 0), \u0000 (0, 0,\n0), and M (0.5, 0, 0) in the k-space. Moreover, to con-\frm the existence of Dirac states, the band structure of\nthe C 4N3H monolayer was also recomputed respectively\nby the more accurate Heyd-Scuseria-Ernzerhof (HSE06)\nfunctional [29] and the spin-orbit coupling (SOC) e\u000bect.\nThe \fnite displacement method, as implemented in the\nphonopy code [30], was employed to calculate the phonon\ndispersion curves of the C 4N3H monolayer. During the\nphonon spectrum calculation, a 4 \u00024\u00021 supercell was em-\nployed and the force constant matrix was determined by\nthe VASP. First-principles molecular dynamics (FPMD)\nsimulations, as also performed in the VASP, were em-\nployed to evaluate the thermal stability of this mono-\nlayer. The initial con\fguration with a 4 \u00024\u00021 supercell\nwas annealed at di\u000berent temperatures of 300, 500 and\n1000 K, respectively. The temperature was controlled\nby the Nos\u0013 e-Hoover thermostat [31]. At each tempera-\nture, FPMD simulations in NVT ensemble lasted for 30\nps with a time step of 2.0 fs. A kinetic energy cuto\u000b\nof 450 eV and the PBE functional were employed in all\nFPMD simulations.\nIII. RESULTS AND DISCUSSION\nA. Geometric structure of the C 4N3H monolayer\nFigures 1(a) and 1(b) respectively present the top and\nside views of the geometric structure for the optimized\nC4N3H monolayer. As shown in Fig. 1(a), this mono-\nlayer has a rhombic primitive cell (represented by the\ngreen dash lines) with the fully relaxed lattice constants\na1=a2= 4.77 \u0017A,\r= 104:5\u000e(angle between the a1and\na2lattice vectors). Obviously, its primitive cell consists\nof four C atoms, three N atoms and one H atom. The\ncorrespondingly optimized lattice constants of the trans-\nformed rectangular conventional cell [denoted by the blue\ndash lines in Fig. 1(a)] with the same symmetry are a\n= 7.54 \u0017A and b= 5.84 \u0017A, respectively. This structure\nhas the symmetry of C14\n2vand belongs to the space group\nofAmm2 . It is obvious that angstrom-scale pores evenly\ndistribute in the structure of the C 4N3H monolayer. In-\ntriguingly, if we link the atoms around the angstrom-scale\npore in its conventional cell one by one, the connecting\nline shows a perfect heart-shape [denoted by the red solid\nline in Fig. 1(a)]. By anatomizing this intriguing struc-\nture, one can \fnd that this monolayer is actually con-\nstructed by using N atoms to link the framework of pyr-\nrole molecules [represented by the pink dash lines in Fig.\n1(a)]. As shown in Fig. 1(a), we label the N atoms in the\nframework of pyrrole molecules as N1, and the ones sited\nat the linking positions as N2. There are also two kinds\nof C atoms, i) the C atoms with the nearest neighbors\nof two C atoms and one N atom (labeled as C1), and\nii) the rest of C atoms which has one C atom and two\nN atoms as the nearest neighbors (labeled as C2). The\nC1{C1 and C1{C2 bond lengths (1.43 and 1.46 \u0017A, re-\nspectively) are only slightly larger than that in graphene\n(1.42 \u0017A, a typical bond length for sp2C{C bonds), and3\nFIG. 1. (a) Top and (b) side views of the optimized geometric\nstructure of the C 4N3H monolayer. The primitive cell and\nconventional cell are respectively denoted by the green and\nblue dashed lines; a1and a2represent the lattice vectors of\nthe primitive cell and \ris the angle between them; aandbare\nthe lattice constants of the conventional cell. The red solid\nline is the connection line among the certain atoms around\nthe angstrom-scale pore in the conventional cell. C1 and C2\ndenote di\u000berent C atoms, and N1 and N2 represent di\u000berent N\natoms. (c) Isosurface of the ELF of C 4N3H monolayer plotted\nwith the value of 0.5.\nare noticeably smaller than 1.54 \u0017A (the standard value of\nC{C bond length for sp3hybridization) [32]. The lengths\nof C1{N2, C2{N1, and C2{N2 bonds in this monolayer\nare respectively about 1.33, 1.38 and 1.31 \u0017A, very simi-\nlar to that in the already-synthesized g-C 3N4(ca. 1.33\nand 1.41 \u0017A) [21] and C 2N-h2D (1.33 \u0017A) [22] which have\nstable structures with sp2hybridization bonds. More\ndetailed information about the structural properties is\nsummarized in Fig. 1 of the Supplemental Material [33].\nIn addition, it is noteworthy that the C 4N3H monolayer\nhas an exactly planar structure, as shown in Fig. 1(b).\nThis exactly planar structure and the features of bond\nlengths and bond angles imply that the chemical bonds\nin this monolayer ought to be covalent bonds with sp2\nhybridization.\nTo con\frm this conjecture and further elucidate the\nbonding nature of the C 4N3H monolayer, we then calcu-\nlated the ELF to analyze its electron distributions. As\nknown, ELF can be described in the form of isosurface in\nreal space with isovalues ranging from 0 to 1. The region\nwith 1 indicates the strong covalent electrons or lone-pair\nelectrons, the region close to 0 implies the area with low\nelectron density, and the region with an isovalue of 0.5 is\nan area with homogeneous electron gas. As shown in Fig.\n1(c), the electron gas is well distributed and delocalized\nat the whole region of this monolayer network, which can\nelectronically stabilize the 2D framework. To highlight\nthe in-plane bonding states, we also plotted the isosurface\nof ELF for this monolayer with an isovalue of 0.8 in Fig.\n2(a). It is found that the ELF localization centers clearly\nFIG. 2. (a) ELF plotted with the value of 0.8. (b) Defor-\nmation charge density of the C 4N3H monolayer. Yellow and\ncyan refer to electron accumulation and depletion regions, re-\nspectively. The isovalue of deformation charge density is 0.3\ne\u0017A\u00003.\nlocate at the middle of the C{C, C{N and N{H bonds,\nindicating that the bonds have strong covalent electron\nstates with \u001b-like sp2hybridization. The \u001bbonds be-\ntween C, N and H atoms can also be evidenced by the\ndeformation electronic density of the C 4N3H monolayer\n[Fig. 2(b)]. The deformation electronic density is de\fned\nas the total electronic density of this monolayer exclud-\ning those of isolated atoms. Clearly, electrons are well\nlocalized over the C{C, C{N and N{H bonds, con\frming\nthe conclusion obtained from the ELF and the conjecture\nwe proposed above. In addition, the remaining valence\nelectrons of carbon and nitrogen atoms form the delocal-\nized\u0019network, like the case of graphene. According to\nthe Bader charge analysis [34, 35], C1, C2, N1, N2 and\nH atoms in the C 4N3H monolayer respectively possess\n+0:55, +1:08,\u00001:28,\u00001:25 and +0:55jejcharge. This\nlevel of charge transfer between C and N atoms is similar\nto that in stable C 2N-h2D monolayer which has been suc-\ncessfully synthesized [22, 36], suggesting that the C 4N3H\nmonolayer is likely to have the similar stability.\nB. Stability of the C 4N3H monolayer\nTo evaluate the feasibility of experimental synthesis\nand the stability of the C 4N3H monolayer, we \frst cal-\nculated its cohesive energy ( Ecoh) de\fned by Ecoh=\n(nCEC+nNEN+nHEH\u0000EC4N3H)=(nC+nN+nH), where\nEC,EN,EHandEC4N3Hare the calculated total energies\nof isolated C, N and H atoms, and C 4N3H monolayer, re-\nspectively; nC,nNandnHare the number of C, N and\nH atoms in the supercell of this monolayer, respectively.\nAccording to our computations, the cohesive energy is\n7.88 eV per atom. This value is evidently larger than\nthat of the already-synthesized borophene (5.87 eV per\natom) [37], silicene (4.01 eV per atom) [38] and phospho-\nrene (4.67 eV per atom) [39], implying the high stability\nand synthetic feasibility of the C 4N3H monolayer from4\nFIG. 3. (a) Phonon dispersion curves and (b) total phonon\nDOS of the C 4N3H monolayer. Inset is the enlarged drawing\nof red rectangle part in the phonon dispersion curves.\nthe viewpoint of energy level.\nThe stability of this monolayer can be further con-\n\frmed by its phonon dispersion curves and phonon DOS.\nAs shown in Fig. 3(a), there is no sign of imaginary\nphonon mode in the phonon spectrum along the highly\nsymmetric points in the entire Brillouin zone. In de-\ntail, there are eight atoms in the primitive cell of the\nC4N3H monolayer, thus its phonon spectrum has twenty-\nfour phonon bands, including three acoustic branches and\ntwenty-one optical branches. The three acoustic branches\nare respectively transverse acoustic (TA) and longitudi-\nnal acoustic (LA) branches corresponding to vibration\nwithin the plane, and the other one (ZA) correspond-\ning to vibration out of plane [inset of Fig. 3(a)]. It can\nbe seen that in contrast to the linear dispersion for the\nTA and LA branches, the frequency of the ZA branch\nshows a quadratic dispersion near the \u0000 point. It is\nworth mentioning that this type of quadratic dispersion\nof ZA branch is a characteristic feature of the phonon\ndispersion curves in monolayered or layered crystals, e.g.,\ngraphene [40], graphite [41] and other layered compounds\n[15, 42, 43]. As shown in Fig. 3(b), the total phonon DOS\nalso reveals that no phonon with imaginary frequency is\nfound in this monolayer, which agrees very well with its\nphonon spectrum. These results demonstrate the good\ndynamic stability of the C 4N3H monolayer.\nMoreover, to further evaluate the thermal stability of\nthe C 4N3H monolayer, we performed FPMD simulations\nusing a 4\u00024\u00021 supercell containing 64 C atoms, 48\nN atoms and 16 H atoms. The initial con\fguration was\nannealed at di\u000berent temperatures of 300, 500 and 1000\nK with a time step of 2 fs. As shown in Figs. 4(a) and\n4(b), the snapshots of the geometry structure at the end\nof 30 ps simulations clearly reveal that the C 4N3H mono-\nlayer can maintain its structural integrity except for some\nthermal \ructuations at the temperature of 300 K. The to-\ntal energy of the simulated system can reach equilibrium\nquickly at 300 K [see Fig. 4(e)], verifying the above re-\nFIG. 4. (a) Top and (b) side views of snapshots for the\nequilibrium structures of the C 4N3H monolayer at the end of\n30 ps FPMD simulations under the temperature of 300 K; (c)\nand (d) are similar respectively to Figs. 4(a) and 4(b), but for\nanother FPMD simulation at the temperature of 500 K. The\ngreen dashed lines denote the 4 \u00024\u00021 supercell used in the\nFPMD simulations. The red arrows and circles display the\nmigration of hydrogen atoms and the break of C{N bonds,\nrespectively. (e) and (f) are the \ructuations of total energies\nwith respect to FPMD simulation times at 300 K and 500 K,\nrespectively.\nsult from the viewpoint of energy level. This result can\nbe understood by the fact that the binding energies of the\nC{C, C{N and C{H bonds are larger than the thermal en-\nergy corresponding to room temperature, consistent with\nother 2D carbon nitride systems [21{23, 44]. Moreover,\nthis slightly distorted structure can restore its initial pla-\nnar con\fguration after complete atomic relaxation.\nThe structure of this monolayer after annealing at 500\nK is shown in Figs. 4(c) and 4(d). Obviously, signi\fcant\natomic rearrangement took place and the basal plane\nsubstantially disordered after thermal annealing [see the\npartial structures pointed by the red arrows and circles\nin Figs. 4(c) and 4(d)]. Further analysis of the FPMD\nsimulations reveals the process of the structure collapse\nat this temperature , as shown in Fig. 4(f) and Fig. 2\nof the Supplemental Material [33]. Moreover, this mono-\nlayer immediately and completely collapsed at a higher\ntemperature of 1000 K. Hence, combined with the above\nresult that the C 4N3H monolayer can maintain its struc-\ntural integrity at room temperature, we can get the con-\nclusion that this monolayer has a melting point between\n300 and 500 K, revealing its decent thermal stability.\nConsidering the importance of the mechanical stabil-\nity of a material for its applications, we also studied the\nmechanical properties of this organic monolayer by ex-\namining its elastic constants. It is known that there are\nfour nonzero elastic constants for a 2D material, which\nareC11,C22,C12(C21) andC66, respectively. These elas-\ntic constants need to satisfy the criteria ( C11C22\u0000C2\n12>5\n0;C66>0) for a mechanically stable 2D sheet [45, 46].\nThe in-plane Youngs modules along a(Ya) and b(Yb) di-\nrections can be expressed as Ya= (C11C22\u0000C12C21)=C22\nandYb= (C11C22\u0000C12C21)=C11. The elastic constants\nof the C 4N3H monolayer computed are C11= 209.8 N\nm\u00001,C22= 138.7 N m\u00001,C12=C21= 71.5 N m\u00001\nandC66= 86.0 N m\u00001. Clearly, these elastic constants\nsatisfy the above criteria, denoting the good mechanical\nstability of this monolayer. Accordingly, YaandYbcalcu-\nlated respectively are 173.1 and 114.3 N m\u00001, indicating\nthat the C 4N3H monolayer is mechanically anisotropic.\nThe Youngs modules of this monolayer are higher than\nthose of already-synthesized phosphorene ( Ya= 25.5 N\nm\u00001andYb= 91.6 N m\u00001) [39, 46], suggesting its good\nmechanical property.\nConsidering that the C 4N3H monolayer has compara-\nble cohesive energy to other already-synthesized 2D ma-\nterials, and dynamic, thermal and mechanical stabilities,\nwe believe that it is viable for its experimental synthesis.\nIt is well known that the condensation reaction is a valid\nmethod for producing exotic materials in polymer chem-\nistry. Most importantly, recent developments in this \feld\nhave already successfully synthesized a series of poly-\nmer materials, for instance, g-C 3N4[20, 21], C 2N-h2D\n[22], C 3N sheets [23], 2D COFs [47, 48], metal-organic\nframeworks [49], etc. These materials show the tremen-\ndous capability of modern chemistry to create novel 2D\nnetworks from custom-designed monomers. Inspired by\nthis, we proposed two hypothetical routes to synthe-\nsize the C 4N3H monolayer based on pyrrole molecules.\nFirstly, one should respectively nitride (or oxidize) pyr-\nrole molecules to the certain products, and then, 2D\nC4N3H material can be synthesized viathe condensation\nreaction of these intermediate products (see Fig. 3 of the\nSupplemental Material [33]). Herein, it should be noted\nthat the real synthetic process must be much more com-\nplicated and di\u000ecult. However, it is worth expecting for\nthe experimental realization of this novel material due\nto its unique electronic properties revealed in the next\nsection.\nC. Electronic structures of the C 4N3H monolayer\nAs a novel material with intriguing structure, its de-\ntailed electronic structure needs to be explored. Thus, we\ncalculated the band structure, total DOS (TDOS) and\nprojected DOS (PDOS) of the C 4N3H monolayer. As\nshown in Fig. 5(a), one can see that this monolayer is a\nsemimetal with the valence and conduction bands meet-\ning in a single point at the Fermi level. The characteris-\ntics of linear bands and degenerate state at this point de-\nnote the appearance of Dirac states in the C 4N3H mono-\nlayer. Di\u000berent from the well-known graphene where\nDirac points locate at the high-symmetry points (K and\nK0points), the Dirac point of the C 4N3H monolayer lo-\ncates on the path from the \u0000 to K point. As a result,\nthere are two symmetry-related Dirac points appearing in\nFIG. 5. (a) Band structure and (b) TDOS and PDOS of\nthe C 4N3H monolayer. The Fermi level is assigned at 0 eV.\nInset in Fig. 5(a) is the enlarged drawing of the bands in\nthe vicinity of the Dirac point, which was calculated by using\nhigh-precision screening parameter of 0.0001 \u0017A\u00001. (c) First\nBrillouin zone with the special kpoints: \u0000 (0, 0, 0), K (0.4,\n0.4, 0), M (0.5, 0, 0), R (0.6, -0.4, 0), and S (0.5, -0.5, 0). b1\nand b2are reciprocal lattice vectors of the k-space, while kx\nand kyare basis vectors of the rectangular coordinate system.\nThe pink lines depict the high symmetric lines connected by\nthe special k points while the black dot represents the position\nof the Dirac point. (d) Distorted Dirac cone formed by the\nvalence and conduction bands in the vicinity of the Dirac\npoint [the region displayed by the green square in Fig. 5(c)].\nthe entire \frst Brillouin zone [only one representative is\nshown in Figs. 5(a) and 5(c), as the two Dirac points are\nrelated by symmetry]. Consistently, its TDOS is zero at\nthe Fermi level [Fig. 5(b)], supporting the presence of the\nDirac point. Figure 5(c) shows the \frst Brillouin zone of\nthe C 4N3H monolayer with orthorhombic ( Amm2 ) struc-\nture, and further demonstrates that neither honeycomb\nstructure nor hexagonal symmetry is prerequisite for the\nexistence of Dirac cones.\nTo obtain deeper insight into the Dirac states of this\norganic monolayer, we further calculated the Dirac cone\nformed by the valence and conduction bands in the vicin-\nity of the Dirac point. As shown in Fig. 5(d), its Dirac\ncone is distinctly distorted, similar to that of some in-\norganic Dirac materials such as phagraphene [13], 6, 6,\n12-graphyne [10] and Pmmn -boronene [12]. The linear\ndispersion curves of energy and momentum in both the\nkxand kydirections around the Dirac point suggest the\nzero e\u000bective mass of the carriers (electrons and holes)6\nnear the Fermi level. To examine the carrier mobility\naround the distorted Dirac cone, we then calculated the\nFermi velocity ( vF) of the C 4N3H monolayer using the\nformula ofvF= (1=~)\u0001(@E=@k ), where@E=@k is the slop\nof valence or conduction band near the Dirac point and ~\nis the reduced Planck0s constant. The slope of the bands\nin the kxdirection is 7.3 eV \u0017A, equivalent to a Fermi\nvelocityvFx= 1:1\u0002106m s\u00001, while in the kydirection,\nthe slop of the bands equal to 6.2 eV \u0017A (vFy= 9:4\u0002105\nm s\u00001) and -2.8 eV \u0017A (vFy= 4:3\u0002105m s\u00001)(see Fig.\n4 of the Supplemental Material [33]). The largest Fermi\nvelocity is comparable to that in graphene and other 2D\ninorganic Dirac materials [10, 12{15], and is roughly one\norder of magnitude larger than that in 2D organic Dirac\nmaterials ever reported [17, 18]. It is noted that Fermi\nvelocities in 2D organic polymers followed an approx-\nimately inverse exponential relation to their pore size,\nlargely independent of the interconnecting oligomer [18].\nThat is to say, the 2D organic Dirac polymer with the\nsmall pore size is favorable to the high Fermi velocity.\nThus, the high Fermi velocity of our organic monolayer\nmay attribute to its angstrom-scale pores. In addition,\nthe anisotropy of the distorted Dirac cone with di\u000berent\nslopes at the Dirac point in the kxand kydirections im-\nplies the direction-dependent electronic properties of the\nC4N3H monolayer, indicating its more \rexible applica-\ntions in contrast with graphene.\nAdditionally, the hybrid HSE06 functional [29] and\nthe SOC e\u000bect were respectively used to recalculate the\nband structure of the C 4N3H monolayer for further con-\n\frming its Dirac states. As shown in Fig. 5(a) of the\nSupplemental Material [33], the dispersion of the valence\nand conduction bands at the Fermi level given by the\nHSE06 functional is very similar to that computed based\non the PBE functional [25] and no band gap can be iden-\nti\fed, indicating that the above predicted intrinsic Dirac\nstates in the C 4N3H monolayer still survive to the hybrid\nHSE06 functional. On the other hand, we also demon-\nstrate that the SOC e\u000bect on the electronic properties\nof this monolayer is negligible and the Dirac cone is still\nwell preserved, which can be seen in Fig. 5(b) of the\nSupplemental Material [33]. This is unsurprising due to\nthe fact that C, N and H are all light elements, and the\nSOC e\u000bect on the nontrivial gap should be very small.\nThen, we explored the physical origin of the Dirac\nstates in the C 4N3H monolayer. Through analyzing the\nPDOS of this monolayer [Fig. 5(b)], one can see that\nboth the occupied and unoccupied peaks near the Fermi\nlevel mostly originate from the 2 pzorbitals of C and N\natoms. The calculated orbital-resolved band structures\nof this monolayer also show the same conclusion that the\nbands near the Fermi level are mainly contributed by\n2pzorbitals of the atoms (see Fig. 6 of the Supplemen-\ntal Material [33]). Furthermore, we also calculated the\nband decomposed charge density at the Dirac point to\nvisualize the above result. Clearly, the electron density\ndistribution presents an obvious characteristic of 2 pzor-\nbitals [see the side views of the isosurfaces plotted in Figs.\nFIG. 6. Top and side views of the isosurfaces of partial charge\ndensities for (a) the highest valence band and (b) the lowest\nconduction band. The isovalue is 0.015 e \u0017A\u00003.\n5(a) and 5(b)]. Moreover, both the top valence band and\nthe bottom of conduction band around the Dirac point\nare mainly contributed by the 2 pzorbitals of C and N2\natoms, while the 2 pzorbital of N1 atoms also makes a\ncontribution to the lowest conduction band. These 2 pz\norbitals overlap, resulting in the formation of an extended\n\u0019-electron conjugation system in the C 4N3H monolayer,\nsimilar to that in graphene. Accordingly, we can take the\nconclusion that this \u0019-electron conjugation system is the\nreason for the emergence of Dirac states in this organic\nmonolayer.\nIV. CONCLUSION\nTo summarize, we designed a novel 2D organic mate-\nrial with evenly distributed heart-shaped angstrom-scale\npores, namely the C 4N3H monolayer. In its unique struc-\nture, every C atom is bonded with three other atoms (C\nor N atoms) via sp2hybridization, giving rise to three\n\u001b-like orbitals placed in its basal plane and one \u0019or-\nbital along Zaxis in the perpendicular direction. These\n\u001band\u0019bonds are responsible for the energy stability\nof the 2D C 4N3H framework. Furthermore, we have\ncon\frmed its dynamic, thermal and mechanical stabil-\nities respectively by its phonon dispersion curves, FPMD\nsimulations and mechanical properties. Band structure\nand TDOS clearly reveal that this organic monolayer is\na semimetal with anisotropic Dirac cones and very high\nFermi velocity which is roughly one order of magnitude\nlarger than that in 2D organic Dirac materials ever re-\nported [17, 18]. Based on the PDOS and the band de-\ncomposed charge density of this monolayer, we demon-\nstrate that the anisotropic Dirac cones originate from the\nextended\u0019-electron conjugation system which is mainly\ncontributed by the 2 pzorbitals of C and N atoms. These\nresults indicate that we successfully designed a 2D or-\nganic Dirac material with high Fermi velocity which can7\nbe comparable to that in 2D inorganic ones. We hope\nthat our \fndings will promote the experimental realiza-\ntion of this novel material and greatly push forward the\nstudy of the 2D organic Dirac materials.\nACKNOWLEDGMENTS\nWe thank Feng Liu of University of Utah for help-\nful discussion. This work was \fnancially supported bythe State Key Program for Basic Research (Grant Nos.\n2014CB921102 and 2017YFA0206304) and NSFC (Grant\nNos. 51572122 and 11304096), China. We are grateful\nto the High Performance Computing Center of Nanjing\nUniversity for doing the numerical calculations in this\npaper on its blade cluster system.\n[1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109\n(2009).\n[2] J. E. Moore, Nature 464, 194 (2010).\n[3] T. O.Wehling, A. M. Black-Scha\u000ber, and A. V. Balatsky,\nAdv. Phys. 63, 1 (2014).\n[4] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nM. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and\nA. A. Firsov, Nature 438, 197 (2005).\n[5] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat.\nphys. 2, 620 (2006).\n[6] C. R. Dean, L.Wang, P. Maher, C. Forsythe, F. Ghahari,\nY. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino,\nT. Taniguchi, K.Watanabe, K. L. Shepard, J. Hone, and\nP. Kim, Nature 497, 598 (2013).\n[7] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and\nZ. Fang, Science 329, 61 (2010).\n[8] K. Asano and C. Hotta, Phys. Rev. B 83, 245125 (2011).\n[9] J. Wang, S. Deng, Z. Liu, and Z. Liu, Natl. Sci. Rev. 2,\n22 (2015).\n[10] D. Malko, C. Neiss, F. Vi~ nes, and A. G orling, Phys. Rev.\nLett. 108, 086804 (2012).\n[11] W. Li, M. Guo, G. Zhang, and Y.-W. Zhang, Phys. Rev.\nB89, 205402 (2014).\n[12] X.-F. Zhou, X. Dong, A. R. Oganov, Q. Zhu, Y. Tian,\nand H.-T. Wang, Phys. Rev. Lett. 112, 085502 (2014).\n[13] Z. Wang, X.-F. Zhou, X. Zhang, Q. Zhu, H. Dong, M.\nZhao, and A. R. Oganov, Nano Lett. 15, 6182 (2015).\n[14] S. Cahangirov, M. Topsakal, E. Akt urk, H. S \u0018ahin, and S.\nCiraci, Phys. Rev. Lett. 102, 236804 (2009).\n[15] H. Zhang, Y. Li, J. Hou, A. Du, and Z. Chen, Nano Lett.\n16, 6124 (2016).\n[16] M. O. Goerbig, J.-N. Fuchs, G. Montambaux, and F.\nPi\u0013 echon, Phys. Rev. B 78, 045415 (2008).\n[17] M. Hirata, K. Ishikawa, K. Miyagawa, M. Tamura, C.\nBerthier, D. Basko, A. Kobayashi, G. Matsuno, and K.\nKanoda, Nat. Commun. 7, 12666 (2016).\n[18] J.-J. Adjizian, P. Briddon, B. Humbert, J.-L. Duvail, P.\nWagner, C. Adda, and C. Ewels, Nat. Commun. 5, 5842\n(2014).\n[19] K. Medjanik, O. Fedchenko, S. Chernov, D. Kutnyakhov,\nM. Ellguth, A. Oelsner, B. Sch onhense, T. R. F. Peixoto,\nP. Lutz, C.-H. Min, F. Reinert, S. D aster, Y. Acremann,\nJ. Viefhaus, W. Wurth, H. J. Elmers, and G. Sch onhense,\nNat. Mater. 16, 615 (2017).\n[20] X. Wang, K. Maeda, A. Thomas, K. Takanabe, G. Xin, J.\nM. Carlsson, K. Domen, and M. Antonietti, Nat. Mater.\n8, 76 (2009).[21] G. Algara-Siller, N. Severin, S. Y. Chong, T. Bj orkman,\nR. G. Palgrave, A. Laybourn, M. Antonietti, Y. Z.\nKhimyak, A. V. Krasheninnikov, J. P. Rabe, U. Kaiser,\nA. I. Cooper, A. Thomas, and M. J. Bojdys, Angew.\nChem. Int. Ed. 53, 7450 (2014).\n[22] J. Mahmood, E. K. Lee, M. Jung, D. Shin, I.-Y. Jeon,\nS.-M. Jung, H.-J. Choi, J.-M. Seo, S.-Y. Bae, S.-D. Sohn,\nN. Park, J. H. Oh, H.-J. Shin, and J.-B. Baek, Nat. Com-\nmun. 6, 6486 (2015).\n[23] S. Yang, W. Li, C. Ye, G. Wang, H. Tian, C. Zhu, P. He,\nG. Ding, X. Xie, Y. Liu, Y. Lifshitz, S.-T. Lee, Z. Kang,\nand M. Jiang, Adv. Mater. 29, 1605625 (2017).\n[24] A. Savin, R. Nesper, S. Wengert, and T. F. F assler,\nAngew. Chem. Int. Ed. 36, 1808 (1997).\n[25] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n[26] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).\n[27] G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169\n(1996).\n[28] P. E. Bl ochl, Phys. Rev. B 50, 17953 (1994).\n[29] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem.\nPhys. 124, 219906 (2006).\n[30] A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106\n(2008).\n[31] S. Nos\u0013 e, Mol. Phys. 52, 255 (1984).\n[32] D. W. Boukhvalov, M. I. Katsnelson, and A. I. Lichten-\nstein, Phys. Rev. B 77, 035427 (2008).\n[33] See Supplemental Material at http://link.aps.org/\nsupplemental/XXX for details on geometric structure,\nstability and electronic structures of the C 4N3H mono-\nlayer.\n[34] G. Henkelman, A. Arnaldsson, and H. J\u0013 onsson, Comp.\nMater. Sci. 36, 354 (2006).\n[35] E. Sanville, S. D. Kenny, R. Smith, and G. Henkelman,\nJ. Comput. Chem. 28, 899 (2007).\n[36] J. Kang, S. Horzum, and F. M. Peeters, Phys. Rev. B\n92, 195419 (2015).\n[37] A. J. Mannix, X.-F. Zhou, B. Kiraly, J. D. Wood, D.\nAlducin, B. D. Myers, X. Liu, B. L. Fisher, U. Santiago,\nJ. R. Guest, M. J. Yacaman, A. Ponce, A. R. Oganov,\nM. C. Hersam, and N. P. Guisinger, Science 350, 1513\n(2015).\n[38] A. Fleurence, R. Friedlein, T. Ozaki, H. Kawai, Y.Wang,\nand Y. Yamada-Takamura, Phys. Rev. Lett. 108, 245501\n(2012).\n[39] L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X.\nH. Chen, and Y. Zhang, Nat. Nanotechnol. 9, 372 (2014).8\n[40] J.-A. Yan, W. Y. Ruan, and M. Y. Chou, Phys. Rev. B\n77, 125401 (2008).\n[41] N. Mounet and N. Marzari, Phys. Rev. B 71, 205214\n(2005).\n[42] H. Pan, Y. Sun, Y. Zheng, N. Tang, and Y. Du, New J.\nPhys. 18, 093021 (2016).\n[43] H. Zabel, J. Phys.: Condens. Matter 13, 7679 (2001).\n[44] A. Du, S. Sanvito, and S. C. Smith, Phys. Rev. Lett.\n108, 197207 (2012).\n[45] S. Zhang, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe, and\nP. Jena, Proc. Natl Acad. Sci. USA 112, 2372 (2015).[46] Y. Wang, F. Li, Y. Li, and Z. Chen, Nat. Commun. 7,\n11488 (2016).\n[47] C. S. Diercks and O. M. Yaghi, Science 355, 923 (2017).\n[48] J. W. Colson, A. R.Woll, A. Mukherjee, M. P. Levendorf,\nE. L. Spitler, V. B. Shields, M. G. Spencer, J. Park, and\nW. R. Dichtel, Science 332, 228 (2011).\n[49] T. Kambe, R. Sakamoto, K. Hoshiko, K. Takada, M.\nMiyachi, J.-H. Ryu, S. Sasaki, J. Kim, K. Nakazato, M.\nTakata, and H. Nishihara, J. Am. Chem. Soc. 135, 2462\n(2013)."    },    {        "title": "0902.2741v1.Spin_density_wave_Fermi_surface_reconstruction_in_underdoped_YBa2Cu3O6_x.pdf",        "content": "Spin-density wave Fermi surface reconstruction in underdoped YBa 2Cu 3O6+x\nN. Harrison\nNational High Magnetic Field Laboratory, Los Alamos National Laboratory, MS E536, Los Alamos, New Mexico 87545\n(Dated: June 4, 2022)\nWe consider the reconstruction expected for the Fermi surface of underdoped YBa 2Cu3O6+xin\nthe case of a collinear spin-density wave with a characteristic vector Q= (\u0019[1\u00062\u000e];\u0019), assuming\nan incommensurability \u000e\u00190.06 similar to that found in recent neutron scattering experiments.\nA Fermi surface possibly consistent with the multiple observed quantum oscillation frequencies is\nobtained. From the low band masses expected using this model as compared with experiment, a\nuniform enhancement of the quasiparticle e\u000bective mass over the Fermi surface by a factor of \u00197\nis indicated. Further predictions of the Fermi surface topology are made, which may potentially be\ntested by experiment to indicate the relevance of this model to underdoped YBa 2Cu3O6+x.\nThe recent discovery of magnetic quantum oscilla-\ntions in high Tcsuperconductors provides a unique op-\nportunity to access the Fermi surface topology and the\nproperties of the quasiparticles undergoing pair forma-\ntion [1, 2, 3, 4, 5, 6, 7, 8, 9]. The reports of multiple\nsmall pockets of carriers [5, 7, 8] and a negative Hall co-\ne\u000ecient [3, 10] in the underdoped systems YBa 2Cu3O6+x\nand YBa 2Cu4O8suggest a possible reconstruction of the\nFermi surface by a density-wave order parameter. In this\npaper, we calculate the Fermi surface topology consider-\ning the case of a collinear spin-density wave with a mod-\nulation vector Q= (\u0019;\u0019[1\u00062\u000e]), where\u000e\u00190.06 is pro-\nvided by recent elastic [11] and inelastic [12] neutron scat-\ntering experiments on underdoped YBa 2Cu3O6+x. In\nthis model, we use the values of \u000eand ratioVs=t\u001c1\n(of the spin modulation potential to the in-plane hop-\nping parameters) which represent the case of a conven-\ntional spin-density wave, rather than stripes of the form\nproposed in Refs. [13, 14].\nWe begin by considering the unreconsructed tight\nbinding approximation\n\"k=\"0+ 2t10[cos(akx) + cos(bky)] + 2t11[cos(akx\n+bky) + cos(akx\u0000bky)] + 2t20[cos(2akx)\n+ cos(2bky)] + 2tccosckz(1)\noriginally proposed for YBa 2Cu3O6+xby Andersen et\nal.[15]. In this approximation, we choose t10=\n\u0000380 meV,t11=t10=\u00000.32 andt20=t10= 0.16 [15, 16],\nwhile the number of holes is tuned by \"0. We intro-\nduce an interlayer hopping term (2 tccosckz) to model the\ndeep corrugation of the otherwise cylindrical Fermi sur-\nface section reported in recent studies [8, 9]. We further\nneglect the ortho-II potential [17], following reports of\nquantum oscillations in non-ortho-II ordered samples [7],\nand the e\u000bect of bilayer splitting [15, 18]. Angle-resolved\nphotoemission spectroscopy experiments on surface K-\ndeposition-treated ortho-II ordered YBa 2Cu3O6:5yield\na Fermi surface whose location in k-space is consistent\nwith these assumptions for Eqn. (1) for hole dopings near\np= 0.1 [19]. We plot the corresponding Fermi surface for\nkz=\u0006\u0019=2cin Fig. 1a.For guidance on the appropriate choice of \u000e, we turn\nto recent neutron scattering experiments on underdoped\nYBa 2Cu3O6+x. Magnetic \feld-dependent elastic scat-\ntering experiments [11] \fnd evidence for long range an-\ntiferromagnetism with \u000e\u00190.055 in samples of similar\ncomposition ( x= 0.45) to those of the lowest compo-\nsition (x= 0.49) in which quantum oscillations are ob-\nserved [9]. Meanwhile, inelastic scattering experiments\nperformed on samples of nominally the same composition\n(x\u00190.5) as those in which multiple Fermi surface pock-\nets are reported [5, 8], \fnd incommensurate magnetic\nexcitations at energies as low as 6 meV at \u000e\u00190.06 [12]\nin the absence of an applied \feld.\nWe therefore consider the possibility that spin-density\nwave ordering occurs near \u000e\u00190.06 in YBa 2Cu3O6:5in\nsu\u000eciently strong magnetic \felds. If it is collinear, the\nFermi surface reconstruction is determined by multiple\ntranslations of \"kby\u0006nQ(each of which we denote\n\"k+nQ). As in Cr [20, 21], this leads to a hierarchy of\ngaps of order 2\u0001 m\u00182Vm\ns=tm\u00001opening at the cross-\ning points of bands \"k+nQand\"k+n\u0006mQ, whereVsis\nthe amplitude of the spin potential and t\u0018t10. Pro-\nvidedVs\u001ct, these gaps become vanishingly small for\nlargem, implying that there is little di\u000berence between\nthe orbits obtained in an incommensurate model with an\nirrational\u000eand those obtained using a commensurate\nmodel with a rational \u000eof similar value. We therefore\nchoose to consider a rational \u000e, since it enables a com-\nplete picture of the reconstructed Fermi surface to be\nobtained from the eigenvalues of a matrix [16]. Given\nthe limited precision of inelastic neutron scattering ex-\nperiments on YBa 2Cu3O6:5[12], we make the convenient\nchoice of\u000e=1\n16= 0.0625, and compute the eigenvalues\nof the 16 \u000216 matrix\nH=0\nBBBBBBB@\"kVsVc::: V cVs\nVs\"k+QVs::: 0Vc\nVcVs\"k+2Q::: 0 0\n..................\nVc0 0 ::: \" k+14QVs\nVsVc 0::: V s\"k+15Q1\nCCCCCCCA:(2)\nAdditional terms ( Vc) are introduced in Eqn. (2) to ac-arXiv:0902.2741v1  [cond-mat.str-el]  16 Feb 20092\nFIG. 1: aUnreconstructed Fermi surface of YBa 2Cu3O6+x\nforp= 0.1 according to Eqn. (1). bReconstructed Fermi\nsurface, as described in the text. The four prominent orbits\nare depicted in color (online). cNotional Fermi surface cor-\nresponding to the eigenvalues of a reduced 3 \u00023 matrix con-\ntaining only the terms in the top-left-hand-corner of Eqn.(2),\nwhich produces the same orbits as in ( b).commodate a possible charge modulation at 2 Q[16],\nwhich often occurs in spin-density wave systems [21].\nFigure 1b shows the in-plane Fermi surface cross-\nsection atkz=\u0006\u0019=2ccorresponding to the 16 eigenval-\nues of Eqn. (2) for the case of a simple collinear spin-\ndensity wave in which Vc= 0. Here, Vsis adjusted\nto a value of 0.15 \u0002t10to yield an electron pocket\n(\u000b) of similar k-space area to that reported experimen-\ntally [10], while \"0is adjusted to conserve the hole \flling\natp= 0.1. The full paramagnetic Brillouin zone shown\nis 16 times larger than the antiferromagnetic Brillouin\nzone. Four primary orbits ( \u000b,\f,\rand\u000e) are expected\nfor this reconstructed Fermi surface, and are indicated\nin color in Fig. 1b. The smallness of the higher order\ngaps 2\u0001m>1implies that these four orbits can be ap-\nproximately reproduced by considering the eigenvalues\nof a reduced 3 \u00023 matrix containing only on the terms\nin the top-right-right-most corner of Eqn. (2). These\nare shown in Fig. 1c. The multitude of very small hole\norbits in Fig. 1b (resulting from the overlap of many\nbands with relative translations of mQwherem > 2)\nare too small to have \flled Landau levels in magnetic\n\felds of the magnitude relevant for quantum oscilla-\ntions [1, 4, 5, 6, 7, 8, 9]. Furthermore, the gaps between\nthem (2\u0001 m\u00182Vm\ns=tm\u00001\u001c2Vs) are small enough to be\ncompletely broken through (having magnetic breakdown\ntunneling probabilities of \u00191 [22]).\nThe corresponding quantum oscillation frequencies\nFi= (Ai=ABZ)h=eab for each of the four primary or-\nbits are plotted as a function of pin Fig. 2a (where \u000e\nandVsare \fxed) to allow for possible discrepances in\nhole doping estimates [1, 23]. The frequencies are cal-\nculated from their k-space areas Aiusing the Onsager\nrelation [22], where ABZis the area of the paramagnetic\nBrillouin zone and aandbare the in-plane lattice con-\nstants. A value of tc\u00198 meV is required to account for\nthe deep corrugation of (the otherwise cylindrical) Fermi\nsurface detected in recent experiments [8, 9, 24], yielding\ntwo (maximum `belly' and minimum `neck') frequencies\nfor each Fermi surface section.\nAn appealling feature of the present Fermi surface\nmodel is that the smallest detectable quantum oscil-\nlation frequency predicted is the electron pocket (re-\nferred to here as the \u000b-pocket following the nomencla-\nture introduced in Ref. [5]), similar to experimental \fnd-\nings [1, 4, 5, 8, 9]. Furthermore, this pocket has the light-\nest e\u000bective mass (see Fig. 2b) and is the only one (of the\nfour) not requiring magnetic breakdown for its observa-\ntion, suggesting that it will dominate both the quantum\noscillation spectrum in amplitude [5, 8] and the Hall coef-\n\fcient in intermediate magnetic \felds [10]. The remain-\ning three prominent orbits ( \f,\rand\u000ein Fig. 1b) require\nvarying degrees of magnetic breakdown to be observed,\nfor which we make a rough estimate of their magnetic3\nFIG. 2: aA comparison of calculated quantum oscillation\nfrequencies (lines) plotted for a range of pwith those ob-\nserved experimentally (circles, with error bars at 2 \u001blevel).\nA \fnite value of tc\u00198 meV yields two parallel lines (i.e.\n`belly' and `neck' frequencies) for each Fermi surface section.\nRed and blue curves and symbols correspond electrons and\nholes respectively. Best agreement is obtained for p= 0.092.\nExperimental values for all the frequencies are taken from\nRef. [8], where the wide magnetic \feld range enables accu-\nrate estimates, with the exception of F\ftaken from Ref. [5].\nbComparison of the calculated band masses mb(lines and\nleft-hand-axis) with those m\u0003reported experimentally (circles\nand right-hand-axis). A renormalization of t10from 390 meV\nto\u001957 meV is required in order to align the calculated and\nexperimental values, suggesting a possible uniform mass en-\nhancement of m\u0003=mb\u00197 across the Fermi surface. E\u000bective\nmass estimates are taken from Refs. [5, 9].\nbreakdown probabilities using\npi\u0019exp\u0012\n\u00004\u00012\niB\n~2!2\nc;iFi\u0013\n: (3)The subscript irefers to a particular orbit of frequency Fi\nand cyclotron frequency !c;i=eB=m b;i, requiring tun-\nneling through a gap 2\u0001 ito be observed [22, 26]. Of\nthe three orbits ( \f,\rand\u000e), the hole pocket denoted \r\nrequires magnetic breakdown through only a third order\ngap of order 2\u0001 3\u00182V3\ns=t2\n10(plus less important higher\norder gaps), giving rise to a tunneling probability of or-\nderp\r\u001898 % forB\u0019\u00160H= 50 T. The frequency\nof\u0019640 T reported by Audouard et al [8, 27] provides\na possible candidate for this orbit (see Fig. 2a). Since\nthe\u000eand\forbits require magnetic breakdown tunnel-\ning through the larger \frst order gap 2\u0001 1\u00182Vs, their\nquantum oscillation amplitudes are expected to be sig-\nni\fcantly weaker than those originating from the \u000band\n\rorbits. The estimated magnetic breakdown probabili-\nties arep\u000e\u00182 % andp\f\u00188 % respectively. Possibly\nconsistent with these greatly reduced tunneling probabil-\nities, weak features of similar frequency ( \u00191130 T and\n\u00191650 T in Fig. 2a) to those in the model are reported\nin Refs. [5, 8], although the harmonics of F\u000bare expected\nto occur nearby in frequency [8].\nWe \fnd that the ratio of m\u0003\n\ftom\u0003\n\u000bin Fig. 2b found\nexperimentally is similar to that of mb;\ftomb;\u000bpre-\ndicted by the model, suggesting a possible uniform en-\nhancement of the quasiparticle e\u000bective mass over the\nFermi surface of \u00197 (relative to the bandstructure es-\ntimate [15]). On considering this renormalization factor\nover the entire Fermi surface, the total summed e\u000bec-\ntive mass of all Fermi surface sections within the an-\ntiferromagnetic Brillouin zone becomes m\u0003\nAFM\u00197me\n(wheremeis the free electron mass), compared to\nm\u0003\nP\u001911mefor the single large orbit of the unrecon-\nstructed paramagnetic Fermi surface. Using these esti-\nmates, we obtain electronic coe\u000ecients of the heat capac-\nity of\rAFM = 1.46 \u0002(m\u0003\nAFM=me) = 10 mJmol\u00001K\u00002[5]\nand\rP= 1.46 \u0002(m\u0003\nP=me) = 16 mJmol\u00001K\u00002for the re-\nconstructed and unreconstructed Fermi surfaces respec-\ntively.\nOne notable caveat in assessing the applicability of the\nspin-density wave model to YBa 2Cu3O6+xpresents it-\nself in Fig. 2a on comparing the experimental frequen-\ncies with those predicted in the model. Best agreement\nis obtained for p= 0.092 [23], which falls short of that\np\u00190.098\u00060.001 estimated from the lattice parameter\nc [1, 23]. While some uncertainty in t20=t10ort11=t10in\nEqn. (1) or our neglect of the ortho-II potential (which\noccurs at 8 Qhere) can likely account for some of the\ndiscrepancy, our neglect of the 2 Qcharge modulation\npotential may be more signi\fcant. On including Vcin\nthe simulations in Fig. 3, the electron pocket shrinks or\nexpands depending on whether Vcis positive or negative,\nsuggesting that improved consistency with the nominal\nhole doping level could be achieved by the simultaneous\nadjustment of VsandVc.\nIn summary, by using a value \u000e\u00190.06 for the in-\ncommensurability parameter found in recent neutron4\nFIG. 3: The Fermi surfaces (shown for1\n4of the paramagnetic\nBrillouin zone) obtained on including a signi\fcant charge\nmodulation potential at 2 Q, withVc=Vsin (a) andVc=\u0000Vs\nin (b).\nscattering experiments, we \fnd that the existence of a\ncollinear spin density wave in suitably strong magnetic\n\felds would yield a reconstructed Fermi surface consis-\ntent with the observed multiple carrier pockets in un-\nderdoped YBa 2Cu3O6+x[5, 8]. Four major orbits are\npredicted, with magnetic breakdown being an important\nfactor for all but the smallest electron pocket.\nA key feature of this model is that the anticipated\nshape of the electron pocket is a rounded rectangle\nwith an aspect ratio >\u00192 and its long axis parallel to\nky(orthogonal to the incommensurate spin modulation\ndirection) \u0000something that can be tested by dual-axis\nangle-dependent magnetic quantum oscillation or magne-\ntoresistance experiments. A small electron pocket that is\nfound to be circular in cross-section (or have an approxi-\nmate fourfold symmetry) in experiments would be unrec-\noncilable with the present collinear density wave model\nand would instead have di\u000berent implications. Such\na pocket would then suggest an evolution of \u000erather\nabruptly to a value signi\fcantly greater than 0.06 in\nstrong magnetic \felds [28], or a helical or spiral spin-\ndensity wave [29], or an entirely di\u000berent form of Fermi\nsurface topology in YBa 2Cu3O6+x(possibly unrelated to\nthe magnetic di\u000braction peaks). If experiments detect\na Fermi surface topology similar to that yielded by the\ncollinear spin density wave model considered here, indica-\ntions would be that such a form of order could be chie\ry\nresponsible for Fermi surface reconstructuion in strong\nmagnetic \felds in underdoped YBa 2Cu3O6+x. Given the\nsmall value of Vsthat we consider in this model, a fur-\nther implication would be that the observed quasiparticle\nmass enhancement has an origin mostly extrinsic to this\nform of collinear density wave.\nThis work is supported by the US Department of En-\nergy, the National Science Foundation and the State\nof Florida. The author acknowledges helpful comments\nfrom S. E. Sebastian.[1] N. Doiron-Leyraud et al. , Nature 447, 565 (2007).\n[2] E.A. Yelland et al. , Phys. Rev. Lett. 100, 047003 (2008).\n[3] A. F. Bangura et al. , Phys. Rev. Lett. 100, 046004\n(2008).\n[4] C. Jaudet et al. , preprint arXiv:0711.3559 (2008).\n[5] S. E. Sebastian et al. , Nature 454, 200 (2008).\n[6] B. Vignolle et al. , Nature 455, 952 (2008).\n[7] J. Singleton, R. D. McDonald, and S. Cox, Physica B\n404, 350 (2009).\n[8] A. Audouard et al. , preprint arXiv:0812.0458 (2008).\n[9] S. E. Sebastian (preprint, 2009).\n[10] D. LeBoeuf et al. , Nature 450, 533 (2007).\n[11] D. Haug et al. , Magnetic \feld enhanced incommensurate\nmagnetism in the underdoped high-temperature super-\nconductor YBa 2Cu3O6:45(preprint, 2008).\n[12] C. Stock et al. , Phys, Rev. B 69, 014502 (2004); C. Stock\net al. , Phys. Rev. B 71, 024522 (2005).\n[13] S. A. Kivelson et al. , Rev. Mod. Phys. 75, 1201 (2003).\n[14] J. M. Tranquada et al. , Nature 375, 561 (1995).\n[15] O. K. Andersen et al. , Phys. Chem. Solids 56, 1573\n(1995).\n[16] A. J. Millis and M. R. Norman, Phys. Rev. B 76, 220503\n(2007).\n[17] I. S. El\fmov, G. A. Sawatsky, and A. Damascelli, Phys.\nRev. B 77, 060504 (2008).\n[18] We consider the simplest case of a spin-density wave that\ncouples bonding and antibonding states, leading to two\nsets of degenerate reconstructed Fermi surfaces in which\nthe e\u000bects of bilayer splitting are averaged out.\n[19] M. A. Hossain et al. , Nature Phys. 4, 527 (2008).\n[20] W. M. Lomer, Proceedings of the International Confer-\nence on Magnetism (Nottingham, UK, 1964).\n[21] E. Fawcett, Rev. Mod. Phys. 60, 209 (1988).\n[22] D. Shoenberg, Magnetic oscillations in metals (Cam-\nbridge University Press, Cambridge 1984).\n[23] R. Liang, D. A. Bonn, and W. N Hardy, Phys. Rev. B\n73, 180505 (2006).\n[24] Because the experimental e\u000bective masses are renormal-\nized by a factor of \u00197 compared to the model (in\nwhicht10=\u0000390 meV), the experimental estimates\noft\u0003\nc\u00191 meV in Refs. [8, 9] must be considered renor-\nmalized down by a similar factor.\n[25] I. Dimov et al. , Phys. Rev. B 78, 134529 (2008).\n[26] For convenience, we have equated the e\u000bective Fermi\nenergy in the standard expression for magnetic break-\ndown [22] with \"F;i=~eFi=mb;ifor a particular orbit.\n[27] In Ref. [8], this orbit has a di\u000berent magnetic \feld-\ndependent attenuation compared to the \u0019450 T and\n\u0019540 T frequencies attributed to the \u000bpocket [9], sug-\ngestive of a di\u000berent e\u000bective mass or scattering rate,\nwhich would normally indicate a di\u000berent Fermi surface\nsection.\n[28] In the present collinear model, an incremental increase in\neitherVsor\u000eworsens the level of agreement between the\narea of the predicted electron pocket and the \u000bfrequency\nreported in experiments. Not until \u000e>\u00190.1 is it possible\nonce again to obtain an electron pocket with an area\ncompatible with experiments using a collinear model [16].\n[29] For a helical spin-density wave, the Fermi surface can be\nobtained by solving a 2 \u00022 matrix [5], which reduces\nthe number of crossing points of the translated bands,5\nyielding an electron pocket in the shape of a rounded\nsquare."    },    {        "title": "2208.10899v1.Local_electron_correlation_effects_on_the_Fermiology_of_the_weak_itinerant_ferromagnet_ZrZn__2_.pdf",        "content": "Local electron correlation effects on the Fermiology of the weak itinerant ferromagnet ZrZn 2\nWenhan Chen,1A. D. N. James,1and S. B. Dugdale1\n1H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom\n(Dated: August 24, 2022)\nThe Fermi surface topology plays an important role in the macroscopic properties of metals. It can be par-\nticularly sensitive to electron correlation, which appears to be especially significant for the weak itinerant ferro-\nmagnet ZrZn 2. Here, we look at the differences in the predicted Fermi surface sheets of this metallic compound\nin its paramagnetic phase for both density functional theory (DFT) and the combination of DFT with dynami-\ncal mean field theory (DFT+DMFT). The theoretical spectral functions evaluated at the Fermi level were used\nalong with calculations of the electron-positron momentum density (also known as the two-photon momenutm\ndensity) in k-space to provide insights into the origin of certain features of the Fermi surface topology. We\ncompare this two photon momentum density to that extracted from the positron annihilation experimental data\n(Phys. Rev. Lett. 92, 107003 (2004)). The DFT+DMFT densities are in better agreement with the experiment\nthan the DFT, particularly with regard to the flat bands around the LandWhigh symmetry points. The exper-\nimental neck around L, which relates to a van Hove singularity, is present in DFT+DMFT but not in the DFT.\nWe find that these flat bands, and as such the Fermi surface topology, are sensitive to the many body electron\ncorrelation description, and show that the positron annihilation technique is able to probe this. This description\nis significant for the observed behavior such as the Lifshiftz transition around the quantum critical point.\nI. INTRODUCTION\nWeak itinerant ferromagnets are a class of materials with\nsmall ordered magnetic moments ( MO) and low Curie tem-\nperatures (TC) [1]. ZrZn 2, which has a C15 cubic Laves\nphase structure with lattice parameter of 7:393 ˚A, is a rare\nexample of a member of this class ( TC\u001928K,MO\u0019\n0:17\u0016Bper formula unit) [2] and one which has been shown\nto host a variety of phenomena including pressure-tuned quan-\ntum criticality and non-Fermi liquid behavior both close to\nand away from the quantum critical point [2–5]. Interest\nin this material was significantly piqued by an apparent co-\nexistence of superconductivity and ferromagnetism in high-\nquality samples (with very low residual resistivities of the or-\nder of 0:62\u0016\ncm ) [6]. Such coexistence, predicted decades\nago by Ginzburg [7], was thought to be due to triplet pairing\nthrough spin fluctuations [8]. On the other hand, the Fulde-\nFerrell-Larkin-Ovchinnikov (FFLO) state [9, 10], influenced\nby the Fermiology, was another candidate to explain this co-\nexistence [11, 12]. P-wave superconductivity was also pre-\ndicted to exist in the paramagnetic phase and it was thought\nthat the reason no superconductivity had been observed was\ndue to sample disorder [13]. However, subsequent investiga-\ntions by Yelland et al. found no evidence of a bulk supercon-\nducting state in ZrZn 2[2], and in fact the previously measured\nsuperconductivity was due to a superconducting surface layer\ninduced by spark erosion [14].\nYamaji et al. pointed out that the metamagnetic transi-\ntions in ZrZn 2around the so-called marginal quantum critical\npoint, accompanied by a discontinuous change in the Fermi\nsurface topology (a Lifshitz transition), belong to an uncon-\nventional class of symmetry-breaking transition [15]. Previ-\nous measurements of the magnetic, transport, and thermody-\nnamic properties strongly support this point [16, 17]. Density\nfunctional theory (DFT) calculations show flat bands at the\nFermi level around the LandWhigh symmetry points which\ngenerate a van Hove singularity responsible for the weak fer-\nromagnetic ordering in the material and play an important rolein the metamagnetic transition in a weak magnetic field, Lif-\nshitz transitions, and non-Fermi-liquid behavior [18]. The\nresulting sensitivity of the predicted Fermi surface to details\nof the calculation can be seen in the literature, as discussed be-\nlow. An early DFT study indicated that a small Fermi surface\npocket is created by the very flat band at L[19]. While later\nresults by Singh and Mazin [11] gave a similar Fermi surface\ntopology with both studies predicting a connected ‘pillow’\nFermi surface sheet around W(described by Singh and Mazin\nas a ‘pancake’), there were differences. Singh and Mazin did\nnot observe this small Fermi surface pocket at L. The Lif-\nshitz transition in this material is tightly connected with the\nvan Hove singularities around LandW.\nThere have been measurements of the ferromagnetic Fermi\nsurface near the quantum critical point at temperatures as low\nas 30mK using the quantum oscillatory de Haas-van Alphen\neffect by Yates et al. [20]. In order to investigate the param-\nagnetic Fermi surface topology away from the quantum crit-\nical point, Major et al. [12] measured the Fermi surface of\nZrZn 2by positron annihilation in the paramagnetic phase (at\nT\u001980K). Interestingly, the positron annihilation experi-\nment on paramagnetic ZrZn 2revealed the existence of an ex-\ntra Fermi surface neck around the Lpoint which was not seen\nin those previous calculations [11, 19]. Note that the positron\nannihilation results were compared to calculations made using\nthe linear-muffin-tin-orbital (LMTO) method which actually\nshowed a neck at the Lpoint. However, since those LMTO\ncalculations were made in the atomic-sphere-approximation\n(ASA), it could be expected that the full-potential calculations\nwould more accurately describe the bands and Fermi surface.\nThis difference strongly indicates that the flat band around L\npoint may in fact be (partially) below the Fermi level as indi-\ncated by a recent DFT+dynamical mean field theory (DMFT)\nstudy [21].\nIn positron annihilation experiments, the positron anni-\nhilates with the electron in the material producing two \r-\nphotons. By measuring the deviation from anti-collinearity\nit is possible to infer two components of the electron-positronarXiv:2208.10899v1  [cond-mat.str-el]  23 Aug 20222\nmomentum distribution, which is also called the two-photon\nmomentum distribution (TPMD). Note that the full three-\ndimensional density can be recovered using tomographic tech-\nniques applied to data measured with different crystallo-\ngraphic orientations (see for example Ref. [22]). The TPMD\nwill include the effects of electron-positron correlations which\ncould potentially obscure the signatures of many-body elec-\ntron correlations. However, it has been shown that the mea-\nsured electron-positron momentum distribution is actually\nstrongly influenced by the electron correlations in the material\nand is not hidden by the electron-positron correlations [23].\nThis may imply that many significant differences between\nthe predictions from DFT and positron annihilation experi-\nments may in fact be due to the limitations of the approxima-\ntion to the exchange-correlation functional used in the DFT\nframework for electrons. Previously, positron annihilation and\nDFT+DMFT has been used to provide insight into the Hub-\nbardUvalue of Ni [24] and electron correlation effects in\nV [25]. It is therefore necessary to include the electron corre-\nlations described beyond the DFT framework in calculations\nof the electron-positron momentum distribution.\nA recent DFT+DMFT study has highlighted that the DMFT\nFermi-liquid-like electron self-energies subtly change the\nFermi surface topology of ZrZn 2[21]. This includes the pre-\ndiction of the existence of a Fermi surface neck which touches\nthe Brillouin zone at L, in agreement with the positron mea-\nsurements. DMFT includes electron correlations beyond that\napproximated within DFT by including the many-body lo-\ncal dynamical electron interactions [26]. These interactions\ncan prove vital in modeling strongly correlated systems where\nthese local electron correlations originating in the more local-\nized (e.g.dandf) orbitals, make a significant contribution\nand give rise to phases such as the Mott insulator [26, 27].\nFor ZrZn 2, the inclusion of the local electron correlations in\nthe partially occupied Zr 4d-states around the Fermi level has\nbeen shown to have a significant influence on the important\nfeatures of the electronic structure and Fermi surface topol-\nogy which is dominated by the flat bands around the L,W,\nandKpoints [21].\nIn this paper, we perform DFT and DFT+DMFT calcula-\ntions which are compared against the positron annihilation\ndata which was previously measured by Major et al. [12]. We\ncompare these theoretical predictions with this experimental\ndata to highlight the importance of electron correlations in-\ncluded in DMFT on the Fermi surface topology. We find that\nthese sheets are mainly influenced by the changes to the flat\nbands around L,W, andKlocated around the Fermi level in-\ndicating the significant impact the local electron correlations\nhave on the macroscopic properties, considering that it is these\nflat bands which influence the Lifshitz transition in this mate-\nrial [15].II. METHODS\nA. DFT+DMFT Calculations\nIn this study, we have used the full potential APW+lo ELK\ncode [28] in combination with the toolbox for research on\ninteracting quantum systems (TRIQS) library [29]. This so-\ncalled ELK+TRIQS DFT+DMFT framework is described in\nRef. [30] together with further discussion of interfacing the\nAPW+lo DFT basis with the DMFT Anderson’s impurity\nmodel found in Ref. [31]. The ZrZn 2DFT calculation was\nconverged on a 32\u000232\u000232Monkhorst-Pack k-mesh of 897\nirreduciblek-points in the first Brillouin zone. The Perdew-\nBurke-Ernzerhof (PBE) generalized gradient approximation\n(GGA) functional [32] was used to approximate the exchange-\ncorrelation interactions within the DFT framework. The DFT\noutputs were interfaced to the TRIQS library by constructing\nWannier projectors, as described in Ref. [30], for all the Zr\n4d-states within a correlated energy window of [\u00008;10]eV\nwhich itself spans the occupied Zn 3dand partially filled Zr\n4dbands; this energy window captures the equivalent states\nused in Ref. [21]. The paramagnetic DMFT calculation was\nimplemented using the continuous time quantum Monte Carlo\n(CT-QMC) solver within the TRIQS/CTHYB application [33]\nwith the Slater interaction Hamiltonian parameterized by the\nHubbard interaction U= 2:5eV and Hund exchange interac-\ntionJ= 0:3eV , as used in Ref. [21]. We approximated the\ndouble counting in the fully localized limit, as implemented\nin Ref. [21]. We used the fully-charge self-consistent (FCSC)\nDFT+DMFT method with an inverse temperature \f=40\neV\u00001(\u0018290 K) and 8:96\u0002108Monte Carlo sweeps for\neach cycle of the impurity solver. The spectral functions\nwere calculated by analytically continuing the DMFT self-\nenergy using the LineFitAnalyzer technique of the maximum\nentropy analytic continuation method implemented within the\nTRIQS/Maxent application [34].\nB. DFT+DMFT Two Photon Momentum Density\nImplementation\nThe two dimensional angular correlation of annihilation ra-\ndiation (2D-ACAR) experiments used positrons from a22Na\nradioisotope source, as described by Major et al. [12]. A\nthermal positron will annihilate with an electron within the\nsample which subsequently results in the emission of two \r-\nphotons which in their rest frame will be directed antipar-\nallel to each other to conserve momentum. In the labora-\ntory frame, however, the \r-photons deviate from being anti-\nparallel mainly due to the momentum of the annihilated elec-\ntron, whose momentum is typically significantly larger than\nthat of the thermalized positron. Since the angular deviation\nis proportional to momentum, the measured deviation gives a\ndirect insight into the electron momentum distribution. The\n\r-photons are recorded by two position-sensitive detectors lo-\ncated either side of the sample chamber, giving two momen-\ntum components. The third component of momentum, which\nis expressed as a Doppler shift in the photon energies, can-3\nnot be measured with the same resolution. Thus the measured\nspectra is a projection of the TPMD along the axis through the\nsample chamber connecting the centre of the two detectors.\nThe connection between the (real space) wave functions\nand the TPMD can be simply understood via the Fourier trans-\nformation [35] (assuming spin degeneracy) as\n\u001a2\r(p) =X\nk;\u0011nk;\u0011\f\f\f\f\f\fZ\nVp\n\r(r) e\nk;\u0011(r) p(r) exp(\u0000ip\u0001r)dr\f\f\f\f\f\f2\n=X\n\u0011nk;\u0011jCk+G;\u0011j2;\n(1)\nwhere p(r)is the positron wave function and  e\nk;\u0011(r)is\nthe Bloch electron wave function with eigenstate index \u0011at\npointkin thek-mesh used. The \r(r)quantity is the electron-\npositron enhancement factor which is designed to describe the\nelectron-positron correlations. If this factor is assumed to be\na constant, then this is referred to as the independent parti-\ncle model. However, different approximations can be adopted\nto improve agreement between experiment, such as the Drum-\nmond enhancement [36, 37] which was used in this study. The\nlast line of Eq. 1 introduces Ck+G;\u0011which is a shorthand for\nthe Fourier coefficients of the combined enhanced electron-\npositron wave function product.\nTo calculate the DFT+DMFT TPMD, we followed a simi-\nlar process as that used to determine the DFT+DMFT electron\nmomentum density (EMD) as described by James et al. [38]\nwhich itself is based on the DFT EMD computation methodol-\nogy implemented by Ernsting et al. [39]. A new set of electron\nwave functions and occupation numbers are determined by\ndiagonalizing the total DFT+DMFT density matrix, derived\nfrom the DFT+DMFT lattice Green’s function by summing\nover the Matsubara frequencies, and which is generally non-\ndiagonal in the Kohn-Sham basis. Further details can be found\nin Refs. [30, 38]. The resulting orthonormal Lowdin-type ba-\nsis electron wave functions and diagonal occupation numbers\ncan be then directly used in Eq. 1 as it is only the electron wave\nfunctions which have been updated to include the many-body\nelectron-electron interactions. Both approximations used to\ngenerate\r(r)and p(r)are unchanged with respect to the\nDFT calculation, although these quantities will likely change\nbetween the DFT and DFT+DMFT calculations as they are\ndependent on the electron density which is updated in each\nFCSC DFT+DMFT cycle. It should be noted that this method\nbenefits from not requiring any analytical continuation of the\nDFT+DMFT quantities. For the DFT and DFT+DMFT TP-\nMDs calculations here, we used a denser Monkhorst-Pack k-\nmesh of 60\u000260\u000260which equates to 5216 irreducible k-\npoints in the first Brillouin zone, and a maximum momentum\ncut-off of 8.5 a.u. at which the TPMD has converged to zero.\nC. The Lock-Crisp-West Theorem\nAlthough the signatures of Fermi surfaces are of course\npresent in the ‘real momentum’-space ( p-space) TPMD, de-veloping a full understanding of the Fermi surfaces through a\ndirect analysis of the TPMD is not easy due to the absence\nof (discrete) translational invariance. However, the TPMD\ninp-space, can be transformed into crystal momentum space\n(k-space) by translating contributions to the TPMD back into\nthe first Brillouin zone through the reciprocal lattice vectors,\nG. This new reduced-TPMD (or RTPMD) can be thought\nof as the occupation number for the electrons as seen by the\npositron for eachk-point summed over the bands. In such a\ndistribution, the Fermi surfaces are now much clearer. The\ntransformation of TPMD to the RTPMD (i.e., from p-space to\nk-space) is based on the so-called Lock-Crisp-West theorem\n(LCW) [40]. As discussed above, and with reference to Eq. 1,\none can directly express the RTPMD, \u001a2\r(k), by summing\nover the reciprocal lattice vectors Gby using\n\u001a2\r(k) =X\n\u0011X\nGnk;\u0011\u000e(p\u0000(G+k))jCk+G;\u0011j2:(2)\nWhile the summation over the reciprocal vectors Gin Eq. 2 is\nin principle infinite, in practice this can be truncated at some\nmomentum cut-off since the TPMD rapidly goes to zero. This\nis because the wave function of the positively charged positron\ndoesn’t overlap much with the most tightly bound (core) elec-\ntrons which would contribute at the highest momentum (lo-\ncalization in real space, means delocalization in momentum\nspace). The delta function ensures conservation of momen-\ntum.\nThe Fermi surface can be identified as the boundary be-\ntween occupied and unoccupied states in k-space. In practice,\nthe locations of the Fermi surfaces are determined by first\nfinding the maximum in the magnitude of the derivatives of\nthe RTPMD. The Fermi surfaces are then contoured in the\nRTPMD at the corresponding isovalues (evaluated at those\nmaxima). In order to facilitate the comparison between ex-\nperiment and calculation (including evaluation on particular\nplanes and along particular directions), and with the benefits\nof better computational resources which permit this, the to-\nmographic reconstruction of the 3D density from the decon-\nvoluted measured projections has been repeated [12, 22, 41–\n43].\nIn the reconstructed experimental RTPMD described\nabove, the related experimental statistical noise \u001b[\u001a(k)]was\nquantified following by that implemented by Samsel et al. [43]\nwithink-space. A set of Mindependent reconstructions were\nperformed on theoretical TPMDs to which (different) random\nnoise was added. The standard error \u001bin\u001a(k)at a particular\nk-point was evaluated as:\n\u001b[\u001a(k)] =\"\n1\nMMX\ni=1[\u001ai(k)\u0000\u001a(k)]2#1\n2\n; (3)\nwhere\u001ai(k)is theithreconstruction and \u001a(k)is the aver-\nage reconstruction. We subsequently used \u001b[\u001a(k)]to rep-\nresent the experimental uncertainty in the usual chi-squared\n‘goodness-of-fit’ parameter to help quantify how well the\nDFT and DFT+DMFT agree with the experimental data.4\nFIG. 1. Electronic structure of paramagnetic ZrZn 2revealed through\nthe band structure from DFT (blue solid lines) and the DFT+DMFT\nspectral functions (shown as a color map). To illustrate the spec-\ntral functions clearly, the logarithm ( log (A(k; !))) is plotted. Note\nthe differences between DFT and DMFT, particularly along the X—\nW—KandW—Lsegments, where the DMFT spectral weight\ngrazes the Fermi level.\nIII. RESULTS\nA. Spectral Function\nFigure 1 shows the DFT bandstructure overlayed onto the\nlogarithm of the DFT+DMFT spectral function. The inclu-\nsion of electron correlations from DMFT clearly introduces\nbroadening to the spectral function, as expected, along with\nvisible changes to the positions of the band centers. The most\nsignificant change in the DFT+DMFT spectral function with\nrespect to the DFT bandstucture is in the bands along the X—\nW—KandW—L. Along these paths it can be seen that the\nflat bands cross the Fermi level at different points in the Bril-\nlouin zone. The bands which dispersed just above the Fermi\nlevel (just grazing it) along these paths in DFT have now, with\nthe inclusion of the electron correlations in the DMFT, been\nshifted downwards so that they cross the Fermi level, resulting\nin the emergence of new features in the Fermi surface. This\nis in good agreement with Ref. [21] with a few exceptions\ndescribed below. The logarithm of DFT+DMFT k-resolved\nspectral function in Fig. 1 shows weakly k-dependent inco-\nherent weight around \u0018–0.6 and 0.8 eV and the DFT band\njust above the Fermi level at Wis shifted down in energy in\nthe DFT+DMFT calculation so that this grazing band which\nin DFT was just above the Fermi level along W—Kis now\nlying partially below the Fermi level in the DFT+DMFT cal-\nculation. We expect this to be the consequence of the interac-\ntion Hamiltonian as well as the lower temperature used in the\nDMFT part of our calculations. In our calculation, the full in-\nteraction Hamiltonian has been used which includes the spin-\nflip and pair-hopping interactions which the density-density\ninteraction Hamiltonian used in Ref. [21] omits. We note that\nthe magnitude of the imaginary part of the real frequency-\nFIG. 2. Contour of spectral functions at !=EFon (a) the plane\nwithkz= 0 centered at \u0000point and (b) the plane that perpendicu-\nlar to reciprocal vector (111) and centered at L. The upper panel of\neach figure is computed from the DFT calculation. The colormap of\nthe upper panel is the DFT spectral functions calculated with a small\nartificial imaginary energy of 0.01 eV . The red dashed lines are the\nDFT Fermi surfaces on those planes. The colormaps of the bottom\nhalf of each figure are computed from the DFT+DMFT calculation.\nThe location of each plane is schematically shown at the top left cor-\nner of the sub-figure.\ndependent self-energy at != 0 is small so that the predicted\nelectronic structure does not deviate too much from the Fermi-\nliquid regime at the temperature used for the DMFT cycle; this\nis similar to the DFT+DMFT results in Ref. [21].\nThe DFT and DFT+DMFT spectral functions in Fig. 2 (a)\nand (b) are evaluated at the Fermi level and calculated on\nthe two dimensional (2D) planes shown by the correspond-\ning insets of Fig. 2 (a) and (b). When both Figs. 2 (a) and\n(b) are taken in conjunction with the DFT band structure and\nDFT+DMFT spectral function in Fig. 1, we see that there are\nclear changes going from DFT to DFT+DMFT in the spec-5\nFIG. 3. Slices through the RTPMD calculation on (a-c) the (001) plane through \u0000and (d-f) the (111) plane through L(see Fig. 2) from the\nDFT calculation (a and d), the DFT+DMFT calculation (b and e) and the experiment (c and f). The colorbar shows the normalized RTPMD\nfor each subfigure.\ntral weight present at the Fermi level which here is evidence\nof the changes in the Fermi surface topology within these\nplanes. Most of the significant changes in the 2D spectral\nfunctions occur in the regions around W,K, andL, where the\npreviously unoccupied DFT flat bands just above the Fermi\nlevel relating to the van Hove singularity have changed in\nDFT+DMFT, now crossing the Fermi level as can be seen in\nFig. 1 along W—LandW—K. Both Figs. 2 (a) and (b)\nshow the appearance of DFT+DMFT spectral weight at W\nwhich indicates that there is now a portion of the correspond-\ning Fermi surface sheet there. In the DFT+DMFT 2D spectral\nfunction shown in Fig. 2 (b), there now exists additional spec-\ntral weight around L(with respect to the DFT 2D spectral\nfunction) which relates to the formation of a large neck in the\nFermi surface topology, in agreement with Ref. [21]. Note\nthat in both Figs. 2 (a) and (b) the DFT Fermi surface (i.e.,\nthe Fermi surface from the DFT energy eigenvalues) in those\nplanes is plotted on top of the DFT 2D spectral function. In\nFig. 2 (b), there is indeed a very small DFT Fermi surface at L\nwhich is in fact a small hole pocket. It is only just visible and\nappears as a dot, being smaller than that seen in Ref. [19]; this\nsmall pocket can also be inferred from the DFT band structure\nnearLalong the path of W—Lin Fig. 1. Interestingly, this\nFermi surface sheet appears to be quite sensitive to how the\nelectron correlations are modeled considering that the Fermi\nsurface topology changes from a hole pocket at Lin DFT to\nan electron neck enclosing Lin the DFT+DMFT results.B. Reduced Two-Photon Momentum Density\nThe Fermi surface can be identified by the loci of wavevec-\ntors which correspond to sharp, discontinuous changes in oc-\ncupation within the Brillouin zone for a non-interacting sys-\ntem at zero temperature. In practice the Fermi surface can\nbe loci of the wave-vectors at which the absolute value of the\ngradient of the occupation function is maximum. Of course,\nat finite temperature and in the presence of electron correla-\ntions, the corresponding change in electron occupation will\nbe smeared. However, the Fermi surface can still be identified.\nHere, we focus our attention on the calculated RTPMDs which\ncan be directly compared to the corresponding experimen-\ntal RTPMDs derived from the data first reported by Major et\nal.[12]. Before inspecting the specific changes in the RTPMD\nbetween DFT and DFT+DMFT, we note that the irreducible\nBrillouin zone chi-squared value from DFT+DMFT is very\nslightly improved (7.4) compared to the corresponding DFT\nvalue of 8.1, although both indicate that there are still signifi-\ncant differences. However, this shows that the inclusion of the\nlocal electron correlations from DMFT are needed for param-\nagnetic ZrZn 2. Figure 3 shows the RTPMD evaluated on the\nsame two 2D planes shown in the insets of Figs. 2 (a) and (b).\nAs one would expect, the changes which occur in the RTPMD\nbetween DFT and DFT+DMFT reflect those seen in the spec-\ntral functions on those 2D planes in Fig. 2. The DFT+DMFT\nRTPMDs are more smeared than the corresponding DFT ones,6\nFIG. 4. The RTPMD plotted along the same high symmetry path that was used in the band structure plot (Fig. 1). The normalized RTPMD\nare shown with solid lines guiding the eye between the data point which are shown as circles. The experimental errors were determined as\ndescribed by Eq. 3 in the main text. The theoretical RTPMDs used a temperature of \u0018290 K, and the inset shows the temperature dependence\nof the DFT RTPMDs along the high symmetry paths where there are the flat bands around the Fermi level.\nFIG. 5. The RTPMD of ZrZn 2along the \u0000—Xdirection for (a)\nthe DFT calculation, (b) the DFT+DMFT calculation, and (c) ex-\nperiment. The experimental errors along \u0000—Xwere determined as\ndescribed by Eq. 3 in the main text. The solid lines show the TPMD\nwhile the absolute value of the gradient is shown with dashed lines.\nThe numbers (27—30) show the corresponding band indices, as de-\nfined in Ref. [12], as discussed in the main text.\nowing to the additional electron correlations. The most note-worthy changes are seen around Win both planes. From\nthe RTPMD along X—Win Figs. 3 (a-c), we see the con-\nsequence of the DFT+DMFT quasiparticle band center cross-\ning the Fermi level around Win the \u0000centered plane. Here,\nwe see that the DFT+DMFT RTPMD is in better agreement\nwith the experimental one at W, which is also clearly seen\nin Fig. 4 in which the RTPMD is plotted along the same\nhigh symmetry path as the band structure in Fig. 1. Even\nthough the DFT+DMFT RTPMD is in very good agreement\nwith the experimental one at W, it would appear that the lo-\ncation of the Fermi surface associated with the quasiparticle\nband needs to be further away from WalongX—Win order\nto agree with the experiment along X—W. Consequently,\nthis would therefore likely cause the global minimum of the\nRTPMD to be at Xas in the experiment instead of that pre-\ndicted in the theoretical RTPMDs. It should be noted that the\ndistribution from X—Wis also sensitive to the positron wave\nfunction as it differs to the ‘pure’ electron occupations along\nthis path, but is relatively insensitive to the electron-positron\nenhancement model used. On the other hand, looking at the\nRTPMD around Win theLcentered plane in Figs. 3 (d-f),\nagain the DFT+DMFT RTPMD better matches the experi-\nmental RTPMD from the 2D-ACAR experiment [12] which\nmeans that the quasiparticle flat band around the Fermi level\nfromW—K(see Fig. 1) does indeed cross the Fermi level.\nHowever, both the DFT and DFT+DMFT predicts a suppres-\nsion of the RTPMD (i.e., the darker regions of Figs. 3 (d) and\n(e)) around the edge of the first Brillouin zone, which is not\nseen in the experiment. These darker regions relate to the pres-\nence of the pillow Fermi surface sheet intersecting this plane7\nFIG. 6. The four Fermi surface sheets of paramagnetic ZrZn 2derived from the RTPMD at the maximum values of the gradient of the RTPMD\nfor DFT (left), DFT+DMFT (middle), and experiment (right). The band indices (27—30) refer to those in Ref. [12], as discussed in the main\ntext.8\nwhen in reality, this is not the case, as seen in the experimental\ndata in Fig. 3 (f); this means that the pillow sheet is smaller\nin size. Regarding the distribution along W—Kshown in\nFig. 4, the DFT RTPMD has a similar shape to the experimen-\ntal one although the magnitude is smaller. On the contrary,\nthe DFT+DMFT RTPMD differs by reducing in magnitude\nfromW—Kwhich leads to the presence of the pillow Fermi\nsurface sheet crossing here. Although the magnitude of the\nRTPMD from DFT+DMFT better agrees with the experimen-\ntal one than that of DFT along W—K, the discrepancy be-\ntween DFT+DMFT and experiment increases along this path\ntoK. This suggests that the quasiparticle flat band around\nthe Fermi level needs to be fully occupied along the path of\nW—Kwhich the experimental RTPMD seems to suggest to\nbe the case. It should be noted that there is an additional con-\ntribution to the RTPMD distribution from W—Kfrom the\npositron wave functions like that seen along X—W. Also,\nthere is a small bump in both theoretical RTPMDs at approx-\nimately midway between W—Kwhich is interestingly due\nto temperature smearing of the electron occupation, as can be\nseen in the temperature dependence of the DFT RTPMD dis-\ntributions shown in the inset of Fig. 4. For the DFT+DMFT\nRTPMD, there is an additional smearing contribution from the\nelectron correlations included in the DMFT.\nIt can be seen in Fig. 4 that the DFT+DMFT RTPMD is in\nbetter overall agreement with the experimental compared to\nthe DFT. Note that the chi-squared calculated along this path\nfor the DFT+DMFT calculation is half that of the DFT calcu-\nlation. This is mostly likely because the changes in the Fermi\nsurface are concentrated around the face of the Brillouin zone,\nmaking it particularly sensitive to changes along a path which\nhas segments on the boundary. Both theoretical RTPMDs pre-\ndict a dip at Lwith the DFT+DMFT being shallower than the\nDFT. Interestingly, the dip at Lin the DFT+DMFT is not as-\nsociated with a Fermi surface hole pocket as one might ex-\npect. Indeed, we see an upward trend in the RTPMD along\nW—Ljust before the dip in both the DFT and DFT+DMFT.\nThis upward trend is due to the temperature smearing of the\nRTPMD, as shown in the inset of Fig. 4, from the DFT band\n(DFT+DMFT quasiparticle band) close to the Fermi level as\nwell as there being an additional smearing contribution from\nthe electron correlations in the DFT+DMFT RTPMD. The dip\natLin the DFT+DMFT RTPMD is likely due to the spec-\ntral weight ‘spilling’ across the Fermi level and as such this\nspillage would consequently reduce the contribution to the\nRTPMD from the degenerate quasiparticle bands at L(see\nFig. 1). This spectral weight effect on the electron occupa-\ntion density has been observed before in momentum densi-\nties [44]. Reducing the temperature will likely suppress the\nhump in the DFT+DMFT just before the dip at LalongW—\nL, similar to that seen in the DFT RTPMD temperature re-\nsults shown in the inset of Fig. 4. Unfortunately, lower tem-\nperature DFT+DMFT calculations are more computationally\nexpensive and may suffer from the sign problem [45], and as\nsuch the DFT+DMFT RTPMD temperature dependence has\nnot been investigated here.\nThere are also notable changes in the DFT+DMFT RTPMD\nalong the other paths in Fig. 4 which brings it closer in agree-ment with the experiment. We highlight the RTPMDs along\n\u0000—Xin Fig. 5 which nicely shows that there is additional\nsmearing in the DFT+DMFT RTPMD with respect to the DFT\nwhich is consistent with effect of the additional local many-\nbody correlations. Also in Fig. 5, we show the gradients of the\nRTPMDs and their maximums agree well with the Fermi sur-\nface locations (represent by the vertical dashes between \u0000—X\nin Fig. 5) as predicted by the spectral functions (or eigenval-\nues in the DFT) within the resolution of the k-mesh. The high-\nlighted band numbers in Fig. 5 correspond to the band nomen-\nclature of Ref. [12]. Here, the DFT+DMFT is in closer agree-\nment with the experiment for bands 27, 28, and 29, but there\nis still a notable discrepancy with the experimental Fermi sur-\nface. On the other hand, the band 30 DFT+DMFT electron\nFermi surface sheet size is slightly worse in agreement with\nthe corresponding smaller experimental sheet compared to the\nDFT. Even with this being the case, the overall agreement with\nthe experimental improves with the inclusion of DMFT.\nFinally, we show each theoretical and experimental Fermi\nsurface sheet extracted from the RTPMDs in Fig. 6. Here,\nall the previously discussed features and changes can be seen\nin each three dimensional Fermi surface sheet. We note that\ncertain regions in the calculated Fermi surface sheets have dis-\ntinct bumpy features on them which is the consequence of the\nk-position of the relatively sharp steps in the RTPMD. This\nwould likely be resolved for a denser k-mesh which would\nproduce a better mesh resolution but this would be more com-\nputationally demanding and have a greater k-mesh resolution\nthan that possible for the experimental data. The theoretical\nband 30 Fermi surface sheet from DFT and DFT+DMFT is\nclearly too large with respect to the experiment. The emer-\ngence of the Fermi surface neck at Lwhich is associated to\nband 29 is observed for the DFT+DMFT and experiments, al-\nthough the experimental neck is smaller than the DFT+DMFT\none. The DFT, on the other hand, predicts no neck and the\ntiny hole Fermi surface pocket sheet seen previously is barely\nvisible here. The topology of the Fermi surface for band 28\nis very similar between the theories and experiment, but the\nexperimental Fermi surface is slightly smaller. Lastly, the\nband 27 ‘pillow’ Fermi sheet at Win DFT are connected\nwhereas in the DFT+DMFT, these pillow Fermi surface sheets\nare disconnected which agrees with the experimental sheets,\nhowever, these experimental ‘pillow’ sheets are smaller than\nthe predicted ones. This reflects the RTPMD along W—K\nshown in both Figs. 3 and 4 along with the previous discus-\nsions about them. Owing to the improved resources available\nfor the new reconstruction, the experimental pillow sheets are\nin fact smaller in size, disconnected, and located around the\nXcentered plane at the Brillouin zone boundary. This dis-\nconnecting part at Wbetween each of the experimental pillow\nFermi surface sheets is highly sensitive to the chosen RTPMD\nFermi surface isovalue, which likely indicates that the band is\nonly just below the Fermi surface. It is clear that these Fermi\nsurface sheets are sensitive to the flat bands around the Fermi\nlevel nearL,W, andKwhich themselves are sensitive to the\nelectron correlation description. Clearly the positron annihi-\nlation data is sensitive to the presence of the many-body elec-\ntron correlations which are not smeared out by the electron-9\npositron correlation contributions.\nIV . CONCLUSIONS\nIn describing the electronic structure of ZrZn 2, the inclu-\nsion of the local electron correlations from DMFT are vital\nas they vastly improve the Fermi surface topology resulting\nin superior agreement with the 2D-ACAR experimental data\n[12]. In order to analyse and compare the features of the Fermi\nsurface from both DFT and DFT+DMFT, we used both the\n2D spectral function (and DFT 2D Fermi surface) and the\nRTPMDs which are in excellent agreement with each other\nas expected. Indeed, the inclusion of DMFT captures certain\nfeatures of the Fermi surface topology, such as the band 29\nFermi surface sheet neck around Lwhich is in fact a hole\npocket in DFT. It is this neck feature which relates to the pre-\nviously noted van Hove singularity at L. With the improved\nagreement from the DFT+DMFT calculations it appears that\nthis feature is dependent on the local electron correlations\nof the Zr 4d electrons. The Fermi surface sheets derived\nfrom the RTPMD show an overall improved agreement with\nthe experimental for the DFT+DMFT compared to the DFT.Along theW—Kpath in the Brillouin zone, the DFT+DMFT\nRTPMD has better agreement with the experimental one, re-\nsulting in the ‘pillow’ Fermi surface sheet to become discon-\nnected which is also seen in the experimental data. Clearly,\nthe flat bands around L,W, andKare sensitive to the de-\nscription of the many-body electron correlations which con-\nsequently affects the Fermi surface topology. There is also a\nclear temperature dependence in the calculated RTPMDs ow-\ning to the temperature smearing of the occupation function of\nthe flat bands in close vicinity to the Fermi level near these\nhigh symmetry points. Finally, the positron annihilation tech-\nnique proves to be a powerful tool to probe the many-body\nelectron correlation effects considering that these are not ob-\nscured by the electron-positron correlation contributions.\nV . ACKNOWLEDGEMENTS\nWenhan Chen acknowledges the funding and support\nfrom the Chinese Scholarship Council (CSC), Grant No.\n201908060087. A. D. N. J. also acknowledges the Doctoral\nPrize Fellow funding and support from the Engineering and\nPhysical Sciences Research Council (EPSRC).\n[1] J. Santiago, C. Huang, and E. Morosan, Itinerant magnetic met-\nals, Journal of Physics: Condensed Matter 29, 373002 (2017).\n[2] E. A. Yelland, S. J. C. Yates, O. Taylor, A. Griffiths, S. M.\nHayden, and A. Carrington, Ferromagnetic properties of ZrZn 2,\nPhys. Rev. B 72, 184436 (2005).\n[3] F. Grosche, C. Pfleiderer, G. McMullan, G. Lonzarich, and\nN. Bernhoeft, Critical behaviour of ZrZn 2, Physica B: Con-\ndensed Matter 206-207 , 20 (1995).\n[4] N. Kabeya, H. Maekawa, K. Deguchi, N. Kimura, H. Aoki, and\nN. K. Sato, Phase diagram of the itinerant-electron ferromagnet\nZrZn 2, physica status solidi (b) 250, 654 (2013).\n[5] R. P. Smith, M. Sutherland, G. G. Lonzarich, S. S. Saxena,\nN. Kimura, S. Takashima, M. Nohara, and H. Takagi, Marginal\nbreakdown of the fermi-liquid state on the border of metallic\nferromagnetism, Nature 455, 1220 (2008).\n[6] C. Pfleiderer, M. Uhlarz, S. M. Hayden, R. V ollmer, H. v.\nL¨ohneysen, N. R. Bernhoeft, and G. G. Lonzarich, Erratum:\nCoexistence of superconductivity and ferromagnetism in the d-\nband metal ZrZn 2, Nature 412, 660 (2001).\n[7] V . Ginzburg, Ferromagnetic superconductors, Journal of Exper-\nimental and Theoretical Physics 4, 153 (1957).\n[8] D. Fay and J. Appel, Coexistence of p-state superconductivity\nand itinerant ferromagnetism, Phys. Rev. B 22, 3173 (1980).\n[9] A. Larkin and Y . N. Ovchinnikov, Quasiclassical method in the\ntheory of superconductivity, Sov Phys JETP 28, 1200 (1969).\n[10] P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-\nexchange field, Phys. Rev. 135, A550 (1964).\n[11] D. J. Singh and I. I. Mazin, Competition of spin fluctuations\nand phonons in superconductivity of ZrZn 2, Phys. Rev. Lett.\n88, 187004 (2002).\n[12] Z. Major, S. B. Dugdale, R. J. Watts, G. Santi, M. A. Alam,\nS. M. Hayden, J. A. Duffy, J. W. Taylor, T. Jarlborg, E. Bruno,\nD. Benea, and H. Ebert, Direct observation of the multisheet\nfermi surface in the strongly correlated transition metal com-pound ZrZn 2, Phys. Rev. Lett. 92, 107003 (2004).\n[13] B. Powell, J. F. Annett, and B. Gy ¨orffy, Competition between\ndisorder and exchange splitting in superconducting ZrZn 2,\nJournal of Physics: Condensed Matter 15, L235 (2003).\n[14] E. A. Yelland, S. M. Hayden, S. J. C. Yates, C. Pfleiderer,\nM. Uhlarz, R. V ollmer, H. v. L ¨ohneysen, N. R. Bernhoeft, R. P.\nSmith, S. S. Saxena, and N. Kimura, Superconductivity induced\nby spark erosion in ZrZn 2, Phys. Rev. B 72, 214523 (2005).\n[15] Y . Yamaji, T. Misawa, and M. Imada, Quantum metamagnetic\ntransitions induced by changes in fermi-surface topology: Ap-\nplications to a weak itinerant-electron ferromagnet ZrZn 2, Jour-\nnal of the Physical Society of Japan 76, 063702 (2007).\n[16] N. Kabeya, H. Maekawa, K. Deguchi, N. Kimura, H. Aoki, and\nN. K. Sato, Non-fermi liquid state bounded by a possible elec-\ntronic topological transition in ZrZn 2, Journal of the Physical\nSociety of Japan 81, 073706 (2012).\n[17] M. Sutherland, R. P. Smith, N. Marcano, Y . Zou, S. E. Row-\nley, F. M. Grosche, N. Kimura, S. M. Hayden, S. Takashima,\nM. Nohara, and H. Takagi, Transport and thermodynamic evi-\ndence for a marginal fermi-liquid state in ZrZn 2, Phys. Rev. B\n85, 035118 (2012).\n[18] P. Igoshev and V . Irkhin, Giant density-of-states van hove sin-\ngularities in the face-centered cubic lattice, Physics Letters A\n438, 128107 (2022).\n[19] M.-c. Huang, H. J. F. Jansen, and A. J. Freeman, Self-consistent\nfull-potential linearized-augmented-plane-wave local-density\nelectronic-structure studies of magnetism and superconductiv-\nity in c15 compounds: ZrZn 2and ZrV 2, Phys. Rev. B 37, 3489\n(1988).\n[20] S. J. C. Yates, G. Santi, S. M. Hayden, P. J. Meeson, and S. B.\nDugdale, Heavy quasiparticles in the ferromagnetic supercon-\nductor ZrZn 2, Phys. Rev. Lett. 90, 057003 (2003).\n[21] S. L. Skornyakov, V . S. Protsenko, V . I. Anisimov, and A. A.\nKatanin, Effect of electronic correlations on the spectral and10\nmagnetic properties of ZrZn 2, Phys. Rev. B 102, 085101\n(2020).\n[22] G. Kontrym-Sznajd, Three-dimensional image reconstruction\nwith application in positron annihilation, physica status solidi\n(a)117, 227 (1990).\n[23] G. Kontrym-Sznajd and H. Sormann, Many-body effects ob-\nserved in the positron annihilation experiment, Acta Physica\nPolonica A 107, 562 (2005).\n[24] H. A. Ceeh, J. A. Weber, P. B ¨oni, M. Leitner, D. Benea,\nL. Chioncel, H. Ebert, J. Min ´ar, D. V ollhardt, and C. Hugen-\nschmidt, Local electron-electron interaction strength in ferro-\nmagnetic nickel determined by spin-polarized positron annihi-\nlation, Scientific Reports 6, 20898 (2016).\n[25] J. A. Weber, D. Benea, W. H. Appelt, H. Ceeh, W. Kreuzpaint-\nner, M. Leitner, D. V ollhardt, C. Hugenschmidt, and L. Chion-\ncel, Electronic correlations in vanadium revealed by electron-\npositron annihilation measurements, Phys. Rev. B 95, 075119\n(2017).\n[26] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dy-\nnamical mean-field theory of strongly correlated fermion sys-\ntems and the limit of infinite dimensions, Rev. Mod. Phys. 68,\n13 (1996).\n[27] K. Held, Electronic structure calculations using dynamical\nmean field theory, Advances in physics 56, 829 (2007).\n[28] J. K. Dewhurst, S. Sharma, L. Nordstr ¨om, F. Cricchio,\nO. Granas, and E. K. U. Gross, The Elk Code (version 6.2.8),\nhttp://elk.sourceforge.net/.\n[29] O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko,\nL. Messio, and P. Seth, TRIQS: A toolbox for research on in-\nteracting quantum systems, Comp. Phys. Commun. 196, 398\n(2015).\n[30] A. D. N. James, E. I. Harris-Lee, A. Hampel, M. Aichhorn,\nand S. B. Dugdale, Wave functions, electronic localization, and\nbonding properties for correlated materials beyond the Kohn-\nSham formalism, Phys. Rev. B 103, 035106 (2021).\n[31] M. Aichhorn, L. Pourovskii, V . Vildosola, M. Ferrero, O. Par-\ncollet, T. Miyake, A. Georges, and S. Biermann, Dynamical\nmean-field theory within an augmented plane-wave framework:\nAssessing electronic correlations in the iron pnictide LaFeAsO,\nPhys. Rev. B 80, 085101 (2009).\n[32] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient\napproximation made simple, Phys. Rev. Lett. 77, 3865 (1996).\n[33] P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet,\nTRIQS/CTHYB: A continuous-time quantum Monte Carlo hy-\nbridisation expansion solver for quantum impurity problems,\nComp. Phys. Commun. 200, 274 (2016).[34] G. J. Kraberger, R. Triebl, M. Zingl, and M. Aichhorn, Maxi-\nmum entropy formalism for the analytic continuation of matrix-\nvalued green’s functions, Phys. Rev. B 96, 155128 (2017).\n[35] J. Rusz and M. Biasini, Nature of f-electrons in CeIn 3: Theo-\nretical analysis of positron annihilation data, Phys. Rev. B 71,\n233103 (2005).\n[36] N. D. Drummond, P. L ´opez R ´ıos, C. J. Pickard, and R. J.\nNeeds, First-principles method for impurities in quantum fluids:\nPositron in an electron gas, Phys. Rev. B 82, 035107 (2010).\n[37] N. D. Drummond, P. L ´opez R ´ıos, R. J. Needs, and C. J. Pickard,\nQuantum monte carlo study of a positron in an electron gas,\nPhys. Rev. Lett. 107, 207402 (2011).\n[38] A. D. N. James, M. Sekania, S. B. Dugdale, and L. Chioncel,\nMagnetic Compton profiles of Ni beyond the one-particle pic-\nture: Numerically exact and perturbative solvers of dynamical\nmean-field theory, Phys. Rev. B 103, 115144 (2021).\n[39] D. Ernsting, D. Billington, T. D. Haynes, T. E. Millichamp,\nJ. W. Taylor, J. A. Duffy, S. R. Giblin, J. K. Dewhurst, and\nS. B. Dugdale, Calculating electron momentum densities and\nCompton profiles using the linear tetrahedron method, Journal\nof Physics: Condensed Matter 26, 495501 (2014).\n[40] D. G. Lock, V . H. C. Crisp, and R. N. West, Positron annihi-\nlation and fermi surface studies: a new approach, Journal of\nPhysics F: Metal Physics 3, 561 (1973).\n[41] S. B. Dugdale, M. A. Alam, H. M. Fretwell, M. Biasini, and\nD. Wilson, Application of maximum entropy to extract fermi\nsurface topology from positron annihilation measurement, Jour-\nnal of Physics: Condensed Matter 6, L435 (1994).\n[42] H. M. Fretwell, S. B. Dugdale, M. A. Alam, M. Biasini,\nL. Hoffmann, and A. A. Manuel, Reconstruction of 3d electron-\npositron momentum densities from 2d projections: Role of\nmaximum-entropy deconvolution prior to reconstruction, Euro-\nphysics Letters (EPL) 32, 771 (1995).\n[43] M. Samsel-Czekała, G. Kontrym-Sznajd, G.and D ¨oring,\nW. Sch ¨ulke, J. Kwiatkowska, F. Maniawski, S. Kaprzyk, and\nA. Bansil, Electron momentum density in Cu 0:9Al0:1, Applied\nPhysics A 76, 87 (2003).\n[44] H. C. Robarts, T. E. Millichamp, D. A. Lagos, J. Laverock,\nD. Billington, J. A. Duffy, D. O’Neill, S. R. Giblin, J. W. Tay-\nlor, G. Kontrym-Sznajd, M. Samsel-Czekała, H. Bei, S. Mu,\nG. D. Samolyuk, G. M. Stocks, and S. B. Dugdale, Extreme\nfermi surface smearing in a maximally disordered concentrated\nsolid solution, Phys. Rev. Lett. 124, 046402 (2020).\n[45] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov,\nM. Troyer, and P. Werner, Continuous-time monte carlo meth-\nods for quantum impurity models, Rev. Mod. Phys. 83, 349\n(2011)."    },    {        "title": "1910.02619v1.Epitaxial_Growth_of_Perovskite_SrBiO__3__Film_on_SrTiO__3__by_Oxide_Molecular_Beam_Epitaxy.pdf",        "content": "Epitaxial Growth of Perovskite SrBiO 3Film on SrTiO 3by Oxide Molecular Beam\nEpitaxy\nFengmiao Li,1, 2,\u0003Bruce A. Davidson,1, 2Ronny Sutarto,3Hyungki Shin,1, 2Chong Liu,1, 2\nIlya El\fmov,1, 2Kateryna Foyevtsova,1, 2Feizhou He,3George A. Sawatzky,1, 2and Ke Zou1, 2\n1Department of Physics and Astronomy, University of British Columbia,\nVancouver, British Columbia, V6T 1Z1, Canada\n2Stewart Blusson Quantum Matter Institute, University of British Columbia,\nVancouver, British Columbia, V6T 1Z4, Canada\n3Canadian Light Source, Saskatoon, Saskatchewan S7N 2V3, Canada\nHole-doped perovskite bismuthates such as Ba 1\u0000xKxBiO 3and Sr 1\u0000xKxBiO 3are well-known\nbismuth-based oxide high-transition-temperature superconductors. Reported thin bismuthate \flms\nshow relatively low quality, likely due to their large lattice mismatch with the substrate and a low\nsticking coe\u000ecient of Bi at high temperatures. Here, we report the successful epitaxial thin \flm\ngrowth of the parent compound strontium bismuthate SrBiO 3on SrO-terminated SrTiO 3(001) sub-\nstrates by molecular beam epitaxy. Two di\u000berent growth methods, high-temperature co-deposition\nor recrystallization cycles of low-temperature deposition plus high-temperature annealing, are de-\nveloped to improve the epitaxial growth. SrBiO 3has a pseudocubic lattice constant \u00184.25 \u0017A, an\n\u00188.8% lattice mismatch on SrTiO 3substrate, leading to a large strain in the \frst few unit cells.\nFilms thicker than 6 unit cells prepared by both methods are fully relaxed to bulk lattice constant\nand have similar quality. Compared to high-temperature co-deposition, the recrystallization method\ncan produce higher quality 1-6 unit cell \flms that are coherently or partially strained. Photoemis-\nsion experiments reveal the bonding and antibonding states close to the Fermi level due to Bi and\nO hybridization, in good agreement with density functional theory calculations. This work provides\ngeneral guidance to the synthesis of high-quality perovskite bismuthate \flms.\nHole-doped perovskite bismuthates Ba 1\u0000xKxBiO 3(x\n= 0.4) and Sr 1\u0000xKxBiO 3(x = 0.6) become superconduct-\ning below \u001830 K and \u001812 K, respectively [1, 2]. Their\nparent compounds BaBiO 3(BBO) and SrBiO 3(SBO)\nhave drawn intense interest [3{16] owing to the high-\ntemperature superconductivity and also because of their\nindirect bandgap semiconductor nature rather than met-\nals as would be expected from the half-\flled Bi 6s band\nassuming the formal 4+ valence of Bi. The most common\nexplanation of the nonmetallic behavior invokes the con-\ncept of charge disproportionation of Bi4+into the more\nstable and closed-shell atomic con\fgurations of Bi3+(6s2)\nand Bi5+(6s0) [3, 4, 17]. This would inevitably result in\na long Bi3+-O and a short Bi5+-O bond lengths ordered\nin the simplest conceivable structure with bond length\nalternation along the three crystallographic axes. Such\nbond disproportionation is observed in neutron and x-ray\ndi\u000braction (XRD) [4, 11] and in the resulting band struc-\nture [6, 8]. However, x-ray photoemission spectroscopy\n(XPS) and density functional theory (DFT) calculations\nshow that the charge di\u000berence between the two inequiv-\nalent Bi sites is trivial, at most a few tenths of one elec-\ntron [8, 10, 18{21]. By DFT calculation, Foyevtsova et\nal.recently demonstrates that the bands straddling the\nFermi level are in fact strongly hybridized Bi 6s with O\n2p molecular orbitals with A 1gsymmetry and contain\nmore O 2p character than Bi 6s character, resulting in\nthe bonding states at \u001810 eV below the Fermi energy and\nthe unoccupied antibonding states with holes in O 2p just\nabove and electrons just below the Fermi level [8], serv-\ning as an alternate explanation for the semiconductingnature. The existence of empty O 2p states just above\nthe chemical potential forming the conduction band was\npreviously mentioned in experiment without reference to\nits molecular orbital and symmetry character [19, 21{\n23]. Further experimental studies are required to com-\npare with the recent DFT calculations. Uncovering the\nelectronic structure in the parent compounds would also\nfacilitate an understanding of the states that lead to su-\nperconductivity when doped with holes.\nBecause of its three-dimensional perovskite structure,\ncleaving of bulk SBO single-crystals does not yield use-\nfully large, \rat areas of known termination. High-quality\nsamples in thin \flm form are desirable not only because\nthey can facilitate the study of electronic structure using\nsurface science probes like photoemission spectroscopy,\nbut also can accommodate the realization of heterostruc-\ntures for \feld-e\u000bect doping without the disorder intro-\nduced by chemical dopants ( e.g.K). A number of reports\non the growth of undoped or hole-doped BBO \flms by\nmolecular beam epitaxy (MBE) and pulsed laser depo-\nsition have recently appeared [9, 10, 16, 24{29]. One\nchallenge is to \fnd perovskite substrates to match BBOs\nlarge pseudocubic lattice parameter \u00184.35 \u0017A. SBO has\na smaller pseudocubic lattice parameter \u00184.25 \u0017A re-\nsulting in a reasonable lattice match to some large\nlattice-parameter perovskite substrates such as BaSnO 3\n(\u00184.12 \u0017A) and LaLuO 3(\u00184.18 \u0017A). However, these two\nsubstrates are not commercially available yet. Film qual-\nity is also limited by the temperature-sensitive sticking\ncoe\u000ecient of Bi, leading to a small growth window to\nobtain cation stoichiometric \flms.arXiv:1910.02619v1  [cond-mat.supr-con]  7 Oct 20192\n300 nm(a) (b) (c) (d)\n(e)\n(f) (g)\n (f) \n-1.9 nm 1.8 nm400 nm \nFIG. 1. (a)-(d) [100] direction RHEED pattern of 2 AL SrO\n(a), 2 UC SBO (b), 4 UC SBO (c) and 20 UC SBO (d) grown\nby the high-temperature codeposition; (e) RHEED intensity\noscillations of the specular and (01) di\u000braction spot after 6 UC\nSBO \flm growth on SrO-terminated STO. (f) AFM image of\n20 UC SBO \flm surface; (g) XRD of the 20 UC SBO \flm\ngrown on STO substrate. \u0003and\u0005label SBO (001) and STO\n(001) di\u000braction, respectively.\nIn this paper, we report on the successful epitaxial\ngrowth of SBO \flm on SrTiO 3(STO) (001) substrate\nby oxide MBE. We have used two growth methods, high-\ntemperature co-deposition or recrystallization cycles of\nlow-temperature deposition plus high-temperature an-\nnealing, to yield \flms with correct stoichiometry, high\ncrystalline quality, and atomically \rat surfaces. Further,\nthe electronic structure of fully relaxed SBO \flms is stud-\nied using angular integrated photoemission spectroscopy\nand it agrees well with the density of states (DOS) in\nDFT.\nThe \flms are grown in an oxide MBE chamber with\na base pressure better than 1.0 \u000210\u000010Torr. During\ngrowth, the \flm surface is monitored by re\rection high-\nenergy electron di\u000braction (RHEED). The Bi and Sr are\nevaporated using low-temperature e\u000busion cells and the\n\rux rate is calibrated with a water-cooled quartz crystal\nmicrobalance before growth. The Sr \rux rate is kept at\n4.4\u00021012atoms \u0001cm\u00002\u0001s\u00001. Reactive atomic oxy-\ngen generated by thermal cracker or plasma is used as\noxygen source. As-received TiO 2-terminated STO sub-\nstrates are annealed at \u0018600\u000eC under an O 2pressure\nof 2.0 \u000210\u00007Torr for 2 hours to remove the surface\ncontamination prior to \flm growth. Two atomic layer(AL) SrO bu\u000ber layers are found necessary to obtain the\nsingle-crystal SBO \flms and are \frst grown on the TiO 2-\nterminated STO surface with the substrate temperature\nat 450\u000eC in 2.0 \u000210\u00007Torr O 2pressure, for both meth-\nods.\nThe sticking coe\u000ecient of Bi depends strongly on\nthe sample temperature and the partial pressure (and\nspecies) of oxygen at the sample surface [25, 30]. To\nobtain the correct Bi oxidation state and \flm stoichiom-\netry, tests have been performed by systematically varying\ngrowth conditions, including substrate temperature, oxy-\ngen pressure, as well as the \rux ratio of Sr and Bi. The\nstoichiometric crystalline \flms reported here are grown\nusing the optimal conditions as follows. SBO \flm is\ngrown by co-deposition of Bi and Sr with about 30%\nexcess Bi at substrate temperature 450\u000eC and oxygen\npressure 2.0 \u000210\u00007Torr (chamber pressure as measured\nby an ion gauge; the local pressure close to the substrate\nwill be higher because the end of nozzle used for oxygen\ndelivery is much closer to sample surface). The 30% ex-\ncess Bi desorbs from the surface due to its volatility at the\ngrowth temperature [25, 30]. Rutherford backscattering\nspectrometry (RBS) collected at the Rutgers University\nLaboratory for Surface Modi\fcation, con\frms that the\nBi/Sr ratio in a \flm grown on MgO(001) substrate using\nthe same conditions is within \u00065% of 1:1. It is found\nthat a bigger than \u00064% change of the optimum F Bi/FSr\nratio leads to an amorphous \flm or a \flm surface with\nislands, with no di\u000braction spot or multiple di\u000braction\nspots along the direction perpendicular to \flm surface\nin RHEED. After \flm growth, the oxygen pressure and\nthermal cracker/plasma power are kept constant until the\nsample temperature falls below 100\u000eC. Note that \flms\ndirectly grown on the TiO 2-terminated surface of STO\nsubstrate always show multiple domains with di\u000berent\norientations in our experiments. The need for the SrO\nlayers is likely due to the tendency of Bi to occupy the\nA-site in perovskite ABO 3structure and react with the\nTiO 2of the substrate to form a stable bismuth titanate\ncompound; the SrO layer functions as a di\u000busion barrier\nto prevent the formation of this phase.\nFigure 1 (a)-(d) shows RHEED patterns after 2 AL\nSrO, 2 unit cell (UC) SBO, 6 UC SBO, and 20 UC\nSBO, respectively, using the high-temperature codeposi-\ntion method of Bi and Sr. The [100] direction of SBO\nis parallel to the [100] of the STO substrate, demon-\nstrating cube-on-cube epitaxial growth. The RHEED\nintensity during the \frst few layers of growth is \frst re-\nduced, likely due to the \flm nonstoichiometry resulted\nfrom the extremely low sticking coe\u000ecient of bismuth at\nthe begining of growth [16]. The RHEED pattern re-\ncovers and its intensity increases after \u00186 UC growth\nindicating improving crystallinity of the SBO phase. As\nshown in Fig. 1 (d), the 20 UC SBO \flm has streaky and\nbright RHEED pattern suggesting an atomically smooth\nand well-ordered \flm surface. The separation of the3\n2(e)\n(f) (g) \n(a) (b) (c) (d) \n200 nm\n-6.3 nm 1.8 nm\n(e)\n(f) \nFIG. 2. (a)-(d) [100] direction RHEED pattern of 2 AL SrO\n(a), 1 UC SBO (b), 4 UC SBO (c) and 6 UC SBO (d) grown\nby the recrystalization method; (e) top panel: substrate tem-\nperature cycle during \flm growth; bottom panel: Sr and Bi\nshutter open/close status; (f) AFM image of 28 UC SBO \flm\nsurface; (g) XRD of the 28 UC SBO \flm grown on STO sub-\nstrate. \u0003and\u0005label SBO (001) and STO (001) di\u000braction,\nrespectively.\ndi\u000bracted streaks re\rects the in-plane lattice parameter\nof \flm in the [100] direction. As shown in Fig. S1 of\nSupplemental Material [31], the pixel spacing of these\nstreaks in the 2 AL SrO bu\u000ber layers and the 2 UC\n\flm surface are similar to that of the substrate, indi-\ncating coherently strained growth. Films more than\n6 UC thickness have \u00184.25 \u0017A in-plane lattice param-\neter, as evidenced by \u001810% reduction in separation of\ndi\u000bracted streaks compared to the substrate. These re-\nsults suggest that the \flm starts to relax from 2 UC\nthickness and becomes fully relaxed at \u00186 UC thick-\nness. Time-dependent RHEED intensity after 6 UC \flm\ngrowth in Fig. 1(e) shows 12 oscillations revealing the\n2-dimensional layer-by-layer growth mode. The atomic\nforce microscope (AFM) of the 20 UC \flm surface in\nFig. 1(f) and Fig. S4 [31] shows atomically \rat steps\nand terraces. X-ray di\u000braction (XRD) experiment was\nperformed on a Rigaku smartlab pXRD machine and a\nstrong (001) di\u000braction peak at 2 \u0012= 20.6ois observed\nin XRD in Fig. 1(g) and Fig. S3 [31], corresponding to\nan out-of-plane lattice parameter \u00184.25 \u0017A that is con-\nsistent with the fully relaxed \flm seen by RHEED. The\nnarrow rocking curve taken at SBO (001) di\u000braction with\n0.1ofull width at half maximum (FWHM), and strong \f-nite thickness oscillations in XRD demonstrates the high\nquality of \flm.\nIn high-temperature codepositioon growth, only a\nsmall window of Bi/Sr ratio allows us to achieve high-\nquality stoichiometric \flms. Intrigued by the growth\nmethod of STO on Si [32] and considering the volatil-\nity of Bi metal and its oxides at high temperature,\nwe developed a new method for perovskite bismuthate\ngrowth, based on cycling between low-temperature de-\nposition and high-temperature crystallization. Low tem-\nperature deposition in atomic oxygen enviroment leads to\nan amorphous SBO layer that avoids Bi re-evaporation\nfrom surface, while high-temperature annealing substan-\ntially promotes crystallization and improves \flm quality.\nAs shown in Fig. 2, in each cycle, atomic layers of Bi\nand Sr are deposited in atomic oxygen environment to\nform an amorphous layer at \u0018200\u000eC, followed by an an-\nnealing step at 470\u000eC. Figure 2(a)-(d) shows the RHEED\npattern after 2 AL SrO, 1 UC SBO, 4 UC SBO and 6 UC\nSBO, respectively. The di\u000bracted streak spacing shows\n1-4 UC \flm is partially strained and becomes fully re-\nlaxed at 8 UC as shown in the Fig. S2 of Supplemental\nMaterial [31], consistent with the obervation in the high-\ntemperature growth method. Note that a much higher\nquality for \flms thinner than 6 UC is obtained in this cy-\ncling method, because low temperature deposition avoids\nthe re-evaporation of Bi from surface leading to a correct\nSr & Bi ratio in the \flm [16]. As shown in Fig. 2(f) and\nFig. S5 [31], steps and terraces are clearly visible in the\nAFM image of the 28 UC \flm surface. The high quality\nof the 28 UC SBO \flm is demonstrated by the strong\n(001) di\u000braction peak in XRD with \fnite thickness os-\ncillations as shown in Fig. 2(g), showing an out-of-plane\nlattice constant of \u00184.25 \u0017A. The rocking curve taken at\nSBO (001) di\u000braction has 0.08\u000eFWHM, similar to that\nseen in the high-temperature codeposition method. We\nconclude that the recrystalization growth method works\nwell for SBO growth, and solves the problem of ultrathin\n\flm nonstoichiometry due to the low sticking coe\u000ecient\nof Bi at the begining of \flm growth [16]. This growth\nmethod could be applied to growth of other Bi-based ox-\nide systems.\nThe chemical composition and electronic structure of\nthe \flms are studied using in situ x-ray photoemis-\nsion spectroscopy (XPS) with monochromatized Al K\u000b\n(1486.6 eV) x-ray source. Figure 3(a)-(c) shows the O1s,\nBi 4f, and Sr 3d core-level spectroscopy of 20 UC SBO\n\flm, respectively. The \flm stoichiometry can be deter-\nmined from the XPS intensities after taking into account\nthe element sensitivity factor which is calibrated by mea-\nsuring STO substrate and Bi 2O3\flm [33], and the result\nshows \flm stoichiometry close to SBO with a \u0018\u00065% er-\nror bar, consistent with the RBS result.\nAs shown in Fig. 3(a), the O 1s spectroscopy with peak\nat\u0018529.8 eV binding energy [Fig. 3(a)] is a little asym-\nmetric, indicating the existence of a small \\satellite\" peak4\n(b)\n (a) (c)\nBinding energy (eV) \nFIG. 3. XPS of O1s (a), Bi4f (b) and Sr3d (c) measured\non the 20 UC SBO \flm grown on Nb-doped STO using the\nhigh-temperature codeposition method.\nat higher binding energy side which may result from the\npresence of O 2p holes interacting with the core 1s hole in\nthe XPS \fnal state [13]. The double peak spliting of Bi\n4f [Fig. 3(b)] and Sr 3d [Fig. 3(c)] is due to spin-orbital\ncoupling. As manifested by the single Bi 4f 7=2peak in\nFig. 3(b), the Bi in our SBO \flm has a single valence.\nHowever, it is di\u000ecult to determine the precise valence\nstate of Bi based on the peak position alone due to the\nvariance of E Fposition in di\u000berent bismuth oxides, to-\ngether with the di\u000berences in the Madelung potentials\nand degrees of covalency. The Bi 4f 7=2and O1s spectra\nshow a shift of \u00181.2 eV to higher binding energy as com-\npared to that in BBO [20]. This could be explained in\npart by the larger indirect band gap in SBO, suggesting\nthat the chemical potential is located below the bottom\nof conduction band [8]. Transport measurement shows\nthe \flm is insulating, indicating the Fermi level located\nin the band gap, close to the conduction band.\nThe valence band measured by XPS on the 20 UC \flm\nsurface is shown in Fig. 4(a). For comparison with the\nXPS result, as shown in Fig. 4(b), total and partial DOS\nare calculated using a P21=nunit cell, a 6 \u00026\u00025 grid of\nkpoints and the Heyd-Scuseria-Ernzerhof (HSE) hybrid\nfunctional in Vienna ab initio simulation package [8, 34,\n35]. The spectral weight in Fig. 4(a) starts to increase\nat\u00181.4 eV, suggesting its \u00181.4 eV band gap and the\n\flm Fermi level is pinned closer to the conduction band\nin band gap [8]. The peak with \u001812 eV binding energy\nin XPS [Fig. 4(a)] corresponds to the bonding state of\nBi 6s and O 2p with the A 1gsymmetry. The spectral\nfeatures between \u001810 eV and \u00182 eV are mainly derived\nfrom O 2p nonbonding states. The antibonding state in\nthe expanded BiO 6octahedron just below the Fermi level\nshown in Fig. 4(b) [8] is seen as a small shoulder at \u00182 eV\nbinding energy in Fig. 4(a).\nIn conclusion, high-quality SBO \flms down to 1 UC are\nsuccessfully prepared on SrO-terminated STO substrate.\nThe successful fabrication of high-quality stoichiometric\nSBO \flm opens the possibility of manipulating the elec-\ntronic states of perovskite bismuthate by interfacial engi-\n/s49/s52 /s49/s50 /s49/s48 /s56 /s54 /s52 /s50 /s48\n/s45/s49/s50 /s45/s49/s48 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50/s40/s98/s41/s32/s32/s66/s105/s110/s100/s105/s110/s103/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s32/s40/s97/s41\n/s32/s116/s111/s116/s97/s108\n/s32/s79/s32/s50/s112\n/s32/s66/s105/s32/s54/s115\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s32\n/s126/s49/s46/s52/s32/s101/s86FIG. 4. (a) Valence band information collected by XPS on\nthe 20 UC SBO \flm grown on Nb-doped STO using the high-\ntemperature codeposition method. (b) total DOS and partial\nDOS for Bi 6s and O 2p calculated with HSE hybrid functional\nDFT [8] and the \u00181.4 eV indirect bandgap was obtained.\nSince the Fermi level position located in SBO band gap is\nuncertain in DFT calculation, peaks at \u0018-2 eV in theory and\n\u00184 eV in experiment are aligned for comparison.\nneering with or without strain applied by the substrates.\nIn addition, low-energy electronic structures of relaxed\nSBO \flm show good agreement with theoretical calcula-\ntion and supports the existence of strong hybridization\nof Bi 6s and O 2p in theoretical calculation that leads to\nits semiconducting nature.\nThe work was supported by Natural Sciences and En-\ngineering Research Council of Canada (NSERC), Canada\nFoundation for Innovation (CFI), Canada First Research\nExcellence Fund (CFREF) and Canadian Institute for\nAdvanced Research (CIFAR). Spectroscopy and part of\n\flm growth were performed at at the Surface Science\nFacility of the REIXS beamline in the Canadian Light\nSource, which is funded by the Canada Foundation for\nInnovation, NSERC, the National Research Council of\nCanada, the Canadian Institutes of Health Research, the\nGovernment of Saskatchewan, Western Economic Diver-\nsi\fcation Canada, and the University of Saskatchewan.\nThe authors also acknowledge the \fnancial support from\nMPI-CPfS (Dresden) for the experiment in the Canadian\nLight Source.\n\u0003fengmiao.li@qmi.ubc.ca5\n[1] R. J. Cava, B. Batlogg, J. Krajewski, R. Farrow,\nL. Rupp Jr, A. White, K. Short, W. Peck, and\nT. Kometani, Nature 332, 814 (1988).\n[2] S. Kazakov, C. Chaillout, P. Bordet, J. Capponi,\nM. Nunez-Regueiro, A. Rysak, J. Tholence, P. Radaelli,\nS. Putilin, and E. Antipov, Nature 390, 148 (1997).\n[3] A. W. Sleight, Physica C 514, 152 (2015).\n[4] D. Cox and A. Sleight, Solid State Commun. 19, 969\n(1976).\n[5] L. Mattheiss and D. Hamann, Phys. Rev. B 28, 4227\n(1983).\n[6] C. Franchini, G. Kresse, and R. Podloucky, Phys. Rev.\nLett. 102, 256402 (2009).\n[7] C. Franchini, A. Sanna, M. Marsman, and G. Kresse,\nPhys. Rev. B 81, 085213 (2010).\n[8] K. Foyevtsova, A. Khazraie, I. El\fmov, and G. A.\nSawatzky, Phys. Rev. B 91, 121114 (2015).\n[9] G. Kim, M. Neumann, M. Kim, M. D. Le, T. D. Kang,\nand T. W. Noh, Phys. Rev. Lett. 115, 226402 (2015).\n[10] N. Plumb, D. Gawryluk, Y. Wang, Z. Risti\u0013 c, J. Park,\nB. Lv, Z. Wang, C. Matt, N. Xu, T. Shang, et al. , Phys.\nRev. Lett. 117, 037002 (2016).\n[11] B. J. Kennedy, C. J. Howard, K. S. Knight, Z. Zhang,\nand Q. Zhou, Acta Cryst. B 62, 537 (2006).\n[12] A. Khazraie, K. Foyevtsova, I. El\fmov, and G. A.\nSawatzky, Phys. Rev. B 97, 075103 (2018).\n[13] S. Balandeh, R. J. Green, K. Foyevtsova, S. Chi,\nO. Foyevtsov, F. Li, and G. A. Sawatzky, Phys. Rev.\nB96, 165127 (2017).\n[14] O. Foyevtsov, S. Balandeh, S. Chi, and G. Sawatzky,\nPhysica B 570, 328 (2019).\n[15] M. Naamneh, M. Yao, J. Jandke, J. Ma, Z. Risti\u0013 c,\nJ. Teyssier, A. Stucky, D. van der Marel, D. Gawryluk,\nT. Shang, et al. , arXiv:1808.06135 (2018).\n[16] M. Zapf, M. St ubinger, L. Jin, M. Kamp, F. Pfa\u000b,\nA. Lubk, B. B uchner, M. Sing, and R. Claessen, Appl.\nPhys. Lett. 112, 141601 (2018).\n[17] C. Varma, Phys. Rev. Lett. 61, 2713 (1988).\n[18] G. Wertheim, J. Remeika, and D. Buchanan, Phys. Rev.\nB26, 2120 (1982).\n[19] M. Hegde, P. Barboux, C. Chang, J. Tarascon,\nT. Venkatesan, X. Wu, and A. Inam, Phys. Rev. B 39,\n4752 (1989).[20] M. Nagoshi, T. Suzuki, Y. Fukuda, K. Ueki, A. Tokiwa,\nM. Kiruchi, Y. Syono, and M. Tachiki, J. Phys. Condens.\nMatter. 4, 5769 (1992).\n[21] Z.-X. Shen, P. Lindberg, B. Wells, D. Dessau, A. Borg,\nI. Lindau, W. Spicer, W. Ellis, G. Kwei, K. Ott, et al. ,\nPhys. Rev. B 40, 6912 (1989).\n[22] S. Salem-Sugui Jr, E. Alp, S. Mini, M. Ra-\nmanathan, J. Campuzano, G. Jennings, M. Faiz, S. Pei,\nB. Dabrowski, Y. Zheng, et al. , Phys. Rev. B 43, 5511\n(1991).\n[23] K. Kobayashi, T. Mizokawa, A. Ino, J. Matsuno,\nA. Fujimori, H. Samata, A. Mishiro, Y. Nagata, and\nF. De Groot, Phys. Rev. B 59, 15100 (1999).\n[24] K. Inumaru, H. Miyata, and S. Yamanaka, Phys. Rev.\nB78, 132507 (2008).\n[25] E. Hellman, E. Hartford, and R. Fleming, Appl. Phys.\nLett. 55, 2120 (1989).\n[26] E. Hellman and E. Hartford, J. Vac. Sci. Technol. B 8,\n332 (1990).\n[27] A. Gozar, G. Logvenov, V. Butko, and I. Bozovic, Phys.\nRev. B 75, 201402 (2007).\n[28] D. Mijatovic, G. Rijnders, H. Hilgenkamp, D. H. Blank,\nand H. Rogalla, Physica C 372, 596 (2002).\n[29] H. G. Lee, Y. Kim, S. Hwang, G. Kim, T. D. Kang,\nM. Kim, M. Kim, and T. W. Noh, APL Mater. 4, 126106\n(2016).\n[30] S. Migita, Y. Kasai, H. Ota, and S. Sakai, Appl. Phys.\nLett. 71, 3712 (1997).\n[31] See Supplemental Material at XXXXXX for details of\nthe evolution of RHEED (01) and (0-1) spacing with \flm\nthickness during high-temperature codepostion and the\ncycling growth, wide-angle XRD scan measured on SBO\n\flm and the step height in AFM image of SBO \flm..\n[32] M. P. Warusawithana, C. Cen, C. R. Sleasman, J. C.\nWoicik, Y. Li, L. F. Kourkoutis, J. A. Klug, H. Li,\nP. Ryan, L.-P. Wang, et al. , Science 324, 367 (2009).\n[33] J. F. Moulder, W. F. Stickle, P. E. Sobol, and K. D.\nBomben, Handbook of X-ray photoelectron spectroscopy\n(Eden Prairie, 1995) p. 241.\n[34] G. Kresse and J. Furthm uller, Comput. Mater. Sci. 6, 15\n(1996).\n[35] J. Paier, R. Hirschl, M. Marsman, and G. Kresse, J.\nChem. Phys. 122, 234102 (2005)."    },    {        "title": "2107.03472v1.Large_Magneto_Electric_Resistance_in_the_Topological_Dirac_Semimetal_alpha_Sn.pdf",        "content": "1 \n Large  Magneto -Electric R esistance in the Topological D irac S emimetal  -Sn \n \nYuejie Zhang,1,2* Vijaysankar Kalappattil,1* Chuanpu Liu,1* Steven S. -L. Zhang,3* Jinjun Ding,1 Uppalaiah  \nErugu,4 Jifa Tian,4 Jinke Tang,4 and Mingzhong Wu1★ \n \n1Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA  \n2School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan, Hubei \n430074, China  \n3Department of Physics, Case Western Reserve University, Cleveland, Ohio, 44106, USA  \n4Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA  \n \n \nAbstract:  The spin -momentum locking of surface states in topological quantum materials can produce  a resistance  \nthat scales linearly with magnetic and electric fields.  Such a bilinear magneto -electric resistance (BMER) effect  offers \na completely new approach for magnetic storage  and magnetic field sensing  applications .  The effect s demonstrated \nso far , however,  are relatively weak or for low temperatures.  Strong room -temperature BMER effects  have  now been \nfound in topological Dirac semimetal -Sn thin films .  The epitaxial -Sn films were grown by sputtering  on silicon  \nsubstrate s.  The y showed BMER  responses that are 106 times larger than previously reported at room temper ature and \nalso larger than that previously reported at low temperatures.  These results represent a major advan ce toward realistic \nBMER application s. The data also made possible the first characterization of the three -dimensional, Fermi -level spin \ntexture of topological surface states in -Sn. \n \nKeywords: Topological Dirac semimetal, topological surface states, spin -momentum locking, and magnetoresistanc e.              \n  2 \n Introduction  \nSurface electronic states in topological quantum materials exhibi t spin -momentum locking .  That is, t he spin \ndirection of the relevant conduction electrons is locked to the momentum  at right angles .  In applicable two-\ndimensional (2D) momentum space, this locking manifests itself as a chiral Fermi contour on which the s pin points \nalong the tangential direction  everywhere, in either a clockwise or counterclockwise manner , as sketched in Fig. 1(a) .  \nSuch a Fermi contour usually takes a circular shape.  It, however, can turn into a hexagonal ly warp ed contour , as in \nFig. 1(b ), if the material structure has three -fold rotation al symmetry.1,2,3,4,5,6,7  As one walks along such a hexagonal  \ncontour, the spin still rotate s in a chiral manner as on the circular contour  in Fig. 1(a) , but now has an out -of-plane  \ncomponent .  Such hexagonal warping exists in thin films of topological materials that have (111) -oriented cubic \nstructures or c-axis-oriented hexagonal structures .  \nThe spin-momentum locking gives rise to two important effects: (1) prohibition of electron backscatt ering and \n(2) highly efficient conversion  between charge and spin currents .  The first effect opens the possibility of dissipation -\nfree information transport.   The second  effect allows on e to use charge currents to control magnetic properties.  Such \ncontro l is a current subject of intense interest .8,9,10,11,12,13,14   \nThere is a third newly reported spin -momentum locking effect : a bilinear magneto -electric resistance (BMER) \nresponse .15,16,17,18  This effect occurs in topological materials with hexagonal warping contours as introduced above.  \nThe material exhibits a resistance that (1) scales linearly with the magnetic field, (2) scales linearly with the electric \nfield or current, and (3) varies with the direction of the electric current relative to the crystallog raphic axes of the \nmaterial.  These kinds of  responses were first observed in the topological insulator Bi 2Se3, with a BMER coefficient \nof 𝜒≈ 0.6 nm2A-1Oe-1.15  Here  𝜒 is the scaled BMER response per unit magnetic field and per unit current density.  \nThe effect was then also observed  for 2D electron gas on surfaces of SrTiO 3 crystals, but with a sub stantially larger \ncoefficient, 𝜒≈ 500 nm2A-1Oe-1.17  These observations were made at relatively low temperatures (60 K for Bi 2Se3 and \n7 K for SrTiO 3).  More recent  work shows that room -temperature BMER responses  are also possible.18  These later  \nresponses were seen in the topological Weyl semimetal WTe 2, but were significantly weaker, with 𝜒≈ 0.001 nm2A-\n1Oe-1 only.    \nA basic  picture of the BMER effect is as follows:16 (1) When an electric field E is applied, in addition to its first -\norder correction to the electron distribution that results in a  charge current, there is a second -order correction to the \ndistribution that results in equal numbers of electrons populated in the surface states with opposite spins and velocities.   \nThis is depicted in Fig. 1(c).  The net effect is a pure spin current Js that depends on the square of the electric field, \nthat is, E2.  (2) When a magnetic field H is applied, the flux  of electrons with opposite spins is no longer balanced.  \nAs a result, the spin current Js is partially converted to a charge current Jc, as sh own in Figs. 1(d) and 1(e).  (3) In Fig. \n1(d), Jc is along E and thereby adds to the ordinary charge current.  The net effect is a lower resistance.  If H is \nreversed, Jc is also reversed.  This gives rise to a higher resistance , as shown in Fig. 1(e).  \n \nFig. 1. Bilinear magneto -electric resistance due to spin -momentum locking.15,16  (a) Spin texture on a circular Fermi contour of \ntopological surface states.  (b) Spin texture on a hexagonal warping Fermi contour.  (c) Electric field E-induced generation of a \npure spin current Js.  (d) Magnetic field H-induced partial conversion of Js to a charge current Jc, giving rise to a lower resistance.  \n(e) In comparison with (d), a flip in H leads to a flip in Jc and a higher resistance.  The short red arrows denote the directions of \nelectron spins.  \n \n \n(c)\nEJs(d)\nE Js\nH\nJc\nLow resistance(e)\nE Js\n H\nJc\nHigh resistance\nkxky  \n    \nkxky  \n(b) (a)3 \n This article reports  a large  room -temperature BMER response with a coefficient that is orders of magnitude larger \nthan that in previous work.  The experiments used a topological Dirac semimetal (TDS) -Sn, in contrast to topological \ninsulators, 2D electron ga s, and Weyl semimetals in previous experiments.  (111) -oriented TDS -Sn thin films in the \n4-6 nm thickness range were grown on single -crystal silicon substrates  by sputtering at room temperature .  \nMeasurements  on these -Sn films  at room temperature  show ed a resistance that scale d linearly with both the magnetic \nfield and the cha rge current and strongly depended  on the current direction relative to the crystalline axes  of the films .  \nThe data show ed a BMER coefficient  of 𝜒≈ 2900 nm2A-1Oe-1.  This coeffic ient is 106 times larger than that obtained \nat room temperature  for the Weyl semimetal WTe 2.18  It is also substantially larger than the previous values measured \nat low temperatures.15,17  Such a giant  effect is most likely due to the low carrier density and spatial asymmetry of the \n-Sn films.   The data also indicate that in the TDS -Sn films , the spins  on the hexagonal contours  lie in -plane  at the \nhexagon vertexes but tilt 30 out-of-plane at the middle points of the hexagon sides.       \nBefore bringing in the key data that inform the effects outlined above, it is useful to establish three important \npoints.  First, the BMER effect s offer a totally new approach for information reading and field sensing technolog ies, \nbut the previous results were weak or  only at low temperature s.  The very large, room temperature -accessible  BMER \nresponses reported here point to the possibility of real a pplications.  This technological significance  is further \nhighlighted by the following facts .  (a) The effects here have been obtained in a single -material, single -element thin \nfilm.  This can significantly simplify device fabrication.  In stark contrast , other known magnetoresistance (MR) \neffects, such as giant  MR, tunnel  MR, and spin-Hall MR, all involve multiple layers of magnetic and non -magnetic \nmaterials.  (b) The -Sn films were grown on silicon , a common industrial substrate.  (c) The film deposition  made \nuse of room -temperature  sputtering , an industry -friendly technique .  Second, this work  show s the technological \npotential for surface states in TDS -Sn.  There  have been major efforts to develop spintronic applications  of surface \nstates in  topological insulators,8-14 but correspond ing efforts for  TDS systems  are still largely unexplored .  Third, t his \nwork also represents the first -ever measurement of three -dimensional (3D) Fermi contours in -Sn.  There have been \nextensive  angle -resolved photoemission spectroscopy (ARPES) measurements on surface and bulk states in -\nSn,19,20,21,22,23,24,25, 26 but none of them have even touched on the 3D aspect of the Fermi -level spin texture . \n \nBasic  Properties of -Sn Thin Films  \n-Sn has a diamond cubic crystal structure.  Unstrained -Sn is a gapless semiconductor in which the  quadratic  \nconduction and valence bands touch each other at the  point near the Fermi level.  In the presence of a tensile strain \nalong the [001] or [111] direction, however, the two bands cross each other near the Fermi level .  This band crossing \nforms  two Dirac points and give rise to a topological Dirac semimetal (T DS) phase.22,27,28  One of such Dirac points \nis shown  in Fig. 2.  The topological nature of the TDS -Sn originates \nfrom the band inversion : The conduction and valence bands (  8+) near \nthe Fermi level are derived from p electr ons, while the s electron -\nderived band (  7−) with opposite parity is below the Fermi level, as \nillustrated  in Fig. 2.  The topological surface states (TSS) bridge the  8+ \ncond uction band and the  7− valence band,22,27 as indicated by the red \ndashed lines.     \nIn this work, (111)-oriented -Sn thin films were grown on silicon \n(Si) substrates  by sputtering .  The film growth details are given in \nSection 1 of th e Supplemental Materials .  The lattice constant of -Sn \n(6.489 Å) is larger than that of Si (5.4307 Å) .  This mismatch yields a \nperpendicular tensile strain in the -Sn films .  This strain pushes the \nfilms into a TDS phase.22,27,28    \nFig. 2. Schematic of the band structure in \ntopological Dirac semimetal -Sn thin films.22,27  \n“TSS” denotes topological surface states.  \n \n \nE\nEF\nk  8+(p)\nTSS\n 8+(p)\n 7−(s)Diracpoint4 \n Figure 3 presents a quick overview of the structural properties of a representative 4-nm-thick -Sn film.  Panel \n(a) presents an X -ray diffraction (XRD) spectrum of the film.  Panel (b) shows the fitting of the -Sn (111) peak in \n(a) to a Voigt function.  The red curve is the fit .  The red dashed line show s the peak position of the Voigt fit , while \nthe gray dashed line shows the theoretically expected position of the peak.  Panels (c) and (d) present 2D and 3D \natomic force microscopy (AFM) surface images, respectively.     \nThe data in Fig. 3 establish four important points .  (1) The XRD spectrum shows a peak for -Sn but no -Sn \npeaks .  This confirms the  phase of the film.  As shown  in the Supplemental Materials, Sn films with a thickness of \n7 nm or larger can exhibit XRD peaks for both the  and  phases, which indicate  the coexistence of the two phases.  \n(2) The -Sn (111) peak  is slightly to the left of the theoretically expected peak.  This is due to the perpendicular \ntensile strain in the film, which , in turn,  is a consequence of the lattice mismatch at the Sn/Si interface.  Analysis y ields \na tensile strain of 0.46%.  It is this strain that makes the -Sn film a TDS material.  (3) The XRD spectrum shows only \na single peak for -Sn.  This is a positive indication of epitaxial growth of the -Sn film.  The (111) orientation is \ncritical.  As explained above, this produces the hexagonal warping Fermi contour in the film , which is essential to the \nBMER response .  (4) The AFM images demonstrate  the granu lar nature of the film .  The grain size is in the 15-20 nm \nrange.  The rms surface roughness is about  0.82 nm.  Such  surface morphological properties are very similar to those \nof the -Sn thin films grown by molecular beam epitaxy.29,30  The significant surface morphology is, in part, \nresponsible for the extremely large BMER response in the film .  This is discussed in more detail below .  \n \nBilinear Magneto -Electric Resistance in -Sn Thin  Films  \nFigure s 4 and 5 present the main results of this work , namely,  the BMER responses obtained on the 4 -nm -Sn \nfilm described above.   As in previous work,15,17,18 the experiments were based on Hall bar structure s and  measurements \nof the second -harmonic resistance in response to an AC current , as discussed in Section 2 in the Supplemental  \nMaterials .  The analysis is based on a drive curren t of the form  𝐼(𝑡)=𝐼0cos(𝜔𝑡), where  𝐼0 is the current amplitude \nand 𝜔 is the frequency .  The total longitudinal  resistance is taken to be comprised of an ordinary resistance 𝑅0 and a \nBMER component 𝐶𝐼(𝑡). Then , the longitudinal voltage in the -Sn film can be evaluated  as \n𝑉=[𝐼(𝑡)][𝑅0+𝐶𝐼(𝑡)]=[𝐼0cos(𝜔𝑡)][𝑅0+𝐶𝐼0cos(𝜔𝑡)]=1\n2𝐶𝐼02+𝑅0𝐼0cos(𝜔𝑡)+1\n2𝐶𝐼02cos(2𝜔𝑡).      (1) \nDivision of Eq. (1) by the current amplitude 𝐼0 gives the longitudinal resistance as  \n𝑉\n𝐼0=1\n2𝐶𝐼0+𝑅0cos(𝜔𝑡)+1\n2𝐶𝐼0cos(2𝜔𝑡).                                                     (2) \nOne can  see that t he coefficient of cos(𝜔𝑡) in Eq. (2) is the ordinary resistance, whereas that of cos(2𝜔𝑡) is one -half \nthe BMER.  Simple second -harmonic signal measurements, therefore, allow for a direct measurement of the BMER \nresponse.  In the following, one uses 𝑅2𝜔 to denote  the second -harmonic resistance  1\n2𝐶𝐼0.  \nFig. 3. Structural properties of a 4 -nm-thick Sn film grown on a (111) Si substrate.  (a) X -ray diffraction (XRD) spectrum.  (b) \nFitting of the XRD -Sn (111) peak to a Voigt function.  (c) Atomic force microscopy (AFM) surface image.  (d) 3D p resentation \nof the AFM image in (c).  \n23.4 23.5 23.6 23.7 23.8 23.93x1034x1035x103Intensity (a.u)\n2q  (degrees)\n24 26 28 30 32103104105106107Intensity (a.u)\n2q  (degrees)\n-Sn (111)\nKαSi (111)\n-Sn\n(111 )-Sn\n(111 )(a) (b) (c) (d)KβSi (111)Stage \n6 nm\n-6 nm50 nm\n100200300400500 nm100200300400500 nm0 6 nm5 \n Figure 4  gives 𝑅2𝜔 as a function of the magnet ic field angle.  Panels (a), (b), and (c) show the 𝑅2𝜔 data measured \nwith rotating the field H in three different planes, as indicated.  The curves in (a) and (b)  are fits to sine and cosine \nfunctions , respectively .  The red and green curves in (c) are t he same as those in (a) and (b) .  They add together to \ngive the blue curve.  Panel (d) shows ∆𝑅2𝜔, the amplitude of the sinusoidal 𝑅2𝜔 vs. angle response, as a function  of \nthe current amplitude  𝐼0 and the field strength H.   \nFive important  results  are evident from the data in Fig. 4 .  (1) The data in panel (a) for field rotation in the xy \nplane indicate that 𝑅2𝜔 takes the maximum and  the minimum at 𝐇||𝐲̂ and 𝐇||(−𝐲̂), respectively.  This is consistent \nwith the picture  depicted in Figs. 1(d) and 1 (e).  It suggests that the in -plane component of the spin vector points along \nthe y axis.  The response in panel (a) also represents a unidirectional magnetoresistance ( UMR ) along the y axis; the \nresistance is the largest for 𝐇||𝐲̂ and is the smallest for  𝐇||(−𝐲̂).  (2) The data in panel (b) show a n UMR along the z \naxis.  This confirms that the spin has a nonzero out -of-plane component.  (3) Considered together, the data in Figs. \n4(a) and 4(b) suggest that the spin vector lies on the yz plane.  This is confirmed by the data in panel (c).  Further, the \ndata show a 𝑅2𝜔 maximum at a field angle of   120 and a minimum at   300.  This indi cates that the spin  vector  \nis tilted about 30 out of the xy plane , because the BMER value is the largest for H parallel to the spin vector and the \nsmallest for H antiparallel to the spin vector .  These results are strongly supported by the nice agreement between the \nexperimental response and the blue curve shown in panel (c).  (4) ∆𝑅2𝜔 scales linearly with both 𝐼0 and H, as shown \nin panel (d).  (5) The data in panel (b) also exclude the possibility in which an out -of-plane temperature gradient \nproduces 𝑅2𝜔 in the film through the Nernst effect .     \nFigure 5  compares the 𝑅2𝜔 vs. field angle responses for the currents applied along two different crystalline axes : \n[112 ] and [11 0], which  correspond to the      and      momentum space lines in Fig. 1(b) , respectively .  The data in \nFig. 5(a) are the sa me as in Fig. 4 , panels (a)-(c). The relevant crystalline axes are defined in Section 1 in the \nSupplemental M aterial .  The data in Fig. 5 (a) show  that when  the current (or the Hall bar length) is applied along     , \nthe spin is in the yz plane and tilts at an angle of about 30  away from the y axis, as discussed above .  In contrast, if \nthe current is along     , 𝑅2𝜔 exhibits a notable angle dependence for field rotation in the xy plane, but remains almost \nzero for field rotation in the xz plane, as shown in Fig. 5(b).  This indicates that the spin points along the y axis and \nhas a zero out -of-plane component.  Such spin orientation is consistent  with the spin texture depicted  in Fig. 1(b) for \na hexagonal Fermi contour.   \nFig. 4. Second -harmonic resistance ( R2) as a function of the magne tic field angle for a 4 -nm -Sn film.  The circles show the data, \nthe curves show sinusoidal fits, and the lines show linear fits.  The field strength H and the current amplitude I0 are indicated.  In \n(c), the red curve shows the fit in (a), the green  curve shows the fit in (b) ; they add together to give the blue curve.  In (d), R2 is \nthe amplitude of the sinusoidal R2 vs. angle response.  \n0 90 180 270 360-15-10-5051015\n0 90 180 270 360-15-10-5051015\n0 90 180 270 360-8-4048Angle q(degree) Angle (degree) Angle (degree)\nR2()R2()\nR2()(a) (b) (c)\n0 4 8 12 1605101520\n0 2 4 6 8 10036912\nR2() R2()\nField H(104Oe)Current I0(10-6A)\n\nq\nq(d)\nI0=110-5AH=9104Oe\nI0=110-5AH=9104Oe\nI0=110-5AH=9104Oe\nI0=110-5AH=9104OeVI\nz\nHyx\n\nVI\nz\nH\nyxq\nVI\nz\nH\nyx\n6 \n Taken together, the above results evidently show the presence of the BMER responses in the 4 -nm -Sn film.  In \nfact, the same responses have been observed in multiple samples.  As an example, Figure 6 gives the data measured \non an -Sn film that is thicker, with a thickness of 5.8 nm.  Figure 6(a) shows the XRD spectrum of the film.  It shows \nan -Sn (111) peak and shows no peaks for -Sn.  Figure 6(b) shows the 𝑅2𝜔 vs. field angle responses for currents \napplied along  [112 ], which  correspond to      in the momentum space.  Figure 6(c) gives the data measured for a \ncurrent direction that was rotated 60  away fr om [112 ] and was therefore along     ′ in the momentum space.  \nThe data in Fig. 6 show three key results.  (1) The fi eld angle dependences in Fig. 6(b) are about the same as those \npresented above for the 4 -nm film.  This consistency supports the results discussed above.  (2) After the current \ndirection was rotated from       to     ′, the  dependence remains the same, but the q respon se flips the sign.  Thi s \nindicate s that a rotation in the current direction did not affect the in -plane component of the spin  vector , but reversed \nthe sign of the out -of-plane component.  (3) ∆𝑅2𝜔 is smaller than that for the 4 -nm -Sn film.  This result, together \nwith the  data in Fig. S3  in the Supplemental Materials, indicates that the BMER decreases with an increase in the fi lm \nthickness.  Possible reasons include tha t as the thickness increase, the phase of a Sn film transforms from pure , to a \nmixture of  and , and then to  dominant.31  This is discussed in detail  in the Supplemental Materials .  \n   \nBilinear Magneto -Electric Resistance Coefficient  \nThe strength of the BMER effect can be characterized by a coefficient17  \n𝜒=2Δ𝑅2𝜔\n𝑅0𝐽0𝐻,                   (3) \n \nFig. 6. BMER responses in a 5.8 -nm -Sn film.  (a) XRD spectrum.  (b) Second -harmonic resistance R2 vs. field angle responses \nmeasured with currents applied along     .  (c) R2 vs. field angle responses measured with currents applied along     ′.  (d) Spin \ntexture on the Fermi contour.     \n \n(b)\n22 24 26 28 30 32104105106Intensity (a.u)\n2q  (degree)\n0 90 180 270 360-2-1012 \n q\n \nField angle (degree)  \n0 90 180 270 360-2-1012\nField angle (degree) \n q\n  (a) (d)-Sn (111)\nKαSi (111)KβSi (111)Stage \nz\nHyx\n\nz\nH\nyxq\nz\nH\nyx\nR2()\nq\n\nR2()\nq\n\n  \n    \nI  ′\n  \n      ′\nI\nkxky  \n30\n30(c) \nFig. 5. Second -harmonic resistance ( R2) as a function of the magnetic field angle for a 4 -nm -Sn film.  The data in (a) were \nmeasured with a Hall bar whose length is along the [112 ] direction of the -Sn film, while those in (b) were measured with a Hall \nbar with its length along the [11 0] direction.   \n0 90 180 270 360-15-10-5051015\nField angle (degree)  \n  q\n  \n0 90 180 270 360-15-10-5051015 \n q\n  \n q\n  \n q\n  \n q\n  \n q\n  \n q\n \nField angle (degree) \n q\n \nz\nHyx\n\nz\nH\nyxq\nz\nH\nyx\nR2()R2()  \n    \nI(a) Current along 112  o      \n    \n  I\n(b) Current along 11 0 o      \n\nq\n\nq\n7 \n where 𝐽0=𝐼0\n𝑤𝑡 (w – Hall bar width, t – film thickness) is the current density.  With th e data in Fig. 4, one obtains 𝜒≈ \n2900 nm2A-1Oe-1.  This coefficient is 106 times larger than the value meas ured previously at room temperature  for the \nWeyl semimetal WTe 2.18 It is also markedly larger than the values measured previously at low temperatures.15,17  \nThe large BMER response in the TDS -Sn films possibly originates from the following facts.  First, the 2D \ncarrier density (n) of the TSS in -Sn films is relatively low.  Previous work has shown that the BMER coefficient 𝜒 \nincreases with a decrease in n in the single relaxation time approximation.16,17 In fact, a 𝜒∝1\n𝑛3 response has been \nobserved for the BMER effect in SrTiO 3 {see Fig. 4(b) in Ref. [ 17]}.  Previous experiments have shown n  1.81012 \ncm-2 for TSS in TDS -Sn films,31 3.01013 cm-2 for TSS in Bi 2Se3 films,32 and 5.3 1013 - 7.71014cm-2 for 2D electron \ngas in SrTiO 3.17  Thus, one can see that n in TDS -Sn films  is more than one order of magnitude smaller than in \neither Bi2Se3 or SrTiO 3. \nSecond, the -Sn film in this work do es not have spatial inversion symm etry.  Specifically, the film bottom \nsurface at the interface with the Si substrate is smooth, while the top surface  is relatively rough , as shown by  the AFM \nimages in Fig. 3.  For this reason, the electrons on the top surface may undergo  more frequent scattering  from \nnonmagnetic disorders than those on the bottom surface.  The net effect is that the measured BMER  responses can be  \nattributed mainly to the spin -momentum locking at the bottom surface.  In the case that the film has inversion  \nsymmetric, the  BMER responses from the top and bottom surfaces would cancel each other and thereby give r ise to \nvery weak overall response s, if not zero.  This partially explains why 𝜒 in Bi 2Se3 films is only 0.6 nm2A-1Oe-1.15   \n \nSpin Texture of Topological Surface States  \nTaken together, the data in Figs. 5 and 6 show the 3D aspect of  the spins of the Fermi -level TSS in TDS -Sn.  \nSpecifically, the spin in the mid dle point of a side of the hexagonal Fermi contour is not exactly along th e hexagon \nside, but tilts 30 out-of-plane.  As one goes along the hexagonal contour to a vertex, the spin rotates into the plane.  \nAs one goes further to the middle point of the nex t hex agon side, the spin tilts -30 out-of-plane.  Such spin rotation \nis illustrated in Fig. 6 (d).  There have been considerable ARPES studies on the band structure of -Sn,19-26 which  \ninclude two  on spin -resolved ARPES measurements.19,20  However, none of those studies  have touched on the 3D \nfeatures of the spins on the Fermi contours of the TSS.      \nThe out -of-plane component (sz) of the spin depends on the properties of the TSS at the Fermi level.  Such \nproperties include the Fermi velocity ( vF), the Fermi wavenumber ( kF), and the strength of the hexagonal warping ( ).  \nIf one denote s ∅𝑘 as the angle between the      direction and the wave vector (the current direction), sz can be evaluated \nas1,15  \n𝑠𝑧=cos(3∅𝑘)\n√[cos(3∅𝑘)]2+(ℏ𝑣𝐹\n𝜆𝑘𝐹2)2,              (4) \nwhere ℏ is the reduced Planck constant.  The term cos(3∅𝑘) dictates the dependence of the  BMER on the current \ndirection presented in Figs. 5 and 6 , as well as  the three -fold symmetry of the chiral spin texture illustrated in Fig. \n6(d).  For the data  in Fig. 5(a) and Fig. 6(b), one has ∅𝑘=0 and 𝑠𝑧=sin30°.  Taken these parameters as well as \n𝑣𝐹=4\nℏ×10−10 eVm and 𝑘𝐹=3.39×108 m-1 from previous experiments,22,31 one can use Eq. (4) to estimate the \nwarping parameter as 2 eV nm3.  This value is about one order of magnitude larger than the values reported for \nBi2Se3.4,6,7,15      \n \nFinal Remarks  8 \n Four  remarks related to the above -presented results  are as follows .  (1) A s discussed above, the BMER effect \nstrongly depends on the properties of the TSS at the Fermi leve l.  As a result , future work is of great interest that \nexplores the effects of the Fermi level on the BMER  response , with an aim to maximize the effect in the TDS -Sn.  \nPossible strategies to tune the Fermi level include doping,19 metallic capping,24 and voltage gating.   \n(2) A lthough this work measured for the first time the spin canting  angle and warping  parameter of the hexagonal \nFermi contour of the TSS in TDS -Sn, the measurements we re somewhat indirect.  It is of great interest to directly \nmap the 3D chiral spin texture at the Fermi level  in the future .   \n(3) The hexagonal warping contour concerned in this work is a consequence of the three -fold symmetry of the \n(111) -Sn films .  In topological insulator -Sn films that are (001) -oriented and therefore do not have three -fold \nsymmetry, there exist also magnetoresistances that scale linearly with the magnetic and electric fields,  as mentioned \nin Ref. [ 33].  However, s uch magnetoresistances have nothing to do with the hexagonal warping, but are associated \nwith electron scattering by spin -orbit structural defects.33   \n(4) T he BMER  response  in this work is also an UMR, as discussed above .  However, it is totally different from \nthe UMR effect in bi-layered structures consisting of a magnetic film and a heavy metal  or topological insulator  thin \nfilm.34,35,36,37,38  They have completely different origins .  Further, a recent work reports an UMR in Ge that res ults \nfrom the interaction of an external  magnetic field and a pseudo -magnetic field associated with spin-splitted subsurface \nstates in Ge.39  This UMR scales linearly with both the magnetic and electric fields, but is independent of the current \ndirection relative to the crystalline axes of Ge.              \n \nAcknowledgment  \nThis work was supported by the U.S. Department of Energy, Office of Science, B asic Energy Sciences (DE -\nSC0018994).  The growth and characterization of -Sn thin film s and the fabrication of the Hall bar structures were \nalso supported by the U.S. National Science F oundation  (EFMA -1641989 ; ECCS ‐1915849 ).  Work at CWRU was \nsupported by  the College of Arts and Sciences at CWRU.   Work at UW was supported by the U.S. National Science \nFoundation (DMR -1710512) and the U.S. Department of Energy, Office of Science, Basic Energy Sciences (DE -\nSC0020074; DE -SC0021281).   The authors thank Carl Patton for valuable suggestions.  \n \n*Y.Z, V.K., C.L. and S.S.-L.Z. contributed equally.  \n★Corresponding author. mwu@colostate.edu      \n \nReferences  \n1 Chen. Y. L. et al.  Experimental realization of a three -dimensional topological insulator Bi 2Te3. Science 325, 178 \n(2009).  \n2 Hsieh D. et al.  A tunable topological insulator in the spin helical Dirac transport regime.  Nature 460, 1101 (2009).  \n3 Fu, L. H exagonal warping effects in the surface states of the topological insulator Bi 2Te3. Phys. Rev. Lett . 103              \n266801 (2009).  \n4 Kuroda, K. et al. Hexagonally deformed F ermi surface of the 3D topological insulator Bi 2Se3. Phys. Rev. Lett . 105, \n076802 (2010).                                                             9 \n                                                                                                                                                                                            \n5 Alpichshev, Z. et al. STM imaging of electronic waves on the surface of Bi 2Te3: topologically protected surface \nstates and hexagonal warping effects. Phys. Rev. Lett . 104, 016401 (2010).  \n6 Wang, Y. H. et al. Observation of a warped heli cal spin texture in Bi2Se3 from circular dichroism angle -resolved                  \nphotoemission spectroscopy. Phys. Rev. Lett . 107, 207602 (2011).  \n7  Nomura, M. et al. Relationship between Fermi surface warping and out -of-plane spin polarization in topol ogical \ninsulators: a view from spin - and angle -resolved photoemission. Phys. Rev. B  89, 045134 (2014) . \n8  Fan, Y. et al. Magnetization switching through giant spin –orbit torque in a magnetically doped topological insulator \nheterostructure. Nat. Mater.  13, 699–704 (2014).  \n9 Han, J. et al. Room -temperature spin –orbit torque switching induced by a topological insulator. Phys. Rev. Lett.  119, \n77702 (2017).   \n10  Wang, Y. et al. Room temperature magnetization switching in topological insulator –ferromagnet heteros tructures \nby spin –orbit torques. Nat. Commun.  8, 1364 (2017).   \n11  Yasuda, K. et al. Current -nonlinear Hall effect and spin -orbit torque magnetization switching in a magnetic \ntopological insulator. Phys. Rev. Lett . 119, 137204 (2017) .  \n12  Li, Y., Ma, Q., Hu ang, S. X. & Chien, C. L. Thin films of topological Kondo insulator candidate SmB 6: strong \nspin–orbit torque without exclusive surface conduction. Sci. Adv.  4, eaap8294 (2018).   \n13  DC, M. et al. Room -temperature high spin –orbit torque due to quantum confinement in sputtered Bi xSe(1-x) films. \nNat. Mater.  17, 800 –807 (2018).   \n14 Khang, N. H. D., Ueda, Y. & Hai, P. N. A conductive topological insulator with large spin Hall effect for u ltralow                                                                     \npower spin –orbit torque switching. Nat. Mater . 17, 808 –813 (2018).  \n15 He, P. et al. Bilinear magnetoelectric resistance as a probe of three -dimensional spin texture in topological s urface \nstates. Nat. Phys.  14, 495 –499 (2018).   \n16 Zhang, S. S. -L., & Vignale, G. Theory of bilinear magneto -electric resistance from topological -insulator surface \nstates. Proc. SPIE  10732 , 1073215 (2018).   \n17 He, P. et al. Observation of out -of-plane spin texture in a SrTiO3 two -dimensional electron gas. Phys. Rev. Lett . \n120, 266802 (2018).   \n18 He, P. et al. Nonlinear magnetotransport shaped by Fermi surface topology and convexity. Nat. Commun. 10, 1290 \n(2019).   \n19 Barfuss, A. et al. Elemental topological insulator with tunable Fermi level: Strained α -Sn on InSb(001). Phys. Rev. \nLett. 111, 157205 (2013).   \n20 Ohtsubo, Y., Le Fèvre, P., Bertran, F. & Taleb -Ibrahimi, A. Dirac Cone with Helical Spin Polarization in Ul trathin \nα-Sn(001) Films. Phys. Rev. Lett. 111, 216401 (2013).   10 \n                                                                                                                                                                                            \n21 Rojas -Sánchez, J. -C. et al.  Spin to charge conversion at room temperature by spin pumping into a new type of \ntopological insulator: α -Sn films. Phys. Rev. Lett. 116, 096602 (2016).   \n22 Xu, C. -Z. et al.  Elemental topological Dirac semimetal: α -Sn on InSb  (111). Phys. Rev. Lett. 118, 146402 (2017).   \n23 Rogalev, V. A. et.al. Double band inversion in α -Sn: Appearance of topological surface states and the role of orbital \ncomposition. Phys. Rev. B 95 , 161117(R) (2017).   \n24 Barbedienne, Q. et al.  Angular -resolved photoemission electron spectroscopy and transport studies of the elemental \ntopological insulator α -Sn. Phys. Rev. B 98, 195445 (2018).   \n25 Rogalev, V. A. et al.  Tailoring the topological surface state in ultrathin α -Sn (111) films. Phys. Rev. B 100, 245144 \n(2019).   \n26 Madarevic, I. et.al. Structural and electronic properties of the pure and stable elemental 3D topological Dirac \nsemimetal α -Sn. APL Mater. 8, 031114 (2020).   \n27 Huang, H. & Liu, F. Tensile strained gray tin: Dirac semimetal for observing negative magnetoresistance with \nShubnikov -de Haas oscillations. Phys. Rev. B 95, 201101 (2017) . \n28 Zhang, D., Wang, H., Ruan , J., Yao, G. & Zhang, H. Engineering topological phases in the Luttinger semimetal α -\nSn. Phys. Rev. B 97, 195139 (2018).   \n29 Madarevic, I. et al.  Structural and electronic properties of the pure and stable elemental 3D topological Dirac \nsemimetal α -Sn. APL  Mater. 8, 031114 (2020).  \n30 Gladczuk , L. et al. Study of spin pumping through α -Sn thin films. Phys. Status Solidi RRL .15, 2100137 (2021) . \n31 Ding, J. et al. Switching of a magnet by spin -orbit torque from a topological Dirac semimetal. Adv. Mater. 2005909 \n(2021).  \n32 Li, P. et al. Magnetization switching using topological surface states. Sci. Adv. 5, eaaw3415 (2019).  \n33 Dyrdal, A., Barnas , J., Fert, A.  Spin -momentum -locking inhomogeneities as a source of bilinear magnetoresistance \nin topological insulators.  Phys. Rev. Lett. 124, 046802 (2020).  \n34 Olejník, K., Novák, V., Wunderlich, J. & Jungwirth, T.  Electrical detection of magnetization reversal without \nauxiliary magnets. Phys. Rev. B 91, 180402(R) (2015) .  \n35 Avci, C. O. et al. Unidirectional spin Hall magnetoresistance in ferromagnet/normal metal bilayers. Nature Phys. \n11, 570 –575 (2015).   \n36 Avci, C. O. et al. Magnetoresistance of heavy and light metal/ferromagnet bilayers. Appl. Phys. Lett. 107, 192405 \n(2015).  \n37 Yasuda, K. et al. Large unidirectional magnetoresistance in a magnetic topological insulator. Phys. Rev. Lett. 117, \n127202 (2016).   11 \n                                                                                                                                                                                            \n38 Zhang, S. S. -L. and Vignale, G. Theory of un idirectional magnetoresistanc e in magnetic heterostructures.  Proc. \nSPIE, 10357 1035707 (2017).  \n39 Guillet, T. et al. Observation of large unidirectional Rashba magnetoresistance in Ge  (111). Phys. Rev. Lett. 124, \n027201 (2020).   \n1 \n Supplemental Materials for  \n  \nLarge Magneto -Electric Resistance in the Topological Dirac Semimetal -Sn \n \nYuejie Zhang,1,2* Vijaysankar Kalappattil,1* Chuanpu Liu,1* Steven S. -L. Zhang,3* Jinjun Ding,1 Uppalaiah Erugu,4 \nJifa Tian,4 Jinke Tang,4 and Mingzhong Wu1★ \n \n1Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA  \n2School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan, Hubei \n430074, China  \n3Department of Physics, Case Western Reserve University, Cleveland, Ohio, 44106, USA  \n4Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA  \n \n \n1. Growth  of Sn Thin Films   \nThe Sn thin films were grown on single -crystal ( 111)-oriented  Si substrates by DC magnetron sputtering.  The \nsubstrates are rinsed sequentially with acetone  and isopropyl alcohol before being loaded into  the sputtering chamber.  \nPrior to sputtering, the chamber has a base pressure of 2.0×10-8 Torr; substrate biasing i s performed that includes \nseveral cycles of Ar ion sputtering of the substrate surface and the post-annealing  of the substrate  at 250 °C.  The Ar \nion sputtering process is aimed at removing the thin oxidized layer on the top of the Si substrate , while  the post -\nannealing process is  to remove the moisture adhe rent on  the substrate  surface .  After removing the oxidized layer and \nthe surface moist ure, t he -Sn deposition is  then carried out at room temperature, at a rate of about 3 nm/min.  The \nsputtering power is set to  a moderate value of  10 W, in order to minimize the heating effect during the deposition.  \nThe major substrate biasing and sputtering control parameters are summarized in Table S1.    \nThe Si substrates were purchased from MTI C orporation, USA.1  The substrates are [111] oriented, have a size \nof 10 mm by 10 mm, and shows a surface roughness smaller than 0.5 mm .  The i n-plane orientation s of the substrates \nare defined by the notches at the corne rs of the substrate s, as indicated in Fig. S1 .  These orientations determine the \ncrystalline orientation s of the -Sn films grown on the substrates, because Si and -Sn both have diamond cubic \ncrystal structures.  The  Si substrates  are n-type and undoped , and have an electric resistivity larger than 1 k cm. \nTable S1. Substrate biasing and sputtering control parameters for Sn film growth.  Substrate \nbiasing  Ar ion sputtering  40 W, 2 min  \nAnnealing  250 °C, 60 min  \nCycles  3 Sputtering  Target -to-substrate distance  6.8 cm  \nSample holder rotation rate  10 rpm  \nAr pressure  3 mTorr  \nAr flow  10 sccm  \nSputtering power  10 W \n  \n2 \n .   \n2. Characterization of Structural Properties of Sn Thin Films   \nThin -film characterization has been performed using a Rigaku smart Lab X-ray diffractometer (XRD ). The XRD \npeaks were measured using two different scans.   A θ-2θ scan is performed first for a broader range (20 to 80) to \nidentify the Si  (111) peak shift from the theoretical value  and adjust  the results for instrumental errors. Following that, \na 2θ scan ( glancing angle XRD) i s performed between 20  and 35 to identify α-Sn and /or β-Sn phases. For  the \nglancing angle XR D (GAXRD), the incident angle i s kept constant at 0.7 .  Thickness calibration for the Sn samples \nis done through  X-ray reflectivity measurements.   By fitting the Kiessig fringes obtained from the Fresnel plot s using \nthe Rigaku smart lab software, film thickness es are  determined within an  accuracy of about 0.1 nm. The  thickness  \nvalues are then  used to calibrate the q uartz crystal thickness monitor in the sputtering chamber.  \nThe s urface morphology of the Sn films is  characterized using a Bruker Innova a tomic force microscopy  (AFM) . \nA surface scan was performed on a 500 × 500 nm surface area, and high -resolution images are taken at 512 × 512 \npixels.  Height deviation from the mean image data plane  is calculated and averaged over about five  trails to obtain \nthe roo t mean square (rms) value of the surface roughness.  \n \n3. Second -Harmonic Resistance Measurements  \nIn order to measure the electric transport properties of  a Sn thin film,  a Hall bar device is fabricated through \nphotolithography and argon ion milling processes.  Figure S 2 illustrates the Hall bar structure.   The central area of the \nHall bar is 300 μm long and 100 μm wide.    \nThe Hal bar device  is connected to a  rotatable sample holde r and installed in a Quantum Design Dynacool Physical \nProperty Measurement S ystem (P PMS). All electrical contacts a re given using indium press -fitting, and all AC \nsecond -harmonic resista nce measurements are performed using the e lectrical transport option (ETO) in the PPMS.  \nAn AC current of 𝐼(𝑡)=𝐼0cos(𝜔𝑡), where 𝐼0 is the amplitude  and  = 21.3 Hz is the frequency, is applied to the \ndevice, and first - and second -harmonic resistances are measured simultaneously while the magnetic field is rotated in \nthe xy, xz, and yz planes.  \n \nFigure  S2. Dimensions of the Sn Hall bar structure.  \n \n \n40 m100 m\n100 m\n60 m 260 m \nFigure  S1. Crystalline axes in the Si substrates used in this work.  \n       \n     \n3 \n  4. Thickness Dependence of Bilinear Magneto -Electric Resistances  \nFigure S3  compares the b ilinear magneto -electric resistance (BMER) responses in four Sn thin film s of different \nthicknesses.  Panels (b) -(e) present the second -harmonic resistance R2 vs. field angle responses for Sn films of \ndifferent thicknesses, respectively.  The field angles are defined in panel (a).  Panels (f) and (g) show R2 vs. current \namplitude I0 and field strength H, respectively, for the four films.  R2 denotes the amplitudes of the R2 vs. field \nangle responses presented in panels (b) -(e).  The XRD spectra of the four films are presented in panel (h).  The red \nand gree n dashed lines in (h) indicate the positions of the -Sn (111) peak and the -Sn (100) peak, respectively.  \nPrior to the discussions of the data shown in Fig. S 3, three notes should be made.  First, the data in panel (b) are \nthe same as those presented in Fig. 5(a) in the main text.  They are repeated to ease the comparison.  Second, among \nthe 12 profiles presented in panels (b) -(e), the blue profile in panel (c) shows a distorted -sinusoidal behavior, while \nthe other 11 all show quasi -sinusoidal responses.  The abnormal response of the blue profile in panel (c) is probably \nan instrumental effect.  Third, the sputtering growth of the four films and the fabrication o f the corresponding Hall bar \n \nFigure  S3. Film thickness dependence of the bilinear magneto -electric resistance effect.  (a) Definition of magnetic field angles.  \n(b)-(e) Second -harmonic resistance R2 vs. field angle responses measured on Sn thin films of different thicknesses, as indicated.  \n(f) R2 vs. I0 responses for four films of different thicknesses, as indicated.  Note that R2 denotes the amplitudes of the R2 vs. \nfield angle responses presented in (b) -(e), and I0 denotes the amplitude of the current.  (g) R2 vs. field streghth H responses for \nthe four films.  Note that in (f) and (g), the symbols show the experimental data, while the lines are linear fits.  (h) The XRD spectra \nof the four films.  The vertical red and green dashed lines indicate the positions of the -Sn (111) and -Sn (100) peaks, respectively.  \n0 3 6 912 15 18-505101520\n j\n q\n f\n0 90 180 270 360-0.4-0.20.00.20.4\n0 90 180 270 360-4-2024(b) 4 nm (e) 15 nm (d) 11 nm\n0 90 180 270 360-8-4048 (c) 7 nm R2()\nField angle (degree)\n24 26 28 30 32104105106107108109\nH=9104OeR2()\nCurrent I0(10-6A)\n0 2 4 6 8 10 12036912\n j\n q\n fI0=110-5A\nField H(104Oe)R2()(h) XRD spectra (f) Current dependence (g) Field dependence\n2q(degree)Intensity ( a.u)\n-Sn (111)\nSi (111)\nβ-Sn (100)\n15 nm\n11 nm\n7 nm\n4 nm\n0 90 180 270 360-16-80816 j\n q\n f\nz\nHyx\nj\nz\nH\nyxq\nz\nH\nyxf\n(a) Definition of field angles\nj\nq\nf\nField angle (degree) Field angle (degree) Field angle (degree)\n4 nm\n7 nm\n11 nm\n15 nm4 nm\n7 nm\n11 nm\n15 nm \n4 \n structures were made under the same conditions at the same time.  The 5.8 -nm-thick film and the corresponding Hall \nbar described in the main text were prepared at a different time and are t herefore not included in Fig. S3  for \ncomparison.   \nThe main res ult shown by the data in Fig. S3  is that an increase in the Sn film thickness (t) gives rise to a notably \nweaker  BMER  effect .  This is particularly evident in panels (f) and (g)  where  R2 for the 15 -nm-thick  film is \nsubstantially smaller than that for the 4 -nm-thick  film.  This result is mainly associated with the fact that at room \ntemperature -Sn is generally more stable than -Sn, and thin films of -Sn can be grown on Si substrates because \nthey both have cubic structures.2  In other words, the realization of -Sn thin films in this work relies on substrate -\nenhanced phase stability.  For this reason, pure  is present only in ultrathin Sn films, while thicker Sn films can host \nboth the  and  phases , or are  dominant .  Suc h thickness -associated phase stability is clearly seen in panel (h) .  \nWith an increase in the film thickness  (t), the -Sn (111) peak becomes weaker gradually  and diminishes at t = 15 nm,  \nwhile the -Sn (100) peak  starts to appear at t = 7 nm and then  beco mes stronger gradually.   Since the BMER effect \nis critically associated with the topological surface states in -Sn, t he gradual transition from  phase to the \ncoexistence of the  and  phases causes a gradual decrease in the BMER  strength .  Note that the absence of -Sn \npeaks in the XRD spectrum of the 15 -nm film does not necessarily mean that the film has pure  phase  and  phase \nis completely absent , but may mean that the  phase is dominant over the  phase.  Future work to check this i s of \ngreat interest.   \nTwo additional results are evident in Fig. S 3.  First, in spite of the weakening of the BMER effect due to the \nthickness increase, the BMER scales linearly with both the magnetic field and the electric current for all the four \nsamples .  This indicates that the responses in the thicker films , though relatively weak,  also result from the BMER \neffect, rather than other magnetoresistance effects.  Second, the data suggest that the ratio of the y component of the \nspin vector at the Fermi le vel to the z component decreases with an increase in t.  The actual reason for this observation \nis unknown currently, but it i s not unreasonable  if one considers that the spin canting strongly depends on the \nproperties of the topological surface states  at the Fermi level , while th ose properties are expected to vary if the film \nundergoes a phase transition from  to a mixture of  and  as well as a change in the strength of the tensil e strain in \n-Sn.  Here the Fermi properties include the Fermi velocit y, the Fermi wave  number, and the warping strength of the \nhexagonal Fermi contour, as shown in Eq. (4) in the main text.    \n \n*Y.Z, V.K., C.L. and S.S. -L.Z. contributed equally.  \n★Corresponding author. mwu@colostate.ed u     \n  \nReferences:   \n1 https://www.mtixtl.com/Si -U-c-101005S1.aspx  \n2 Ding, J. et.al. Switching of a magnet by spin -orbit torque from a topological Dirac semimetal. Adv. Mater. 2005909 \n(2021).                                                             "    },    {        "title": "1302.6437v1.A_Molecular_Platinum_Cluster_Junction__A_Single_Molecule_Switch.pdf",        "content": "arXiv:1302.6437v1  [cond-mat.mes-hall]  26 Feb 2013A MolecularPlatinumClusterJunction: A Single-MoleculeS witch\nLindaA. Zotti,∗,†EdmundLeary,†,‡Maria Soriano,†,¶Juan CarlosCuevas,§andJuanJose Palacios†\nDepartamento de Físicade laMateriaCondensada, Universid ad Autónomade Madrid, 28049 Madrid, Spain, Inst\nMadrileno Estudios Avanzados Nanociencia IMDEA,E-28049 M adrid, Spain, Departamento de FísicaAplicada,\nUniversidad de Alicante, 03690 Alicante, Spain, andDepart amento de FísicaTeórica de laMateriaCondensada,\nUniversidad Autónoma de Madrid, E-28049 Madrid, Spain\nReceived August 22, 2018; E-mail: linda.zotti@uam.es\nAbstract: Wepresent atheoretical studyoftheelectronic transport\nthrough single-molecule junctions incorporating a Pt 6metal cluster\nbound within an organic framework. We show that the insertio n of\nthismoleculebetweenapairofelectrodesleadstoafullyat omically-\nengineerednano-metallicdevicewithhighconductanceatt heFermi\nlevel and two sequential high on/off switching states. The o rigin of\nthis property canbe traced back tothe existenceof aHOMOwhi ch\nconsists of two degenerate and asymmetric orbitals, lying c lose in\nenergy to the Fermi level of the metallic leads. Their degene racy is\nbroken when the molecule is contacted to the leads, giving ri se to\ntwo resonances which become pinned close to the Fermi level a nd\ndisplay destructiveinterference.\nMolecular metal clusters have recently started being ex-\nplored in breakjunction type single-molecule-conductanc e experi-\nments,1–3whilst previous studies have also investigated clusters in\nthe STM-STS configuration which mantains a tunnel gap betwee n\nthe tip and the molecule.4,5Clusters have fewer metal atoms than\nmetal nanoparticles but a more precisely defined compositio n and\nstructure and as such they are better suited for comparative studies\nbetween theory and experiments. Bare metal clusters have be en\nsuccessfully deposited and studied on various surfaces.6,7Such\nclustershoweverrequireverycleanconditionsforstudydu etotheir\nhighreactivity. Inorder toovercome this fact,and studymo lecules\nunder the normal ambient conditions of the breakjunction ex peri-\nment, the metal atoms must be encapsulated in a ligand sphere to\navoid unwanted chemical reactions with the atmosphere. Lig and\nstabilized metal clusters (in particular those incorporat ed in an or-\nganic framework) have therefore been suggested as componen ts of\ndata storage devices where they would act as nano-capacitor s due\nto their redox properties.8Transport measurements on individual\nmolecular metal clusters are, however, still rare. Metal at oms have\nbeen successfully incorporated into organic frameworks as metal\ncomplexes for single-molecule experiments, as well as chai ns of\natoms.9–12Regarding metal clusters, the Mn 12structure has been\ninvestigatedasthefunctionalcoreinsinglemoleculemagn ets1and\ntheoretical studies about its transport properties have be en carried\nout.13,14However, despite the vast range of molecular clusters\nknown, only a few have been analyzed in this context. This is\nespeciallytrueregardingthetheoryoftheelectronictran sport.15,16\nInref.,2single-molecule experiments on\n[Pt6(µ−PtBu2)4(CO)4(S(CH2)4SH)2)3](fromnowon(SC 4S)2Pt6)\nwere carried out using the STM based I(s) technique.17,18The\nstructure ofthisPtclustermakes itideal forthese kindofm easure-\n†Universidad Autónoma deMadrid\n‡InstMadrileno Estudios AvanzadosNanocienciaIMDEA,E-28 049Madrid,\nSpain\n¶Universidad de Alicante\n§Universidad Autónoma deMadridments as the ligands which act as binding groups (the alkanet hiol\nchains) are oriented trans to each other, producing a linear wire.\nWithout these chains, it would be difficult to determine a pre cise\npoint of attachment between the molecule and the electrodes . The\npresence of these groups not only provides robust thiol anch ors,\nbut also increases the total length of the molecule in one defi ned\ndirection, allowing measured break-off distances to asses s accu-\nrately the orientation of the molecule in the junction (the n eed of\nincorporating themetal clusterina wirehas alsobeensugge sted in\na theoretical study of the electronic transport through a Mn 12clus-\nter14). Intheexperimentalstudy,itwassuggestedthattheprese nce\nof the platinum cluster increases the conductance compared to an\nalkane chain of equivalent length due to there being Pt state s in\nclose proximity to the Fermi level. These were claimed to cre ate\nan indentation in the potential barrier, giving a molecular analog\nof an inorganic double tunneling barrier, with the Pt 6unit acting\nas a well and the alkane chains acting as barriers so to decoup le\nthe central unit from the leads. This was, however, a specula tive\nargument whichwasnever fullyproven, andneither was thena ture\nof these states identified. Finding out whether this asserti on could\nbetrueisimportantbecause themolecular double tunneling barrier\nconfiguration could be the basis for creatingmolecular swit ches,19\nanalogous of conventional electronic components. This is u lti-\nmately one of the main aims of molecular electronics.20In this\nstudy presented here, we carried out first principle calcula tions\nto find out if we could corroborate the assumptions made in the\nexperimental work. We will show that the HOMO of (SC 4S)2Pt6\nconsists of two degenerate levels which get pinned to the Fer mi\nlevel when the molecule is placed between two gold electrode s.\nHowever, these two levels do not originate simply from the Pt 6\nunit, but contain states localized on the ligands also. More over,\ninterestinginterference features appear inthetransmiss ioncurve at\nthe Fermilevel.\nWe optimized all geometries with Turbomole 6.1,21using the\nBP86functional22andthedef-SVPbasisset.23Webuiltthemolec-\nular junctions by placing the molecules (previously relaxe d in the\ngas phase) between two20 gold atom clusters and then relaxed the\nmolecule and the 4(3) innermost gold atoms on each side in a to p\n(hollow)geometry. Thesubsequenttransportcalculations werecar-\nried out with ANT,24which is built as an interface to Gaussian.25\nThis code computes the electronic transmission using noneq uilib-\nrium Green’s function techniques in the spirit of the Landau er for-\nmalism,employing parametrizedtight-bindingBethelatti cesinthe\nelectrode description.26For these calculations, we used the PBE\nfunctional,27a lanl2DZ basis set for Pt atoms and the four inner-\nmost Au atoms on each side,28while we used a CRENBS basis\nset29for all the other Au atoms. A 6-31++G basis set was chosen\nforC, S,P,O,and sto-3g forH.\nWe first studied the molecule in the gas phase. The optimized\ngeometry is shown in 1. It consists of a Pt 6cluster sandwiched\nbetween two butanedithiol (four carbon atoms) chains. The c en-\n1Figure1. Centralmoietyof(SC 4S)2Pt6(a)andoptimized geometry ofthe\nwhole molecule with thetwo degenerate HOMO orbitals (b-c).\ntral cluster consists in turn of two orthogonal Pt 3triangles, con-\nfirmedbycrystallographic structuraldeterminationonthe carbonyl\nsubstituted compound.30Each Pt 3triangle contains two bridging\nphosphine groups and the remaining sides are joined togethe r to\nform a Pt tetrahedral cored. The tetrahedral core also conta ins four\nCO ligands. The molecule is slightly bent, with the angle bet ween\nthe alkyl chains less than 180o. In the second and third panel of\n1 we show the HOMO. It consists of two doubly occupied levels\nwhich are degenerate and the corresponding orbitals of whic h are\nrelated to each other by a 4-fold improper rotation. It is mai nly\nlocalized on the two innermost S atoms and on the two neighbor -\ning Pt atoms. Notice that these frontier orbitals are differ ent from\nthe HOMO shown inref30due tothe presence of the butanedithiol\nligands. We also studied a C 18H36alkyl chain terminated with S\natoms (from now on C 18), as it was claimed2to have the same\nlength as (SC 4S)2Pt6but a much lower conductance. Such a com-\nparisonwouldhelpus decidewhetherthepresence ofthePtcl uster\nprovidesasignificantchange intheconductance withrespec ttothe\nalkylchainalone. Wefirstcheckedthatthelengthofthischa inpro-\nvides a reasonable comparison to the cluster molecule. The l ength\nof C18is 24.9 Å, which is halfway between the through space S to\nS distance (23.9 Å) and the length measured by adding the 3 com -\nponents (2 chains + Pt core, 25.8 Å ) in (SC 4S)2Pt6. In C18, the\nHOMO is localizedon the Satoms, whilethe HOMO-1is delocal-\nizedthroughout the whole chain20.\nAfteranalyzing the molecules inthe gas phase, we proceeded to\nstudy junctions containing (SC 4S)2Pt6and C18bound to the gold\nclusters in a top and a hollow geometry (2). 3 shows the transm is-\nsion curves for all four cases. In the case of C 18, the energy align-\nmentseemstobeconsiderablyaffectedbythebindinggeomet ry. In\nthe hollow geometry, the HOMO-1 (which is delocalized throu gh-\noutthewholechainandgivesthemaincontributiontothetra nsmis-\nsion20) is shifted down in energy compared to the top position (as\nhas alsobeen observed for other thiolated molecules31),yielding a\nFigure 2. (Color online) Optimized geometries for (SC 4S)2Pt6and C18\nembedded between Au clusters.\ndifference in the conductance of one order of magnitude betw een\nthe two geometries. Notice alsothe presence of the HOMO (loc al-\nizedontheSatoms)veryclosetotheFermilevel,clearlyvis ibleas\na bump in the transmission curve for the top geometry. In the c ase\nof (SC4S)2Pt6, the alignment of the HOMO states does not show a\nparticular dependence on the binding geometry as they are pi nned\nat the Fermi level in both cases, but rather their splitting s eems to\nbe affected. The degeneracy of the HOMO states is broken in th e\njunction as the symmetry has changed upon geometry relaxati on.\nWe stress that the two peaks close to the Fermi level do not ari se\nsimply from the Pt 6unit as suggested in ref.,2but rather from or-\nbitals which contain both S and Pt contribution, which howev er do\nnot spread over allPtatoms, as discussed above.\nFermi level pinning has been widely studied but its nature is\nstill under discussion (see ref32and references therein). It is also\nwell known that molecular HOMO-LUMO gaps are usually un-\nderestimated in DFT and that this affects the conductance va l-\nues in DFT-based transmission calculations. In order to gai n in-\nsight intothe level alignment inour calculations and tound erstand\nwhetheritisreliable,weevaluatedtheionizationpotenti al(IP)and\nelectron affinity (EA) in the gas phase for the two molecules, as\nE(N)−E(N−1)andE(N+1)−E(N), respectively, where Nis\nthe total electron number. These quantities (based on total -energy\ndifferences) are expected to be more reliable than the HOMO a nd\nLUMO assigned by the Kohn-Sham states (at -4.27 and -3.07 eV\nfor(SC 4S)2Pt6andat-5.3and-0.08eVforC 18,respectively). The\ncalculated IP and EA in the gas phase were found to be at -5.66\nand -1.70 eV for (SC 4S)2Pt6, and -7.4 and 0.45 eV for C 18, re-\nspectively. Molecular HOMO-LUMO gaps are expected tonarro w\nwhenmolecules approach metalelectrodes due tothe screeni ngef-\nfect.33Inthecaseof(SC 4S)2Pt6,sincetheIPisveryclosetotheAu\nFermi level (which is at -5.0 eV), it is plausible that contac ting the\nmolecule to the electrode raises the IP until such a point tha t it be-\ncomes pinned to the Fermi level. This is accompanied by a char ge\ntransfer from the molecule onto the metal, confirmed by 1.47 a nd\n4.3 e positive charge found on the molecule in the top and holl ow\nposition, respectively. Notice that the transferred charg e originates\n2Figure 3. Transmission as a function of energy for all studied molecul ar\njunctions.\nmostly from the S atoms directly connected to the leads. Base d\non the proximity of the IP to the gold Fermi level, we believe t hat\nthe energy alignment of the HOMO states appearing in our tran s-\nmissioncurvesfor(SC 4S)2Pt6isrobustandunaffectedbycommon\nDFT failures, providing a realistic picture of the experime ntal sce-\nnario. Ontheotherhand,inthecaseofC 18,theIPcalculatedinthe\ngasphaseisatamuchlowerenergythanthepeakcorrespondin g to\ntheHOMO(localizedontheSatoms)inthetransmissioncurve for\nthe Au-C 18-Au junction, especially for the top binding geometry.\nThissuggeststhatitshouldalignatlowerenergy;ifso,the conduc-\ntance values of C 18would be significantly lower than those which\nwe computed.\nA further significant result is the presence of interference res-\nonances in the transmission curves, which are clearly visib le in 3.\nTheinterestininterferenceeffectsinmolecularjunction sissteadily\ngrowing34–37andthisstemsfromthefactthattheymodifythether-\nmoelectricpropertiesofthejunctions20andthattheensuingdipsin\nthe transmission curves could give rise to large on/off rati os in fu-\nture molecule-based electronic devices.36Interference effects have\nrecently been detected experimentally in molecular juncti ons.38In\nourcase,theyoriginatefromtheexistenceoftwopossiblep athways\n(thetwoHOMOstates). Forthe topposition, wefindasimilarf ea-\nture to that produced in ref.39Interferences yielding peaks at the\nFermi level have already been predicted,40,41however their occur-\nrence from two degenerate levels pinned at the Fermi level, t o the\nbestofourknowledge,israre. Inordertoshedlightontheor iginof\ntheseinterferencefeatures,webuiltasimplemodelasdepi ctedin4,\nwhereweconsider twolevelsbothconnectedtotheleads. The yare\nrelatedtothetwoHOMOlevelsof(SC 4S)2Pt6,whicharedegener-\nate and orthogonal in the gas phase. However, when the molecu le\nis placed between two electrodes, the symmetry and conseque ntly\nthe degeneracy is broken due to the geometrical readjustmen t of\nthemolecule. Thus,inourmodel,weconsidertwostateswhic hare\nno longer eigenstates of the system and are coupled by a non-z ero\nhopping matrixelement t.\nIn this model, the Hamiltonian of the central (molecular) pa rt\ntakes the following form:\nHC=\nε0t2t1t14\nt2ε2t t1\nt1tε1t2\nt14t1t2ε0\n\nwhereε1andε2aretheenergiesoftwodegeneratemolecularlevels,\nFigure4. (a)Schematicrepresentation (black)ofthetoymodeldescr ibing\nthe interference effects close to the Fermi energy and of the double tunnel\nbarrier (blue). (b) The corresponding transmission as a fun ction of energy\nfor the case of the top position (left panel) with the followi ng parameter\nvalues:Γ=0.05 eV,ε0=−0.7 eV,ε1=ε2=−0.12 eV,t1=0.009 eV,\nt2=0.005 eV,t=0.14 eV,t14=0.003 eV, and for the hollow geometry\n(right panel) with the parameter values: Γ=0.05 eV,ε0=−0.7 eV,ε1=\nε2=−0.06 eV,t1=0.005 eV,t2=0.002 eV,t=0.03 eV,t14=0.000145\neV.\ncoupled to each other via t.ε0is the energy of the interface levels,\ncoupledto ε1andε2throught1andt2. Noticethat t1mustbebigger\nthant2, in order to account for the asymmetry of each orbital. The\ntransmissionis givenby\nT(E)=4ΓLΓR|G14|2(1)\nwhere\nG=[E−HC−Σ]−1(2)\nandΣ=ΣL+ΣR, beingΣLandΣRthe self energies for left and\nright lead, respectively. These two matrices have all terms equal to\n0 butΣL11=iΓLandΣR44=iΓR, beingΓLandΓRthe coupling to\nleftand rightlead,respectively. Inourcase, ΓL=ΓR=Γas weare\nconsidering symmetric junctions.\nAs anexample, inthe inset of 4 we show the transmission curve\ngivenbythismodelforachosensetofparameters(listedint hecap-\ntion) which can reproduce the shape of the DFT-calculated tr ans-\nmission curve for both hollow and top position. This proves t hat\nthe nature of the features appearing in the transmission cur ves at\nthe Fermi level is in the interference between the two levels of the\ncentralmoietyandtwolevelslocalizedattheinterfacebet weenthis\ncentralpartandtheleads. Inthecaseofasymmetricjunctio ns,such\nas top-hollow, the transmission curve is expected to resemb le that\ncorrespondingtothetop-topcase(seetheSupportingInfor mation).\nAs mentioned above, the presence of a resonance immediately\nfollowed by an antiresonance increases the on/off ratio in g ating\nexperiments. OnecansimplyviewthisbyimaginingthattheF ermi\nlevel (located at 0 eV) is shifted to more negative values alo ng the\nenergy axis: by doing this, in the top geometry, for instance , the\nFermilevelwillcross a firstsharppeak ( on) and, aftera verysmall\nshift, a sharp dip ( off). This sequence is then repeated by shift-\n3ing the Fermi level further so as to cross the second resonanc e-\nantiresonance couple. Indeed, we propose that the low bias c on-\nductance of this system should be measured experimentally a s a\nfunction of the gate voltage, in order to test the breaking of the de-\ngeneracy.\nFinally, we compare our results with the experiments. In ref .,2\na conductance of 3 ×10−5G0was measured for (SC 4S)2Pt6, while\nthe conductance of C 18was extrapolated from the experimental at-\ntenuation (beta) value measured for a series of shorter comp ounds\nas 6×10−10G0. It was argued that the alkyl chains in (SC 4S)2Pt6\nsimply act as spacers as the frontier orbitals lie far from th e Fermi\nlevel, while the frontier orbitals of the central unit creat e a barrier\nindentation and, consequently, raise the conductance by ap proxi-\nmately 6 orders of magnitude compared to C 18. In our results, it\nis true that precisely at the very Fermi level the conductanc e of\n(SC4S)2Pt6is definitely higher than C 18; however, at other ener-\ngies immediately below or above it, the conductance is compa ra-\nble or even lower, especially as a result of the destructive i nterfer-\nence. This could potentially explain some non-linearities observed\nin some of the I-V curves recorded experimentally (see the Su p-\nporting Information of ref2). It is also worth adding that, due to\nthe uncertainty about the precise length of the Pt molecule, exactly\ncomparing the conductances of (SC 4S)2Pt6and C18is somehow\nambiguous, since, in our equilibrium geometries, C 18is straight\n(withonlyslightdefects)whereastheclustermoleculeisb ent,with\nthe Pt6cluster slightly out of the Au-Au axis. In the experiments\nthough, the Pt molecule is probably straighten, due to the pu lling\nstress applied. Regardless these uncertainties, the overa ll picture\nsupports thenaive ideabasedontheexperimental finding,co nfirm-\ning the presence of resonances at the Fermi level arising fro m the\npresence of the central additional moiety inthe molecular c luster.\nIn conclusion, we have theoretically studied single molecu le\njunctionsincorporatingaPt 6cluster,showingthatitactsinananal-\nogous fashion to a double tunneling barrier due to two (origi nally\ndegenerate) states which align at the Fermi level. These sta tes do\nnot,however, stemfromthePtunitalone,butspecificallyfr omtwo\napical Pt atoms and their neighboring S atoms. This gives ris e to\nquantum interference effects due to multiple electronic pa thways\nthrough the molecule. Due to the pinning, these effects shou ld be\neasily detectable experimentally. Despite the seemingly a pparent\ncomplexity of the structure, all the physical properties ar ise from\nthese four atoms alone, highlighting the delicate relation ship be-\ntween molecular structure and electrical properties. We pr opose\nthat further chemical synthesis combined with theoretical calcula-\ntionsshouldbeperformedinordertoexploremorecombinati onsof\nmetal clusters with organic moieties. This could provide an alter-\nnatedesignstrategyformoleculardevicesotherthanthrou ghpurely\norganicmolecules due tothewealthofdifferent structures possible\nvia inorganic chemistry. Recent progress in implementing a gate\nin single-molecule based devices42–45allows us to predict for the\nstudied molecule, with its controlled binding to the electr odes and\nwithitstwolevelsalignedattheFermilevel,thatitcouldb eusedin\nathree-terminaldeviceinwhichonlyasmallgatevoltagewo uldbe\nenough topass through twohigh/low sequential switchingst ates.\nAcknowledgement This research was supported by the Co-\nmunidad de Madrid through the project NANOBIOMAGNET\nS2009/MAT1726,bytheGeneralitatValencianathroughthep roject\nPROMETEO2012/011 and by the Spanish MICINN under the\nGrant Nos. FIS2010-21883 and CONSOLIDER CSD2007-0010.\nEL was funded by the EU through the ELFOS Network (FP7-\nICT2009-6). We thank the CCC of Universidad Autónoma de\nMadrid forcomputational resources.References\n(1) Heersche, H. B.; de Groot, Z.; Folk, J. A.; van der Zant, H. S. J.;\nRomeike, C.; Wegewijs, M. R.; Zobbi, L.; Barreca, D.; Tondel lo, E.;\nCorni,A. Phys.Rev.Lett. 2006,96,206801.\n(2) Leary, E.; van Zalinge, H.; Higgins, S. J.; Nichols, R. J. ; de Biani, F. F.;\nLeoni, P.; Marchetti, L.; Zanello, P. Phys. Chem. Chem. Phys. 2009,11,\n5198.\n(3) Boardman,B.M.;Widawsky,J.R.;Park,Y.S.;Schenck,C. L.;Venkatara-\nman, L.; Steigerwald, M. L.; Nuckolls, C. J. Am. Chem. Soc. 2011,133,\n8455.\n(4) Soldatov, E. S.; Gubin, S. P.; Maximov, I. A.; Khomutov, G . B.;\nKolesov, V. V.; Sergeev-Cherenkov, A. N.; Shorokhov, V. V.; Su-\nlaimankulov, K.S.;Suyatin,D.B. Microelectron. Engn. 2003,69,536.\n(5) Gubin,S.P.;Gulayev,Y.V.;Khomutov,G.B.;Kislov,V.V .;Kolesov,V.V.;\nSoldatov, E. S.; Sulaimankulov, K. S.; Trifonov, A. S. Nanotechnology\n2002,13,185.\n(6) Yasumatsu, H.; Hayakawa, T.; Kondow, T. Chem. Phys. Lett. 2010,487,\n279.\n(7) Isomura,N.;Wu,X.;Watanabe,Y. Journ.Chem.Phys. 2009,131,164707.\n(8) Femoni, C.; Iapalucci, M. C.; Kaswalder, F.; Longoni, G. ; Zacchini, S.\nCoordin. Chem.Rev. 2006,250,1580.\n(9) Park, J.; Pasupathy, A. N.; Goldsmith, J. I.; Chang, C.; Y aish, Y.;\nPetta, J. R.; Rinkoski, M.; Sethna, J. P.; Abruna, H. D.; McEu en, P. L.;\nRalph,D.C. Nature2002,417,722.\n(10) Ruben, M.; Landa, A.; Lörtscher, E.; Riel, H.; Mayor, M. ; Görls, H.; We-\nber,H.B.;Arnold, A.;Evers,F. Small2008,4,2229.\n(11) Mayor, M.; von Hänisch, C.; Weber, H. B.; Reichert, J.; B eckmann, D.\nAngew. Chem. 2002,41,1183.\n(12) Georgiev,V.P.;McGrady,J. E. J.Am.Chem.Soc. 2011,133, 12590.\n(13) Renani,F.R.;Kirczenow,G. Phys.Rev.B 2011,84,180408.\n(14) Barraza-Lopez, S.; Park, K.; García-Suárez, V.; Ferre r, J.J. Appl. Phys.\n2009,105,07E309.\n(15) Gerasimov, Y. S.; Shorokhov, V. V.; Soldatov, E. S.; Sni girev, O. V. Proc.\nof SPIE2010,7521,75210U–1.\n(16) Gerasimov, Y. S.; Shorokhov, V. V.; Maresov, A. G.; Sold atov, E. S.; Sni-\ngirev, O.V. J.Commun.Technol.El. + 2011,56,1483.\n(17) Haiss, W.; van Zalinge, H.; Higgins, S. J.; Bethell, D.; Höbenreich, H.;\nSchiffrin,D.J.; Nichols, R.J. J.Am.Chem.Soc 2003,125, 15294.\n(18) Nichols, R. J.; Haiss, W.; Higgins, S. J.; Leary, E.; Mar tin, S.; Bethell, D.\nPhys.Chem.Chem.Phys. 2010,12,2801.\n(19) Leary, E.; Higgins, S. J.; van Zalinge, H.; Haiss, W.; Ni chols, R. J. Chem.\nComm.2007,38,3939.\n(20) Cuevas, J. C.; Scheer,E. Molecular electronics: An introduction to theory\nand experiment ; WorldScientific,2010.\n(21) Ahlrichs,R.;Bär,M.;Häser,M.; Horn,H.;; Kölmel,C. Chem.Phys.Lett.\n1989,162,165.\n(22) Perdew,J. P. Phys.Rev.B 1986,33,8822.\n(23) Schäfer,A.;Horn,H.; Ahlrichs,R. J.Chem.Phys. 1992,97,2571.\n(24) Palacios, J. J.; Jacob, D.; Perez-Jimenez, A. J.; Fabia n, E. S.; Louis, E.;\nVerges, J. A. ALACANT ab initio quantum transport package, see URL:\nhttp://alacant.dfa.ua.es\n(25) Frish, M. J.; Trucks, G. W.; et al., H. B. S. GAUSSIAN 03, Revision\nB.01,Gaussian. Inc.,Pittsburg 2003,\n(26) Jacob,D.;Palacios,J. J. Journ. Chem.Phys. 2011,134, 044118.\n(27) Perdew,J. P.;Burke,K.;Ernzerhof,M. Phys.Rev.Lett. 1997,78, 1396.\n(28) Wadt,W.R.;Hay,P.J. J. Chem.Phys. 1985,82,284.\n(29) Ross, R. B.; Powers, J. M.; Atashroo, T.; Ermler, W. C.; L aJohn, L. A.;\nChristiansen, P.A. J.Chem.Phys. 1990,93,6654.\n(30) deBiani,F.F.;Ienco,A.;Laschi,F.;Leoni,P.;Marche tti,F.;Marchetti,L.;\nMealli, C.;Zanello,P. J.Am.Chem.Soc. 2005,127,3076.\n(31) Zotti, L. A.; Kirchner, T.; Cuevas, J. C.; Pauly, F.; Huh n, T.; Scheer, E.;\nErbe,A.Small2010,6,1529.\n(32) Bokdam,M.; Cakir,D.; Brocks,G. Appl. Phys.Lett. 2011,98,113303.\n(33) García-Lastra,J.M.;Rostgaard,C.;Rubio,A.;Thyges en,K.S.Phys.Rev.\nB2009,80, 245427.\n(34) Bergfield,J. P.; Solomon, G. C.; Stafford,C.A.; Ratner ,M. A.Nano Lett.\n2011,11,2759.\n(35) Markussen, T.;Stadler,R.;Thygesen,K.S. Nano Lett. 2010,10, 4260.\n(36) Nozaki, D.; Sevincli, H.; Avdoshenko, S. M.; Gutierrez , R.; Cuniberti, G.\narXiv:1203.5269v1\n(37) Kaliginedi, V.; Moreno-García, P.; Valkenier, H.; Hon g, W.; García-\nSuárez,V.M.; Buiter,P.;Otten, J.L.H.; Hummelen, J.C.;La mbert,C.J.;\nWandlowski, T. J.Am.Chem.Soc. 2012,134,5262.\n(38) Guedon, C.M.; Valkenier,H.; ; Markussen, T.; Thygesen , K. S.; Humme-\nlen,J. C.;van derMolen,S.J. Nat.Nanotechnol. 2012,7,305.\n(39) Satanin,A.M.; Hedin,E.R.;Joe, Y.S. Phys. Lett.A 2006,349,45.\n(40) Finch, C.M.; García-Suárez,V.M.; Lambert, C.J. Phys. Rev. B 2009,79,\n033405.\n(41) Wei,Z.;T.Li,;Jennum,K.;Santella,M.;Bovet,N.;Hu, W.;Nielsen,M.B.;\nBjørnholm, T.; Solomon, G. S.; Laursen, B. W.; Nørgaard, K. Langmuir\n2012,28,4016.\n(42) Park, H.; Park, J.; Lim, A. K. L.; Anderson, E. H.; Alivis atos, A. P.;\nMcEuen,P.L. Nature2000,407, 57.\n(43) Zhang, J.; Kuznetsov, A. M.; Medvedev, I. G.; Chi, Q.; Al brecht, T.;\nJensen,P.S.;Ulstrup, J. Chem.Rev. 2008,108,2737.\n(44) Leary, E.; Higgins, S. J.; van Zalinge, H.; Haiss, W.; Ni chols, R. J.; Ny-\ngaard,S.;Jeppesen,J.O.;Ulstrup, J. J.Am.Chem.Soc. 2008,130,12204.\n(45) Meded, V.; Bagrets, A.; Fink, K.; Chandrasekar,R.; Rub en, M.; Evers, F.;\nBernand-Mantel, A.; Seldenthuis, J. S.; Beukman, A.; van de r Zant, H.\nS.J.Phys.Rev.B 2011,83,245415.\n4"    },    {        "title": "1912.10580v3.Electronic_correlation_driven_orbital_polarization_transitions_in_the_orbital_selective_Mott_compound_Ba__2_CuO___4_δ__.pdf",        "content": "arXiv:1912.10580v3  [cond-mat.str-el]  16 Jun 2021Electronic correlation-driven orbital polarization tran sitions in the orbital-selective\nMott compound Ba 2CuO4−δ\nYu Ni1, Ya-Min Quan2, Jingyi Liu1, Yun Song1,∗and Liang-Jian Zou2†\n1Department of Physics, Beijing Normal University, Beijing 100875, China and\n2Key Laboratory of Materials Physics, Institute of Solid Sta te Physics,\nChinese Academy of Sciences, P. O. Box 1129, Hefei 230031, Ch ina\n(Dated: June 17, 2021)\nThe electronic states near the Fermi level of recently disco vered superconductor Ba 2CuO4−δ\nconsist primarily of the Cu dx2−y2andd3z2−r2orbitals. We investigate the electronic correlation\neffect and the orbital polarization of an effective two-orbit al Hubbard model mimicking the low-\nenergy physics of Ba 2CuO4−δin the hole-rich regime by utilizing the dynamical mean-fiel d theory\nwith the Lanczos method as the impurity solver. We find that the hole-overdope d Ba2CuO4−δ\nwith 3d8(Cu3+) is in the orbital-selective Mott phase (OSMP) at half-filli ng, and the typical two-\norbital feature remains in Ba 2CuO4−δwhen the electron filling approaches ne∼2.5, which closely\napproximates to the experimental hole doping for the emerge nce of the high- Tcsuperconductivity.\nWe also obtain that the orbital polarization is very stable i n the OSMP, and the multiorbital\ncorrelation can drive orbital polarization transitions. T hese results indicate that in hole-overdoped\nBa2CuO4−δthe OSMP physics and orbital polarization, local magnetic m oment and spin or orbital\nfluctuations still exist. We propose that our present result s are also applicable to Sr 2CuO4−δand\nother two-orbital cuprates, demanding an unconventional m ultiorbital superconducting scenario in\nhole-overdoped high- Tccuprates.\nI. INTRODUCTION\nThe involvement of two Cu 3 dorbitals in the super-\nconducting (SC) states in the recently discovered high-\nTcsuperconducting(HTSC) compound Ba 2CuO4−δwith\nTc=73 K,1as well as the early discovered compound\nSr2CuO4−δwithTc=95 K,2–4greatly challenges the\nprevailing single-orbital scenario in conventional HTSC\ncuprates. In the previous cuprates La 2CuO4, and\nYBa2Cu3O6,etc., the parent phases of these undoped\ncompounds are charge transfer insulator or Mott insu-\nlator, where the active Cu 3 dx2−y2orbital is singly oc-\ncupied and the ground state is N´ eelantiferromagnetic\ninsulator. Once holes are doped into the O 2 pxand 2py\norbitals, the strong O 2 p-Cu 3dhybridization and large\ncharge transfer gap form the Zhang-Rice singlet,5and an\neffective single-orbital t−Jmodel is proposed for de-\nscribing the low-energy physics of doped cuprates.6Such\nan effective single-orbital scenario addressed many ex-\nperimental results,7demonstrating the reasonability of\nthe model. The essential electronic states in Ba 2CuO4−δ\ndo not fall into this scenario: first, neither Ba 2CuO3\nnor Ba 2CuO4is a charge transfer insulator, instead, its\ncharge transfer gap is rather small; second, both the Cu\n3d3z2−r2and 3dx2−y2orbitals appear near the Fermi en-\nergy in SC Ba 2CuO4−δ.1This suggests that Ba 2CuO4−δ\nis a multiorbital superconductor. This scenario which is\ncompletely different from that of the well-known t−J\nmodel brings about the assumptions of two SC dome\nphases and orbital selective superconductivity.8,9\nThe detail inspections to the electronic properties of\nBa2CuO4−δwill provide new insight or even new sce-\nnario, especially the electronic states of SC Ba 2CuO4−δlie in Ba 2CuO4and in Ba 2CuO3. It is well-known that\nLa2CuO4and derivative cuprates are strongly correlated\nsystems. We expect that Ba 2CuO4−δis a correlated sys-\ntem, though it can be viewed as a hole-overdoped com-\npound. Recently, Liu et alproposed that Ba 2CuO3is\nan antiferromagnetic insulator10and should be the par-\nent phase; however, Maier et alsuggested that Ba 2CuO4\nshould be the parent phase.8To resolve such a dispute,\nit is crucial to clarify the role of the electronic correlation\nin Ba2CuO4−δin the hole-rich regime. Correspondingly,\none may also ask what the role of Hund’s rule coupling\nplays in such a multiorbital system and how the quan-\ntumphasesevolvewithincreasingholeconcentration.8–12\nThese issues are important since orbital and magnetic\nfluctuations are closely related to the ground-state mag-\nnetism.\nOn the other hand, we notice that compared with\nantiferromagnetically insulating Ba 2CuO3, Ba2CuO4ex-\nhibitsaparamagneticallymetallicgroundstate,10though\nit is stoichiometric 3 d7configuration. How can such a\nparamagnetically metallic phase be stable in an integer-\nfilling correlated electron system? At present it is not\nclear that what role the electronic correlation plays in\nthe paramagneticallymetallicgroundstatesofBa 2CuO4.\nMeanwhile, the orbital polarizationcharacterof the hole-\noverdoped Ba 2CuO4−δ, which is essential for SC pair-\ning symmetry, is also profoundly affected by the elec-\ntronic correlation. These facts urge us to clarify the role\nof electronic correlations in the two-orbital compound\nBa2CuO4−δ, as well as Sr 2CuO4−δ.\nIn this paper we use the dynamical mean-field theory\n(DMFT)13–15with the Lanczos method as its impurity\nsolver to investigate the influences of Coulomb correla-2\ntion and hole doping on the electronic states in a two-\norbital Hubbard model, which is applicable for the com-\npressed Ba 2CuO4−δcompound. Our results suggest that\nBa2CuO4−δcompound has a typical two-orbital charac-\nter, even when the electron filling approaches ne∼2.5,\nwhich is very close to the optimal hole doping of the\nhigh-Tcsuperconducting phase.1We demonstrate that\nthe hole-overdoped Ba 2CuO4−δis in the orbital-selective\nMott phase (OSMP) at half-filling with two electrons in\nthe two egorbitals, which is regarded as the the par-\nent phase of Ba 2CuO4−δcompound. Our results also\nshow that the orbital polarization is extremely stable in\nthe OSMP region, providing a direct evidence for the\noccurrence of the OSMP. The orbital polarization tran-\nsitions can be driven by the multiorbital correlations in\nthe Ba 2CuO4−δ, especially in the hole-rich region.\nThis paper is organized as follows. In Sec. II we\nintroduce a two-orbital Hubbard model for the hole-\noverdoped Ba 2CuO4−δ, and we explain the numerical\nmethod adopted in our study: the DMFT approach with\ntheLanczos solver. In Sec. III we demonstrate the ef-\nfects of electronic correlation and hole doping on the\nelectronic states by analyzing the phase diagrams of the\nhole-overdoped Ba 2CuO4−δ. The principal findings of\nthis paper are summarized in Sec. IV.\nII. MODEL AND METHOD\nOnaccountofthecrossingtotheFermienergyforboth\nthe two bands formed from the Cu 3 dx2−y2and 3d3z2−r2\norbitals of the compressed Ba 2CuO4−δcompound, we in-\nvestigate the electronic states described by a two-orbital\nHamiltonian H=Ht+HI, where the tight-binding (TB)\nHamiltonian Htreads8\nHt=−/summationdisplay\nij/summationdisplay\nlσt(ij)\nlld†\nilσdjlσ−/summationdisplay\nij/summationdisplay\nl/negationslash=l′,σt(ij)\nll′d†\nilσdjl′σ\n+/summationdisplay\nilσ(ǫl−µ)d†\nilσdilσ. (1)\nd†\nilσ(dilσ) is an electron creation (annihilation) operator\nfor orbital l(=1 fordx2−y2and 2 for d3z2−r2) at site i\nwith spin σ.t(ij)\n11/22andt(ij)\n12represent the intraorbital\nand interorbital hoppings between sites iandj, respec-\ntively.ǫlrepresents the on-site energy of orbital l, and\nthe crystal-field splitting is given as ǫd=ǫ1−ǫ2.µde-\nnotes the chemical potential.\nThe interaction Hamiltonian HIis exactly the same as\nthe correlation part of the standard two-orbital Hubbard\nmodel,16,17\nHI=U\n2/summationdisplay\nilσnilσnil¯σ+/summationdisplay\ni,l<l′,σσ′(U′−δσσ′JH)nilσnil′σ′\n+JH\n2/summationdisplay\ni,l/negationslash=l′,σd†\nilσd†\nil¯σdil′¯σdil′σTABLE I: Model parameters of the TB Hamiltonian of\nBa2CuO4−δat half-filling in eV.8\non-site 1sthop-2ndhop-3rdhop-\nenergy (ǫ)ping (t)ping (t′)ping (t′′)\norbitaldx2−y2-0.222 0.504 -0.067 0.130\norbitald3z2−r20.661 0.196 0.026 0.029\ninter-orbital 0-0.302 0-0.051\n+JH\n2/summationdisplay\ni,l/negationslash=l′,σσ′d†\nilσd†\nil′σ′dilσ′dil′σ, (2)\nwhereU(U′) corresponds to the intraorbital (interor-\nbital) interaction, and JHis the Hund’s rule coupling.\nFor the systems with spin rotation symmetry, we have\nU=U′+2JH.\nBa2CuO4−δcan be viewed as a hole-overdoped com-\npound. The nominal2(1- δ) holesper Cu aredoped in the\nBa2CuO4−δcompound,1giving a relation between δand\nthe hole concentration xhasδ= 1−xh/2. Accordingly,\nthe copper valence can be expressed as Cu2+xhfor a hole\nconcentration xh.8At half-filling with δ= 0.5, there are\ntwo electrons in the dx2−y2andd3z2−r2orbitals of Cu3+\nin Ba2CuO4−δ.\nBased on the DFT-calculated band structures of the\ncompressed Ba 2CuO4−δcompound with hole concentra-\ntionxh= 1,8the model parameters of the TB Hamilto-\nnianHtin Eq. (1) aregivenin Table I, including the hop-\npingparametersforthe 1 st(t), 2nd(t′), and3 rd(t′′)near-\nest neighbors, as well as the on-site energy ǫl. Through\nthe Fourier transformation, Htis changed to\nH0(k) =/summationdisplay\nkσ/summationdisplay\nll′{ξll′(k)+[ǫl−µ]δll′}d†\nlσ(k)dl′σ(k),(3)\nwith\nξ11/22(k) = 2t11/22(coskx+cosky)\n+4t′\n11/22coskxcosky\n+2t′′\n11/22(cos2kx+cos2ky),(4)\nand\nξ12(k) =ξ21(k)\n= 2t12(coskx−cosky)\n+2t′′\n12(cos2kx−cos2ky).(5)\nThe energyof the d3z2−r2orbitalvariesas ǫ2= 0.661−\n5(1−xh)with decreasingholeconcentration xh, whilethe\nenergyǫ1of the orbital dx2−y2is kept as a constant.8For\nthe conditions with xh<1, the chemical potential µhas\nbeen adjusted to keep an electron filling ne= 1 + 2δ\n(ne= 3−xh) for the hole-overdoped Ba 2CuO4−δ.\nWe map the lattice Hamiltonian on to an impurity\nmodel with fewer degrees of freedom,\nHimp=/summationdisplay\nmlσεmlσc†\nmlσcmlσ+/summationdisplay\nlσ(ǫl−µ)d†\nlσdlσ3\n+/summationdisplay\nll′mσVll′mσ/parenleftBig\nd†\nlσcml′σ+c†\nml′σdlσ/parenrightBig\n+Himp\nI(6)\nwith\nHimp\nI=U\n2/summationdisplay\nlσnlσnl¯σ+/summationdisplay\nl<l′,σσ′(U′−δσσ′JH)nlσnl′σ′\n+JH\n2/summationdisplay\nl/negationslash=l′,σd†\nlσd†\nl¯σdl′¯σdl′σ\n+JH\n2/summationdisplay\nl/negationslash=l′,σσ′d†\nlσd†\nl′σ′dlσ′dl′σ, (7)\nwherec†\nmlσ(cmlσ) denotes the creation(annihilation) op-\nerator for the bath lattice of orbital l,εmlσdenotes the\nenergy of the m-th environmental bath of orbital l, and\nVll′mσrepresents the coupling between the orbital lof\nthe impurity site and environmental bath of orbital l′.\nTheGreen’sfunctionofthetwo-orbitalimpuritymodel\ncan be expressed as a 2 ×2 matrix,\nGimp(iωn) =/parenleftbigg\nG11(iωn)G12(iωn)\nG21(iωn)G22(iωn)/parenrightbigg\n.(8)\nThe Green’s function Gimpat zero-temperature is calcu-\nlate by the Lanczos solver18–20. We choose a bath size\nnb= 3 in our calculations. It has been proved that the\ncritical values of the OSMT in a two-orbital Hubbard\nmodel are almost the same when the bath size is taken as\nnb≥3 in the DMFT calculations with Lanczos solver.21\nIn theLanczos procedure,18the diagonal matrix ele-\nments of the Green’s function Gllare expressed as\nGll(ω) =G(+)\nll(ω)+G(−)\nll(ω), (9)\nwhere\nG(+)\nll(ω) =/angbracketleftBig\nφ0/vextendsingle/vextendsingle/vextendsingledld†\nl/vextendsingle/vextendsingle/vextendsingleφ0/angbracketrightBig\nω−a(+)\n0−b(+)2\n1\nω−a(+)\n1−b(+)2\n2\nω−a(+)\n2−...,(10)\nand\nG(−)\nll(ω) =/angbracketleftBig\nφ0/vextendsingle/vextendsingle/vextendsingled†\nldl/vextendsingle/vextendsingle/vextendsingleφ0/angbracketrightBig\nω+a(−)\n0−b(−)2\n1\nω+a(−)\n1−b(−)2\n2\nω+a(−)\n2−....(11)\n|φ0/an}bracketri}htis the groundstate. a(+)\nnandb(+)\nnarethe elementsof\ntridiagonal form of the Hamiltonian matrix, constructed\nfrom the initial state d†\nl|φ0/an}bracketri}ht//radicalbigg/angbracketleftBig\nφ0/vextendsingle/vextendsingle/vextendsingledld†\nl/vextendsingle/vextendsingle/vextendsingleφ0/angbracketrightBig\n, anda(−)\nn\nandb(−)\nnare correspondingly obtained by another initial\nstatedl|φ0/an}bracketri}ht//radicalbigg/angbracketleftBig\nφ0/vextendsingle/vextendsingle/vextendsingled†\nldl/vextendsingle/vextendsingle/vextendsingleφ0/angbracketrightBig\n. The off-diagonalelementsG12andG21canbealsoobtainedbythe Lanczosmethod,\nbased on the relation\nG12(iωn) =G21(iωn)\n=1\n2/bracketleftBigg\nG1+2,1+2(iωn)−/summationdisplay\nlGll(iωn)/bracketrightBigg\n,(12)\nwhereG1+2,1+2is a combined Green’s function,\nG1+2,1+2=/an}bracketle{t/an}bracketle{td1+d2|d†\n1+d†\n2/an}bracketri}ht/an}bracketri}ht, which is calculated in\ntheLanczosmethod by replacing the operators d†\nlanddl\nasd†\n1+d†\n2andd1+d2, respectively.\nThe Weiss function of the impurity model can be ob-\ntained through the parameters of the impurity Hamilto-\nnian by\nGimp\n0(iωn)−1= (iωn+µ)I−∆(iωn),(13)\nwhereGimp\n0(iωn) and∆(iωn) are 2×2 matrices, and\nsymbolIdenotes the identity matrix. The hybridization\nmatrix∆(iωn) is defined as\n∆(iωn) =/parenleftbigg\n∆11(iωn) ∆12(iωn)\n∆21(iωn) ∆22(iωn)/parenrightbigg\n,(14)\nwith\n∆l1l2(iωn)≡/summationdisplay\nmlVl1lmVl2lm\niωn−εml. (15)\nWe calculate the 2 ×2 self-energy matrix Σimp(iωn) of\nthe impurity Hamiltonian by the Dyson equation\nΣimp(iωn) =Gimp\n0(iωn)−1−Gimp(iωn)−1,(16)\nand the 2 ×2 lattice Green’s function matrix Glat(iωn)\nis obtained by\nGlat(iωn) =1\nN/summationdisplay\nkG(iωn,k)\n=1\nN/summationdisplay\nk1\niωnI−H0(k)−Σimp(iωn),\n(17)\nwhereH0is the matrix representation of Eq. (3).\nWe build the DMFT self-consistent loop with\nGimp(iωn) =Glat(iωn) to determine the parameters εml\nandVll′m. Analytic continuation is also performed to\nobtain the real frequency Green’s function G(ω).13\nWe calculate the orbital-resolved spectral density of\neach orbital by\nAl(ω) =−1\nπImG(lat)\nll(ω+iη), (18)\nwhereηis anenergybroadeningfactor. Then, the orbital\nprojected optical conductivity can be expressed approx-\nimately as\nσl(ω) =π/integraldisplay∞\n−∞dǫDl(ǫ)/integraldisplay∞\n−∞dω′\n2πAl(ω′)Al(ω′+ω)\n×n(l)\nf(ω′)−n(l)\nf(ω′+ω)\nω, (19)4\nU (eV) 1.0 2.0 3.0 4.0 n\n0.5 1.0 1.5 Z\n0.0 0.5 1.0 \ndx2-y2   orbital \nd3z2-r2 orbital Metal O\nS\nM\nP\nInsulator (a) \n(b) \nFIG. 1: (Color online) Orbital-resolved quasiparticle wei ght\nZl(a) and electron occupation nl(b) as a function of inter-\nactionUwhenJH= 0.25Ufor Ba 2CuO4−δat half-filling.\nBetween the metallic (yellow) and insulating (gray) phases ,\nan OSMP (orange) occurs in a narrow interaction region with\n3.55 eV≤U <3.75 eV.\nwherenf(ω) is the Fermi function, and Dlrepresents the\ndensity of states (DOS) of the TB Hamiltonian. We ne-\nglect the vertex correction to the current operator, and\nthe off-diagonal elements are also neglected in our calcu-\nlations.\nTo explore possible orbital polarization and orbital or-\ndering in Ba 2CuO4−δ, we also calculate the local orbital\nsquared moment/angbracketleftbig\nT2\nz/angbracketrightbig\nof the two orbitals by22\n/angbracketleftbig\nT2\nz/angbracketrightbig\n=/an}bracketle{t(ˆnl1−ˆnl2)2/an}bracketri}ht, (20)\nfrom which we could obtain the evolution of orbital po-\nlarization with increasing electronic correlation.\nIII. RESULTS AND DISCUSSIONS\nWe study the electronic states in Ba 2CuO4−δas the\nhole doping varies from intermediate doping region to\nhighly overdoped region, and we also pay close attention\nto the optimal hole-doping region around the electron\nfillingne∼2.5, where the high- Tcsuperconductivity oc-\ncurs in Ba 2CuO4−δ. Firstly, we study the orbital selec-\ntive Mott transition (OSMT) in the strongly overdoped\nsystem for 3 d8(Cu3+) with two electrons in the two eg\norbitals, i.e., at half-filling with ne= 2, which is regarded\nas the the parent phase of Ba 2CuO4−δcompound.\nA. OSMT at half-filling\nHalf filled Ba 2CuO4−δwithne= 2 displays promi-\nnent OSMP character under the correlated effect. InU=3.5 eV A( /g90)\n0.00.10.20.3\nU=3.6 eV A( /g90)\n0.00.10.2\nU=3.8 eV \n/g90-2 -1 0 1 2 A( /g90)\n0.00.10.2 U=3.5 eV /g86/g11/g90/g12 \n0.0 0.2 0.4 0.6 \ndx2-y2     orbital \nd3z2-r2 orbital \nU=3.6 eV /g86/g11/g90/g12 \n0.000.050.10\nU=3.8 eV \n/g900 1 2 3 /g86/g11/g90/g12 \n0.0 0.2 0.4 (a)  (b) \n(c) (d) \n(e) (f) \nFIG. 2: (Color online) Effect of interaction Uon the orbital-\nresolved spectral density Al(ω) (left panel) and the corre-\nsponding orbital-dependent optical conductivity σl(ω) (right\npanel) of the two bands of Ba 2CuO4−δwithJH= 0.25Uat\nhalf-filling. The Fermi energy is denoted by a blue dotted\nline, and the energy broadening is given as η=0.05 eV.\nFig. 1(a) we present the orbital-dependent quasiparticle\nweightZl,Zl= (1−∂\n∂ωReΣl(ω)|ω=0)−1,23as a function\nof the intraorbital interaction UwhenJH= 0.25Uat\nhalf-filling. When U <3.55 eV, the two-band system is\nmetallic with finite quasiparticle weights of the two or-\nbitals. By contrast, the insulating phase with zero Zlis\nstable for both orbitals when U >3.75 eV. An OSMP\noccurs between the metallic and insulating phases with\n3.55 eV≤U <3.75 eV, in which the narrow d3z2−r2\nband behaves insulating, while the wide dx2−y2band is\nstill metallic. The U-dependence of the electron occupa-\ntionsnlare shown in Fig. 1(b). We find that the elec-\ntrons transfer from the wide dx2−y2band to the the nar-\nrowd3z2−r2band with increasing Uat half-filling. Both\nbands become singly occupied ( nl= 1) after the OSMT,\nindicating that the two electrons in the two Cu egor-\nbitals distribute uniformly in both the OSMP and insu-\nlating phase, as observed in other degenerate two-orbital\nsystems.24–27\nFig. 2 shows the orbital-resolved spectrum Al(ω) and\noptical conductivity σl(ω) obtained in different phases.\nIn the metallic phase with U= 3.5 eV, both bands have\na finite spectral weight at the Fermi level, and there is\na Drude peak in the optical conductivity accordingly, as\nshown in Fig. 2(a) and 2(b) respectively. At U=3.6 eV\na small resonance peak can be found in the DOS of the\nmetallic wide dx2−y2band, accompanied with a small\nDrude peak in its optical conductivity. Meanwhile, a\nMott gap opens around the Fermi level in the narrow\nd3z2−r2band, and the Drude weight is zero for its opti-\ncal conductivity, demonstrating the well-defined OSMP\ncharacter.28–30Insulating phase appears at U=3.8 eV,5\nU (eV) 2.0 3.0 4.0 5.0 n\n0.51.01.5Z\n0.00.51.0\ndx2-y2  orbital  \nd3z2-r2 orbital Insu- \nlator (a) \n(b) Metal \nFIG. 3: (Color online) Quasiparticle weight (a) and electro n\noccupation (b) as a function of interaction UwhenJH=\n0.125Uat half-filling. A direct transition from metallic phase\nto insulating phase is found.\nwhere a Mott gap opens in the DOS and there is no\nDrude peak in the optical conductivity for both bands as\nshown in Fig. 2(e) and 2(f), respectively.\nDecreasing the Hund’s rule coupling to JH= 0.125U,\nthe Mott transitions in the two bands occur simulta-\nneously at U=Uc1=Uc2= 4.6 eV, as shown in\nFig. 3(a). This finding is in agreement with the early re-\nsult that alarge JHpromotesthe OSMT athalf-filling by\nstrongly suppressing the coherence scale to block the or-\nbitalfluctuations.28,31–33Fig.3(b)showsthatbothbands\nbecome half-filled after the Mott transition.\nWe construct the U-JHphase diagram in Fig. 4.\nOne observes that there exists a narrow region of the\nOSMP between the weakly-correlatedmetallic phase and\nstrongly-correlated Mott insulating phase when JH>\n0.2U, which is getting to broaden with increasing JH.\nIt is obvious that large JHis beneficial to the occurrence\nof OSMP. The OSMP vanishes when JH<0.2Ubecause\nCoulomb correlation and Hund’s rule coupling are infe-\nrior to the crystal-field splitting, in agreement with the\npreviousresults.31–34In theregionof JH>0.2U, thesys-\ntem undergoes the transitions of a metallic phase to an\nOSMP and of an OSMP to an insulating phase as Uin-\ncreases. Because Ba 2CuO4−δat half-filling is at least an\nintermediate correlated system,8it should be an OSMP\ncompound, or near the edge of the OSMP.\nThere have been two possibilities regarding the par-\nent compound of the SC Ba 2CuO4−δ. One candidate is\nBa2CuO4,8and the other one is Ba 2CuO3.10Our study\nsuggests an alternative possibility that the half-filled\nBa2CuO3.5is the parent compound. Increasing the elec-\ntron filling by removing some oxygens from Ba 2CuO3.5,\nhigh-Tcsuperconductivity emerges in Ba 2CuO4−δwhen\nne∼2.5.1JH/U 0.10 0.15 0.20 0.25 0.30 0.35 0.40 U (eV) \n2.83.23.64.04.44.8\nInsulator \nOSMP Metal \nFIG. 4: (Color online) Phase diagram of Ba 2CuO4−δat half-\nfilling. The critical values of Mott transitions in the narro w\nand wide bands decrease with increasing JH/U, leading the\nOSMP region to be wider accordingly. The OSMT vanishes\nifJH/Uis smaller than 0.2.\nU2.0 2.5 3.0 3.5 4.0 4.5 5.0 JH\n0.00.20.40.60.81.00.2 \n0.4 \n0.6 \n0.8 \n1.0 \nU2.0 2.5 3.0 3.5 4.0 4.5 5.0 JH\n0.00.20.40.60.81.00.2 \n0.4 \n0.6 \n0.8 \n1.0 (a) \n(b) =\u0014\n=\u0015\nFIG. 5: (Color online) Effects of Hund’s rule coupling JHand\ninteraction Uon quasiparticle weight Zlof the wide band\ndx2−y2(upper panel) and the narrow band d3z2−r2(lower\npanel) for Ba 2CuO4−δwhenne∼2.5.\nB. Hole overdoping effect\nFocussing on the optimal hole doping for the occur-\nrence of the high- Tcsuperconductivity, in Fig. 5 we\npresent the evolution of the orbital-resolvedquasiparticle\nweightZlwith increasingmultiorbital interactions Uand\nJHin Ba2CuO4−δwhen the electron filling neis around\n2.5. Although the two bands are both good metal with6\nU=2.4 \nJH=0.5 A( /g90)\n0.00.40.8\nU=4.8 \nJH=1.5 \n/g90-2 -1 0 1 2 A( /g90)\n0.00.40.8dx2-y2  orbital \nd3z2-r2 orbital (a) \n(b) \nJH/U 0.1 0.2 0.3 0.4 0.5 n\n0.00.51.01.52.02.5\nU=2.4 \nU=3.6 \nU=4.8 (c) dx2-y2         d3z2-r2\nFIG. 6: (Color online) Orbital-resolved spectral density Al(ω)\nof Ba 2CuO4−δwithne∼2.5 for different interactions: (a)\nU=2.4 eV and JH=0.5 eV and (b) U=4.8 eV and JH=1.5 eV.\nThe energy broadening is η= 0.05 eV. (c) Orbital-resolved\nelectron occupation nlas a function of JH/Ufor different\ninteractions: U=2.4 eV, 3.6 eV and 4.8 eV when the electron\nfillingneis around 2.5.\nlargeZlin the weakly correlated region, a significant dif-\nference between the quasiparticleweight distributions for\nthe two bands can be found in the strongly correlated re-\ngion. It is obvious that the decline of the quasiparticle\nweight of the narrow d3z2−r2band is mainly driven by\nthe intraorbital interaction U. On the other hand, a bad\nmetallic character appears in the wide dx2−y2band when\nthe Hund’s rule coupling is strong enough. The two or-\nbitalsofBa 2CuO4−δdisplaydifferentcorrelationfeatures\nin the strongly correlated region when ne∼2.5.\nWe present the spectral density Al(ω) and electron\noccupation nlat different JH/Uin Fig. 6. In Fig. 6\n(a) both bands have finite spectral weight at Fermi level\nand the system displays prominent metal character when\nU=2.4 eV and JH=0.5 eV. With increasing correlations,\nwe find two soft gaps in the DOS of both orbitals when\nU=4.8 eV and JH=1.5 eV shown in Fig. 6(b), indicating\nthat the system becomes a bad metal in the strongly cor-\nrelated region. We show the orbital-dependent electron\noccupation in Fig. 6(c) when the electronfilling is around\n2.5. Differentfromtheresultsofhalf-filledsystemsshown\nin Fig. 1(b) and Fig. 3(b), electrons prefer to occupy the\nnarrowd3z2−r2band. It is worth noticing that both JH\nandUtend to uniformly distribute electrons within the\ntwo orbitals, and a finite Hund’s rule coupling can make\nthe wide dx2−y2band to be around half-filled, indicating\nthat both orbitals have significant contributions. There-\nfore, our results give strong evidences of the two-orbital\ncharacter in Ba 2CuO4−δwhenne∼2.5, which corre-\nsponds to the experimental hole-dopingconcentrationfor\nthe occurrence of the high- Tcsuperconductivity.\nTo find out the influence of the hole doping in\nBa2CuO4−δ, weextend ourstudy toawidedopingregion\nwith 2.0≤ne≤2.5. In Fig. 7 we present the spectral\ndensityAl(ω) and optical conductivity σl(ω) for differ-/g71=0.65 A( /g90)\n0.00.10.20.3ne=2.04 \n/g90-2 -1 0 1 2 A( /g90)\n0.00.20.4ne=2.04 /g86/g11/g90/g12 \n0.00.20.40.6\nne=2.30 \n/g900 1 2 3 4 5 /g86/g11/g90/g12 \n0.00.51.0\norbital dx2-y2\norbital d3z2-r2(a)  (b)\n(c) (d)ne=2.30 \nFIG. 7: (Color online) The orbital-resolved spectral densi ty\nAl(ω) (left panel) and the corresponding orbital-dependent\noptical conductivity σl(ω) (right panel) on different electron\nfillingneof Ba2CuO4−δwhenJH= 0.25UandU= 3.6 eV.\nThe energy broadening is η= 0.05 eV.\nent electron filling newhenJH= 0.25UandU= 3.6 eV,\nwhere the system should be in an OSMP at half-filling\n(ne= 2) based on the phase diagram shown in Fig. 4.\nWhen we change the electron filling to ne= 2.04, a Mott\ngap still opens in the narrow d3z2−r2band and its optical\nconductivity is zero at ω= 0 correspondingly, as shown\nin Fig. 7(a) and 7(b). Also, the wide dx2−y2band has a\nfinite resonance peak and a large Drude peak, indicating\nthat the wide-band is in a metallic phase. This indicates\nthat an OSMP also occurs near half-filling. When the\nelectron filling is changed to ne= 2.3, the finite spec-\ntral weights at Fermi level and the large Drude peaks\nfor both bands indicate that Ba 2CuO4−δtransfers to a\nmetallic phase, as shown in Fig. 7(c) and 7(d).\nne2.0 2.1 2.2 2.3 2.4 2.5 n\n0.00.40.81.21.62.0\nJH=0.2U \nJH=0.3U \nJH=0.4U U=3.6 \nne2.0 2.1 2.2 2.3 2.4 2.5 JH/U \n0.00.10.20.30.40.5\nMetalOS \nMP Insulator \ndx2-y2  d 3z2-r2(a) (b)\nFIG. 8: (Color online) (a) Phase diagram of Ba 2CuO4−δfrom\nthe intermediate hole-doping region to the overdoped regio n\nwith 2.0≤ne<2.5 when U=3.6 eV. (b) Orbital-resolved\nelectron occupation as a function of nefor different JH. The\ngreen dotted line indicates the half-filling of an orbital wi th\nn= 1. The OSMP occurs in a hole-rich regime 2 .0≤ne≤\n2.1 for a not weak JH, but the Mott insulating phase can\nonly occur at half-filling with ne= 2.0 when the Hund’s rule\ncoupling is larger as JH>0.3U.7\nIn the phase diagram shown in Fig. 8(a), an OSMP\nis found in a highly overdoped region near half-filling,\nwhich looks like an OSMP peninsula in the sea of metal-\nlic phase. If the electron filling neis more than 2.1,\nBa2CuO4−δwithU=3.6 eV can only be a metal no mat-\nter how large the Hund’s rule coupling is. On the other\nhand, the Mott insulating phase occurs only at half-\nfilling with a strong Hund’s rule coupling as J >0.3U.\nThe critical value Jc\nHfor the OSMT takes the mini-\nmum value at ne≈2.04. Besides, the disappearance\nof the OSMP when JH≤0.2Uprovides further evidence\nthat the Hund’s rule coupling can promote the OSMT,35\neven away from half-filling. Although the difference be-\ntween the electron occupancies of the two orbitals in-\ncreases with increasing ne, as shown in Fig. 8(b), the\nwidedx2−y2band is still approximately half-filled with\nn1≈0.95 when ne= 2.5 andJH= 0.3U. Our calcula-\ntions demonstrate that Ba 2CuO4−δcompound displays a\ntypical two-orbital character from the intermediate hole-\ndoping region to the overdoped region, including the op-\ntimal doping ne∼2.5 for the occurrence of the high-\nTcsuperconductivity. In the next subsection, we detect\nthe correlation driven orbital polarization transitions in\nBa2CuO4−δ, resulting from the electron transferbetween\nthe two orbitals.\nU0 1 2 3 4 5 <Tz2>\n0.00.20.40.60.8\nJH=0.25U \nJH=0.125U \nFIG. 9: (Color online) Udependence of the squared moment/angbracketleftbig\nT2\nz/angbracketrightbig\nfor different Hund’s rule coupling JH= 0.25U(black\nline) and JH= 0.125U(red line) at half-filling.\nC. Orbital Polarization\nIn Fig. 9 we present the local orbital squared moment/angbracketleftbig\nT2\nz/angbracketrightbig\nas a function of the intraorbital interaction Uat\nhalf-filling. The model Hamiltonian employed in our cal-\nculations is orbitally asymmetric, in which a largeorbital\npolarizationcan existin the metallicphase. Basedonthe\nobtained phase diagram shown in Fig. 4, we find that\nthe orbital squared moments are finite when the system\nis metallic or in the OSMP, whereas a zero squared mo-\nment/angbracketleftbig\nT2\nz/angbracketrightbig\n=0 corresponds to the insulating phase. As aresult, the Udependence of the squared moment/angbracketleftbig\nT2\nz/angbracketrightbig\ndisplays a stair-step profile with a quickly drop point,\nwhich corresponds to the happening of the Mott transi-\ntion.\nIt is worth noticing that there exists a platform in\nthe/angbracketleftbig\nT2\nz/angbracketrightbig\n-Ucurve within the interaction region 3.55 eV ≤\nU <3.75 eV, indicating that the orbital polarization is\nespecially stable in the OSMP when JH= 0.25U. Be-\ncause the narrow d3z2−r2band with localized electron\nkeeps half-filled in the OSMP, the itinerant electrons in\nthe wide dx2−y2can not transfer to the lower Hubbard\nsubband of the narrow-orbital, leading the orbital polar-\nization to be fixed. Therefore, the orbital polarization\ncan also provide strong evidence for the occurrence of\nthe OSMP.\nne2.0 2.1 2.2 2.3 2.4 2.5 <T z2>\n0.00.40.8\nJH=0.1U \nJH=0.2U \nJH=0.3U \nJH=0.4U \nJH/U 0.0 0.1 0.2 0.3 0.4 0.5 <T z2>\n0.00.40.8\nne=2.0 \nne=2.1 \nne=2.2 \nne=2.4 (a) (b)\nFIG. 10: (Color online) Orbital polarization in the hole-\noverdoped Ba 2CuO4−δwhenU= 3.6 eV. (a) Local orbital\nsquared moment as a function of nefor different Hund’s rule\ncouplings. (b) Comparing the JHdependencies of/angbracketleftbig\nT2\nz/angbracketrightbig\nfor\ndifferent ne.\nThe effect of the electron filling on the local orbital\nsquared moments is presented in Fig. 10 for Ba 2CuO4−δ\nwhen 2.0≤ne≤2.5. As shown in Fig. 10(a), the lo-\ncal orbital squared moments slightly increase with the\nincreasing of electron filling when the system is in the\nmetallic phase for ne>2.2. Near the half-filling region,\nthe Hund’s rule coupling plays an essential role:/angbracketleftbig\nT2\nz/angbracketrightbig\nde-\ncreases with increasing neforJH= 0.1U, because more\nelectrons transfer to the low energy wide dx2−y2band\nwhen the system has a large crystal-field splitting and a\nsmallJH.36When the Hund’s rule coupling increases to\nJH= 0.3U, an OSMT happens when ne≤2.1, and the\nsquared moments keep almost unchanged in the OSMP.\nThis can also be seen in Fig. 10(b).\nTotally speaking, the orbital polarizationis suppressed\nby the Hund’s rule coupling in the metallic phase, but it\nbecomes almost constant in the OSMP, as one sees in\nFig. 10(a) and 10(b). As a result, the orbital polariza-\ntions in the OSMP or near the edge of the OSMP may\nbe helpful for the occurrence of orbital-selective super-\nconductivity in Ba 2CuO4−δcompound8. Meanwhile, the\norbital polarization in metallic phase does not lead to\norbital order due to the absence of lattice distortion.8\nD. Discussion\nThus far, we may expect that in the SC Ba 2CuO4−δ,\nthere exist two types of electrons: the narrow-band elec-\ntrons near the edge of the Mott localized state, and the\nwide-band electrons, resemble to an earlier two-band hy-\npothesis by Xiang et al.37Intuitively the multiorbital\nmodel for HTSC cuprates is more reasonable than the\nsingle-orbital model: in conventional BCS superconduc-\ntors, the ionic background and phononic vibrations pro-\nvidetheSC pairingforcefieldofCooperpairs,andpaired\ncarriers are responsible for carrying the supercurrent; in\ncontrast,the carriersofthe single-orbital t−Jmodelplay\nduplicate roles, they not only create spin fluctuations to\nprovide a SC pairing force but also carry a supercurrent.\nThus, the single-orbital t−Jmodel leads to a dilemma:\nthe creation of the pairing force and carrying of the su-\npercurrent are competitive; the more the carriers partici-\npate in spin fluctuations, the less the carriers participate\nin carrying the supercurrent, and vice versa .38–40As a\ncomparison, a multiorbital superconductor could avoid\nsuch a difficulty: the electrons in one or two orbitals\ncan contribute spin or orbital fluctuations, and elec-\ntrons in another one or two orbitals contribute SC pairs\nand carry supercurrent. In the same time, the multior-\nbital and OSMP characters of the compressed compound\nBa2CuO4−δ, as well as of the compound Sr 2CuO4−δ, im-\nply that the spin fluctuations along with the orbital fluc-\ntuationsmayenhancetheSCpairingforceandgreatlylift\nTcin Ba2CuO4−δand Sr 2CuO4−δ, resembling to multi-orbital high- Tcironpnictide superconductors. Thus one\ncould understand why the SC critical temperatures of\nBa2CuO4−δand Sr 2CuO4−δare about 70 and 90 K, sig-\nnificantlylargerthanthoseofLa 2CuO4−δ, whichisabout\n30-40 K.\nIV. CONCLUSIONS\nIn summary, we study the orbital selectivity of the ef-\nfective two-orbital Hubbard model of Ba 2CuO4−δcom-\npound byusing the dynamicalmean-field theorywith the\nLanczosmethod as the impurity solver. We demonstrate\nthat Ba 2CuO4−δis an OSMP compound at half-filling or\nis near the edge of the OSMP in the optimal hole-doping\nregion, and a stable orbital polarization can be observed\nin the OSMP regime. These suggest that a local mag-\nnetic moment and spin or orbital fluctuations still exist,\nandtheOSMPandtheorbitalpolarizationaresignificant\nfeatures of the hole-overdoped Ba 2CuO4−δ. Our results\nare also applicable to Sr 2CuO4−δand other two-orbital\ncuprates. Regardingthehalf-filledBa 2CuO3.5asthepar-\nent phase, our work provides a new perspective to under-\nstand the physics in the superconducting Ba 2CuO4−δ.\nAcknowledgments\nThis work is supported by the National Natural Sci-\nence Foundation of China (NSFC) under the Grant Nos.\n11474023, 11774350 and 11174036.\n∗yunsong@bnu.edu.cn\n†zou@theory.issp.ac.cn\n1W.M. Li, J. F. Zhao, L. P. Cao, Z. Hu, Q. Z. Huang, X.\nC.Wang, Y. Liu, G. Q. Zhao, J. Zhang, Q. Q. Liu, R. Z.\nYu, Y. W. Long, H.Wu, H. J. Lin, C. T. Chen, Z. Li,\nZ. Z. Gong, Z. Guguchia, J. S. Kim, G. R. Stewart, Y.\nJ. Uemura, S. Uchida, and C. Q. Jin, PNAS 116, 12156\n(2019).\n2Q. Q. Liu, H. Yang, X. M. Qin, Y. Yu, L. X. Yang, F. Y.\nLi, R. C. Yu, C. Q. Jin, and S. Uchida, Phys. Rev. B 74,\n100506(R) (2006).\n3W. B. Gao, Q. Q. Liu, L. X. Yang, Y. Yu, F. Y. Li, C. Q.\nJin, and S. Uchida, Phys. Rev. B 80, 094523 (2009).\n4P. D. Han, L. Chang, and D. A. Payne, Physica C 228,\n129 (1994).\n5F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759(R)\n(1988).\n6K. J. von Szczepanski, T. M Rice, and F. C Zhang, Euro-\nphys. Lett. 8, 797 (1989).\n7N. P. Armitage, P. Fournier, and R. L. Greene, Rev. Mod.\nPhys.82, 2421 (2010).\n8T. Maier, T. Berlijn, and D. J. Scalapino, Phys. Rev. B\n99, 224515 (2019).\n9K. Jiang, X. Wu, J. Hu, and Z. Wang, Phys. Rev. Lett.\n121, 227002 (2018).\n10K. Liu, Z.-Y. Lu, and T. Xiang, Phys. Rev. Mater. 3,044802 (2019).\n11Z. Wang, S. Zhou, W. Chen, and F.-C. Zhang, Phys. Rev.\nB101, 180509(R) (2020).\n12Y. Li, S. Du, Z.-Y. Weng, and Z. Liu, Phys. Rev. Mater.\n4, 044801 (2020).\n13A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,\nRev. Mod. Phys. 68, 13 (1996).\n14G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O.\nParcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865\n(2006).\n15G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin,\nA. E. Antipov, M.I. Katsnelson, A.I. Lichtenstein, A.N.\nRubtsov, and K. Held, Rev.Mod. Phys. 90, 025003 (2018).\n16A. M. Ole´ s, Phys. Rev. B 28, 327 (1983).\n17A. Koga, N. Kawakami, T. M. Rice, and M. Sigrist, Phys.\nRev. B72, 045128 (2005).\n18E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).\n19M. Caffarel and W. Krauth, Phys. Rev. Lett. 72, 1545\n(1994).\n20M. Capone, L. de’ Medici, and A. Georges, Phys. Rev. B\n76, 245116 (2007).\n21Y. Niu, J. Sun, Y. Ni, J. Liu, Y. Song, and S. Feng, Phys.\nRev. B100, 075158 (2019).\n22F. Lu, W.-H. Wang, and L.-J. Zou, Phys. Rev. B 77,\n125117 (2008).\n23C. A. Perroni, H. Ishida, and A. Liebsch, Phys. Rev. B 75,9\n045125 (2007).\n24E. Jakobi, N. Bl¨ umer, and P. van Dongen, Phys. Rev. B\n80, 115109 (2009).\n25A. Koga, N. Kawakami, T. M. Rice, and M. Sigrist, Phys.\nRev. Lett. 92, 216402 (2004).\n26V. I. Anisimov, I. A. Nekrasov, D. E. Kondakov, T. M.\nRice, and M. Sigrist, Eur. Phys. J. B 25, 191 (2002).\n27Y. K. Niu, J. Sun, Y. Ni, and Y. Song, Physica B 539, 106\n(2018).\n28J. Sun, Y. Liu, and Y. Song, Acta Phys. Sin. 64, 247101\n(2015).\n29T. A. Costi and A. Liebsch, Phys. Rev. Lett. 99, 236404\n(2007).\n30L. de’ Medici, J. Mravlje, and A. Georges, Phys. Rev. Lett.\n107, 256401 (2011).\n31A. Georges, L. de’ Medici, and J. Mravlje, Annu. Rev.Condens. Matter Phys. 4, 137 (2013).\n32A. Liebsch, Phys. Rev. Lett. 95, 116402 (2005).\n33L. de’ Medici, Phys. Rev. B 83, 205112 (2011).\n34E. Jakobi, N. Bl¨ umer, and P. van Dongen, Phys. Rev. B\n87, 205135 (2013).\n35L. de’ Medici, S. R. Hassan, M. Capone, and X. Dai, Phys.\nRev. Lett. 102, 126401 (2009).\n36J. Zaanen and A. M. Ole´ s, Phys. Rev. B 48, 7197 (1993).\n37T. Xiang, H. G. Luo, D. H. Lu, K. M. Shen, and Z. X.\nShen, Phys. Rev. B 79, 014524 (2009).\n38T. Jarlborg, B. Barbiellini, R. S. Markiewicz, and A. Ban-\nsil, Phys. Rev. B 86, 235111 (2012).\n39C. de Graaf and F. Illas, Phys. Rev. B 63, 014404 (2000).\n40J. Zaanen, G. A. Sawatzky and J. W. Allen, Phys. Rev.\nLett.55, 418 (1985)."    },    {        "title": "2107.14340v1.Interplay_between_superconductivity_and_non_Fermi_liquid_at_a_quantum_critical_point_in_a_metal__VI__The__γ__model_and_its_phase_diagram_at__2___γ_3_.pdf",        "content": "Interplay between superconductivity and non-Fermi liquid at a\nquantum-critical point in a metal. VI. The \rmodel and its phase\ndiagram at 2<\r < 3\nShang-Shun Zhang,1Yi-Ming Wu,1Artem Abanov,2and Andrey V. Chubukov1\n1School of Physics and Astronomy and William I. Fine Theoretical Physics Institute,\nUniversity of Minnesota, Minneapolis, MN 55455, USA\n2Department of Physics, Texas A&M University, College Station, USA\n(Dated: August 2, 2021)\n1arXiv:2107.14340v1  [cond-mat.supr-con]  29 Jul 2021Abstract\nIn this paper, the sixth in series, we continue our analysis of the interplay between non-Fermi\nliquid and pairing in the e\u000bective low-energy model of fermions with singular dynamical interaction\nV(\nm) = \u0016g\r=j\nmj\r(the\rmodel). The model describes low-energy physics of various quantum-\ncritical metallic systems at the verge of an instability towards density or spin order, pairing of\nfermions at the half-\flled Landau level, color superconductivity, and pairing in SYK-type models.\nIn previous Papers I-V we analyzed the \rmodel for\r\u00142 and argued that the ground state is an\nordinary superconductor for \r <1, a peculiar one for 1 <\r < 2, when the phase of the gap function\nwinds up along real frequency axis due to emerging dynamical vortices in the upper half-plane of\nfrequency, and that there is a quantum phase transition at \r= 2, when the number of dynamical\nvortices becomes in\fnite. In this paper we consider larger 2 < \r < 3 and address the issue what\nhappens on the other side of this quantum transition. We argue that the system moves away from\ncriticality in that the number of dynamical vortices becomes \fnite and decreases with increasing\n\r. The ground state is again a superconductor, however a highly unconventional one with a non-\nintegrable singularity in the density of states at the lower edge of the continuum. This implies that\nthe spectrum of excited states now contains a level with a macroscopic degeneracy, proportional\nto the total number of states in the system. We argue that the phase diagram in variables ( T;\r)\ncontains two distinct superconducting phases for \r <2 and\r >2, and an intermediate pseudogap\nstate of preformed pairs.\nI. INTRODUCTION.\nThis paper continues our studies of the interplay between non-Fermi liquid (NFL) and\nsuperconductivity for itinerant fermions near a quantum-critical point (QCP) towards charge\nor spin order. The key interaction between fermions in this situation is mediated by soft\nbosonic order parameter \ructuations. When soft bosons are slow compared to electrons (e.g.,\nwhen bosons are Landau-overdamped collective modes of fermions), the low-energy physics\nis described by an e\u000bective dynamical model with 4-fermion interaction V(\n)/1=j\nj\r. At\na QCP, when order parameter propagator is massless, this form holds down to \n = 0.\nThe model with V(\n)/1=j\nj\rhas been nicknamed the \r-model. The exponent \r\nhas particular values for a growing number of speci\fc microscopic realizations: \r= 0+\n2for 3D QC-systems and for pairing of quarks, mediated by gluon exchange, \r= 1=3 for a\nsystem near a nematic QCP and for fermions at a half-\flled Landau level, \r= 1=2 near an\nantiferromagnetic QCP, \r= 0:68 for Sachdev-Ye-Kitaev (SYK) model of Nfermions coupled\nto equal number of bosons, \r= 1 for pairing by propagating bosons, \r= 2 for phonon-\nmediated pairing at vanishing Debye frequency, etc. Microscopic models with varying \r\nhave also been proposed. We listed and discussed some microscopic models in the \frst\npaper of the series (Paper I). In all cases, the same interaction, mediated by low-energy\nbosons, gives rise to fermionic self-energy, which accounts for NFL behavior in the normal\nstate, and at the same time serves as glue that binds fermions into pairs. The two tendencies\n(NFL and SC) are intertwined as they come from the same interaction, and compete with\neach other: a fermionic self-energy makes fermions incoherent and reduces the tendency to\npairing, while if bound pairs develop, they provide a feedback on the self-energy, which at\nlowest frequencies recovers the Fermi liquid form, i.e., fermions become propagating rather\nthan di\u000busive excitations.\nFor non-SYK systems, in each case SC emerges in a particular momentum channel, e.g,\nin ad\u0000wave channel near an antiferromagnetic QCP. However, once the pairing symmetry\nis incorporated and momentum integration in the formulas for the fermionic self-energy and\nthe pairing vertex is carried out, the e\u000bective low-energy model for di\u000berent microscopic\nrealizations becomes the same one, speci\fed only by the value of \r. The sign of V(\n) is\nattractive, i.e., if fermions were free, the ground state would necessarily be a superconductor.\nIn previous papers (Refs. [1{5]), which we refer to as Papers I-V, we treated \ras a\nparameter and analyzed the interplay between NFL and pairing for \r\u00142. In this paper we\nconsider\r >2. For convenience of a reader, we list some results of previous works, which\nform the base for the analysis in this paper.\n•For any\r >0, the ground state is a superconductor, i.e., superconductivity wins the\ncompetition with a NFL. However, in distinction to the pairing of coherent fermions in\na Fermi liquid, the pairing of incoherent fermions is a threshold phenomenon, and in an\nextended\rmodel with di\u000berent magnitudes of V(\n) in the particle-hole and particle-\nparticle channels ( V(\n) andV(\n)=N, respectively), there exists a \r\u0000dependent crit-\nicalNcr>1 separating a SC state for N < Ncr(including the original model with\nN= 1) and a NFL ground state for N >Ncr.\n3•In another crucial distinction from pairing in a Fermi liquid, the gap equation at a\nQCP atT= 0 has an in\fnite set of solutions \u0001 n(!), wherenruns between 0 and 1.\nAt zero frequency, \u0001 n(0)\u0018\u0016ge\u0000Anare all \fnite (\u0016 gis electron-boson coupling and Ais a\n\r\u0000dependent number). However, the n-th solution changes sign ntimes along the the\nMatsubara axis, !\u0011!m. The solutions are then topologically distinct as each zero of\n\u0001n(!m) is a center of a dynamical vortex on the upper complex plane of frequency.\nThen= 0 solution is sign-preserving and its structure along the Matsubara axis is\nsimilar to a conventional gap function in a Fermi liquid with attraction. The n=1\nsolution has an in\fnitesimally small magnitude and is the solution of the linearized\ngap equation. We presented the exact proof that the solution of the linearized gap\nequation exists along with the solutions of the non-linear gap equation. Away from a\nQCP, only a \fnite number of solution remains, and above a certain deviation from a\nQCP only the conventional n= 0 solution survives.\n•Each solution from the in\fnite set at QCP evolves with Tand vanishes at a separate\nTc;n. The largest Tc;0\u0018\u0016g, At largen,Tc;n/e\u0000An. We presented strong numerical\nevidence for the existence of the set of critical temperatures and showed that the\ncorresponding eigenfunctions change sign ntimes along the Matsubara axis.\n•For\r < 2, the set is discrete, and the largest condensation energy at T= 0 and\nthe highest Tcis for then= 0 solution. In this respect, the ground state is still a\n\\conventional\" superconductor in the sense that \u0001 0(!m) is a regular, sign-preserving\nfunction of the Matsubara frequency. Phase \ructuations of \u0001 0(!m) are weak in the\nsame parameter by which soft bosons are slow modes compared to fermions. However,\nas\rincreases towards 2, the other solutions become progressively more relevant.\nNamely, the spectrum of the condensation energy Ec;nbecomes more dense and Ec;n\nwithn >0 come closer to Ec;0. Simultaneously, the frequency range, where \u0001 n(!m)\nchanges sign ntimes, shifts to progressively smaller !m/(2\u0000\r), while at larger\nfrequencies all \u0001 n(!m) nearly coincide with \u0001 0(!m).\n•At\r= 2, a critical behavior emerges: all \u0001 n(!m) with \fnite nbecome undistinguish-\nable from \u0001 0(!m) at any!m>0, while the solutions with n!1 form a continuum\nspectrum \u0001 \u0018(!m). A continuous \u0018is the product of nand 2\u0000\r, and its value is deter-\nmined by how the double limit n!1 and\r!2 is taken. This is similar to how a\n4continuous phonon spectrum emerges in the thermodynamic limit from a discrete set\nof energy levels. The condensation energy Ec;\u0018also becomes a continuous function of\n\u0018. A visual picture is that an in\fnite set of Ec;napproaches Ec;0at\r!2 and touches\nit at\r= 2\u00000 (Fig. 1(a) and 1(b)). This creates a branch of gapless \\longitudinal\"\n\ructuations. We argued that these \ructuations destroy phase coherence at any T >0\nand give rise to pseudogap behavior at 0 <T <T p, whereTp\u0018\u0016gis a would be tran-\nsition temperature if the solutions with n>0 didn't exist. Away from a QCP, when\na pairing boson has a gap !D,Tc/!D. This last result applies to electron-phonon\npairing at small !D.\n•Extra information about the critical behavior emerges at \r!2, comes from the\nanalysis of the gap equation on the real axis. Here, V(\n)/ei\u0019\r=2is complex and\nhence \u0001 0(!) is also complex. For \r <1, ReV(\n)/cos\u0019\r=2 is positive (attractive),\nand Re\u0001 0(!) is a regular, sign-preserving function of !. The corresponding density of\nstates (DOS) vanishes at ! <\u0001 and is non-zero for larger frequencies, as is expected\non general grounds for the case when the pairing boson is massless. For \r >1, ReV(\n)\nchanges sign. We found that in this situation there appears a \fnite frequency range\nwhere the phase \u00110(!) of \u0001 0(!) =j\u00010(!)jei\u00110(!)winds up by 2 \u0019m, wheremis an\ninteger. The value of mincreases in increments of one at \r > 1, and the increase\naccelerates as \rapproaches 2. As the consequence, the DOS develops a set of maxima\nand minima in the range where the phase winds up. We extended \u0001 0(z) to complex\nzin the upper half-plane and traced the phase winding 2 \u0019mto the emergence of\nmvortices at complex z; each vortex moves from the lower to the upper frequency\nhalf-plane as \rincreases, leaving a 2 \u0019phase winding along the real axis. At \r= 2,\nthe number of vortices becomes in\fnite and the frequency range, where \u00110(!) winds\nup, extends to an in\fnity, where \u0001 0(z) develops an essential singularity. Its presence\nis a must as otherwise an extension from an in\fnite set of vortex points would give\n\u00010(z) = 0. In explicit form, the gap function along the real frequency axis at \r= 2 is\n\u00010(!)\u0018!=sin\u001e(!+i\u000e), where\u001e(x) is an increasing function of the argument [6{8].\nThe DOS for such \u0001 0(!) consists of a set of \u000e-functional peaks at frequencies where\nsin\u001e(!) =\u00061. This is qualitatively di\u000berent from a continuum DOS for \r <2. Away\nfrom a QCP (i.e., for a non-zero !D), the continuum remains, but with sharp maxima\n5FIG. 1. Condensation energy ( Ec;n) of each topologically distinct solution labeled by integer\nn= 0;1;2;:::, where (a) \r <2, (b)\r= 2, and (c) \r >2.\nand nearly zero DOS between the maxima. The other \u0001 \u0018(z) from a continuum set\nat\r= 2 also have an in\fnite number of vortices, likely at the same z, and essential\nsingularity at z=1. This clearly indicates that (i) the \r= 2 model is indeed critical\nand (ii) there is an ultimate connection between criticality and topology.\nIn this paper, we show that the new phase develops on the other side of the critical point,\nat\r > 2. We present evidence for this from calculations on the Matsubara axis and on\nthe real axis. On the Matsubara axis, we \fnd the spectrum of condensation energies Ec;n\nagain becomes a discrete one at \r >2 andEc;0is the largest. In simple words, condensation\nenergiesEc;nwithn>0 approach Ec;0as\rincreases towards 2, merge with Ec;0and form a\ncontinuous spectrum at \r= 2, and then bounce back at \r >2, re-creating the gap between\nEc;0and otherEc;n(Fig. 1 (c)). At a \frst glance it looks that the system behavior at \r >2\nis a mirror copy of that at \r < 2. However, we show that there is one crucial di\u000berence:\nfor\r > 2 the condensation energy Ec;0behaves di\u000berently from other Ec;n. This can be\nseen most explicitly in the extend \rmodel with reduced interaction in the pairing channel\nrelative to the one in the particle-hole channel. For \r < 2, allEc;n, including the one for\nn= 0, vanish simultaneously once the pairing interaction reduces below a certain threshold.\nFor\r > 2,Ec;nwithn > 0 all vanish at the threshold, while Ec;0remains \fnite, i.e., the\nn= 0 solution of the gap equation exists below the threshold for all other n. We see the\nthis last behavior in the original model: Ec;nwithn > 0 vanish at \rcr\u00192:81, whileEc;0\nremains \fnite.\n6We present another evidence for the decoupling between the n= 0 solution and the\nsolutions with n > 0, this time from the analysis of the gap function \u0001 n(z) in the upper\nhalf-plane of complex frequency, z=!0+i!00. We recall that for \r <2, \u0001n(z) with alln\nhave the same number of zeros (centra of 2 \u0019vortices) away from the Matsubara axis. The\nnumber of zeros, m, is \fnite and increases with \r. The analysis of the exact solution for\n\u00011(z) shows that these mvortices are part of an in\fnite set of vortices, which crosses into\nthe lower frequency half-plane at larger zand eventually approaches the \\critical\" line in\nthe lower half-plane, at the angle \u0019(1=2\u00001=\r), counted from the real axis. We show this\nin detail in Fig.13. There is an essential singularity at the end of this line. At \r= 2, the\ncritical line is along the real axis, what causes the special behavior of \u0001 n(!) with allnand\nessential singularity at !=1. It turns out that at \r > 2, the critical line for all n > 0\ngradually continues into the upper half-plane, and each \u0001 n(z) possesses an in\fnite number\nof zeros (vortex points), whose positions approach this line at z!1 . Forn= 0, the critical\nline bounces back into the lower half-plane, and, as a result, \u0001 0(z) has a \fnite number of\nzeros in the upper half-plane and show regular behavior at z!1 along any direction in\nthe upper half-plane.\nWe next focus on the n= 0 solution and search for qualitative changes in the physical\nproperties of the system between \r <2 and\r >2. For this we analyze the form of \u0001 0(!)\non the real axis and use it to obtain the DOS. We recall that for \r <2, the DOS, N(!) is\na continuous function of !at frequencies above the gap. We show that for \r >2, the DOS\nagain forms a gapped continuum, but there is a non-integrable singularity (an \\in\fnite\"\npeak) at the lower end of the continuum. The prefactor for this singular term increases with\n\r, initially as \r\u00002, i.e., the weight of the \\in\fnite peak\" increases with \r.\nWe extend the analysis of \u0001 0(!) andN(!) to the case when a boson has a \fnite mass,\nwhich we label !Dby analogy with the phonon case. We show that the \\in\fnite\" peak\nsurvives in a \fnite range of !D, i.e., the new structure is stable against small perturbations\nand occupies a \fnite region in the phase diagram. This state is a superconductor with a\nnon-zero super\ruid sti\u000bness \u001asatT= 0, as we explicitly show, yet it is qualitatively di\u000berent\nfrom a superconductor at \r <2. In essence the total area of the peak, divided by the total\nnumber of states, can be regarded as the \\order parameter\" of the new state.\nThe phase diagrams for the \rmodel in variables ( T;\r) at!D= 0 and in ( !D;\r) at\nT= 0 are shown in Figs. 2 and 3. To obtain these phase diagrams, we combined the\n7FIG. 2. Phase diagram on T\u0000\rplane at a QCP ( !D= 0), where the model parameter 0 \u0014\r <3.\nFIG. 3. Phase diagram on !D\u0000\rplane atT= 0K, where the model parameter 0 \u0014\r <3.\nresults for\r >2 with the results of previous papers from the series, Refs. [1{5]. At T= 0,\nthe two superconducting phases SC I and SC II merge at the critical \r= 2. SC I is a\nsuperconducting phase with conventional properties, and SC II is the new state, which we\ndiscuss in this paper, with an \\in\fnite\" peak in the DOS. At a \fnite temperature, there is\nan intermediate regime between the two ordered phases, where long range superconducting\norder is destroyed by \\longitudinal\" gap \ructuations, associated with the presence of an\nin\fnite set of low-lying states with n>0. In this regime, fermions form bound pairs, which,\nhowever, remain incoherent and do not superconduct. The observables in this regime display\npseudogap behavior, e.g., fermionic spectral function has a peak at the gap value, but the\nspectral weight below the gap remains \fnite.\n8The structure of the paper is the following. In the next section, we brie\ry review the \r\nmodel and present the gap equations along the Matsubara and the real frequency axis. In\nSec. III, we analyze the gap equation along the Matsubara axis and show that for \r >2 it\nstill has an in\fnite number of topologically distinct solutions, \u0001 n(!m), withn= 0;1;2::, like\nfor smaller \r. We present the exact solution of the linearized gap equation, \u0001 1(!m) and\nthen use it to obtain discrete solutions of the non-linear gap equation, \u0001 n(!m) (Sec. III A 3).\nIn Sec. III A 2, we discuss the structure of the n= 0 solution, \u0001 0(!m). In Sec. III B, we\nextend the model to non-equal interactions in particle-hole and particle-particle channels,\ntaking special care to avoid introducing unphysical divergencies. We show that for \r <2,\nall \u0001n(!m) disappear once the pairing interaction drops below a certain threshold, while for\n\r > 2, the solutions with all n > 0 disappear at the threshold, while the n= 0 solution\nsurvives. In Sec. III C, we extend the gap equation from Matsubara axis into the upper\nhalf-plane of frequency and show that the distinction between n= 0 andn>0 can be seen\nby analyzing the structure of dynamical vortices. In Sec. IV, we analyze the gap function\n\u00010(!) along the real axis, particularly its form near the frequency !0, where \u0001 0(!) =!. We\n\frst present, in Sec.IV A, an approximate treatment, in which we replace the integral gap\nequation by the di\u000berential one and keep only the lowest derivatives of \u0001( !). We show that\nat!\u0018!0, \u00010(!) is entirely real and \u0001 0(!)=!\u00001 scales as ( !0\u0000!\u0000i\u000e)4. In Sec. IV A 1,\nwe obtain the DOS for this \u0001 0(!) and show that it has an in\fnite peak (a non-integrable\nsingularity) at !0. In Sec. IV B, we present more accurate treatment, in which we include\nhigher-order derivatives of \u0001( !). We show that the form of \u0001( !) near!0get modi\fed,\nyet the DOS still has an in\fnite peak. In Sec. IV C, we show that this non-integrable\nsingularity can be extracted directly from the integral gap equation. In Sec. V, we extend\nthe analysis to \fnite mass of a boson and show that the in\fnite peak survives in a \fnite\nrange of the mass. We summarize our results in Sec. VI, combine them with earlier results\nfor smaller \r, and present the phase diagram of the \r-model. The phase diagram in ( T;\r)\nplane contains two di\u000berent superconducting phases and intermediate regime of preformed\npairs with pseudogap behavior of observables.\nSome technical details of calculations are moved to the Appendices. Throughout the\npaper we use !mfor fermionic frequency along the Matsubara axis (a continuous variable at\nT= 0 and a discrete one at a \fnite T,!m=\u0019T(2m+ 1)),!for fermionic frequency along\nthe real axis, and z=!0+i!00,!00>0, for complex frequency in the upper half-plane.\n9II. MODEL AND ELIASHBERG EQUATIONS\nThe\r-model is an e\u000bective model that describes low-energy fermions with dynamical\ninteraction V(\nm)/1=j\nmj\r. This model is obtained from an underlying model of itinerant\ndispersion-full fermions with interaction mediated by a soft boson near a charge or spin QCP,\nafter one integrates over momenta in the expressions for the fermionic self-energy and the\npairing vertex. When collective bosons are slow modes compared to fermions (e.g., when\nthey are Landau overdamped by fermions), the momentum integration factorizes between the\none transverse to the Fermi surface, which involves only fermionic propagators, and the one\nalong the Fermi surface, which involves the bosonic propagator between points on the Fermi\nsurface and converts it into the local propagator. At a QCP, the local bosonic propagator is\nmassless, and its frequency dependence is singular, 1 =j\nmj\r. The dimensionless interaction,\nmediated by this boson, is then V(\nm) = \u0016g\r=j\nmj\r, where \u0016gis the e\u000bective fermion-boson\ncoupling constant. The exponent \ris determined by the type of the underlying microscopic\nmodel. We refer a reader to Paper I for the list of speci\fc examples [1].\nThe interaction V(\nm) is sign-preserving on the Matsubara axis and singular at \n m!0.\nIt gives rise to two competing e\u000bects: (i) a NFL behavior in the normal state and (ii) an\nattraction in one or more pairing channels (chosen within the original model with momentum\nand frequency-dependent interaction). The two trends are described by coupled equations\nfor the fermionic self-energy \u0006( !m) and the pairing vertex \b( !m) (see Papers I-IV for the\nexact forms of these equations). One can replace these two equations by the equation for\nthe pairing gap \u0001( !m) = \b(!m)=(1 + \u0006(!m)=!m) and the inverse quasiparticle residue\nZ(!m) = 1 + \u0006(!m)=!m. One advantage of using \u0001 instead of \b is that the equation for\n\u0001(!m) can be expressed solely in terms of \u0001( !m0). In explicit form, the non-linear gap\nequation is\n\u0001(!m) = \u0016g\r\u0019TX\n!m0\u0001(!m0)\u0000\u0001(!m)!m0\n!mp\n(!m0)2+ \u00012(!m0)1\nj!m0\u0000!mj\r; (1)\nAnother advantage of using \u0001 instead of \b is that a potentially singular contribution from\nV(\nm!0), i.e., from !m0!!m, is eliminated by vanishing numerator. The cancellation\nholds both at a \fnite Tand atT= 0. At a \fnite T, the would be divergent contribution\ncomes from the term with m0=min the summation over discrete m0. It vanishes, because\nthe numerator vanishes exactly at m=m0, and this holds even if we keep a small mass\nin the bosonic propagator in intermediate calculations. We note in passing that the term\n10FIG. 4. The integration contour for C(!) in Eq. (15). The contour bypasses the point \n m= 0,\nwhere the interaction V(!) is singular andR\nd\n\nImV(!) diverges.\nwithV(0) describes thermal \ructuations, whose role for the pairing parallels that of non-\nmagnetic impurities. The cancellation of the thermal contribution can then be viewed as\na realization of the Anderson theorem. At T= 0, the integralR\nd!0=j!\u0000!0j\ris singular\nfor\r >1, but the singular behavior is eliminated as the expansion of the numerator yields\ncompensating ( !\u0000!0)2. The frequency integral then remains convergent as long as \r <3,\nwhich we consider here.\nAt the onset of the pairing, when \u0001( !m) is in\fnitesimally small, the gap equation reduces\nto\n\u0001(!m) = \u0016g\r\u0019TX\n!0m\u0012\u0001(!0\nm)\n!0\nm\u0000\u0001(!m)\n!m\u0013sgn(!m)\nj!0\nm\u0000!mj\r: (2)\nAt zero temperature, one can replace the sum over !0\nmin (1) and (2) by the integral\n\u0019TP\n!0m!(1=2)R\nd!0\nm.\nThe gap equation on the real axis is obtained by applying spectral representation to\nEq. (1) [see Refs. [6, 7, 9] and Papers I, IV and V for details]. It takes the form\n\u0001(!)B(!) =A(!) +C(!); (3)\nwhereD(!) = \u0001(!)=!, the functions A(!),B(!) andC(!) are given by Eqs. (14), (15) in\nSec. IV.\nIII. SOLUTION OF THE GAP EQUATION ALONG THE MATSUBARA AXIS\nIn this Section, we present two sets of results. First, we show that at T= 0, there\nexists an in\fnite number of topologically distinct solutions of the non-linear gap equation.\n11We label these solutions as \u0001 n(!m), where an integer nindicates how many times \u0001 n(!m)\nchanges sign along the positive Matsubara axis. We recall that we previously found that an\nin\fnite discrete set of solutions exists for 0 <\r < 2 (Papers I-IV) and becomes continuous\nat\r= 2 (paper V). Here we show that the set again becomes a discrete one for \r >2. In\nsimple words, condensation energies Ec;nwithn>0 come closer to Ec;0as\rapproaches 2,\n\\touch\" it\r= 2, where the condensation energy becomes a continuous function, and then\npull back at larger \r, leavingEc;0the largest and separated by the gap from other Ec;n.\nSecond, we show that the behavior of \u0001 0(!m) before and after \\touching\" is qualitatively\ndi\u000berent. Namely, for \r <2, \u0001 0(!) disappears simultaneously with other \u0001 n(!m) once the\npairing interaction drops below some critical value. For \r > 2, \u0001 0(!) remains non-zero\nwhen all other \u0001 n(!m) vanish. To demonstrate this explicitly, we extend the model and\nintroduce a parameter M, which distinguishes between the strength of the interaction in the\nparticle-particle and the particle-hole channel ( M= 1 in the original model). For \r < 2,\n\u0001n(!m) with alln, including n= 0, vanish at M <Mcr(\r). For\r >2, \u0001n(!m) withn>0\nstill vanish at M < Mcr(\r), but \u0001 0(!m) remains \fnite down to M= 0 and vanishes there\nin a highly non-trivial manner. Later, in Sec. IV, we analyze the gap function on the real\naxis and show that the n= 0 solution does change qualitatively compared to that for \r <2\nand yields qualitatively di\u000berent structure of the density of states.\nA. Discrete set of \u0001n(!m)for\r >2\n1. Solution of the linearized gap equation\nWe begin by showing that the solution of the linearized gap equation at T= 0 still exists\nfor\r > 2, like for smaller \r. We label this solution \u0001 1(!m) as the corresponding gap\nfunction changes sign an in\fnite number of times as a function of !m.\nAtT= 0, the linearized gap equation (2) reads\n\u00011(!m) =\u0016g\r\n2Z1\n\u00001d!m0\u0012\u00011(!m0)\n!m0\u0000\u00011(!m)\n!m\u0013sgn(!m0)\nj!m0\u0000!mj\r: (4)\nCandidate solutions of this equation can be identi\fed analytically at frequencies much larger\nand much smaller than \u0016 g. At large!m\u001d\u0016g, one can pull out 1 =j!mj\rfrom the integral and\nobtain \u00011(!m)/1=j!mj\r. At small !m\u001c\u0016g, the solution is a combination of two power-\n12FIG. 5. The parameter \f, which sets the periodicity of the logarithmic oscillation of \u0001 1(!m) at\nthe smallest !mregime, as a function of \r, where\rc'2:81.\nlaws \u0001(!m)/j!mja1;2. Substituting this form into (4) we \fnd the condition on a:\nZ1\n\u00001dxjxja\u0000sgn(x)\njx\u00001j\r= 0: (5)\nFor\r\u0014\rcr'2:81,a1;2are complex-conjugated numbers, \r=2\u0006i\r\f, where\fis determined\nfrom\n1\u0000\r\n2\u0000(\r\n2+i\f\r)\u0000(\r\n2\u0000i\f\r)\n\u0000(\r)\u0012\n1 +cosh(\u0019\r\f)\ncos(\u0019\r=2)\u0013\n= 1: (6)\n(\rcris the solution of this equation for \f= 0). We plot \f=\f(\r) in Fig. 5. The gap function\n\u00011(!m) =j!mj\r=2\u0000\nCj!ji\r\f+C\u0003j!j\u0000i\r\f\u0001\noscillates as a function of log j!mjas (C=jCjei\u001e)\n\u00011(!m\u001c\u0016g) =jCjj!mj\r=2cos(\flogj!mj+\u001e); (7)\nwhere\u001eis a free phase factor in this approximation. The infrared behavior is the same as we\npreviously found for smaller \r. It is tempting to use \u001eas a tool that allows one to smoothly\nconnect the limits of large and small !m. There is no guarantee that this is possible as the\ngap equation is integral rather than di\u000berential. In Papers I-V we went a step further and\nobtained the exact solution of the linearized gap equation at T= 0. It reproduces 1 =j!mj\r\nbehavior at large !mand log-oscillations at small !mwith some particular \u001e. This eventually\nallowed us to obtain a discrete set of solutions of the non-linear gap equation, \u0001 n(!m), in\nwhich \u00011(!m) is the smallest member. Here, we borrowed computational technique from\nPapers I-V and obtained the exact solution \u0001 1(!m) for\r >2 (up to\rcr= 2:81). The exact\nsolution again matches with analytical high-frequency and small-frequency forms, with some\n\r\u0000dependent parameter \u001e. We show \u0001 1(!m) for representative \r= 2:5 in Fig. 6. Note that\nbecause log-oscillations extend down to !m= 0, \u00011(!m) changes sign an in\fnite number\nof times, what justi\fes labeling it as n=1solution.\n13FIG. 6. The gap function \u0001 1(!m) along the Matsubara axis for \r= 2:5. The inset shows the\nlogarithmic oscillation in the infrared limit.\n2. Sign-preserving solution.\nWe now consider the opposite limit { the sign-preserving, n= 0 solution of the non-linear\ngap equation. We obtained this solution numerically and show the results in Fig. 7. In Fig.\n7 (a), we show \u0001 0(!m) for several representative 2 <\r < 3. We see that the \u0001 0(!m) has a\n\fnite value at !m= 0 and monotonically decreases with increasing !m. This is similar to the\nbehavior of \u0001 0(!m) at smaller \r. In Fig. 7 (b), we show \u0001 0(0) vs\r. For a generic \rbetween\n2 and 3, \u0001 0(0)\u0018\u0016g. At\r!3, \u0001 0(0) diverges logarithmically (Ref. [10]). For completeness,\nin Fig. 7 (c) and (d) we show the corresponding onset temperature for the pairing Tp;0and\nthe ratio \u0001 0(0)=Tp;0. The results are consistent with what has been reported earlier [10, 11].\nAt large frequencies, \u0001 0(!m) scales as 1 =j!mj\r. This form can be straightforwardly extracted\nfrom the gap equation in the same way as for the n=1solution, by pulling out 1 =j!mj\r\nfrom the integrand. For the n= 0 solution, this gives\n\u00010(!m) =Q\r;0\u0012\u0016g\nj!mj\u0013\r\n; (8)\nwhere\nQ\r;0=Z1\n0d!0\nm\u00010(!0\nm)p\n\u00012\n0(!0\nm) + (!0\nm)2: (9)\nSubstituting \u0001 0(!m)/1=j!mj\r, we \fnd that the integral is ultra-violet convergent, what\njusti\fes pulling out 1 =j!mj\r. For a generic \rbetween 2 and 3, the frequency integral in (9)\n14FIG. 7. (a) The numerical solution of the gap function \u0001 0(!m) for 2< \r < 3 at temperature\nT= 10\u00003\u0016g\u001cTp;0. (b) The amplitude of \u0001 0(0), (c) the onset temperature Tp;0and (d) the ratio\n\u00010(0)=Tp;0as a function of \r2(2;3). The dashed vertical line indicates the critical \rcabove which\nthen\u00151 solutions do not exist. Correspondingly, it appears as a kink in the curve of \u0001 0(0) and\nthe ratio \u0001 0(0)=Tp;0.\nconverges at !0\nm\u0018\u00010(0)\u0018\u0016g, henceQ\r;0is of order \u0016g. We showQ\r;0in Fig. 8. We see that\nit is indeed of order \u0016 g.\n3. Discrete set of solutions.\nFor\r <2, we demonstrated in Papers I-IV that \u0001 1(!m) and \u0001 0(!m) are the two end\npoints of an in\fnite discrete set of solutions \u0001 n(!m). A gap function labeled by nchanges\nsignntimes along the positive Matsubara axis. The set becomes continuous at \r= 2 (Paper\nV). Here we show that an in\fnite set of \u0001 n(!m) exists also for \r >2, but again becomes\ndiscrete.\nTo demonstrate this, we search for the solution of the non-linear gap equation by expand-\n15FIG. 8.Q\r;0, de\fned in Eq. (9), as a function of \r2(2;3). It is obtained from the numerical\nsolution of the non-linear gap equation shown in Fig. 7. The dashed vertical line has the same\nmeaning as in Fig. 7. A kink in this curve also appears around this critical \r.\ning to in\fnite order in \u0001( !m0) in Eq. (1). This yields\n\u0001(!m) =1X\nj=0\u000f2j+1\u0001(2j+1)(!m); (10)\nwhere \u0001(1)(!m) = \u00011(!m) from Eq. (7), \u000fis a parameter, which we adjust to get a solution.\nThe two limits we considered earlier correspond to an in\fnitesimally small \u000f, when \u0001(!m) =\n\u000f\u00011(!m), and to some \fnite \u000f=\u000f0for then= 0 solution.\nIn general, the conditions on \u000fare obtained by substituting \u0001( !m) from Eq. (10) into\nEq. (1), solving iteratively for \u0001(2j+1)in terms of \u0001(2j0+1)andj0<j, and requiring that the\nseries converge. For a BCS superconductor, the solution exists only for a single value of \u000f.\nFor the\r-model with \r\u00142, the solutions exist for a discrete set of \u000fnfor\r < 2 and for\narbitrary 0 <\u000f<\u000fmaxfor\r= 2.\nFor\r > 2, we \fnd that the solutions exist for a discrete set of \u000fn, of which \u000f0is the\nlargest. This is very similar to the case \r < 2. The details of the calculations are rather\ninvolved and we moved them to Appendix E.\nWe also compute the condensation energy for di\u000berent solutions using the expression for\nthe free energy in the \r-model in Paper I. The set of condensation energies Ec;nis discrete,\nand, as one could expect, the largest condensation energy is for the n= 0 solution. This\n1600.511.522.53012345Tp,1/g10-3Tp,1/¯gFIG. 9. The onset temperature for the pairing Tp;1as a function of \r. It is a non-monotonic\nfunction of \rand vanishes at \r=\rcr= 2:81.\nagain is very similar to what we previously found for \r <2. We illustrate this in Fig. 1 (c).\nB. Decoupling of the n= 0solution from the set\nSo far, our results for \r > 2 agree with those for \r < 2. In both cases, there exists a\ndiscrete set of \u0001 n(!m) with integer n, ranging from 0 to 1, and the condensation energy\nEc;0is the largest.\nWe now show that the analogy is only partially correct, and there is one crucial feature\non which the two cases di\u000ber qualitatively. Namely, we argue that for \r <2 the solutions\nwith allnbehave as one set, while for \r > 2, then= 0 solution decouples from the set\nand behaves di\u000berently from the other solutions with n > 0. What we mean here is that\nfor smaller \r, all \u0001n(0) disappear simultaneously once we extend the model and reduce the\nstrength of the pairing interaction below a certain value (more on this below). For \r >2, the\nsolutions with n>0 disappear under the same conditions, but the one with n= 0 survives.\nThis distinction can be seen already in the original \rmodel. As we said before, the solution\nwithn=1exists only up to \rcr= 2:81 [12]. If the solutions form a single set, the solutions\nwith non-in\fnite nshould disappear at the same \rcr. This can be veri\fed by computing\n17the corresponding onset pairing temperatures Tp;n. In Fig. 9 we plot Tp;1as a function of \r.\nWe see that it vanishes at \rcr, as we anticipated. We veri\fed that Tp;2vanishes as well, this\nleaves little doubt that all Tp;nwithn>0 vanish at \rcr. Then, at T= 0, all \u0001 n(!m) with\nn >0 vanish simultaneously at \rcr. However, we see from Fig. 6 (b) and (c) that Tp;0and\nthe gap function \u0001 0(!m) atT= 0 remain \fnite at this \r, the only signature of \rcrin these\n\fgures is a kink in the \rdependence of Tp;0and of \u0001 0(0). Clearly then, the n= 0 solution\ndecouples from the set of \u0001 n(!m) withn>0.\n1. Extended \rmodel\nTo see this more clearly and also to understand the di\u000berence between \r <2 and\r >2,\nwe extend the \rmodel in the same way as in Papers IV and V, by introducing a parameter\nM6= 1, which separates the pairing interaction and the one in the particle-hole channel.\nThe original \r-model, in which both interactions are V(\nm), corresponds to M= 1. We\nintroduceM6= 1 in such a way that the pairing interaction gets weaker at M < 1. The\nextension has to be done carefully to avoid emerging singularities fromR\nd!m0=j!m\u0000!m0j\r,\nwhich cancel out in the gap equation at M= 1 (see Eq. (1)).\nWe already used this extension for di\u000berent purposes in Papers IV and V. There, we\nderived the modi\fed gap equation:\nD(\u0016!m) \n\u0016!m+1\u0000M\n2Zd\u0016!0\nm\nj\u0016!m\u0000\u0016!0\nmj\r \nsign\u0016!mp\n1 +D2(\u0016!m)\u0000sign\u0016!0\nmp\n1 +D2(\u0016!0\nm)!!\n=\n1\n2Zd\u0016!0\nm\nj\u0016!m\u0000\u0016!0\nmj\rD(\u0016!0\nm)\u0000D(\u0016!m)p\n1 +D2(\u0016!0\nm)sign\u0016!0\nm (11)\nwhereD(\u0016!m) = \u0001(!m)=!m, \u0016!m=!m=\u0016gMand \u0016gM= \u0016g=M1=\r. AtM= 1, Eq. (11) reduces\nto Eq. (1)\nThe extended model has the same structure of solutions as the original one: there is a\ndiscrete set of solutions \u0001 n(!m) for\r <2 and\r >2 and a continuous set for \r= 2. The end\npoint, \u00011(!m) is the solution of the linearized gap equation. Like for the original model,\nat small!m, \u00011(!m)/j!mj\r=2cos(\flogj\u0016!mj\r+\u001e). The parameter \fmust be real, which\nrestrictsMtoM\u0015Mcr(\r). The critical value is\nMcr(\r) =1\u0000\r\n2\u00002(\r\n2)\n\u0000(\r)\u0012\n1 +1\ncos(\u0019\r=2)\u0013\n: (12)\n18FIG. 10. The critical value of the parameter M(see text) as a function of \r. At\r\u00142,Mcris\nnon-zero and the same for all solutions with n\u00150. At\r > 2, criticalMis zero for the n= 0\nsolution (red dots) and is \fnite (and the same) for all other solutions with n\u00151.\nWe plotMcrvs\rin Fig. 10. The solution with n=1exists in the blue area in this \fgure.\nThe boundary crosses M= 1 at\rcr= 2:81, as we found earlier.\nWe obtained numerically the onset temperatures for the pairing Tp;n(M). For\r\u00142 we\nfound that all Tp;nvanish at the same M=Mcr. This implies at T= 0, \u0001n(!m) with all\nn, including n= 0, vanish upon approaching the critical line Mcr(\r) from above. We show\nthe behavior of \u0001 0(!m) in Fig. 12 (a) and illustrate this result in Fig. 11 (a). For \r= 2,\nMcr= 0. The set is continuous, and all gap functions from the set vanish upon approaching\nMcr= 0 from above (see Fig. 11 (b)).\nFor\r >2, the result is di\u000berent. The onset temperatures Tp;nwithn>0 still vanish at\nMcr>0, along with the corresponding \u0001 n(!m) atT= 0. However, Tp;0and \u0001 0(!m) remain\n\fnite atMcr(see Fig. 11 (c) for illustration). We show the numerical results for \u0001 0(!m) at\ndi\u000berentMin Fig. 12 (a) for representative \r= 2:5 (Mcr= 0:192). This clearly shows that\nfor\r >2 the solution with n= 0 decouples from the set of solutions with n\u00151. A non-zero\n\u00010(!m) exists down to M= 0 (the red dashed line in Fig. 10), where it vanishes in a rather\npeculiar way: \u0001 0(0) gradually tends to zero at M!0, while the full function \u0001 0(!m)\nremains \fnite (see Fig. 12 (b)) and at M= 0+ becomes the end point of a continuum of\nsolutions (see Appendix F for details).\n19FIG. 11. Onset pairing temperature ( Tp;n) of each topologically distinct solution labeled by integer\nn= 0;1;2;:::, where (a) \r <2, (b)\r= 2, and (c) \r >2.\nFIG. 12. Solution of the extended model, Eq. (11), for various values of Min the vicinity of the\ncritical point, where (a) \r= 1:5 and (b)\r= 2:5.\nC. Disparity between gap functions with n= 0andn>0in the upper frequency\nhalf-plane\nWe now present complimentary evidence for qualitative di\u000berence between the gap func-\ntions withn= 0 andn>0, by extending \u0001 n(!m) from the Matsubara axis into the upper\nhalf-plane of frequency. To obtain \u0001 n(z), wherez=!0+i!00and!00>0, we \frst obtained\n\u0001n(!) along the real axis by solving the gap equation (3) and then used Cauchy relation\n\u0001n(z) =2\n\u0019Z1\n0d!!\u000100\nn(!)\n!2\u0000z2; (13)\n20FIG. 13. Schematic plot of the locations of dynamical vortices on the complex frequency plane for\n\u0001n(z): (a)\r <2, alln\u00150, (b)\r >2,n>0, and (c),\r >2,n= 0. Vortices along the Matsubara\naxis for the gap functions with n>0 are not shown.\nWe will discuss the gap function along the real axis in the next section. Here, we focus on\nzeros of \u0001 n(!) away from the Matsubara axis, i.e., at the dynamical vortices at complex z.\nIn Paper IV we showed that the vortices appear at \r > 1. For\r < 2, the number of\nvortices is \fnite and the same for all n, including n= 0, which is another evidence that\ngap functions with all nare members of the same set. The structure of vortices can be\nunderstood by extending the exact solution for n=1to complex z. This analysis shows\n(see Appendix D for details) that vortices are located above the line in the lower frequency\nhalf-plane, at the angle \u0019=2\u0000\u0019=\r, counted from the real axis, see Fig. 13 (a). (to obtain\nthis, we allowed zto move into the lower frequency half-plane). As \rapproaches 2 from\nbelow, more vortices cross from the lower to the upper frequency half-plane. At \r= 2, the\nline coincides with the real axis, and the number of vortices in the upper frequency half-\nplane becomes in\fnite. Again, this behavior holds for all n\u00150, and our numerical analysis\nin Paper V shows that even the locations of vortices are the same for all n. The set of\nvortices ends up at an essential singularity at z=1. Its presence is crucial as otherwise the\nextension from an in\fnite set of vortex points would give rise to zero gap function everywhere\nin the upper half-plane, including the Matsubara axis.\nFor\r >2, our numerical analysis shows di\u000berent behavior, schematically shown in panels\n(b) and (c) in Fig. 13. Namely, \u0001 n(z) withn>0 still possess an in\fnite number of vortices\nabove the direction in the upper half-plane, speci\fed by the angle \u0019=2\u0000\u0019=\r, see Fig. 13 (b).\nHowever, for the n= 0 solution, the corresponding axis bounds back into the lower half-\n21FIG. 14. The plot of the phase of the gap function \u0001 n(z) =j\u0001n(z)jei\u0011n(z)in the upper half-plane\nof complex frequency z=!0+i!00for representative \r= 2:7. The positions of the vortices are\nmarked by red dots. Left panel, n= 0 (obtained by using the approximated approach based on\nEq. (17)). There is a \fnite number of vortices in the upper half-plane. Right panel - n=1. The\nnumber of vortices is in\fnite; all vortices are located above the direction speci\fed by the angle\n\u0019=2(\r\u00002)=\rcounted from the real axis.\nplane, and as the consequence, the number of vortices in the upper half-plane of frequency\nbecomes \fnite, Fig. 13 (c). In Fig. 14, we present the numerical results for \u0001 0(z) and \u00011(z)\nwhich show this behavior for representative \r= 2:7. The result for \u0001 1(z) was obtained\nby analytical continuation of the exact solution on the Matsubara axis to complex zin the\nupper frequency half-plane. We clearly see that for \r >2, the solution with n= 0 decouples\nfrom other solutions with n > 0, like we found above in the Matsubara axis analysis. We\nalso note that for \r >2, \u0001n(z) withn>0 are non-zero because of essential singularity at\nthe end point of the set of vortices at z=1. Without it, an analytic continuation from\nthe in\fnite set of vortex points into the upper half-plane would give \u0001 n>0(z) = 0. That\nall non-zero \u0001 n>0(z) emerge at \r >2 as a multi-valued extension from the same essential\nsingularity is fully consistent with our earlier results that these solutions form the set with\nthe same qualitative behavior of all members. The n= 0 solution, on the other hand, does\nnot come from an essential singularity and therefore is not a part of the set (at large z,\n\u00010(z)\u00181=jzj\reverywhere in the upper half-plane).\n22IV. GAP EQUATION ALONG THE REAL FREQUENCY AXIS\nWe now address the issue of whether there are any qualitative di\u000berences in the behavior\nof observables at T= 0 between \r <2 and\r >2. In both cases the condensation energy\nis the largest for the n= 0 solution, so we focus on the form of \u0001 0(z). For de\fniteness, we\nfocus on the original \rmodel with M= 1. We found earlier in this paper that the forms of\n\u00010(!m) on the Matsubara axis at \r <2 and\r >2 are very similar. In both cases, the gap\nhas a \fnite value at zero frequency, decreases monotonically with !m, and scales as 1 =j!mj\r\nat the largest !m. Vortex structure at complex zis also similar { in both cases there is\na \fnite number of vortices in the upper frequency half-plane. Below we analyze the gap\nfunction along the real axis. We show that its form changes qualitatively between\r <2 and\n\r >2, and the change in \u0001 0(!) leads to new feature in the DOS for \r >2: the appearance\nof a bound state inside the gap with degeneracy proportional to the total number of particles\nin the system. This bound state shows up as a \u000e-functional peak in the DOS with an in\fnite\nweight in the thermodynamic limit. We show later that this feature is robust against weak\nperturbations, in particular it survives when a pairing boson is massive, as long as the mass\nvalue is below a \fnite threshold.\nWe now analyze the gap function \u0001 0(!). At the smallest and the largest frequencies,\n\u00010(!) can be obtained by a direct rotation from the Matsubara axis, i.e., by replacing i!m\nby!+i0+. This yields \u0001 0(!)\u0019\u00010(0) (1\u0000O((!=\u0016g)2) +:::) atj!j\u001c\u0016gand \u0001 0(! > 0)\u0019\nQ0;\rei\u0019\r=2=(!+i0+)\ratj!j\u001d \u0016grespectively. However, to obtain the form of \u0001 0(!) at\nintermediate !\u0015\u0016gone has to solve the non-linear gap equation in real frequencies, Eq. (3).\nThis equation contains three functions of frequency, A(!),B(!), andC(!). The functions\nA(!) andB(!) can be expressed in terms of the gap function on the Matsubara axis:\nA(!) =\u0019TX\n!m>0D(!m)p\n1 +D2(!m)\u0012\u0016g\r\n(!m+i!)\r+\u0016g\r\n(!m\u0000i!)\r\u0013\n;\nB(!) = 1 +i\u0019\n!TX\n!m>01p\n1 +D2(!m)\u0012\u0016g\r\n(!m+i!)\r\u0000\u0016g\r\n(!m\u0000i!)\r\u0013\n;(14)\nwhere, we remind, D(!m) = \u0001(!m)=!m. For then= 0 solution, \u0001( !m) = \u0001 0(!m). Using\nthe fact that \u0001 0(!m) is monotonically decreasing function of frequency, one can verify that\nat!\u0015\u0016g,A(!) andB(!) can be well approximated by A(!)'Q\r;0(\u0016g=j!j)\rcos\u0019\r\n2and\nB(!)'1. The function C(!) on the other hand is not expressed in terms of \u0001( !m). The\n23functionC(!) depends on the gap function\nC(!) =i\u0016g\rsin(\u0019\r\n2)Zj!j\n0+d\n\n\rD(j!j\u0000\n)\u0000D(j!j)p\n1\u0000D2(j!j\u0000\n): (15)\nThe 0+in the lower limit of the integral in (15) implies that a special care is needed to\nproperly treat the limit \n !0, as the integrand in (15) is of order 1 =\n\r\u00001at small \n,\nandR\nd\n=\n\r\u00001is infra-red divergent. The divergence is eliminated by slightly shifting the\nintegration contour into the upper half-plane of frequency, as shown in Fig. 4. Such shift\nis necessary to satisfy the Kramers-Kronig (KK) relation for the interaction on the real\naxis:V(!) = (1=\u0019)R\ndxV00(x)=(x\u0000!\u0000i\u000e) (see Appendix A). For practical purposes, the\ncorrect result for C(!) is obtained by integrating in (15) along the real axis down to an\nin\fnitesimally small but \fnite \u000fand subtracting from the integral\n\u00001\n\r\u000021\n\u000f\r\u00002dD(!)\nd!p\n1\u0000D2(!)(16)\nThe gap equation with C(!) given by (15) is an integral equation, even with the analytic\nexpressions for A(!) andB(!). In Papers IV and V we converted this equation into an\napproximate di\u000berential equation by Taylor expanding the integrand in (15). We use the\nsame approach here. We present the results in two steps. First, we restrict with the lowest-\norder derivatives, like we did in Paper IV for \r < 2 and show that a qualitatively new\nbehavior emerges for \r > 2. Then we present the results for the gap function at \r > 2,\nobtained by expanding to an in\fnite order in derivatives. We show that the new feature,\ndetected in the \frst procedure, remains.\nLike in Papers IV and V, we follow Refs. [6, 7] and express the gap function as \u0001 0(!) =\n!=sin[\u001e0(!)], where\u001e(!) is in general a complex function of frequency. At \r= 2,C(!)\ncontains only the term with _\u001e0. The equation on \u001e0then reduces to _\u001e0= (2=\u0019\u0016g2)(!+\nQ2;0(\u0016g=!)2sin\u001e0), whereQ2;0is given by Eq. (9). The solution of this equation is a mono-\ntonically increasing realfunction\u001e0(!) (Ref. [6]) This leads to rather peculiar behavior of\n\u00010(!) and the DOS consisting of a set of \u000e-functions at !pwhere\u001e0(!p) =\u0019=2 +p\u0019(pis\nan integer). For \r6= 2, there appear an in\fnite number of other terms with the derivatives\nof\u001e0, all with coe\u000ecients O(\r\u00002). To get some physics insight, below we \frst consider\nthe toy model, in which keep only one of the leading new term - the one with _\u001e2\n0. This\ntoy model already shows that the system behavior at \r >2 is qualitatively di\u000berent from\nthat at\r <2. Then we consider the actual gap equation and sum up series of terms with\n24FIG. 15. Distinct behaviors of the solution \u001e0(!) around!=!0(vertical blue line) for (a) \r= 2:1\nand (b)\r= 1:9, which are obtained by solving Eq. (17) numerically.\nhigher-order derivatives and higher powers of _\u001e0. We show that the structure of the gap\nfunction changes a bit, compared with the toy model, but qualitative di\u000berence between\n\r <2 and\r >2 holds.\nA. Expansion to order _\u001e2\n0(!)\nFor convenience, we keep \rclose to 2 and expand to \frst order in \r\u00002. Expanding in\nthe integrand for C(!) to order _\u001e2\n0, we express the gap equation as\n_\u001e0+!\u000e_\u001e2\n0tan\u001e0=2\n\u0019\u0016g\r\u0012\n!\r\u00001\u0000Q\r;0\u0016g\r\n!2ei\u0019\r=2sin\u001e0\u0013\n; (17)\nwhere\u000e= (\r\u00002)=2.\nWe are interested in the behavior of \u001e0(!) at!\u0015\u0016g, where our approximations for A(!)\nandB(!) are valid. The boundary condition for (17) can be set at some initial \u001e0< \u0019= 2\nat, e.g.,!= \u0016g. Extending from this to larger !, we see that \u001e0(!) increases with !and\nremains real as long as \u001e0remains smaller than \u0019=2. At these frequencies, the Q\r;0term is\nsmaller than !and can be safely neglected. Eq. (17) then becomes the quadratic equation\non_\u001e0. Solving it and choosing the solution that matches the boundary condition, we obtain\n_\u001e0=\u00001\n2\u000e!tan\u001e0\"\n1\u0000s\n1 +8\u000etan\u001e0\n\u0019\u0012!\n\u0016g\u0013\r#\n: (18)\nA simple analysis of this equation shows that, as we anticipated, the behavior of \u001e0(!)\nat\r < 2 and at\r > 2 is qualitatively di\u000berent. Indeed, at \r < 2, when\u000e < 0,\u001e0(!)\nbecomes complex at (8 =\u0019)j\u000ejtan\u001e0(!=\u0016g)\r= 1, before \u001e0reaches\u0019=2. For a complex \u001e0,\n25FIG. 16. (a) Solution of the di\u000berential equation Eq. (17) at \r= 2:7, where the Q0;\rterm has been\nconsidered. Gray lines are asymptotic behaviors at large frequency given in Eq. (21). (b) shows\nthe gap function \u0001 0(!) =!=sin\u001e0(!) and (c) the phase \u00110(!) = Arg[\u0001 0(!)].\ntan\u001e0is non-singular, and \u001e0(!) evolves smoothly with !{ its real part increases up to\nsome value and then saturates when Q\r;0term becomes relevant, while Im \u001e0(!) increases\nlogarithmically at large !, such thatQ\r;0(\u0016g\r=!2) sin\u001e0\u0019!\r\u00001. The high-frequency behavior\nyields \u0001 0(!)/1=!\r.\nFor\r >2,\u000e>0. Now\u001e0remains real all the way up to a frequency, !0, where\u001e0=\u0019=2\nand tan\u001e0diverges. An elementary analysis of (18) shows that _\u001e0then vanishes upon\napproaching this point. Expanding at !\u0014!0, we obtain from (18)\n\u001e0=\u0019\n2\u00001\n2\u0019\u000e!\r\u00002\n0\n\u0016g\r(!\u0000!0)2\u00001\n8\u0019\u000e2!\r\u00003\n0\n\u0016g\r(!\u0000!0)3+:::: (19)\nWe see that \u001e0now approaches \u0019=2 horizontally. One can verify that Eq. (19) also holds for\n!\u0015!0, this requires one to choose another branch of the solution of the quadratic equation\non_\u001e0.\nWe see that \u001e0initially increases with !and approaches \u0019=2 quadratically, and then\nbends back to smaller values. We veri\fed this result by solving the full Eq. (17) numerically.\nIn Fig. 15, we show numerical results for \r= 1:9 and\r= 2:1. We see that, in the \frst case,\nRe\u001e0(!) increases monotonically and Im \u001e0(!) emerges before Re \u001e0(!) reaches\u0019=2. In\nthe second case, \u001e0(!) remains real and varies quadratically near !0, where\u001e0(!0) =\u0019=2.\nAt! > ! 0,\u001e0(!) decreases and remains real, until it reaches \u001e0= 0 at some !1> ! 0.\n26At around this frequency,\n\u001e0'r\n2\n\u000e!1(!1\u0000!\u0000i0+) +4\n3\u0019!\r\u00001\n1\n\u0016g\r(!1\u0000!) +:::: (20)\nWe again veri\fed this behavior by solving numerically the full di\u000berential equation (17).\nWe show the result in Fig. 16 (a). We clearly see that Im \u001e0(!) emerges at a frequency\n!1> ! 0. As!increases above !1, Im\u001e0increases in amplitude. When it becomes large\nenough, tan \u001e0approaches\u0000i, and the solution of Eq.(17) without the Q\r;0term becomes\nRe\u001e0\u0018Im\u001e0\u0018\u0000p\n1=(\u0019\u000e\u0016g\r)!\r=2. This behavior is clearly reproduced in Fig. 16 (a). As\n!increases further, jIm\u001e0jcontinue increasing, hence sin \u001e0increases and above a certain\nfrequency, the Q\r;0term becomes comparable to the !\r\u00001term. At even larger frequencies,\nthe balance between these two terms holds, and we obtain\nRe\u001e0=\u00002m\u0019\u0000(\r\u00001)\u0019\n2;\nIm\u001e0=\u0000log2\u0016g\nQ0;\r\u0000(1 +\r) log!\n\u0016g: (21)\nwheremis the number of the additional vortices in the \frst quadrant. This yields \u0001 0(!)/\n1=!\r, as it should be.\nWe plot Re\u0001 0(!) and Im\u0001 0(!) in Fig. 16 (b) and the phase \u00110(!) of \u0001 0(!) =\nj\u00010(!)jei\u00110(!)in Fig. 16 (c). The phase undergoes two slips by 2 \u0019at positive !, con-\nsistent with the presence of two vortices in the \frst quadrant of the complex plane of\nfrequency (see Fig. 14 (a)).\n1. Density of states\nThe density of single-electron states is de\fned as N(!) = (\u0000N0=\u0019)ImGl(!), whereN0is\nthe DOS in the normal state and Gl(!) is the (retarded) single-electron Green's function,\nintegrated over the dispersion:\nGl(!) =\u0000i\u0019s\n!2\n!2\u0000\u00012\n0(!); (22)\nIn terms of \u001e0(!),N(!) =N0Rep\n\u0000tan2\u001e0.\nOne can easily verify that the DOS vanishes at small frequencies, as expected for a super-\nconductor with a \fnite gap, and is non-zero at frequencies !>! 1, where Im \u001e0(!) is \fnite.\n27FIG. 17. The density of states N(!) in the approximate approach, described by Eq. (17). We set\n\r= 2:1.\nIt is tempting to call !1a spectral gap, by analogy with a BCS/Eliashberg superconductor.\nFor\r <2, there are no other features in the DOS, although there is a structure inside the\ncontinuum. For \r >2, there is also a continuum above !1, but in addition, there is a level\ninside the continuum, at !=!0, where tan \u001e0diverges and an imaginary part appears once\nwe shift!in the upper frequency half-plane by an in\fnitesimally small amount. Moreover,\nbecauseN(!)\u00181=(!\u0000!0+i0+)2, the integral of the DOS over a narrow range around !0\ndiverges. The prefactor for the divergent term scales as \u000e, the capacity of the level is pro-\nportional to \r\u00002. We show the result of numerical evaluation of the DOS for representative\n\r= 2:1 in Fig. 17. We clearly see that the DOS has a continuum, which starts at !1, and\nan in-gap state at !0<! 1with the \\in\fnite\" weight, comparable to the total weight of the\ncontinuum. We will see below that the DOS for the actual model also contains an \\in\fnite\"\npeak, but it is located at the lower end of the continuum and shows up as a non-integrable\nsingularity.\nB. Equation for \u001e0(!)with derivatives to all orders\nWe now analyze whether the results from the previous section survive if we add higher-\norder derivative. These higher-order derivatives appear in combination with higher powers\nof tan\u001e0, which diverges at !=!0. It is then a'priori unclear whether the macroscopically\ndegenerate level at !0survives once we include higher-order terms. We show below that it\n28FIG. 18. Evolution of DOS obtained by solving Eq. (24) for di\u000berent \rapproaching 2 from above.\nThe red dotted line is the analytic result, Eq. (28).\ndoes survive.\nThe analysis is rather involved and we present the details in Appendix C. There are\ntwo types of terms in the expansion of C(!) in the derivatives of \u001e0(!): terms with higher\npowers of _\u001e0, combined with higher powers of tan \u001e0, and terms with higher derivatives\nof\u001e0, see Eq. (C2). We argue in Appendix C that the terms with higher derivatives are\nirrelevant, but the terms with higher powers of _\u001e0must be kept. These last terms form\nseries inX=!_\u001e0tan\u001e0in the form\nC(!) =\u0016g\r\n!\r\u00002sin\u0019\r\n2\n2\u0000\rD(!)_\u001e0\u0014\n1 +\r\u00002\n2(3\u0000\r)X\u0000\r\u00002\n2(4\u0000\r)X2+\r\u00002\n2(5\u0000\r)X3+:::\u0015\n:(23)\nThe series in Eq. (23) sum up into Hypergeometric function 2F1(1;2\u0000\r;3\u0000\r;\u0000X). Substi-\ntuting into the gap equation and again neglecting the Q\r;0term, we obtain the di\u000berential\nequation on \u001e(!) in the form\n1\n2_\u001e0\u0014\n1 +2F1(1;2\u0000\r;3\u0000\r;\u0000X)\u0015\n=2\n\u0019\u0016g\r!\r\u00001: (24)\nAt largeX, the asymptotic expansion of a Hypergeometric function yields 2F1(1;2\u0000\r;3\u0000\n\r;\u0000X)\u0019X\r\u00002\u0000(3\u0000\r)\u0000(\r\u00001). Substituting into (24) and solving for \u001e0(!) near!0, we\nobtain\n\u001e0(!)'\u0019\n2\u00004\n\u0019\u0012!0\n\u0016g\u0013\rB\r(1\u0000!=! 0\u0000i0+)\r\u00001\n1 +B\r(1\u0000!=! 0\u0000i0+)\r\u00002; (25)\nB\r=1\n(\r\u00001)\r\u00001\u0000(3\u0000\r)\u0000(\r\u00001): (26)\nWe see that \u001e0(!) approaches \u0019=2 with zero derivative, albeit the exponent is smaller than\n292 and reaches this value only at \r!3. ForD0(!) = \u0001 0(!)=!, this yields\nD0(!) =1\nsin\u001e0(!)= 1 +A\u0012!0\u0000!\u0000i0+\n!0\u00132(\r\u00001)\n(27)\nwhereA=O(1). The new element, compared to our approximate analysis in the previous\nsection, is that now Im \u001e0(!) develops immediately above !0.\n1. Density of states\nWe now show that the vanishing of _\u001e(!) at!0gives rise to a non-integrable singularity\nin the DOS at !=!0. In explicit form, the DOS for \u001e0(!) from Eq. (25) is\nN(!)\nN0=\u0016g\r\n!0\u0019sin[\u0019(1\u0000\r)]\n4B\r\u0002(!\u0000!0)\n(!\u0000!0)\r\u00001; (28)\nwhere \u0002(x) is the unit step function. In distinction from the toy model, there is no gap\nbetween the macroscopically degenerate level and the continuum, but still,R1\n!0d!N (!)\ndiverges at the lower limit, i.e., the DOS contains a non-integrable singularity at !=!0+0.\nIn practice, this implies that the number of states within a tiny interval above !0is a \fnite\nfraction of the total number of electrons Ntotin the system. Because N(!)/(\r\u00002), the\nfraction initially increases linearly with \r\u00002.\nWe show the DOS for several \rin Fig. 18. We see that the DOS vanishes below !0,\nforms a continuum above this frequency, and displays an \\in\fnite\" peak at the boundary\n(a non-integrable singularity). At the highest frequency, N(!) approaches the DOS of the\nnormal state, N0.\nThe non-integrability of the singularity at !=!0+ 0 means that the total number of\nstates in an arbitrary small range above !0diverges. By physical reasons, the divergence\nmust be regularized by extending the model in a proper direction. We show below that\na \fnite!Ddoes not provide the regularization | the singularity remains a non-integrable\none up to some \fnite value of !D. One way to regularize the divergence is to keep the\ntotal number of states large but \fnite, by imposing the limits of integration over fermionic\ndispersion, and require that the total number of states in the \\in\fnite\" peak is a fraction,\nproportional to \r\u00002, of the total number of states in the band. From this perspective,\nthe \\in\fnite\" peak should be viewed as a level with a macroscopic degeneracy. A further\ninvestigation of the \\in\fnite\" peak is clearly called for.\n30FIG. 19. Solution of the modi\fed gap equation Eq. (24) around !0, where (a) \r= 2:3 and (b)\n\r= 2:1. Gray dotted line shows the solution at \r= 2, i.e.,\u001e0(!) =!2=\u0019without imaginary part.\nWe note in passing that the analysis, presented here, can be extended to smaller \rbetween\n1 and 2. For these \r, the DOS still diverges at !=!0+ 0 with the fractional exponent\n\r\u00001, but the singularity is now integrable.\n2. Continuity at \r= 2 + 0\nWe see from Eq. (25) that at \r= 2+0, the frequency dependence of \u001e0(!) becomes!2=\u0019\n(which corresponds to taking \r!2 keeping!away from!0), like at\r= 2 (see Refs. [5{7]).\nWe show the numerical solution of Eq. (24) in Fig. 19. We see that the behavior of \u001e0(!) at\n\r >2 continuously approaches that at \r= 2: Im\u001e0(!) gradually gets smaller and Re \u001e0(!)\napproaches !2=\u0019. Simultaneously, the maxima in the continuum in the DOS get sharper\nand at\r= 2 + 0 evolve into a discrete set of \u000e-functional peaks, see Fig. 18. We emphasize\nthat the continuity at \r!2 does not hold in our approximate treatment in the previous\nsubsection and emerges only after we sum up in\fnite series in _\u001e0tan\u001e0.\nOn a more closer look, we \fnd that the analysis at \r!2 + 0 needs extra care. In this\nlimit, the series in Xyields\nC(!) =\u0019\u0016g\r\n2!\r\u00002D(!)_\u001e0\u0014\n1 +\u000e\u0012\nX\u0000X2\n2+X3\n3+:::\u0013\u0015\n=\u0019\u0016g\r\n2!\r\u00002D(!)_\u001e0[1 +\u000elog (1 +X)] (29)\nwhere, we remind, \u000e= (\r\u00002)=2. Substituting into the gap equation and restricting to\n31!\u0018!0, we obtain\n_\u001e0h\n1 +\u000elog\u0010\n1 +!_\u001e0tan\u001e0\u0011i\n=2\n\u0019\u0016g\r!\r\u00001: (30)\nSolving this equation, we \fnd that \u001e0(!)\u0019(2=\u0019\r)(!=\u0016g)\rup to an exponentially short\ndistance to !0, and within this distance\n\u001e0(!)'\u0019\n2+2\n\u0019\u000e\u0012!0\n\u0016g\u001321\u0000!=! 0\nlog(1\u0000!=! 0\u0000i0+): (31)\nWe see that \u001e0(!) still approaches \u0019=2 with zero derivative, but vanishes only logarithmically.\nThe Im\u001e0(!) does develop immediately above !0like at larger \r, but in the immediate\nvicinity of !0, Im\u001e0(!) is parametrically small compared to Re \u001e0(!) by 1=jlog!\u0000!0j.\nIf we were to neglect Im \u001e0(!), we would \fnd that Re \u001e0(!) monotonically increases with\n!, as (!=\u0016g)2=\u0019, and just \rattens in exponentially small regions near the frequencies where\n\u001e0(!) =\u0019=2 +p\u0019.\nFinally, we consider the terms with higher derivatives, like \u001e0,...\n\u001e0, etc. For de\fniteness,\nlet's restrict to \r\u00152 and compare these terms with ( \r\u00002)_\u001elogX. Each term with a higher\nderivative gets renormalized by series in X. We evaluate the series in Appendix C. At large\nX, which we are interested in, the series for each term have the same asymptotic form and\nreduce each prefactor by 1 =2. Including these terms with rescaled prefactors, we \fnd that\nthe last term in Eq. (29) changes to\n1 +\u000e[log(1 +X) +K] (32)\nwhere\nK=!\n\u001e(1)\n0\u00121\n2!\u001e(2)\n0\u0000!\n3!2\u001e(3)\n0+!2\n4!2\u001e(4)\n0+:::\u0013\n; (33)\nwhere\u001e(m)\n0is them-th derivative of \u001e0(!). We assume and then verify that at large X,\ni.e., at!\u0019!0, the inclusion of the Kterm only changes the prefactor for the second term\nin Eq. (31). To see this, we assume that at !slightly below !0,\u001e0(!) =\u0019=2 +Q(!0\u0000\n!)=log (1\u0000!=! 0) withQ= 4!\r\u00001\n0=[\u0019(\r\u00002)\u0016g\r], and compute the series for Kusing this\nform of\u001e0. A straightforward analysis then yields\nK=\u00001\n2 logXF(X); (34)\n32where\nF(X) = 2X1X\nm=0(\u00001)mXm\n(n+ 1)2(n+ 2)\n=\u00002 (Li 2(\u0000X) + log(1 + X)(1 + 1=X)\u00001) (35)\nand Li 2(\u0000X) is a polylogarithm. At large X, Li 2(\u0000X)\u0019(\u00001=2) log2X. Substituting into\nEq. (34), we obtain K\u0019\u0000(1=2) logX. Substituting into Eq. (32), we see that the log X\ndependence survives, only the prefactor drops by a factor of 2. Then Eq. (31) remains valid,\nwith extra 2 in the prefactor for the second term.\nC. Extraction of the non-integrable singularity in the DOS directly from the\nintegral gap equation.\nWe next show that the non-integrable singularity in the DOS can be obtained directly\nfrom the integral equation (3). For this, we \frst rewrite this equation in the form, which\ntakes care of the regularization of the formal divergence of the integral for C(!) in Eq. (15):\n\u0001(!) =1\n2Zd!m\n(\u0000(i!m\u0000!)2)\r=2\u0001(!m)\u0000\u0001(!)i!m\n!p\n!m2+ \u00012(!m)+ (36)\nsin(\u0019\r=2)\n\r\u00002 \n_D(!)Z!\n0_g(\n)d\n(!\u0000\n)\r\u00002\u0000(2\u0000\r)Z!\n0d\nD(\n)\u0000D(!) + (!\u0000\n)_D(!)\n(!\u0000\n)\rg(\n)!\n;\nwhere\ng(!) =1p\nD2(!)\u00001: (37)\nWe take as an input the evidence from the numerical analysis that at small !, \u0001(!) is real,\nand that there exists !0, at which \u0001( !0) =!0. At this point, we have\n\u0001(!0) =!0; D (!0) = 1:\nLet's assume that for !just below!0we have\nD(!) = 1 +A(!0\u0000!)\u000b; \u000b> 0;\nwhereAis some real positive constant.\nThe integral over !min Eq. (36) is completely regular at !!!0, the dangerous terms\nare the ones coming from the upper limit of integration over !in the last term. We then\n33write!=!0\u0000\u000f!, and \n =!0\u0000\u000f!\u0000\u000f\n, assume that both \u000f!and\u000f\nare small, and consider\nthe contribution from the upper limit. Then\ng(\n)\u00191p\n2A1\n(\u000f!+\u000f\n)\u000b=2;\ng0(\n)\u0019\u000b\n21p\n2A1\n(\u000f!+\u000f\n)\u000b=2+1: (38)\nExpressing \u000f\n=x\u000f!, we then obtain the dangerous contribution to the gap equation in the\nform\n\u000f1+\u000b=2\u0000\r\n!p\nAp\n2\u0012\u000b2\n2Z\n0dx\nx\r\u000021\n(1 +x)\u000b=2+1\u0000(2\u0000\r)Z\n0dx(1 +x)\u000b\u00001 +x\u000b\nx\r(1 +x)\u000b=2\u0013\n:\nIn order for this term to be \fnite, we must have\n\u000b\u00152(\r\u00001)>2:\nBy continuity, we expect \u000b= 2 at\r= 2 (see previous section). Invoking this argument,\nwe \fnd\u000b= 2(\r\u00001). This is exactly the same form as we obtained by summing up Taylor\nseries, Eq. (27).\nThe function g(\n) must be analytic in the upper half plane. Adding + i0 to!and\nsubstituting \u000f!+\u000f\n=!0\u0000\n\u0000i0 into Eq. (38), we obtain\ng(\n)\u00191p\n2A1\n(!0\u0000\n\u0000i0)\r\u00001:\nThe imaginary part of gis proportional to the density of states. We have\nN(\n)/1p\n2AIm1\n(!0\u0000\n\u0000i0)\r\u00001=8\n<\n:0; if \n<! 0\n1p\n2Asin(\u0019(\r\u00001))\n(\n\u0000!0)\r\u00001;if \n>! 0:\nThis is the same expression as Eq. (28).\nV. FINITE !D\nIn this section, we examine whether the state with an \\in\fnite\" peak in the DOS is stable\nwith respect to perturbation imposed by a small but \fnite mass of the pairing boson. On\nthe Matsubara axis, a \fnite mass of the boson changes the interaction to\nV(\nm) =\u0016g\r\n[\n2\nm+!2\nD]\r=2: (39)\n34A \fnite!Deliminates the solutions with large n, leaving only a \fnite number of the gap\nfunctions. The number of remaining solutions decreases as !Dincreases, and beyond some\nthreshold only the n= 0 solution survives. At the same time, the form of \u0001 0(!m) is only\nweakly a\u000bected by !Dboth for\r <2 and\r >2.\nOn the real axis, the e\u000bect from !Don then= 0 solution is far stronger for \r > 2,\nbut still it does not a\u000bect the physics qualitatively as long as !Dremains below a \fnite\nthreshold. To demonstrate this, we analyze how a \fnite !Da\u000bects the gap, re-expressed via\n\u001e0(!) related to the gap function via \u0001 0(!) =!=sin\u001e0(!). One can easily verify that in the\ngap equation D(!)B(!) =A(!) +C(!), the terms B(!) andA(!) are only weakly a\u000bected\nby!D, as long as !Dremains much smaller than \u0016 g, butC(!) changes substantially because\na \fnite!Dimposes a lower frequency cuto\u000b on Im V(\n) = Im \u0016g\r=(!2\nD\u0000(\n +i0+)2)\r=2.\nWe analyze the e\u000bect of a \fnite !Din two steps, like in Sec. IV. Namely, we \frst restrict\nwith only _\u001e2\n0tan\u001e0term in the expansion of C(!) in the derivatives of \u001e0(!), and then\ninclude the series of higher-order terms. The equation on \u001e0(!) to order _\u001e2\n0tan\u001e0in the\npresence of !Dhas been derived in Paper V for \r\u00192. Combining that expression with\nEq. (17) we obtain\n_\u001e0+_\u001e2\n0tan\u001e0\u0002\u0012\r\u00002\n2!\u0000\r\n4!D\u0013\n=2\n\u0019\u0016g\r\u0012\n!\r\u00001\u0000Q\r;0\u0016g\r\n!2ei\u0019\r=2sin\u001e0\u0013\n; (40)\nAs before, we will be interested in !\u0019!0, where\u001e0(!) reaches\u0019=2, and neglect the Q\r;0\nterm.\nWe see from Eq. (40) that for \r > 2, the two terms in the prefactor for _\u001e2\n0tan\u001e0have\nopposite signs and hence compete. The competition sets a characteristic frequency\n!c=\r\n2(\r\u00002)!D: (41)\nIts relevance becomes clear once we solve Eq. (40) for _\u001e0in terms of tan \u001e0:\n_\u001e0=\u00001 +q\n1 +4(\r\u00002)\n\u0019!\r(1\u0000!c\n!) tan\u001e0\n(\r\u00002)!(1\u0000!c\n!) tan\u001e0(42)\nAt!D= 0,!calso vanishes. In this situation, \u001e0(!) remains real up to !0, where\u001e0(!0) =\n\u0019=2, and approaches this frequency quadratically, as \u0019=2\u0000(!0\u0000!)2(!\r\u00002\n0=(\u0019(\r\u00002)\u0016g\r)).\nThis gives rise to the appearance of a macroscopically degenerate level in the DOS at !=!0.\nEq. (42) shows that this behavior holds as long as !Dis smaller than some critical value\n351.01.52.02.53.03.50246810\nω/gN(ω)/No1.01.52.02.53.03.5-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)/uni03C9D=0.78¯g\n1.01.52.02.53.03.5-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)\n1.01.52.02.53.03.50246810\nω/gN(ω)/No/uni03C9D=0.96¯g\n1.01.52.02.53.03.5-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)\n1.01.52.02.53.03.5-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)\n1.01.21.41.61.82.02.22.40246810\nω/gN(ω)/No1.01.21.41.61.82.02.22.4-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)/uni03C9c=0.99¯g\n1.01.52.02.53.03.5-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)\n1.01.52.02.53.00246810\nω/gN(ω)/No1.01.52.02.53.0-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)ϕo'(ω)ϕo''(ω)1.01.52.02.53.0-1.0-0.50.00.51.01.5\nω/gϕo(ω)ϕo'(ω)ϕo''(ω)ϕo'(ω)ϕo''(ω)ωD=0.99¯gFIG. 20. Numerical results for \u001e0(!) and the DOS N(!) for a \fnite !D. As long as !Dis small,\nthe \\in\fnite\" peak in N(!) survives. Once !Dexceeds a threshold value, it gets absorbed into the\ncontinuum. To get the DOS in panel (c), we solved Eq. (40) near !0, where it is valid, obtained\nIm\u001e0(!), and smoothly extended it to larger !.\n!c. At larger !D, Im\u001e0(!) emerges before \u001e0reaches\u0019=2, and an the bound state gets\nabsorbed into the continuum. We see therefore that a \fnite!D>!cis required to destroy\nthe \\in\fnite\" peak. We show this behavior in Fig.20.\nTheO(!D) term inC(!) also contains the combination !D\u001e0. As long as !Dis below\nthe threshold at !cand\u001e0(!) approaches \u0019=2 quadratically, this term only shifts !0by a\nsmall amount.\nWe now include into C(!) series of terms with higher powers of _\u001e0tan\u001e0. We present\ncomputational details in Appendix C and here quote the result. For simplicity, we again\nfocus on\rnear 2. The equation on \u001e0(!) near!0becomes\n_\u001e0\u0012\n1\u0000Y\n2(1 +Y)+\r\u00002\n2logX\u0013\n=2\n\u0019\u0016g\r!\r\u00001(43)\nwhereX=!0_\u001e0tan\u001e0is the same as before, and Y=!D_\u001e0tan\u001e0. At\r= 2, the imaginary\npart of\u001e0(!) emerges at ! <! 0, for which Y=O(1). At!D=Y= 0,\u001e0(!) remains real\nup to!=!0and approaches !0from below with vanishing derivative, which gives rise to\nan \\in\fnite\" peak in the DOS. When both !Dand\r\u00002 are \fnite, the analysis of Eq. (43)\n36shows that the \\in\fnite\" peak in the DOS survives as long as !D<!\u0003\nc, where\n!\u0003\nc=!0e\u00002\n\r\u00002 (44)\nWe see that the critical value of !Dis still \fnite, although exponentially small for \rslightly\nabove 2.\nA \fnite!Dalso introduces series of terms with higher-order derivatives. The series\nhold in!m\u00001\nD\u001e(m)\n0(m\u00152). We show the calculations in Appendix C and here quote the\nresults. The prefactors form series in Xand at large Xrescale the prefactor for each\nterm by 1=2. To understand potential relevance of these term, we use the same strategy\nas in the previous Section and evaluate the series in !m\u00001\nD\u001e(m)\n0using the solution \u001e0(!) at\n!D= 0. The series then hold in powers of Yand at large Yreplace\u0000_\u001e0=2 in Eq. (43)\nby\u0000_\u001e0=2 (log (!0=!D)=logX). At large X, the ratio of the two logarithms is small, hence\nsumming up series of terms with higher powers of !Dand higher derivatives of \u001e0reduces the\noverall e\u000bect from a \fnite !D. This rea\u000erms that the \\in\fnite\" peak in the DOS survives\nin a \fnite interval of !D.\nVI. PHASE DIAGRAM OF THE \rMODEL\nThe key result of our analysis is the realization that at T= 0, the superconducting state\nat\r >2 is qualitatively di\u000berent from the one at \r <2. We label these two superconducting\nstates as SC II and SC I, respectively. In both cases, the gap function in the upper half-plane\nof frequency is \u0001 0(z), wherez=!0+i!00.\nA superconducting state for \r < 2 (SC I) does by itself evolve with \rfrom BCS-like\nbehavior for \r <1 to the novel behavior for 1 <\r < 2, in which dynamical vortices cross,\none by one, into the upper half-plane of frequency, and the phase winding of \u0001 0(!) along the\nreal axis increases by 2 \u0019each time a new dynamical vortex moves into the upper half-plane.\nFrom topological perspective, there is then a cascade of topological transitions at a set of\ndiscrete\rbetween 1 and 2. Yet, the behavior of the DOS, N(!), is conventional for all\n\r <2 in the sense that N(!) is non-zero and is a continuous function of !at frequencies\nabove the spectral gap. There are no divergencies in N(!) at\r <2, yet a set of maxima\nand minima develops inside the continuum when \rbecomes close to 2.\nThe number of dynamical vortices becomes in\fnite at \r= 2. At this \r, several things\n3711.21.41.61.822.22.42.62.8301234510-3\n00.10.20.3s/EFEs/Tp00.20.40.60.8100.0050.010.0150.020.025=1.5=2=2.5\n00.20.40.60.8100.050.10.150.2\n=1.5=2=2.5ωD/¯gωD/¯gρs/EF\nλEρs/Tp(a)(b)\n(c)\nγFIG. 21. Numerical results for the super\ruid sti\u000bness \u001asfor di\u000berent \rand!D. (a):\u001asin units\nof Fermi energy EF, as a function of !Dfor di\u000berent \r. For\r >1, the sti\u000bness vanishes as !\r\u00001\nD.\nWe veri\fed this dependence analytically. (b) The sti\u000bness in units of Tp=\u0015E, whereTpis the onset\ntemperature of the pairing and \u0015E= \u0016g\r=(EF!\r\u00001\nD) is Eliashberg parameter, which measures the\nstrength of corrections to side vertices in the diagrams for the self-energy and the pairing vertex\n(in commonly accepted language, \u0015Fmeasures the strength of vertex corrections to Eliashberg\ntheory). The parameter \u0015Ehas to be smaller than (roughly) one. We see that \u001as\u0015E=Tptends to\na \fnite value at !D= 0. This result implies that vertex corrections from only n= 0 state do not\ndestroy superconducting order up to T\u0014Tp. (c) Comparable analysis of \u001as=EFand\u001as\u0015E=Tpat\n!D!0.\n38happen: (i) an essential singularity necessarily develops at !!1 as without it an extension\nfrom an in\fnite array of vortex points would give a zero gap everywhere in the upper\nhalf-plane; (ii) an in\fnite number of other solutions of the gap equation, \u0001 n(z), become\ndegenerate with \u0001 0(z) for allz > 0, the solutions with n!1 form a continuous set of\n\u0001\u0018(z), (iii)N(!) becomes the in\fnite set of \u000e-functions at particular !p.\nAt\r >2 (SC II), the number of dynamical vortices in \u0001 0(z) again becomes \fnite, and\nthere is a cascade of topological transitions at a set of discrete \rbetween 2 and 3, when\na vortex leaves the upper half-plane of frequency, in apparent mirror symmetry to what\nhappens for 1 <\r < 2. However, we argue that beyond this, SC II and SC I are qualitatively\ndi\u000berent states. Speci\fcally, we found that \u0001( !) approaches !0, where \u0001 0(!0) =!0, with\nzero derivative, as \u0001 0(!0)=!0= 1+O(!0\u0000!)2(\r\u00001). At!=!0, the gap function has a branch\ncut, and the density of states N(!) develops a non-integrable singularity (an \\in\fnite\" peak)\nat the lower edge of the continuum. We conjectured that if we keep the total number of\nstates \fnite by imposing a \fnite cuto\u000b on fermionic dispersion, the total weight under the\npeak will be proportional to the total number of states in the system. This \\in\fnite\" peak\nholds for all \r > 2 (its degeneracy contains \r\u00002 in the prefactor) and can be viewed as\nan order parameter that distinguishes between SC I and SC II. It is very likely that there\nexists another, possibly topological characteristic, which distinguishes SC II from SC I.\nWe analyzed the e\u000bect of a \fnite mass of the pairing boson, !D, and found that for \r >2\na \fnite!D>!\u0003\ncis needed to transform SC II into SC I. This analysis leads to T= 0 phase\ndiagram in ( !D;\r) plane, shown in Fig. 3. There is a single transition line between SC I,\nwhich holds for all !D\u00150 at\r <2 and for!D> !\u0003\ncat\r >2, and SC II, while exists at\n\r >2 in the interval 0 \u0014!D\u0014!\u0003\nc.\nWe next consider the phase diagram in the ( T;\r) plane at!D= 0. We argued in Paper V\nthat for\r= 2, massless \\longitudinal\" \ructuations, associated with the continuum spectrum\nof condensation energies, destroy superconducting order at any \fnite T, although the onset\ntemperature Tp=Tp;0for the emergence of a non-zero \u0001 0(z) is of order \u0016 g. At\r < 2,\nlongitudinal \ructuations are gapped, and it is natural to expect that Tcbecomes non-zero.\nWe argued that Tcincreases gradually with 2 \u0000\r, and the di\u000berence between TpandTcholds\nfor all\r < 2 and vanishes only at \r= 0. In between TpandTc, the system displays the\npseudogap behavior associated with the formation of fermionic pairs without global phase\ncoherence (preformed pairs). For \r >2, longitudinal \ructuations become gapped, and it is\n39natural to assume that Tcagain becomes \fnite and increases towards Tpas\r >2 increases.\nWe veri\fed this last point by computing numerically the superconducting sti\u000bness \u001as\natT!0 (the prefactor in F=\u001asR\ndrr2\u00110(r), where\u00110(r) is the phase of the order\nparameter \u0001 0(r) = \u0001 0ei\u00110(r). We show the results for di\u000berent \rand!Din Fig. 21. In the\ncalculations, we only included the n= 0 solution, i.e., we neglected \ructuation corrections\nfrom the solutions with other n.\nAt small!D, the sti\u000bness, expressed in units of EF, rapidly decreases with increasing\n\r. Taken at a face value, this would imply that the strength of phase \ructuations rapidly\nincreases with \r. One has to be careful here, however, because our analysis, based on\nthe analysis of modi\fed Eliasberg equations, is valid as long as corrections to Eliashberg\ntheory are small. These corrections come from the renormalizations of side vertices in\nthe diagrams for fermionic self-energy and the pairing vertex and hold in powers of the\nEliashberg parameter \u0015E. For an electron phonon problem (the case \r= 2 in our notations),\nthe Eliashberg parameter is \u0015E=a2\u0016g2=(EF!D), wherea2=O(1) (see, e.g., Paper V\nand Ref.[13]). To keep it small at small !D=\u0016g, one need to simultaneously increase EF.\nThe sti\u000bness, re-expressed in units of the onset temperature for the pairing, Tp, scales as\n\u001as=Tp\u00181=\u0015E. Then, as long as \u0015E\u00141, the ratio \u001as=Tpdoes not become small at small !D,\nwhich implies that phase \ructuations from the n= 0 solution alone cannot substantially\nreduce the actual Tccompared to Tp.\nFor\r6= 2, the Eliashberg parameter is, up to a prefactor, \u0015E=a\r\u0016g\r=(EF!\r\u00001\nD). In\npanels (b) and (c) of Fig. 21 we plot \u001asin units ofTp=\u0015E, withTptaken from [14]. We see\nthat this ratio remains \fnite at !D!0 for all\r >1, where one needs to adjust EFat small\n!Dto keep\u0015Esmall. This implies that within Eliashberg theory, phase \ructuations from\nthen= 0 state do not destroy superconducting order even at !D!0. Moreover, if the\nprefactora\rweakly depends on \r, the ratio\u001as=(Tp=\u0015E) =\u001as\u0015E=Tpactually increases with\nincreasing\r, i.e., for\u0015E= 1, which is at the boundary of applicability of the Eliashberg\ntheory, the ratio \u001as=Tpactually increases with \r, i.e., phase \ructuations become weaker.\nA more subtle question is whether for \r >2, the order below Tcis SC II. In our approx-\nimate analysis, Eq. (17), the order remains SC II up to some critical temperature, which\nis natural to be associated with Tc. Indeed, at a \fnite Tand\r\u00192, the prefactor for the\n40_\u001e2\n0tan\u001e0term, which plays the crucial role in distinguishing between SC I and SC II, is\n\r\u00002\n2!\u0000T; (45)\nsee Ref. [6]. The SC II state then holds as long as this coe\u000ecient is positive for !\u0019!0,\nwhich holds at T < ((\r\u00002)=2)!0. In this respect, Tplays the same role as a \fnite mass\nof a boson \feld, !D. When we include in\fnite series in the derivatives of \u001e0, the analysis\nbecomes more involved, but we still can identify a characteristic temperature T\u0003\nc, which\nseparates the behavior at higher T, when Im\u001e0(!) develops before it would \ratten due to\n(\r\u00002) logXterm in (29), and at smaller T, when\u001e0\rattens up before temperature e\u000bects\nbecome relevant. This scale is\nT\u0003\nc\u0018!0e\u00002\n\r\u00002; (46)\nIt is similar to !\u0003\ncin (44). This T\u0003\ncis exponentially small, but \fnite, hence, the order SC\nII survives in a \fnite range of T. Whether T\u0003\nccoincides with the actual Tc, is beyond the\nscope of our analysis. Assuming that it does, we arrive at the \\symmetric\" phase diagram\nshown in Fig. 2, with two distinct ordered phases SC I and SC II, and the pseudogap phase\nin between.\nThere is one caveat that needs to be addressed in further studies. In the discussion above\nwe assumed that \u0001( !) does not acquire an imaginary part for frequencies below !0. At a\n\fniteT, one generally expects that the DOS becomes non-zero for all !, in which case the\nnon-integrable singularity in N(!) gets regularized. A more careful extension of the present\napproach to \fnite Tis needed to address this issue.\nVII. CONCLUSIONS\nIn this paper, the sixth in the series, we analyzed the interplay between non-Fermi liquid\nand pairing in the e\u000bective low-energy model of fermions with singular dynamical interaction\nV(\nm) = \u0016g\r=j\nmj\r(the\rmodel). The model describes low-energy physics of various\nquantum-critical metallic systems at the verge of an instability towards density or spin\norder as well as pairing of fermions at the half-\flled Landau level, color superconductivity,\nand pairing in SYK-type models (see Paper I for the list of microscopic models). In previous\npublications, Paper I-V, we analyzed the physics of the model with \r\u00142. The key outcome\nof those studies was that a peculiar quantum-critical behavior develops within this space\n41of critical models as the exponent \rapproaches \r= 2. The critical behavior is with a\ntopological twist, as the number of dynamical vortices in the upper half-plane of frequency\ntends to in\fnity at \r= 2. In this paper we consider the \r-model with exponents 2 <\r < 3\nand address the issue what happens on the other side of the quantum transition. We argue\nthat the system moves away from criticality, e.g., the number of dynamical vortices becomes\n\fnite and decreases with increasing \r. This is similar to what happens when \rdecreases\nfrom\r= 2. Our key result, however, is the discovery that superconducting order for\n\r > 2 is qualitatively di\u000berent from that for \r < 2 (we labeled these states as SC II and\nSC I, respectively). Speci\fcally, we found that for \r > 2, the DOS has a non-integrable\nsingularity at the lower edge of the gapped continuum. In physical terms, this implies that\nthe spectrum of excited states contains a level with macroscopic degeneracy proportional\nto the total number of states in the system. We obtained the phase diagram at T= 0 in\nvariables (!D;\r), where!Dis the mass of a pairing boson, and argued that for \r >2, the\nSC II state exists in a \fnite range of !D(Fig. 3). We conjectured that SC II state survives\nat a \fniteTand rationalized the phase diagram in Fig. 2 in variables ( T;\r) for!D= 0.\nThe phase diagram contains two distinct superconducting phases SC I and SC II and an\nintermediate state with preformed pairs and pseudogap behavior of observables.\nFrom physics perspective, the appearance of an \\in\fnite\" peak can be understood using\nthe same reasoning as in Ref. [6], as a bound state between an excitation and an o\u000b-diagonal\npairing \feld that this excitation can modify via the self-energy. Indeed, we \fnd that the\nself-energy \u0006( !) becomes singular at the lower end of the continuum, where \u0001( !) =!, i.e.,\nat this frequency the e\u000bective potential, acting on a fermion in a superconductor, is in\fnite.\nA fermion in an in\fnite potential undergoes a self-trapping that generally leads to bound\nstates. This argument however, does not immediately explains why we get a non-integrable\nsingularity.\nThe emergence of the non-integrable singularity may be related to the fact that for\n\r >2, the gap equation on the real axis contains a formally divergent contribution, which\nneeds to be regularized. The divergence comes from the interaction V(\n) in the limit of\nzero frequency transfer \n !0. The interaction V(\n!0) scatters with vanishingly small\nfrequency transfer and in this respect acts on electrons in the same way as impurities.\nThe contribution from V(0) that cancels out without regularization, is analogous to the\ncontribution from non-magnetic impurities, while the one, which cancels out only after\n42regularization, is analogous to the contribution from magnetic impurities. In this respect,\nthere may be a similarity between our bound state and Yu-Shiba-Rusinov in-gap bound\nstate in the DOS of a superconductor in the presence of magnetic impurities [15{17].\nFinally, the very fact that the leading order in the expansion in X=!_\u001e0tan\u001e0captures\nthe divergence in the DOS, but does not capture the power-law singularity at the edge of\nthe continuum, is similar to the situation in the X-ray Fermi edge and Kondo problems (see\ne.g., [18{22] and references therein). From this perspective, one might think that e\u000bects\nsimilar to the orthogonality catastrophe [23] are also at play in the \r-model despite that\nthis model is for a clean system.\nWe call for more e\u000borts to establish physical interpretation of the non-integrable singu-\nlarity in the DOS for \r >2.\nACKNOWLEDGMENTS\nWe thank I. Aleiner, B. Altshuler, E. Berg, D. Chowdhury, L. Classen, R. Combescot,\nK. Efetov, R. Fernandes, A. Finkelstein, E. Fradkin, A. Georges, S. Hartnol, S. Karchu, S.\nKivelson, I. Klebanov, A. Klein, R. Laughlin, S-S. Lee, G. Lonzarich, D. Maslov, F. Mar-\nsiglio, I. Mazin, M. Metlitski, W. Metzner, A. Millis, D. Mozyrsky, C. Pepan, V. Pokrovsky,\nN. Proko\fev, S. Raghu, S. Sachdev, T. Senthil, D. Scalapino, Y. Schattner, J. Schmalian,\nD. Son, G. Tarnopolsky, A-M Tremblay, A. Tsvelik, G. Torroba, Y. Wang E. Yuzbashyan,\nand J. Zaanen for useful discussions of this and previous works (Papers I-V). The work\nby Y.M.W., S.-S.Z, and A.V. C. was supported by the NSF DMR-1834856. Y.-M.W, S.-\nS.Z.,and A.V.C also acknowledge the hospitality of KITP at UCSB, where part of the work\nhas been conducted. The research at KITP is supported by the National Science Foundation\nunder Grant No. NSF PHY-1748958.\nAppendix A: KK transformation for the interaction\nIn this Appendix, we discuss the subtlety with expressing the gap equation on the real\naxis, Eq. (3), in terms of C(!), given by Eq. (15). Taken at a face value, the integral in the\n43r.h.s. of (15) contains the piece\n\u0000i\u0016g\rsin\u0019\r\n2dD(!)\nd!p\n1\u0000D2(!)Z!\n0+d\n1\n\n\r\u00001; (A1)\nFor\r >2, the integral formally diverges and has to be properly regularized.\nWe went back to the computational steps, involved in the derivation of the gap equation\non the real axis, and traced the divergence in the integral for C(!) to the divergence in the\nKK relation for the interaction on the real axis. Speci\fcally, on the real axis,\nV(\n) =\u0012\u0016g\nj\nj\u0013\r\u0010\ncos\u0019\r\n2+isin\u0019\r\n2sgn\n\u0011\n(A2)\nThe derivation of C(\n) uses the KK relation expressing V0(\n) in terms of V00(\n):\nV0(!) =1\n\u0019PZ1\n\u00001V00(x)\nx\u0000\n=2\n\u0019PZ1\n0V00(x)x\nx2\u0000\n2(A3)\nwherePstands for principle value. Rescaling xby \n we \fnd that for V(\n) from (A2), this\nrelation is satis\fed if\n2\n\u0019Z1\n0dy\ny\r\u000011\ny2\u00001= cot\u0019\r\n2(A4)\nFor\r < 2, this relation holds, as one can easily verify, but for \r > 2, the integral in the\nl.h.s. of (A4) diverges.\nWe argue that to avoid the divergence and satisfy the KK relation for all \r, one has\nto modify the integration contour to the one shown in Fig. 4, which by-passes y= 0 by\nmoving slightly into the upper half-plane of frequency. Indeed, integrating over the contour\nby standard means, we \fnd that the integral in the l.h.s. of Eqn. (A4) gets modi\fed to\n2\n\u0019\u0014Z1\n\u000f=!dy\ny\r\u000011\ny2\u00001+\u0010!\n\u000f\u0011\r\u000021\n\r\u00002\u0015\n(A5)\nfor 2<\r < 4. The remaining integral is\nZ1\n\u000f=!dy\ny\r\u000011\ny2\u00001=\u0000Z1\n\u000f=!dy\ny\r\u00001+Z1\n0dyy(3\u0000\r)\ny2\u00001\n=\u0000\u0010!\n\u000f\u0011\r\u000021\n\r\u00001\u0000tan(\r\u00001)\u0019\n2(A6)\nSubstituting into (A5), we \fnd that the divergent term cancels out, and the KK relation\nis satis\fed. For \r > 4, the subleading term in (A6) also diverges, and the integral over a\nhalf-circle near z= 0 has to be computed by including ( \u000f=!)2terms (and higher powers for\n44even larger \r > 6). We veri\fed that the subleading divergent terms also cancel out, i.e.,\nintegrating over the modi\fed contour one does satisfy the KK relation (A3) for all \r. One\ncan also check that the other KK relation\nV00(!) =\u00001\n\u0019PZ1\n\u00001V0(x)\nx\u0000\n=\u00002\n\u0019PZ1\n0V0(x)\nx2\u0000\n2(A7)\nis also satis\fed for all \r, despite that the integral in the r.h.s. of (A7) formally diverges for\n\r >1. The Cauchy relation between V(\nm) andV00(\n):V(\nm) = (1=\u0019)R1\n0dxV00(x)x=(x2+\n\n2\nm) is also satis\fed for the integration contour as in Fig. 4.\nIn practical terms, bending of the integration contour to by-pass the z= 0 point is\nequivalent to just cancelling out the divergent terms in the KK transformation. For C(!),\nthis implies thatR!\n0+d\n=\n\r\u00001has to be evaluated as\nZ!\n\u000fd\n\n\r\u00001\u00001\n\r\u000021\n\u000f\r\u00002=\u00001\n\r\u000021\n!\r\u00002(A8)\nUsing this procedure, one obtains that the prefactor for the _\u001e0term in the gap equation\nevolves smoothly through \r= 2.\nAppendix B: The gap function along the real axis\nWhen the critical boson becomes massive, the Eliashberg equantion along the Matsubara\naxis takes the following form\n\u0001(!m) = \u0016g\r\u0019TX\n!0m\u0001(!0\nm)\u0000\u0001(!m)!0\nm\n!mp\n(!0\nm)2+ \u00012(!0\nm)1\n[(!0\nm\u0000!m)2+!2\nD]\r=2; (B1)\nwhere!D>0 is the mass of the intermediate boson. In this section, we make the analytic\ncontinuation of the above equation to the real axis.\nTo that end, we use the spectral representation of the interaction \u001f(!m) = (1=\u0019)R\nd!\u001f00(!)=(!\u0000\ni!m), where\u001f00(!) is the imaginary part of the interaction along the real axis\n\u001f(!) =\u0016g\r\n(!D\u0000!\u0000i\u000e)\r=2(!D+!+i\u000e)\r=2; (B2)\nwhere\u000eis an in\fnitesimal positive number. Noting that Arg[( !D\u0000!\u0000i\u000e)(!D+!+i\u000e)] =\n\u0000\u0019sign!\u0002(j!j\u0000!D), we have\n\u001f0(!) =\u0016g\r\n(j!j2\u0000!2\nD)\r=2\u0002(!D\u0000j!j) +\u0016g\r\n(j!j2\u0000!2\nD)\r=2cos(\u0019\r\n2)\u0002(j!j\u0000!D); (B3)\n45\u001f00(!) =\u0016g\r\n(j!j2\u0000!2\nD)\r=2sin(\u0019\r\n2)sign!\u0002(j!j\u0000!D): (B4)\nWith this representation, the gap equation can be rewritten as\n\u0001(!m) =Z1\n\u00001d!\u001f00(!)8\n<\n:TX\n!0m\u0001(!0\nm)\u0000\u0001(!m)!0\nm\n!mp\n(!0\nm)2+ \u00012(!0\nm)1\n!\u0000i(!m\u0000!0\nm)9\n=\n;: (B5)\nNow we make the analytic continuation i!m!z, while keeping the terms within the brace\nbracket analytic on the upper complex plane:\nTX\n!0m\u0001(!0\nm)\u0000\u0001(!m)!0\nm\n!mp\n(!0\nm)2+ \u00012(!0\nm)1\n!\u0000i(!m\u0000!0\nm)\n!TX\n!0m\u0001(!0\nm)p\n(!0\nm)2+ \u00012(!0\nm)1\n!\u0000z+i!0\nm\u0000\u0001(!m)\n!mTX\n!0m!0\nmp\n(!0\nm)2+ \u00012(!0\nm)1\n!\u0000z+i!0\nm\n\u00001\n2\u0001(z\u0000!)p\n\u0000(z\u0000!)2+ \u00012(z\u0000!)\u0012\ntanh!\u0000z\n2T\u0000coth!\n2T\u0013\n+1\n2\u0001(z)\nzz\u0000!p\n\u0000(z\u0000!)2+ \u00012(z\u0000!)\u0012\ntanh!\u0000z\n2T\u0000coth!\n2T\u0013\n: (B6)\nThe additional terms except that from the replacement i!m!zensure that the extended\nfunction of zgets rid of the pole at z=!+i!0\nm(j!j>!D). The gap equation on the upper\ncomplex plane takes the form\nzD(z) =\u0019TX\n!0m\u0001(!0\nm)p\n(!0\nm)2+ \u00012(!0\nm)\u001f(!0\nm+iz)\u0000iD(z)\u0019TX\n!0m!0\nmp\n(!0\nm)2+ \u00012(!0\nm)\u001f(!0\nm+iz)\n\u00001\n2Z1\n\u00001d!\u001f00(!)\u0001(z\u0000!)\u0000(z\u0000!)D(z)p\n\u0000(z\u0000!)2+ \u00012(z\u0000!)\u001a\ntanh!\u0000z\n2T\u0000coth!\n2T\u001b\n; (B7)\nwhereD(z) = \u0001(z)=zandV(\u0000iz) = (\u0016g2=(!2\nD\u0000z2))\r=2. In a compact form, we have\nzD(z)B(z) =A(z) +C(z);\nwhere\nA(z) =\u0019TX\n!0m>0D(!0\nm)p\n1 +D2(!0\nm)(\u001f(!0\nm+iz) +\u001f(!0\nm\u0000iz)); (B8)\nB(z) = 1 +i\u0019T\nzX\n!0m>01p\n1 +D2(!0\nm)(\u001f(!0\nm+iz)\u0000\u001f(!0\nm\u0000iz)); (B9)\nC(z) =\u00001\n2Z1\n\u00001d!\u001f00(!)\u0001(z\u0000!)\u0000(z\u0000!)D(z)p\n\u0000(z\u0000!)2+ \u00012(z\u0000!)\u001a\ntanh!\u0000z\n2T\u0000coth!\n2T\u001b\n:(B10)\n46Below we consider the real axis where we replace zby!+i\u000e. At zero temperature, using\nthe spectral representation of the interaction \u001f(!), the above functions reduce to\nA(!) =1\n2Z1\n0d!mD(!m)p\n1 +D2(!m)(\u001f(!m+i!) +\u001f(!m\u0000i!)) (B11)\nB(!) = 1 +i\n2zZ1\n0d!m1p\n1 +D2(!m)(\u001f(!m+i!)\u0000\u001f(!m\u0000i!)) (B12)\nC(!) =i\n2Zj!j\n0d\n\u001f00(\n)D(!\u0000\n)\u0000D(!)p\n1\u0000D2(!\u0000\n): (B13)\nOnce we obtained D(!m) by solving the Eliashberg equation along the Matsubara axis, A(!)\nandB(!) are known functions.\nAppendix C: Expansion of C(!)\nWe evaluate the integral for C(!) in (15) by Taylor-expanding the integrand in powers\nof internal \n, integrating each term in the expansion, and summing up the series. This\nprocedure is inspired by the fact that only one term in the series survives at \r= 2. However,\naway from this \r, an in\fnite number of terms appear with the same prefactor ( \r\u00002), and\none has to sum up in\fnite series.\n1. At a QCP\nWe \frst consider the case at a QCP and perform the integral over \n at each order of the\nexpansion:\nZ!\n0d\n\n\r\nn=!n+1\u0000\r\nn+ 1\u0000\r;n= 1;2;:::: (C1)\n47The infrared divergence for n= 1 is avoided using the trick discussed in Appendix A. The\nexpansion of C(!) is then given by a di\u000berential form\nC(!) =\u0016g\r\n!\r\u00002sin\u0019\r\n2\n2\u0000\rD(!)(\n_\u001e+\r\u00002\n2(3\u0000\r)!h\ntan\u001e_\u001e2+\u001ei\n\u0000\r\u00002\n6(4\u0000\r)!2h\n(2 + 3 tan2\u001e)_\u001e3+ 3 tan\u001e_\u001e\u001e+...\n\u001ei\n+\r\u00002\n24(5\u0000\r)!3h\u0000\n11 tan\u001e+ 12 tan3\u001e\u0001_\u001e4+\u0000\n12 + 18 tan2\u001e\u0001_\u001e2\u001e+ 3 tan\u001e\u001e2\n+4 tan\u001e(_\u001e)(...\n\u001e) +\u001e(4)i\n\u0000\r\u00002\n120(6\u0000\r)!4\"\n\u0000\n16 + 75 tan2\u001e+ 60 tan4\u001e\u0001_\u001e5+\u0000\n110 tan\u001e+ 120 tan3\u001e\u0001_\u001e3\u001e\n+\u0000\n20 + 30 tan2\u001e\u0001_\u001e2...\n\u001e+ 5\u0000\n6 + 9 tan2\u001e\u0001_\u001e\u001e2+ 5 tan\u001e_\u001e\u001e(4)\n+ 10 tan\u001e(\u001e)(...\n\u001e) +\u001e(5)#\n+:::)\n: (C2)\nThe order of this expansion is equal to the number of derivatives with respect to !(denoted\nasM). The leading order M= 1 survives at \r\u00002. All the higher order terms are proportional\nto the small parameter \r\u00002. Clearly, the small-\n expansion is not equivalent to a small-\n(\r\u00002) expansion.\nAs we are mainly interested in the gap function around !0, where\u001e=\u0019=2 and tan\u001e=1,\nwe choose the highest power of tan \u001ein the coe\u000ecients of each di\u000berential term in Eq. (C2).\nKeeping only the \frst derivative terms gives rise to\nC(!) =\u0016g\r\n!\r\u00002sin\u0019\r\n2\n2\u0000\rD(!)_\u001e\u0014\n1 +\r\u00002\n2(3\u0000\r)X\u0000\r\u00002\n2(4\u0000\r)X2+\r\u00002\n2(5\u0000\r)X3+:::\u0015\n;(C3)\nnamely Eq. (23) in the main text, where X=!tan\u001e_\u001e. This leads to the gap equation in\nEq. (24), which has been analyzed in Sec. IV B.\nNow we examine the e\u000bect of terms with higher derivatives (e.g., \u001e;...\n\u001e, etc.) on the\nsolution around !0by evaluating these terms using the above approximate solution. To\nsimplify the discussion, we consider the case \r= 2 + 0+as an example, where the solution\nat!slightly below !0reads\u001e=\u0019=2+Q(!0\u0000!)=log(1\u0000!=! 0) withQ= 4!\r\u00001\n0=[\u0019(\r\u00002)\u0016g)\r]\n48(see Eq. (31). For practical reasons, we consider a subset whose contribution to C(!) is\n\u0000\u0016g\rsin\u0019\r\n2\n!\r\u00002D0(!)\"\n1\n2!!\u001e\u0012\n1\u00001\n2X+1\n2X2\u00001\n2X3+:::\u0013\n\u00001\n3!2!2...\n\u001e\u0012\n1\u00002\n3X+3\n4X2\u00004\n5X3+:::\u0013\n+1\n4!3!3....\n\u001e\u0012\n1\u00003\n4X(1\u00006\n5X+8\n6X2+:::)\u0013\n;\n\u0000:::#\n; (C4)\nin which the coe\u000ecients of the n-th derivative \u001e(n)are formed by series in X. In the limit\nofX!1 , which corresponds to !!!0, the series in Xin each term sums up to 1 =2.\nEvaluating the di\u000berentials, _\u001e=Q=logX,!n\u00001\n0\u001e(n\u00152)=\u0000(n\u00002)!QXn\u00001=log2X, one\nobtains the sum\n1\n2!!\u001e\u00001\n3!2!2...\n\u001e+1\n4!3!3....\n\u001e\u0000:::\n=\u0000Q\nlog2X1X\nm=0(\u00001)mXm+1\n(m+ 1)2(m+ 2): (C5)\nIncluding these contributions to C(!), we obtain the modi\fed gap equation\n_\u001e\u0014\n1 +\r\u00002\n2[log(1 +X) +K]\u0015\n=2!\r\u00001\n\u0019\u0016g\r; (C6)\nwhereKandF(X) are the same functions de\fned in Eqs. (34), (35) of the main text. At\nlargeX,F(X)\u0019log2XandK\u0019\u0000(1=2) logX. Therefore, the leading order term log( X)\nnear!0drops by a factor of 2. The only change of the functional form of \u001e(!) near\u0019=2 is\nan extra factor of 1 =2.\n2. Away from a QCP\nNext, we consider the e\u000bect of a \fnite but small mass ( !D>0) of the critical boson. We\nredo the integral over \n in the presence of a \fnite !D:\nZ!\n!Dd\n(\n2\u0000!2\nD)\r=2\nn=!n+1\u0000\r\nD\n2B1\u0000(!D\n!)2\u0012\n1\u0000\r\n2;\r\u0000n\u00001\n2\u0013\n; (C7)\n49whereBz(a;b) refers to the incomplete Beta function. The divergence at \n = !Dat\r >2\nis again avoided using the trick discussed in Appendix A. Near \r= 2, this integral depends\non the ratio between !n\u00001\nDand\r\u00002, i.e.,\nZ!\n!Dd\n(\n2\u0000!2\nD)\r=2\nn=1\n2\u0000\r!n\u00001\nD+O((2\u0000\r)0): (C8)\nThe function C(!), however, is regular because 1 =(2\u0000\r) is cancelled out by the small factor\nsin(\u0019\r=2) from the interaction function. Subtracting the contribution at !D= 0 and keeping\nonly the leading order in \r\u00002, we obtain the modi\fcation to C(!) due to a \fnite mass\n\u0016g\r\n!\r\u00002sin\u0019\r\n2\n2\u0000\rD(!)\"\n\u00001\n2_\u001e\u0000\nY\u0000Y2+Y3\u0000Y4+:::\u0001\n\u0000\u00121\n2!!D\u001e\u00001\n3!!2\nD...\n\u001e+1\n4!!3\nD....\n\u001e+:::\u0013\n\u0002\u0012\n1\u0000Y+3\n2Y2\u00002Y3+5\n2Y4+:::\u0013#\n; (C9)\nwhereY=!Dtan\u001e_\u001e.\nIgnoring the second and higher order derivatives, we obtain the gap function Eq. (43) for\na \fnite mass of the boson. Its e\u000bect on the gap function has been analyzed in Sec. V of the\nmain text.\nThen, we verify that these neglected terms do not a\u000bect the solution around !0using\nthe same strategy for a QCP case, namely, by evaluating them explicitly using the solution\nEq. (31) near !0. The series formed by Ysums up to 1 =2 whenY!1 at!0. The sum\nover terms including higher-order di\u000berentials reads\n1\n2!!D\u001e\u00001\n3!!2\nD...\n\u001e+1\n4!!3\nD....\n\u001e+:::\n=\u0000Q\nlog2XY1X\nm=0(\u00001)m\n(m+ 1)(m+ 2)Ym\n=\u0000Q\nlog2X\u00121 +Y\nYlog(1 +Y)\u00001\nY\u0013\n: (C10)\nNear!0, whereX;Y! 1 , the above sum reduces to \u0000QlogY=logXasymptotically.\nAdding their contributions to C(!), the modi\fed gap equation takes the form\n_\u001e\u0014\n1\u0000log(!0=!D)\n2 logX+\r\u00002\n2log(X)\u0015\n=2\n\u0019\u0016g\r!\r\u00001: (C11)\n50Without including these higher order terms, the second term log( !0=!D)=(2 logX) becomes\na constant 1 =2. At large X, the ratio of the two logarithms is small, hence summing up\nseries of terms with higher powers of !Dand higher derivatives of \u001e0reduces the overall\ne\u000bect from a \fnite !D.\nAppendix D: The n=1solution on the upper complex plane\nThen=1solution along the Matsubara axis is given analytically by the same expression\nas for\r\u00142, and we refer to Refs. [1, 4, 5] for details. Its analytic continuation towards to\nthe upper complex plane of frequency is obtained by a rotation of frequency axis, i!m!\nz=!0+i!00=jzjei , which gives rise to\n\u00011(z) =Z1\n\u00001dke\u0000\u0012ke\u0000iIk\u0000iklogyz\np\ncosh(\u0019(k\u0000\f)) cosh(\u0019(k+\f)): (D1)\nwhereyz= (jzj=\u0016g)\r,\u0012= (\u0019=2\u0000 )\r, and\nbk=e\u0000i(Ik+klog (\r\u00001))\n[cosh(\u0019(k\u0000\f)) cosh(\u0019(k+\f))]1=2: (D2)\nHere\nIk=1\n2Z1\n\u00001dk0logj\u000fk0\u00001jtanh\u0019(k0\u0000k); (D3)\n\u000fk0=1\u0000\r\n2\u0000\u0000\r\n2(1 + 2ik0)\u0001\n\u0000\u0000\r\n2(1\u00002ik0)\u0001\n\u0000(\r)\u0012\n1 +cosh\u0019\rk0\ncos\u0019\r=2\u0013\n; (D4)\nand\f >0 is the solution of \u000f\f= 1. This extension is limited to the region \u0000\u0019=\r < \u0000\u0019=2<\n\u0019=\rwhere the integral giving rise to \u0001 1(z) is convergent. The critical axis  =\u0019=2\u0006\u0019=\r\nis on the lower complex plane when \r < 2, and rotates to the upper plane when \r > 2.\nAlong the critical axis, the behavior of \u0001 1(z) is very similar to that along the real axis at\n\r= 2, where the phase \u00111(z) = Arg(D1(z)) winds up to in\fnity as jzj!1 , while the\namplitude follows a power-law increase \u0018jzj\r=2=(\r\u00001). The phase winding is attributed to\nthe existence of an array of in\fnite vortices that line up along the critical axis as jzj!1 .\nConsequently, there are only a \fnite number of vortices on the upper half-plane when \r <2\nbut in\fnite number of vortices when \r\u00152. This evolution has been shown schematically\nby Fig. 13 in the main text. Representative examples are provided in Fig. 14 (b).\n51Appendix E: A discrete set of solutions of the non-linear gap equation\nHere we present the details of the analysis of a discrete set of solutions \u0001 n(!m). We\ndepart from the solution of the linearized gap equation and expand the solution of the full\nnon-linear gap equation in powers of \u0001 as\n\u0001(!m) =1X\nj=0\u000f2j+1\u0001(2j+1)(!m); (E1)\nwhere \u0001(1)(!m) = \u00011(!m). We then solve iteratively for \u0001(2j+1)in terms of \u0001(2j0+1)and\nj0<j.\nThe gap equation must be satis\fed at each order of \u000f, which imposes the following\nequation\n!mD(2j+1)(!m)\u0000\u0016g\r\n2Z1\n\u00001d!0\nm\u0000\nD(2j+1)(!0\nm)\u0000D(2j+1)(!m)\u0001sign(!0\nm)\nj!0\nm\u0000!mj\r=K(2j+1)(!m);\n(E2)\nwithj= 0;1;2;:::. The source term K(2j+1)(!m) is built from the gap function of a lower\norder 1\u0014j0<j. For example, the \frst two orders are given by\nK(0)(!m) = 0; (E3)\nK(3)(!m) =\u0000\u0016g\r\n4Z1\n\u00001d!0\nmsgn(!0\nm)\nj!0\nm\u0000!mj\r\u0000\nD(1)(!0\nm)\u0000D(1)(!m)\u0001\nD(1)2(!0\nm): (E4)\nSinceK(0)(!m) = 0, the leading order is given by the solution of the linearized gap equation\nD(1)(!m) =D1(!m): (E5)\nAt!\u001c\u0016g, there is\nD(1)(!m)!2sgn(!m)\u0012j!mj\n\u0016g\u0013\u000e\ncosf(!m); (E6)\nwhere\u000e= (\r\u00002)=2 and\nf(!m) =\flogj!mj\r\n\u0016g\r+\u001e: (E7)\nWe note that the term !mD(0)(!m) in the gap equation is irrelevant for the small frequency\nbehavior. This holds true for each subleading order to be discussed.\nProvided the leading order j= 0 solved, one can compute the source term at the next or-\nder,K(3)(!), and then search for the induced solution D(3)(!m). For the smallest frequency,\n52K(3)(!) is free from the ultra-violet details, and thus fully determined by the asymptotic\nform ofD(1)(!) in Eq. (E6). One can continue this process to higher orders, which is sum-\nmarized as a two-step iterative procedure. (1) Once we solved the solution at orders j0<j,\nwe \frst compute the source term at order j:\nK(j)(!m) = \u0016g\u0012j!mj\n\u0016g\u0013(j\u00001\n2)(\r\u00002)\u00001jX\nr=0ei(2r+1)f(!m)I(2j+1)\n2r+1+c:c:; (E8)\nwhereI(2j+1)\n2r+1is determined from the lower-order solutions. For j= 1, we use the j= 0\nsolution and obtain\nI(3)\n1=\u00001\n4I(3\u000e+i\f\r;2\u000e+ 2i\f\r)\u00001\n2I(3\u000e+i\f\r;2\u000e); (E9)\nI(3)\n3=\u00001\n4I(3\u000e+ 3i\f\r;2\u000e+ 2i\f\r): (E10)\nHere we have de\fned the integrals\nI(a;b) =Z1\n\u00001dx\njx\u00001j\r\u0000\njxja\u0000sign(x)jxjb\u0001\n=B(\r\u00001\u0000a;1 +a) +B(\r\u00001\u0000b;1 +b)\n+\u0019csc(\u0019\r)\n\u0000(\r)\u0012\u0000(1 +a)\n\u0000(2\u0000\r+a)\u0000\u0000(1 +b)\n\u0000(2\u0000\r+b)+\u0000(\r\u00001\u0000a)\n\u0000(\u0000a)\u0000\u0000(\r\u00001\u0000b)\n\u0000(\u0000b)\u0013\n;(E11)\nwhereB(x;y) is the Beta function. The convergence of this integral requires \r < 3 and\n\u00001<Re[a];Re[b]<\r\u00001. On order j= 1, it requires \r <3; on an arbitrary order j >1,\nit requires\r <2 + 2=(2j\u00001).\n(2) The source term in Eq. (E8) leads to the induced solution at order j:\nD(j)(!m)'2sgn(!m)\u0012j!mj\n\u0016g\u0013(2j+1)\u000ejX\nr=0Q(2j+1)\n2r+1cos\u0010\n(2r+ 1)f(!m) +\u001e(2j+1)\n2r+1\u0011\n:(E12)\nwhere\nQ(2j+1)\n2r+1exp[i\u001e(2j+1)\n2r+1] =\u00002I(2j+1)\n2r+1=J(2j+1)\n2r+1; r= 0;1;:::;j: (E13)\nand\nJ(2j+1)\n2r+1=I((2j+ 1)\u000e+i\f\r(2r+ 1);0): (E14)\nThe integrals J(2j+1)\n2r+1is convergent under the same condition as I(2j+1)\n2r+1.\nTo apply the above iterative procedure for any given \r > 2, however, we must stop at\na \fnite order j\u00181=(\r\u00002), above which, the gap function cannot be satis\fed because the\ndivergence in both integrals I(2j+1)\n2r+1andJ(2j+1)\n2r+1cannot be cancelled out from the equation.\n53The divergence indicates the gap equation at the low-frequency limit depends on the gap\nfunction at the higher frequency, which in turns depends on the parameter \u000f. In other words,\n\u000fenters the gap equation at each order by renormalizing the divergence. To satisfy the gap\nequation, only a discretized set of \u000fis possible, indicating that the solutions form an in\fnite\nand discrete set.\nAppendix F: Behavior of \u00010(!m)in the extended \r-model at M!0\nThe numerical solution in Fig. 12 (b) for \r >2 shows that \u0001 0(\u0016!m), where \u0016!mis a properly\nnormalized frequency, vanishes at M= 0 in a rather peculiar way: the gap function at zero\nfrequency, \u0001 0(0), gradually decreases as Mgets smaller and vanishes at M= 0, however\nthe full function \u0001 0(\u0016!m) remains \fnite at M= 0+ and scales as \u0016 !at small frequencies.\nIn this section, we analyze the behavior of \u0001 0(\u0016!m) analytically and argue that at M= 0+,\nthere exists a one-parameter continuous set \u0001 0;\"(\u0016!m), speci\fed by a parameter \", which runs\nbetween\"min= 0+ and a \fnite \"max. All \u0001 0;\"(\u0016!m) vanish at \u0016 !m= 0 and scale linearly with\n\u0016!mat small frequencies, but the slope is proportional to \". AsMapproaches zero from the\npositive side, the gap function \u0001 0(\u0016!m) approaches \u0001 0;\"max(\u0016!m), while asMapproaches zero\nfrom the negative side, the gap function is in\fnitesimally small and approaches \u0001 0;\"min(\u0016!m),\nThe gap function with \"minis the solution of the linearized gap equation. At small\nfrequencies, \u0001 0;\"min(\u0016!m) is the sum of two power-laws (\u0016 !m)a1;2. AtM!0,a1approaches 1\nanda2approaches \r\u00001>1, hence (\u0016!m)a1is much larger, hence \u0001 0;\"min(\u0016!m) is linear in \u0016 !m\nat small frequencies. Like we said, the numerical solution of the non-linear gap equation at\nM!0 also shows linear dependence of the gap function on frequency at small \u0016 !m. Based\non this analogy, we assume that at M!0, there is a set of gap functions \u0001 0;\"(\u0016!m), which\nat small \u0016!mare all linear in \u0016 !matM= 0+ and at vanishingly small but \fnite Mbehave\nas \u0001 0;\"(\u0016!m) =\"j\u0016!mj1+\u000esign(\u0016!m), where\u000escales with M.\nTo determine the two parameters \"and\u000e, we substitute this trial function into the\nmodi\fed gap equation in Eq. (11). In the infrared limit, the bare \u0016 !mterm in the l.h.s. is\n54irrelevant, and ignoring it we rewrite Eq. (11) as\nZd\u0016!0\nm\nj\u0016!m\u0000\u0016!0\nmj\r \njD(\u0016!m)jp\n1 +D2(\u0016!m)\u0000jD(\u0016!0\nm)jp\n1 +D2(\u0016!0\nm)!\n=MD(\u0016!m)Zd\u0016!0\nm\nj\u0016!m\u0000\u0016!0\nmj\r \nsign(\u0016!m)p\n1 +D2(\u0016!m)\u0000sign(\u0016!0\nm)p\n1 +D2(\u0016!0\nm)!\n: (F1)\nSubstituting the trial function into this equation, expanding to order \u000f3, and evaluating the\nintegrals which turn out to be convergent in in both infrared and ultra-violet limits, we\nobtain at vanishing \u000e\n\u000e\n2I(\r)\u0000\n1\u0000\"2+O(\"4)\u0001\n=M\n\r\u00001+O(M\"2); (F2)\nwhere\nI(\r) =\u0000Z1\n0dxlogx\u00121\nj1\u0000xj\r+1\n(1 +x)\r\u0013\n=1\n\r\u00001\u0012\nH(\r\u00002)\u0000H(1\u0000\r) +\u0019\nsin\u0019\r\u0013\n: (F3)\nandH(x) is the Harmonic number, analytically continued from H(n) =Pn\nk=11=k. In the\ntwo limitsI(\r)'\u00192(\r\u00002)=2 near\r= 2 andI(\r)'1=(3\u0000\r) near\r= 3.\nWe see that \u000e/M, as we anticipated. Eq. (F2) sets one condition on two parameters,\n\u000eand\"and therefore allows for a continuous set of solutions. Taking the limit M!0 and\nkeeping\u000e=M =\u000b>0 as a constant, we obtain \"as a function of \u000b:\n\"=s\n1\u00002\n(\r\u00001)I(\r)1\n\u000b(F4)\nAs\u000bvaries between ( \r\u00001)I(\r)=2 and1, the amplitude parameter \"changes continuously\nfrom\"min= 0+ to\"max= 1. The gap function with \"minis the solution of the linearized\ngap equation, which is also the only solution one can obtain by approaching M= 0 from\nnegativeM, while the solution with \"maxis \u0001 0(\u0016!m) that we obtained numerically by solving\nthe non-linear gap equation at M!0 coming from positive M.\n[1] A. Abanov and A. V. Chubukov, Interplay between superconductivity and non-fermi liquid\nat a quantum critical point in a metal. i. the \rmodel and its phase diagram at T= 0: The\ncase 0<\r < 1, Phys. Rev. B 102, 024524 (2020).\n55[2] Y.-M. Wu, A. Abanov, Y. Wang, and A. V. Chubukov, Interplay between superconductivity\nand non-fermi liquid at a quantum critical point in a metal. ii. the \rmodel at a \fnite Tfor\n0<\r < 1, Phys. Rev. B 102, 024525 (2020).\n[3] Y.-M. Wu, A. Abanov, and A. V. Chubukov, Interplay between superconductivity and non-\nfermi liquid behavior at a quantum critical point in a metal. iii. the \rmodel and its phase\ndiagram across \r= 1, Phys. Rev. B 102, 094516 (2020).\n[4] Y.-M. Wu, S.-S. Zhang, A. Abanov, and A. V. Chubukov, Interplay between superconductivity\nand non-fermi liquid at a quantum critical point in a metal. iv. the \rmodel and its phase\ndiagram at 1 <\r < 2, Phys. Rev. B 103, 024522 (2021).\n[5] Y.-M. Wu, S.-S. Zhang, A. Abanov, and A. V. Chubukov, Interplay between superconductivity\nand non-fermi liquid at a quantum critical point in a metal. v. the \rmodel and its phase\ndiagram. the case \r= 2, Phys. Rev. B 103, 024522 (2021).\n[6] R. Combescot, Strong-coupling limit of eliashberg theory, Phys. Rev. B 51, 11625 (1995).\n[7] A. Karakozov, E. Maksimov, and A. Mikhailovsky, The investigation of eliashberg equations\nfor superconductors with strong electron-phonon interaction, Solid State Communications 79,\n329 (1991).\n[8] F. Marsiglio and J. P. Carbotte, Gap function and density of states in the strong-coupling\nlimit for an electron-boson system, Phys. Rev. B 43, 5355 (1991), for more recent results see F.\nMarsiglio and J.P. Carbotte, \\Electron-Phonon Superconductivity\", in \\The Physics of Con-\nventional and Unconventional Superconductors\", Bennemann and Ketterson eds., Springer-\nVerlag, (2006) and references therein; F. Marsiglio, Annals of Physics 417, 168102-1-23 (2020).\n[9] F. Marsiglio, M. Schossmann, and J. P. Carbotte, Iterative analytic continuation of the electron\nself-energy to the real axis, Phys. Rev. B 37, 4965 (1988).\n[10] Y.-M. Wu, A. Abanov, and A. V. Chubukov, Pairing in quantum critical systems: Transition\ntemperature, pairing gap, and their ratio, Physical Review B 99, 014502 (2019).\n[11] T.-H. Lee, A. Chubukov, H. Miao, and G. Kotliar, Pairing mechanism in hund's metal su-\nperconductors and the universality of the superconducting gap to critical temperature ratio,\nPhysical review letters 121, 187003 (2018).\n[12] We note in passing that at \r=\rcr, the two power-law solutions merge into a single j!j\r=2,\nbut at this point another solution \u0001 1(!m)/j!mj\r=2logj!mjemerges, as can be veri\fed\nby using he identityR1\n\u00001dxjxj\r=2\u00001logjxj=jx\u00001j\r= 0. As a result, the low-frequency\n56\u00011(!m)/j!mj\r=2logj!mj=!\u0003still contains a free parameter !\u0003that allows one to match\nthis low-frequency form with \u0001 1(!m)/1=j!mj\rat high frequencies.\n[13] A. V. Chubukov, A. Abanov, I. Esterlis, and S. A. Kivelson, Eliashberg theory of phonon-\nmediated superconductivity { when it is valid and how it breaks down, Annals of Physics 417,\n168190 (2020).\n[14] Y. Wang, A. Abanov, B. L. Altshuler, E. A. Yuzbashyan, and A. V. Chubukov, Superconduc-\ntivity near a quantum-critical point: The special role of the \frst matsubara frequency, Phys.\nRev. Lett. 117, 157001 (2016).\n[15] L. Yu, Bound state in superconductors with paramagnetic impurities, Acta Phys. Sin 21, 75\n(1965).\n[16] H. Shiba, Classical spins in superconductors, Progress of theoretical Physics 40, 435 (1968).\n[17] A. Rusinov, Superconductivity near a paramagnetic impurity, JETP Lett.(USSR)(Engl.\nTransl.);(United States) 9(1969).\n[18] P. NOZI \u0012ERES and C. T. DE DOMINICIS, Singularities in the x-ray absorption and emission\nof metals. iii. one-body theory exact solution, Phys. Rev. 178, 1097 (1969).\n[19] D. V. Khveshchenko and P. W. Anderson, Fermi-edge singularities in x-ray spectra of strongly\ncorrelated fermions, Phys. Rev. B 61, 1658 (2000).\n[20] M. Hentschel and F. Guinea, Orthogonality catastrophe and kondo e\u000bect in graphene, Phys.\nRev. B 76, 115407 (2007).\n[21] I. A\u000feck and A. W. Ludwig, The fermi edge singularity and boundary condition changing\noperators, Journal of Physics A: Mathematical and General 27, 5375 (1994).\n[22] G. D. Mahan, Excitons in metals: In\fnite hole mass, Phys. Rev. 163, 612 (1967).\n[23] P. W. Anderson, Infrared catastrophe in fermi gases with local scattering potentials, Phys.\nRev. Lett. 18, 1049 (1967).\n57"    },    {        "title": "2003.12715v2.Field_induced_SU_4__to_SU_2__Kondo_crossover_in_a_half_filling_nanotube_dot__spectral_and_finite_temperature_properties.pdf",        "content": "arXiv:2003.12715v2  [cond-mat.mes-hall]  6 Oct 2020Field-induced SU(4) to SU(2) Kondo crossover in a half-filli ng nanotube dot:\nspectral and finite-temperature properties\nYoshimichi Teratani,1Rui Sakano,2Tokuro Hata,3Tomonori\nArakawa,3,4Meydi Ferrier,3,5Kensuke Kobayashi,3,6and Akira Oguri7,8\n1Department of Physics, Osaka City University, Sumiyoshi-k u, Osaka, 558-8585, Japan\n2The Institute for Solid State Physics, the University of Tok yo, Kashiwa, Chiba 277-8581, Japan\n3Department of Physics, Osaka University, Toyonaka, Osaka 5 60-0043, Japan\n4Center for Spintronics Research Network, Osaka University , Toyonaka, Osaka 560-8531, Japan\n5Laboratoire de Physique des Solides, CNRS, Universit´ e Par is-Sud,\nUniversit´ e Paris Saclay, 91405 Orsay Cedex, France\n6Institute for Physics of Intelligence and Department of Phy sics,\nThe University of Tokyo, Tokyo 113-0033, Japan\n7Department of Physics, Osaka City University, Sumiyoshi-k u, Osaka, 558-8585, Japan\n8Nambu Yoichiro Institute of Theoretical and Experimental P hysics, Sumiyoshi-ku, Osaka, 558-8585, Japan\n(Dated: October 7, 2020)\nWe study finite-temperature properties of the Kondo effect in a carbon nanotube (CNT) quantum\ndot using the Wilson numerical renormalization group (NRG) . In the absence of magnetic fields,\nfour degenerate energy levels of the CNT consisting of spin a nd orbital degrees of freedom give\nrise to the SU(4) Kondo effect. We revisit the universal scali ng behavior of the SU(4) conductance\nfor quarter- and half-filling in a wide temperature range. We find that the filling dependence of\nthe universal scaling behavior at low temperatures Tcan be explained clearly with an extended\nFermi-liquid theory. This theory clarifies that a T2coefficient of conductance becomes zero at\nquarter-filling whereas the coefficient at half-filling is fini te. We also study a field-induced crossover\nfrom the SU(4) to SU(2) Kondo state observed at the half-fille d CNT dot. The crossover is caused\nby the matching of the spin and orbital Zeeman splittings, wh ich lock two levels among the four at\nthe Fermi level even in magnetic fields B. We find that the conductance shows the SU(4) scaling\nbehavior at µBB < k BTSU(4)\nKand it exhibits the SU(2) universality at µBB≫kBTSU(4)\nK, where\nTSU(4)\nKis the SU(4) Kondo temperature. To clarify how the excited st ates evolve along the SU(4) to\nSU(2) crossover, we also calculate the spectral function. T he results show that the Kondo resonance\nwidth of the two states locked at the Fermi level becomes shar per with increasing fields. The spectral\npeaks of the other two levels moving away from the Fermi level merge with atomic limit peaks for\nµBB/greaterorsimilarkBTSU(4)\nK.\nPACS numbers: 72.15.Qm, 73.63.Kv, 75.20.Hr\nI. INTRODUCTION\nQuantum dots provide an ideal testbed to investigate\nstrong correlationsbetween the electrons in localized lev-\nels and the conduction electrons in reservoirs. Kondo\neffect1,2is a typical many-body phenomenon that occurs\nalso in quantum dots having local spin degrees of free-\ndom. TheKondoeffectinquantumdotshasbeenstudied\ntheoretically3,4and experimentally.5,6In addition to the\nspin degrees of freedom, carbon nanotube (CNT) quan-\ntum dots have also the orbital (valley) degrees of free-\ndom, corresponding to clockwise and counter clockwise\norbitals around the nanotube axis.7Four energy levels\nconsisting of the spin and orbital degrees of freedom give\nrise to the SU(4) Kondo effect in the case where the four\nlocalized states are degenerate.8–14A number of exper-\niments for non-equilibrium transport have observed the\nSU(4) Kondoeffect.15–20Perturbationssuchasspin-orbit\ncoupling∆ SO, valleymixing∆ K,K′andmagneticfields B\nbreak the SU(4) symmetry. Effects of such perturbations\non the Kondo state are theoretically studied for instance,\nusing Wislon’s numerical renormalization group (NRG)approach21–23which has been extended to explore trans-\nport coefficients and spectral functions with very high\naccuracy.12,24–26\nThe main purpose of the present paper is to clarify\nthe finite temperature properties of the Kondo effect in\nCNT dots. In the first half of this paper, we study the\nscaling behavior of the SU(4) conductance at quarter-\nand half-filling. Although the scaling behavior has been\nstudied,12,16,27–29we revisit it with an extended micro-\nscopicFermi-liquidtheorywhichdescribestransportphe-\nnomena at low-temperatures.30–34It has recently been\nshown that the T2coefficientCTfor the conductance\nis determined in terms of five Fermi-liquid parameters:\nelectron filling, two linear susceptibilities, and two non-\nlinear susceptibilities which are defined with respect to\nthe equilibrium ground state. Calculating these parame-\nters using the NRG, we successfully explain the filling de-\npendence ofthe scalingwith the description. Specifically,\nwe explore the filling dependence of a T2coefficientCT\nand find that CTbecomes zero at quarter-filling whereas\nit is finite at half-filling.\nIn the second half of this paper, we examine a field2\ninduced crossover from SU(4) to SU(2) Kondo state at\nhalf-filling nanotube dot.35,36At the valley where the\ncrossover occurs, the SU(4) Kondo resonance emerges\nin the absence of magnetic field, because ∆ K,K′and\n∆SOare smaller than the SU(4) Kondo energy scale,\nkBTSU(4)\nK. The field induced crossover is different from\nthe other crossover occurring at quarter-filling.27,28,37–43\nSpecifically, this crossover at half-filling occurs in a situ-\nation where two localized levels among the four remain\nthe Fermi level even in magnetic fields while the other\ntwo levels move away from the Fermi level. The situ-\nation realizes if the spin Zeeman splitting coincides the\norbital splitting. Such coincidence can reasonably occur\nin real CNT dots.\nIn the previous work, we have studied the crossover\noccurring in this situation, by calculating quasi-particle\nparameters such as phase shift δ, wave function renor-\nmalization factor Zand Wilson ratio R.36We havefound\nthat the applied magneticfields enhance the electron cor-\nrelations. For instance, as magnetic fields increase, the\nrenormalization factor Zdecreases from the SU(4) value\nto the SU(2) value and thus the Kondo energy scale TK\ndecreases. Our NRG results are in good agreement with\nthe experimental observations.35\nIn this paper, we calculate the temperature depen-\ndence of the conductance in magnetic fields to clarify the\ncrossover in a wide range of temperature T. We show\nthat a temperature scale T∗around which the conduc-\ntance shows log Tdependence decreases with increasing\nmagnetic fields. This decrease of T∗becomes clearer in a\nstrong Coulomb interaction case and agree with the field\ndependenceof Z. Wealsoexaminethescalingbehaviorof\nthe conductance. Whereas the conductance follows the\nSU(4) scaling at µBB < k BTSU(4)\nK, it shows the SU(2)\nscaling atµBB≫kBTSU(4)\nK.\nIn addition to the conductance, we calculate the level\nresolved spectral functions in magnetic fields. The com-\nponent for the doubly degenerate levels shows that the\nKondo resonance width becomes sharper with increasing\nmagnetic fields. This field dependence of the width cor-\nresponds to that of T∗. Spectral weights of the other two\nstates transfer from the Fermi level, and the peaks merge\nwith atomic limit peaks.\nThis paper is organized as follows. In the next section,\nwe describe the microscopic Fermi-liquid theory and the\nNRG approach to CNT dots. In Sec. III, we examine the\nscaling behavior of the SU(4) conductance at quarter-\nand half-filling. We discuss how the quasi-particle pa-\nrameters evolve during the field induced crossoverin Sec.\nIV. We present the NRG results of conductance and dis-\ncuss the influence of magnetic fields on the temperature\ndependence of conductance in Sec. V. The spectral func-\ntions in increasing magnetic fields are shown in Sec. VI.\nSummary is given in Sec. VII.II. FORMULATION\nTransport properties of carbon nanotube quantum\ndots are determined by a linear combination of four one-\nparticle levels, consisting of the spin ( ↑,↓) and valley (K,\nK’) degreesoffreedom. The structureofthese four states\nstaying near the Fermi level depend on the inter-valley\nscattering, the spin-orbitcoupling, and the Zeeman split-\ntingsofthespinandorbitaldegreesoffreedom. Weintro-\nduce the Anderson model for the CNT dot in this section\nusing a diagonal form for these local levels. We provide a\nmore microscopic description specific to a real CNT dot,\nin which experiments havemeasured the currentand cor-\nresponding noise with high accuracy. The renormalized\nparametersthat characterizethe low-energyFermi-liquid\nproperties and details of the NRG calculations are also\ndescribed in this section.\nA. Anderson model for CNT quantum dots\nWe start with an Anderson impurity model for a CNT\ndot, which has N= 4 internal degrees of freedom and is\nconnected to two noninteracting leads:\nH=H0\nd+HU+Hc+HT, (1)\nH0\nd=N/summationdisplay\nm=1ǫmd†\nmdm,HU=U/summationdisplay\nm<m′ndmndm′,(2)\nHc=/summationdisplay\nν=L,RN/summationdisplay\nm=1/integraldisplayD\n−Ddεεc†\nν,εmcν,εm (3)\nHT=/summationdisplay\nν=L,RN/summationdisplay\nm=1vν/parenleftbig\nψ†\nν,mdm+d†\nmψν,m/parenrightbig\n, (4)\nψν,m≡/integraldisplayD\n−Ddε√ρccν,ε,m, ndm≡d†\nmdm.(5)\nHere,d†\nmanddmare the creation and annihilation op-\nerators, respectively, for an electron with energy εm\nin them-th discrete one-particle eigenstate of the dot\n(m= 1,2,...,N). We shall also call mthe “flavour” in\nthe following. The Coulomb interactions Ubetween the\nelectrons occupying the dot levels are assumed to be in-\ndependent of m, assumingthat the intra-andinter-valley\nCoulomb repulsions to be identical. We also assume that\nHund’s rule coupling can be neglected. This is consis-\ntent with the CNT dot in which the field-induced SU(4)\nto SU(2) Kondo crossover has been observed19, and also\nwith the other nanotube dots.15Hcdescribes conduction\nbands in the leads on the left and right. ν=Landν=R\nrespectively denote the left and right lead. The conduc-\ntion electrons are assumed to carry the flavour index m,\nand the continuous energy states in the bands are nor-\nmalized such that {cν,εm, c†\nν′,ε′m′}=δ(ε−ε′)δνν′δmm′.\nThe Fermi level is chosen to be εF= 0, at the center of\nthe flat conduction bands with the width 2 D.HTde-\nscribes charge transfer between the dot and leads. We3\nassume that the tunneling matrix element vνis indepen-\ndent of flavour m, which can also be justified for a class\nof CNT dots.19,35In this situation, the resonance width\ndue to these hybridizations becomes ∆ = ∆ L+∆Rwith\n∆ν≡πρcv2\nν, andρc= 1/2Dthe density of states of the\nconduction band. Unless other wise stated, we set the\nBoltzmann constant kBto unity i.e., kB= 1.\nThis Hamiltonian Hhas the SU( N) symmetry when\nall the impurity energies are equal ǫm≡ǫd. This is be-\ncauseHTthat describes charge transfer preserves the\nflavour index m, and the Coulomb interaction is deter-\nmined essentially by the total number of impurities elec-\ntrons/hatwideNd≡/summationtext\nmndmasHU= (U/2)/hatwideNd(/hatwideNd−1). In\nAppendix A, we explain the symmetry of this Hamilto-\nnian in more detail.\nB. Fermi-liquid parameters\nThe equilibrium retarded Green Gr\nm(ω) is an useful\ntool to study transport coefficients of quantum dots.\nGr\nm(ω) is given by\nGr\nm(ω,T)≡ −i/integraldisplay∞\n0dtei(ω+i0+)t/angbracketleftbig\n{dm(t), d†\nm(0)}/angbracketrightbig\n(6)\n=1\nω−ǫm+i∆−Σm(ω,T), (7)\nAm(ω,T)≡ −1\nπImGr\nm(ω,T). (8)\nHere,/an}bracketle{tO/an}bracketri}ht ≡Tr/bracketleftbig\nOe−H/T/bracketrightbig\n/Ξ with Ξ ≡Tre−H/Tis the\nthermal average of an observable O.\nThe phase shift δmis a primary parameter that char-\nacterizes the Fermi-liquid ground state. It is determined\nby the value of the self-energy Σ m(ω,T) atω=T= 0,\nδm=π\n2−tan−1/bracketleftbiggǫm+ Σm(0,0)\n∆/bracketrightbigg\n.(9)\nThe Friedel sum rule also relates the phase shift δmto\nthe occupation number /an}bracketle{tndm/an}bracketri}ht,44\n/an}bracketle{tndm/an}bracketri}ht=∂Ω\n∂ǫmT→0− −− →δm\nπ. (10)\nHere, Ω = −Tlne−H/Tis the free energy. The phase\nshift also determines the value of the spectral function at\nω=T= 0,\nAm(0,0) =sin2δm\nπ∆. (11)\nThe linear-susceptibilities χm1,m2are also important\nparameters that determine the Fermi-liquid properties:\nχm1,m2≡/integraldisplay1/T\n0dτ/an}bracketle{tδndm1(τ)δndm2/an}bracketri}ht.(12)\nHere,δndm≡ndm−/an}bracketle{tndm/an}bracketri}htis the fluctuation of the oc-\ncupation number, and δndm(τ) = eτHδndme−τH. Thederivative of the self-energy with respect to the impurity\nlevel also gives the susceptibilities,\nχm1,m2=−∂2Ω\n∂ǫm1∂ǫm2=−∂/an}bracketle{tndm1/an}bracketri}ht\n∂ǫm2\nT→0− −− →Am1(0,0)/parenleftbigg\nδm1,m2+∂Σm1(0,0)\n∂ǫm2/parenrightbigg\n.\n(13)\nThe Ward-Takahashi identities relate the linear-\nsusceptibilities to the wave function renormalization fac-\ntorZmand the vertex function Γ mm′;m′m(0,0;0,0),45,46\nχm,m=Am(0,0)\nZm,1\nZm≡1−∂Σm(ω,0)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,(14)\nχm,m′=−Am(0,0)Am′(0,0)Γmm′;m′m(0,0;0,0).(15)\nΓmm′;m′m(ω,ω′;ω′,ω) is them/ne}ationslash=m′component of the\nvertex correction, defined at T= 0 for the causal Green’s\nfunctions. Note that the intra-level components for m=\nm′vanish at zero frequencies, Γ mm;mm(0,0;0,0) = 0,\nbecause of the fermionic antisymmetrical properties, i.e.,\nthe Pauli exclusion principle. The renormalized level po-\nsition/tildewideǫmand corresponding resonance width /tildewide∆mare de-\ntermined by Zm,\n/tildewideǫm≡Zm/bracketleftbig\nǫm+Σm(0)/bracketrightbig\n,/tildewide∆m≡Zm∆.(16)\nThe residual interaction between quasi-particles /tildewideUm,m′\nis also an essential Fermi-liquid parameter:47\n/tildewideUm,m′≡ZmZm′Γmm′;m′m(0,0;0,0).(17)\nThe Wilson ratio Rm,m′corresponds to a dimensionless\nresidual interaction,48and it generallydepends on mand\nm′:\nRm,m′≡1 +/radicalBig\n/tildewideAm/tildewideAm′/tildewideUm,m′.(18)\nHere,/tildewideAm≡Am(0,0)/Zmis the density of states of the\nquasi-particles. Using Eqs.(14)-(17), Rm,m′can also be\nexpressed in terms of the linear susceptibilities,\nRm,m′−1 =−χm,m′\n√χm,mχm′,m′.(19)\nwe calculate the renormalization factor Zmand residual\ninteraction/tildewideUm,m′using the NRG, and deduce the Wil-\nson ratioRm,m′from Eq. (18). We note that the linear\nsusceptibility χm,mdetermines the T-linear specific heat\nCdotdue to the impurity,\nCdot=γT, γ =π2\n3/summationdisplay\nmχm,m.(20)\nFurthermore, fluctuations of the impurity electron filling\nare described by the chargesusceptibility χc≡/summationtext\nmχc,m,4\nwhich can also be expressed in terms of /tildewideAmand/tildewideUm,m′\nusing Eqs. (14) and (15):\nχc,m≡ −/summationdisplay\nm′∂/an}bracketle{tndm/an}bracketri}ht\n∂εm′=/summationdisplay\nm′χm,m′\n=/tildewideAm(0)\n1−/summationdisplay\nm′(/negationslash=m)/tildewideUm,m′/tildewideAm′(0)\n.(21)\nThe last line shows that the residual interaction /tildewideUm,m′\nreduces the free-quasiparticle contributions given by the\nfirst term in the right-hand side.\nC. Conductance and non-linear susceptibilities\nThe conductance gtotthrough a quantum dot can be\nexpressed in the Landauer form,49–51\ngtot=N/summationdisplay\nm=1gm, (22)\ngm=e2\nh4∆L∆R\n∆L+∆R/integraldisplay∞\n−∞dω/parenleftbigg\n−∂f(ω)\n∂ω/parenrightbigg\nπAm(ω,T).\n(23)\nHere,f(ω) = 1/(eω/T+ 1) is the Fermi distribu-\ntion function. It has recently been clarified that low-\ntemperature behavior of conductance up to order T2can\nbe determined completely by an extended Fermi-liquid\ntheory.31–33This formulation is also applicable to the\nmultilevel Anderson impurity model at arbitrary elec-\ntron fillings, and the low-temperature expansion can be\nexpressed in the following form, for symmetric tunneling\ncouplings ∆ L= ∆R,33,34\ngm=e2\nh/bracketleftBig\nsin2δm+cT,m/parenleftbig\nπT/parenrightbig2+···/bracketrightBig\n.(24)\nHere, the first term in the right-hand side corresponds to\nthe ground-state value that is determined by the trans-\nmission probability sin2δm. The coefficient cT,mfor the\nterm of order T2consists of two parts:31–34\ncT,m=π2\n3/bracketleftBig\nwT,m+θT,m/bracketrightBig\n, (25)\nwT,m=−cos2δm/parenleftbigg\nχ2\nm,m+2/summationdisplay\nm′(/negationslash=m)χ2\nm,m′/parenrightbigg\n,(26)\nθT,m=sin2δm\n2π/parenleftbigg\nχ[3]\nm,m,m+/summationdisplay\nm′(/negationslash=m)χ[3]\nm,m′,m′/parenrightbigg\n.(27)\nHere,wT,mrepresents the two-body contributions which\nare determined by the linear susceptibilities χm,m′. The\nother partθT,mrepresents the three-body contributions,\ndetermined by the non-linear susceptibilities,33,34\nχ[3]\nm1,m2,m3≡ −/integraldisplayβ\n0dτ1/integraldisplayβ\n0dτ2/angbracketleftbig\nTτδndm3(τ3)δndm2(τ2)δndm1/angbracketrightbig\n.\n(28)Here,Tτis the imaginary time ordering operator. This\ncorrelation function corresponds also to the derivative of\nlinear susceptibilities:\nχ[3]\nm1,m2,m3=−∂3Ω\n∂ǫm1∂ǫm2∂ǫm3=∂χm2,m3\n∂ǫm1.(29)\nWe note that χ[3]\nm1,m2,m3vanishes in the case at which\nthe system has both the particle-hole and time-reversal\nsymmetries.\nD. NRG approach to the spectral function and\ntransport coefficients\nThe NRG has successfully been applied to multi-\norbitalquantum dots including CNT dots since a seminal\nwork of Izumida et al.10,11,37,40,52–55Using the NRG, we\ncalculate the renormalized parameters such as the phase\nshifts of electrons δ, the wavefunction renormalization\nfactorZ, and the Wilson ratio R.56–58The present work\nuses the NRG to calculate not only the renormalized pa-\nrameters, but also the linear conductance gand the spec-\ntral function Am(ω,T).21–23\nThe key approximation of the NRG is the logarith-\nmic discretization of the conduction band. A dimen-\nsionless parameter Λ ( >1) controls the discretization.\nThe noninteracting part of the discretized Hamiltonian\nH0\nd+HT+Hcistransformedintoaone-dimensionaltight-\nbiding chain with exponentially decaying hopping matrix\nelementstn∼DΛ−n/2. Then, the total Hamiltonian H\nincluding the interactions can be diagonalized iteratively\nby adding the states on the tight-biding chain, starting\nfromtheimpuritysite. Owingtotheexponentialdecayof\ntn, high energy states can be discarded at each successive\nstep without affecting low-lying energy states so much.\nAlthough this iteration itself is an artificial procedure,\nit can be interpreted as a process to probe lower energy\nscale, step by step, stating from high-energy scale.21We\nbriefly explain the process of the NRG iteration in Ap-\npendixB.Furthermore,wecandeducethequasi-particles\nparameters Zm,/tildewideǫm, and/tildewideUm,m′, from asymptotic behav-\niorsofthelow-lyingeigenvaluesnearthe NRGfixedpoint\n(see Appendix C).22,23,48\nIn the present work, we explore the CNT dots in a case\nwherethe four-folddegeneracyis notcompletely lifted by\nthe magnetic field but still a double degeneracy remains\nfor the reason which will be discussed in more detail in\nSec. IIIB.\nOur NRG code uses the U(1) ⊗[SU(2)⊗U(1)]⊗U(1)\nsymmetries to explore the SU(4) to SU(2) Kondo\ncrossover. The U(1) ⊗U(1)⊗U(1)⊗U(1) symmetries are\nused to examine how perturbations, i.e., the valley mix-\ning and spin-orbit interaction, affect the crossover be-\ncause the perturbations break the SU(2) symmetry. The\nNRG calculations are carried out keeping typically the\nlowest 4100 states in the truncation procedure, choosing\nthe discretization parameter to be Λ = 6 .0. Further-\nmore, the spectral function and temperature-dependent5\nconductance59,60are obtained using the complete Fock-\nspace basis algorithm,24–26together with the Oliveira z\naveraging .59,61For obtaining the spectral functions, we\nalso calculate the higher-order correlation function in or-\nder to directly deduce the self-energy Σ m(ω,T).62These\nsupplemental techniques for the dynamical correlations\nfunctions are also described in Appendix E.\nIII. FIELD-INDUCED SU(4) TO SU(2)\nCROSSOVER OF KONDO SINGLET STATE\nA. Microscopic description for the CNT-dot levels\nThe Hamiltonian Hdefined in Eqs. (1)–(4) are de-\nscribed using a representation in which the dot part H0\nd\nhas already been diagonalized. However, to see a micro-\nscopic background, the other basis set using the spin ( ↑,\n↓) and valley (K,K′) degrees of freedom is more suitable.\nThe valley degrees of freedom capture a magnetic mo-\nmentalongtheCNTaxisbecauseofthecylindricalgeom-\netry of the CNT. This orbital moment couples to an ex-\nternal magnetic field parallel to the CNT axis.7,28,37,63,64\nThe four levels of the CNT dot are also coupled each\notherthrough the spin-orbitinteraction ∆ SOandthe val-\nley mixing ∆KK′. The dot part of the Hamiltonian can\nbe expressedin the following form, using the dot-electron\noperatorψ†\nd:ℓsfor orbitalℓ(= K,K′) and spins(=↑,↓),\nH0\nd≡/summationdisplay\nℓ,ℓ′/summationdisplay\ns,s′ψ†\nd:ℓsH0\nd:ℓs,ℓ′s′ψd:ℓ′s′=ψ†\ndH0\ndψd.(30)\nThe matrixH0\nd≡ {H0\nd:ℓs,ℓ′s′}is given by7,28,37\nH0\nd=εd1s⊗1orb+∆KK′\n21s⊗τx+∆SO\n2σz⊗τz−− →M·/vectorb,\n(31)\n− →M≡ −1\n2gs/vectorσ⊗1orb−gorb1s⊗τz/vector ez. (32)\nHere,σj={σj\nss′}andτj={τj\nℓℓ′}forj=x,y,zare the\nPauli matrices for the spin and the valley pseudo-spin\nspaces, respectively. Correspondingly, 1s={δss′}and\n1orb={δℓℓ′}are the corresponding unit matrices. In\na finite magnetic field /vectorb≡µB/vectorB, whereµBis the Bohr\nmagneton, both the spin and orbital moments contribute\ntothemagnetization− →M. Theg-factorforthespinis gs=\n2.0. The orbital moment couples to the magnetic field\nalongthenanotubeaxis, andtheorbitalZeemansplitting\nis given by ±gorb|/vectorb|cosΘ. Here, gorbis the orbital Land´ e\nfactor and Θ is the angle of the magnetic field relative to\nthe nanotube axis, which is chosen as the z-axis with/vector ez\nthe unit vector. The one-particle Hamiltonian of the dot\nlevelsH0\ndcan be diagonalized via the unitary transform\nwith the matrix Ud= (u1,u2,u3,u4), constructed with\nthe eigenvectors,\nH0\ndum=ǫmum,u†\nm·um′=δmm′.(33)Theoperator dmthatannihilatesanelectronintheeigen-\nstate with energy ǫmcan be expressed in a linear combi-\nnation ofψd:ℓs’s via this transform with Ud.\nFor ∆SO= ∆KK′= 0, the spin component of the mag-\nnetization becomes parallel to the field /vector eΘ= cosΘ/vector ez+\nsinΘ/vector exwhile the orbital component is in the direction\nalong the nanotube axis (Θ ≤π/2). Then, the eigenval-\nues ofH0\ndcan be written as ǫ1=εd−(gorbcosΘ+gs/2)b,\nǫ2=εd−(gorbcosΘ−gs/2)b,ǫ3=εd+ (gorbcosΘ−\ngs/2)b,andǫ4=εd+(gorbcosΘ+gs/2)b. Theeigenstates,\nform= 1,2,3 and 4, correspond to |K′↓/vectorb/an}bracketri}ht,|K′↑/vectorb/an}bracketri}ht,|K↓/vectorb/an}bracketri}ht\nand|K↑/vectorb/an}bracketri}ht, respectively, with ↑/vectorband↓/vectorbthe spin defined\nwith respect to the direction along the field /vectorb. Therefore,\nthe thermal average of− →Mcan be written in the form\n/an}bracketle{tψ†\nd− →Mψd/an}bracketri}ht=Morb/vector ez+Ms/vector eΘ, (34)\nMorb=gorb/bracketleftBig\n/an}bracketle{tnd1/an}bracketri}ht−/an}bracketle{tnd4/an}bracketri}ht+/an}bracketle{tnd2/an}bracketri}ht−/an}bracketle{tnd3/an}bracketri}ht/bracketrightBig\n,(35)\nMs=gs\n2/bracketleftBig\n/an}bracketle{tnd1/an}bracketri}ht−/an}bracketle{tnd4/an}bracketri}ht−/an}bracketle{tnd2/an}bracketri}ht+/an}bracketle{tnd3/an}bracketri}ht/bracketrightBig\n.(36)\nIn more general cases, the matrix for− →Mwith respect\nto one-particle eigenvector um, can be expressed in the\nfollowing forms using the Feynman theorem,\n/vectorKm≡u†\nm− →Mum=−u†\nm∂H0\nd\n∂/vectorbum=−∂ǫm\n∂/vectorb.(37)\nThethermalaverageofthemagnetization− →Mcanbewrit-\nten in terms of these matrix elements,\n− →M ≡/angbracketleftBig\nψ†\nd− →Mψd/angbracketrightBig\n=/summationdisplay\nm/vectorKm/an}bracketle{tnd,m/an}bracketri}ht.(38)\nTherefore, the magnetic susceptibility χµν\nM≡∂Mµ/∂bν,\nwhich is not isotropic for nano-tube dots, can be ex-\npressed in the following form,\nχµν\nM=/summationdisplay\nm∂Kµ\nm\n∂bν/an}bracketle{tnd,m/an}bracketri}ht+/summationdisplay\nm,m′Kµ\nm∂ǫm′\n∂bν∂/an}bracketle{tnd,m/an}bracketri}ht\n∂ǫm′\n=/summationdisplay\nm∂Kµ\nm\n∂bν/an}bracketle{tnd,m/an}bracketri}ht+/summationdisplay\nm,m′Kµ\nmKν\nm′χm,m′.\n=/summationdisplay\nm∂Kµ\nm\n∂bν/an}bracketle{tnd,m/an}bracketri}ht+/summationdisplay\nmKµ\nmKν\nm/tildewideAm\n−/summationdisplay\nm/negationslash=m′Kµ\nmKν\nm′/tildewideAm/tildewideAm′/tildewideUmm′. (39)\nHere, the last term in the right-hand side represents the\ncontributions of the residual interaction, or the vertex\ncorrections. Since the Wilson ratio Rm,m′given in Eq.\n(18) depends on the residual interaction, Rm,m′relates\nto the magnetic susceptibilities.\nB. CNT level structure & field-induced crossover\nIn recentexperimentsreportedin Refs.19and 35, non-\nlinearcurrentandcurrentnoiseweremeasuredforaCNT6\ndot with the orbital Land´ e factor gorb≈4 at finite mag-\nnetic fields with an angle Θ ≈75◦. These values of gorb\nand Θ imply that the magnitude of the orbital Zeeman\nsplitting becomes almost the same as the spin Zeeman\nsplitting in this particular situation,\ngorbcosΘ≈1\n2gorb= 1. (40)\nThis situation can take place for CNT dots as the orbital\nLand´ e factor gorbdepends significantly on the diameter\nof nanotube and takes a value around gorb∼10.7In the\ncase where this matching of the orbital and spin Zeeman\nsplittings is satisfied, the energy level of the dot has a\ndouble degeneracy which remains unlifted in magnetic\nfields:\nǫ1=εd−2b, ǫ2≡ǫ3=εd, ǫ4=εd+2b.(41)\nIn this case the occupation numbers of the degeneracy\nbecome the same /an}bracketle{tnd2/an}bracketri}ht=/an}bracketle{tnd3/an}bracketri}ht. Thus, both the orbital\nand spin magnetizations are determined by the occupa-\ntion numbers of the other two levels m= 1 andm= 4:\nMorb=gorbM14andMs=gs\n2M14, with\nM14≡ /an}bracketle{tnd1/an}bracketri}ht−/an}bracketle{tnd4/an}bracketri}ht. (42)\nWe note that ∆ SOand ∆ KK′are less important for the\nexamined CNT dot.19,35,36In this situation, the system\nhas an SU(2) rotational symmetry defined with respect\nto the degeneratestatesin the middle, and the U(1) sym-\nmetry that conserves the sum of the occupation numbers\nn2+n3, in addition to the other two U(1) symmetries\ncorresponding to n1andn4for the levels m= 1 and\nm= 4, respectively. Therefore, the SU(4) symmetry\nthat the total Hamiltonian has at zero field breaks down\nto the U(1) m=1⊗[SU(2)⊗U(1)]m=2,3⊗U(1)m=4symme-\ntry at finite magnetic fields. This SU(2) symmetric part\nplays a central role in the field-induced SU(4) to SU(2)\nKondo crossover, occurring at half-filling point Nd= 2.\nAt this point, due to the matching condition given in Eq.\n(41), the Hamiltonian His invariant under an extended\nelectron-hole transformation described by Eq. (A5).\nC. Comparison of NRG and experiments results:\ngate-voltage dependence at finite BorT\nWe have shown in the previous work35,36that the\nlevel scheme, given in Eq. (41), nicely explains the field-\ninduced SU(4) to SU(2) Kondo crossover observed at\nhalf-fillingNd= 2 where two electrons occupy the lo-\ncal levels of the CNT dot. The Coulomb interaction for\nthis CNT dot is estimated to be U≈6 meV, and the hy-\nbridization energy is ∆ ≡∆L+∆R≈0.9 meV. Its asym-\nmetric factor is 4∆ L∆R/(∆L+ ∆R)2= 0.92. The en-\nergy scales the interaction as U/(π∆) = 2.0. The SU(4)\nKondo temperature for the scaled UisTSU(4)\nK/∆ = 0.41\nwhich corresponds to TK= 4.3 K in a real scale. These\nvalues ofUand ∆ are much larger than the value of the/s50/s53 /s50/s54 /s50/s55/s48/s49/s50/s51/s52\n/s100/s97/s115/s104/s101/s100/s32/s108/s105/s110/s101/s115/s58/s32/s78/s82/s71/s99/s111/s110/s100/s117/s99/s116/s97/s110/s99/s101/s32 /s40/s101/s50\n/s47/s104/s41\n/s86\n/s103/s32/s40/s86/s41/s115/s111/s108/s105/s100/s32/s108/s105/s110/s101/s115/s58/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116/s115/s103\n/s111/s114/s98/s32/s99/s111/s115/s32 /s113/s32 /s61/s32/s49\n/s85/s47/s40 /s112/s68/s41 /s32/s61/s32/s50/s46/s48\n/s68 /s32/s61/s32/s48/s46/s57/s32/s109/s101/s86/s68\n/s75/s44/s75/s39/s32/s61 /s32/s68\n/s83/s79/s32/s61/s32/s48\n/s32/s48\n/s32/s50/s46/s48\n/s32/s52/s46/s48\n/s32/s49/s48/s46/s48/s66/s32/s40/s84/s41/s40/s97/s41\n/s50/s53 /s50/s54 /s50/s55/s48/s49/s50/s51/s52\n/s66/s32/s40/s84/s41/s99/s111/s110/s100/s117/s99/s116/s97/s110/s99/s101/s32 /s40/s101/s50\n/s47/s104/s41\n/s86\n/s103/s32/s40/s86/s41/s32/s48\n/s32/s50/s46/s48\n/s32/s52/s46/s48\n/s32/s49/s48/s46/s48/s68 /s32/s61/s32/s48/s46/s57/s32/s109/s101/s86/s85/s47/s40 /s112/s68/s41 /s32/s61/s32/s50/s46/s48/s68\n/s75/s44/s75/s39/s32/s61 /s32/s68\n/s83/s79/s32/s61/s32/s48/s46/s50/s32/s109/s101/s86\n/s103\n/s111/s114/s98/s32/s99/s111/s115/s32 /s113/s32 /s61/s32/s48/s46/s55/s40/s98/s41\nFIG. 1. (a) and (b) plot the zero temperature conductance\nas functions of experimental gate voltages Vgfor four values of\nmagnetic fields: B= 0,2,4, and 10 T. In the CNT dot where\nthe experiments observed the crossover, the experimental Vg\nrelates to theoretical the theoretical ǫdvia a linear relation:\nVg= (0.8ǫd/U+27.18) in units of volt. The asymmetric fac-\ntor in Eq. (23) and the Coulomb interaction are respectively\n4∆L∆R/∆2= 0.92 andU/(π∆) = 2.0 for each two figure.\nHere, ∆ ≡∆L+ ∆R= 0.9meV. In (a), the dashed lines\nand solid lines respectively represent NRG and experimenta l\nresults. The NRG results plotted in (a) are for parameters:\n∆SO= ∆K,K′= 0,gorbcosθ= 1. The experimental results\nhave been obtained at T= 16 mK,19,35which is much lower\nthanTSU(4)\nK= 4.3K,theKondotemperaturefor thehalf-filled\ncase (at Vg≃26 mV). (b) plots only NRG results by solid\nlines. The parameters for the results are ∆ SO= ∆K,K′= 0.2\nmeV,gorbcosθ= 0.7.\nvally mixing ∆ KK′and than that of the spin-orbit in-\nteraction ∆ SO. The experimental values of ∆ KK′and\n∆SOare ∆KK′≃∆SO≈0.2 meV. We note that these\nvalues depend on the detailed structures of carbon nan-\notube samples. Furthermore, it has also estimated in\nthe experiment that the orbital Zeemann coupling as\ngorbcosθ≃0.7,35which is not so far from the match-\ning condition gorbcosθ= 1, described in Eq. (40).\nWe fist of all discuss the magnetoconductance for the\ncasegorbcosθ= 1 and ∆ KK′= ∆SO= 0, at which an7\n/s115/s111/s108/s105/s100/s32/s108/s105/s110/s101/s115/s58/s32/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116/s115\n/s100/s97/s115/s104/s101/s100/s32/s108/s105/s110/s101/s115 /s58/s32/s78/s82/s71\n/s50/s53 /s50/s54 /s50/s55/s48/s49/s50/s51/s52\n/s39/s39/s50/s39/s39 /s39/s39/s49/s39/s39/s84\n/s78/s82/s71/s61/s32/s49/s46/s55/s32/s75 /s32/s84\n/s78/s82/s71/s61/s32/s48/s75/s84\n/s101/s120/s61/s32/s50/s46/s48/s32/s75 /s32/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s50/s46/s48 /s84/s83/s85/s40/s52/s41\n/s75/s32/s61/s32/s52/s46/s51/s75/s99/s111/s110/s100/s117/s99/s116/s97/s110/s99/s101/s32 /s40/s101/s50\n/s47/s104/s41\n/s86\n/s103/s32/s40/s86/s41/s84\n/s78/s82/s71/s32/s61/s32/s51/s46/s56/s32/s75/s84\n/s101/s120/s61/s32/s52/s46/s53/s32/s75\n/s32/s84\n/s101/s120/s61/s49/s54/s32/s109/s75\n/s39/s39/s51/s39/s39\nFIG. 2. Zero-field, B= 0, conductance at finite tempera-\ntures: experimental (solid line) and NRG(dashedline) resu lts\nare plotted vs Vg. The experimental data has been obtained\natT= 16 mK, 2 K, and 4.5 K.19For NRG calculations,\nslightly lower temperatures T= 0 K, 1.7 K, and 3.8 K are\nchosen. The other parameters are the same as those used for\nFig. 1. The numbers “3”, “2”, and “1” shown in these two\nfigures represent the electron filling Ndat corresponding Vg.\nidealSU(4)toSU(2)Kondocrossoveroccurs. Figure1(a)\ncompares the NRG results of the conductance at T= 0\nwith the experimental results obtained at T= 16 mK.\nThe values of Uand ∆ for the NRG calculations are the\nexperimental values. The comparisons which have been\ndone also in Ref. 35 show that the NRG results nicely\nagree with the experimental results. We can clearly see\nin this figure that the Kondo ridge emerges near half-\nfillingVg≃26V in the absence and presence of magnetic\nfields. Its height reduces from the SU(4) value 3 .68 e2/h\nto the SU(2) value 1 .84 e2/has magnetic field increases.\nThis reduction of the height implies that the observed\nSU(2)behavioriscausedbythedoublydegeneratestates,\nwhich are labeled as m= 2 andm= 3 in Eq. (41),\nare shifted towards the Fermi level in the half-filled case\nwhereǫd=−3U/2. Furthermore, the additional sub-\npeaks, which emerge outside the Kondo ridge for large\nmagnetic fields B/greaterorsimilar4 T, can also be regarded as the res-\nonances corresponding to the other two non-degenerate\nlevels, labeled as m= 1 andm= 4. Note that the Kondo\nridgescorrespondingto the 1 /4and 3/4fillings arenot so\npronounced at B= 0 because the Coulomb interaction\nfor this CNT dot U/(π∆) = 2.0 is not very large.\nWe also examine the case where the matching con-\ndition described in Eq. (40) cannot be satisfied, taking\nthe parameters such that ∆ KK′= ∆SO= 0.2 meV and\ngorbcosθ= 0.765. At zero field b= 0, the level structure\nis given by\nǫ1=ǫ2=ǫd−1\n2/radicalBig\n∆2\nKK′+∆2\nSO,(43)\nǫ3=ǫ4=ǫd+1\n2/radicalBig\n∆2\nKK′+∆2\nSO.(44)The valueofthe gapdue tothe spin-orbitinteractionand\nvalley mixing is/radicalbig\n∆2\nKK′+∆2\nSO≃0.28meV, which is\nsmaller than the resonance width. Figure 1(b) shows the\nNRG results, obtained taking the other parameters the\nsame as those for Fig. 1(a), i.e. U/(π∆) = 2.0, ∆ = 0.9\nmeV. The broad peak emerges also in Fig. 1(b) although\nits peak value 3 .14e2/his smaller than that the value\n3.68e2/hwhich is for the case of ∆ KK′= ∆SO= 0 and\ngorbcosθ= 1. The flat structure is still preserved in\nsmall magnetic fields up to B≈5T. This preserva-\ntion shows that the SU(4) to SU(2) Kondo crossover is\nrobust against the perturbations. The larger magnetic\nfieldsB/greaterorsimilar5Tsplit the peak into two peaks. Further-\nmore, other two sub peaks grow with increasing B, and\nthus the four distinct peaks emerge. Figure 1(b) shows\nthat the magnetoconductance for B= 10 T has the four\npeaks.\nOur NRG results for the CNT dots, reported so far,\nwere restricted to ground-state properties. The present\nwork sheds light also on the finite-temperature and dy-\nnamic properties of the Kondo crossover. Figure 2 com-\npares the experimental results35of the zero-field conduc-\ntance measured at T= 16 mK, 2 K, and 4.5 K with the\ncorresponding NRG results, calculated for slightly lower\ntemperatures T= 0 K, 1.7 K, and 3.8 K to demonstrate\nhowthesecomparisonsworkatthebest. Weseeareason-\nableagreementbetweenthetheoreticalresultsandexper-\nimental results. The height of the Kondo ridge emerging\nnear half-filling Vg≃26Vdecreases as temperature in-\ncreases. At temperatures of order T∼TSU(4)\nK= 4.3\nK, Four peaks corresponding to the Coulomb oscillation\nemerge. This agreement also indicates that the theory\nof the SU(4) Kondo effect can explain the experimental\nresults of the conductance.\nD. Scaling behavior of SU( 4) conductance at\nquarter and half-filling\nIn this section, we examine scaling behavior of the\nSU(4)conductanceasfunctionsoftemperatureespecially\nat quarter-filling Nd= 1 and half-filling Nd= 2. In Fig.\n2, the valley is quarter and half-filled at gate voltages,\nVg≃25VandVg≃26V, respectively. The first value\nofVgcorresponds to the theoretical value ǫd/U=−1/2,\nand the second one corresponds to ǫd/U=−3/2.\nSince theT2coefficientcT,mof conductance given in\nEq.(25)dependsonlinearandnon-linearsusceptibilities,\nwe discuss properties of these susceptibilities to exam-\nine the scaling. In the SU( N) symmetric case at which\ntheNimpurity levels are degenerate ǫm≡ǫd, the lin-\near susceptibilities have only two independent elements:\na diagonal element χm,mand the off-diagonal element\nχm,m′form/ne}ationslash=m′. The diagonal element determines a\ncharacteristic energy scale T∗,\nT∗≡1\n4χm,m. (45)8\nIn the particle-hole symmetric case ǫd/U→ −(N−1)/2,\nit correspondsthe renormalizedwidth of the Kondo reso-\nnanceasT∗→(π/4)/tildewide∆. Atanyarbitraryelectronfillings,\nT∗scales the impurity specific heat defined in Eq. (20)\nasCdot=Nπ2\n12(T/T∗).\nIn the SU(N) symmetric case, the Wilson ratio Rde-\nfined in Eq. (18) also takes a simplified form R−1 =\n−χm,m′/χm,m, wherethe flavourindexhas been dropped\nin the left-hand side. The Wilson ratio takes the maxi-\nmum value in the strongcouplinglimit U→ ∞atinteger\nfilling points:\nRmax\nSU(N)−1U→∞, Nd→integer− −−−−−−−−−−−− →1\nN−1.(46)\nThis is because the charge fluctuations are suppressed in\nthis limit, and the charge susceptibility defined in Eq.\n(21) vanishes: χc,m=χm,m+(N−1)χm,m′→0.\nThe non-linear susceptibilities have three independent\nelements in the SU( N) symmetric case for N >2:\nχ[3]\nm,m,m,χ[3]\nm,m′,m′, andχ[3]\nm,m′,m′′, wherem/ne}ationslash=m′,m′/ne}ationslash=\nm′′, andm/ne}ationslash=m′′. We note that the derivative of the di-\nagonal linear susceptibility χm,mwith respect to ǫdcan\nbe related to the first two elements, as\n∂χm,m\n∂ǫd=χ[3]\nm,m,m+ (N−1)χ[3]\nm,m′,m′.(47)\nWe next revisit the temperature dependence of con-\nductance for the SU( N) Anderson model, which has been\npreviously studied in detail by Anders et al,12taking in\na recent Fermi-liquid viewpoint.31–34The low-energy ex-\npansion, given in Eqs.(24) and (25), takes the following\nform in the SU( N) symmetric case,\ngtot=Ne2\nh/bracketleftBigg\nsin2δ−CT/parenleftbiggπT\nT∗/parenrightbigg2\n+···/bracketrightBigg\n,(48)\nCT≡π2\n48/parenleftbig\nWT+ ΘT/parenrightbig\n. (49)\nThe two-body contributions WTand three-body contri-\nbutions Θ T, defined respectively in Eqs. (26) and (27),\ncanberewritteninthe followingformusingalsoEq.(47),\nWT≡ −/bracketleftBig\n1 + 2 (N−1)(R−1)2/bracketrightBig\ncos2δ,(50)\nΘT=sin2δ\n2π1\nχ2m,m∂χm,m\n∂ǫd. (51)\nFigures 3(a) and 3(b) show the temperature de-\npendence of gtotat half-filling and quarter-filling, re-\nspectively. We choose four values of the interaction,\nU/(π∆) = 2.0,3.0,4.0,and 5.0. The first value is the\nexperimental value for the valley where the SU(4) Kondo\neffect occurs. In other valleys, the experimental values\nofU/(π∆) can be larger than 2 .0, and we also consider\nthe larger interaction cases. The temperatures are scaled\nby the Kondo energy scales T∗defined in Eq. (45) foreachNdandU. In each of the two figures, we find\nthat the scaled conductance curves collapse into a sin-\ngle curve over a wide range of temperatures T/lessorsimilarT∗for\nU/(π∆)/greaterorsimilar3.0, and thus the conductance shows the uni-\nversality for each filling.\nTo clarify the filling dependence of the universality,\nwe replot the curves of quarter and half filling in Fig.\n3(c). For the two curves, we choose the largest Uamong\nthe four,U/(π∆) = 5.0. We find that whereas these\ntwo curves almost overlap each other around T≃T∗,\nthe conductance of quarter-filling is slightly larger than\nthatofhalf-fillingatlow-temperatures T <T∗, especially\naroundT≃0.1T∗. The inset of Fig. 3(c) clearly shows\nthe different behavior depending on the filling.\nThis filling dependence of the scaling can be explained\nby the Fermi-liquid theory31–34. We have recently clar-\nified using also the NRG that the derivative of the di-\nagonal susceptibility |∂χm,m/∂ǫd|becomes much smaller\nthanχ2\nm,min awideregionofelectronfillings1 /lessorsimilarNd/lessorsimilar3\nforlargeinteractions U≫∆.34Fromthisresultitfollows\nthat the three-body contributions Θ Tgiven by Eq. (51)\nalmostvanishinthesameregion,andthe T2conductance\nin this case is determined by the two-body contributions,\nas\nCT≃ −/parenleftbig\nπ2/48/parenrightbig/bracketleftbig\n1 + 6(R−1)2/bracketrightbig\ncos(πNd/2).(52)\nThus,CTbecomes 0 at quarter filling Nd= 1, whereas it\nis finiteCT=/parenleftbig\nπ2/48/parenrightbig\n[1+6(R−1)2] at half-filling Nd=\n2. Therefore,the conductanceat Nd= 1persiststhezero\ntemperature value at 0 ≤T/T∗≃0.1 as shown in the\ninset of Fig. 3(c). At Nd= 2, the coefficient approaches\nC∞\nT= 5π2/144≃0.3427 atU→ ∞. This is because the\nWilson ratio for N= 4 saturates to the strong coupling\nlimit value Rmax\nSU(4)−1→1/3. Already at U/(π∆) = 3.0,\nthe coefficient is given by CT≃0.34, which is very close\nto the strong-coupling value C∞\nT.9\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s97/s41\n/s84/s47 /s84/s42/s104/s97/s108/s102/s45/s102/s105/s108/s108/s105/s110/s103/s44\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s103\n/s116/s111/s116/s47/s103\n/s48/s32/s32/s50/s46/s48\n/s32/s32/s51/s46/s48\n/s32/s32/s52/s46/s48\n/s32/s32/s53/s46/s48/s85/s47/s40 /s112/s68 /s41/s32\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s98/s41/s103\n/s116/s111/s116/s47/s103\n/s48/s85/s47/s40 /s112/s68 /s41/s32\n/s113/s117/s97/s114/s116/s101/s114/s45/s102/s105/s108/s108/s105/s110/s103/s44\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s49/s47/s50\n/s84/s47 /s84/s42/s32/s32/s50/s46/s48\n/s32/s32/s51/s46/s48\n/s32/s32/s52/s46/s48\n/s32/s32/s53/s46/s48\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s46/s48/s49 /s48/s46/s49 /s49/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48\n/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s53/s46/s48/s113/s117/s97/s114/s116/s101/s114/s45/s102/s105/s108/s108/s105/s110/s103\n/s84/s47 /s84/s42/s104/s97/s108/s102/s45/s102/s105/s108/s108/s105/s110/s103/s103\n/s116/s111/s116/s47/s103\n/s48\n/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s53/s46/s48/s113/s117/s97/s114/s116/s101/s114/s45/s102/s105/s108/s108/s105/s110/s103\n/s84/s47 /s84/s42/s104/s97/s108/s102/s45/s102/s105/s108/s108/s105/s110/s103/s40/s99/s41\nFIG. 3. SU(4) conductance curves are plotted as functions\nof temperature T. (a) and (b) show the results at half-filling\nand quarter-filling, respectively. In these figures, the con duc-\ntance curves are plotted for four values of the interaction,\nU/(π∆) = 2.0,3.0,4.0,and 5.0. (c) shows the curves of half-\nfilling and quarter-filling for the largest value U/(π∆) = 5.0.\nThe inset of (c) is an enlarged view for 0 .01≤T/T∗≤1.\nThis inset clearly shows that the T2coefficient CTgiven in\nEq. (49) vanishes at quarter-filling whereas that at half-fil ling\ndoes not. In (a)-(c), the x-axis is scaled by the Kondo tem-\nperature T∗≡1/(4χm,m). The values of T∗/∆ at half-filling\nare 0.41,0.29,0.20,and 0.13 forU/(π∆) = 2.0,3.0,4.0,and\n5.0, respectively. Similarly, the values at quarter-filling a re\n0.82,0.57,0.37,and 0.23. The y-axis is normalized by the\nzero temperature values of conductance, g0= (4e2/h)sin2δ.\nAt half-filling, sin2δ≡1 for any value of U, and at quarter-\nfilling, sin2δ= 0.56,0.55,0.54,and 0.54.10\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s77\n/s49/s52/s40/s97/s41/s83/s105/s110/s50\n/s100\n/s50/s61/s83/s105/s110/s50\n/s100\n/s51/s61/s49\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s49/s57/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s52/s49/s32/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s83/s105/s110/s50\n/s100\n/s49/s61/s83/s105/s110/s50\n/s100\n/s52\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s50/s46/s48/s83/s105/s110/s50\n/s100\n/s109/s32/s38/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s56/s45/s52/s48/s52/s56\n/s101/s52/s40/s72/s70/s41/s32/s61/s32/s50/s98/s32/s43/s32/s85/s47/s50\n/s101/s49/s40/s72/s70/s41/s32/s61/s32/s45/s50/s98/s32/s45/s32/s85/s47/s50/s101/s52/s40/s85/s61/s48/s41/s32/s61/s32/s50/s98\n/s109/s32/s61/s32/s50/s44/s51\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s109/s32/s61/s32/s52\n/s109/s32/s61/s32/s49/s101\n/s109/s47/s68\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s126\n/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s50/s46/s48/s40/s98/s41\n/s101/s49/s40/s85/s61/s48/s41/s32/s61/s32/s45/s50/s98\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s90\n/s109\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s49/s57/s32 /s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s52/s49/s32/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32/s109/s32/s61/s32/s49/s44/s52\n/s109/s32/s61/s32/s50/s44/s51\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s85/s47/s40 /s112/s68/s41 /s32/s61/s32/s50/s46/s48/s40/s99/s41\n/s90\n/s50/s32 /s32/s48/s46/s50/s52 /s32\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s82\n/s50/s44/s51/s32/s45/s32/s49/s32 /s32/s48/s46/s57/s54/s32/s40/s50/s44/s51/s41/s32\n/s40/s49/s44/s52/s41 /s32/s82\n/s109/s44/s109/s39/s32/s45/s49/s32\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s40/s100/s41\n/s40/s49/s44/s50/s41/s32/s61/s32/s40/s49/s44/s51/s41/s32/s61/s32/s40/s50/s44/s52/s41/s32/s61/s32/s40/s51/s44/s52/s41\n/s82\n/s49/s44/s50/s32/s62/s32 /s82\n/s49/s44/s52/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s50/s46/s48 /s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s101/s41\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s85\n/s109/s44/s109/s39/s32/s47 /s40/s112/s68/s41 /s32 /s126/s85/s47/s40 /s112/s68/s41 /s32/s61/s32/s50/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50\n/s40/s50/s44/s51/s41/s32\n/s40/s49/s44/s50/s41/s32/s61/s32/s40/s49/s44/s51/s41/s32/s61/s32/s40/s50/s44/s52/s41/s32/s61/s32/s40/s51/s44/s52/s41/s40/s49/s44/s52/s41 /s32\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s102/s41\n/s98/s47/s85/s32/s61/s32/s48/s46/s48/s54/s53/s32/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s98/s47/s85/s85/s47/s40 /s112/s68/s41 /s32/s61/s32/s50/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50\n/s40/s50/s44/s51/s41/s32\n/s40/s49/s44/s50/s41/s32/s61/s32/s40/s49/s44/s51/s41/s32/s61/s32/s40/s50/s44/s52/s41/s32/s61/s32/s40/s51/s44/s52/s41/s40/s49/s44/s52/s41 /s32\n/s85\n/s50/s44/s51/s47/s40/s112/s68/s41 /s32 /s32/s48/s46/s50/s51 /s32 /s126/s126/s85\n/s109/s44/s109/s39/s32/s47 /s40/s112/s68/s41 /s32\nFIG. 4. (a) sin2δmand magnetization M14=/angbracketleftnd1/angbracketright − /angbracketleftnd4/angbracketright, (b) renormalized level position /tildewideεm, (c) renormalization factor\nZm, (d) Wilson ratio Rm,m′−1, and (e) residual interaction /tildewideUm,m′are plotted as functions of magnetic field bat half-filling\nεd/U=−3/2 forU/(π∆) = 2.0. the x axis in (a)-(e) is scaled by the SU(4) Kondo temperatu reTSU(4)\nK= 0.41∆ = (0 .065U)\ndetermined at b= 0. In (f) the axis is scaled by Ufor examining the behavior of /tildewideUm,m′at larger magnetic fields b > TSU(4)\nK.\nThe dot levels ǫmare chosen in a such way that is described in Eq. (41). In (b), t he dashed lines indicate the bare Zeemann\nsplitting, and the dash-dotted lines indicate the mean-fiel d splitting εHF\n1=−(2b+U/2) andεHF\n4= (2b+U/2). In a similar way,\nthe dashed lines in (c)-(f) indicate the SU(2) symmetric val ues ofZ2→0.23,R2,3→1.96, and /tildewideU2,3/π∆→0.026, respectively.\nIV. EVOLUTION OF QUASI-PARTICLES\nALONG THE FIELD-INDUCED CROSSOVER\nThe Fermi-liquid parameters for renormalized quasi-\nparticles describe low energy properties of quantum dots.\nThe phase shift δm, which is the primary parameter,\ncorresponds to zero temperature transmission probabil-\nity. The renormalization factor Zmand the Wilson ra-\ntioRm,m′are also important parameters to examine the\nhigher-order properties of the Fermi-liquid states. We\ndescribe how these and related parameters evolve along\na field induced crossover from the SU(4) to SU(2) Kondo\nsinglet state. The crossover occurs for the dot levels de-\nfined in Eq. (41). Specifically, we consider the half-filled\ncase corresponding to the point Vg≃26 V in the middle\nof the Kondo ridge seen in Fig. 1. The center of the dot\nlevelsis chosento be εd=−(3/2)U, and thus the average\nnumber of electrons in the dot levels conserves in a way\nsuch that /an}bracketle{tnd2/an}bracketri}ht=/an}bracketle{tnd3/an}bracketri}ht= 1/2 and/an}bracketle{tnd1/an}bracketri}ht+/an}bracketle{tnd4/an}bracketri}ht= 1 atfinite magnetic fields.\nWe examine two different values for the Coulomb in-\nteraction in the following: (i) U/(π∆) = 2.0 and (ii)\nU/(π∆) = 4.0. The first one, (i), simulates the situation\nof the CNT dot, in which the field-induced crossover has\nbeen observed and the parameters have been estimated\nasU≈6 meV and ∆ ≈0.9 meV.19,35We can see more\nclearly the renormalization effects due to strong correla-\ntions in the second case (ii).\nA. Fermi-liquid parameters for the real CNT dot\nFirst of all, we consider the case U/(π∆) = 2.0 that is\nestimated by the recent experiments. The NRG results\nfor this case are shown in Fig.4. Figure 4(a) shows the\ntransmission probability Tm(0) = sin2δmand magneti-\nzationM14≡ /an}bracketle{tnd1/an}bracketri}ht − /an}bracketle{tnd4/an}bracketri}ht, as a function of magnetic\nfieldbat half-filling. The degenerate levels, m= 2 and11\nm= 3, keep their positions just on the Fermi level for fi-\nnite magneticfields, andshowtheunitarylimit transport\nsin2δm= 1 asδ2=δ3=π/2. The magnetic field partly\nlifts the degeneracy and the other two states, m= 1 and\nm= 4. For these orbitals, sin2δmdecreases as magnetic\nfield increases. The magnetization M14, which in the\npresent case is determined by the occupation number or\nthese two levels, increases as the magnetic field increases.\nIt saturates to M14→1 in the limit of b→ ∞, and\nthe charge fluctuations are suprressed as /an}bracketle{tnd,1/an}bracketri}ht →1 and\n/an}bracketle{tnd,4/an}bracketri}ht →0.\nFigure 4(b) shows the renormalizedresonancelevel po-\nsition/tildewideεmas a function of magnetic field b. The two-fold\ndegenerate states at the center, /tildewideε2=/tildewideε3= 0, remain\njust on the Fermi level at arbitrary magnetic fields. The\nother two levels, /tildewideε1and/tildewideε4move away from the Fermi\nlevel asbincreases. Slopes of them are steeper than\nthose for the noninteracting electrons 2 b(dashed line).\nIn the large field limit b→ ∞, the renormalized level\npositions approach the one described in the mean-field\ntheory, i.e., εHF\n1=−(2b+U/2) andεHF\n4= (2b+U/2).\nThese asymptotic form can be obtained as follows, sub-\nstituting the mean values /an}bracketle{tnd1/an}bracketri}ht= 1,/an}bracketle{tnd4/an}bracketri}ht= 0 and\n/an}bracketle{tnd2/an}bracketri}ht=/an}bracketle{tnd3/an}bracketri}ht= 1/2 into the dot-part of the Hamilto-\nnian withεd=−(3/2)U;\nH0\nd+HU= 2b(nd4−nd1)−3U\n2(nd2+nd3+nd1+nd4)\n+U/bracketleftBig\nnd2nd3+nd1nd4+(nd2+nd3)(nd1+nd4)/bracketrightBig\nb→∞− −−− →U/bracketleftbigg\nnd2nd3−1\n2(nd2+nd3)/bracketrightbigg\n+/parenleftbigg\n2b+U\n2/parenrightbigg/parenleftBig\nnd4−nd1/parenrightBig\n+const.(53)\nHere, the Coulomb interaction between the orbitals m=\n2 and 3 is kept undecoupled. This asymptotic Hamil-\ntonian also shows that the symmetric SU(2) Anderson\nmodel describes the Fermi-liquid properties of these two\norbitals.\nThe magnetic field dependence of the wavefunction\nrenormalization factors Zmplotted in Fig. 4(c) more\nclearly shows the crossover. At finite magnetic fields,\nonly two of the four Zm’s become independent: Z2=Z3\nandZ1=Z4because ofthe particle-holesymmetry given\nin Eq. (A5). The first one is for the degenerate levels re-\nmaining on the Fermi level, and the second one is for the\nlevels moving away from the Fermi level. At zero field,\nwhere the system has the SU(4) symmetry, these two fac-\ntorsforthe different orbitalsbecome identical eachother:\nZ2=Z1=ZSU(4)= 0.52 forU/(π∆) = 2.0. Substitut-\ning this SU(4) value into Eq. (45) gives the SU(4) Kondo\nenergy scale TSU(4)\nK/∆ = 0.41. Many-body effects signif-\nicantly renormalize Z2from the SU(4) value as magnetic\nfield increases. In the limit of b→ ∞, it approaches the\nSU(2) symmetric value ZSU(2)= 0.23, which determinesthe SU(2) Kondo energy scale TSU(2)\nK/∆ = 0.19. The\nmany-body effects become less important for Z1with\nincreasing field, and Z1approaches the non-interacting\nvalueZ1→1 for the large magnetic field.\nIn order to clarify the many-body effects between elec-\ntrons occupying the different orbitals, we also examine\nthe orbital dependent Wilson ratio Rm,m′and corre-\nsponding residual interaction /tildewideUm,m′. Figures 4(d) and\n4(e) respectively show Rm,m′−1 and/tildewideUm,m′as func-\ntions ofb/TSU(4)\nK. Magnetic field dependences of /tildewideUm,m′\nare plotted also in Fig. 4(f), where the magnetic field\nis scaled by Uto examine behaviors of /tildewideUm,m′at larger\nfieldsb≫TSU(4)\nK. Owing to the particle-hole symmetry,\nonly three of the six /tildewideUm,m′are independent: /tildewideU2,3,/tildewideU1,4,\nand/tildewideU1,2=/tildewideU1,3=/tildewideU2,4=/tildewideU3,4. Correspondingly, three\nindependent parameters of the Wilson ratios, R2,3,R1,4,\nandR1,2, can be deduced from Eq. (18):\nR2,3−1 =1\nZ2/tildewideU2,3\nπ∆, (54)\nR1,4−1 =sin2δ1\nZ1/tildewideU1,4\nπ∆, (55)\nR1,2−1 =/radicalBigg\nsin2δ1\nZ11\nZ2/tildewideU1,2\nπ∆. (56)\nAmong the three independent parameters of Rm,m′and\n/tildewideUm,m′,R2,3−1and/tildewideU2,3arefor the doublydegenerateor-\nbitals onthe Fermilevel. At zerofield, R2,3and/tildewideU2,3take\nthe SU(4) values R2,3−1 = 0.31 and/tildewideU2,3/(π∆) = 0.16\nforU/(π∆) = 2.0, andR2,3−1 already approaches very\nclosely to the value for the infinite Coulomb interaction:\nRmax\nSU(4)−1≡1/3. These parameters continuously evolve\nfrom the SU(4) values to the SU(2) symmetric values:\nRSU(2)−1 = 0.96 and/tildewideU2,3/(π∆) = 0.23.\nWe also discuss the field dependence of the other pa-\nrameters,R1,2,R1,4,/tildewideU1,2, and/tildewideU1,4./tildewideU1,2decreases from\nthe SU(4) value with increasing magnetic field, and the\ncorrespondingWilson ratio R1,2−1 decreasesto the non-\ninteracting value 0. In contrast to /tildewideU1,2,/tildewideU1,4increases\nfrom the zero field value and becomes larger than /tildewideU2,3\nand/tildewideU1,2forb>TSU(4)\nK. It further increases at the larger\nmagnetic field regions b≫TSU(4)\nKas shown in Fig. 4(f).\nThis field dependence of /tildewideU1,4is similar to that of /tildewideUfor\na single orbital Anderson model66,67, although/tildewideU1,4does\nnot approach to the bare value U. We briefly discuss\nthe field dependence of /tildewideUand of the other Fermi liquid\nparameters for the single orbital Anderson model in Ap-\npendix D. This enhancement of /tildewideU1,4does not result in\nthe enhancement of R1,4. In fact, R1,4−1 as well as\nR1,2−1 decreases to 0 since the factor sin2δ1goes to 0.\nWe note that R1,2is slightly largerthan R1,4at arbitrary\nbin this case of U/(π∆) = 2.0.12\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s97/s41/s83/s105/s110/s50\n/s100\n/s50/s61/s83/s105/s110/s50\n/s100\n/s51/s61/s49\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32\n/s83/s105/s110/s50\n/s100\n/s49/s61/s83/s105/s110/s50\n/s100\n/s52\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s52/s46/s48/s83/s105/s110/s50\n/s100\n/s109/s32/s38/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s77\n/s49/s52\n/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s50/s32\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s48/s50/s32\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s101\n/s52/s40/s72/s70/s41/s32/s61/s32/s50/s98/s32/s43/s32/s85/s47/s50\n/s101\n/s49/s40/s72/s70/s41/s32/s61/s32/s45/s50/s98/s32/s45/s32/s85/s47/s50/s109/s32/s61/s32/s50/s44/s51\n/s109/s32/s61/s32/s52\n/s109/s32/s61/s32/s49/s101\n/s109/s47/s68\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s126/s40/s98/s41\n/s85/s47/s40 /s112/s68/s41 /s32/s61/s32/s52/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s90\n/s109\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s48/s50/s32/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s50/s32/s109/s32/s61/s32/s49/s44/s52\n/s109/s32/s61/s32/s50/s44/s51\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s40/s99/s41\n/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s52/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32\n/s90\n/s50/s32 /s32/s48/s46/s48/s50/s54\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s50/s44/s51/s41/s32\n/s40/s49/s44/s52/s41 /s32/s82\n/s109/s44/s109/s39/s32/s45/s49/s32\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s40/s100/s41\n/s40/s49/s44/s50/s41/s32/s61/s32/s40/s49/s44/s51/s41/s32/s61/s32/s40/s50/s44/s52/s41/s32/s61/s32/s40/s51/s44/s52/s41/s82\n/s49/s44/s50/s32/s60/s32 /s82\n/s49/s44/s52/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s52/s46/s48\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s50/s44/s51/s41/s32/s40/s49/s44/s52/s41 /s32\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s52/s46/s48/s40/s101/s41\n/s40/s49/s44/s50/s41/s32/s61/s32/s40/s49/s44/s51/s41/s32/s61/s32/s40/s50/s44/s52/s41/s32/s61/s32/s40/s51/s44/s52/s41/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32\n/s32\n/s85\n/s50/s44/s51/s32/s47 /s112/s68 /s32 /s32/s48/s46/s48/s50/s54 /s32 /s126/s85\n/s109/s44/s109/s39/s32/s47 /s112/s68 /s32 /s126\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s50/s51/s52\n/s40/s102/s41/s40/s49/s44/s52/s41 /s32\n/s126 /s126 /s126/s98/s47/s85/s32/s61/s32/s48/s46/s48/s49/s54/s32/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s98/s47/s85/s126/s85/s47 /s40/s112/s68/s41 /s32/s61/s32/s52/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32/s32/s85\n/s109/s44/s109/s39/s32/s47 /s112/s68 /s32\n/s85\n/s49/s44/s50/s32/s60/s32 /s85\n/s50/s44/s51/s32/s60/s60 /s32/s85\n/s49/s44/s52/s85\n/s50/s44/s51/s32/s47 /s112/s68 /s32 /s32/s48/s46/s48/s50/s54 /s32 /s126\nFIG. 5. (a) sin2δmand magnetization M14=/angbracketleftnd1/angbracketright − /angbracketleftnd4/angbracketright, (b) renormalized level position /tildewideεm, (c) renormalization factor\nZm, (d) Wilson ratio Rm,m′−1, and (e) residual interaction /tildewideUm,m′are plotted as functions of magnetic field bat half-filling\nεd/U=−3/2 forU/(π∆) = 4.0. The x axis in (a)-(e) is scaled by the SU(4) Kondo temperatu reTSU(4)\nK= 0.2∆ = (0 .016U)\ndetermined at b= 0. The axis in (f) is scaled by Ufor examining the behavior of /tildewideUm,m′at larger magnetic fields. The dot\nlevelsǫmare chosen in a such way that is described in Eq. (41). In (a), s in2δmandMdforU/(π∆) = 2.0 are also plotted by\ndashed lines to compare them with those for the present case U/(π∆) = 4.0. Similarly, /tildewideεmforU/(π∆) = 2.0 are plotted in (b).\nThe dash-dotted lines indicate the mean-field splitting εHF\n1=−(2b+U/2) andεHF\n4= (2b+U/2). In the limit of b→ ∞,Z2,\nR2,3−1, and/tildewideU2,3approach the SU(2) values for U/(π∆) = 4.0:Z2→0.026,R2,3→1.99, and /tildewideU2,3/π∆→0.026.\nB. Fermi-liquid parameters for larger U\nWe next consider a strong coupling case, taking the\nCoulomb repulsion to be U/(π∆) = 4.0, which is twice as\nlargeastheonestudiedintheabove. Forthiscase,effects\nof the interactions on the field-induced SU(4) to SU(2)\nKondo crossover emerges in a pronounced way. Such a\nsituation is also realistic because the experimental values\nofUand ∆ depend on individual quantum dots and on\nthe valleys to be measured.\nFigure 5(a) plots ground-state values of Tm(0) =\nsin2δmand ofM14as functions of magnetic fields\nb/TSU(4)\nKfor bothU/(π∆) = 4.0 andU/(π∆) = 2.0.\nThe results for U/(π∆) = 4.0 andU/(π∆) = 2.0 are\nplotted with solid lines and dashed lines, respectively.\nThe energy scale depends on the coupling constant as\nTSU(4)\nK/∆ = 0.2 forU/(π∆) = 4.0 andTSU(4)\nK/∆ = 0.41\nforU/(π∆) = 2.0. We see that sin2δmandMdof\nU/(π∆) = 4.0 show almost same bdependences as those\nofU/(π∆) = 2.0, and thus they show the universality.The universal behavior is determined by the bdepen-\ndence of a single parameter δ1(=π−δ4). Renormalized\nlevels/tildewideεmforU/(π∆) = 4.0 plotted in Fig. 5(b) show\nthe different bdependence from those for U/(π∆) = 2.0.\nSpecifically,/tildewideε1and/tildewideε4stay closer to the Fermi level than\nthose forU/(π∆) = 2.0. However, this different bdepen-\ndence does not affect the universalbehaviorof δ1because\na ratio of/tildewideεmto/tildewide∆mdetermines the phase shift δm, i.e.,\nδm= cot−1(/tildewideεm//tildewide∆m).\nFigures 5(c) and5(d)showthe renormalizationfactors\nZm=/tildewide∆m/∆ and the Wilson ratios Rm,m′, respectively.\nAs in theU/(π∆) = 2.0 case, the quasi-particle parame-\ntersZ2andR23for the doubly degenerate states at the\nFermi level continuously evolve from the SU(4) value to\nthe SU(2) value as bvaries from 0 to ∞. At zero field,\nthese parameters take the SU(4) values: ZSU(4)= 0.25\nandRSU(4)−1 = 0.33 forU/(π∆) = 4.0. Note that\nthe Wilson ratio is almost saturated to the maximum\npossible value Rmax\nSU(4)−1≡1/3 at zero field. In the\nopposite limit b→ ∞, these parameters for the two-13\nfold degenerate states ( m= 2,3) approach those for the\nsymmetric SU(2) Anderson model: ZSU(2)→0.026 and\nRSU(2)\n23−1→0.99 for the same U. These results show\nthat the strong Coulomb interaction significantly affects\nthe renormalization factor Z2or/tildewide∆2.Z2determines the\nenergy scale for large field as TSU(2)\nK= 0.02∆ with Eq.\n(45). The quasi-particle parameters Z1,R1,2andR1,4,\nfor the states movingawayfrom the Fermi levelapproach\nthe noninteracting value in the limit of b→ ∞; i.e,\nZ1→1,R12→1, andR14→1. Notably, R1,4becomes\nlarger than R1,2forU/(π∆) = 4.0 at finite b. This is\nquite different what we have found for the smaller inter-\naction case U/(π∆) = 2.0.\nIn order to clarify this difference, we plot the residual\ninteractions/tildewideUm,m′as functions ofmagnetic fields in Figs.\n5(e) and 5(f). The magnetic fields of Figs. 5(e) and 5(f)\nare respectively scaled by TSU(4)\nKandU. In these figures,\nespeciallyinFig.5(f), wecanseethat /tildewideU1,4becomesmuch\nlarger than the other two residual interactions /tildewideU1,2and\n/tildewideU2,3asbincreases. This field dependence of /tildewideU1,4clearly\nexplainwhythecorrespondingWilsonratio R1,4becomes\nlarger than R1,2. We also note that /tildewideU2,3for the doubly\ndegenerate levels approach the SU(2) symmetric value\n/tildewideU2,3/(π∆)→0.026.\nAll these results discussed in this section indicate that\nthe quantum fluctuations and many-body effects are en-\nhanced for large magnetic fields as the number of active\nchannel decreases from 4 to 2.36. We have also shownthe enhancement of the fluctuations are more clearly\nseen for strong interactions by comparing the results for\nU/(π∆) = 4.0 to those for U/(π∆) = 2.0.\nV. TEMPERATURE DEPENDENCE OF\nMAGNETOCONDUCTANCE\nThe above discussions about the Fermi-liquid parame-\nters have mainly focused on the zero temperature prop-\nerties of the crossover. The results show that the quasi-\nparticles are strongly renormalized as the ground state\nundergoes the crossover from the SU(4) Kondo state to\nthe SU(2) Kondo state.\nWe also study finite temperature properties of the\ncrossover in this section by calculating each component\nof the magnetoconductance gmform= 1,2,3,4 and the\ntotal conductance gtot. At half-filling, εd=−3\n2U, only\ntwo components are independent: g2=g3andg1=g4\ndue to the level structure described in Eq. (41). The\nfinite-temperature conductance, defined in Eq.(23), de-\npends on the excited states whose contributions enter\nthrough the spectral function Am(ω,T). We calculate\ntheT-dependent Am(ω,T), using the NRG with some\nextended methods for dynamic correlation functions de-\nscribed in Sec. IID and Appendix E, to obtain gm. We\nexamine two different interactions, U/(π∆) = 2.0 and\n4.0, also for these components of the conductance as-\nsuming symmetric couplings ∆ L= ∆R= ∆/2.\nA. Conductance for U/(π∆) = 2.0at half-filling\nFigures 6(a)-(d) plot the total conductance gtotand\nthe components g2=g3andg1=g4as functions of the\ntemperature for six values of magnetic fields, b/TSU(4)\nK=\n0.0,0.25,0.5,1.0,2.0,and 4.0. The SU(4) Kondo energy\nscale, determined at b= 0forU/(π∆) = 2.0, isestimated\nto beTSU(4)\nK= 0.41∆ as mentioned in Sec. IVA.\nFigure 6(a) shows that the total counductance at b= 0\nlogarithmically increases around T∼TSU(4)\nK. This log-\narithmic temperature dependence is a hallmark of the\nSU(4) Kondo effect. gtotincreases to the unitary-limit\nvalue 4e2/has temperature goes down to T→0. As\nthe magnetic fields increase, the conductance at low-\ntemperatures T≪TSU(2)\nKdecreases from the SU(4) uni-\ntary limit value 4 e2/hto the SU(2) one 2 e2/h. We can\nalso see that in a temperature range of 0 .1TSU(2)\nK/lessorsimilarT/lessorsimilar\nTSU(4)\nKthe conductance curve deforms continuously into\nthe curve for the SU(2) symmetric case. The SU(2)\nKondo state emerges for b→ ∞where the characteris-\ntic energy scale becomes TSU(2)\nK= 0.19∆. Therefore, the\nfinite temperature conductance also shows the crossover\nbehavior. To examine how the magnetoconductance gtotevolves with increasing bin more detail, we discuss the\ntwo components, g2andg1.\nFigure 6(b) plots the component g2which represents\ncontributions of the state m= 2 andm= 3 remaining\nat the Fermi level. This component g2decreases as b\nincreases in the temperature range of 0 .1TSU(2)\nK/lessorsimilarT/lessorsimilar\nTSU(4)\nK. In order to investigate the behavior of g2in this\nrange in more detail, we introduce an energy scale T∗\n2:\nT∗\n2≡π\n4/tildewide∆2,/tildewide∆2=Z2∆. (57)\nAside from a numerical factor π/4, this energy T∗\n2corre-\nsponds to the width /tildewide∆2(=/tildewide∆3) of the Kondo resonance\nform= 2 andm= 3, locked at the Fermi level even at fi-\nnite magnetic fields. We use this energy scale to examine\nthe scaling behavior. T∗\n2coincides with TSU(4)\nK= 0.41∆\natb= 0, and with TSU(2)\nK= 0.19∆ in the opposite limit\nb→ ∞. The inset of Fig. 6(b) shows the energy scale\nT∗as functions of b. We can see that T∗\n2decreases from\nTSU(4)\nKtoTSU(2)\nKwith increasing b. Correspondingly, a\nregion where g2shows the log Tdependence moves to-\nwards a low temperature side as bincreases. Although\ng2decreases with increasing bat the finite temperatures,\nit approaches the unitary limit e2/hforT→0 for arbi-14\n/s48/s46/s48/s48/s49 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s50/s51/s52\n/s84/s47 /s68/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s98/s32/s61/s32 /s44/s83/s85/s40/s50/s41/s32/s75/s111/s110/s100/s111/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111\n/s84/s83/s85/s40/s50/s41\n/s75/s84/s83/s85/s40/s52/s41\n/s75/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s52/s49\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s49/s57\n/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s50/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32/s103\n/s116/s111/s116/s97/s108/s32/s40/s101/s50\n/s47/s104/s41/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48\n/s32\n/s40/s97/s41\n/s85/s47/s50 /s85/s47/s50/s43/s50/s98/s48/s46/s48/s48/s49 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s49 /s50 /s51 /s52/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s84/s42 /s50\n/s47/s68/s32\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s84/s47 /s68/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s52/s49/s40/s98/s41\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s49/s57/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s50/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s103\n/s50/s32/s61/s32/s103\n/s51/s32/s40/s101/s50\n/s47/s104/s41/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48\n/s32/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111\n/s98/s32/s61/s32 /s44/s32/s83/s85/s40/s50/s41/s32/s75/s111/s110/s100/s111\n/s84/s83/s85/s40/s50/s41\n/s75/s84/s83/s85/s40/s52/s41\n/s75/s85/s47/s50\n/s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48\n/s32/s32/s165/s40/s99/s41\n/s98/s32/s61/s32 /s44/s32/s83/s85/s40/s50/s41/s32/s75/s111/s110/s100/s111/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111/s103\n/s50/s32/s61/s32/s103\n/s51/s32/s40/s101/s50\n/s47/s104/s41\n/s84/s47/s84/s42\n/s50/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s50/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s52/s49\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s49/s57\n/s48/s46/s48/s48/s49 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s101\n/s52/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s40/s100/s41\n/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s50/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s32/s103\n/s49/s32/s61/s32/s103\n/s52/s32/s40/s101/s50\n/s47/s104/s41\n/s84/s47 /s68/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111\n/s84/s83/s85/s40/s50/s41\n/s75/s84/s83/s85/s40/s52/s41\n/s75/s85/s47/s50/s43/s50/s98/s126\nFIG. 6. Temperature dependence of the linear conductance fo rU/(π∆) = 2.0 are plotted for six values of magnetic fields\nb/TSU(4)\nK= 0.0,0.25,0.5,1.0,2.0,and 4.0 at half-filling εd=−3U/2. (a) shows the total conductance gtot=/summationtext4\nm=1gm. The\nconductance consists of two components, i.e., g2=g3andg1=g4. (b) and (c) show the first one g2, and (d) shows the second\noneg1. (a)-(c) also show the results for the SU(2) symmetric case b y the symbols (+). In (a), (b), and (d), the x-axis is\nnormalized the bare resonance width ∆, and the axis in (c) is n ormalized by a field dependent energy scale T∗\n2= (π/4)Z2∆.\nThe inset of (b) shows T∗\n2as functions of b/TSU(4)\nK. Atb= 0,T∗\n2takes the SU(4) symmetric value, TSU(4)\nK/∆ = 0.41. In the\nopposite limit b=∞, it takes SU(2) value, TSU(2)\nK/∆ = 0.19 which is indicated by the dashed line. The vertical arrows at the\nbottom of the panels indicate TSU(2)\nK,TSU(4)\nK,U/2,/tildewideε4andU/2+2b; specifically the last two, ε4andU/2+2b, are defined with\nrespect to b/TSU(4)\nK= 4.0.\ntrarymagneticfields. Thisisbecausethe matchingofthe\nspin and orbital Zeeman splitting locks the phase shifts\natδ2=δ3=π/2 even for magnetic fields.\nAnother important aspect of the crossover is the scal-\ning behaviorofthe conductance. In Ref. 28, Mantelli and\nhiscoworkersexamineeffectsofthe spin-orbitinteraction\non the scaling behavior at quarter filling, εd=−1\n2U. We\nexaminehowthemagneticfieldaffectsthescalingathalf-\nfilling,εd=−3\n2U. To explore the scaling behavior, we\nrescale temperatures by the field dependent energy scale\nT∗\n2defined in Eq. (57) and replot g2for the six values\nofbas functions of the rescaled temperatures T/T∗\n2in\nFig. 6(c). The six curves are almost overlapped eachother in this figure. At low fields b/lessorsimilarTSU(4)\nK, the curves\nremain the same curve as the SU(4) universal one over\na wide temperature range. At high fields b≫TSU(4)\nK,\nthe curves collapse into the SU(2) universal curve. At\nhalf-filling points ǫd=−N−1\n2U, the Wilson ratio deter-\nmines theT2coefficientCTfor the SU( N) conductance:\nCT= (π2/48)/bracketleftbig\n1 + 2(N−1)(R−1)2/bracketrightbig\n. Substituting\nthe Wilson ratio for the SU(4) case RSU(4)−1 = 0.32\nand for the SU(2) case RSU(2)−1 = 0.96 into the for-\nmula ofCTyields theT2coefficients CTof each case\nforU/(π∆) = 2.0:CSU(4)\nT≃0.33 andCSU(2)\nT≃0.59.\nSinceCSU(4)\nT< CSU(2)\nT, the conductance for N= 4 is15\nlarger than that for N= 2 at the low-temperature re-\ngionsT/T∗\n2/lessorsimilar0.1. Figure 6(c) shows this magnitude\nrelation of the conductance, and thus demonstrates that\nthe scaling behavior depends on the number of orbitals\nNand the Wilson ratio R.\nFigure6(d) showsthe other component g1(=g4) which\ncorrespond to the contributions of the other two state\nmoving away from the Fermi level. At low temperatures\nT/lessorsimilarTSU(4)\nK, thesecomponentsdecreaseas bincreasesand\neventually vanish at the high magnetic fields b≫TSU(4)\nK.We can also see that g1has a peak at large mange fields\nb/lessorsimilar0.5TSU(4)\nK. The emergent peak is caused by thermal\nexcitations from (to) the renormalizedlevel /tildewideε1(/tildewideε4) which\nsituates deep inside (far above) the Fermi level for large\nfields as shown in Fig. 5(b). Furthermore, tor the large\nfields, the level structure of the CNT dot approaches the\nmean-field levels described in Eq. (53), and the atomic-\nlimit peak also emerges at U/2+ 2b[see also Appendix\nF].\nThese results obtained for U/(π∆) = 2.0 show a rather\nmoderate evolution of the crossover and the Kondo en-\nergy scaleT∗\n2asTSU(2)\nKis only half of TSU(4)\nK.\nB. Conductance for U/(π∆) = 4.0at half-filling\nWe next examine the conductance for a case of strong\ninteraction U/(π∆) = 4.0 to see more clearly the field-\ninduced crossoveratfinitetemperatures. In thiscase, the\ncharacteristic energy scale for the SU(2) case is signifi-\ncantly suppressed TSU(2)\nK= 0.02∆, which becomes much\nsmaller than the SU(4) energy scale TSU(4)\nK= 0.2∆, i.e.,\nthe difference is about one order of magnitude.\nFigure 7(a) shows that the total conductance gtotfor a\ntemperature region 0 .1TSU(2)\nK/lessorsimilarT/lessorsimilarTSU(4)\nKdecreases as\nmagnetic field increases. Since the characteristic energy\nscaleT∗\n2defined in Eq. (57) in this case becomes much\nsmaller than that for U/(π∆) = 2.0, the region where\nthe crossover occurs moves towards a low-temperature\nregion,T/lessorsimilarTSU(4)\nK= 0.2∆. Furthermore, the shoulder\nstructures emerging in the high temperature region are\nmore pronounced.\nFigure 7(b) clearly shows that the curves of g2evolve\nfrom the SU(4) curve to the SU(2) curve during the\ncrossover. Specifically, the energy scale T∗\n2around which\ng2showslogTdependencedecreaseswithincreasingmag-\nneticfield. TheinsetofFig.6(b)showsthesuppressionof\nT∗\n2: it decreases from TSU(4)\nK= 0.2∆ toTSU(2)\nK= 0.02∆.\nThe scaling behavior of g2in Fig. 6(c) also becomes\nclear because of this suppression. In a wide range of tem-\nperatures, we can see that the scaled results collapse into\ntwo different universal curves, i.e., the SU(4) curve for\nsmall fields b/TSU(4)\nK/lessorsimilar0.25, and SU(2) curve for large\nfieldsb/TSU(4)\nK/greaterorsimilar1. Since the Wilson ratios for N= 4\nandN= 2 are respectively saturated to the maximum\npossible values Rmax\nSU(4)−1 = 1/3 andRmax\nSU(2)−1 = 1,\ntheT2coefficients CTfor eachNare also saturated:\nCSU(4)\nT≃0.34 andCSU(2)\nT≃0.62. Figure 6(c) clearly\nshows that g2forb= 0 is larger than that for b→ ∞at\nT <T∗\n2because ofCSU(4)\nT<CSU(2)\nT.\nFurthermore, we can recognize that a broad peak\nemerges for b≫TSU(4)\nKatT≈U/2. It corresponds\nto an thermal energy needed to add an electron or a\nhole to the degenerate states. Thus, this atomic limitpeak and the quasi-particle excitation peak in Fig. 7(d)\natT≈2b+U/2 yield the shoulder structure of gtotat\nthe high temperatures, as shown in Fig. 7(a).\nVI. SPECTRAL PROPERTIES ALONG THE\nFIELD-INDUCED CROSSOVER\nSpectral functions at finite magnetic fields also reflect\nthe crossover from the SU(4) to SU(2) Kondo states. In\nadditiontothe KondoresonanceneartheFermilevel, the\nZeemann splitting causes a shift of the atomic-limit peak\nto±(2b+U/2). The spectral functions for the doubly\ndegenerate states remain the same A2(ω,T) =A3(ω,T)\nfor finite magnetic fields owing to the dot level structure\ngiven in Eq. (41). Following relations additionally hold\nin the particle-hole symmetric case εd=−(3/2)U,\nA2(ω,T) =A2(−ω,T), A1(ω,T) =A4(−ω,T).\n(58)\nThe second relation shows that A4is a mirror image of\nA1, and we discuss A1andA2in the following. We\nexamine the spectral functions at T= 0, and hence\nwe drop the second argument of the functions, namely,\nAm(ω)≡Am(ω,T= 0),(m= 1,2,3,4). As in the pre-\nvious sections, we consider the two cases for the inter-\naction: (i) U/(π∆) = 2.0 and (ii)U/(π∆) = 4.0. The\nspectral function obtained by the NRG procedure is a set\nof discreteδfunctions. To obtain a continuous spectrum,\nthe logarithmic Gaussian function is used (see Appendix\nE3).\nA. Spectral function for U/(π∆) = 2.0\nFigure8(a)showsthetotalspectralfunction Atot(ω) =/summationtext4\nm=1Am(ω) for five different values of magnetic fields\nandU/(π∆) = 2.0. At zero magnetic field, we can see\nthat a single SU(4) Kondo resonance peak emerges on\nthe Fermi level ω= 0. As the magnetic field increases in\nthe range of 0 < b <∞, the height of the Kondo peak\ndecreases from 4 to 2 in units of π∆ since the resonance16\n/s48/s46/s48/s48/s49 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s50/s51/s52\n/s85/s47/s50 /s85/s47/s50/s43/s50/s98/s98/s47/s84/s83/s85/s40/s52/s41\n/s75/s103\n/s116/s111/s116/s97/s108/s32/s40/s101/s50\n/s47/s104/s41\n/s84/s47 /s68/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48\n/s32\n/s98/s32/s61/s32 /s44/s32/s83/s85/s40/s50/s41/s32/s75/s111/s110/s100/s111/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s52/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50\n/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s50\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s48/s50\n/s84/s83/s85/s40/s52/s41\n/s75/s84/s83/s85/s40/s50/s41\n/s75/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111/s40/s97/s41\n/s48/s46/s48/s48/s49 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s84/s42 /s50/s47/s68/s32\n/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s84/s47 /s68/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s50/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s40/s98/s41/s103\n/s50/s32/s61/s32/s103\n/s51/s32/s40/s101/s50\n/s47/s104/s41/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48\n/s32\n/s85/s47/s50/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111\n/s98/s32/s61/s32 /s44/s32/s83/s85/s40/s50/s41/s32/s75/s111/s110/s100/s111\n/s84/s83/s85/s40/s52/s41\n/s75/s84/s83/s85/s40/s50/s41\n/s75/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s52/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s48/s50\n/s48/s46/s49 /s49 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s52\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s40/s99/s41/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111/s103\n/s50/s32/s61/s32/s103\n/s51/s32/s40/s101/s50\n/s47/s104/s41\n/s84/s47/s84/s42\n/s50/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48\n/s32 /s44/s83/s85/s40/s50/s41/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s98/s32/s61/s32 /s44/s32/s83/s85/s40/s50/s41/s32/s75/s111/s110/s100/s111/s84/s83/s85/s40/s52/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s50\n/s84/s83/s85/s40/s50/s41\n/s75/s47/s68 /s32/s61/s32/s48/s46/s48/s50\n/s48/s46/s48/s48/s49 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s84/s47 /s68/s101\n/s52/s85/s47/s40 /s112/s68 /s41/s32/s61/s32/s52/s46/s48\n/s101\n/s100/s47/s85/s32/s61/s32/s45/s51/s47/s50/s98/s47/s84/s83/s85/s40/s52/s41\n/s75\n/s40/s100/s41/s103\n/s49/s32/s61/s32/s103\n/s52/s32/s40/s101/s50\n/s47/s104/s41/s32/s32/s48/s46/s48/s48\n/s32/s32/s48/s46/s50/s53\n/s32/s32/s48/s46/s53/s48\n/s32/s32/s49/s46/s48/s48\n/s32/s32/s50/s46/s48/s48\n/s32/s32/s52/s46/s48/s48\n/s85/s47/s50/s43/s50/s98/s98/s32/s61/s32/s48/s44/s32/s83/s85/s40/s52/s41/s32/s75/s111/s110/s100/s111\n/s84/s83/s85/s40/s52/s41\n/s75/s84/s83/s85/s40/s50/s41\n/s75/s126\nFIG. 7. Temperature dependence of the linear conductance fo rU/(π∆) = 4.0 are plotted for six values of magnetic fields\nb/TSU(4)\nK= 0.0,0.25,0.5,1.0,2.0,4.0 at half-filling εd=−3U/2. (a) shows the total conductance gtot=/summationtext4\nm=1gm. The\nconductance consists of two components, i.e., g2=g3andg1=g4. (b) and (c) show the first one g2, and (d) shows the second\noneg1. (a)-(c) also show the results for the SU(2) symmetric case b y the symbols (+). In (a), (b), and (d), the x-axis is\nnormalized the bare resonance width ∆, and the axis in (c) is n ormalized by characteristic energy scales T∗\n2. The inset of (b)\nshowsT∗\n2as functions of b/TSU(4)\nK. Atb= 0,T∗\n2takes the SU(4) symmetric value, TSU(4)\nK/∆ = 0.20. In the opposite limit\nb=∞, it takes SU(2) value, TSU(2)\nK/∆ = 0.02 which is indicated by the dashed line. The vertical arrows at the bottom of the\npanels indicate TSU(2)\nK,TSU(4)\nK,U/2,/tildewideε4andU/2 + 2b; specifically the last two, ε4andU/2 + 2b, are defined with respect to\nb/TSU(4)\nK= 4.0.\npeak positions for m= 1 andm= 4 move away from\nthe Fermi level, leaving the other positions for m= 2\nandm= 3 just on the Fermi level, as shown in Fig. 4(a).\nThis field dependence of the peak positions results in de-\nforming the peak shape of the SU(4) Kondo resonance\ninto that of the SU(2) Kondo resonance on the Fermi\nlevel. Furthermore, two sub peaks emerge at higher en-\nergies, i.e., ±(2b+U/2).\nFigure 8(b) plots A2(=A3) for four values of b.\nA2(=A3) shows such deformation of the peak shape.\nThe deformation of A2is not so clear in the case of\nU/(π∆) = 2.0 sinceTSU(2)\nKis only half as large as TSU(4)\nK.Nevertheless, the inset of Fig. 8(b) which is an enlarged\nview around the Fermi level shows that the resonance\nwidth on the Fermi level becomes sharper. The sharp-\nening of the width leads to the decrease of the spectral\nweight. As the magnetic field increases, A2also develops\ntwo sub peaks at ω=±U/2 to compensate the decrease\nof the weight. The two peak positions correspond to the\nexcitation energies on adding an electron or hole to the\ndot. In Appendix F, we provide analytic expressions of\nthe spectral functions in the atomic limit vν→0, where\ntheCNTdotisdisconnectedfromthemetallicleads. The\nremaining degenerate states turn into the SU(2) Kondo17\n-8-6-4-20246801234\n/Atota)U)=2.0\nd3/2)U\nTKSU4)=0.41\nTKSU2)=0.19b/TKSU4)\n0\n0.5\n1\n2\n4\n-U/2-2b U /2+2bU/2 -U/2\n-8-6-4-2024680.20.40.60.8\n01\nω/ΔπΔA2=πΔA3\n-0.5 0.5 -1 0 10.20.40.60.8\n01\nUπΔ2.0\nεd3/2)UTKSU(4)=0.41Δ\nTKSU \u00002)=0.19Δ\nU/2 -U/2b/TKSU\u00014)\n0\n2\n4\n∞,SU2)b)\n-8-6-4-2024680.20.40.60.8\n01\n/A1\n-U/2-2bϵ1(c)\nU/()=2.0\nd=-(3/2)Ub/TKSU(4)\n00 \u0002 \u0003\n1\n2\n4TKSU(4)=0.41Δ\nTKSU(2)=\u0004 \u0005 \u0006 \u0007Δ\nFIG. 8. Zero temperature spectral functions for U/(π∆) =\n2.0 are plotted for five values of magnetic fields, b/TSU(4)\nK=\n0.0,0.5,1.0,2.0,4.0athalf-filling εd=−3U/2: (a)Atot(ω) =/summationtext4\nm=1Am(ω), (b)A2(ω), and (c) A1(ω). Vertical arrows at\nthe bottom of the panels indicate the points ω=±U/2 and\n±(2b+U/2) where peaks emerge in the atomic limit. The\npeaks of ω=±(2b+U/2) are for the largest value of bamong\nthe five, b/TSU(4)\nK= 4.0. The position of the renormalized\nresonance level /tildewideε1for the same value of bare also shown in\nthe bottom.state in the limit of b→ ∞. indicating that the states\nundergo the crossover from the SU(4) to SU(2) Kondo\nstate.\nFigure 8(c) shows the other component A1(ω), which\ncorresponds to the component of the level going down\nfrom the Fermi level. Note that A4(ω) =A1(−ω), as\nmentioned. We can see that the spectral weight transfers\nto the negative frequency region ω<0, and an evolution\nofitsresonancepeakpositionshowsgoodagreementwith\nthe field dependence of /tildewideε1presented in Fig. 4(a). This\ntransfer leads to the development of the sub peaks and\ndecrease of the Kondo peak of Atot. With increasing\nmagnetic fields, the resonance peak at ω=/tildewideε1merges\nwith the atomic-limit peak at ω=−U/2−2bwhich shifts\nfrom the zero field position −U/2 in the presence of b.\nThis shift of the atomic-limit peak, which we discuss in\nthe Appendix F, results from the descent of the energy\nlevelε1described in Eq. (41). For much larger fields\nb≫TSU(4)\nK, the curve of A1approaches the Lorentzian\nform. Therefore, the quasiparticle state of m= 1 is un-\nrenormalizedfrom the correlatedKondostate to the bare\nstate.\nB. Spectral function for U/(π∆) = 4.0\nIn order to investigate the effects of the strong interac-\ntion on the spectral functions, we next discuss the spec-\ntral functions for U/(π∆) = 4.0. Figures 9(a)-(c) re-\nspectively show results of Atot,A2andA1. We com-\npare these results with the corresponding results for the\nweak interaction case in Fig. 8, Atotfor the strong in-\nteraction in Fig. 9(a) shows a similar trend as that for\nU/(π∆) = 2.0 in Fig. 8(a). However, the width of\nresonance for U/(π∆) = 4.0 is smaller than that for\nU/(π∆) = 2.0 in arbitrary magnetic fields, because U\nis larger.\nThe component A2in Fig. 9(b) more clearly shows\nthe narrowing of the resonance width than that for\nU/(π∆) = 2.0, becauseTSU(2)\nKis smaller than TSU(4)\nKby\none order of magnitude in this case i.e., TSU(2)\nK= 0.02∆\nandTSU(4)\nK= 0.2∆. The narrowing of the width leads\nto a loss of the spectral weight around the Fermi level,\nwhich is compensated by an enhancement of the atomic-\nlimit peak at ±U/2.\nThe atomic limit peak around −U/2−2bofA1is\nbroader in the strong interaction case shown in Fig 9(c)\nthan in the weak interaction case, because the quasi-\nparticle resonance position /tildewideε1presented in Fig. 5(b)\nstill remains around the Fermi level at the higher fields,\nb≫TSU(4)\nK. Owing to this remaining, the quasiparticle\nstate is still renormalized even at the higher fields, and\nthus the shape of A1differs from the Lorentzian form.18\n-10 -50\b10012\n\t4\nω/ΔπΔA\nt\n\u000b(a)U/(πΔ)=4.0\nεd=-( \f/2)U\nTKSU(4)=0.2Δ\nTKSU(2)=0.02Δb/TKSU(4)\n0\r \u000e \u000f\n1\n2\n4\n-U/2-2b U /2+2bU/2 -U/2\n-10 -\u00100\u0011100.20.40.60.8\n01\nω/ΔπΔA2=πΔA\n\u0012\n- \u0013 \u0014 \u0015 \u0016 \u0017 \u0018 -1 0 10.20.40.60.8\n01\nU/(πΔ)=4.0\nεd=-( \u0019/2)UTKSU(4)=0.2Δ\nTKSU(2)=0.02Δ(b)\nU/2 -U/2b/TKSU(4)\n0\n2\n4\n∞,SU(2)\n-10 -\u001a0\u001b100.20.40.60.8\n01\nω/ΔπΔA1\n-U/2-2bϵ1( \u001c)U/(πΔ)=4.0\nd=-(3/2)U\nTKSU(4)=0.2Δ\nTKSU(2)=0.02Δb/TKSU(4)\n0\u001d \u001e \u001f\n1\n2\n4\nFIG. 9. Spectral functions for U/(π∆) = 4.0 are plotted for\nfive values of magnetic fields, b/TSU(4)\nK= 0.0,0.5,1.0,2.0,4.0\nat half-filling εd=−3U/2: (a)Atot(ω) =/summationtext4\nm=1Am(ω), (b)\nA2(ω), and (c) A1(ω). Vertical arrows at the bottom of the\npanels indicate the points ω=±U/2 and−(2b+U/2) where\npeaks emerge in the atomic limit. The peaks of ω=±(2b+\nU/2) are for the largest value of bamong the five, b/TSU(4)\nK=\n4.0. The position of the renormalized resonance level /tildewideε1for\nthe same value of bare also shown in the bottom.VII. SUMMARY\nWehavestudiedtheKondoeffectinacarbonnanotube\nquantum dot in a wide range of temperature and mag-\nnetic field using the numerical renormalization group.\nIn the first half of the present paper, we have studied\nfinite temperature properties of the SU(4) Kondo state\nby calculating the finite temperature conductance in a\nwide range of electron filling Nd. The NRG results nicely\nagree with the experimental results in the wide range,\nsupporting an emergence of the SU(4) Kondo resonance\nat low-temperatures, T/lessorsimilarTSU(4)\nK. Furthermore, we have\nprecisely examined the temperature dependence of con-\nductance especially at two fixed values of Nd: quarter-\nfillingNd= 1 and half-filling Nd= 2. The obtained\nresults show that the scaled conductance of Nd= 1 is\nlarger than that of Nd= 2 at the low-temperatures. A\nmicroscopic Fermi-liquid theory, which is extended to ar-\nbitraryNd, successfully explains such different behavior\nof the conductance depending on Nd. The theory shows\nthat aT2coefficientCTfor the conductance vanishes at\nNd= 1. In contrast to the quarter-filling case, CTdoes\nnot become zero at half-filling, but saturates to a strong\ncoupling limit value. Thus, the universality depends on\nthe electron filling. We expect that this filling depen-\ndence of the universality can be observed.\nInthesecondhalfofthepresentpaper, wehaveinvesti-\ngatedhowmagneticfieldsaffectthegroundstateandalso\nexcited states in the course of the SU(4) to SU(2) Kondo\ncrossover. Our previous papers show that quasiparticle\nstates remaining the Fermi level are renormalized as the\nnumber of active levels decreases from four to two. The\nother two states become unrenormalized in the course\nof the crossover. The present paper has shown that the\nrenormalization of the quasiparticle states more clearly\nappear in a strong interaction case because a character-\nistic energy scale T∗\n2clearly decreases from the SU(4)\nKondo energy scale TSU(4)\nKto the SU(2) Kondo energy\nscaleTSU(2)\nK.\nThe finite temperature conductance in the magnetic\nfields also shows such decrease of the energy scale. In\naddition, the scaling behavior at half-filling shows that\nthe excited states undergo the crossover. Specifically,\nas soon as the magnetic fields bbecome comparable to\nTSU(4)\nK, the SU(4) universality is lost, and for the much\nlarger fields, b≫TSU(4)\nK, the SU(2) universality emerges.\nFurthermore, the NRG results for both SU(2) and SU(4)\nsymmetric cases indicate that the Wilson ratio and the\nKondo energy scale determine the low-temperature be-\nhavior of half-filled quantum dots.\nWe have also calculated total spectral function and\ntheir components in magnetic fields. The obtained spec-\ntral function shows that the resonance states remaining\non the Fermi level become sharper as the magnetic field\nincreases, showing a good agreement with the field de-\npendence of the corresponding renormalized resonance\nwidth. Furthermore, a spectral weight of the other two19\nstates transfers towards the higher frequency region, be-\ncause the Zeeman splitting shifts the two peak positions\nupward and downward from the Fermi level. Such trans-\nfer results in the emergence of two sub-peaks whose po-\nsitions approach atomic-limit peak positions.\nAn interesting future work is to explore how the\nthermoelectric transport of multilevel quantum dots de-\npends on the electron filling42,68,69. We have already\nhad the low-temperature expansions for the thermal\nconductance,34and the progress along this line will be\ndiscussed elsewhere.\nACKNOWLEDGMENTS\nThis work was supported by Grant-in-Aid for JSPS\nFel- 1035 lows Grant No. JP18J10205 and JSPS KAK-\nENHI Grant 1036 Nos. JP18K03495, JP19H00656,\nJP19H05826, JP16K17723, 1037 and JP19K14630, JST\nCREST Grant No. JPMJCR1876, 1038 the French pro-\ngram Agence nationale de la Recherche 1039 DYMESYS\n(ANR2011-IS04-001-01), and Agence nationale 1040 de\nla Recherche MASH (ANR-12-BS04-0016).\nAppendix A: Symmetries of Hamiltonian\nIn the case where the impurity level is degenerate\nǫm≡ǫd, the Hamiltonian Hdefined in Eq. (1) has the\nSU(N) symmetry. This is also owing to the fact that\nthe tunneling matrix element and the Coulomb interac-\ntion do not depend on the flavour m. It can also be\nconfirmed that Hcommutes with each component of the\nSU(N) generators:\nJµ≡1\n2N/summationdisplay\nm,m′=1\nd†\nmλµ\nm,m′dm′+/summationdisplay\nν=L,R/integraldisplayD\n−Ddεc†\nν,ε,mλµ\nm,m′cν,ε,m\n.\n(A1)\nHere,λµforµ= 1,2,···,N2−1 are the Gell-mann\nmatrices that satisfy the commutation relations,\n/bracketleftbig\nλi,λj/bracketrightbig\n= 2iN2−1/summationdisplay\nk=1fijkλk, (A2)\nwithfijkthe structure factor. The Hamiltonian Hhas\nalso an U(1) totsymmetry and commutes with the total\nnumber operator Qtot=/summationtextN\nm=1Qm. Here,Qmis the\nnumber operator for flavour m:\nQm=\nndm+/summationdisplay\nν=L,R/integraldisplayD\n−Ddεc†\nν,ε,mcν,ε,m\n,(A3)\nform= 1,2,···,N. Perturbations that lift the degen-\neracy of the impurity level lower the symmetry from\nSU(N)⊗U(1)totto U(1)m=1⊗U(1)m=2⊗U(1)m=3⊗U(1)m=4asHstill commutes with each number oper-\natorQm.\nFor carbon nanotube dots, the two states of m= 2 and\nm= 3 remain degenerated even in magnetic fields when\nthe matching condition Eq. (40) holds. In the condition,\nHcommutes with the SU(2)-pseudospin operator,\nSµ≡1\n2/summationdisplay\nm,m′=⇑,⇓\nd†\nmσµ\nm,m′dm′+/summationdisplay\nν=L,R/integraldisplayD\n−Ddεc†\nν,ε,mσµ\nm,m′cν,ε,m\n.\n(A4)\nacting on the two states. Here, σµforµ=x,y,andzare\nthePaulimatrices. Thepseudospins ⇑and⇓respectively\ndenote the state of m= 2 andm= 3. Thus, the Hamil-\ntonian has the U(1)m=1⊗[SU(2)⊗U(1)]m=2,3⊗U(1)m=4\nsymmetries. The SU(2) symmetry plays a central role in\nthe field-induced SU(4) to SU(2) Kondo crossover.\nAt half-filling point ǫd/U=−3/2, the Hamiltonian H\nis invariant under an extended electron-hole transforma-\ntion:\nd†\n1⇒h4, d†\n2⇒h3, d†\n3⇒h2, d†\n4⇒h1,(A5)\nand correspondingly, c†\nν,εm,m⇒ −fν,−εm′,m′for\n(m,m′) = (1,4),(2,3),(3,2),(4,1). Here, hmand\nfν,εm′,m′annihilate hole in the dot and the conduction\nbands, respectively.\nAppendix B: Overview of NRG iterations\nWe provide a brief overview of an NRG iteration. The\nNRG Hamiltonian Hfor an Anderson impurity model is\na semi-infinite tight-binding chain with the imuprity at\na left end. Specifically,\nHNRG=H0\nd+HU+Hchain+HT, (B1)\nH0\nd=N/summationdisplay\nm=1ǫmd†\nmdm,HU=U/summationdisplay\nm<m′ndmndm′,(B2)\nHchain=∞/summationdisplay\nn=0N/summationdisplay\nm=1tn/parenleftbig\nf†\nn,mfn+1,m+ H.c./parenrightbig\n,(B3)\nHT=VN/summationdisplay\nm=1/parenleftbig\nd†\nmfm,0+ H.c./parenrightbig\n. (B4)\nHere, Λ is a logarithmic discretization parameter. A hy-\nbridization matrix element Vcouples the impurity site\nto a 0th site of the chain. Similarly, a hopping element\ntncouples the nth site to ( n+ 1)th site. The explicit\nforms ofVandtnare respectively\nV\nD=/radicalbigg\n2∆AΛ\nπD, (B5)\ntn\nD=/parenleftbig\n1+Λ−1/parenrightbig/parenleftbig\n1−Λ−n−1/parenrightbig\n2 (1−Λ−2n−1)1/2(1−Λ−2n−3)1/2Λ−n/2.\n(B6)20\nNote that the logarithmic discretization correctsthe tun-\nnelingmatrixelement V.AΛrepresentssuchacorrection\nand can be expressed as,\nAΛ=1\n2/parenleftbigg1+1/Λ\n1−1/Λ/parenrightbigg\nlnΛ. (B7)\nThis correction AΛapproaches 1 in the continuum limit,\nlim\nΛ→1AΛ= 1. (B8)\nThe hopping element tnexponentially decays with in-\ncreasingnand its asymptotic form is\ntn\nD∼1+Λ−1\n2Λ−n/2. (B9)\nTo iteratively diagonalize the Hamiltonian HNRG, we in-\ntroduce a Hamitonian HLwhich describe the chain ends\nat anLth site. The explicit form of HLis\nHL= Λ(L−1)/2/bracketleftbig\nH0\nd+HU+HL\nchain+HT/bracketrightbig\n,(B10)\nHL\nchain=L−1/summationdisplay\nn=0N/summationdisplay\nm=1tn/parenleftbig\nf†\nn,mfn+1,m+ H.c./parenrightbig\n.(B11)\nΛ(L−1)/2is multiplied to the right-hand side of Eq. (B10)\nto makes the hopping element Λ(L−1)/2tL−1be order of\n1. Eigenvalues of HLalso become order of 1 due to the\nmultiplication. The full Hamiltonian HNRGis recovered\nas the limit of\nHNRG= lim\nL→∞Λ−(L−1)/2HL.(B12)\nHere, the prefactor corresponds to the energy scale of\nlow-lying excited states of HL:\nTL\nD= Λ−(L−1)/2. (B13)\nTo run the iteration, we deduce a recurrence relation for\nHLfrom Eq. (B10),\nHL+1= Λ1/2HL+ ΛL/2tLN/summationdisplay\nm=1/parenleftBig\nf†\nL,mfL+1,m+ H.c./parenrightBig\n.\n(B14)\nUsing this relation, we can construct a matrix for HL+1\nfrom eigenvalues and corresponding eigenstates for the\nHamiltonian HL. The iterative procedure yields the\neigenenergies EL(ω) at eachLth step. we keep only the\nNKeigenstates with the lowest energies and discard the\nhigher energy states because a dimension of HLincreases\nwith increasing L.\nFigure 10 plots low-lying eigenenergies of a CNT dot\nas fucntions of even Lwith typical parameters: Λ =\n6.0,NK= 4100,U/(π∆) = 4.0,ǫd/U=−3/2, and\nb/TSU(4)\nK= 1.0. Here, the bare resonance width is\n∆/D= 1/(100π). The SU(4) temperature TSU(4)\nK/∆ =0.2determined at b= 0scalesthe magneticfield b. In the\nvalue ofb/TSU(4)\nK= 1.0, the Fermi-liquid state of the dot\nis in the middle of the SU(4) to SU(2) Kondo crossover\nas shown in Fig. 5(a). Specifically, Fig. 5(a) shows the\nmagnetization Mdis still not saturated to 1 at the value\nofb. Figure 10 shows that the energies do not depend on\nLin the region of L/greaterorsimilar20. At the last iteration L= 30,\nthe energy scale TLgiven in Eq. (B13) is much smaller\nnot only than TSU(4)\nKbut also than the SU(2) Kondo\ntemperature TSU(2)\nK/∆ = 0.02:TL/∆≈1.6×10−9,\nTL/TSU(4)\nK≈8.2×10−9, andTL/TSU(2)\nK≈1.0×10−7.\nThe value of TSU(2)\nKis determined at b→ ∞. Therefore,\nthe low-lying energy states approach the Fermi-liquid\nstates forLgreater than 20.\n◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼\n◼◼◼◼◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n0102030\n ! \"\n1 # $\n2 % &\n0123\nChainLengthLEnergy/DU/ 'πΔ)=4.0,ϵd/U=-3/2,b/TKSU(4)=1.0\nFIG. 10. NRG energy flow of a CNT dot as functions of\nthe even chain length L. A CNT dot has N= 4 internal\ndegrees of freedom. We keep 4100 eigenvalues at each NRG\nstep and plot the lowest 100 values in this figure. The log-\narithmic discretization parameter is Λ = 6. Values of U,\nǫd, and magnetic field bfor the calculation are U/(π∆) = 4.0,\nǫd/U=−3/2, andb/TSU(4)\nK= 1.0, respectively. The impurity\nresonance width is ∆ /D= 1/(100π). The SU(4) temperature\ndetermined at b= 0 isTSU(4)\nK/∆ = 0.2.\nAppendix C: Calculation method of Fermi-liquid\nparameters\nWe show how to calculate Fermi-liquid parameters\nsuch as phase shifts δm, renormalization factors Zm,\nand residual interactions /tildewideUm,m′. Let us consider a non-\ninteracting case of the NRG Hamiltonian, i.e., HNRGat\nU= 0 since a single particle and hole excitation from the\nground state determine the phase shifts and the renor-\nmalization factors. In such a case, a matrix form of the21\n◼\n◼\n◼\n◼\n◼\n◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n□□□□□□□□□□ □ □ □ □ □●●\n●●●●● ● ● ● ● ● ● ● ●\n○○○○○○○○○ ○ ○ ○ ○ ○ ○▲\n▲\n▲\n▲\n▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲\n△△△△△△△ △ △ △ △ △ △ △ △\n01020300.20.40.60.8\n01\nChainLengthLOccupationnumbersU) *πΔ+ ,4.0,ϵd/U= -3/2,b/TKSU.4461.0●:<n1,p\n>\n○: 7n1,h\n8■:9n2,p\n:\n□: ;n2,h\n?▲:@n4,p\nA\n△: Bn4,h\nC\n◼\n◼\n◼\n◼\n◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◆\n◆\n◆\n◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆◆◆\n01020300.20.40.60.8\n01\nChainLengthLRenormalizationfactorsUD EπΔ)=4.0,ϵd/U=-3/2,b/TKSU(4)=1.0\nZ1\nZ2\nFIG. 11. (a) shows flows of occupation numbers: /angbracketleftn1,p/angbracketright,\n/angbracketleftn1,h/angbracketright,/angbracketleftn2,p/angbracketright,/angbracketleftn2,h/angbracketright,/angbracketleftn4,p/angbracketright, and/angbracketleftn4,h/angbracketright. (b) shows those of\nrenormalization factors: Z1andZ2. The x-axis is the NRG\nchain length Lin each figure. In (a), Following six symbols,\n•,◦,/squaresolid,/square,/trianglesolid, and△respectively denote the values of /angbracketleftn1,p/angbracketright,\n/angbracketleftn1,h/angbracketright,/angbracketleftn2,p/angbracketright,/angbracketleftn2,h/angbracketright,/angbracketleftn4,p/angbracketright, and/angbracketleftn4,h/angbracketright. In (b), •and/squaresolid\nrespectively denote the values of Z1andZ2. The parameters\nareU/(π∆) = 4.0,ǫd/U=−3/2, andb/TSU(4)\nK= 1.0 for\neach figure. These parameters are the same as those for Fig.\n10. The SU(4) Kondo temperature is TSU(4)\nK/∆ = 0.2 for the\nvalues of Uandǫd. Eq. (41) describes the magnetic filed b\ndependence of the four dot levels ǫ1,ǫ2,ǫ3, andǫ4.\nHamiltonian HLis\nHL\nU=0= Λ(L−1)/2\nǫmV 0\nV0t0\n0t00t1\n.........\ntL−20tL−10 tL−10\n.\n(C1)\nTheGreen’sfunction Gd(ω)attheimpuritysite( i=−1)\nis the (1,1) element of the resolvent of this Hamiltonian\nmatrix,\nGm(ω)≡/parenleftbig\nω1− HL\nU=0/parenrightbig−1\n1,1\n=1\nω−ǫmΛ(L−1)/2−V2Λ(L−1)g00,m(ω).(C2)\nAn eigenvalue EofHLis a pole of Gd. Specifically, the\nvalue is a solution of the below equation,\nE−ǫmΛ(L−1)/2−V2Λ(L−1)g00,m(E) = 0.(C3)◼\n◼\n◼◼\n◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◆\n◆\n◆◆\n◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆●\n●\n●●\n●●●●● ● ● ● ● ● ●▲\n▲\n▲\n▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲\n0102030\nF G H\nI J K\n012\nChainLengthLU\n2,3/(πΔ)\nLMNOPQ R\nSTUVWX Y\nZ[\\]^_ `▲\n▲▲▲▲▲▲\n81012141618200.10.20.30.4U/(πΔa b4.0,ϵd/Ud e3/2,b/TKSU(4)f1.0\n■:U\n2,3ppg:U\n2,3hhh:U\n2,3ph\n▲:U\n2,3hp(a)\n◼\n◼\n◼\n◼◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼ ◼◆\n◆\n◆\n◆◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆●\n●\n●\n●●●●●● ● ● ● ● ● ●▲\n▲\n▲\n▲\n▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲\n0102030\ni j k\nl m n\n012\nChainLengthLU\n1,2/(πΔ)\n◼◼◼◼◼◼◼◆\n◆\n◆◆◆◆◆●\n●●●●●●▲\n▲\n▲▲▲▲▲\n81012141618200.10.20.3\n0U/(πΔ)=4.0,ϵd/U=-3/2,b/TKSU(4)=1.0\n■:U\n1,2pp\n●:U\n1,2hh\n◆:U\n1,2ph\n▲:U\n1,2hp(b)\n◼\n◼\n◼◼◼◼◼◼◼ ◼ ◼ ◼ ◼◼◼◆\n◆\n◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆◆◆●\n●\n●\n●●●●● ●● ● ● ●●●▲\n▲\n▲\n▲▲▲▲▲▲ ▲ ▲ ▲ ▲▲▲\n0102030\no p q\nr s u\n012\nChainLengthLU\n1,4/(πΔ)\n◼◼◼◼◼◼◼\n◆◆◆◆◆◆◆\n●●●●●●●\n▲▲▲▲▲▲▲\n81012141618200.10.20.30.4U/(πΔ)=4.0,ϵd/U=-3/2,b/TKSU(4)=1.0\n■:U\n1,4pp\n●:U\n1,4hh\n◆:U\n1,4ph\n▲:U\n1,4hp(c)\nFIG. 12. Residual interactions /tildewideUm,m′are plotted as functions\nof the chain length L. (a), (b), and (c) respectively show\n(m,m′) = (2,3), (1,2), and (1 ,4) element of /tildewideUm,m′. The\nparticle-hole symmetry given by Eq. (A5) imposes following\nrelations to the residual interactions: /tildewideU1,2=/tildewideU1,3=/tildewideU2,4=\n/tildewideU3,4. The parameters are U/(π∆) = 4.0,ǫd/U=−3/2, and\nb/TSU(4)\nK= 1.0 for every three figure. The SU(4) Kondo tem-\nperature is TSU(4)\nK/∆ = 0.2 for the values of Uandǫd. Fol-\nlowing four symbols: /squaresolid,•,/diamondsolid, and/trianglesolidrespectively represent\n/tildewideUpp\nm,m′,/tildewideUhh\nm,m′,/tildewideUph\nm,m′, and/tildewideUhp\nm,m′in each figure. The insets\nshow an enlarged view in the range of 8 ≤L≤20.\nIn Eqs. (C2) and (C3), g00,m(ω) is the Green’s function\nat a sitei= 0 for a chain which starts from the zeroth\nsite and ends at the Lth site. A Hamiltonian HL\n0de-\nscribes such a chain. g00,m(ω) is the (1,1) element of the22\n◼◼◼◼◼\n◼◼◼◼◼ ◼ ◼ ◼ ◼ ◼\n◆◆◆◆\n◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆\n●●●●\n●●●● ● ● ● ● ● ●●\n01020300.20.40.60.81.2\n01\nChainLengthL\nvwxyz{ ratios\nU/(πΔ | }4.0,ϵd/U ~ 3/2,b/TKSU(4)1.0R2,3\n1\nR1,2\n1R1,4\n1\nFIG. 13. Wilson ratios R2,3−1,R1,4−1, andR1,2−1 are\nplotted as functions of the chain length Lfor the parameters:\nU/(π∆) = 4.0,ǫd/U=−3/2, andb/TSU(4)\nK= 1.0. Each three\nsymbol, i.e., /squaresolid,•, and/diamondsolidrepresents values of R2,3−1,R1,4−1,\nandR1,2−1, respectively. The SU(4) Kondo temperature is\nTSU(4)\nK/∆ = 0.2 for the values of Uandǫd.\nresolvent of HL\n0,\ng00,m(ω)≡/parenleftbig\nω1− HL\n0/parenrightbig−1\n1,1. (C4)\nA ratio of the impurity level position ǫmto its width ∆\nis deduced from Eq. (C3),\nǫm\n∆=E\n∆Λ−(L−1)\n2−2AΛ\nπlim\nL→∞ΛL−1\n2Dg00,m(E).\n(C5)\nWe here use the relation Eq. (B5) between the hybridiza-\ntionVand ∆.\nWe henceforth discuss interacting cases, i.e., cases of\nU/ne}ationslash= 0, by extending the Eq. (C5) to such cases. Let\nEp,m(L) denote an excitation energy on adding an elec-\ntron to the ground state. Similarly, we define Eh,m(L)\nas an excitation energy on adding a hole to the ground\nstate. The NRG iteration procedure calculates Ep,m(L)\nandEh,m(L) as a function of the chain length L. We\nnote thatEp,m(L)>0 andEh,m(L)<0. Here we set\nthe energy of the ground state as 0. Replacing the non-\ninteractingeigenvalue Ein Eq.(C5)to thecorresponding\nEp,m(L) orEh,m(L) yields ratios of renormalized level\nposition/tildewideǫmto its width/tildewide∆m,\n/tildewideǫm,p\n/tildewide∆m,p=Ep,m(L)\n/tildewide∆m,pΛ−(L−1)\n2−2AΛ\nπΛL−1\n2Dg00,m(Ep,m(L)),\n(C6)\n/tildewideǫm,h\n/tildewide∆m,h=Eh,m(L)\n/tildewide∆m,hΛ−(L−1)\n2−2AΛ\nπΛL−1\n2Dg00,m(Eh,m(L)).\n(C7)\nHere,/tildewideǫm,pand/tildewideǫm,hare the renormalized resonance\nlevel positions calculated from Ep,mandEh,m, respec-\ntively. Similarly, /tildewide∆m,pand/tildewide∆m,hare the renormalized\nline widths from Ep,mandEh,m. The first terms in theright hand of the equations can be neglected in the limit\nofL→ ∞becauseEp,m(L) andEh,m(L) are of order 1.\nThus, the ratios become\n/tildewideǫp(h),m\n/tildewide∆p(h),m=−2AΛ\nπlim\nL→∞ΛL−1\n2Dg00,m/parenleftbig\nEp(h),m(L)/parenrightbig\n.\n(C8)\nWe obtain occupation numbers /an}bracketle{tndm,p/an}bracketri}htand/an}bracketle{tndm,h/an}bracketri}htby\nsubstituting the ratios into the Friedel sum rule given by,\n/an}bracketle{tnm,p(h)/an}bracketri}ht=1\n2−tan−1/parenleftBigg\n/tildewideǫp(h),m\n/tildewide∆p(h),m/parenrightBigg\n.(C9)\nFurthermore, subtracting Eq. (C7) from Eq. (C6) gives\nthe renormalized level width /tildewide∆m,\n/tildewide∆m=π\n2AΛEp,m(L)−Eh,m(L)\nΛ(L−1)Dg00,m(Ep,m(L))−Λ(L−1)Dg00,m(Eh,m(L)).\n(C10)\nFigure 11(a) plots the occupation numbers of the CNT\ndot as functions of the NRG chain length L. The param-\neters areU/(π∆) = 4.0,ǫd/U=−3/2, andb/TSU(4)\nK=\n1.0 for the plots. The SU(4) Kondo temperature is\nTSU(4)\nK/∆ = 0.2 for the values of Uandǫd. Figure\n11(a) clearly shows that /an}bracketle{tn1,p/an}bracketri}htfrom the particle exci-\ntation and /an}bracketle{tn1,h/an}bracketri}htfrom the hole excitation converge to a\nsame value /an}bracketle{tn1/an}bracketri}ht= 0.84 with increasing L./an}bracketle{tn2,p/an}bracketri}htand\n/an}bracketle{tn2,h/an}bracketri}htalso converge to the half-filled value /an}bracketle{tn2/an}bracketri}ht= 0.5.\nSince the extended electron-hole symmetry described by\nEq. (A5) imposes relations between the occupation num-\nbers:/an}bracketle{tn4,p/an}bracketri}ht= 1−/an}bracketle{tn1,h/an}bracketri}htand/an}bracketle{tn4,h/an}bracketri}ht= 1−/an}bracketle{tn1,p/an}bracketri}ht,/an}bracketle{tn4,p/an}bracketri}ht\nand/an}bracketle{tn4,h/an}bracketri}htconverge to a value 1 −/an}bracketle{tn1/an}bracketri}ht= 0.16. The val-\nues of/an}bracketle{tnd,1/an}bracketri}htand/an}bracketle{tnd,4/an}bracketri}htgive a value of the magnetization\nMd=/an}bracketle{tnd,1/an}bracketri}ht−/an}bracketle{tnd,4/an}bracketri}ht= 0.67. Figure 5(a) shows this value\nofMdat the point of b/TSU(4)\nK= 1.0.\nFigure 11(b) plots the chain length Ldependence of\nrenormalization factors Zm, which are ratios of /tildewide∆mto\n∆, i.e.,Zm=/tildewide∆m/∆. We note that the electron-hole\nsymmetryalsomakestwoofthefourfactorsindependent:\nZ1=Z4andZ2=Z3This figure shows that values\nofZ1andZ2respectively converge to Z1= 0.34 and\nZ2= 0.14. Since the value of Z2is about ten times\nas large as the SU(2) limit value 0 .026, the Fermi-liquid\nstate atb/TSU(4)\nK= 1.0 is in the middle of the crossover\nbetween the SU(4) and SU(2) Kondo state.\nThe NRG iteration also yields higher excitation ener-\ngies from the ground state. Ek,p,mandEk,h,mrespec-\ntively represent such electron and hole excitation ener-\ngies. We note that Ek,p,m>0 andEk,h,m<0. They\ndescribe a free quasiparticle Hamiltonian,\nH0=N/summationdisplay\nm=1/summationdisplay\nk/parenleftBig\nEk,p,mp†\nk,mpk,m−Ek,h,mh†\nk,mhk,m/parenrightBig\n.\n(C11)23\nHere, operators p†\nk,mandh†\nk,mrespectively create the\nquasiparticle with Ek,p,mand the quasihole with Ek,h,m.\nCorrespondingly, pk,mandhk,mannihilate the quasipar-\nticle and quasihole. The ground state |0/an}bracketri}htis defined such\nthatpk,m|0/an}bracketri}ht= 0 andhk,m|0/an}bracketri}ht= 0. A linear combination\nofp†\nk,mandhk,mexpressesthe operator d†\nmwhich creates\nthe impurity electron,\nd†\nm=/summationdisplay\nk/parenleftBig\np†\nk,mφk,p,m(−1) +hk,mφk,h,m(−1)/parenrightBig\n.\n(C12)\nTo calculate Wilson ratios Rm,m′defined in Eq. (18),\nwe consider a residual interaction term,\nHU\nL= Λ(L−1)/2/summationdisplay\nm<m′/tildewideUm,m′N[ndmndm′].(C13)\nHere,/tildewideUm,m′is the strength of the residual interaction\nbetween the quasiparticles. N is the normal ordering\noperator. We examine how the interaction affects the\nlow-lying many-particle states to a first order perturba-\ntionin/tildewideUm,m′. Thisordercalculationaccuratelydescribes\nthe effects of the interaction on the low-energy states be-\ncause the term vanishes in the limit of L→ ∞. For the\nexplicit calculation, we introduce a two-particle excited\nstate from the ground state:\n|A/an}bracketri}ht=p†\n1,mp†\n1,m′|0/an}bracketri}ht, (C14)\nwherem/ne}ationslash=m′. The NRG iteration calculates a corre-\nsponding eigenvalue Epp(L). The first order calculation\nof/tildewideUm,m′also givesEpp,\nEpp(L) =/an}bracketle{tA|H0+HU\nL|A/an}bracketri}ht\n= 2Ep,m(L) +/tildewideUm,m′Λ−L−1\n2|β1,p,m|2|β1,p,m′|2.\n(C15)\nIn this equation, |β1,p,m|2is given by\n|β1,p,m|2=Λ(L−1)/2\n1−/tildewideV2mΛ(L−1)dg00,m(E)\ndE/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nE=E1,p,m,\n(C16)\nwhere/tildewideV2\nm= 2D/tildewide∆mAΛ/π. TheLdependent/tildewideUm,m′(L) is\ndeducible from Eq. (C15) since the NRG iteration pro-\ncedure determines Epp(L) in the left hand of the equa-\ntion. We can alternatively consider two hole excitations\nto calculate/tildewideUm,m′. Considering particle-hole excitations\nand hole-particle excitations also yields /tildewideUm,m′. Here, the\nword “particle-hole excitations” denote one particle ex-\ncitation with mflavour and one hole excitation with m′\nflavour. The word “hole-particle excitations” similarly\ndenote one hole and one particle excitation with mand\nm′. Thus,/tildewideUm,m′is calculable by four different ways./tildewideUpp\nm,m′(L),/tildewideUhh\nm,m′(L),/tildewideUph\nm,m′(L), and/tildewideUhp\nm,m′(L)denotethe\nresidual interaction deduced from the two-particle, two-\nhole, particle-hole and hole-particle excitation energies,\nrespectively.\nFigures 12(a)-(c) respectively plot Ldependence of\n/tildewideU2,3,/tildewideU1,2, and/tildewideU1,4which are calculated by the four\nways. The three figures show that /tildewideUpp\nm,m′,/tildewideUhh\nm,m′,/tildewideUph\nm,m′,\nand/tildewideUhp\nm,m′convergea same value /tildewideUm,m′for each (m,m′).\nEach inset clearly shows such convergence of the four\nvalues. The converged values are /tildewideU2,3/(π∆) = 0.13,\n/tildewideU1,2/(π∆) = 0.05, and/tildewideU1,4/(π∆) = 0.35.\nThe Wilson ratios Rm,m′are deducible from the ob-\ntained values of δm=/an}bracketle{tndm/an}bracketri}ht/π,Zm, and/tildewideUm,m′using\nEqs. (54)-(56). Figure 13 plots R2,3−1,R1,4−1, and\nR1,2−1 as functions of L. The four values of /tildewideUpp\nm,m′,\n/tildewideUhh\nm,m′,/tildewideUph\nm,m′, and/tildewideUhp\nm,m′are averaged to calculate the\nratiosRm,m′−1. Each of the three ratios converges for\nL/greaterorsimilar20. The converged values are R2,3−1 = 0.92,\nR1,2−1 = 0.11, andR1,4−1 = 0.24.R2,3becomes\nlarger than the other two ratios because the correspond-\ning impurity levels of R2,3remain the Fermi level.\nAppendix D: Fermi-liquid parameters for single\norbital Anderson impurity\nWebrieflydiscusshowthe Fermiliquid stateofthe sin-\ngle Anderson impurity evolves with increasing magnetic\nfield. The field dependence of the Fermi liquid param-\neters have been discussed also in Refs. 66 and 67. Fig-\nures 14(a) and 14(b) plot NRG results of the Fermi liq-\nuid parameters and the residual interaction, respectively.\nIn the NRG calculations, the spin-dependent impurity\nlevel is chosen such that εd,σ=εd+ sgn(σ)b, and the\ncenter of the impurity level is locked at the half-filling,\nεd=−U/2. Figure 14(a) shows that the transmission\nprobability T(0) = sin2δdecreases as soon as magnetic\nfields become comparable with the Kondo temperature\nTSU(2)\nK, and correspondingly, the induced magnetization\nis rapidly saturated to 1, i.e., Md→1. In contrast,\nZandR−1 vary more slowly than T(0) andMdwith\nthe scales of U. The inset of Fig.14(a) shows that /tildewide∆\nandR−1 are still renormalized for small magnetic fields\nb/lessorsimilarTSU(2)\nK. For large magnetic fields b≫TSU(2)\nK, these\nparameters approach the non-interacting values, Z→1\nandR−1→1.\nThe residual interaction plotted in Fig. 14(b) also\nvaries from a zero field value 4 TSU(2)\nK= 0.23π∆ with\nincreasingb. For small magnetic fields b/lessorsimilar0.2U,/tildewideUis\nenhanced and its value becomes larger than the bare\nCoulomb interaction U. As magnetic fields further in-\ncrease, it decreases from the enhanced value to the bare\nvalueU.24\n++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++\n+\n¡¢£¤¥¦§¨©ª«¬®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×Ø ÙÚÛÜÝÞß��áâãäåæçèéêëìíîïðñòóô õö÷ øùú ûü ýþ ÿ \u0001\u0000 \u0002 \u0003\u0004 \u0005 \u0006●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★\n0.20.40.60.8 0 10.20.40.60.8\n01\nb/UFermi\nl\u0007\b\t\n\u000bp\f\r\u000e\u000f\u0010\u0011\u0012\u0013\u0014++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++\n+◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★\n012340.20.4\n0 \u0015 \u0016\n\u0017 \u0018 \u0019\n01\nb/TKR-1\nMdZ\nSin2δ\nR-1Md\n\u001a\nSin2δU/(πΔ)=2.0\nϵd/U=-1/2(a)\u001b \u001c\n\u001d\n\u001e\n\u001f\n \n!\n\"\n#\n$\n%\n&\n'\n()\n*\n+\n,\n-\n.\n/\n1\n2\n3\n4\n5\n6\n7\n8\n9\n:\n;\n<\n=\n>\n?\n@\nA\nB\nC\nD\nE\nF\nG\nH\nI\nJ\nK\nL\nM\nN\nO\nP\nQ\nR\nS\nTU\nV\nW\nX\nY\n[\n\\\n]\n^\n_\n`\na\nb\ncd\ne\nf\ng\nh\ni\nj\nk\nm\nn\no\nq\nr\ns\nt\nu\nv\nw\nx\ny\nz\n{|}~\n¡\n0.20.40.60.8 0 101234\nb/UU/()¢ £\n¤\n¥\n¦\n§\n¨\n©\nª\n«\n¬\n\n®\n¯\n°\n±\n²\n³\n´\nµ\n¶\n·\n¸\n¹\nº\n»\n¼\n½\n¾\n¿\nÀ\nÁ\nÂ\nÃ\nÄ\nÅ\nÆ\nÇ\nÈ\nÉ\nÊ\nË\nÌ\nÍ\nÎ\nÏ\nÐ\nÑ\nÒ\nÓ\nÔ\nÕ\nÖ\n×\nØ\nÙ\nÚ\nÛ\nÜ\nÝ\nÞ\nß\nà\ná\nâ\nã\nä\nå\næ\nç\nè\né\nê\në\nì\ní\nî\nï\nð\nñ\nò\n0123401234\nb/TKU/()=2.0\nϵd/U=-1/2(b)\nFIG. 14. (a) Magnetic field dependence of Fermi liquid pa-\nrameters for single orbital Anderson impurity at the electr on-\nhole symmetric point εd/U=−1/2 forU/(π∆) = 2 .0.\nAt this point, a total occupation number is locked at one,\ni.e.,/angbracketleftnd↓/angbracketright+/angbracketleftnd↑/angbracketright= 1 at arbitrary magnetic fields, and\nthus/angbracketleftndσ/angbracketrightcan be expressed in terms of the magnetization\nMd≡ /angbracketleftnd↑/angbracketright − /angbracketleftnd↓/angbracketrightas follows: /angbracketleftndσ/angbracketright= (1 + sgn( σ)Md)/2.\nFurthermore, sin2δ↑= sin2δ↓andZ↑=Z↓. (b) Residual\ninteraction /tildewideUas a function of b. The x-axis is scaled by the\nCoulomb interaction U. Each inset shows an enlarged view\nof the region around b= 0. In the insets, the axis is scaled\nby the Kondo temperature TSU(2)\nK= 0.19∆ = (0 .03U).\nAppendix E: NRG for dynamical correlations\n1. Spectral function and self-energy\nIn this work, the spectral function Am(ω) has been\ncalculated using the “ complete Fock-space basis set ”, de-\nveloped by Andes et al.24,25and by Weichselbaum and\nvon Delft.26In this approach, contributions of the high\nenergy states which are discarded at the NRG steps can\nbe recovered to form a complete basis for the Wilson’s\nNRG chain, by carrying out the backward iteration. The\nmerit of this approach is that the sum rule for the spec-\ntral weights can be fulfilled.\nWe have also employed the method due to Bulla, Hew-\nson, and Pruschke:62we have calculated not only Gm(ω)but also the higher-order Green’s function Fm(ω)\nFm(ω) =−i/integraldisplay∞\n0dtei(ω+i0+)t\n×/summationdisplay\nm′(/negationslash=m)/angbracketleftbig/braceleftbig\nndm′(t)dm(t), d†\nm(0)/bracerightbig/angbracketrightbig\n.(E1)\nThen, the self-energy can be determined directly through\nthe relation Σ m(ω) =UFm(ω)/Gm(ω). The final form\nof the Green’s function has been obtained from Σ m(ω)\nandthe noninteractingGreen’s function G0\nm(ω) using the\nDyson equation given in Eq. (7). The merit to treat the\nself-energy as an input is that the fully analytic expres-\nsion which is not affected by the logarithmic discretiza-\ntion can be used for G0\nm(ω).\n2. The z averaging\nThe size ofthe Hilbert space to be diagonalizedat each\nNRG step increasesas the number ofconduction electron\nchannels increases. To ensure the accuracy of the NRG\ncalculation, alargeΛisused forquantumimpurities with\na number of internal degrees of freedom.\nOliveira and Oliveira found that thermodynamic aver-\nages which are calculated for large Λ show an artificial\noscillation at low temperatures,59,61and they proposed\nthez averaging for removingsuch an artificial oscillation.\nThe parameter zslides a set of discretization points from\nthat of the standard Wilson chain,22\n±Λ−n→ ±Λ−(n+1−z), n= 0,1,2,···,(E2)\nwith 0≤z≤1.Forz= 1, it coincides with the stan-\ndard Wilson chain. The discretized conduction band can\nbe transformed into a zdependent Wilson chain\nHc⇒∞/summationdisplay\nn=04/summationdisplay\nm=1tn(z)/parenleftBig\nf†\nn,mfn+1,m+f†\nn+1,mfn,m/parenrightBig\n.(E3)\nThe hopping matrix element tn(z) that can be deter-\nmined using the Hauseholder algorithm summarized in\nRefs. 59 and 61. We have carried out NRG calculations\nfor some fixed values of z, and calculate expectation val-\nues using the obtained eigenstates. Then, an average is\ntaken over “ z” for two different values z= 0.5 and 1,\nwhich is enough to eliminate the artificial oscillations in\nour case.\n3. Logarithmic-Gaussian function to broaden\ndiscrete spectral function\nWe have calculated the spectral function Amusing the\nLehman representation given in Eq. (F1). The result-\ning function is a set of discrete δfunctions at frequencies\nωn. Replacingthe δfunction tothe logarithmic-Gaussian\nfunction makes Amcontinuous since the NRG energy25\nscaleTLgiven in Eq. (B13) exponentially falls of with\nincreasing the chain length L. The function to broaden\na particle excitation peak at ωn>0 is\nX(ω) =e−b2/4\nbωn√πexp/braceleftBigg\n−/parenleftbiggln(ω/ωn)\nb/parenrightbigg2/bracerightBigg\n,(E4)\nwhich is defined for ω>0. Correspondingly, a hole exci-\ntation peak at ωn<0 is broadened by\nY(ω) =e−b2/4\nb|ωn|√πexp/braceleftBigg\n−/parenleftbiggln|ω/ωn|\nb/parenrightbigg2/bracerightBigg\n,(E5)\nforω <0. We set a broadening parameter b= 1.1 for\nthe spectral functions plotted in Figs. 8 and 9. Note that\nthe spectral weight is conserved because the broadening\nfunctions are normalized to 1.\nAppendix F: Spectral function in atomic limit\nWe consider the atomic limit in order to show how the\nspectralweightofthe impuritystatesevolvesasmagnetic\nincreases at high-energies in the case that the Zeemann\nsplittings of the impurity levels are given by Eq. (41).\nAt zero temperature, the flavour m-resolved single-\nparticle spectral function can be written in the Lehmann\nrepresentation as,\nAm(ω) =1\nMM/summationdisplay\ni=1/summationdisplay\nn/bracketleftBigg\n/vextendsingle/vextendsingle/an}bracketle{tn|d†\nm|ΨGS,i/an}bracketri}ht/vextendsingle/vextendsingle2δ/parenleftBig\nω−(En−EGS)/parenrightBig\n+|/an}bracketle{tn|dm|ΨGS,i/an}bracketri}ht|2δ/parenleftBig\nω+(En−EGS)/parenrightBig/bracketrightBigg\n.\n(F1)\nHere,|n/an}bracketri}htandEnare the eigenstate and eigenenergy of\nHamiltonian H, respectively. The ground state |ΨGS,i/an}bracketri}ht\nwith the energy EGScan generally be degenerate, and\nthe summation over irepresents an average over M-fold\ndegenerate states.\nIn the atomic limit, the CNT dot whose eigen energies\nare defined is Eq. (41) is disconnected from the leads\n(vν= 0 forν=L,R), and there remains two-fold degen-\neracy for the ground state at half-filling εd=−3U/2,\n|ΨGS,2/an}bracketri}ht=d†\n1d†\n2|0/an}bracketri}ht, (F2)\n|ΨGS,3/an}bracketri}ht=d†\n1d†\n3|0/an}bracketri}ht. (F3)Either of the two one-particle states, m= 2 or 3 sit-\nuated on the Fermi level, is occupied, and the lowest\none-particle level with the energy ε1is occupied while\nthe highest one with ε4is empty. Therefore, the ground\nenergy for these two-electron states becomes\nEGS=ε1+ε2+U= 2εd−2b+U.(F4)\nWe next consider a single-particle excitation to add an\nelectron into the level of m= 4, and a single-hole excita-\ntion to remove the electron occupying the m= 1 level,\n|Ψp4/an}bracketri}ht=d†\n4|ΨGS,i/an}bracketri}ht, E p4= 3εd+3U,(F5)\n|Ψh1/an}bracketri}ht=d1|ΨGS,i/an}bracketri}ht, E h1=εd. (F6)\nThe excitation energies from the ground state i= 2,3 to\nthese two states are given by\nEp4−EGS=εd+2b+2U= 2b+U\n2,(F7)\nEGS−Eh1=εd−2b+U=−2b−U\n2.(F8)\nTherefore, the spectral weights of these processes are\ngiven by\nA4(ω) =δ/parenleftbigg\nω−/parenleftbigg\n2b+U\n2/parenrightbigg/parenrightbigg\n,\nA1(ω) =δ/parenleftbigg\nω+/parenleftbigg\n2b+U\n2/parenrightbigg/parenrightbigg\n.(F9)\nThese weights shift towards high-energy region from the\nusual atomic limit position ±U/2. We have observed the\ncorresponding shifts of the spectral weight in the NRG\nresults shown in Fig. 8(c) and Fig. 9(c) although these\natomic peaks merge with the resonance peak which also\nmoves away from the Fermi level as bincreases.\nThe other single electron (hole) excitation from the\nground state |ΨGS,2/an}bracketri}htcorresponds to an addition of an\nelectron to the level m= 3 (annihilation of an electron\nfromm= 2). The similar excitations also occur from\n|ΨGS,3/an}bracketri}ht. The peaks corresponding to these excitations\nappear atω=±U/2 in the spectral functions for m= 2\nand 3,\nA2(ω) =A3(ω) =1\n2δ/parenleftBig\nω−U\n2/parenrightBig\n+1\n2δ/parenleftBig\nω+U\n2/parenrightBig\n.(F10)\nThese two peaks are equivalent to the Hubbard peaks for\nthe SU(2) symmetric case.\n1J. Kondo, Progress of Theoretical Physics 32, 37 (1964).\n2A. C. Hewson, The Kondo Problem to Heavy Fermions\n(Cambridge University Press, Cambridge, U.K., 1993).3L. I. Glazman and M. ´E. Ra˘ ıkh,\nJETP Lett. 47, 452 (1988).\n4T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988).26\n5D. Goldhaber-Gordon, J. G¨ ores, M. A. Kast-\nner, H. Shtrikman, D. Mahalu, and U. Meirav,\nPhys. Rev. Lett. 81, 5225 (1998).\n6W. Van der Wiel, S. De Franceschi, T. Fujisawa, J. Elzer-\nman, S. Tarucha, and L. Kouwenhoven, Science 289, 2105\n(2000).\n7E. A. Laird, F. Kuemmeth, G. A. Steele, K. Grove-\nRasmussen, J. Nyg˚ ard, K. Flensberg, and L. P. Kouwen-\nhoven, Rev. Mod. Phys. 87, 703 (2015).\n8S. Sasaki, S. De Franceschi, J. M. Elzerman, W. G. van der\nWiel, M. Eto, S. Tarucha, and L. P. Kouwenhoven,\nNature405, 764 (2000).\n9M.-S. Choi, R. L´ opez, and R. Aguado,\nPhys. Rev. Lett. 95, 067204 (2005).\n10W. Izumida, O. Sakai, and Y. Shimizu,\nJournal of the Physical Society of Japan 67, 2444 (1998).\n11L. Borda, G. Zar´ and, W. Hofstetter, B. I. Halperin, and\nJ. von Delft, Phys. Rev. Lett. 90, 026602 (2003).\n12F. B. Anders, D. E. Logan, M. R. Galpin, and G. Finkel-\nstein, Phys. Rev. Lett. 100, 086809 (2008).\n13I. Weymann, R. Chirla, P. Trocha, and C. P. Moca,\nPhys. Rev. B 97, 085404 (2018).\n14Y. Nishikawa, A. C. Hewson, D. J. G. Crow, and J. Bauer,\nPhys. Rev. B 88, 245130 (2013).\n15A. Makarovski, A. Zhukov, J. Liu, and G. Finkelstein,\nPhys. Rev. B 75, 241407(R) (2007).\n16A. J. Keller, S. Amasha, I. Weymann, C. P. Moca,\nI. G. Rau, J. A. Katine, H. Shtrikman, G. Zar´ and, and\nD. Goldhaber-Gordon, Nat. Phys. 10, 145 (2014).\n17P. Jarillo-Herrero, J. Kong, H. S. J. van der Zant,\nC. Dekker, L. P. Kouwenhoven, and S. De Franceschi,\nNature434, 484 (2005).\n18K. J. G. G¨ otz, D. R. Schmid, F. J. Schupp,\nP. L. Stiller, C. Strunk, and A. K. H¨ uttel,\nPhys. Rev. Lett. 120, 246802 (2018).\n19M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. Dela-\ngrange, R. Weil, R. Deblock, R. Sakano, A. Oguri, and\nK. Kobayashi, Nat. Phys. 12, 230 (2016).\n20T. Hata, R. Delagrange, T. Arakawa, S. Lee, R. De-\nblock, H. Bouchiat, K. Kobayashi, and M. Ferrier,\nPhys. Rev. Lett. 121, 247703 (2018).\n21K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n22H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson,\nPhys. Rev. B 21, 1003 (1980).\n23H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson,\nPhys. Rev. B 21, 1044 (1980).\n24F. B. Anders and A. Schiller,\nPhys. Rev. Lett. 95, 196801 (2005).\n25R. Peters, T. Pruschke, and F. B. Anders,\nPhys. Rev. B 74, 245114 (2006).\n26A. Weichselbaum and J. von Delft,\nPhys. Rev. Lett. 99, 076402 (2007).\n27M. R. Galpin, F. W. Jayatilaka, D. E. Logan, and F. B.\nAnders, Phys. Rev. B 81, 075437 (2010).\n28D. Mantelli, C. P. Moca, G. Zar´ and, and M. Grifoni, Phys-\nica E77, 180 (2016).\n29P. Roura-Bas, L. Tosi, A. A. Aligia, and K. Hallberg,\nPhys. Rev. B 84, 073406 (2011).\n30C. Mora, P. Vitushinsky, X. Leyronas, A. A. Clerk, and\nK. Le Hur, Phys. Rev. B 80, 155322 (2009).\n31C. Mora, C. P. Moca, J. von Delft, and G. Zar´ and,\nPhys. Rev. B 92, 075120 (2015).\n32M. Filippone, C. P. Moca, A. Weichselbaum, J. von Delft,\nand C. Mora, Phys. Rev. B 98, 075404 (2018).33A. Oguri and A. C. Hewson,\nPhys. Rev. B 97, 035435 (2018).\n34Y. Teratani, R. Sakano, and A. Oguri, (2020),\narXiv:2001.08348v1.\n35M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. Dela-\ngrange, R. Deblock, Y. Teratani, R. Sakano, A. Oguri,\nand K. Kobayashi, Phys. Rev. Lett. 118, 196803 (2017).\n36Y. Teratani, R. Sakano, R. Fujiwara, T. Hata,\nT. Arakawa, M. Ferrier, K. Kobayashi, and A. Oguri,\nJ. Phys. Soc. Jpn. 85, 094718 (2016).\n37D. R. Schmid, S. Smirnov, M. Marga´ nska, A. Dirnaichner,\nP. L. Stiller, M. Grifoni, A. K. H¨ uttel, and C. Strunk,\nPhys. Rev. B 91, 155435 (2015).\n38V. Lopes, R. A. Padilla, G. B. Martins, and E. V. Anda,\nPhys. Rev. B 95, 245133 (2017).\n39C. A. B¨ usser and G. B. Martins,\nPhys. Rev. B 75, 045406 (2007).\n40J. S. Lim, M.-S. Choi, M. Y. Choi, R. L´ opez, and\nR. Aguado, Phys. Rev. B 74, 205119 (2006).\n41Y. Kleeorin and Y. Meir, Phys. Rev. B 96, 045118 (2017).\n42P. Roura-Bas, L. Tosi, A. A. Aligia, and P. S. Cornaglia,\nPhys. Rev. B 86, 165106 (2012).\n43D. Krychowski and S. Lipi´ nski,\nThe European Physical Journal B 91, 8 (2018).\n44D. C. Langreth, Phys. Rev. 150, 516 (1966).\n45H. Shiba, Prog. Theor. Phys. 54, 967 (1975).\n46A. Yoshimori, Prog. Theor. Phys. 55, 67 (1976).\n47A.C.Hewson,J. Phys.: Condens. Matter 13, 10011 (2001).\n48A. Hewson, A. Oguri, and D. Meyer, Eur. Phys. J. B 40,\n177 (2004).\n49R. Landauer, Philos. Mag. 21, 863 (1970).\n50Y. Meir and N. S. Wingreen,\nPhys. Rev. Lett. 68, 2512 (1992).\n51S. Hershfield, J. H. Davies, and J. W. Wilkins,\nPhys. Rev. B 46, 7046 (1992).\n52W. Izumida, O. Sakai, and S. Suzuki, J. Phys. Soc. Jpn.\n70, 1045 (2001).\n53A. Oguri, Phys. Rev. B 85, 155404 (2012).\n54Y. Nishikawa, D. J. G. Crow, and A. C. Hewson,\nPhys. Rev. B 82, 115123 (2010).\n55K. M. Stadler, A. K. Mitchell, J. von Delft, and A. We-\nichselbaum, Phys. Rev. B 93, 235101 (2016).\n56P. Nozi` eres, J. Low Temp. Phys. 17, 31 (1974).\n57K. Yosida and K. Yamada,\nProgress of Theoretical Physics Supplement 46, 244 (1970).\n58K.Yamada,Progress of Theoretical Physics 53, 970 (1975).\n59W. C. Oliveira and L. N. Oliveira,\nPhys. Rev. B 49, 11986 (1994).\n60T. A. Costi, Phys. Rev. B 64, 241310(R) (2001).\n61M. Yoshida, M. A. Whitaker, and L. N. Oliveira,\nPhys. Rev. B 41, 9403 (1990).\n62R. Bulla, A. C. Hewson, and T. Pruschke,\nJournal of Physics: Condensed Matter 10, 8365 (1998).\n63W. Izumida, R. Okuyama, and R. Saito,\nPhys. Rev. B 91, 235442 (2015).\n64W. Izumida, R. Okuyama, A. Yamakage, and R. Saito,\nPhys. Rev. B 93, 195442 (2016).\n65Supplenental Material of Ref. 35 at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLet t.118.196803.\n66A. C. Hewson, J. Bauer, and W. Koller,\nPhys. Rev. B 73, 045117 (2006).\n67J. Bauer and A. C. Hewson,\nPhys. Rev. B 76, 035119 (2007).\n68T. A. Costi, Phys. Rev. B 100, 161106 (2019).27\n69A. Svilans, M. Josefsson, A. M. Burke, S. Fahlvik, C. Theland er, H. Linke, and M. Leijnse,\nPhys. Rev. Lett. 121, 206801 (2018)."    },    {        "title": "1208.5192v1.Observation_of_Temperature_Induced_Crossover_to_an_Orbital_Selective_Mott_Phase_in_A___x__Fe___2_y__Se__2___A_K__Rb__Superconductors.pdf",        "content": "Observation of Temperature-Induced Crossover to an Orbital-Selective Mott Phase in\nAxFe2\u0000ySe2(A=K, Rb) Superconductors\nM. Yi,1, 2D.H. Lu,3R. Yu,4S. C. Riggs,1, 2J.-H. Chu,1, 2B. Lv,5Z. Liu,1, 2M. Lu,1, 6Y.-T. Cui,1\nM. Hashimoto,3S.-K. Mo,7Z. Hussain,7C. W. Chu,5I.R. Fisher,1, 2Q. Si,4and Z.-X. Shen1, 2\n1Stanford Institute of Materials and Energy Sciences, Stanford University, Stanford, CA 94305, USA\n2Departments of Physics and Applied Physics, and Geballe Laboratory\nfor Advanced Materials, Stanford University, Stanford, CA 94305, USA\n3Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA\n4Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA\n5Department of Physics, Texas Center for Superconductivity,\nUniversity of Houston, Houston TX 77204, USA\n6National Laboratory of Solid-State Microstructures and Department of Materials Science and Engineering,\nNanjing University, Nanjing 210093, P. R. China\n7Advanced Light Source, Lawrence Berkeley National Lab, Berkeley, CA 94720, USA\n(Dated: October 9, 2018)\nIn this work, we study the A xFe2\u0000ySe2(A=K, Rb) superconductors using angle-resolved photoe-\nmission spectroscopy. In the low temperature state, we observe an orbital-dependent renormalization\nfor the bands near the Fermi level in which the dxybands are heavily renormliazed compared to the\ndxz/dyzbands. Upon increasing temperature to above 150K, the system evolves into a state in which\nthedxybands have diminished spectral weight while the dxz/dyzbands remain metallic. Combined\nwith theoretical calculations, our observations can be consistently understood as a temperature in-\nduced crossover from a metallic state at low temperature to an orbital-selective Mott phase (OSMP)\nat high temperatures. Furthermore, the fact that the superconducting state of A xFe2\u0000ySe2is near\nthe boundary of such an OSMP constraints the system to have su\u000eciently strong on-site Coulomb\ninteractions and Hund's coupling, and hence highlight the non-trivial role of electron correlation in\nthis family of iron superconductors.\nPACS numbers: 74.25.Jb, 74.70.-b, 79.60.-i\nElectron correlation remains a central focus in the\nstudy of high temperature superconductors. The\nstrongly correlated cuprate superconductors are under-\nstood as doped Mott insulators (MI) [1] while the iron-\nbased superconductors (FeSCs) have been found to be\nat most moderately correlated [2{4]. Even though the\nlow energy electronic structures of di\u000berent FeSCs fam-\nilies share the common Fe 3 dbands, there are system-\natic variations in their physical properties, such as or-\ndered magnetic moment and e\u000bective mass [5]. In gen-\neral, electron correlation is the weakest in iron phosphides\nwith relatively low mass renormalization [2], and moder-\nate in the more extensively studied iron arsenides [2, 3].\nThe Fe(Te,Se) chalcogenide family, in comparison, seems\nto harbor stronger correlation as inferred from larger\nordered moment, yet metallic resistivity is still ob-\nserved [6]. The newest iron chalcogenide superconduc-\ntors, AxFe2\u0000ySe2(A=alkali metal) [7{11] (AFS) hints at\neven stronger correlation with a large observed moment\nof 3.3\u0016B[12] and insulating transport behavior in the\nphase diagram.\nAnother important factor in understanding the FeSCs\nlies in their multi-orbital nature. In such a system,\norbital-dependent behavior as well as competition be-\ntween inter- and intra-orbital interactions could play a\ncritical role in determining their physical properties. One\nexample is the orbital anisotropy that onsets with thein-plane symmetry breaking structural transition as ob-\nserved in the underdoped arsenides [13]. Another exam-\nple is the di\u000berent pairing symmetry that could arise con-\nsidering the relative strength of inter- and intra-orbital\ninteractions extensively studied theoretically [14]. Theo-\nretical models have considered correlation e\u000bects in the\nbad metal regime in terms of an incipient Mott pic-\nture [5, 15], and the proximity to the Mott transition\nmay be orbital-dependent even for orbitally-independent\nCoulomb interactions [16{19]. What arises from the\nmodel is an orbital selective Mott phase (OSMP), in\nwhich some orbitals are Mott-localized while others re-\nmain itinerant. First introduced in the context of the\nCa2\u0000xSrxRuO 4system, an OSMP may result from both\nthe orbital-dependent kinetic energy and the combined\ne\u000bects of the Hund's coupling and crystal level split-\ntings [20, 21]. An OSMP links naturally with mod-\nels of coexisting itinerant and localized electrons that\nhave been proposed to compensate for the shortcom-\nings of both strong coupling and weak coupling ap-\nproaches [22, 23]. However, to date, there has been no\nexperimental evidence for OSMP in any FeSC.\nIn this paper, we present angle-resolved photoemission\nspectroscopy (ARPES) data from two superconducting\nAFSs, KxFe2\u0000ySe2(KFS) and Rb xFe2\u0000ySe2(RFS), with\nTCof 32K and 31K, respectively, as well as insulating\nand intermediate dopings (see SI). We observe the su-arXiv:1208.5192v1  [cond-mat.supr-con]  26 Aug 20122\nFIG. 1: Measured electronic structure of K xFe2\u0000ySe2. (a) Fermi surface mapping by integrating 20meV window about EF.\nGreen lines outline the 2-Fe Brillouin zone. (b),(d) Spectral images and (c),(e) second derivatives taken along the \u0000-X direction\nusing light polarizations as marked. (f)-(g), equivalent of that of (b)-(c) for Ba(Co 0:07Fe0:93)2As2. (h) Schematic of the dominant\norbital characters of the two electron pockets of the same size near X point, with one of the pockets (dotted) imploded for\nclarity. (i) LDA calculations [27] for KFS with the dominant orbital characters labeled. (j)-(l), Spectral images taken across\nthe X-point under di\u000berent polarizations and cut directions. Guides to eye for the observable bands are overlaid, with colors\nindicating the dominant orbital characters-blue: dxy; red:dxz; green:dyz. All data taken at 30K, with 47.5eV photons except\n(d), (e), and (k), which were taken with 26eV photons.\nperconducting AFSs undergoing a temperature-induced\ncrossover from a metallic state in which all three t2g\norbitals-dxy,dxzanddyzare present near the Fermi level\n(EF) to a state in which the dxybands has diminished\nspectral weight while the dxz/dyzbands remain metal-\nlic. In addition, the intermediate doping shows stronger\ncorrelation than the superconducting doping, as seen in\nthe further renormalization of the dxz/dyzbands and the\nmuch more suppressed dxyintensity, while the insulating\ndoping has no spectral weight near EFin any orbitals.\nFrom comparison with our theoretical calculations, the\nensemble of our observations are most consistent with\nthe understanding that the presence of strong Coulomb\ninteractions and Hund's coupling places the supercon-\nducting AFSs near an OSMP at low temperatures, and\ncrosses over into the OSMP via raised temperature, while\nthe intermediate and insulating compounds are on the\nboundary of the OSMP and in the MI phase, respec-\ntively, suggesting the importance of electron correlation\nin this family of FeSCs.\nThe low temperature electronic structure of\nKxFe2\u0000ySe2is shown in Fig. 1. The Fermi surface\n(FS) of KFS consists of large electron pockets at the\nBrillouin zone (BZ) corner-X point, and a small electron\npocket at the BZ center-\u0000 point (Fig. 1(a)), consistent\nwith previous ARPES reports [24{26]. For the crys-\ntallographic 2-Fe unit cell, LDA calculations predict\ntwo electron pockets at the X point [27] (Fig. 1(i)).\nWhile FS map appears to show only one electron\npocket at X, measurements under di\u000berent polarizations(Fig. 1(j)-(l)) reveal the expected two electron bands\nwith nearly degenerate Fermi crossings ( kF) but dif-\nferent band bottom positions-a shallower one around\n-0.05eV and a deeper one that extends to the top of\nthedxz/dyzhole-like bands at -0.12eV. The Luttinger\nvolume of the two electron pockets gives \u00180.16 electrons\nper Fe. Considering that the C 4symmetry of the\ncrystal dictates degeneracy of the dxz/dyzelectron band\nbottom and hole band top at X, the shallower electron\nband that is not degenerate with the hole-like band at\nhigher binding energy is most likely of dxycharacter,\nand the deeper one dxzalong \u0000-X and dyzalong the\nperpendicular direction. This observed orbital character\nassignment seems to contradict the LDA prediction\nof the shallower electron band as dxz/dyzand the\ndeeper one dxyin FeSC [28] (Fig. 1(i)), as observed\nin Co-doped BaFe 2As2(Fig. 1(f)-(g)). However, this\nassignment can be understood if we consider the KFS\nband structure as a whole. Three \flled hole bands are\nobserved near the \u0000 point (Figs. 1(d)-(e)), where the\ntwo lower ones can be identi\fed as dxz/dyzand the\nhigher one dxy. Interestingly, the dxyband is far more\nrenormalized (\u0018a factor of 10) compared to LDA than\nthedxz/dyzbands (\u0018a factor of 3), indicating stronger\ncorrelation for the dxyorbital. This is in contrast to\nthe Co-doped BaFe 2As2band structure, in which all\norbitals are renormalized by roughly the same factor\n(\u00182) compared to LDA. Hence, our assignment of\ndxycharacter to the shallower electron band is consistent\nwith strong orbital-dependent renormalization, which3\nFIG. 2: Temperature dependence of electron bands near X. (a) Spectral images taken with 26eV photons along cut1. (b)\nSpectral images and (c) corresponding energy distribution curves (EDCs) taken with 47.5eV photons along cut2. Polarizations\nare as marked. Spectral images for selected temperatures are shown for both geometries. To rule out a trivial thermal\nbroadening for the diminishing spectral weight, we arti\fcially introduce a 210K thermal broadening to the 30K data and plot\nthem next to the 210K data for comparison.\nbrings the deeper dxyelectron band at X predicted\nby LDA to be shallower than the dxz/dyzband. This\norbital-dependent renormalization behavior also emerges\nfrom our theoretical calculations as will be discussed\nlater.\nAs the electronic structure at EFis dominated by the\nlarge electron pockets at X, we focus on these features.\nFig. 2 shows a temperature-dependent study of these\nbands taken in two polarization geometries. In Fig. 2(a),\nthe matrix element for dxyis stronger than dxz. At low\ntemperatures, dxyband is clearly resolved with weaker\nintensity for the dxzband. With increasing temperature,\nthe spectral weight of the dxyband noticeably diminishes,\neventually revealing the remaining dxzband at high tem-\nperatures. In Fig. 2(b), the deeper dyzband has more\nintensity than the corresponding dxzband in Fig. 2(a)\nwhile the presence of the dxyband is still very noticeable\nfrom the increased intensity where they signi\fcantly over-\nlap, as well as from the energy distribution curves (EDCs)\nshown in Fig. 2(c). With increasing temperature, again,\nthe spectral weight of the dxyband diminishes, leaving\nonly thedyzband at high temperatures, which clearly\nhas a deeper band bottom than the shallow dxyband\n(Fig. 2(c)). In addition, we have arti\fcially introduced a\n210K thermal broadening to the 30K spectra as shown in\nthe last column of Fig. 2. The clear contrast to the 210K\ndata rules out a trivial thermal broadening as an origin\nfor the observed diminishing of dxyspectral weight.\nTo capture this behavior quantitatively, we analyze the\ntemperature dependence of the EDCs at the X-point,\nstacked in Fig. 3(c). At all temperatures, there is a largehump background corresponding to the large hole-like\ndispersion at high binding energy. At low temperatures,\nthere is an additional peak around -0.05eV correspond-\ning to thedxyband bottom. We \ft these EDCs with a\nGaussian background for the large hump feature and a\nLorentzian peak for the dxyband. The integrated spec-\ntral weight for the \ftted dxypeak is plotted in Fig. 3(d),\nwhich decreases toward zero with increasing temperature,\nseen as a non-trivial drop between 100K and 200K. As\nan independent check against trivial thermal e\u000bect, we\nchoose small regions in the raw spectral image (marked in\nFig. 3(a)-(b)) dominated by dxy(blue),dyz(green), and\nmixeddxyanddxz/dyz(magenta/cyan) characters and\nplot their integrated intensities as a function of tempera-\nture (Fig. 3(e)). The spectral weight from dxy-dominated\nregion rapidly decreases, consistent with the \ftted result\nin Fig. 3(d), while that of dyz-dominated region does not\ndrop in a similar manner. The regions of mixed orbitals\nshow a slower diminishing spectral weight compared to\nthat fordxy, re\recting the combined contributions from\nbothdxz/dyzanddxyorbitals. Although this method\nhas the uncertainty of small leakage of other orbitals into\nthe chosen regions, which is the cause of the \fnite resid-\nual value for the dxycurve, the contrasting behavior of\nthedxyversusdxz/dyzorbitals is clearly demonstrated.\nA temperature cycle test was performed to exclude the\npossibility of sample aging (see SI). Measurements on the\nsister compound RFS reveal similar behavior (SI), sug-\ngesting the generality in this family of superconductors.\nThis observation of a selected orbital that loses coher-\nent spectral weight while the others remain metallic is4\nFIG. 3: Quantitative analysis of the temperature dependence. (a),(b) 90K spectral image from Fig. 2(a),(b). (c) EDCs at\nthe X point from (a) for selected temperatures. Curves are o\u000bset for clarity. Each curve is \ftted to a Gaussian (dotted line)\nfor the background feature at high binding energy and a Lorentzian (shaded region) for the feature around -0.05eV. (d) The\nintegrated Lorentzian spectral weight are plotted as a function of temperature. (e) Temperature-dependence of the averaged\nintensity in the regions marked by colored boxes in (a)-(b), with background (dotted box of the same energy window) for each\nbox subtracted. The resulting temperature-dependent curve is then normalized by the initial value. The blue (green) region has\ndominant spectral weight from the dxy(dyz) band, whereas the magenta (cyan) region is of mixed dxyanddxz(dyz) characters.\nreminiscent of a crossover into an OSMP in which se-\nlected orbitals become Mott localized while others remain\nmetallic.\nTo further understand this phenomenon, we perform\ntheoretical calculations based on a \fve-orbital Hubbard\nmodel to study the metal-to-insulator transition (MIT)\nin the paramagnetic phase using a slave-spin mean-\feld\nmethod [29, 30]. At commensurate electron \flling n=6\nper Fe (corresponding to Fe2+of the parent FeSC), we\n\fnd the ground state of the system to be a metal, an\nOSMP or a MI depending on the intra-orbital repulsion\nU and the Hund's coupling J. Furthermore, the MIT can\nbe triggered by increasing temperature (Fig. 4(a)) due\nto the larger entropy of the insulating phase. At a \fxed\ninteraction strength (within a certain range, Fig. 4(a)),\nthe system goes from a metal to an OSMP and then to\na MI with increasing temperature. The MI phase is sup-\npressed by electron doping. By contrast, the OSMP can\nsurvive, as shown in Fig. 4(b) for n=6.15, which roughly\ncorresponds to the \flling of the superconducting state\nfrom ARPES measurements. From the evolution of the\norbitally resolved quasiparticle spectral weight, Z\u000b, as\na function of the temperature (Fig. 4(c)), we show that\nthe OSMP corresponds to the dxyorbital being Mott\nlocalized (Z= 0) and the rest of the 3 dorbitals remain-\ning delocalized ( Z > 0). This result originates primarily\nfrom a combined e\u000bect of the orbital dependence of the\nprojected density of states and the interplay between the\nHund's coupling and crystal level splitting (see SI and\nRef. 31).\nTo compare with the ARPES measurements, we have\ncalculated the quasiparticle spectral function A(k,E)\nin the 2-Fe BZ. At low temperature and in the non-\ninteracting limit U=0 (Fig. 4(d)), the electronic structureof the model agrees well with that from LDA, with the\ndxyband deeper than the dxz/dyzband at X. This order\nswitches with su\u000eciently strong interaction. At U=3.75\neV (Fig. 4(e)), the dxy-dominated bands are pushed\nabove their dxz/dyzcounterparts by the strongly orbital-\ndependent mass renormalization, as re\rected in the or-\nbital dependent quasiparticle spectral weights (Fig. 4(c)).\nThe mass renormalization is the largest for the dxyorbital\n(\u001810), and smaller for the dxz/dyzorbitals (\u00183), which\nis compatible with the low-temperature ARPES results\n(Fig. 1(e)). The temperature-induced crossover to the\nOSMP is clearly seen from the suppression of the spectral\nweights in the dxyorbital that accompanies the reduction\nof the weights in the other orbitals (Fig. 4(c),(e)-(g)), in\nagreement with the ARPES results (Fig. 3).\nThe temperature-induced nature of the crossover con-\nstrains these AFS superconductors to be very close to the\nboundary of the OSMP in the zero temperature ground\nstate, which is also the superconducting state. The best\nagreement between theory and experiments is achieved at\nU\u00183.75 eV. While the absolute value of this interaction\nis sensitive to the parameterization of the crystal levels\nand Hund's coupling, it is instructive to make a qualita-\ntive comparison with the case of the iron pnictides. The\nenhanced correlation e\u000bects for AFS tracks the reduction\nof the width of the (U=0) dxyband, which is about 0.7\nof its counterpart in 1111 iron arsenides.\nOne known concern for the AFS materials is the exis-\ntence of mesoscopic phase separation into superconduct-\ning and insulating regions [32], which would both con-\ntribute spectral intensity in ARPES data. From mea-\nsurement on an insulating RFS sample (Fig. S4(d)), we\nsee negligible spectral weight and no well-de\fned disper-\nsions within 0.1eV from EF, as expected for an insula-5\nFIG. 4: Theoretical calculation based on a \fve-orbital Hubbard model. (a),(b) Slave-spin mean-\feld phase diagrams of the\n\fve-orbital Hubbard model for KFS at J/U=0.15 and two electron densities. OSMP and MI refer to orbital-selective Mott phase\nand Mott insulator, respectively. (c) The evolution of the orbital-resolved quasiparticle spectral weight Z \u000bwith temperature at\nn=6.15, U=3.75eV, and J/U=0.15. (d)-(g) show the quasiparticle spectral functions along the \u0000-X direction at n=6.15. The\ncolor curves highlight the dominant orbital of the band, with dxz/dyzin red anddxyin blue.\ntor. Hence, the insulating regions in the superconduct-\ning compounds do not contribute spectral weight to the\nnear-EFenergy range, in which the temperature-induced\ncrossover is observed. We have also measured a KFS\nsample whose resistivity is intermediate between super-\nconducting and insulating (Fig. S4(b)), and was previ-\nously proposed to be semiconducting containing both\nmetallic and insulating phases [33]. Interestingly, its\ndxz/dyzbands, which must come from the metallic phase,\nappear further renormalized by a factor of 1.3 compared\nwith those of the superconducting compounds. In addi-\ntion, we resolve small but \fnite spectral weight for a very\nshallowdxyelectron band near the X-point (Fig. S5(c)).\nAs expected, the peak position is even closer to EF, con-\nsistent with additional renormalization for the shallow\nband near X, which is harder to discern with tempera-\nture dependent study. These observations are consistent\nwith the OSMP picture in that that the metallic phase in\nthis KFS compound is likely even closer to the boundary\nof OSMP at low temperatures from the mass-diverging\nbehavior of the dxybands. For the same interaction\nstrength, calculation from our model also identi\fes the\nlow temperature ground state of the superconducting, in-\ntermediate, and insulating phases to be located close to\nan OSMP, just at the boundary of an OSMP, and in a\nMI phase, respectively (see SI).\nWhile we cannot completely rule out alternative ex-\nplanations for the observations presented above, the\nconsistency of the totality of the observations-includingstrongly orbital dependent band renormalization for\ndxyversusdxz/dyzat both \u0000 and X points in the low\ntemperature metallic state, the non-trivial temperature-\ndependent spectral weight change for only the dxyband,\nsystematic doping dependence of the related e\u000bects in the\nintermediate and insulating compounds-and the theoret-\nical calculations makes this understanding a most likely\nscenario, suggesting that the superconductivity in this\nAFS family exists in close proximity to Mott behavior.\nWe thank V. Brouet, W. Ku, B. Moritz and I. Mazin\nfor helpful discussions. ARPES experiments were per-\nformed at the Stanford Synchrotron Radiation Light-\nsource and the Advanced Light Source, which are both\noperated by the O\u000ece of Basic Energy Science, U.S.\nDepartment of Energy. The work at Stanford is sup-\nported by DOE O\u000ece of Basic Energy Science, Division\nof Materials Science and Engineering, under contract DE-\nAC02-76SF00515. The work at Rice has been supported\nby NSF Grant DMR-1006985 and the Robert A. Welch\nFoundation Grant No. C-1411. The work at Houston is\nsupported in part by US Air Force O\u000ece of Scienti\fc Re-\nsearch contract FA9550-09-1-0656, and the state of Texas\nthrough the Texas Center for Superconductivity at the\nUniversity of Houston. MY thanks the NSF Graduate\nResearch Fellowship Program for \fnancial support.6\n[1] P.A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys.\n78, 17-85 (2006).\n[2] D.H. Lu et al: , Nature 455, 81 (2008).\n[3] W.L. Yang W. L. etal: , Phys. Rev. B 80, 014508 (2009).\n[4] M.M. Qazilbash etal: , Electronic correlations in the iron\npnictides. Nat. Phys. 5, 647 (2009).\n[5] Z.P. Yin, K. Haule, and G. Kotliar, Nat. Mat. 10, 932-935\n(2011).\n[6] T.J. Liu et al: , Nat. Mat. 9, 718-720 (2010).\n[7] J. Guoet al: , Phys. Rev. B 82, 180520(R) (2010).\n[8] A. Krzton-Maziopa etal: , J. Phys.: Condens. Matter 23,\n052203 (2011).\n[9] C.-H. Li, B. Shen, F. Han, X. Zhu, and H.-H. Wen, Phys.\nRev. B 83, 184521 (2011).\n[10] M.-H. Fang et al: , Europhys. Lett. 94, 27009 (2011).\n[11] H.-D. Wang et al: , Europhys. Lett. 93, 47004 (2011).\n[12] W. Bao et al: , Chin. Phys. Lett. 28, 086104 (2011).\n[13] M. Yi, et al: , PNAS 108, 6878 (2011).\n[14] A.V. Chubukov, Mat. Phys. 3, 52 (2012).\n[15] Q. Si, and E. Abrahams, Phys. Rev. Lett. 101, 076401\n(2008).\n[16] R. Yu, and Q. Si, Phys. Rev. B 84, 235115 (2011).\n[17] R. Yu, J.-X. Zhu, and Q. Si, Phys. Rev. Lett. 106, 186401\n(2011).\n[18] Y. Zhou, D.-H. Xu, F.-C. Zhang, and W.-Q. Chen, Eu-\nrophys. Lett. 95, 17003 (2011).\n[19] L. Craco, M.S. Laad, and S. Leoni, Phys. Rev. B 84,\n224520 (2011).\n[20] V. Anisimov et al: , Eur. Phys. J. B 25, 191 (2002).\n[21] L. de' Medici, S.R. Hassan, M. Capone, and X. Dai, Phys.\nRev. Lett. 102, 126401 (2009).\n[22] Y.-Z. You, F. Yang, S.-P. Kou, and Z.-Y. Weng, Phys.\nRev. B 84, 054527 (2011).\n[23] S. J. Moon et al: , Phys. Rev. B 81, 205114 (2010).\n[24] Y. Zhang et al: , Nat. Mat. 10, 273-277 (2011).\n[25] T. Qian et al: , Phys. Rev. Lett. 106, 187001 (2011).\n[26] D. Mou et al: , Phys. Rev. Lett. 106, 107001 (2011).\n[27] I.A. Nekrasov, and M.V. Sadovskii, JETP Lett. 93, 166-\n169 (2011).\n[28] S. Graser et al: , Phys. Rev. B 81, 214503 (2010).\n[29] R. Yu, and Q. Si, Phys. Rev. B 86, 085104 (2012).\n[30] L. de'Medici, L., A. Georges, and S. Biermann, Phys.\nRev. B 72, 205124 (2005).\n[31] R. Yu, and Q. Si, unpublished (2012).\n[32] W. Liet al: , Nat. Phys. 8, 126-130 (2012).\n[33] F. Chen et al: , Phys. Rev. X 1, 021020 (2011).\nSUPPLEMENTARY INFORMATION\nMaterials and methods\nSingle crystals of KFS were grown by the following\nmethod. Precursor FeSe was synthesized using Se powder\n(Alfa 99.999%) and Fe powder (99.998%) in a 1:1 molar\nratio. The reagents were weighed and placed in a 2mL\nalumina crucible. The quartz tube was sealed after be-\ning \rushed with argon and evacuated. The sealed quartz\ntube was placed in a furnace and heated from room tem-perature to 1050\u000eC in 12 hours. The furnace remained\nat 1050\u000eC for 12 hours, then was shut o\u000b and cooled to\nroom temperature. Single crystals of KFS were obtained\nby a self-\rux method with mixtures of K (Alfa, 99.95%),\nand FeSe in molar ratios of 1:3 respectively. Potassium\nhas a low boiling point so a small amount of additional\nK needs to be incorporated into the growth during syn-\nthesis. The reagents were weighed and placed in a 2mL\nalumina crucible, which was then sealed in a 2mL quartz\ntube after being \rushed with argon and evacuated. The\n2mL quartz tube was then placed into a larger 5ml quartz\ntube and sealed after again being \rushed with argon and\nevacuated. The double sealed quartz tube technique is\nemployed because potassium attacks quartz. The sealed\nquartz tubes were placed in a furnace and heated from\nroom temperature to 1040\u000eC over the course of 4 hours.\nAfter dwelling for 2 hours, the furnace was cooled to\n850\u000eC. The quartz tube was then removed from the fur-\nnace, rotated 180\u000eC, and spun in a centrifuge for a few\nseconds to separate the self-\rux from the single crystals.\nCrystals with dimensions up to approximately 3mm x\n3mm x 50mm could readily be extracted from the cru-\ncible. The crystals have a platelike morphology, with\nthe c-axis perpendicular to the plane. Two types of\nKFS were studied, one superconducting with TC on-\nset of 32K and chemical composition as K 0:76Fe1:72Se2,\nand the other non-superconducting with composition as\nK0:76Fe1:78Se2. Single crystals of RFS were grown as de-\nscribed elsewhere [1]. Two types of RFS were studied:\none superconducting with T Conset of 31K and chemi-\ncal composition Rb 0:93Fe1:70Se2; and the other insulating\nwith composition Rb 0:90Fe1:78Se2.\nAs the crystals are sensitive to air, all sample prepa-\nrations were preformed inside a nitrogen-\rushed glove\nbox. ARPES measurements were carried out at beam-\nline 10.0.1 of the Advanced Light Source and beamline\n5-4 of Stanford Synchrotron Radiation Lightsource using\nSCIENTA R4000 electron analyzers. The total energy\nresolution used was 25 meV or better and the angular\nresolution was 0.3\u000eC. Single crystals were cleaved in situ\nat low temperatures and measured in an ultra high vac-\nuum chamber with a base pressure better than 3 x10\u000011\nTorr.\nDiscussion of orbital assignment\nThe common band structure for FeSC in the Brillouin\nzone (BZ) corresponding to the two-Fe unit cell consists\nof three hole bands near the \u0000 point and two electron\nbands at the X point [2]. Under C 4rotational symmetry,\nthedxzanddyzbands are degenerate at the in-plane\nhigh symmetry points of the BZ-\u0000 and X points. At\nthe \u0000 point, this means that the dxzanddyzhole bands\nmust be degenerate. Hence we have assigned the two\nlower degenerate hole bands in Fig. 1e to dxz/dyzand the7\nhigher one to dxy. At the X point, C 4symmetry dictates\nthe bottom of the dxz/dyzelectron band and the top of\nthedxz/dyzhole band be degenerate. Since the shallow\nelectron band is separated from the hole band by more\nthan 70meV, it is unlikely to be the dxz/dyzelectron\nband, but is instead the dxyband.\nThere is in principle the possibility of an alternative\nunderstanding. As described in Ref. [5, 6], the alternat-\ning arsenic atoms about the iron plane may induce parity-\nswitching of certain orbitals when folding from the 1-Fe\nBZ to the 2-Fe BZ, which would swap the orbital charac-\nters of thedxzelectron band and the dxyelectron band at\nX along \u0000-X direction, making the originally dxzelectron\nband to be more observable under an odd polarization\n(Fig. 1c). Under this understanding, the shallow band is\nstill thedxzband rather than the dxyband. However, for\nthis understanding to hold, the aforementioned degener-\nacy of the dxz/dyzbands must be lifted to account for\nthe 70meV gap between the shallow electron band and\nthe hole band at X.\nThere are two mechanisms that could lift this degen-\neracy. One is spin-orbit coupling. However, in this com-\npound, the magnitude of the spin-orbit coupling is not\nstrong enough to cause such a large gap. Moreover, the\ne\u000bect of the spin-orbit coupling is stronger at the \u0000 than\nat the X point [3], and should cause a larger gap between\nthedxz/dyzhole bands at \u0000, which is not observed. The\nsecond mechanism is in-plane symmetry breaking, which\ncauses anisotropic shift of the dxz/dyzbands and hence\nlift the degeneracy of these two orbitals, as has been ob-\nserved in the orthorhombic state of BaFe 2As2[4]. Up to\ndate, there has not yet been any report of in-plane sym-\nmetry breaking in the AFS superconductors. In addition,\none should note that the orthorhombicity causes the ex-\nistence of natural twinning in the crystals. Hence in an\nunstressed crystal, the ARPES signal is a combination of\ndispersions from both types of domains, and would see\nbands from \u0000-X and \u0000-Y directions simultaneously, as\nwas observed in BaFe 2As2[4]. For the microscopically\nphase separated AFS superconductors, such twinning ef-\nfect should be even more noticeable than BaFe 2As2, but\nis not observed in the dispersions of AFS superconduc-\ntors.\nAnother factor that should be considered is the com-\nparison between \u0000 and X. The \rat dxyhole band ob-\nserved at \u0000 is renormalized by a factor of \u001810 compared\nto bare LDA while the dxz/dyzhole bands are renor-\nmalized by a factor of \u00183. When such kind of orbital-\ndependent renormalization is considered for the bands at\nX, it would bring the dxyelectron band to be shallower\nthan thedxz/dyzelectron bands, which is more consis-\ntent with the assignment of the shallower electron band\ntodxy. Further support for this assignment comes from\nthe general consideration that a stronger mass renormal-\nization would imply a stronger temperature dependence\nof the corresponding quasiparticle spectral weight, as hasbeen shown in the main text.\nTo summarize, while the possibility in principle exists\nfor other unknown mechanisms and alternative explana-\ntions, given the totality of the observations and consid-\nerations given above, the understanding of the shallow\nelectron band as dxyseems to be the most consistent\noverall.\nTemperature cycle test\nTo test that the disappearance of the dxyorbital spec-\ntral weight with raised temperature is not due to sample\naging, we have performed a temperature cycle test in\nwhich we cleaved the sample at 10K and took measure-\nments as temperature is raised up to 300K and cooled\nback down again. Spectra images for selected tempera-\ntures as well as the EDC at X point in this temperature\ncycle are shown in Fig. S1, where we see that the spec-\ntral weight for the dxyorbital dominated shallow elec-\ntron band diminishes at higher temperatures, and recov-\ners when cooled back down. This shows that the observed\ntemperature-induced crossover is not due to any surface\ne\u000bects and is a robust bulk phenomenon.\nResults on Rb xFe2\u0000ySe2\nWe have done similar measurements on superconduct-\ning RFS as discussed in the main text for superconduct-\ning KFS. Fig. S2 summarizes the main results. RFS has\nsimilar electronic structure as KFS, including Fermi sur-\nface, band dispersions and their orbital characters, specif-\nically there is a shallow electron band at X that is dxyand\na deep electron band that is dxz/dyz. Fig. S2d-f shows\nthe temperature dependence of the electron bands, equiv-\nalent to Fig. 2(a) for the KFS. Here again we see that\nthe shallow dxyband is present at low temperatures, and\nits spectral weight diminishes with raised temperature.\nSimilar quantitatively analysis is also done on the RFS.\nFig. S2(g) shows the EDC at X point taken in the geom-\netry of Fig. S2(d)-(f), taken from 10K to 200K. Similar\nto the case of KFS, the \u0018-0.05eV hump shaded by blue is\nthedxyband bottom, whose spectral weight diminishes\nwith raised temperature. The EDCs are again \ftted with\na Gaussian background and a Lorentzian peak, whose\nintegrated intensity as a function of temperature is plot-\nted in Fig. S2h. The method of integrating intensity in\nboxed regions is also done on RFS. Here two regions were\nchosen (Fig. S2d): blue box for a region dominated by\ndxyband, and magenta box for a region of mixed dxyand\ndxz. The background for each region is taken as a box of\nsame energy window away from dispersions, marked by\ndotted boxes in Fig. S2(d). The integrated intensity of\nthese two regions as a function of temperature is plotted\nin Fig. S2(i), showing the diminishing trend of dxyregion8\nFIG. S1: Temperature cycle test on K xFe2\u0000ySe2(a) Spectral\nimages across X point measured at selected temperatures in\nthe temperature cycle from 10K to 300K to 40K. Polarization\ngeometry and photon energy is the same as that in Fig. 2(a).\n(b) EDC at X point taken in the temperature cycle, where the\npeak around -0.05eV is the dxyband bottom, whose spectral\nweight diminishes up to 300K and recovers when cooled back\ndown.\nand the slower trend of the mixed region. These behav-\niors in RFS are very similar to those in KFS in the main\ntext.\nOrbital-selective Mott phase of the \fve-orbital\nHubbard model for K xFe2\u0000ySe2\nThe \fve-orbital Hubbard model is given by H=\nHkin+Hint, whereHkinandHintrespectively denote the\nkinetic and the on-site interaction parts of the Hamilto-\nnian.Hintcontains the intra- and inter-orbital Coulomb\nrepulsion, as well as the Hund's rule coupling and the pair\nhoppings.[10] The corresponding coupling strengths are\nrespectively U,U0, andJ, which satisfy U0=U\u00002J. [7]\nFor simplicity, we consider only the density-density inter-\nactions and neglect the spin-\rip and pair-hopping terms.The results including these terms are qualitatively the\nsame. The kinetic part Hkinis a tight-binding Hamil-\ntonian, and is conveniently speci\fed in the momentum\nspace. In FeSCs, each unit cell contains two Fe ions.\nHence, ideally the tight-binding Hamiltonian must be de-\n\fned in the BZ corresponding to this two-Fe unit cell.\nHowever, the lattice symmetry of the FeSC system al-\nlows us to work in an unfolded BZ corresponding to one-\nFe unit cell. Following Ref. [8] and notice that the ions\nare invariant under the transformation PzTxandPzTy,\nwhereTx(y)is the translation along x(y) direction by one\nFe-lattice spacing, and Pzrefers to the re\rection about\nthe Fe plane, we may de\fne a pseudocrystal momentum\n~kin the extended one-Fe BZ. This pseudocrystal mo-\nmentum and the conventional momentum kare related\nby~k=k+Qindxzanddyzorbitals (where Q= (\u0019;\u0019)),\nbut~k=kin other orbitals. In the extended BZ, the\ntight-binding Hamiltonian reads\nHkin=X\n~k\u000b\f\u001b[\u0018\u000b\f(~k) + (\u0001\u000b\u0000\u0016)\u000e\u000b\f]dy\n~k\u000b\u001bd~k\f\u001b:(1)\nHere\u0018\u000b\f(~k) is the hopping matrix in the momentum\nspace associated with orbitals \u000band\f, \u0001\u000bis the on-site\nenergy re\recting the crystal \feld splitting, \u0016is the chem-\nical potential, and \u000e\u000b\fis the Kronecker's delta function.\nThe expression of \u0018\u000b\f(~k) is given in Ref. [9], and it has\nthe same form as appeared in the appendix of Ref. [2].\nWe adopt the tight-binding parameters of Ref. [9], where\nthey are obtained by \ftting the LDA band structure of\nKFS. To better \ft the LDA results, we have further tuned\nseveral hopping parameters by hand from their values in\nRef. [9]. Using the same notation as in Ref. [2], the tight-\nbinding parameters used in this paper for KFS are listed\nin Table I. The chemical potential corresponding to the\nelectron \flling n=6.15 is\u0016=-0.365 eV.\nThe \fve-orbital Hubbard model is studied using the re-\ncently developed U(1) slave-spin mean-\feld method [10].\nIn this method, a slave quantum S=1/2 spin is introduced\nto carry the charge degree of freedom, and the spin of the\nelectron is carried by a fermionic spinon. This approach\ndetermines the quasiparticle spectral weight Z\u000bin each\norbital. An orbital \u000bis delocalized when Z\u000b>0, but\nbecomes Mott localized if Z\u000b= 0.\nTo compare with the ARPES data, the coherent part\nof the orbital resolved spectral function A\u000b(k;E) in the\nfolded BZ corresponding to the two-Fe unit cell is fur-\nther calculated by convoluting the slave-spin and spinon\nGreen's functions and writing them in the Lehmann rep-\nresentation via the following formula\nA\u000b(k;E) =X\n\u000b\u0017Z\u000bjU\u000b\u0017\nkj2\u000e(E\u0000\"\u0017k): (2)\nHere,\"\u0017kandU\u000b\u0017\nkare respectively the \u0017's eigenenergy\nand eigenvector of the hopping matrix \u0018\u000b\f(k). This al-9\nFIG. S2: (a) Fermi surface measured at 10K, with integration window of 20meV about Fermi level. (b) Spectral image and\n(c) second derivative measured along \u0000-X. Blue (green) curves trace the observed dxy(dyz) dispersions observed under this\npolarization. (a)-(c) were measured with 47.5eV photons. (d)-(f) Spectral images measured across X in the same geometry\nas that in Fig. 2(a) for selected temperatures. (g) EDC at X measured in the same geometry as that for (d)-(f) for selected\ntemperatures. Each curve is \ftted to a Gaussian (dotted lines) for the background feature around -0.15eV and a Lorentzian\n(shaded region) for the feature around -0.05eV. (h) The integrated Lorentzian spectral weight is plotted as a function of\ntemperature. (i) Temperature-dependence of the intensity in the regions marked by colored boxes in (d). The blue region has\ndominant spectral weight from the dxyband, whereas the magenta region is of mixed dxyanddxz.\n\u000b= 1\u000b= 2\u000b= 3\u000b= 4\u000b= 5\n\u0001\u000b -0.36559 -0.36559 -0.56466 -0.05096 -0.91583\nt\u000b\u000b\n\u0016\u0016=x \u0016 =y \u0016 =xy \u0016 =xx \u0016 =xxy \u0016 =xyy \u0016 =xxyy\n\u000b= 1 -0.11475 -0.38868 0.20881 -0.04557 -0.00866 -0.03143 0.01899\n\u000b= 3 0.32523 -0.09783 -0.00537\n\u000b= 4 0.20633 0.09682 -0.07525 -0.02189 0.00423\n\u000b= 5 -0.0427 0.01117 0.00177 -0.01349\nt\u000b\f\n\u0016\u0016=x \u0016 =xy \u0016 =xxy \u0016 =xxyy\n\u000b\f= 12 0.10161 -0.02017 0.03273\n\u000b\f= 13 -0.31447 0.06225 0.0103\n\u000b\f= 14 0.13785 -0.03105 0.0104\n\u000b\f= 15 -0.04825 -0.10096 -0.01204\n\u000b\f= 34 -0.04795\n\u000b\f= 35 -0.30966 -0.01498\n\u000b\f= 45 -0.08359 -0.00766\nTABLE I: Tight-binding parameters of the \fve-orbital model for K xFe2\u0000ySe2. Here we use the same notation as in Ref [2]. The\norbital index \u000b=1,2,3,4,5 correspond to dxz,dyz,dx2\u0000y2,dxy, andd3z2\u0000r2orbitals, respectively. The units of the parameters\nare eV.\nlows us to determine the orbital character of each band\nnear the Fermi level. In Fig. S3, we show the spectral\nfunctions near the Fermi level in the \fve-orbital model\nat n=6.15 and T=10 K for three di\u000berent U values. Com-\npared to the U=0 band structure, the bands are stronglyrenormalized by the interactions. Moreover, with increas-\ning U, the spectral weight of the bands with a dxyorbital\ncharacter is reduced (Fig. S3(b)) and eventually goes\naway (Fig. S3(c)). The bands with the dxz/dyzchar-\nacter, on the other hand, still have nonzero spectral10\nFIG. S3: Calculated quasiparticle spectral weight functions for various U. (a) Orbitally resolved quasiparticle spectral weight\nfunctions along the \u0000 to X direction of the Brillouin zone at J/U=0.15, n=6.15, T=10 K, and three di\u000berent U values (in three\ncolumns). The total spectral weights contributed from all \fve orbitals are shown in the \frst row; the spectral weights from the\ndxyanddxz+dyzorbitals are respectively shown in the second and the third row.\nweights. This behavior clearly indicates an interaction\ndriven transition to an OSMP.\nSeveral factors favor stabilizing the OSMP.[12] Firstly,\nfrom the orbitally-projected density of states of the non-\ninteracting bands, the width of the dxyorbital is nar-\nrower than that of the other Fe 3d orbitals, especially\nthedxz/dyzorbitals. This factor recalls the mechanism\nfor the OSMP initially proposed for the Ca 2\u0000xSrxRuO 4\nsystem [13]. Secondly, when the Hund's coupling is suf-\n\fciently strong compared to the (nonzero) splitting be-\ntween thedxyanddxz/dyzorbitals, the high-spin con\fg-\nuration is favored, and, due to the crystal level splitting,\nthedxyorbital is non-degenerate and located at a higher\nenergy than other orbitals. As a result, for a range of\ndensities, the dxyorbital is kept at half-\flling while the\ndxz/dyzorbitals are more than (though still close to) half\n\flled. The non-degenerate dxyorbital has a lower repul-\nsion threshold for the Mott transition than the doubly de-\ngeneratedxz/dyzorbitals within a wide range of Hund's\ncoupling. This picture has some connection with the one\nstudied in a di\u000berent regime for a three-orbital model\naway from half-\flling [14]. As a combined e\u000bect of these\nfactors, the dxyorbital is more strongly localized than\nthedxz/dyzorbitals.Comparison of insulating, intermediate and\nsuperconducting compounds\nHere, we compare the measured electronic structure\nof three kinds of phases in KFS and RFS. The four\ncompounds shown in Fig. S4 have compositions deter-\nmined by energy dispersive X-ray spectroscopy to be (a)\nK0:76Fe1:72Se2, (b)K0:76Fe1:78Se2, (c) Rb 0:93Fe1:70Se2,\nand (d) K 0:90Fe1:78Se2. From resistivity data shown,\n(a) and (c) are superconducting, (d) is insulating, and\n(b) shows a behavior somewhat between superconducting\nand insulating, which we shall call intermediate. Its re-\nsistivity is most like the compound suggested to be semi-\nconducting in a previous report [15], which suggests that\nboth insulating and metallic regions exist in these sam-\nples. All compounds were measured in the same experi-\nmental geometry. The two superconducting samples ex-\nhibit similar dispersions as discussed in previous section.\nThe insulating sample shows negligible spectral weight\ntowardsEFand no well de\fned dispersions near EF, as\nexpected of an insulator. The interesting case is the inter-\nmediate sample, which has resolvable dispersions, but dif-\nferent from that of the superconducting samples. Firstly,\nwe see that the most well-resolved band is the dyzband,\nwhich is traced in green in the second derivative plot in\nthe bottom panel. Its bandwidth is approximately renor-\nmalized by a factor of 1.3 compared to its equivalent in\nthe superconducting samples. Secondly, from the EDC\ntaken at the X point from both the superconducting KFS\nand the intermediate KFS (Fig. S5(b)-(c)), we see that11\nFIG. S4: Comparison of superconducting, intermediate and insulating compounds (a) Resistivity (top), ARPES spectral images\non the \u0000-X high symmetry cut (middle) and its second derivative plot (bottom) for superconducting K 0:76Fe1:72Se2, taken at\n10K. Same measurements are shown for (b) intermediate K0:76Fe1:78Se2at 10K, (c) superconducting Rb 0:93Fe1:70Se2at 10K,\nand (d) insulating K 0:90Fe1:78Se2at 50K. All ARPES data were taken with 26eV photons in the same polarization geometry\nas that for Fig. 1(d).\nFIG. S5: Comparison of superconducting and intermediate\nKFS (a) Calculated orbital-resolved quasiparticle spectral\nweight for U=3.75eV and T=30K as a function of electron\n\flling, n, for dxz/dyz(red squares) and dxy(blue triangles).\n(b) Measured EDC at X (black solid curve) from supercon-\nducting KFS from Fig. S4(a), \ftted to a Gaussian plus a\nLorentzian, plotted in red and blue, respectively. (c) Same as\n(b) but for the intermediate KFS sample from Fig. S4b, with\nthe EDC from the superconducting sample renormalized by\n1.3 in energy for comparison (gray dotted curve). Blue arrows\npoint to the positions of the dxyband bottom.\nthe peak around -0.05eV indicating the dxyelectron band\nbottom in the superconducting sample (Fig. S5(b)) be-\ncomes a smaller but discernible shoulder closer to EFin\nthe intermediate sample (Fig. S5(c)). Even after a renor-malization of 1.3 to account for the increased renormal-\nization of the dyzband (gray dotted line in Fig. S5(c)) as\ndiscussed above, we see that the dxyband in the inter-\nmediate compound still needs a further renormalization\ngoing from the superconducting to the intermediate sam-\nple (Fig. S5(c)). Also, the relative spectral weight of the\ndxyorbital to that of dyzis much more reduced in the\nintermediate sample. Both the further renormalization\nof the bandwidth and the reduction of spectral weight\nof thedxyband indicate that the intermediate sample\nis even closer to the OSMP than the superconducting\ncompounds at low temperatures.\nAssuming the same interaction strengths, the various\nphases in the KFS and RFS compounds can be under-\nstood in terms of an interplay between the electron dop-\ning and vacancy order. In the vacancy disordered case,\nour calculation (Fig. S5(a)) identi\fes a doping-induced\ntransition to an OSMP near n=6.02 per Fe. Hence the\nundoped system with n=6 is already in an OSMP. The\nvacancy order further drives it through a Mott transition\nin all orbitals to a Mott insulator [16, 17]. This accounts\nfor the absence of spectral weights near EFin the in-\nsulating RFS. For the superconducting samples and the\nintermediate KFS compound, the phases giving ARPES\nsignals are likely to be vacancy disordered. We therefore\ninterpret them as corresponding to two vacancy disor-\ndered phases at two di\u000berent electron densities (n=6.15-\n6.25 for the superconducting samples, and n=6.05-6.10\nfor the metallic phase in the intermediate KFS). In addi-\ntion, we emphasize that the high temperature state of the12\nsuperconducting region is intrinsically di\u000berent from the\nvacancy-ordered insulating phase. Rather, the supercon-\nducting, intermediate, and insulating phases likely have\nincreasing correlation as they may be located close to an\nOSMP, just at the boundary of an OSMP, and in a Mott\ninsulating phase, respectively.\n[1] M. Gooch et al: , Phys. Rev. B 84, 184517 (2011).\n[2] S. Graser, T.A. Maier, P.J. Hirschfeld, and D.J.\nScalapino, New J. Phys. 11, 025016 (2009).\n[3] I.I. Mazin. private communications.\n[4] M. Yiet al: , PNAS 108, 6878 (2011).\n[5] C.-H. Lin et al: , Phys. Rev. Lett. 107, 257001 (2011).\n[6] V. Brouet et al: , arXiv:1205.4513.[7] C. Castellani, C.R. Natoli, and J. Ranninger, Phys. Rev.\nB 18, 4945 (1978).\n[8] X.-G. Wen, and P.-A. Lee, Phys. Rev. B 78, 144517\n(2008).\n[9] R. Yuet al: , arXiv:1103.3259.\n[10] R. Yu, and Q. Si, Phys. Rev. B 86, 085104 (2012).\n[11] R. Yu, and Q. Si, Phys. Rev. B 84, 235115 (2011).\n[12] R. Yu, and Q. Si, unpublished (2012).\n[13] V. Anisimov et al: , Eur. Phys. J. B 25, 191 (2002).\n[14] L. de' Medici, S.R. Hassan, M. Capone, and X. Dai, Phys.\nRev. Lett. 102, 126401 (2009).\n[15] F. Chen et al: , Phys. Rev. X 1, 021020 (2011).\n[16] R. Yu, J.-X. Zhu, and Q. Si, Phys. Rev. Lett. 106, 186401\n(2011).\n[17] L. Craco, M.S. Laad, and S. Leoni, Phys. Rev. B 84,\n224520 (2011)."    },    {        "title": "2205.03172v1.Mapping_the_energy_time_landscape_of_spins_with_helical_X_rays.pdf",        "content": "Mapping the energy-time landscape of spins with helical X-rays\nN. Pontius1, J. K. Dewhurst2, C. Sch ussler-Langeheine1, S. Jana3, C.\nv. Kor\u000b Schmising3, S. Eisebitt3, S. Shallcross3, and S. Sharma3\u0003\n1 Helmholtz-Zentrum Berlin f ur Materialien und Energie GmbH,\nAlbert-Einstein-Stra\u0019e 15, 12489 Berlin, Germany\n2 Max-Planck-Institut fur Mikrostrukturphysik Weinberg 2, D-06120 Halle, Germany and\n3 Max-Born-Institute for Non-linear Optics and Short Pulse Spectroscopy, Max-Born Strasse 2A, 12489 Berlin, Germany\n(Dated: May 9, 2022)\nUnveiling the key mechanisms that determine optically driven spin dynamics is essential both to\nprobe the fundamental nature of ultrafast light-matter interactions, but also to drive future tech-\nnologies of smaller, faster, and more energy e\u000ecient devices. Essential to this task is the ability to\nuse experimental spectroscopic tools to evidence the underlying energy- and spin-resolved dynamics\nof non-equilibrium electron occupations. In this joint theory and experimental work, we demon-\nstrate that ultrafast helicity-dependent soft X-ray absorption spectroscopy (HXAS) allows access to\nspin-, time- and energy speci\fc state occupation after optical excitation. We apply this method to\nthe prototype transition metal ferromagnet cobalt and \fnd convincing agreement between theory\nand experiment. The richly structured energy-resolved spin dynamics unveil the subtle interplay\nand characteristic time scales of optical excitation and spin-orbit induced spin-\rip transitions in this\nmaterial: the spin moment integrated in an energy window below the Fermi level \frst exhibits an\nultrafast increase as minority carriers are excited by the laser pulse, before it is reduced as spin-\rip\nprocess in highly localized, low energy states start to dominate. The results of this study demon-\nstrate the power of element speci\fc transient HXAS, placing it as a potential new tool for identifying\nand determining the role of fundamental processes in optically driven spin dynamics in magnetic\nmaterials.\nI. INTRODUCTION\nThe macroscopic spin and charge state of solids can\nbe manipulated by light pulses on few femtosecond1,2to\npicosecond time scales3{6. A wealth of fundamental new\nphenomena have been uncovered in this \\femtoscape\",\nincluding all-optical magnetization reversal7,8and laser-\ndriven spin transport9{11, e\u000bects that carry profound im-\nplications for technological application. Progress in the\n\felds of femtomagnetism, ultrafast valleytronics12and\nultrafast spintronics, ultimately rely on the ability of ex-\nperiments to decode the complex and non-equilibrium\npopulation of spin and charge on ultrafast time scales.\nParticularly important for building a microscopic under-\nstanding of such processes is the ability to obtain, from\nspectroscopy experiments, a state and element-selective\nview of the spin and charge dynamics, i.e., at what ener-\ngies, and from which atoms, is spin optically excited to\nand from during the time evolving magnetic state.\nSuch charge and spin excitations are determined by\nthe pump pulse parameters and the nature of the mag-\nnetic material, as encoded in the spin-resolved density of\nstates (DOS). In a magnetic material possessing nearly\nfull majority bands the dominant available optical exci-\ntations are from minority states below to minority states\nabove the Fermi level ( EF), a process that leaves the\nmacroscopic magnetic state unchanged. Concurrently\noccurring spin-\rip transitions will, however, reoccupy\nthese minority states leading to the observed ultrafast\ndemagnetization13,14. In alloys and hetero-structures dif-\nferences between the local DOS of distinct elemental\nspecies may themselves result in the laser induced trans-fer of spin from one atomic site to the other, adding a\nfurther layer of complexity to the spin dynamics2,11,15{18.\nAssessing the role these basic principles of optical ex-\ncitation and spin \rips play in the complex landscape\nof ultrafast demagnetization is of crucial importance for\nunderstanding, and possibly controlling, the remarkable\nphysics of ultrafast magnetization dynamics.\nTo unravel the complex pathways of non-equilibrium\ncharge within a site-, energy- and spin dependent\nlandscape during ultrafast demagnetization, requires\ntechniques capable of resolving these degrees of free-\ndom. Some of this information can be extracted from\nspin and time dependent angle-resolved photoemission\nspectroscopy19{22, which essentially measures the tran-\nsient electronic spectrum. However, such experiments\nare not in general element speci\fc and, in addition,\nare strongest when probing bands and momentum di-\nrections on surface/interface and around the Fermi en-\nergy, not when providing a global picture (i.e. fully en-\nergy resolved spectrum of the entire bulk of the solid) of\nspin and charge excitation. Magnetic circular dichroism\n(MCD) experiments, on the other hand, provide an ele-\nment speci\fc tool that readily o\u000bers access to transient\nmagnetization23,24separated into the contribution from\nspin and orbital moments25{30. Such experiments typi-\ncally employ the XMCD contrast at \fxed photon energy\nto probe magnetization dynamics, with recently a few ex-\nperiments investigating spectral changes in absorption as\na possible route to probe energy dependent magnetiza-\ntion dynamics16,31{34. While intriguing and as yet unex-\nplained e\u000bects have been observed, a direct link to tran-\nsient energy and spin-dependent electron occupations has\nnot been established.arXiv:2205.03172v1  [cond-mat.mtrl-sci]  6 May 20222\nIn this joint theory and experimental work, we demon-\nstrate that transient helicity-dependent X-ray absorption\nspectroscopy (HXAS), the \\raw data\" of XMCD exper-\niments that is typically discarded, encodes detailed in-\nformation of the energy- spin- and species-resolved elec-\ntronic transitions that occur both during the optical ex-\ncitation as well as in the subsequent relaxation processes.\nIn order to extract this information we provide a method\nbased on well founded assumptions of the underlying op-\ntical transitions, requiring for its facilitation the ab-initio\ndetermination of 4 parameters per material system. Em-\nploying this tool and with an example of Co, we unveil\nthe energy and spin resolved dynamics of laser pumped\ndemagnetization, revealing the pathways in energy-spin\nspace in which the ultrafast loss of moment occurs. We\n\fnd a richly structured energy resolved spin dynamics\nwith, remarkably, the spin moment below the Fermi level\n\frst exhibiting an increase after the laser pulse (due to\nexcitation of minority charge), before falling as spin-\rip\nprocess in highly localized low energy states dominate.\nIn contrast, the magnetic moment above Fermi energy\nshows a continuous increase in magnitude (but with the\nmoment pointing in the opposite direction) from the out-\nset of the laser pulse.\nII. HXAS, XMCD AND GOING BEYOND\nWhen circular polarized X-ray or XUV photons are\ntransmitted through a magnetic material the absorption\nstrength of excitations from spin-orbit split core levels\nto the valence bands depends on the sign of the helic-\nity, i.e. the material shows circular dichroism. A speci\fc\nexample of the absorption of circular polarized X-rays is\ntheL3resonance (excitations from the 2 p3=2core level\nto the 3dvalence band) in ferromagnetic Cobalt, illus-\ntrated in Fig. 1a. This response function represents a\nmeasure of magnetic moment of the material: light of\npositive helicity primarily couples to spin-up (majority\nspin) electrons (leading to a response function we call\nH+(E)), while light of negative helicity couples predom-\ninantly to spin-down (minority spin) electrons (leading\nto a response function we call H\u0000(E))27. The absorp-\ntion probability for a particular spin channel depends\nadditionally on the available unoccupied valence states\nof the respective spin. This allows for \\counting\" of\navailable valence states of speci\fc spin, which is a direct\nmeasure of the spin polarization in the case of isolated\natoms. To obtain this spin polarization from H+(E) and\nH\u0000(E) for a solid one uses the normalized di\u000berence be-\ntween both response functions, which yields the X-ray\ncircular dichroism (XMCD) spectrum, together with the\nXMCD sum-rules (i.e. integration of this spectrum over\ntheL3and theL2resonances in case of 3 delements like\nCo)25{27,35. Element speci\fcity of the XMCD response\ncomes from the fact that the energies of these core levels\nstrongly vary with the atomic species and can therefore\nbe energetically separated.\nFigure 1. (a) Schematic illustration of helicity dependent\nX-ray absorption (HXAS) for the L 3absorption resonance\nin Cobalt in which electrons are promoted from the 2 p3=2\ncore level into the spin polarized 3 dunoccupied valence lev-\nels. Light of positive helicity yields the yellow ( H+) XAS,\nnegative helicity the blue ( X\u0000) XAS. (b) Helicity dependent\ntransient X-ray absorption. Laser induced changes to the spin\ndependent state occupation \u0001DOS, shown for majority (\u0001 u)\nand minority (\u0001 d) channels, are re\rected in changes to the\nhelicity dependent X-ray absorption \u0001XAS, with these two\nconnected through the spin dependent excitation probabili-\ntiesA,B,A0, and B0. Integration over the coloured energy\nregions yields the time, energy, and spin resolved changes in\nstate occupation and XAS, shown on the right hand side. (c)\nIllustration of the spin-orbit spin \rip process in ferromagnetic\nCobalt after optical excitation. The stronger optical laser ex-\ncitation of the minority electrons creates localized holes be-\nlowEF, enhancing subsequent spin-orbit coupling induced\ntransitions, in turn leading to an increased spin \rip rate and\ndemagnetization at early times.\nIn the transient case occupations of bands above and\nbelow the Fermi energy evolve as charge is excited by\nthe laser pump pulse (compare Fig. 1b). Subsequent dy-\nnamical processes such as spin-orbit scattering, reduce\nthe local spin moments and so further change these oc-\ncupations. The XMCD spectra in combination with the\nsum rules35can then be employed to obtain the resulting\ndynamics of the magnetic moment. This procedure has\nbeen enormously successful, and has been used to map\nearly time spin dynamics in a number of magnets includ-\ning species resolved magnetic order in multi-component\nmagnets8,28{30. However, what is not readily accessible,\nyet is of great importance, is information concerning en-\nergy and time dependent changes of a speci\fc spin chan-\nnel (and not just the integrated di\u000berence as represented\nby the magnetic moment).\nIn obtaining magnetization dynamics from the XMCD\nspectra, the individual response functions to helical light,\nH+andH\u0000, are discarded. As we will now show, hid-3\nden within H+andH\u0000lies detailed information on spin\nand energy resolved changes in the electronic structure\nof dynamical magnetic systems.\nThe helicity dependent absorption can be written as:\nH\u0006(E;t) =H\u0006\n\"(E;t) +H\u0006\n#(E;t)\n==X\ni;f\"\np\u0006\ni;f;\"(t)(ni;\"\u0000nf;\")(t)\n\u0001\u000fif;\"(t)\u0000E+i\u0011\n+p\u0006\ni;f;#(t)(ni;#\u0000nf;#)(t)\n\u0001\u000fif;#(t)\u0000E+i\u0011#\n(1)\nwhere in the \frst line we have decomposed the helicity\ndependent absorption into the spin projected responses\nH\u0006\n\u001b(E;t), and in the second line give the corresponding\nlinear response formulae. In these latter equations iand\nfare joint indices of state and k-point and represent the\ninitial and the \fnal state respectively, \"and#refer to\nthe majority and minority spins, and p\u0006\nifare the transi-\ntion probabilities for transitions from state itofwith\nleft/right circularly polarized light. \u000fiis the energy and\nnithe occupation of the i'th state. In the simplifying\nlimit of a two state model, Eq. 1 can be written as\na(t)u(E;t) +b(t)d(E;t) =H+(E;t) (2)\na0(t)u(E;t) +b0(t)d(E;t) =H\u0000(E;t);\ni.e. in a way that separates the transition element infor-\nmation from that of state occupation and DOS (with\na(t) =p+\nif\",b(t) =p+\nif#,a0(t) =p\u0000\nif\",b0(t) =p\u0000\nif#,\nandu(E;t) ==(ni\"\u0000nf\")=(\u0001\u000fif;\"\u0000E+i\u0011)), and\nd(E;t) ==(ni#\u0000nf#)=(\u0001\u000fif;#\u0000E+i\u0011)). The latter\ntwo quantities, u(E;d) andd(E;d), represent the DOS\nand occupation from which transient spin resolved charge\ncan be extracted.\nFor a realistic solid such a decomposition cannot be rig-\norously made. However, as we now show ab-initio theory\nallows us to obtain an approximate scheme providing a\ncorresponding decomposition for solids. To this end we\nconsider the spin projected helicity dependent XAS spec-\ntra (H\u0006\n\u001b(E;t)), i.e. the XAS spectra with majority or\nminority spin transitions blocked, see Fig. 2. We now\nmake the assumption (to be numerically veri\fed later)\nthat an approximation to the average value over a band\nof the ratios of transition probabilities p\u0006\nif\u001b=\u0010P\n\u001bp\u0006\nif\u001b\u0011\ncan be obtained by integration over the L3 edge the ratio\nH\u0006\n\u001b=H\u0006. With these ratios in hand an approximation to\nEq. 2 can be written as\nAu(E;t) +Bd(E;t) =H+(E;t)R\ndEH+(E;0)(3)\nA0u(E;t) +B0d(E;t) =H\u0000(E;t)R\ndEH\u0000(E;0);where we have employed a second assumption in neglect-\ning any time dependence of these ratios (again, to be val-\nidated later by numerical calculation). They thus may be\nobtained from the ground state as:\nA=Z\ndEH+\n\"(E;0)=Z\ndEH+(E;0); (4)\nB=Z\ndEH+\n#(E;0)=Z\ndEH+(E;0);\nA0=Z\ndEH\u0000\n\"(E;0)=Z\ndEH\u0000(E;0);\nB0=Z\ndEH+\n#(E;0)=Z\ndEH\u0000(E;0):\nwhere the energy integration is again over the L3 edge.\nIn this way we can identify the parts of the response\nfunction that come primarily from the bands and their\noccupation (the u(E;t) andd(E;t)) separately from the\nparts that comes from the transition probabilities ( A,B,\nA0andB0). From the transient-HXAS response func-\ntions of laser pumped systems Eq. 3 can then be solved\ntransiently to obtain u(E;t) andd(E;t), a procedure\nschematically illustrated in Fig.1b. This will therefore\nprovide us with the required information: changes in ma-\njority (u) and minority (d) charge as a function of energy\nand time.\nIn Fig. 2 we shown these response functions and their\nenergy integration over the L3 edge for Cobalt. The cal-\nculated values of these parameters of Co found from these\nresponse functions are A= 0:846,B= 0:154,A0= 0:646\nandB0= 0:355. For the corresponding values for Fe and\nNi see 36. In the following sections we now describe the\ntheoretical and experimental method used to determine\nthe dynamics of magnetization from the HXAS spectra.\nIII. THEORETICAL METHODOLOGY\nTo calculate transient HXAS for laser pumped Co we\nhave used the fully ab-initio state-of-the-art time depen-\ndent density functional theory (TDDFT)37, that rigor-\nously maps the computationally intractable problem of\ninteracting electrons to the Kohn-Sham (KS) system of\nnon-interacting fermions in a \fctitious potential, which\ncan be solved by modern computing clusters. The time-\ndependent KS equation is:\ni@ jk(r;t)\n@t=\"\n1\n2\u0012\n\u0000ir+1\ncAext(t)\u00132\n+vs(r;t)\n+1\n2c\u001b\u0001Bs(r;t) +1\n4c2\u001b\u0001(rvs(r;t)\u0002\u0000ir)#\n jk(r;t)\n(5)\nwhere Aext(t) is a vector potential representing the ap-\nplied laser \feld. It is assumed that the wavelength of4\n774 776 778 780 782 784\nEnergy(eV)00.0050.010.015absorption (arb. units)H+\ntotal\nH+\nH+\nFigure 2. Helicity dependent x-ray absorption spectra from\nright circularly polarized light for Cobalt for the L edge.\nThe response function is calculated in three di\u000berent ways:\nby blocking all majority 2 pstates ( H+\nmin), by blocking all\nminority 2 p-states ( H+\nmaj) and by including all transitions\nfrom core levels ( H+\ntotal, the full response), with the cor-\nresponding dashed lines the energy integration of these re-\nsponse functions. The parameters for extracting spin re-\nsponse functions from the response functions of circularly po-\nlarized light are given by the ratios of the integrated quantitiesR\ndE H\u0006\n\u001b(E)=R\ndE H\u0006(E) over the L3 edge.\nthe applied laser is much greater than the size of a unit\ncell and the dipole approximation can be used, i.e. the\nspatial dependence of the vector potential is disregarded.\nThe KS potential vs(r;t) =vext(r;t) +vH(r;t) +vxc(r;t)\nis decomposed into the external potential vext, the classi-\ncal electrostatic Hartree potential vHand the exchange-\ncorrelation (XC) potential vxc. Similarly, the KS mag-\nnetic \feld is written as Bs(r;t) =Bext(t) +Bxc(r;t)\nwhere Bext(t) is an external magnetic \feld and Bxc(r;t)\nis the exchange-correlation (XC) magnetic \feld. \u001bare the\nPauli matrices. The \fnal term of Eq. (5) is the spin-orbit\ncoupling term. In the fully non-collinear spin-dependent\nversion of this theory13,38the orbitals,  are two compo-\nnent Pauli spinors and using these orbitals magnetisation\ndensity can be calculated as:\nm(r;t) =X\nj \u0003\nj(r;t)\u001b j(r;t); (6)\nthe integral of this vector \feld over space gives the mag-\nnetic moment as a function of time (M(t)).\nIn order to calculate the response function at various\ntimes, linear response formalism of the TD-DFT is the\nused37,39,40. This method of calculating the transient re-\nsponse function has been studied extensively41{43, and\ndemonstrated to be in very good agreement with experi-\nment. For the response function calculations a broaden-\ning of 0.9 eV was used. This choice of smearing is based\non ground-state GW calculations, being the average of\nthe width of p 1=2and p 3=2states. Ground-state GWcalculations were also used to determine the position of\nthe 2p-states and the KS states were then scissors cor-\nrected. These scissors corrected states are used for per-\nforming response function calculations. For performing\ntheseGW calculations a \fnite temperature (a tempera-\nture of 500 K was used), all electron, spin-polarized GW\nmethod44is used, where the spectral function on the real\naxis is constructed using a Pade approximation. Spin-\norbit coupling was included in the GW calculations and\na Matsubara cuto\u000b of 12 Ha was used.\nIn the present work we have used adiabatic local den-\nsity approximation for XC potential and all calcula-\ntions are performed using the highly accurate full poten-\ntial linearized augmented-plane-wave method45, as im-\nplemented in the ELK38,46code. A face centred cubic\nunit cell with lattice parameter of 3 :53/RingA was used for Co.\nThe Brillouin zone was sampled with a 20 \u000220\u000220 k-\npoint mesh. For time propagation the algorithm detailed\nin Ref. 38 was used with a time-step of 2 :42 atto-seconds.\nThe \fnal magnetization value converges with the above\nmentioned computational parameters to 1.61 \u0016B. The\ntheoretical pump pulse used in the present work has cen-\ntral frequency of 1.55 eV, duration of 24 fs and incident\n\ruence of 6.7 mJ/cm2. In comparison to the experimen-\ntal pulse this is of shorter duration, a situation motivated\nby several factors: (a) non-inclusion of nuclear dynamics\nprevents us from describing physics at longer time scales\nand (b) while the method proposed above is state-of-the-\nart, there is a price to pay in computational burden that,\nwith present computer power, precludes the calculation\nof longer pulses.\nIV. EXPERIMENTAL METHOD\nThe ultrafast HXAS experiments were conducted at\nthe BESSY II Femtoslicing facility (beamline UE56/1-\nZPM and DynaMaX end-station) at Helmholtz-Zentrum\nBerlin47. Femtosecond X-ray pulses of 100 fs temporal\nwidth and elliptical polarization are generated via femto-\nslicing from electron bunches stored in the BESSY II\nelectron storage ring by interaction with intense infrared\nlaser pulses. Laser pulses from a second laser ampli\fer\nsystem (pump-laser) intrinsically synchronized with the\nfs-X-ray pulses are used as pump-pulses for sample exci-\ntation (wavelength 800 nm and pulse width 50 fs). The\ne\u000bective temporal resolution of this experimental setup\nis 120 fs. The experiment is operated at a repetition\nrate of 6 kHz, where only every second fs-X-ray pulse\nis synchronized to a pump-pulse (i.e. pump-laser run-\nning at 3 kHz). This allows for alternatingly probing the\nexcited sample and the sample ground state on a shot-\nto-shot basis, providing excellent normalization opportu-\nnities. X-ray and pump-laser beams enclose an angle of\nabout 1\u000ewhen incident on the sample. The sample was a\nPt(3 nm)/Co(15 nm)/Al(300 nm) layer stack deposited\non a 200 nm thin Si 3N4membrane by magnetron sputter-\ning, which allowed measurement of XAS spectra across5\nthe Co L 3resonance in transmission geometry. The X-\nray photons hit the sample at 43\u000eincidence angle to sur-\nface normal. The transmitted intensity is detected by\nan avalanche photo diode screened from the laser light\nby an Al membrane47. The relative orientation between\nX-ray helicity and magnetization was swapped by inver-\nsion of an external magnetic \feld (0.1 T \feld amplitude)\npointing parallel to the X-ray beam. The X-ray absorp-\ntion spectra were recorded quasi-simultaneously by \feld\ninversion for every single photon energy on a ten sec-\nond basis. The energy resolution of the UE56/1-ZPM\nbeamline achieved at the Co L 3absorption edge is ap-\nproximately 1.6 eV.\nTo determine time zero of the pump-probe experiment\nfor the recorded transient X-ray spectra with highest pos-\nsible precision, a MoSi multi-layer reference sample was\nused48,49. To correct for slow drifts of time-zero we em-\nployed the following measurement procedure: Time zero\nwas determined on the MoSi multi-layer by repeated de-\nlay scans with a net acquisition time of 1 hour. The\ndata statistics of this measurement allows determination\nof time zero in an interval smaller than \u000615 fs. Subse-\nquently, X-ray spectra for a set of delays in a \u0006150 fs\ninterval around nominal time zero from the Co sample\nwere recorded during a 2.5 hours acquisition time. These\ntwo types of measurements were alternatingly repeated.\nThe e\u000bective delay for any recorded X-ray spectrum was\n\fnally recalculated by linear interpolation between the\ntwo time-zero values from the measurements on the MoSi\nmulti-layer before and after the XAS acquisitions. Such\nobtained distribution of e\u000bective delays and associated\nX-ray spectra were \fnally binned in e\u000bective delay win-\ndows of 50 fs width ranging from -150 to 200 fs delay,\naveraged, and evaluated.\nV. RESULTS\nEmploying the method proposed above we now use\nCo as an example to examine the dynamics of spin and\ncharge upon laser pumping. A pump laser pulse drives\ndemagnetization of Co, a process re\rected in the HXAS\nspectra. These changes are shown in Fig. 3(a) and 3(d)\nwhere we show the di\u000berence between the unpumped and\npumped HXAS spectra obtained by theoretical calcula-\ntions (20 fs after laser pumping) and from experiment\n(50 fs after laser pumping) respectively. Negative val-\nues indicate an increase and positive numbers a decrease\nin the HXAS signal after laser induced changes in the\nstate occupation (note that a decrease in absorption is\nequivalent to an increase in state-occupation). As ex-\npected, both experiments and theory show a decrease in\nthe state-occupation below and an increase in the state-\noccupation above Fermi energy (which is set to 0 eV in\nFig. 3(a) and (d)), representing the excitation of charge\nfrom below to above the Fermi energy due to the laser\npump. The amount of change varies with x-ray helicity,\nsignifying spin-dependent pump e\u000bects.\nFigure 3. Di\u000berence between the helicity dependent x-ray\nabsorption spectra (HXAS) before pumping and after laser\npumping showing (a) theoretical data and (d) experimental\ndata. The theoretically calculated transient change in the (b)\nmajority and (c) minority spins is obtained by using Eqs. 3\nand integrating within the various energy windows, as indi-\ncated by di\u000berent colours in panels shown in (a). For the ex-\nperiment the integration is performed in the windows shown\nin (d), and the correspondingly obtained transient majority\nand minority spins shown in panels (e) and (f) respectively.\nThe pump laser pulse of central wave length 800 nm is shown\nin grey for theory in panels (b) and (c), while the Gaussian en-\nvelope of the experimental pulse is shown in (e) and (f) in grey.\nNote that the theoretical calculations were performed using\na pump pulse of 24 fs duration while in the experiment the\nlaser pulse full width at half maximum is approximately 50 fs.\nDespite this one can see excellent qualitative agreement be-\ntween theory and experiment. In particular the turning point\nof the minority charge, that represents the point at which\nspin-\rip induced transitions dominate over spin preserving\noptical transitions, is seen for both theory and experiment to\noccur close to half cycle of the pump pulse.\nIn order to examine changes of majority and minority\ncharge as a function of time and energy, we now solve6\nEqs. 3 to obtain uanddand integrate these changes\nof the spin dependent occupation in the energy windows\nindicated by the shaded rectangles, shown in panels (a)\nand (d) of Fig. 3. These windows (width and position)\nare chosen to capture the dynamics of charge loss be-\nlow the Fermi energy, the dynamics of charge gain above\nthe Fermi energy, and the interesting region around the\nFermi energy within which more complex behavior can\ntake place.\nRemarkably, we see that the integrated change in u\nandd(i.e state occupation) obtained from theory very\nclosely follows changes in the corresponding experimen-\ntal data(Fig. 3 b, c, e, f), analyzed using exactly the same\nparameters A,B,A0,B0as derived from and used in an-\nalyzing the theoretical data. Below (green) the Fermi\nenergy both majority and minority state occupations are\nlost and gained respectively, while for the window en-\nclosing the Fermi energy (yellow) majority is lost and\nminority gained. This indicates not just optical excita-\ntion of charge but also a large number of spin \rips (also\nsee Fig. 4 for further details) in this energy range, leading\nto an increase in the minority occupation and decrease in\nthe majority spins. This leads to demagnetization of Co.\nIn the window above the Fermi energy (blue) a strong\ngain in minority charge carriers is seen both in theory\nand experiment. In our early time theoretical calcula-\ntions a much smaller gain in majority is seen, re\recting\nthe nearly full occupation of majority states in the high\nenergy window above EF, while for the longer times used\nin experiment almost no change in majority charge above\nEFis seen. We note that both in experiment and theory\nwe \fnd charge excitations outside the energy window of\n\u00061.55 eV around the Fermi energy indicating a contri-\nbution from non-linear two-photon excitation processes.\nTo prove this in experiment would entail a full probe of\n\ruence and energy dependent spectra.\nAn interesting feature o\u000bering microscopic insight into\nthe spin-\rip process is the turning point in the minority\ncharge of the low energy window (green). This turn-\ning point precisely demarks the time at which spin-\rip\nprocesses (which will increase minority charge) begin to\ndominate over optical excitations (that reduce minority\ncharge) in the low energy window. These results indi-\ncate that the spin-\rip process play a very important role\nin the energy region of low-energy states, as is indicated\nby the Hamiltonian which shows that the SOC term is\nstrongest for such low energy localized states as the rvs\nterm is largest for these states13.\nDespite two di\u000berent pump pulse lengths used in the-\nory and experiment we see that the qualitative behavior\nof the spin dependent charge corresponds overall very\nwell between theory and experiment in each energy win-\ndow. This highlights the universality of the underlying\nphysics of laser pumped demagnetizaton, as well as the\nrobustness of the procedure used to extract spin- and\nenergy- resolved excited charge proposed in the present\nwork.\nWe note that the energy corresponding to the Fermilevel in the experimental data in Fig. 3 is not given in\na natural way. We de\fne zero energy in the latter case\nin analogy to theory at the zero crossing point of the\ntransientuspectrum (see Fig. 3). The energy resolution\nof this procedure may be inaccurate by about 0.2 eV; this,\nhowever, does not e\u000bect any of our conclusions given that\nthe width of our energy window is much broader than\nthis.\n-3 -2 -1 0 1\nenergy (eV)-0.3-0.2-0.100.10.2change in occupied DOS-0.100.1\nMinorityMajority\nt=-10fs\nt=0fs\nt=10fst=25fs(a)-40 -20 0 20 40 60 80time (fs)\n-0.2-0.100.10.2\nchange in charge (e)(b) minority above\nmajority above\nmajority below\nminority below\n-40 -20 0 20 40 60 80\ntime (fs)-0.2-0.10\nM(t)-M(0) (µΒ) (c)\nabove EFbelow EF\nTotal\nFigure 4. (a) Change in transient occupied spin projected\nDOS. Results are shown at various times after laser pump-\ning. Pump pulse induced (b) change in majority and minority\ncharge integrated below and above the ground-state Fermi en-\nergy and (c) change in total spin magnetic moment and spin\nmoment integrated below and above the Fermi energy. The\npump pulse has central wave length of 800 nm, full width half\nmaximum of 24 fs and incidence \ruence of 6.7mJ/cm2.\nThe HXAS thus encodes a great deal of information of\nthe underlying excitation processes in ultrafast spin dy-\nnamics, well beyond that which can be obtained directly\nfrom MCD spectra, which provides only the dynamics\nof the local moments. While attempts have been made\nin previous works to infer signatures of such excitation\nprocesses from MCD/MLD34,41spectra, it is only by sys-\ntematically disentangling the spin degrees of freedom in\nHXAS, as we do here, that this information can be ob-\ntained. To do this the only extra ingredient required are\nthe parameters A,B,A0andB0in Eqs. 3. As these\ncan be obtained ab-initio from the ground state, then\nthis procedure can be preformed for all elemental and\nmulti-component magnets of interest (values of these pa-\nrameters for Fe, Co and Ni are given in this work).\nHaving experimental access to energy and spin resolved\ninformation via HXAS o\u000bers new opportunities to probe\nin-depth the physics of the excitation process, in par-\nticular energy and time resolved changes in magnetiza-\ntion. To explore this physics further in Fig. 4(a) we show\nchanges in the state-density for majority and minority\nelectrons as a function of time. One observes a broad loss\nof majority states from below the Fermi energy, and a cor-\nresponding large gain over and energy range of \u00181:5 eV7\nabove the Fermi energy. Closer examination unveils a\nsubtle interplay of spin orbit and optically induced tran-\nsitions, in particular the large peak in the minority den-\nsity of states di\u000berence observed 10 fs but not at 25 fs\ndue to \flling of these states by spin-orbit induced tran-\nsitions from majority to minority. It is this transient\npeak that is responsible for the turning point in the in-\ntegrated transient spin resolved DOS occupations seen\nin both theory and experiment above, and revealed also\nin the integrated minority charge below Fermi energy,\nsee Fig. 4b. These energy resolved spin channels below\nand above the Fermi energy also show clear evidence of\nthe extent of spin \rips as the increase in minority charge\nabove the Fermi energy is signi\fcantly greater than the\ndecrease in minority charge from below the Fermi energy.\nIn a similar way we can also energy resolve the dynam-\nics of magnetic moments, At early times minority elec-\ntrons undergo optical excitation from below the Fermi en-\nergy, and the magnetic moment below the Fermi energy\n(orange curve in Fig. 4c) shows a corresponding increase .\nAt\u001820 fs spin-\rip processes begin to dominate, driving\na reduction in the majority carriers and hence a reduction\nin the moment below the Fermi level. At the same time,\nthis excitation of minority charge from below to above\nthe Fermi level leads to a negative change in the moment\nto develop above the Fermi level (pink). However, the\nmagnitude of the moment above the Fermi energy also\nstarts to saturate as spin-\rips dominate the physics of\ndemagnetization. These results again point towards the\nphysics seen in Fig. 1c and 3; the spin \rip process are\nnot just around the Fermi level but are strongly energy\ndependent and show signatures well below the Fermi en-\nergy.\nVI. THEORETICAL VALIDATION OF HXAS\nDISENTANGLEMENT\nOne of the major approximations in solving Eqs. 3 was\nthe assumption of very weak time dependence of the tran-\nsition probabilities. This not only reduces the computa-\ntional task, most importantly it allows calculation and\ntabulation of A;B;A0andB0from static ground-state\ncalculations for the analysis of future experiments. While\nwe have justi\fed this assumption post-hoc by agreement\nbetween theory and experiment, within the theoretical\nframework itself this approximation can be justi\fed. In\norder to do this in Fig. 5 we present the changes in HXAS\nand DOS occupations 20 fs after laser pumping. The\nDOS is broadened by a Gaussian (FWHM 0 :82 eV) in\norder to have the same spectral width as the HXAS re-\nsponse function. Evidently, the two sets of data show\nthe same trend in that occupation of both majority and\nminority spins decreases below and increases above the\nFermi energy, while at the Fermi level the two curves\ncross zero. Closer inspection of the helicity dependent\nresponse function, Fig. 5(a), and the transient DOS oc-\ncupation, Fig. 5(d), reveals slight di\u000berences arising from\nFigure 5. Internal theoretical consistency of HXAS and DOS\nderived spin, energy, and time resolved state occupation. (a)\nHXAS change 20 fs after laser pumping and (d) the di\u000ber-\nence in the occupied density of states 20 fs after pumping.\nCharge in (b) majority and (c) minority spins obtained by\nusing Eqs. 3 and integration in various energy windows in-\ndicated by the di\u000berent colours in panel (a). Change in (e)\nmajority and (f) minority spins obtained from integrating the\ndi\u000berence in DOS occupations in the corresponding energy\nwindows shown in panel (d). Note that the energy windows\nfor integration are exactly the same width and at the same\nposition with respect to the Fermi energy in panels (a) and\n(d). As can be seen, both routes to deriving transient state\noccupations yield very similar results, validating the method\nproposed for disentangling spin, energy, and time resolved\nstate occupation from the HXAS spectra. The A-\feld of the\npump pulse has an incidence \ruence of 6.7mJ/cm2.\nthe fact that the response functions contain information\nboth of changes in occupation as well as the respective\ntransition probabilities. Integrated changes in the occu-8\npation in three di\u000berently positioned energy windows are\nshown in panels 5(b) and (e) for majority and (c) and\n(f) for minority spins. In this case, we see that the in-\ntegrated charge from HXAS very closely follows changes\nin the corresponding state density. In both the positive\n(blue) and negative (green) energy windows both major-\nity and minority states are lost and gained respectively,\nwhile for the window enclosing the Fermi energy (yellow)\nmajority is lost and minority gained. This close agree-\nment between the changes in spin projected occupations\nobtained from DOS and HXAS calculations provides a\ndemonstration within the theoretical framework that the\nextraction from the HXAS of pure spin response func-\ntions using coe\u000ecients tabulated from the ground state\nis indeed a good approximation.\nVII. DISCUSSION\nStrong light-matter interaction in magnetic systems\ncreates a highly non-equilibrium state of matter exhibit-\ning remarkable phenomena such as all-optical switching\nand OISTR. Exploring this diversity of phenomena de-\nmands experimental probes able to precisely track the\nmicroscopic processes underlying ultrafast spin dynam-\nics. In the present work we have proposed and veri-\n\fed a simple extension of the well established XMCD\ntechnique, allowing a transient spin- energy- and species-\nresolved probe of ultrafast demagnetization.\nThis extension is based upon disentangling the spin de-\ngrees of freedom that are, generally, woven in a complex\nway into the response of a magnetic system to circularly\npolarized light. We show that, by employing parameters\nobtained ab-initio , the experimental HXAS spectra may\nbe inverted to give spin speci\fc response functions that,\nin turn, provide a window into the spin-energy landscape\nin which ultrafast demagnetization occurs.\nUsing the example of elemental cobalt we resolve the\nmagnetization dynamics into energy regions that vividly\nillustrate the wide energy range on which the spin-orbit\nand optical transitions that drive demagnetization occur.\nSpeci\fc quantitative features, such as turning points in\nspin and energy resolved charge, further reveal criticalpoints at which the balance between spin preserving opti-\ncal transitions and spin-orbit induced spin-\rips changes.\nThe ab-initio procedure employed to disentangle the\nHXAS spectra is shown to be valid within the theoreti-\ncally obtained HXAS spectra and state density, with the\nqualitative agreement between theory and experiment\n(which employ lasers pulses of signi\fcantly longer dura-\ntion than those that can be treated theoretically) further\nbolstering the correctness of the spin projection of the\nHXAS spectra. As the parameters encoding spin speci\fc\ntransitions required to preform this transformation are,\nto an excellent approximation, time independent, they\ncan be obtained from the ground state and so this pro-\ncedure can, in principle, be applied to multi-component\nmagnetic materials of almost any complexity. The ex-\ntension of HXAS that we propose is therefore of great\napplicability.\nPromising future experimental applications of this \\ex-\ntended HXAS\" method include probing spin- and energy-\nresolved dynamics in all-optical spin switching magnets.\nDirect access to the transient spin- and energy resolved\ncharge is expected to provide invaluable insight into still\ncontroversial features of the dynamics of spin switching,\ne.g. which moments rotate, at what times, and the in-\nplane versus out-of-plane components of magnetization.\nUnderstanding the still unclear microscopic world under-\npinning such well established phenomena may provide\nthe basis for enriched exploration and control of the fron-\ntiers of ultrafast spin dynamics.\nVIII. ACKNOWLEDGEMENTS\nSharma, CvKS, SE, NP and CSL would like to\nthank for funding by the Deutsche Forschungsgemein-\nschaft (DFG, German Research Foundation) { Project-\nID 328545488 { TRR 227 (projects A04, A02 and A03).\nShallcross would like to thank DFG for funding through\nSPP 1840 QUTIF Grant No. SH 498/3-1. The au-\nthors acknowledge the North-German Supercomputing\nAlliance (HLRN) for providing HPC resources that have\ncontributed to the research results reported in this paper.\n\u0003sharma@mbi-berlin.de\n1J. Y. Bigot, M. Vomir, and E. Beaurepaire, Nat. Phys. 5,\n515 (2009).\n2F. Siegrist, J. A. Gessner, M. Ossiander, C. Denker,\nY. P. Chang, M. C. Schroder, A. Guggenmos, Y. Cui,\nJ. Walowski, U. Martens, J. K. Dewhurst, U. Kleineberg,\nM. M unzenberg, S. Sharma, and M. Schultze, Nature 571,\n240 (2019).\n3A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys.\n82, 2731 (2010).\n4K. Carva, P. Bal\u0013 a\u0014 z, and I. Radu, Chapter 2 -\nLaser-Induced Ultrafast Magnetic Phenomena , edited byE. Br uck, Handbook of Magnetic Materials, Vol. 26 (Else-\nvier, 2017) pp. 291{463.\n5A. V. Kimel, A. M. Kalashnikova, A. Pogrebna, and A. K.\nZvezdin, Phys. Rep. 852, 1 (2020).\n6S. Jeppson and R. Kukreja, APL Mater. 9, 100702 (2021).\n7C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n99, 047601 (2007).\n8I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH. A. D urr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W.\nChantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing,\nand A. V. Kimel, Nature 472, 205 (2011).9\n9M. Battiato, K. Carva, and P. M. Oppeneer, Phys. Rev.\nLett.105, 027203 (2010), arXiv:1106.2117.\n10A. Eschenlohr, M. Battiato, P. Maldonado, N. Pontius,\nT. Kachel, K. Holldack, R. Mitzner, A. F ohlisch, P. M.\nOppeneer, and C. Stamm, Nat Mater 12, 332 (2013).\n11J. K. Dewhurst, P. Elliott, S. Shallcross, E. K. U. Gross,\nand S. Sharma, Nano Letters 18, 1842 (2018).\n12C. Jin, E. Y. Ma, O. Karni, E. C. Regan, F. Wang, and\nT. F. Heinz, Nat. Nanotechnol. 13, 994 (2018).\n13K. Krieger, J. K. Dewhurst, P. Elliott, S. Sharma, and\nE. K. U. Gross, Journal of Chemical Theory and Compu-\ntation 11, 4870 (2015).\n14G. P. Zhang and W. H ubner, Phys. Rev. Lett. 85, 3025\n(2000).\n15M. Hofherr, S. H auser, J. K. Dewhurst, P. Tengdin,\nS. Sakshath, H. T. Nembach, S. T. Weber, J. M. Shaw,\nT. J. Silva, H. C. Kapteyn, M. Cinchetti, B. Rethfeld,\nM. M. Murnane, D. Steil, B. Stadtm uller, S. Sharma,\nM. Aeschlimann, and S. Mathias, Science Advances 6\n(2020), 10.1126/sciadv.aay8717.\n16F. Willems, C. von Kor\u000b Schmising, C. Str uber, D. Schick,\nD. W. Engel, J. K. Dewhurst, P. Elliott, S. Sharma, and\nS. Eisebitt, Nat. Commun. 11, 1 (2020).\n17P. Tengdin, C. Gentry, A. Blonsky, D. Zusin, M. Ger-\nrity, L. Hellbr uck, M. Hofherr, J. Shaw, Y. Kvashnin,\nE. K. Delczeg-Czirjak, M. Arora, H. Nembach, T. J. Silva,\nS. Mathias, M. Aeschlimann, H. C. Kapteyn, D. Thonig,\nK. Koumpouras, O. Eriksson, and M. M. Murnane, Sci.\nAdv.6, eaaz1100 (2020).\n18E. Golias, I. Kumberg, I. Gelen, S. Thakur, J. Gordes,\nR. Hosseinifar, Q. Guillet, J. K. Dewhurst, S. Sharma,\nC. Sch u\u0019ler-Langeheine, N. Pontius, and W. Kuch, Phys\nRev Lett 126, 107202 (2021).\n19H.-S. Rhie, H. A. D urr, and W. Eberhardt, Phys. Rev.\nLett.90, 247201 (2003).\n20R. Gort, K. B uhlmann, S. D aster, G. Salvatella, N. Hart-\nmann, Y. Zemp, S. Holenstein, C. Stieger, A. Fognini,\nT. U. Michlmayr, T. B ahler, A. Vaterlaus, and Y. Acre-\nmann, Phys. Rev. Lett. 121, 87206 (2018).\n21S. Eich, M. Pl otzing, M. Rollinger, S. Emmerich, R. Adam,\nC. Chen, H. C. Kapteyn, M. M. Murnane, L. Plucinski,\nD. Steil, B. Stadtm uller, M. Cinchetti, M. Aeschlimann,\nC. M. Schneider, and S. Mathias, Sci. Adv. 3, 1 (2017).\n22B. Frietsch, A. Donges, R. Carley, M. Teichmann,\nJ. Bowlan, K. D obrich, K. Carva, D. Legut, P. M. Op-\npeneer, U. Nowak, and M. Weinelt, Sci. Adv. 6, 1 (2020).\n23C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast,\nK. Holldack, S. Khan, C. Lupulescu, E. F. Aziz, M. Wiet-\nstruk, H. A. D urr, and W. Eberhardt, Nat. Mater. 6, 740\n(2007).\n24C. Boeglin, E. Beaurepaire, V. Halt\u0013 e, V. L\u0013 opez-Flores,\nC. Stamm, N. Pontius, H. A. D urr, and J.-Y. Bigot, Na-\nture465, 458 (2010).\n25B. T. Thole, P. Carra, F. Sette, and G. van der Laan,\nPhys. Rev. Lett. 68, 1943 (1992).\n26P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys.\nRev. Lett. 70, 694 (1993), arXiv:0000135489.\n27J. St ohr and H. C. Siegmann, Magnetism , Vol. 152\n(Springer Berlin Heidelberg, Berlin, Heidelberg, 2006) pp.\n1{822.\n28C. Stamm, N. Pontius, T. Kachel, M. Wietstruk, and\nH. A. D urr, Phys. Rev. B 81, 104425 (2010).\n29N. Bergeard, V. L\u0013 opez-Flores, V. Halt\u0013 e, M. Hehn,\nC. Stamm, N. Pontius, E. Beaurepaire, and C. Boeglin,Nat. Commun. 5, 3466 (2014).\n30M. Hennecke, I. Radu, R. Abrudan, T. Kachel, K. Holl-\ndack, R. Mitzner, A. Tsukamoto, and S. Eisebitt, Phys.\nRev. Lett. 122, 157202 (2019).\n31D. J. Higley, A. H. Reid, Z. Chen, L. L. Guyader, O. Hell-\nwig, A. A. Lutman, T. Liu, P. Shafer, T. Chase, G. L.\nDakovski, A. Mitra, E. Yuan, J. Schlappa, H. A. D urr,\nW. F. Schlotter, and J. St ohr, Nat. Commun. 10, 1 (2019).\n32M. Hennes, B. R osner, V. Chardonnet, G. S. Chiuzba-\nian, R. Delaunay, F. D oring, V. A. Guzenko, M. Hehn,\nR. Jarrier, A. Kleibert, M. Lebugle, J. L uning, G. Mali-\nnowski, A. Merhe, D. Naumenko, I. P. Nikolov, I. Lopez-\nQuintas, E. Pedersoli, T. Savchenko, B. Watts, M. Zan-\ngrando, C. David, F. Capotondi, B. Vodungbo, and E. Jal,\nAppl. Sci. 11, 325 (2020).\n33B. R osner, B. Vodungbo, V. Chardonnet, F. D oring, V. A.\nGuzenko, M. Hennes, A. Kleibert, M. Lebugle, J. L uning,\nN. Mahne, A. Merhe, D. Naumenko, I. P. Nikolov, I. Lopez-\nQuintas, E. Pedersoli, P. R. Ribi\u0014 c, T. Savchenko, B. Watts,\nM. Zangrando, F. Capotondi, C. David, and E. Jal, Struct.\nDyn.7(2020), 10.1063/4.0000033.\n34L. Le Guyader, D. J. Higley, M. Pancaldi, T. Liu, Z. Chen,\nT. Chase, P. W. Granitzka, G. Coslovich, A. A. Lutman,\nG. L. Dakovski, W. F. Schlotter, P. Shafer, E. Arenholz,\nO. Hellwig, M. L. M. Lalieu, B. Koopmans, A. H. Reid,\nS. Bonetti, J. St ohr, and H. A. D urr, Appl. Phys. Lett.\n120, 032401 (2022), arXiv:2110.15615.\n35C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith,\nG. Meigs, E. Chaban, G. H. Ho, E. Pellegrin, and F. Sette,\nPhysical Review Letters 75, 152 (1995), publisher: Amer-\nican Physical Society.\n36\\Ni:A= 0:76,B= 0:24,A0= 0:528, and B0= 0:472. Fe:\nA= 0:84,B= 0:16,A0= 0:63, and B0= 0:37,\".\n37E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997\n(1984).\n38J. K. Dewhurst, K. Krieger, S. Sharma, and E. K. U.\nGross, Computer Physics Communications 209, 92 (2016).\n39S. Sharma, J. K. Dewhurst, and E. K. U. Gross, \\Op-\ntical response of extended systems using time-dependent\ndensity functional theory,\" in First Principles Approaches\nto Spectroscopic Properties of Complex Materials , edited\nby C. Di Valentin, S. Botti, and M. Cococcioni (Springer\nBerlin Heidelberg, Berlin, Heidelberg, 2014) pp. 235{257.\n40S. Sharma, J. K. Dewhurst, A. Sanna, and E. K. U. Gross,\nPhys. Rev. Lett. 107, 186401 (2011).\n41F. Willems, C. von Kor\u000b Schmising, C. Str uber, D. Schick,\nD. W. Engel, J. K. Dewhurst, P. Elliott, S. Sharma, and\nS. Eisebitt, Nature Communications 11, 871 (2020).\n42J. K. Dewhurst, F. Willems, P. Elliott, Q. Z. Li, C. v. K.\nSchmising, C. Str uber, D. W. Engel, S. Eisebitt, and\nS. Sharma, Phys. Rev. Lett. 124, 077203 (2020).\n43S. Sharma, S. Shallcross, P. Elliott, S. Eisebitt, C. v. Ko-\nr\u000b Schmising, and J. K. Dewhurst, Appl. Phys. Lett. 120,\n062409 (2022).\n44L. Hedin, Phys. Rev. 139, A796 (1965).\n45D. J. Singh, Planewaves Pseudopotentials and the LAPW\nMethod (Kluwer Academic Publishers, Boston, 1994).\n46J. K. Dewhurst, S. Sharma, and et al., (Jan. 14 2018 ).\n47K. Holldack, J. Bahrdt, A. Balzer, U. Bovensiepen,\nM. Brzhezinskaya, A. Erko, A. Eschenlohr, R. Follath,\nA. Firsov, W. Frentrup, L. Le Guyader, T. Kachel,\nP. Kuske, R. Mitzner, R. M uller, N. Pontius, T. Quast,\nI. Radu, J.-S. Schmidt, C. Sch u\u0019ler-Langeheine, M. Sper-\nling, C. Stamm, C. Trabant, and A. F ohlisch, J. Syn-10\nchrotron Radiat. 21, 1090 (2014).\n48D. Schick, S. Eckert, N. Pontius, R. Mitzner, A. F ohlisch,\nK. Holldack, and F. Sorgenfrei, Struct. Dyn. 3, 054304\n(2016).49S. Jana, S. Muralidhar, J. \u0017Akerman, C. Sch u\u0019ler-\nLangeheine, and N. Pontius, Struct. Dyn. 8, 044304\n(2021)."    },    {        "title": "1606.01375v1.Mott_transitions_in_the_Periodic_Anderson_Model.pdf",        "content": "Mott transitions in the Periodic Anderson Model\nDavid E. Logan, Martin R. Galpin and Jonathan Mannouch\nDepartment of Chemistry, Physical and Theoretical Chemistry,\nOxford University, South Parks Road, Oxford, OX1 3QZ, United Kingdom\n(Dated: September 11, 2018)\nThe periodic Anderson model (PAM) is studied within the framework of dynamical mean-\feld\ntheory, with particular emphasis on the interaction-driven Mott transition it contains, and on resul-\ntant Mott insulators of both Mott-Hubbard and charge-transfer type. The form of the PAM phase\ndiagram is \frst deduced on general grounds using two exact results, over the full range of model pa-\nrameters and including metallic, Mott, Kondo and band insulator phases. The e\u000bective low-energy\nmodel which describes the PAM in the vicinity of a Mott transition is then shown to be a one-band\nHubbard model, with e\u000bective hoppings that are not in general solely nearest neighbour, but decay\nexponentially with distance. This mapping is shown to have a range of implications for the physics\nof the problem, from phase boundaries to single-particle dynamics; all of which are con\frmed and\nsupplemented by NRG calculations. Finally we consider the locally degenerate, non-Fermi liquid\nMott insulator, to describe which requires a two-self-energy description. This is shown to yield a\nnumber of exact results for the associated local moment, charge, and interaction-renormalised levels,\ntogether with a generalisation of Luttinger's theorem to the Mott insulator.\nPACS numbers: 71.10.-w, 71.10.Fd, 71.10.Hf, 71.30.+h\nI. INTRODUCTION\nThe periodic Anderson model (PAM) is one of the clas-\nsic models of strongly correlated electron systems.1Mini-\nmalist by design, but physically richer than the canonical\none-band Hubbard model, the PAM is a two-band model\nwith the two orbitals per lattice site coupled by a lo-\ncal one-electron hybridization. One band is localised but\ncorrelated, with on-level interaction U, the other uncorre-\nlated but itinerant. While traditionally considered in the\ncontext off-electron systems1{ heavy fermion materials\n{ the correlated orbitals can equally refer to the localised\nd-orbitals of transition metal oxide and related materials,\nenabling access also to the basic physics of these systems.\nWithin the powerful framework of dynamical mean-\n\feld theory (DMFT),2{7which is formally exact in the\nlimit of large coordination number and known to cap-\nture well many properties of real materials,8numerous\nfacets of the PAM have been extensively studied over\nmany years.6,7,9{31The model is known to contains a di-\nverse range of phases,6,7including the metal, Kondo and\nband insulators { all Fermi liquids since they are adia-\nbatically connected to the non-interacting limit { as well\nas a Mott insulator phase and hence an underlying Mott\nmetal-insulator transition.24Due to the two-band nature\nof the model, the resultant Mott phase also exempli\fes\nthe Zaanen-Sawatzky-Allen (ZSA) scheme,32distinguish-\ning between Mott-Hubbard or charge-transfer insulators\naccording respectively to whether \u0001 &Uor \u0001.U, with\n\u0001 the charge transfer energy { the one-electron energy\ngap between the uncorrelated and correlated local levels.\nIn the present paper we report a detailed DMFT study\nof the PAM, with particular emphasis on understanding\nthree related aspects of it: (a) its phase diagram over\nessentially the full range of underlying model parameters;\n(b) the connection between the Mott transition in thePAM and that occurring in a one-band Hubbard model;\nand (c) the non-Fermi liquid Mott insulator phase itself.\nSince the paper is quite wide ranging we begin with an\noverview of it, and some of the issues to be addressed.\nA. Overview\nThe model and DMFT background is \frst summarised\nin sec. II. The basic parameter space for the PAM is of\ncourse large compared e.g. to a nearest neighbour Hub-\nbard model. In addition to the one-electron hopping\nt\u0003, which connects and broadens the uncorrelated con-\nduction band levels (with level energies \u000fc), we take its\n`bare' parameters to be \u000fc;U;V and\u0011f; withUthe local\nCoulomb repulsion for the correlated levels with energies\n\u000ff, andVthe one-electron hybridization coupling local\nc- andf-levels (for speci\fcity we refer to the correlated\norbitals asf-levels). As for an Anderson impurity model,\nthe asymmetry of the correlated level is embodied in20,33\n\u0011f= 1 + 2\u000ff=U, which controls whether the level is in a\nlocal moment, mixed-valent or empty-orbital regime.\nSec. III begins with two exact results which play an\nimportant role in the subsequent analysis. We \frst iden-\ntify a general inequality that must be satis\fed by any\ninsulator { Mott, Kondo or band { relating \u000fcandV\nto the interaction-renormalised f-level energy \u000f\u0003\nf. The\nsecond result relates to Fermi liquid phases, encompass-\ning metallic, Kondo and band insulator phases. It is has\nbeen known for some time,20and relates the total charge\nper site to\u000fc;Vand\u000f\u0003\nf. Conjoined with a series of sim-\nple physical arguments, these results are shown in secs.\nIII B, III C to enable the general form and structure of\nthe PAM phase diagram to be deduced, as a function of\n\u000fc=t\u0003,U=t\u0003and\u0011f, encompassing all phases of the model.\nFull numerical renormalization group (NRG) calcula-arXiv:1606.01375v1  [cond-mat.str-el]  4 Jun 20162\ntions of the phase boundaries are also given. They con-\n\frm the deductions made and, for the Mott transition in\nparticular, are seen to provide clear examples of the ZSA\nclassi\fcation scheme.32\nThe resultant behaviour nonetheless also raises a num-\nber of questions. For example, on progressively deplet-\ning the conduction band occupancy by raising its cen-\nter of gravity relative to the Fermi level { by ramping\nup\u000fc=t\u0003\u001d1 { the critical U=t\u0003for the Mott tran-\nsition asymptotically vanishes; which begs an explana-\ntion, given that the Mott transition is a paradigm of\nstrong correlations. More generally, the question nat-\nurally arises24as to what extent the Mott transition in\nthe PAM is similar to that occurring in a simpler one-\nband Hubbard model. We consider these and other basic\nmatters in the remainder of the paper.\nWhile it has previously been thought that the low-\nenergy properties of the PAM close to a Mott transition\ncannot be described generally in terms of a one-band\nHubbard model,24,30one central result of the present\nwork is to show (sec. IV) that the e\u000bective low-energy\nmodel which describes the PAM in the vicinity of the\nMott transition is in fact precisely a single-band Hub-\nbard model. The one-electron hoppings in this e\u000bective\nHubbard model are in general long-ranged, decaying ex-\nponentially with the topological distance between sites\n(providing a connection to models hitherto studied in the\nnon-interacting, tight-binding limit34); and with an e\u000bec-\ntive range found to be controlled by \u000fc=t\u0003. For the par-\nticular regime \u000fc=t\u0003\u001d1 { relevant close to the transition\nto a Mott-Hubbard insulator in the ZSA scheme32{ the\ne\u000bective hopping, t\u0003e\u000b, is in practice nearest neighbour\n(NN) only, but with t\u0003e\u000bstrongly reduced from the bare\nPAM hopping t\u0003; which underlies the narrowness of elec-\ntron bands anticipated in the vicinity of the transition\nto Mott-Hubbard insulators.32Away from this asymp-\ntotic regime however, and in particular close to the Mott\ntransition to charge-transfer insulators, the long-ranged\nnature of the hoppings is central to the problem.\nIn sec. IV D we also touch brie\ry on the Kondo lattice\nmodel (KLM), pointing out that, while commonly used\nas a proxy model in regimes where the correlated lev-\nels are almost singly occupied, the rich physics of Mott\ntransitions in the PAM is simply absent in the KLM.\nGiven the mapping to an e\u000bective Hubbard model, we\nrevisit the phase diagram in sec. IV E. That the e\u000bective\nmodel is naturally speci\fed by fewer parameters than\nthe PAM, is \frst shown on general grounds to imply a\nscaling collapse of phase boundaries for the Mott transi-\ntion (whereby the hybridization Ve\u000bectively scales out\nof the problem); as is indeed veri\fed in NRG calculations.\nPhysical arguments are then used to obtain simple ana-\nlytical estimates for the transition to a Mott insulator {\nincluding the approach to it from both the electron- and\nhole-doped sides (in regimes where both are possible) {\nas well as that for the metal to band insulator transitions.\nThese too are found to agree well with NRG results for\nthe full PAM.Single-particle dynamics in the metal close to the Mott\ntransition are considered in sec. IV F, to exemplify typical\nbehaviour associated with Mott transitions to both Mott-\nHubbard and charge-transfer insulators. The approach\nto the transition in either case is characterised by a van-\nishing low-energy scale, embodied in a low-energy Kondo\nresonance which narrows progressively as the transition is\napproached, and in terms of which the f- andc-electron\nspectra exhibit universal scaling. Scaling spectra are\nhowever found to be quite distinct in the Mott-Hubbard\nand charge-transfer regimes, as too are the associated\nbandwidth scales (`narrow' versus `broad' bands), and\nthe character of the insulating gaps.\nSec. V is concerned with the paramagnetic Mott in-\nsulator phase per se . This presents a greater theoretical\nchallenge than the metal, since the Mott insulator is not\nadiabatically connected to the non-interacting limit and\nas such is not a Fermi liquid. The electron spin degrees of\nfreedom { which in the metallic phase are quenched com-\npletely by the Kondo e\u000bect { are incompletely quenched\nin the Mott insulator, resulting in a locally degenerate\nground state with a residual local magnetic moment. To\nhandle this degenerate Mott insulator requires a two-self-\nenergy description.35,36This enables a number of exact\nresults to be obtained for the local moment, charge, and\nassociated renormalised levels; which are also shown to\nbe well captured by NRG calculations.\nFinally, in sec. V B we consider the standard Luttinger\nintegralIL,35{37the vanishing of which throughout the\nmetal is tantamount to Luttinger's theorem,6and re\rects\nthe Fermi liquid nature of that phase. While this result\nnaturally does not hold in the Mott insulator, a gener-\nalisation of it is proven, showing that ILhas constant\nbut now non-zero magnitude throughout the entire Mott\nphase. We have recently shown this result holds also for\nlocal moment phases of a wide range of quantum im-\npurity models,35and for the one-band Hubbard model\nwithin DMFT;36suggesting its ubiquity as a hallmark of\nlocally degenerate ground states, and indicative of their\nperturbative continuity to the uncoupled atomic limit.\nII. MODEL AND BACKGROUND\nThe model is given in standard notation by\nH=Hc+Hf+Hhyb (1a)\nHc=X\ni;\u001b\u000fccy\ni\u001bci\u001b\u0000tX\n(i;j);\u001bcy\ni\u001bcj\u001b (1b)\nHf=X\ni;\u001b\u0010\n\u000ff+1\n2Ufy\ni\u0000\u001bfi\u0000\u001b\u0011\nfy\ni\u001bfi\u001b (1c)\nHhyb=VX\ni;\u001b\u0010\nfy\ni\u001bci\u001b+ h:c:\u0011\n(1d)\nwith local \u001b-spin number operators ^ nc\u001b=cy\ni\u001bci\u001band\n^nf\u001b=fy\ni\u001bfi\u001b.Hcrepresents the non-interacting conduc-\ntion band. Its nearest neighbour hopping tis rescaled3\nwithin DMFT as6,7t=t\u0003=(2pZc) with coordination\nnumberZc!1 ; and the only relevant property of the\nc-band energy dispersion is \u001a0(\u000f), the free ( V= 0) den-\nsity of states (DoS) for \u000fc= 0. This we take to be of\nstandard compact, semicircular form\n\u001a0(\u000f) =2\n\u0019t\u0003p\n1\u0000(\u000f=t\u0003)2 (2)\nwith band halfwidth1\n2W=t\u0003, corresponding formally\nto a Bethe lattice. Hfspeci\fes the correlated levels,\nwhileHhybhybridizes the c- andf-levels via a local one-\nelectron matrix element V. While referring to the corre-\nlated orbitals as f-levels, we remind the reader that they\ncould equally refer to the d-levels of transition metal sys-\ntems (with e.g. ligand p-levels for the conduction band).\nWitht\u0003as the basic energy unit, the model is charac-\nterised by four `bare' parameters, \u000fc=t\u0003,\u000ff=t\u0003,U=t\u0003and\nV=t\u0003. Equivalently and more usefully, as employed in the\nfollowing, it can be parameterised by \u000fc=t\u0003,U=t\u0003,V=t\u0003\nand\u0011f, with thef-level asymmetry \u0011fgiven by20,33\n\u0011f= 1 +2\u000ff\nU: (3)\nFor\u000ffin the range\u0000U < \u000ff<0 (\u0011EF, the Fermi\nlevel),\u0011flies in the range\u00001<\u0011f<1. Here an uncou-\npledf-level is singly occupied only, thus carrying a local\nmoment. For obvious physical reasons this is often the\nregime of primary interest in the full system, although\nwe will not restrict ourselves to it in part because (see\nsecs. III \u000b) Mott insulators are not con\fned to j\u0011fj<1.\n1. DMFT equations\nThe localc- orf-electron charge, nc=h^nci=P\n\u001bh^nc\u001bi\nornf=h^nfi, is of course related to the retarded propa-\ngatorGc(!) ($Gc(t) =\u0000i\u0012(t)hfci\u001b(t);cy\ni\u001bgi) orGf(!),\nby\n1\n2n\u0017=\u00001\n\u0019ImZ0\n\u00001d! G\u0017(!) :\u0017=c;f (4)\nwith!= 0 the Fermi level. The local propagators\nG\u0017(!)\u0011G\u0017;ii;\u001b(!) are naturally independent of both site\niand spin\u001bfor the homogeneous paramagnetic phases\nwe consider.\nWithin DMFT the propagator is given by20\nG\u0017(!) =Z1\n\u00001d\u000f \u001a 0(\u000f)G\u0017(\u000f;!) (5)\nwith\nGc(\u000f;!) =\"\n!+\u0000\u000fc\u0000V2\n!+\u0000\u000ff\u0000\u0006f(!)\u0000\u000f#\u00001\n(6)and\nGf(\u000f;!) =\u0014\n!+\u0000\u000ff\u0000\u0006f(!)\u0000V2\n!+\u0000\u000fc\u0000\u000f\u0015\u00001\n(7a)\n=1\n!+\u0000\u000ff\u0000\u0006f(!)+V2\n[!+\u0000\u000ff\u0000\u0006f(!)]2Gc(\u000f;!)\n(7b)\nwhere \u0006f(!) = \u0006R\nf(!)\u0000i\u0006I\nf(!) is thef-level interaction\nself-energy (purely local, independent of \u000f) and!+=\n!+i0+.\nWith\u001a0(\u000f) as in eq. 2, the G\u0017(!) may be written equiv-\nalently as\nGc(!) =h\n!+\u0000\u000fc\u0000V2\n!+\u0000\u000ff\u0000\u0006f(!)\u00001\n4t2\n\u0003Gc(!)i\u00001\n(8a)\nGf(!) =h\n!+\u0000\u000ff\u0000\u0006f(!)\u0000V2\n!+\u0000\u000fc\u00001\n4t2\u0003Gc(!)i\u00001\n(8b)\n\u0011h\n!+\u0000\u000ff\u0000\u0006f(!)\u0000Sf(!)i\u00001\n(8c)\nwith local c-level Feenberg self-energy38Sc(!) =\n1\n4t2\n\u0003Gc(!) (i.e.Sc(!) =P\njt2Gc;jj;\u001b(!) with sites j\nNN to site i), and local f-level Feenberg self-energy\nSf(!) =V2[!+\u0000\u000fc\u00001\n4t2\n\u0003Gc(!)]\u00001. Eq. 8 embodies\nthe fact that within DMFT any lattice-fermion model\nreduces to a self-consistent quantum impurity model;6,7\nfor it is precisely that for a 2-level Anderson impurity\nmodel in which the hybridization function coupling\nthec-level to the bath is Sc(!), while that coupling\nthef-level isSf(!), and where the S\u0017(!) must be\ndetermined self-consistently.\nWe also touch here on particle-hole (ph) asymmetry,\nwhich enters the problem in two distinct ways:\n(a) Conduction band asymmetry, embodied in \u000fc; which\ndetermines the centre of gravity of the free conduction\nband relative to the Fermi level, as re\rected in its density\nof statesdc\n0(!) =\u001a0(!\u0000\u000fc). Under a ph-transformation,\nHcitself (eq. 1b) is ph-symmetric at the point \u000fc= 0.\n(b)f-level asymmetry, embodied in \u0011f(eq. 3), with \u0011f=\n0 at the ph-symmetric point of Hfitself (\u000ff=\u0000U=2).\nUnder a ph-transformation it is readily shown that\nH(\u000fc;\u0011f)\u0011H(\u0000\u000fc;\u0000\u0011f) (for given U;V). Hence the full\nmodel is ph-symmetric only at the point \u000fc= 0 =\u0011f;\nand only\u0011f\u00150 need be considered. Likewise, with\nthe\u000fc;\u0011fdependence temporarily explicit, ^ n\u0017(\u000fc;\u0011f) +\n^n\u0017(\u0000\u000fc;\u0000\u0011f) = 2. The total charge\nnt:=nc+nf (9)\nthus satis\fes\nnt(\u000fc;\u0011f) +nt(\u0000\u000fc;\u0000\u0011f) = 4: (10)4\nFor the self-energy,\n\u0006R\nf(!;\u000fc;\u0011f) =U\u0000\u0006R\nf(\u0000!;\u0000\u000fc;\u0000\u0011f) (11a)\n\u0006I\nf(!;\u000fc;\u0011f) = \u0006I\nf(\u0000!;\u0000\u000fc;\u0000\u0011f) (11b)\nunder a ph-transformation, as employed below.\nIII. METALS VSINSULATORS\nLeaving aside for the moment band insulators with\nnt= 0 or 4, the PAM is well known to contain three\ndistinct phases:9{31a metal (M), a Kondo insulator (KI)\nand a Mott insulator (MI). Insulating states are char-\nacterised by a \fxed total charge throughout the phase:\nnt= 2 for the KI and, for the MI, nt= 1 or 3.\nSince metals vsinsulators are distinguished at the\nmost basic level by the single-particle spectra D\u0017(!) =\n\u00001\n\u0019ImG\u0017(!) at the Fermi level, we \frst ask what can be\ninferred on general grounds for D\u0017(!= 0).\nThe imaginary part of the self-energy at the Fermi level\nvanishes generically in all phases, \u0006I\nf(!= 0) = 0, re-\n\recting the Fermi liquid character of the metal and the\ngapped nature of the insulators. With this, the spectra\nD\u0017(!= 0) follow simply from eqs. 5-7 as\nDc(0) =\u001a0\u0010\n\u0000\u000fc+V2\n\u000f\u0003\nf\u0011\n(12a)\nDf(0) =V2\n\u000f\u0003\nf2\u001a0\u0010\n\u0000\u000fc+V2\n\u000f\u0003\nf\u0011\n: (12b)\nHere,\n\u000f\u0003\nf=\u000ff+ \u0006R\nf(0) =\u0000U\n2(1\u0000\u0011f) + \u0006R\nf(0) (13)\nis the interaction-renormalised f-level energy; with \u000f\u0003\nf\n(\u0011\u000f\u0003\nf(\u000fc;\u0011f) for givenU;V) satisfying (from eq. 11a)\n\u000f\u0003\nf(\u000fc;\u0011f) =\u0000\u000f\u0003\nf(\u0000\u000fc;\u0000\u0011f): (14)\nSince\u001a0(\u000f) is compact, with band edges at \u000f=\u0006t\u0003\n(eq. 1), it follows (eq. 12) that the renormalised levels for\nany insulating phase necessarily satisfy\n\f\f\f\u0000\u000fc+V2\n\u000f\u0003\nf\f\f\f> t\u0003=1\n2W : insulators ;(15)\nwith the system metallic otherwise. We will invoke this\ncondition many times in the following.\nA. Fermi liquid phases\nThe condition eq. 15 for an insulator is general, but\ndoes not distinguish between Kondo and Mott insulators.\nWith that in mind we turn to the Fermi liquid regimes\nof the model { those adiabatically connected to the non-\ninteracting limit. A key result here20is that the total\ncharge is related to the renormalised level by39\n1\n2nt=Z\u0000\u000fc+V2\n\u000f\u0003\nf\n\u00001d\u000f \u001a 0(\u000f) +\u0012(\u0000\u000f\u0003\nf) (16)\n0.00.51.0\n-0.15 -0.1 -0.05 0.0 0.050.00.51.0\n0 30.00.51.0\n\u0019\n2t<Dc.!/\n!_t<\u0019\n2\u0000V_\u000fc\u00012t<Df.!/U_t<= 0:14\nU_t<= 0:155\nU_t<= 0:165\nIncreasingUFIG. 1. Illustrative single-particle spectra for the PAM,\n\u0019\n2t\u0003(V=\u000fc)2Df(!) and\u0019\n2t\u0003Dc(!),vs!=t\u0003. Shown for \fxed\n\u000fc=t\u0003= 3,V=t\u0003= 0:4 and\u0011f= 0, on increasing U=t\u0003towards\nand through the Mott transition: U=t\u0003= 0:14 and 0:155 in\nthe metal, and 0 :165 in the MI (the critical Uc=t\u0003'0:160).\nOn approaching the transition a low-energy metallic Kondo\nresonance arises, centred on the Fermi level, here lying at\nthe lower edge of the upper `Hubbard band'. The resonance\nnarrows progressively, vanishing on the spot at the transition,\nto leave a MI with a fully formed gap. Eq. 18 for the metallic\nphase is obeyed accurately by the NRG results. Lower inset:\nc-spectrum at higher energies, centred on !=\u000fcwith band\nhalfwidtht\u0003(there are no higher energy features in the f-\nspectrum).\n(with\u0012(x) the unit step function). This follows20from\neqs. 4-7, combined with the central fact that the Lut-\ntinger integral35,37\nIL:= ImZ0\n\u00001d! Gf(!)@\u0006f(!)\n@!(17)\nvanishes throughout a Fermi liquid phase, IL= 0, re\rect-\ning perturbative continuity to the non-interacting limit.\nFrom the perspective of the e\u000bective quantum impurity\nmodel, eq. 16 amounts to a Friedel sum rule.35,40\nFrom the condition eq. 15, the only non-trivial insu-\nlator that can arise from eq. 16 is clearly the nt= 2\nKI (which is perturbatively connected to the U= 0\nhybridization-gap insulator, and is thus a Fermi liquid).\nForj\u0000\u000fc+V2=\u000f\u0003\nfj<t\u0003by contrast the system is neces-\nsarily metallic, with D\u0017(0)6= 0 (eq. 12) and non-integral\nnt. There is thus no hint of a MI phase here; indeed from5\neq. 16,nt= 1 (or 3) arises only for \u000fc=V2=\u000f\u0003\nf, for which\n(eq. 12)Dc(0) =\u001a0(0) = (V=\u000fc)2Df(0), or equivalently\n\u0019\n2t\u0003Dc(0) = 1 =\u0019\n2t\u0003\u0012V\n\u000fc\u00132\nDf(0) :nt= 1;3 (18)\n{ corresponding to a metal. This re\rects the fact that the\nMI is not adiabatically connected to the non-interacting\nlimit, whence the Luttinger theorem IL= 0 does not\nhold. To handle this non-Fermi liquid insulator requires\na two-self-energy description,35,36considered in sec. V\nEq. 18 does of course capture the approach to the\nMott transition from the metallic , Fermi liquid side.\nHere, thef-level spectrum Df(!) contains a low-energy\nKondo resonance centred on the Fermi level (re\recting\na metallic Kondo e\u000bect). The width of the resonance is\ncharacterised by a low-energy scale !L(proportional to\nthe quasiparticle weight Z= [1\u0000(@\u0006R\nf(!)=@!)!=0]\u00001),\nwhich decreases continuously and vanishes as as the tran-\nsition is approached from the metal, nt!1; with the\nmetallic phase spectra at the Fermi level given by eq. 18.\nAt the transition the Kondo resonance then collapses `on\nthe spot', to leave the MI with a fully formed spectral\ngap. An explicit example is given in \fg. 1, showing NRG\nresults for both f- andc-level single-particle spectra as\nthe Mott transition is approached and crossed (we return\nto and discuss this \fgure several times in later sections).\nAlthough eq. 16 is con\fned to Fermi liquid phases, it\nproves important in understanding the behaviour of the\nPAM phase diagram, to which we now turn.\nB. Phase diagram: overview\nIn considering the phase diagram, our main interest\nis its dependence on \u000fc,Uand\u0011f, in particular for the\nM/MI (Mott) transition (where the hybridization Vac-\ntually scales out of the problem as shown in sec. IV E 1).\nIts qualitative behaviour can in fact be deduced on gen-\neral grounds, using little more than results already given.\nFig. 2 shows the schematic phase boundary in the\n(U=t\u0003;\u000fc=t\u0003)-plane, for f-level asymmetry \u0011f= 0 (\u000ff=\n\u0000U=2), with the boundaries between the KI/M and\nM/MI indicated. For \u0011f= 0 all phase boundaries are\nsymmetric under \u000fc$\u0000\u000fc(eq. 10), as indicated, so only\n\u000fc\u00150 need be considered in the following.\nThe key features of the phase diagram are as follows:\n(a) On increasing the interaction Ufor any \fxed \u000fc>t\u0003,\nor on increasing \u000fcfrom 0 for any \fxed U > 0, three\nsuccessive phases arise: a KI with nt= 2, followed by\na metal in which ntvaries continuously from 2 to 1,\nand \fnally a MI phase throughout which nt= 1. Each\nphase occupies a \fnite fraction of the parameter space,\ni.e. none is `special' in the sense of requiring parametric\n\fne-tuning. No further phases arise here (e.g. a metal\nwithnt<1, or a band insulator with nt= 0).\n(b) The phase boundaries decrease monotonically with\nincreasingU(as follows under the natural assumption\n-101\u000fc_t<\nU_t< KI (nt= 2)M (2>nt>1)\nM (2<nt<3)MI (nt= 1)\nMI (nt= 3)FIG. 2. Schematic phase boundary in the ( U=t\u0003;\u000fc=t\u0003)-\nplane, shown for asymmetry \u0011f= 0. Kondo insulator (KI),\nmetallic (M) and Mott insulator (MI) phases are indicated.\nThe KI/M phase boundary corresponds to a total charge nt=\nnf+nc= 2, and the M/MI boundary to nt= 1 for\u000fc>0\nandnt= 3 for\u000fc<0. Note that the MI phase arises only\nforj\u000fcj=t\u0003>1 (as is generic for j\u0011fj<1). For\u0011f= 0 the\nphase boundaries are precisely symmetric under \u000fc$\u0000\u000fc, as\nindicated.\nthatnt\u0011nt(\u000fc;U) in the M phase decreases monotoni-\ncally on increasing \u000fcorU).\n(c) In the strong coupling limit U!1 ,\u000fcfor the M/MI\nphase boundary asymptotically approaches t\u0003=1\n2W,\nwhile\u000fcfor the KI/M phase boundary vanishes.\n(d) In weak coupling U!0 by contrast, \u000fcfor both\nphase boundaries diverges.\nThe behaviour above is readily understood as follows:\n(i)Since\u000ff=\u0000U=2, it vanishes in the non-interacting\nlimit (where \u000f\u0003\nf\u0011\u000ff). Hence from eq. 16,39nt= 2\nforall\u000fc, i.e. the system is a KI along the entire \u000fc-\naxis forU= 0 (which gives the origin of (d) above).\nThe system is likewise a KI along the entire U-axis\nfor\u000fc= 0, as follows from eq. 16 on noting from\neq. 14 that \u000f\u0003\nf(\u000fc= 0;\u0011f= 0) = 0. Neither phase\nboundary can therefore intersect the axes; a KI thus\npersists for a \fnite range of U,\u000fcaway from them.\n(ii)Now consider U! 1 , remembering that \u000ff=\n\u0000U=2 and\u000ff+U= +U=2. Here the f-levels are\nasymptotically singly occupied only (with doubly-\noccupied or empty f-levels precluded). Hence nf=\n1, andnc\u0011n0\ncis equivalently that for the free con-\nduction band ( Vis obviously asymptotically inop-\nerative asU!1 ); i.e.\nnt= 1 +n0\nc :U!1 (19a)\n1\n2n0\nc=Z\u0000\u000fc\n\u00001d\u000f \u001a 0(\u000f) (19b)\nFrom this, n0\nc= 1 and hence nt= 2 (a KI) arises\nonly for\u000fc= 0. By contrast, n0\nc= 0 and hence6\nnt= 1 (a MI) arises for all \u000fc>t\u0003(the band edges\nof\u001a0(\u000f) occur at\u0006t\u0003). These give the U! 1\nasymptotes for the KI/M and M/MI boundaries in\n(c) above (as shown in \fg. 2).\n(iii) Consider now any given U > 0, as\u000fc! 1 .\nHere the conduction band is `projected out', so\nthec-level charge ncobviously vanishes and the\nhybridization V\u00110 is inoperative. The cor-\nrelatedf-levels are thus free and singly-occupied\nonly,nf= 1; whence nt= 1, corresponding to a\nMI. This is why no further phases arise in \fg. 2, as\nnoted in (a) above.\nOne further observation bears note here. As argued\nabove, the critical \u000fc(U) for the Mott transition exceeds\nt\u0003for all \fnite U, and indeed diverges as U!0. In\notherwords the metal persists to values of \u000fcfar in excess\nof the Fermi level, where one might naively expect the\nconduction band to empty and the system to be insulat-\ning; and the corresponding critical U=t\u0003for the transition\nasymptotically vanishes, i.e. appears in `weak coupling'\n{ somewhat counterintuitively for a transition regarded\nas an archetype of strong correlations. This is striking,\nsuggests that the transition is a fairly subtle a\u000bair, and\ninvites an explanation. We provide it in secs. IV \u000b.\n1. Renormalised levels\nThe phase boundaries correspond of course to spe-\nci\fc values of nt. Eq. 16 thus generically determines the\nrenormalised levels \u000f\u0003\nfalong them; or more precisely, in\nthe case of the M/MI boundary, as the transition is ap-\nproached from the M side (remembering that eq. 16 holds\nstrictly for FL phases). Note from eq. 16 that nt\u00142 for\n\u000f\u0003\nf\u00150, whence\u000f\u0003\nf\u00150 for\u000fc\u00150. For the M/MI border,\nwherent!1+, eq. 16 requires \u0000\u000fc+V2=\u000f\u0003\nf= 0, i.e.\n\u000f\u0003\nf=V2\n\u000fc: M=MI boundary ; \u000fc\u0015t\u0003: (20)\nFor the KI/M border, eq. 16 gives \u0000\u000fc+V2=\u000f\u0003\nf=t\u0003, i.e.\n\u000f\u0003\nf=V2\n\u000fc+t\u0003: KI=M boundary ; \u000fc\u00150:(21)\nReferring then to \fg. 2, on increasing the interaction for\nany \fxed\u000fc>t\u0003the renormalised level thus progressively\nincreases from ( \u000ff\u0011)\u000f\u0003\nf= 0 forU= 0, to\u000f\u0003\nf=V2=(\u000fc+\nt\u0003) at the KI/M boundary, to \u000f\u0003\nf=V2=\u000fcas the M/MI\ntransition is approached from the metal.\nResults for \u000fc\u00140 naturally follow in a directly anal-\nogous way, giving \u000f\u0003\nf=\u0000V2=(j\u000fcj+t\u0003) for the KI/M\nboundary, and \u000f\u0003\nf=\u0000V2=j\u000fcjfor the M/MI border. For\n\u0011f= 0 in particular (\fg. 2), these re\rect the symmetry\n\u000f\u0003\nf(\u000fc;0) =\u0000\u000f\u0003\nf(\u0000\u000fc;0) for given U(eq. 14).41\nWe return to these results in sec. V B (\fg. 10), after \u000f\u0003\nf\nin the MI phase has been considered.\nFIG. 3. NRG-determined phase boundaries, shown in the\n(U=t\u0003;\u000fc=t\u0003)-plane for di\u000berent values of the f-level asymme-\ntry\u0011f= 0;1\n4;1\n2;3\n4;1;5\n4(from bottom to top). V=t\u0003= 0:4 is\n\fxed, and only \u000fc=t\u0003\u00150 is shown for clarity. The charge nt\nis colour coded as indicated. The transition to the nt= 1\nMott insulator is shown as a solid line, the KI/M transition\nas dashed, and the metal to nt= 0 band insulator transition\nis the dotted line (as relevant to the \u0011f=5\n4case).\nC. Phase diagram: take two\nWhile we have thus far focused on \u0011f= 0, the same\nessential characteristics above arise also for j\u0011fj<1 (al-\nthough for \u0011f6= 0 the phase boundaries are obviously\nno longer strictly symmetric under \u000fc$\u0000\u000fcat \fxedU).\nHere,\u000ff=\u00001\n2U(1\u0000\u0011f)<0 and\u000ff+U=1\n2U(1+\u0011f)>0\nfor allU > 0, so the arguments given in (ii)and(iii)\nabove again go through. Likewise, by the same reason-\ning as in (i)above, the system is a KI along the entire\n\u000fc-axis forU= 0 (so for \u000fc= 0 in particular, nt= 2 (KI)\nfor both the U= 0 andU!1 asymptotes).\nThere is a further notable feature in \fg. 2, occur-\nring generally for j\u0011fj<1: the Mott transition to the\nnt= 1 insulator arising for \u000fc>0 occurs solely from the\nelectron-doped side, i.e. nt= 1 +\u000ewith\u000e!0+ [and\ncorrespondingly for \u000fc<0, the transition to the nt= 3\nMI occurs only from the hole-doped side, nt= 3+\u000ewith\n\u000e!0\u0000]. This re\rects the fact, see (iii) above, that the\nnt= 1 MI persists as \u000fc!1 for anyU > 0.\nThat situation changes for asymmetries j\u0011fj>1,\nthef-level mixed-valence regime. For \u0011f>1,\n\u000ff=\u00001\n2U(1\u0000\u0011f) and\u000ff+Uboth exceed 0 (the Fermi7\nlevel). Hence, following the arguments in (iii) above, as\n\u000fc!1 for any given U > 0, thef-levels become free\nbut are now unoccupied, with nf= 0 (=nc): the system\nis thus asymptotically an nt= 0 band insulator [or an\nnt= 4 band insulator from the obvious argument for\n\u000fc!\u00001 ]. For this reason the nt= 1 MI sector of the\nphase diagram is then surrounded on the higher- \u000fcside\nby a metal with nt<1 (which for larger \u000fcundergoes\na transition to the nt= 0 band insulator); and on the\nlower-\u000fcside by a metal with nt>1. In this case the\nMott transition can thus be approached from either\nthe electron-doped or the hole-doped sides, nt!1\u0006\nrespectively; although we add that, in contrast to the\nMott transition in the one-band Hubbard model, the\ntwo are not in any way related by a ph-transformation.\nThat the characteristics deduced in the sections above\nare indeed found in NRG calculations is seen clearly in\n\fg. 3, where resultant phase diagrams are shown in the\n(U=t\u0003;\u000fc=t\u0003)-plane for six representative values of the f-\nlevel asymmetry \u0011f. NRG phase boundaries between\nmetallic and insulating phases are determined36by ap-\nproaching a transition from the metal, and locating the\npoints where the total charge nt!1, 2 or 0 for the metal\nto MI, KI and band insulator phases respectively. Note\nthat the\u0011f= 5=4 case in particular shows all possible\nphases { metals with nt?1, MI, KI and band insulators.\nThe results above also provide concrete examples of\nZaanen-Sawatzky-Allen (ZSA) diagrams,32with Mott in-\nsulators classi\fed as either of `Mott-Hubbard' (MH) or\n`charge-transfer' (CT) type; according respectively to\nwhetherU < \u0001 orU > \u0001, with \u0001 =j\u000fc\u0000\u000ffj=\nj\u000fc+1\n2U(1\u0000\u0011f)jthe charge transfer energy. Fig. 4\nshows an NRG phase diagram for \u0011f= 0, including the\n(crossover) line \u0001 = Useparating MH and CT insula-\ntors, viz\u000fc;cr=\u000ff+U\u00111\n2U. For transition metal (TM)\nmaterials, oxides or halides of the lighter TMs tend to\nbe MH insulators, while those of the heavier TMs are\nmore typically of CT type.32Metallic TM compounds\nwith \u0001\u001dUare `d-band metals' (again usually for the\nlighter TMs), characterised by narrow bands and heavy\nelectrons/holes;32while those with \u0001 \u001cUare `p-band\nmetals', with light carriers and broad bands.\nIn the following sections, within the periodic Anderson\nmodel under study, we aim to shed some light on how and\nwhy these and concomitant characteristics arise.\nIV. MAPPING TO A HUBBARD MODEL\nIt has hitherto been thought that the low-energy prop-\nerties of the metallic phase close to the Mott transition\ncannot be interpreted generally in terms of a single-band\nHubbard model.24,30Here, however, we show that the\ne\u000bective low-energy model which describes the PAM in\nthe vicinity of the Mott transition, is in fact a one-band\nHubbard model, with long-ranged hopping between the\nsites; speci\fcally hoppings that connect all pairs of sites\n024\n0 1 2 3 4 5\n\u000fc_t<\nU_t<KIMMI\n\u0001=UA\nCBMH\nCTFIG. 4. NRG phase diagram for \u0011f= 0 (withV=t\u0003= 0:4),\nshowing the crossover line with charge transfer energy \u0001 = U\n(i.e.\u000fc=1\n2U) separating Mott-Hubbard and charge-transfer\ninsulators; amounting to a particular example of a Zaanen-\nSawatzky-Allen diagram.32Discussion in text. Points A, B\nand C indicate metals with \u0001 \u001dU, \u0001\u001cUand \u0001'U\nrespectively; considered as representative examples in secs.\nIV F (single-particle dynamics for point A are shown in \fgs.\n1 and 7, and for point B in \fg. 8).\nrather than simply nearest neighbours, and which de-\ncay exponentially with the topological distance between\nthem. The plausibility of generating an e\u000bective one-\nband model is intuitive, since on integrating out virtual\nexcitations to high-lying c-levels one expects to generate\ne\u000bective hoppings between f-levels. There are in fact\nseveral ways to obtain these results, and here we do so\nby direct analysis of the underlying propagators. We also\nfocus in the following on the Mott transition for nt= 1,\nwhich (secs. III B,III C) requires \u000fc> t\u0003(thent= 3\ncase obviously follows in the same way by considering\n\u000fc<\u0000t\u0003, and adds nothing new).\nA. One-band Feenberg self-energy\nTo this end we \frst point out a general condition that\nmust be satis\fed by the Feenberg self-energy38for a one-\nband model. Within DMFT, the local propagator for any\none-band model (here denoted generically by G(!)) is of\nform\nG(!) =Z1\n\u00001d\u000f~\u001a(\u000f)\n!+\u0000\u000f0\u0000\u0006(!)\u0000\u000f(22)\nwith\u000f0the site-energy, \u0006( !) the (local) self-energy, and\n~\u001a(\u000f) the non-interacting DoS for the one-band model.\nThe corresponding Feenberg self-energy S(!) is (by its\nde\fnition) related to G(!) by\nG(!) =\u0002\n!+\u0000\u000f0\u0000\u0006(!)\u0000S(!)\u0003\u00001(23)8\n(e.g.S(!) =1\n4~t2\n\u0003G(!) for a NN Hubbard model on a\nBethe lattice, with hopping ~t\u0003). Eqs. 22,23 are general;\nwhat distinguishes di\u000berent one-band models is of course\nthe associated DoS ~ \u001a(\u000f). From eqs. 22,23 it follows that\nG(!) =Z1\n\u00001d\u000f~\u001a(\u000f)\n1\nG(!)+S(!)\u0000\u000f: (24)\nHenceS\u0011S(G(!)); i.e. the !-dependence of the one-\nband Feenberg self-energy arises solely from its depen-\ndence onG(!), with no explicit !-dependence otherwise.\nThis necessary condition for a one-band model is ex-\nploited below.\nB. E\u000bective one-band model\nSince the Mott transition occurs generally only for \u000fc>\nt\u0003(secs. III B,III C), it is this regime on which we focus\nin the following. Consider \frst the propagator gc(!) for\nthe free (V= 0) PAMc-band, given (eq. 8a) by\ngc(!) =\u0002\n!+\u0000\u000fc\u00001\n4t2\n\u0003gc(!)\u0003\u00001: (25)\nThe corresponding spectrum dc\n0(!) =\u001a0(!\u0000\u000fc) is non-\nzero forj!\u0000\u000fcj< t\u0003; so since\u000fc> t\u0003,dc\n0(!= 0) in\nparticular vanishes and gc(0) is pure real,\n1\n2t\u0003gc(0) =\u0000\u000fc\nt\u0003+s\u0012\u000fc\nt\u0003\u00132\n\u00001 (26a)\n\u000fc=t\u0003\u001d1\u0018 \u0000t\u0003\n2\u000fc: (26b)\nWe seekGf(!) in the vicinity of the Fermi level, and\nto that end consider the f-level Feenberg self-energy\nSf(!) =V2[!+\u0000\u000fc\u00001\n4t2\n\u0003Gc(!)]\u00001(eq. 8). Separate\nGc(!) =gc(!) +\u000eGc(!) (27)\nwheregc(!) is pure real for !\u00190, so the spectral weight\ninGc(!) resides entirely in \u000eGc(!). Hence, using eq. 25,\nSf(!) =V2gc(!)\n1\u00001\n4t2\u0003gc(!)\u000eGc(!): (28)\nWe thus consider \u000eGc(!), and \frst de\fne the auxilliary\nfunctions\n~gc(!) =\u0002\n!+\u0000\u000fc\u00001\n4t2\n\u0003Gc(!)\u0003\u00001\ngf(!) =\u0002\n!+\u0000\u000ff\u0000\u0006f(!)\u0003\u00001:\nThen eqs. 8 give the identity\nGc(!) = ~gc(!) + [V~gc(!)]2Gf(!): (29)\nSeparating Gc(!) as in eq. 27, and noting that ~ gc(!) =\ngc(!)=[1\u00001\n4t2\n\u0003gc(!)\u000eGc(!)], eq. 29 gives a simple cubic\nequation for t\u0003\u000eGc(!),\n1\n16(t\u0003gc)2[t\u0003\u000eGc(!)]3+1\n16t\u0003gc([t\u0003gc]2\u00008) [t\u0003\u000eGc(!)]2\n+ (1\u00001\n4[t\u0003gc]2)t\u0003\u000eGc(!) = (Vgc)2t\u0003Gf(!)\n(30)wheregc\u0011gc(!). In the trivial uncoupled limit of V= 0,\nthe correct physical root is of course \u000eGc(!) = 0.\nEq. 30 shows that \u000eGc(!) { and hence Sf(!) (eq. 28)\n{ is in general a function of Gf(!)andgc(!) (which is\nan explicit function of !). By the argument given in sec.\nIV A,Gf(!) does not therefore generically reduce to an\ne\u000bective one-band model, just as one expects. There are\nhowever two important exceptions, as now considered.\n1.\u000fc\u001dt\u0003\nFor\u000fc\u001dt\u0003, the free conduction band (centred on !=\n\u000fc) lies so far above the Fermi level ( != 0) that for\nall!\u001c\u000fc\u0000t\u0003{ and in particular for energies around\nthe Fermi level { gc(!) is independent of !and given\nbygc(0)\u0018\u00001=\u000fc(eq. 26b). In this case eq. 30 gives to\nleading order\n\u000eGc(!) =\u0012V\n\u000fc\u00132\nGf(!) (31)\n(where only the \fnal term on the left side of eq. 30 is\nrelevant); showing that for all !\u001c\u000fc\u0000t\u0003, the spectral\ndensity ofGc(!) is entirely controlled by that of Gf(!).\nEq. 28 then yields asymptotically\nSf(!) =\u0000V2\n\u000fc+1\n4\u0012\nt\u0003V2\n\u000f2c\u00132\nGf(!);\nthe!-dependence of which is encoded solely in that of\nGf(!). The e\u000bective low-energy model is thus one-band,\nwith eq. 8c reducing to\nGf(!) =\u0014\n!+\u0000(\u000ff\u0000V2\n\u000fc)\u0000\u0006f(!)\u00001\n4t2\n\u0003e\u000bGf(!)\u0015\u00001\n(32)\nwhere\nt\u0003e\u000b=t\u0003\u0012V\n\u000fc\u00132\n: (33)\nThis corresponds precisely to an e\u000bective NN one-band\nHubbard model for the f-levels alone, with a site-energy\n\u000f0=\u000ff\u0000V2=\u000fc(naturally renormalised from the `bare'\n\u000ffdue to an induced level-shift from the c-band), and\nan e\u000bective NN hopping t\u0003e\u000b. Since the shortest con-\nnected path between two NN f-levels is a 3-step path {\nproceeding by virtual hopping to and between the inter-\nvening pair of c-levels { the third-order perturbative form\neq. 33 fort\u0003e\u000bis physically quite obvious.\n2. General case: vicinity of the Mott transition\nThe second case where the e\u000bective low-energy model\nfor the PAM is of one-band form, is quite generally in the\nvicinity of the Mott transition for any\u000fc> t\u0003, as now9\nshown. Here the f-spectrum in the metallic phase con-\ntains a Kondo-like resonance, characterised by the low-\nenergy scale !Lwhich vanishes at the transition. The\nscaling regime of the transition corresponds to consider-\ning any \fnite ~ !=!=!Lin the formal limit !L!0;\nand as such encompasses the entire energy-dependence\nof the underlying Kondo resonance. In this regime, the\npropagators G\u0017and self-energy \u0006 fare functions solely\nof ~!, although we continue to denote them by G\u0017(!) and\n\u0006f(!). By contrast, `bare' factors of !\u0011!L~!can of\ncourse be neglected; e.g. eq. 8b,8c becomes\nGf(!) =\u0014\ni0+\u0000\u000ff\u0000\u0006f(!)\u0000V2\ni0+\u0000\u000fc\u00001\n4t2\u0003Gc(!)\u0015\u00001\n(34a)\n=\u0002\ni0+\u0000\u000ff\u0000\u0006f(!)\u0000Sf(!)\u0003\u00001; (34b)\nand likewise gc(!)\u0011gc(0). Separating Gc(!)\u0011gc(0) +\n\u000eGc(!), and running through the argument given in eqs.\n27\u000b, leads again to the cubic eq. 30 for t\u0003\u000eGc(!), but\nnow withgc\u0011gc(0); such that \u000eGc(!)\u0011\u000eGc(Gf(!)) is\na function of Gf(!) alone. Hence the ~ !-dependence of\n\u000eGc(!), and in consequence of\nSf(!)\u0011V2gc(0)\n1\u00001\n4t2\u0003gc(0)\u000eGc(Gf(!)); (35)\narises solely from that of Gf(!). By the argument given\nin sec. IV A, the e\u000bective low-energy model in the vicinity\nof the Mott transition is thus generally of one-band form.\nAs in eqs. 22,24, to determine what that e\u000bective model\nis requires us to ascertain the corresponding DoS ~ \u001a(\u000f).\nThis we now consider.\nC. Long-ranged Hubbard model\nIn the ~!-scaling regime of the Mott transition discussed\nabove, thef-propagator is given by (see eqs. 5,7a)\nGf(!) =Z1\n\u00001d\u000f\u001a0(\u000f)\ni0+\u0000\u000ff\u0000\u0006f(!)\u0000V2\ni0+\u0000\u000fc\u0000\u000f(36)\n(with\u001a0(\u000f) from eq. 2); which again we emphasise en-\ncompasses the ~ !-scaling regime of the entire Kondo res-\nonance. Now de\fne\nt\u0003e\u000b=t\u0003[Vgc(0)]2(=4V2\nt\u0003y2) (37a)\ny=1\n2t\u0003gc(0); (37b)\nthe physical signi\fcance of which will become clear below;\nwhereyis a function solely of \u000fc=t\u0003, from eq. 26a. Since\ngc(0) satis\fes eq. 25, it follows that ( i0+\u0000\u000fc)gc(0) =\n1+[1\n2t\u0003gc(0)]2, whenceV2[i0+\u0000\u000fc\u0000\u000f]\u00001=t\u0003eff\n2y[1+y2\u00002y\u000f\nt\u0003]\u00001. Eq. 36 thus becomes\nGf(!) =\nZ1\n\u00001d\u000f\u001a0(\u000f)\ni0+\u0000(\u000ff+t\u0003eff\n2y)\u0000\u0006f(!)\u0000t\u0003eff\n2y\u0010\n1\n(1+y2)\u00002y\u000f\nt\u0003\u00001\u0011\n(38)\nNow simply change integration variables from \u000fto\u000f0,\nde\fned by\n\u000f0=t\u0003e\u000b\n2y \n1\n(1 +y2)\u00002y\u000f\nt\u0003\u00001!\n;\nand with\u000f\u0011\u000f(\u000f0) given on inversion by\n\u000f=t\u0003\n2y \n[1 +y2]\u00001\n1 +2y\nt\u0003eff\u000f0!\n=t\u0003\u0015(\u000f0)(39)\nand\u0015(\u000f0) thus de\fned. With this (using t\u0003e\u000b=2y=\nV2gc(0) from eq. 37), eq. 38 reduces to\nGf(!) =Z1\n\u00001d\u000f~\u001a(\u000f)\ni0+\u0000(\u000ff+V2gc(0))\u0000\u0006f(!)\u0000\u000f\n(40)\nwhere\n~\u001a(\u000f) =1\n\u0010\n1 +2y\nt\u0003eff\u000f\u001122\n\u0019t\u0003e\u000bp\n1\u0000\u00152(\u000f): (41)\nEq. 41 is precisely the density of states for a one-band\ntight-binding model on a ( Zc!1 ) Bethe lattice with\nexponentially decreasing hopping amplitudes, as consid-\nered in Ref. [34]; such that the nthNN hopping ampli-\ntude,t(n)\n\u0003e\u000b, is\nt(n)\n\u0003e\u000b=t\u0003e\u000byn\u00001=t\u0003\u00122V\nt\u0003\u00132\nyn+1(42)\nwith a nearest neighbour ( n= 1) hopping amplitude of\nt\u0003e\u000b=t\u0003[Vgc(0)]2(as in eq. 37a).\nEq. 40 is thus the local propagator for a one-band Hub-\nbard model with this long-ranged hopping; and an e\u000bec-\ntive site-energy \u000f0(eq. 22) given on comparison of eqs.\n22,40 by\n\u000f0=\u000ff+V2gc(0) (43a)\n=\u000ff+2V2\nt\u0003y=\u000ff+t\u0003e\u000b\n2y(43b)\n(with1\n2t\u0003gc(0) =ypure real for all \u000fc> t\u0003, eq. 26).\nPhysically, \u000f06=\u000ffre\rects the renormalization of the\nsite-energy from the `bare' \u000ffdue to level-repulsion from\nthec-band. Note that the general case considered here\nreduces to the results of sec. IV B 1, in the regime \u000fc=t\u0003\u001d10\n1 where the low-energy Hubbard model becomes in e\u000bect\npure NN, with \u000f0=\u000ff\u0000V2=\u000fc.\nNow consider brie\ry the non-interacting e\u000bective Hub-\nbard model's spectrum ~ \u001a(\u000f), eq. 41, which is clearly ph-\nasymmetric in general (~ \u001a(\u000f)6= ~\u001a(\u0000\u000f)).34Its band edges\nin particular occur at \u000f=W+and\u000f=\u0000W\u0000, with\nW+6=W\u0000in general. These are given simply by\nW\u0007=[2\u0006y]\n[1\u0006y]2t\u0003e\u000b\n2; (44)\nwithW\u0000\u0015W+(sincey=1\n2t\u0003gc(0)2(\u00001;0) for all\n\u000fc=t\u00032(1;1), eq. 26a). For the case \u000fc=t\u0003\u001d1, where\ny!0 and\u0015(\u000f)!\u000f=t\u0003e\u000b(eq. 39),W\u0007both approach\nt\u0003e\u000b. The DoS ~ \u001a(\u000f) then asymptotically approaches\n~\u001a0(\u000f) =2\n\u0019t\u0003e\u000bp\n1\u0000(\u000f=t\u0003e\u000b)2 :\u000fc\nt\u0003\u001d1 (45)\n{ a symmetric semicircular spectrum ( cfeq. 2), but with\narenormalised band halfwidth, t\u0003e\u000b, indicative of the\npurely NN Hubbard model arising in this limit.\nD. Qualitative consequences: \u000fc=t\u0003\u001d1\nBefore pursuing the general mapping to a one-band\nlong-ranged Hubbard model in the vicinity of a Mott\ntransition, we consider some qualitative physical conse-\nquences of it for the regime \u000fc=t\u0003\u001d1 (sec. IV B 1); as\nthus relevant close to the transition to a Mott-Hubbard\ninsulator in the ZSA scheme32(sec. III C and \fg. 4). Four\npoints in particular should be noted:\n(i)Sincegc(!) is pure real, eqs. 27,31 show that the c-\nandf-band spectra are asymptotically related by\nDc(!) =\u0012V\n\u000fc\u00132\nDf(!): (46)\nEq. 46 conforms as it must to the general result eq. 18 for\nthe Fermi level ( != 0) spectra at the Mott transition.\nBut it holds generally for all!\u001c\u000fc\u0000t\u0003(sec. IV B 1);\nwhere the!-dependences of Dc(!) andDf(!) are thus\nthe same, with Dc(!) simply `cut down' from Df(!) by\nthe factor ( V=\u000fc)2\u001c1. This behaviour is indeed ob-\nserved in NRG calculations; being apparent even e.g. for\n\u000fc=t\u0003= 3, as shown in \fg. 1.\n(ii)For\u000fc=t\u0003\u001d1, the entire free ( V= 0)c-band DoS {\ndc\n0(!) =\u001a0(!\u0000\u000fc), centred on !=\u000fc{ lies far in excess\nof the Fermi level and as such is `empty'. This is natu-\nrally re\rected in the full c-band spectrum Dc(!), where\nthe great bulk of its spectral weight lies within \u0018\u0006t\u0003\nof!=\u000fc(as seen e.g. in the inset to \fg. 1). Eq. 46\nnevertheless shows that, due to the local hybridization\nVbetween the high-lying c-level and the f-level, the full\nc-band spectrum Dc(!) acquires a small spectral weight\nfor energies !\u001c\u000fc\u0000t\u0003which encompass the Fermi level;\nthereby enabling a metallic state to persist in a regime\nwhere one might naively expect otherwise.(iii) As one anticipates near the transition to a Mott-\nHubbard insulator,32thef- andc-bands in the general\nvicinity of the Fermi level are narrow . The natural energy\nscale for these bands is not however the bare hopping t\u0003,\nbut rather the e\u000bective hopping t\u0003e\u000b\u0018t\u0003(V=\u000fc)2\u001ct\u0003.\nThis too is evident in \fg. 1, for which We\u000b= 2t\u0003e\u000b'\n0:04t\u0003(some two orders of magnitude less than the scale\nW= 2t\u0003associated with dc\n0(!)). Likewise, the low-\nenergy `coherence' scale !Lcharacteristic of the Fermi\nliquid Kondo resonance in the f- andc-electron spectra\nis not\u0018Zt\u0003, but rather !L\u0018Zt\u0003e\u000b\u001cZt\u0003(providing\na natural explanation for the marked di\u000berence observed\nbetweenZt\u0003and the coherence scale in Ref. [30]).\n(iv) Since eq. 46 holds in particular for all !around\nand below the Fermi level, the c- andf-level charges\nare directly related by nc= (V=\u000fc)2nf. Hence, since\nnt=nc+nf= 1 at the Mott transition,\nnf= 1\u0000\u0012V\n\u000fc\u00132\n; nc=\u0012V\n\u000fc\u00132\n(47)\nhold asymptotically for \u000fc=t\u0003\u001d1. This behaviour is con-\n\frmed by our NRG calculations. It also underscores the\nfact24,30that, for the transition to a Mott-Hubbard insu-\nlator (\u0001\u001dUin the ZSA scheme32), thef-level charge\nitself is very close, but not exactly equal, to one.\n1. Kondo lattice model\nFollowing on from the above, we point out that while\nthe physics captured here is inherent to the periodic An-\nderson model, it is quite di\u000berent from that arising in\na Kondo lattice model (KLM). In the latter case f-level\nelectrons are replaced by strictly localised spins, which\nare purely exchange-coupled to an otherwise free c-band,\nand hence generate scattering of c-band electrons. But\nif the freec-band is empty { as arises e.g. for \u000fc\u001dt\u0003\n{ then there are no electrons to scatter, and the system\nis insulating. Indeed by this reasoning one expects the\nKLM to be insulating for any \u000fc>t\u0003; in marked contrast\nto the PAM (e.g. \fg. 2) where a metallic phase persists\nup to critical \u000fc's far in excess of t\u0003. This re\rects the fact\nthat the procedure of secs. IV B,IV C generates an e\u000bec-\ntive one-electron hopping betweenf-level electrons in the\nPAM, via higher-order (and indeed fundamentally di\u000ber-\nent) virtual processes to those arising in the conventional\nSchrie\u000ber-Wol\u000b transformation employed to map a PAM\nonto a KLM; and does so moreover without the constraint\nthatnfis precisely unity (which itself would preclude a\nnon-trivial Mott transition, for which nt=nf+nc= 1 is\nrequired). Put more bluntly, the rich physics of the Mott\ntransition in the PAM is simply absent in the KLM.\nE. Phase diagram\nGiven the mapping to an e\u000bective one-band Hubbard\nmodel in the vicinity of a Mott transition, we revisit the11\n02468\n0.0 0.1 0.2 0.3\n02468\n0 1 2 3 4 5\n\u000fc_t<\nU_t<\u000fc_t<\nU=Ut<_V21 + 2_UV_t<= 0:4\nV_t<= 0:2\nFIG. 5. NRG results for the PAM Mott transition, for\n\fxed\u0011f= 0, shown for two di\u000berent hybridization strengths\nV=t\u0003= 0:4 (circles) and 0 :2 (diamonds). Inset shows the\ncritical\u000fc=t\u0003vsU=t\u0003, while the main \fgure shows \u000fc=t\u0003vs\n~U=Ut\u0003=V2. When scaled in this way the phase boundary\nis seen to be independent of the hybridization (sec. IV E 1) to\nwithin numerical accuracy. The \fgure also shows comparison\nto eq. 52 (solid line); which is asymptotically exact for \u000fc=t\u0003\u001d\n1, but captures the phase boundary very well over essentially\nthe full\u000fc=t\u0003range.\nphase diagram of the PAM in the light of it. As in pre-\nvious sections we naturally focus on the Mott transition\nfornt= 1 (requiring as such \u000fc=t\u0003>1 and\u0011f\u00150).\n1. Phase boundary: scaling collapse\nThe e\u000bective one-band Hubbard model onto which the\nPAM maps in the vicinity of the Mott transition, is of\ncourse determined by itsthree intrinsic parameters: (a)\nU=t\u0003e\u000b; (b)y=1\n2t\u0003gc(0) (eq. 26a), which controls the ra-\ntio of thenthNN to the NN hopping amplitudes t(n)\n\u0003e\u000b=t\u0003e\u000b\n(eq. 42); and (c) the asymmetry\n\u0011e\u000b:= 1 +2\u000f0\nU; (48)\ngiven by (see eq. 43b)\n\u0011e\u000b=\u0011f+t\u0003e\u000b\nUy(49)\nwith\u0011fthef-level asymmetry (eq. 3).\nNotice then that for given \u0011f, the asymmetry \u0011e\u000bis\nentirely determined by U=t\u0003e\u000bandy; whence the ef-\nfective Hubbard model depends solely on U=t\u0003e\u000bandy\n(\u0011y(\u000fc=t\u0003)). This is one parameter fewer than for the\nPAM itself, which for given \u0011fis speci\fed by U=t\u0003,\u000fc=t\u0003\nandV=t\u0003; and this parametric redundancy clearly im-\nplies a scaling collapse of the phase boundary for the\nMott transition. The origin of this is clear from eq. 37:\nVenters the e\u000bective Hubbard model parameters solely\nthrought\u0003e\u000b/V2=t\u0003. Hence the critical U=t\u0003e\u000bas afunction of \u000fc=t\u0003(which determines y) must be indepen-\ndent of the hybridization; or equivalently \u000fc=t\u0003vsUt\u0003=V2\nmust beV-independent. NRG results for the M/MI tran-\nsition in the PAM are shown in \fg. 5 for two di\u000berent val-\nues ofV=t\u0003, with the inset showing the phase boundaries\nas\u000fc=t\u0003vsU=t\u0003. The resultant scaling collapse is seen in\nthe main \fgure, showing the same data vs~U=Ut\u0003=V2\n(and we attribute the small deviation from perfect col-\nlapse to numerical inaccuracies in our NRG calculations).\n2. Mott transition phase boundaries\nTo gain further insight we consider now a physically\nintuitive, if in general approximate, estimate for the\nMott transition boundaries, by simple consideration of\nthe spectrum D(!) of the underlying e\u000bective one-band\nHubbard model. The already apparent richness of the\nphase behaviour will be seen to re\rect the fact that the\ne\u000bective model has long-ranged (rather than solely NN)\nhoppings, as embodied in the non-trvial y\u0011y(\u000fc=t\u0003)-\ndependence of the problem.\nD(!) in the MI phase consists of lower and upper Hub-\nbard bands (HBs) centred on !=\u000f0and!=\u000f0+U\nrespectively (with site-energy \u000f0given by eq. 43). Model\nthe HBs simply as non-interacting bands centred on\n!=\u000f0,\u000f0+U. The upper edge of the lower Hub-\nbard band (LHB) then occurs at !=\u000f0+W+, and\nthe lower edge of the upper Hubbard band (UHB) at\n!=\u000f0+U\u0000W\u0000; withW\u0007given by eq. 44. Now adopt\nthe familiar simple estimate that the system is a MI pro-\nvided the Fermi level EF\u00110 lies above the upper edge\nof the LHB and below the lower edge of the UHB; i.e.\nprovided\u000f0+W+\u00140\u0014\u000f0+U\u0000W\u0000. Recalling that\n\u000ff=\u00001\n2U(1\u0000\u0011f), and using eq. 37a for t\u0003e\u000b, this con-\ndition is readily shown to reduce to\n~U(\u0011f\u00001)\u00002\n(\u000fc\nt\u0003+ 1)\u00140\u0014~U(\u0011f+1)\u00002\n(\u000fc\nt\u0003\u00001)(50)\nwhere\n~U=Ut\u0003\nV2(51)\nis thus de\fned (such that the entire V-dependence of\nthe phase boundary is encoded on ~U, as established on\ngeneral grounds in sec. IV E 1 above).\nThe right inequality in eq. 50 is the condition for the\nlower edge of the UHB to lie above or precisely at the\nFermi level. The latter corresponds to the nt= 1+\n(`electron-doped') Mott transition, the critical \u000fcfor\nwhich (denoted by \u000fc>) is thus given by\n\u000fc>\nt\u0003= 1 +2\n(\u0011f+ 1) ~U:nt= 1+ (52)\n(holding for the full range \u0011f\u00150). The left inequality\nin eq. 50 is by contrast the condition for the upper edge12\nof the LHB to lie below or at the Fermi level, the latter\ncorresponding to an nt= 1\u0000(`hole-doped') Mott transi-\ntion. But for \u0011f\u00141, the left hand side of this inequality\nis necessarily negative for all U\u00150. Annt= 1\u0000Mott\ntransition cannot therefore occur for \u0011f<1, as argued\ngenerally in sec. III C and con\frmed by NRG (\fg. 3).\nFor\u0011f>1, bothnt= 1\u0000andnt= 1+ transitions can\nthen occur. The critical \u000fc\u0011\u000fc>fornt= 1+ is again\ngiven by eq. 52, while \u000fc\u0011\u000fc<for thent= 1\u0000Mott\ntransition follows from the left side equality in eq. 50 as\n\u000fc<\nt\u0003=\u00001 +2\n(\u0011f\u00001)~U:nt= 1\u0000:(53)\nThe Mott insulating region in the ( ~U;\u000fc=t\u0003)-plane is thus\nbounded by the lines \u000fc>and\u000fc<(\u0015\u000fc>); which, since\neq. 50 must be satis\fed, requires \u000fc\u0015\u000fc\u0003and ~U\u0014~U\u0003,\ngiven (on equating \u000fc>=\u000fc<, eqs. 52,53) by\n\u000fc\u0003\nt\u0003=\u0011f;~U\u0003=2\n(\u00112\nf\u00001): (54)\nThe asymmetry of the e\u000bective Hubbard model fol-\nlows from eqs. 49,37a,51 as \u0011e\u000b=\u0011f\u00004jyj=~U; whence\nthe asymmetries for the two Mott transition boundaries\n(denoted by \u0011e\u000b?) are given using eqs. 52,53 by\n\u0011e\u000b?=\u0011f\u00002(\u0011f\u00061)(x\u00071)h\nx\u0000p\nx2\u00001i\n(55)\nwherex=\u000fc=t\u0003. For\u000fc=t\u0003\u001d1 in particular, this yields\n\u0011e\u000b?\u000fc=t\u0003\u001d1\u0018 \u0007 1 (56)\nindependently of \u0011f. The\u0011e\u000b's for thent=\u00061 Mott\ntransitions are in otherwords the maximum that are pos-\nsible36for a purely NN Hubbard model (which arises\nasymptotically for \u000fc=t\u0003\u001d1, sec. IV B 1); and in which\nregime the metallic phase Kondo resonance in D(!) in-\ndeed sits at the edges of the gap36directly adjacent to\nthe Hubbard bands. For this reason, we expect the above\nestimates for the Mott transition phase boundaries to be\nasymptotically exact for \u000fc=t\u0003\u001d1. Forx=\u000fc=t\u0003!1+\nby contrast (relevant only to \u0011e\u000b>as explained above),\neq. 55 gives \u0011e\u000b>!\u0011f, and the asymmetries of the e\u000bec-\ntive Hubbard model and the PAM itself then coincide.\nThe simple predictions above are compared to NRG\ncalculations in \fgs. 5,6. Eq. 52 is compared in \fg. 5 to\nthe NRG-determined phase boundary for \u0011f= 0. While\nwe expect it to be asymptotically exact for \u000fc>=t\u0003\u001d1,\nit clearly describes very well the phase boundary over\nessentially the full range; re\recting no doubt the fact that\nit also captures correctly the ~U!1 asymptote\u000fc>=t\u0003\n(sec. III B), and that the metallic Kondo resonance in\npractice lies close to the lower edge of the UHB even for\nfairly small \u000fc=t\u0003, see e.g. \fg. 1.\nFig. 6 shows NRG results for \u0011f= 1:25, and contains\nthe full range of phases { metal, Mott insulator, band\ninsulator, Kondo insulator. The simple estimates eqs.\n0123\n0 5 10\n\u000fc_t<\nU=Ut<_V2KIMMIBI\nnt= 1FIG. 6. NRG results for PAM phase boundaries in the\n(~U;\u000fc=t\u0003)-plane ( ~U=Ut\u0003=V2), for \fxed \u0011f= 1:25 (with\nlong-dash line as guide to eye). Phases shown are metal, Mott\ninsulator (MI, nt= 1), band insulator (BI, nt= 0), Kondo\ninsulator (KI, nt= 2). Both nt= 1+ (electron-doped) and\nnt= 1\u0000(hole-doped) Mott transitions occur for \u0011f>1,\nas discussed in text (and the dotted line simply shows the\nlocation of nt= 1 in the M phase). The estimates eqs. 52,53\nfor the two Mott boundaries are shown (short-dash lines), and\nbecome asymptotically exact for \u000fc=t\u0003\u001d1. The exact result\neq. 59 for the metal/BI phase boundary is also indicated (solid\nline); NRG points are barely distinguishable from it.\n52,53 for the two Mott insulator boundaries are indicated\n(short-dashed lines). While not wholly quantitative,\nparticularly in the vicinity of ( ~U\u0003;\u000fc\u0003) ('(3:56;1:25)\nhere), they nevertheless capture the behaviour well and,\nas anticipated, appear to become asymptotically exact\nfor\u000fc=t\u0003\u001d1.\nFinally, we return to a point noted in sec. III B, and ev-\nident e.g. in \fgs. 2,3: for \u000fc=t\u0003\u001d1, the critical U=t\u0003(s)\nfor the Mott transition asymptotically vanish, and ap-\npear as such to be in `weak coupling'. From eqs. 52,53\nfor example, the critical U=t\u0003for the two Mott transitions\n(call themU?=t\u0003) follow as\nU?\nt\u0003=\u0012V\nt\u0003\u001322\n(\u0011f\u00061)1\n[\u000fc\nt\u0003\u00071]:nt= 1\u0006(57)\nand for\u000fc=t\u0003\u001d1 indeed vanish asymptotically. While\ncorrect, as pointed out in sec. IV D the relevant scale\nwith which to compare the interaction Uis the e\u000bective\nhoppingt\u0003e\u000b, which likewise vanishes asymptotically for\n\u000fc=t\u0003\u001d1 (eq. 33). Expressed in these terms, the asymp-\ntotic behaviour is clearly\nU?\nt\u0003e\u000b=2\n(\u0011f\u00061)\u000fc\nt\u0003:\u000fc\nt\u0003\u001d1 (58)\nand thus grows with increasing \u000fc=t\u0003, indicative of the\nstrongly correlated nature of the Mott transition.13\n3. Metal/band insulator\nThe phase boundary between the metal and the band\ninsulator can also be obtained simply and exactly. Recall\nthat as a result of Luttinger's theorem the total charge\nntis given by eq. 16, with \u000f\u0003\nf=\u000ff+ \u0006R\nf(!= 0) the\nrenormalised level. The boundary to the nt= 0 band\ninsulator obviously requires \u000f\u0003\nf>0, and corresponds to\n\u0000\u000fc+V2=\u000f\u0003\nf=\u0000t\u0003(i.e. to the lower band edge of \u001a0(\u000f) in\neq. 16). But for nt= 0 there are no electrons in the sys-\ntem and the problem is thus e\u000bectively non-interacting.\nHence\u000f\u0003\nf\u0011\u000ff, so the boundary to the band insulator is\n\u000fc=t\u0003= 1 +V2=(t\u0003\u000ff). But\u000ff=1\n2U(\u0011f\u00001) (and the\nrequirement \u000f\u0003\nf=\u000ff>0 is equivalently just that \u0011f>1\nis required for the metal to nt= 0 band insulator to\narise). Noting eq. 51 de\fning ~U, the phase boundary for\nthe metal to band insulator is thus\n\u000fc\nt\u0003= 1 +2\n(\u0011f\u00001)~U:nt= 0: (59)\nThis is also compared to NRG results for \u0011f= 1:25 in\n\fg. 6, which are in essence indistinguishable from it.\nF. Single-particle dynamics\nWe touch now on single-particle dynamics in the metal\nclose to the Mott transition, to illustrate typical be-\nhaviour associated with the two classes of insulators\nin the ZSA32scheme (sec. III C) { Mott-Hubbard and\ncharge-transfer insulators. We do so with reference to\nthe phase diagram \fg. 4 (for \u0011f= 0), where point A is\nrepresentative of behaviour close to a MH insulator, and\npoint B to a CT insulator.\nSingle-particle dynamics in the vicinity of point A have\nbeen shown in \fg. 1, and their qualitative characteristics\ndiscussed in sec. IV D (and sec. III A). The great bulk\nof thec-band spectrum lies far above the Fermi level\n(\fg. 1 inset), and is e\u000bectively a passive spectator to\nthe Mott transition. The `action' in Dc(!) around the\nFermi level is instead generated by hybridization Vto\nthe high-lying c-level, leading to characteristically nar-\nrow Hubbard bands with widths controlled by the e\u000bec-\ntive hopping t\u0003e\u000b\u001ct\u0003. The low-energy Kondo resonance\n{ occurring in both Dc(!) andDf(!) { lies at the lower\nedge of the upper Hubbard band (\fg. 1), narrows progres-\nsively as the transition is approached and its Kondo scale\n!Lvanishes; and collapses on the spot as the transition is\ncrossed, to yield the MH insulator with a \fnite spectral\ngap of order \u0001 g'U\u00002t\u0003e\u000b. Since!Lvanishes continu-\nously on approaching the transition from the metal, the\nspectra exhibit scaling as a function of !=! L; as shown\nexplicitly in \fg. 7 for both the c- andf-spectra,42which\nare seen to have rather asymmetric scaling resonances.\nTypical dynamics close to the Mott transition to a CT\ninsulator are illustrated in \fg. 8 (for point B of \fg. 4,\n\u000fc=t\u0003= 0:85 andU=t\u0003= 4); again showing c- andf-\nlevel spectra as\u0019\n2t\u0003Dc(!) and\u0019\n2t\u0003(V=\u000fc)2Df(!)vs!=t\u0003.\n0.00.51.0\n-10 -5 0 5 100.00.51.0\n0.00.51.0\n-0.004 0.0 0.004\n\u0019\n2t<Dc.!/\n!_!L\u0019\n2\u0000V_\u000fc\u00012t<Df.!/U_t<= 0:15\nU_t<= 0:155\nU_t<= 0:159\nU_t<= 0:1595\n!_t<FIG. 7. Scaling of dynamics in the metal close to the\ntransition to a Mott-Hubbard insulator (see point A in \fg.\n4 and \fg. 1). Shown for the same parameters as \fg. 1, with\nU=t\u0003's as indicated. As the transition is approached and the\nlow-energy scale !Lvanishes, the spectra exhibit scaling col-\nlapse as a function of !=!L.42Upper panel :f-band spectrum\n\u0019\n2t\u0003(V=\u000fc)2Df(!)vs!=!L.Lower panel :c-band spectrum\n\u0019\n2t\u0003Dc(!)vs!=!L. Inset:c-spectrum vs!=t\u0003.\nThey have clear di\u000berences from, but important similar-\nities to, those close to a MH insulator (\fg. 1). The rele-\nvant bands are now by contrast broad, with their widths\ndictated by the bare hopping t\u0003; and { in marked contrast\nto eq. 46 close to a MH insulator { the !-dependences\nof thec- andf-spectra are quite di\u000berent. Hubbard\nbands are for example evident in the f-spectrum (cen-\ntred around !'\u00061\n2U= 2t\u0003), but are barely apparent\nin thec-spectrum, which is instead dominated by the\nnearly-free semicircular spectrum centred on !'\u000fcwith\nhalfwidth\u0018t\u0003.\nThe physics at low-energies around the Fermi level\nremains however characterised by the Kondo-like reso-\nnances evident in both spectra. These again lie close to\nthe lower edge of the upper Hubbard band in Df(!), nar-\nrow progressively as the low-energy scale !Lvanishes on\napproaching the transition (e.g. by increasing \u000fctowards\nits critical value\u0018t\u0003); and again collapse on the spot as\nthe transition is crossed, to yield the CT insulator whose\n\fnite spectral gap \u0001 gis now controlled by the charge\ntransfer energy \u0001 = \u000fc\u0000\u000ff(rather than by U), and is of\norder \u0001g'\u0001\u00002t\u0003('t\u0003for the case considered).\nThe insets to \fg. 8 show the low-energy Kondo reso-14\n0.00.51.0\n-2 0 20.00.51.01.5\n01\n-4 0 401\n-4 0 4\n\u0019\n2t<Dc.!/\n!_t<\u0019\n2\u0000V_\u000fc\u00012t<Df.!/\n106!_t<106!_t<\nFIG. 8. Typical single-particle dynamics close to the\ntransition to a charge-transfer insulator (point B in \fg. 4:\n\u000fc=t\u0003= 0:85,U=t\u0003= 4, with\u000ff=t\u0003=\u00002 since\u0011f= 0). Upper\npanel :f-spectrum shown as\u0019\n2t\u0003(V=\u000fc)2Df(!)vs!=t\u0003.Lower\npanel :c-band spectrum\u0019\n2t\u0003Dc(!)vs!=t\u0003.Insets : spectra\non low-energy scales; since !L'10\u00006t\u0003here, these amount\nin practice to the scaling forms of the spectra. See text for\ndiscussion.\nnances of both f- andc-level spectra, for \u000fc=t\u0003= 0:85 in\nthe metal close to the transition, where ntis very close\nto one and the low-energy scale !L'10\u00006t\u0003(su\u000eciently\nsmall that the resonances shown amount in practice to\ntheir scaling forms). The Fermi level spectra indeed ac-\ncurately satisfy eq. 18, but the resonances in the f- andc-\nspectra are clearly di\u000berent; that for the f-level being rel-\natively weakly asymmetric (with its maximum displaced\nslightly above the Fermi level to !'!L), while that for\nthec-level is markedly asymmetric.\nFinally, we add that single-particle spectra for point C\nof \fg. 4 { corresponding to the crossover region where the\ncharge transfer energy \u0001 'U{ are qualitatively similar\nto those shown in \fg. 8 close to a CT insulator; and for\nthat reason are omitted here.\nV. MOTT INSULATOR\nThus far we have said little about the Mott insulator\nphase itself. We turn to it now.\nIn the Fermi liquid phases the electron spin degrees\nof freedom are completely quenched, re\rected in a non-degenerate ground state with e.g. a vanishing T= 0 en-\ntropy. By contrast, the MI phase within DMFT { be it\nfor the Hubbard model or the PAM { is characterised by\na residual entropy of kBln 2 per site;6re\recting incom-\nplete Kondo-like quenching of electron spins, and hence\nan unquenched local moment per site (denoted by ~ \u0016).\nTo handle the locally doubly-degenerate MI phase re-\nquires a two-self-energy (TSE) description, recently con-\nsidered in the context of both a range of quantum impu-\nrity models35and the one-band Hubbard model within\nDMFT.36We draw on this work in the following, and con-\n\fne ourselves below to a brief summary of key features;\nemphasising that the TSE description used here is exact\n(it also underlies the local moment approach43;19{21but\nits use there, while providing a rather successful descrip-\ntion of e.g. the PAM,19{21,26,29is in general approximate).\nWithin a TSE description,35,36the local propaga-\ntorsG\u0017(!) with\u0017=c;f, are expressed as G\u0017(!) =\n1\n2[G\u0017A\u001b(!) +G\u0017B\u001b(!)].G\u0017A\u001b(!) refers to the propaga-\ntor for local moment ~ \u0016= +j~\u0016j, whileG\u0017B\u001b(!) refers to\nthat for ~\u0016=\u0000j~\u0016j.44From the invariance of Hunder spin\nexchange it follows that G\u0017A\u001b(!) =G\u0017B\u0000\u001b(!), such that\nG\u0017(!) is correctly rotationally invariant (independent of\n\u001b). Hence\nG\u0017(!) =1\n2\u0002\nG\u0017A\"(!) +G\u0017A#(!)\u0003\n:\u0017=c;f (60)\nenabling one to consider solely the A-type propagators,\nas employed in the following. The G\u0017A\u001b(!) are given\nin terms of the corresponding two-self-energies \u0006 fA\u001b(!)\n(= \u0006R\nfA\u001b(!)\u0000i\u0006I\nfA\u001b(!)) by ( cfeqs. 8)\nGcA\u001b(!) =h\n!+\u0000\u000fc\u0000V2\n!+\u0000\u000ff\u0000\u0006fA\u001b(!)\u00001\n4t2\n\u0003Gc(!)i\u00001\n(61a)\nGfA\u001b(!) =h\n!+\u0000\u000ff\u0000\u0006fA\u001b(!)\u0000Sf(!)i\u00001\n; (61b)\nwith the local c- andf-level Feenberg self-energies again\ngiven precisely as in eq. 8 by Sc(!) =1\n4t2\n\u0003Gc(!) and\nSf(!) =V2[!+\u0000\u000fc\u00001\n4t2\n\u0003Gc(!)]\u00001(re\recting the fact\nthat the nearest neighbours to any site are equally prob-\nablyA-type (~\u0016= +j~\u0016j) orB-type (~\u0016=\u0000j~\u0016j)).\nAs detailed in Ref. 35 it is the two-self-energies\n\u0006fA\u001b(!) (and not the conventional single self-energy\n\u0006f(!)) that are directly calculable from many-body per-\nturbation theory in the degenerate MI, as functional\nderivatives of a Luttinger-Ward functional. In con-\nsequence, a Luttinger theorem holds for the two-self-\nenergies and associated propagators, namely\nILA\u001b= ImZ0\n\u00001d! GfA\u001b(!)@\u0006fA\u001b(!)\n@!= 0:(62)\nThe standard Luttinger theorem applicable to the Fermi\nliquid phase, IL= 0 (withILgiven by eq. 17 in terms of\nthe single self-energy \u0006 f(!)), does not by contrast hold in\nthe MI phase; we consider and determine it explicitly in15\nsec. V B 1. \u0006 f(!) nonetheless remains de\fned in the MI\njust as in eq. 8; comparison of which with eqs. 60,61 gives\nan exact relation between \u0006 fand the two-self-energies\nf\u0006fA\u001bg(which we omit here because no essential use is\nmade of it in the following). Finally, as discussed in Refs.\n35,36, we add that both the \u0006 fA\u001b(!) and \u0006f(!) may be\ncalculated using NRG (as will be employed below).\nA. Total site charge and moment\nThe charges n\u0017are as ever given from eq. 4 (with G\u0017(!)\nin the MI from eq. 60); with the total local charge nt=\nnc+nf, and local moment j~\u0016j, given by\nnt=X\n\u0017=c;f(\u00001)\n\u0019ImZ0\n\u00001d!\u0002\nG\u0017A\"(!) +G\u0017A#(!)\u0003\n(63a)\nj~\u0016j=X\n\u0017=c;f(\u00001)\n\u0019ImZ0\n\u00001d!\u0002\nG\u0017A\"(!)\u0000G\u0017A#(!)\u0003\n:(63b)\nIn parallel to nt, the momentj~\u0016jis a `total' site local\nmoment, in general receiving contributions from both\nf- andc-levels (naturally so, given the local hybridiza-\ntionVcoupling the levels). We focus on the total site\ncharge/moment { rather than the separate contributions\nfromf- andc-levels { because it is these about which\nexact statements can be made, as below; this is directly\nanalogous to the fact that eq. 16 for Fermi liquid phases,\nwhich has played an important role in our analysis of the\nmetal, refers likewise to the total local charge nt.\nWhile the charge is of course \fxed at nt= 1 or 3\nthroughout the MI phase, the local moment j~\u0016jis not.\nBy way of example, \fg. 9 shows typical NRG results for\nj~\u0016jas a function of \u000fc=t\u0003, for a \fxed interaction U=t\u0003= 2\n(with\u0011f= 0,V=t\u0003= 0:4); taking as such a cut through\nthe phase diagram of \fg. 4. As occurs also for the\nusual one-band Hubbard model,36the moment is non-\nzero throughout the MI, and vanishes discontinuously at\nthe Mott transition (re\recting the fact that the critical\nlineU\u0011Uc(\u000fc) for the transition is a line of \\ Uc2's\" in\nstandard terminology6). Local moments are in fact ex-\ntremely well-developed in the insulator, for although j~\u0016j\nis seen to deviate steadily from its saturation value of 1,\nit remains strikingly close to it throughout the MI phase\n{ much more so than in the one-band Hubbard model\n(e.g. \fg. 2 of Ref. 36). An explanation for this behaviour\nis given below, see eqs. 71,74.\nThe essential results for ntandj~\u0016jare expressed in\nterms of interaction-renormalised levels associated with\nthe two-self-energies, de\fned by ( cfeq. 13 for the Fermi\nliquid phases)\n\u000f\u0003\nfA\u001b=\u000ff+ \u0006R\nfA\u001b(0): (64)\nThese results arise directly from eqs. 63, conjoined with\n00.9980.9991\n0 1 2 3 4 501\n0 1 2 3 4 5\nð\u0016ð\n\u000fc_t<KI M MIFIG. 9. NRG results (diamonds) for the local moment j~\u0016jvs\n\u000fc=t\u0003, shown for \fxed interaction U=t\u0003= 2 with\u0011f= 0 and\nV=t\u0003= 0:4. The moment is non-zero throughout the MI phase\n(arising for \u000fc=t\u0003&1:1, \fg. 4), vanishes discontinuously at the\nMott transition (inset), and is zero in both the (Fermi liquid)\nM and KI phases. Throughout the MI j~\u0016jdeviates from, but\nremains remarkably close to, its saturation value of 1 (main\n\fgure) for reasons explained in text. The leading asymptotic\nbehaviour ofj~\u0016jgiven by eq. 71 is also shown (solid line), and\nis seen to capture the NRG results; its ultimate asymptotic\nform, eq. 74, is also shown for comparison (dotted line).\nthe Luttinger theorem ILA\u001b= 0, and are\nnt=X\n\u001b\"\n\u0012\u0000\n\u0000\u000f\u0003\nfA\u001b\u0001\n+\u0012 \n\u0000\u000fc+V2\n\u000f\u0003\nfA\u001b\u00001\n4t2\n\u0003GR\nc(0)!#\n(65)\nand\nj~\u0016j=X\n\u001b\u001b\"\n\u0012\u0000\n\u0000\u000f\u0003\nfA\u001b\u0001\n+\u0012 \n\u0000\u000fc+V2\n\u000f\u0003\nfA\u001b\u00001\n4t2\n\u0003GR\nc(0)!#\n\u0000J\n(66)\n(with\u001b=\u0006for\"=#-spins), where\nJ=1\n4t2\n\u00031\n\u0019ImZ0\n\u00001d!\u0000\nGcA\"(!)\u0000GcA#(!)\u0001@Gc(!)\n@!:\n(67)\nThey hold throughout the nt= 1 or 3 MI phases, and\nare exact; their derivation is outlined in Appendix A.\nEq. 65 in particular is the analogue, for the MI, of its\ncounterpart eq. 16 for the Fermi liquid phases.\n1. Renormalised levels \u000f\u0003\nfA\u001b\nThe spin-dependent renormalised levels \u000f\u0003\nfA\u001bare char-\nacteristic of the MI phase (just as the renormalised level\n\u000f\u0003\nf, eq. 13, is characteristic of Fermi liquid phases); and\nan important element of eqs. 65,66 is that they imply\nstrong bounds on the \u000f\u0003\nfA\u001bthroughout the MI phase.16\nTo see this, let us focus on the nt= 1 MI. Recalling\nthatj~\u0016j2(0;1) in the MI, consider \frst the situation\ndeep in the MI phase. Speci\fcally, it is helpful to have\nin mind the regime \u0011f2[0;1), where for any interac-\ntionUthent= 1 MI persists to arbitrarily large values\nof\u000fcand the moment j~\u0016jasymptotically approaches 1\n(i.e.J \u001c 1). Sincent= 1, it follows from eq. 65 that\nthree of the four step functions must vanish; and since\nj~\u0016jcannot be negative it follows from eq. 66 that both\nstep functions with \u001b=#must vanish. In other words,\nthe only consistent possibilities are either \u0012(\u0000\u000f\u0003\nfA\") = 1\nor\u0012(\u0000\u000fc+V2=\u000f\u0003\nfA\"\u00001\n4t2\n\u0003Gc(0)) = 1, with all other step\nfunctions vanishing. From this it follows that the \u001b=\"\nrenormalised level necessarily lies above the Fermi level,\n\u000f\u0003\nfA#>0 (68)\n(together with \u000fc+1\n4t2\n\u0003Gc(0)>0), and that\n\u000f\u0003\nfA\"< \u000f\u0003\nfA#: (69)\nBut since three of the four step functions must vanish\nbecausent= 1, it follows that \u000f\u0003\nfA#cannot change sign\nthroughout the MI phase; otherwise (from eq. 66) j~\u0016j\nwould decrease by 2 (and thus contradict j~\u0016j>0). Eqs.\n68,69 thus hold throughout the entirent= 1 MI phase\n(we illustrate them in sec. V B, see \fg. 10).\n2. Asymptotic behaviour of the local moment\nWe turn now to the leading corrections to the local mo-\nment (in the regime \u0011f2[0;1), as above), embodied in\nJ(eq. 67), such that j~\u0016j= 1\u0000J. Note \frst that there\nare two distinct decoupled limits where the moment is\nfully saturated/spin-polarised: (a) V= 0 (butt\u00036= 0),\nwhere thef-levels decouple completely from the conduc-\ntion band (thatJvanishes here is seen directly from eq.\n67 on noting that GcA\"(!) =GcA#(!) forV= 0, eq. 61a);\n(b)t\u0003= 0 (butV6= 0), the `atomic limit' where sites de-\ncouple from each other (with J= 0 self-evident from\neq. 67). As outlined in Appendix B, Jcan be obtained\nexactly to leading order in V(O(V2)), but without con-\nstraint on either \u000fc=t\u0003or\u000ff=t\u0003=\u0000U\n2t\u0003(1\u0000\u0011f). Denoting\nby~\u0001 the dimensionless charge transfer energy,\n~\u0001 = (\u000fc\u0000\u000ff)=t\u0003 (70a)\n=1\nt\u0003\u0002\n\u000fc+1\n2U(1\u0000\u0011f)\u0003\n; (70b)\nthe local moment is given asymptotically by\nj~\u0016j= 1\u00002\u0012V\nt\u0003\u00132h\nY(~\u0001)i22\n4\u0012\n1\u00001\n~\u00012\u0013\u00001\n2\n\u000013\n5(71)\nwhere\nY(~\u0001) =\u0000~\u0001 +p\n~\u00012\u00001~\u0001\u001d1\u0018 \u00001\n2~\u0001:(72)Aside from the explicit factor of ( V=t\u0003)2, note thatj~\u0016jis\na function solely of the charge transfer energy ~\u0001 eq. 70\n(itself dependent on \u000fc=t\u0003,U=t\u0003and\u0011f).\nRecall (sec. III C) that the ZSA classi\fcation32divides\nMott insulators into Mott-Hubbard (MH) insulators for\n\u0001&Uand charge-transfer (CT) insulators for \u0001 .U,\ni.e. (eq. 70b)\n\u000fc?1\n2U(1 +\u0011f) : MH =CT: (73)\nWell into the MH regime ( \u000fc\u001dU),~\u0001\u0018\u000fc=t\u0003from eq.\n70b, with\u000fc=t\u0003\u001d1 required for the Mott transition to\noccur here; in other words, ~\u0001\u001d1. Likewise, su\u000eciently\nfar into the CT regime ( \u000fc\u001cU),~\u0001\u0018U\n2t\u0003(1\u0000\u0011f) from\neq. 70b, with U=t\u0003\u001d1 for the Mott transition to occur\nin this case; again, ~\u0001\u001d1. Since ~\u0001\u001d1 in either case,\nthe large- ~\u0001 asymptotic behaviour of eq. 71 dominates,\nviz\nj~\u0016j~\u0001\u001d1\u00181\u00001\n4\u0012V\nt\u0003\u001321\n~\u00014= 1\u00001\n4V2t2\n\u0003\u0002\n\u000fc+1\n2U(1\u0000\u0011f)\u00034:\n(74)\nThe high(4th)-order character of this leading perturba-\ntive correction is the natural reason why j~\u0016jis so well-\ndeveloped throughout the entire MI phase, see e.g. \fg. 9\n[for the standard one-band Hubbard model by contrast,\nthe corresponding leading correction is second order in\nt\u0003=U\u001c1,viz36j~\u0016j\u00001 = (t\u0003=2U)2, with the moment\naccordingly less fully developed].\nThe results eqs. 71,74 are compared to NRG calcula-\ntions ofj~\u0016jin \fg. 9, and the leading asymptotic behaviour\neq. 74 seen clearly to emerge with increasing \u000fc=t\u0003. The\n`full' asymptotic form eq. 71 is also seen to capture re-\nmarkably well the NRG results over the full \u000fc=t\u0003-range\nin the Mott insulator. Indeed one might turn the lat-\nter comment on its head, to note the rather impressive\naccuracy with which NRG can capture deviations from\nsaturation on the order of 10\u00004.\nB. Renormalised levels \u000f\u0003\nf, and Luttinger theorem\nIn considering the MI phase, we have naturally focused\non the renormalised levels \u000f\u0003\nfA\u001bgiven in terms of the two-\nself-energies. As in the metallic Fermi liquid phase, how-\never, one can also consider the ( \u001b-independent) renor-\nmalised levels de\fned in terms of the single self-energy\nand given by eq. 13, \u000f\u0003\nf=\u000ff+ \u0006R\nf(0).\nAs has been exploited throughout, the ph-symmetry\nH(\u000fc;\u0011f)\u0011H(\u0000\u000fc;\u0000\u0011f) enables consideration solely\nof\u0011f\u00150. Here, as we have seen, both nt= 1 and\nnt= 3 Mott insulators arise (now unrelated under a ph-\ntransformation); with the nt= 1 MI occurring only for\n\u000fc=t\u0003>1, and thent= 3 MI only for \u000fc=t\u0003<\u00001. In ad-\ndition, we have the general condition j\u0000\u000fc+V2=\u000f\u0003\nfj>t\u0003\n(eq. 15) for an insulating phase to occur. The condi-\ntion\u000fc>t\u0003, conjoined with eq. 15, implies directly that17\n-0.10.00.1\n0.12 0.14 0.16 0.18 0.2\n\u000f<\nf_t<\nU_t<M MIV2_\u000fcV2_.\u000fc*t</\u000f<\nf\n\u000f<\nf\u000f<\nfA\n\u000f<\nfA~\nFIG. 10. NRG results for renormalised levels, shown vsU=t\u0003\nfor \fxed\u000fc=t\u0003= 3,V=t\u0003= 0:4 and\u0011f= 0 (where the critical\nUc=t\u0003'0:160). On approaching the nt= 1 Mott transition\nfrom the metal, \u000f\u0003\nfindeed tends to the limit \u000f\u0003\nf=V2=\u000fc(eq.\n20), and then jumps discontinuously on entering the MI, while\nremaining>0 (eq. 75). Approaching the transition from the\nMI, the renormalised levels \u000f\u0003\nfA#and\u000f\u0003\nfA\"remain \fnite, with\n\u000f\u0003\nfA#>0 and\u000f\u0003\nfA\"<\u000f\u0003\nfA#as required (eqs. 68,69).\n\u000f\u0003\nf>0 for thent= 1 MI, with the following bounds on\nthe renormalised levels\n0<\u000f\u0003\nf<V2\n\u000fc+t\u0003or 0<V2\n\u000fc\u0000t\u0003<\u000f\u0003\nf:nt= 1 (75)\naccording to whether \u0000\u000fc+V2=\u000f\u0003\nf>t\u0003or<\u0000t\u0003respec-\ntively. Likewise, for the nt= 3 MI the condition \u000fc<\u0000t\u0003\ntogether with eq. 15 implies \u000f\u0003\nf<0, with corresponding\nbounds\u000f\u0003\nf<V2=(\u000fc+t\u0003)<0 orV2=(\u000fc\u0000t\u0003)<\u000f\u0003\nf<0.\nAs shown in sec. III B 1, on approaching either the\nnt= 1 ornt= 3 Mott transition from the metallic phase,\nthe renormalised level \u000f\u0003\nfapproaches the limiting value\n\u000f\u0003\nf=V2=\u000fc(see eq. 20). From the above it thus follows\nthat, while sgn( \u000f\u0003\nf) remains \fxed as the Mott transition is\ncrossed, a discontinuous jump occurs in the renormalised\nlevel\u000f\u0003\nf. NRG results exemplifying this behaviour are\ngiven in \fg. 10, which also shows the calculated renor-\nmalised levels \u000f\u0003\nfA#,\u000f\u0003\nfA\"(sec. V A 1).\n1. Luttinger theorem for IL\nLuttinger's theorem IL= 0 holds throughout the\nFermi liquid phases, with the usual Luttinger integral\nILgiven by eq. 17. It does not however hold in the MI,\nso what can be deduced about ILin this phase?\nConsideration of the two-self-energies, and associated\nrenormalised levels \u000f\u0003\nfA\u001b, is mandatory to say anything\nabout the local moment j~\u0016j(including its mere existence).\nBut that is not the case for the total charge nt. In thiscase one can repeat the analysis of sec. V A, working with\nthe single self-energy \u0006 f(!) instead of thef\u0006fA\u001b(!)g{\nbut without assuming a vanishing Luttinger integral IL.\nThe resultant analogue of eq. 65 for ntis then\n1\n2nt=h\n\u0012(\u0000\u000f\u0003\nf)+\u0012\u0010\n\u0000\u000fc+V2\n\u000f\u0003\nf\u00001\n4t2\n\u0003Gc(0)\u0011i\n\u00001\n\u0019IL(76)\nwherent= 1 or 3 for the MI phase. Gc(0) itself\nis moreover a function of \u000f\u0003\nf: eq. 8a gives Gc(0) =\n[\u0000\u000fc\u00001\n4t2\n\u0003Gc(0) +V2=\u000f\u0003\nf]\u00001, and hence a quadratic for\nGc(0), from which it is easily shown that the sign of\n\u0000\u000fc+V2=\u000f\u0003\nf\u00001\n4t2\n\u0003Gc(0) is necessarily that of \u0000\u000fc+V2=\u000f\u0003\nf.\nEq. 76 then gives\n1\n\u0019IL=h\n\u0012(\u0000\u000f\u0003\nf) +\u0012\u0010\n\u0000\u000fc+V2\n\u000f\u0003\nf\u0011i\n\u00001\n2nt: (77)\nUnder a ph-transformation where ( \u000fc;\u0011f)!(\u0000\u000fc;\u0000\u0011f),\nthe renormalised level changes sign, \u000f\u0003\nf!\u0000\u000f\u0003\nf(eq. 14),\nand the total charge nt!4\u0000nt(eq. 10). From\neq. 77, the Luttinger integral thus inverts under a ph-\ntransformation, IL!\u0000IL. As usual, we can thus focus\non thent= 1 or 3 MIs arising for \u0011f\u00150. Here, as shown\nin sec. V B, \u000f\u0003\nf>0 for thent= 1 MI, while \u000f\u0003\nf<0 for\nthent= 3 MI. From eq. 77 it follows that, for both the\nnt= 1 and 3 MIs, the Luttinger integral is given by\nIL= +\u0019\n2:\u0000\u000fc+V2\n\u000f\u0003\nf>t\u0003 (78a)\n=\u0000\u0019\n2:\u0000\u000fc+V2\n\u000f\u0003\nf<\u0000t\u0003 (78b)\n(noting also the bounds eq. 15 required for an insulator).\nThe magnitude of the Luttinger integral is thus \u0019=2\nthroughout the MI phases, independent of interaction\nstrength (or indeed of any underlying model parameters,\nprovided only the system is a MI). This generalises to\nthe non-Fermi liquid MI the familiar Luttinger theorem\nIL= 0 applicable to the Fermi liquid phases; and, since it\nencompasses the V= 0 limit where the f-levels decouple\ncompletely from the c-band, suggests it re\rects perturba-\ntive continuity to that limit (akin to the fact that IL= 0\nfor Fermi liquid phases re\rects adiabatic continuity to the\nnon-interacting limit U= 0). The same result is more-\nover found for the local moment phases of a wide range\nof quantum impurity models per se ,35as well as for the\none-band NN Hubbard model within DMFT;36suggest-\ning its ubiquity as a hallmark of the locally degenerate\nground states arising in both the local moment phases of\nquantum impurity models and, relatedly, the MI phases\nof lattice-fermion models within DMFT.\nACKNOWLEDGMENTS\nWe are grateful to the EPSRC for \fnancial support un-\nder grants EP/L015722/1 and EP/N01930X/1; and the18\nwork is compliant with EPSRC Open Data requirements.\nWe thank A. Mitchell for helpful discussions about sev-\neral aspects of this work. One of us (DEL) also expresses\nhis warm thanks for the hospitality of the Physics De-\npartment, Indian Institute of Science, Bangalore, where\npart of this work was completed; and to H. R. Krishna-\nmurthy, T. V. Ramakrishnan and N. S. Vidhyadhiraja for\nstimulating discussions. DEL would also like to acknowl-\nedge, with gratitude, numerous enlightening discussions\nabout correlated electron systems over many years, with\nhis late friend and sometime collaborator, Thomas Pr-\nuschke.\nAppendix A: Total charge and local moment in\nMott insulator phase\nHere we outline the derivation of eqs. 65,66 for the total\nsite chargentand local moment j~\u0016jin the MI phase. The\npropagators GcA\u001b(!) andGfA\u001b(!) are given by eqs. 61;\nfrom which follows an identity relating the two:\nGfA\u001b(!) =\n1\n[!+\u0000\u000ff\u0000\u0006fA\u001b(!)]\u0014\n1 +V2\n[!+\u0000\u000ff\u0000\u0006fA\u001b(!)]GcA\u001b(!)\u0015\n(A1)\nEqs. 61a andA1 thus give\nX\n\u001bl\u001b(GcA\u001b(!) +GfA\u001b(!)) =X\n\u001bl\u001b\u0002\n!+\u0000\u000ff\u0000\u0006fA\u001b(!)\u0003\u00001\n+X\n\u001bl\u001b\u0012\n1 +V2\n[!+\u0000\u000ff\u0000\u0006fA\u001b(!)]2\u0013\nGcA\u001b(!)\n(A2)\nwherel\u001bhere denotes either 1 or \u001b(and with\u001b=\u0006for\n\"=#spins, as usual).\nThe following is simply an identity,\n\u0012\n1 +V2\n[!+\u0000\u000ff\u0000\u0006fA\u001b(!)]2\u0013\nGcA\u001b(!) =\n@\n@!ln\u0012\n!+\u0000\u000fc\u0000V2\n!+\u0000\u000ff\u0000\u0006fA\u001b(!)\u00001\n4t2\n\u0003Gc(!)\u0013\n+h\nGfA\u001b(!)\u00001\n[!+\u0000\u000ff\u0000\u0006fA\u001b(!)]i@\u0006fA\u001b(!)\n@!\n+GcA\u001b(!)1\n4t2\n\u0003@Gc(!)\n@!\n(A3)\nwhere the left side of eq. A3 appears in eq. A2 (and eq.A1 has been used). With it, eq. A2 gives\n(\u00001)\n\u0019ImX\n\u001bl\u001bZ0\n\u00001d!(GcA\u001b(!) +GfA\u001b(!)) =\n(\u00001)\n\u0019ImX\n\u001bl\u001bZ0\n\u00001d!\"\n@\n@!ln\u0000\n!+\u0000\u000ff\u0000\u0006fA\u001b(!)\u0001\n+@\n@!ln\u0012\n!+\u0000\u000fc\u0000V2\n!+\u0000\u000ff\u0000\u0006fA\u001b(!)\u00001\n4t2\n\u0003Gc(!)\u0013\n+GcA\u001b(!)1\n4t2\n\u0003@Gc(!)\n@!#\n(A4)\non using the Luttinger theorem eq. 62, ILA\u001b= 0 (which\nholds for either spin \u001bthroughout the degenerate MI\nphase). With l\u001b= 1, eq. A4 gives nt, and withl\u001b=\u001bit\ngivesj~\u0016j, see eqs. 63.\nThe two ln-derivative terms in eq. A4 can be evaluated\nexplicitly, using \u0006I\nfA\u001b(0) = 0 and(\u00001)\n\u0019ImGc(!= 0) =\nDc(0) = 0, as appropriate to the Mott insulator. With\nthis, eq. A4 becomes\n(\u00001)\n\u0019ImX\n\u001bl\u001bZ0\n\u00001d!(GcA\u001b(!) +GfA\u001b(!)) =\nX\n\u001bl\u001b \n\u0012\u0000\n\u0000\u000f\u0003\nfA\u001b\u0001\n+\u0012\u0010\n\u0000\u000fc+V2\n\u000f\u0003\nfA\u001b\u00001\n4t2\n\u0003Gc(0)\u0011\n+(\u00001)\n\u0019ImZ0\n\u00001d! GcA\u001b(!)1\n4t2\n\u0003@Gc(!)\n@!!\n(A5)\nwhere\u000f\u0003\nfA\u001bare the renormalised levels, eq. 64. With l\u001b=\n\u001b, eq. A5 gives directly eqs. 66,67 for the moment j~\u0016j.\nNow consider l\u001b\u00111, such that the left side of eq. A5\ngivesnt(eq. 63a). SinceP\n\u001bGcA\u001b(!) = 2Gc(!) (eq. 60),\nX\n\u001b(\u00001)\n\u0019ImZ0\n\u00001d! GcA\u001b(!)1\n4t2\n\u0003@Gc(!)\n@!\n=1\n4t2\n\u0003(\u00001)\n\u0019Im\u0010\n[Gc(0)]2\u0011\n= 0(A6)\n(asDc(0) = 0 and Gc(!=\u00001) = 0). With this, eq. A5\nyields directly eq. 65 for nt.\nAppendix B: Asymptotic behaviour of j~\u0016j.\nWe outline the origins of eqs. 71,74 for the asymptotic\nbehaviour of the local moment j~\u0016j= 1\u0000J(withJgiven\nby eq. 67), in the regime \u0011f2[0;1) considered.\nEq. 61a for GcA\u001b(!) may be expanded to leading order\ninV2, to give:\nGcA\"(!)\u0000GcA#(!) =V2 1\n[!+\u0000\u000fc\u00001\n4t2\u0003Gc(!)]2\n\u0002\u00141\n[!+\u0000\u000ff\u0000\u0006fA\"(!)]\u00001\n[!+\u0000\u000ff\u0000\u0006fA#(!)]\u0015(B1)19\nGiven the explicit V2factor here, to obtain Jto leading\nV2order thus requires \u0006 fA\u001b(!) andGc(!) forV= 0.\nThis is simple, because for V= 0 thef- andc-levels\ndecouple. Since the f-levels are then free, the \u0006 fA\u001b(!)\nare thus given by35\u0006fA\"(!) = 0 and \u0006 fA#(!) =U; while\n[!+\u0000\u000fc\u00001\n4t2\n\u0003Gc(!)]\u00001forV= 0 reduces simply to\ngc(!) (see eq. 25). Hence, writing \u000ff=\u00001\n2U(1\u0000\u0011f) and\n\u000ff+U=1\n2U(1 +\u0011f),\nGcA\"(!)\u0000GcA#(!) =V2[gc(!)]2\n\u0002\u00141\n!++1\n2U(1\u0000\u0011f)\u00001\n!+\u00001\n2U(1 +\u0011f)\u0015(B2)\nis exact to leading order in V2, but for any\u000fc=t\u0003,Uand\n\u0011fthroughout the MI phase.Recall that \u000fc=t\u0003>1 is required for the nt= 1 MI. In\nconsequence, gc(!) is pure real for all !\u00140 (\u0011EF), and\ngiven by (eq. 25)\n1\n2t\u0003gc(!) =(!\u0000\u000fc)\nt\u0003+\"\u0012!\u0000\u000fc\nt\u0003\u00132\n\u00001#1=2\n:(B3)\nEmploying eq. B2 in eq. 67 for Jand performing the\n!-integration, gives\nJ=\u00001\n4V2t2\n\u0003\u0012\n[gc(!)]2@gc(!)\n@!\u0013\n!=\u00001\n2U(1\u0000\u0011f):(B4)\nEvaluation of eq. B4 using eq. B3 then gives directly eq.\n71 forj~\u0016j= 1\u0000J.\n1A. C. Hewson, The Kondo Problem to Heavy Fermions\n(Cambridge University Press, Cambridge, 1993).\n2W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324\n(1989).\n3E. M uller-Hartmann, Z. Phys. B 74, 507 (1989); 76, 211\n(1989).\n4A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 (1992).\n5M. Jarrell, Phys. Rev. Lett. 69, 168 (1992).\n6A. Georges, G. Kotliar, W. Krauth, and M. J. Rosenberg,\nRev. Mod. Phys. 68, 13 (1996).\n7T. Pruschke, M. Jarrell, and J. K. Freericks, Adv. Phys.\n44, 187 (1995).\n8G. Kotliar and D. Vollhardt, Phys. Today 57, 53 (2004);\nG. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko,\nO. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78,\n865 (2006).\n9H. Schweitzer and G. Czycholl, Phys. Rev. Lett. 67, 3724\n(1991).\n10M. Jarrell, H. Akhlaghpour, and T. Pruschke, Phys. Rev.\nLett. 70, 1670 (1993); M. Jarrell, Phys. Rev. B 51, 7429\n(1995).\n11S. J. Sun, M. F. Yang, and T. M. Hong, Phys. Rev. B 48,\n16127 (1993).\n12N. Grewe, T. Pruschke, and H. Keiter, Z. Phys. B: Con-\ndens. Matt. 71, 75 (1988); T. Pruschke and N. Grewe, 74,\n439 (1989).\n13M. J. Rozenberg, Phys. Rev. B 52, 7369 (1995).\n14M. J. Rozenberg, G. Kotliar, and H. Kajueter, Phys. Rev.\nB54, 8452 (1996).\n15A. N. Tahvildar-Zadeh, M. Jarrell, and J. K. Freericks,\nPhys. Rev. B 55, R3332 (1997); Phys. Rev. Lett. 80, 5168\n(1998); A. N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke,\nand J. K. Freericks, Phys. Rev. B 60, 10782 (1999).\n16T. Pruschke, R. Bulla, and M. Jarrell, Phys. Rev. B 61,\n12799 (2000).\n17N. S. Vidhyadhiraja, A. N. Tahvildar-Zadeh, M. Jarrell,\nand H. R. Krishnamurthy, Europhys. Lett. 49, 459 (2000).\n18S. Burdin, A. Georges, and D. R. Grempel, Phys. Rev.\nLett. 85, 1048 (2000).\n19V. E. Smith, D. E. Logan, and H. R. Krishnamurthy,\nEur. Phys. J. B 32, 49 (2003); N. S. Vidhyadhiraja, V. E.\nSmith, D. E. Logan, and H. R. Krishnamurthy, J. Phys.:Condens. Matter 15, 4045 (2003); A. Gilbert, N. S. Vid-\nhyadhiraja, and D. E. Logan, 19, 106220 (2007).\n20N. S. Vidhyadhiraja and D. E. Logan, Eur. Phys. J. B 39,\n313 (2004).\n21D. E. Logan and N. S. Vidhyadhiraja, J. Phys.: Condens.\nMatter 17, 2935 (2005); N. S. Vidhyadhiraja and D. E.\nLogan, 17, 2959 (2005).\n22C. Grenzebach, F. B. Anders, G. Czycholl, and T. Pr-\nuschke, Phys. Rev. B 74, 195119 (2006).\n23N. S. Vidhyadhiraja, Europhys. Lett. 77, 36001 (2007).\n24G. Sordi, A. Amaricci, and M. J. Rozenberg, Phys. Rev.\nLett. 99, 196403 (2007); Phys. Rev. B 80, 035129 (2009).\n25A. Amaricci, G. Sordi, and M. J. Rozenberg, Phys. Rev.\nLett. 101, 146403 (2008).\n26D. Parihari, N. S. Vidhyadhiraja, and D. E. Logan, Phys.\nRev. B 78, 035128 (2008).\n27S. Burdin and V. Zlati\u0013 c, Phys. Rev. B 79, 115139 (2009).\n28A. Benlagra, T. Pruschke, and M. Vojta, Phys. Rev. B\n84, 195141 (2011).\n29P. Kumar and N. S. Vidhyadhiraja, J. Phys.: Condens.\nMatter 23, 485601 (2011).\n30A. Amaricci, L. Medici, G. Sordi, M. J. Rozenberg, and\nM. Capone, Phys. Rev. B 85, 235110 (2012).\n31\u0014Z. Osolin, T. Pruschke, and R. \u0014Zitko, Phys. Rev. B 91,\n07510 (2015).\n32J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev.\nLett. 55, 418 (1985).\n33P. W. Anderson, Phys. Rev. 124, 41 (1961).\n34M. Eckstein, M. Kollar, K. Byczuk, and D. Vollhardt,\nPhys. Rev. B 71, 235119 (2005).\n35D. E. Logan, A. P. Tucker, and M. R. Galpin, Phys. Rev.\nB90, 075150 (2014).\n36D. E. Logan and M. R. Galpin, J. Phys.: Condens. Matter\n28, 025601 (2016).\n37J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417\n(1960); J. M. Luttinger, 119, 1153 (1960); 121, 942\n(1961).\n38E. Feenberg, Phys. Rev. 74, 206 (1948); E. N. Economou,\nGreen's Functions in Quantum Mechanics (Springer,\nBerlin, 1983).\n39The case\u000f\u0003\nf= 0 in eq. 16 is to be understood as the limit\n\u000f\u0003\nf!0 (givingnt= 2 whether \u000f\u0003\nf!0+ or 0\u0000).20\n40D. C. Langreth, Phys. Rev. 150, 516 (1966).\n41They further imply that as U!1 (where\u000fc!0 for the\nKI/M transition), the renormalised level changes discon-\ntinuously across \u000fc= 0, from\u000f\u0003\nf=\u0000V2=t\u0003for\u000fc= 0\u0000to\n+V2=t\u0003for\u000fc= 0+.\n42!Lis de\fned in practice as the right ( !>0) half-width at\nhalf-maximum in Dc(!).43D. E. Logan, M. P. Eastwood, and M. A. Tusch, J. Phys.:\nCondens. Matter 10, 2673 (1998); 9, 4211 (1997); M. T.\nGlossop and D. E. Logan, 14, 6737 (2002); N. L. Dickens\nand D. E. Logan, 13, 4505 (2001).\n44Local moments of ~ \u0016=\u0006j~\u0016jare realised physically by\nvanishing local magnetic \felds h= 0\u0006respectively (see\n[35,36]), with the \feld acting on either the f- orc-level, or\nboth."    },    {        "title": "1903.09949v1.Probing_the_Melting_of_a_Two_dimensional_Quantum_Wigner_Crystal_via_its_Screening_Efficiency.pdf",        "content": "arXiv:1903.09949v1  [cond-mat.mes-hall]  24 Mar 2019Probing the Melting of a Two-dimensional Quantum Wigner Cry stal\nvia its Screening Efficiency\nH. Deng1, L.N. Pfeiffer1, K.W. West1, K.W. Baldwin1, L.W. Engel2, and M. Shayegan1\n1Department of Electrical Engineering, Princeton Universi ty\n2National High Magnetic Field Laboratory, Tallahassee, Flo rida\n(Dated: March 26, 2019)\nOne of the most fundamental and yet elusive collective phase s of an interacting electron system\nis the quantum Wigner crystal (WC), an ordered array of elect rons expected to form when the\nelectrons’ Coulomb repulsion energy eclipses their kineti c (Fermi) energy. In low-disorder, two-\ndimensional (2D) electron systems, the quantum WC is known t o be favored at very low tempera-\ntures (T) and small Landau level filling factors ( ν), near the termination of the fractional quantum\nHall states. This WC phase exhibits an insulating behavior, reflecting its pinning by the small but\nfinite disorder potential. An experimental determination o f aTvsνphase diagram for the melting\nof the WC, however, has proved to be challenging. Here we use c apacitance measurements to probe\nthe 2D WC through its effective screening as a function of Tandν. We find that, as expected, the\nscreening efficiency of the pinned WC is very poor at very low Tand improves at higher Tonce\nthe WC melts. Surprisingly, however, rather than monotonic ally changing with increasing T, the\nscreening efficiency shows a well-defined maximum at a Twhich is close to the previously-reported\nmelting temperature of the WC. Our experimental results sug gest a new method to map out a Tvs\nνphase diagram of the magnetic-field-induced WC precisely.\nIn a Wigner crystal (WC) [1], one of the earliest pre-\ndicted many-body phases of an interacting electron sys-\ntem, the dominance of electrons’ Coulomb repulsion en-\nergy over their kinetic energy forces them into a periodic\narray with long-range order. In a two-dimensional elec-\ntron system (2DES), a quantum WC has long been ex-\npected to form at low temperature ( T/lessorsimilar1 K) and high\nmagnetic field ( B) when the electrons occupy the lowest\nLandau level and their kinetic energy is quenched [2–4].\nThere is alsosome experimental evidence, albeit often in-\ndirect, fortheformationofsuchamagnetic-field-induced,\nquantum WC in very high-mobility (low-disorder) GaAs\n2DESsnearthe Landaulevelfillingfactor ν= 1/5[5–19].\nThe main conclusion of these studies is that the WC, be-\ning pinned by the ubiquitous residual disorder, manifests\nin DC transport as an insulating phase with non-linear\ncurrent-voltage (I-V) characteristics [7–10], and exhibits\nresonances in its high-frequency (microwave) AC trans-\nportwhichstronglysuggestcollectivemotionsoftheelec-\ntrons [5, 9, 11–13, 15, 16]. In a recent bilayer experi-\nment, a high-density layer hosting a composite fermion\nFermi sea around ν= 1/2 was used to directly probe\nthe microstructure of the WC forming in an adjacent,\nlow-density layer [18].\nA very fundamental property of the magnetic-field-\ninduced WC is its melting temperature vs filling fac-\ntor phase diagram. Probing the melting of such a WC,\nhowever, has been challenging. Different experimental\napproaches have strived to determine the WC melting\nphase diagram, but all the techniques face their own lim-\nitations. One set of measurements showed kinks in the\nArrhenius plots of resistance vs 1 /Twhich were used to\nextract a phase diagram [11]; however, such kinks were\nnot reported by other groups [6–8]. The I-V measure-\nments used the disappearance of the I-V non-linearity\nat high temperatures to extract a melting temperaturefor the WC, but the non-linearity often disappears very\ngradually and varies significantly from sample to sam-\nple [7–10]. The microwave resonance measurements also\nshow broad resonancepeaks at high Tand a rather grad-\nual evolution with temperature, making it difficult to pin\nthe transition precisely [16].\nHere, we probe the 2D WC through measuring, as\na function of Tandν, the capacitance between a top\nand a bottom gate that sandwich the 2DES. Monitor-\ning this capacitance provides a direct measure of the\nscreening efficiency of the 2DES. Similar measurements\nhave demonstrated various properties of the 2DES such\nas its compressibility [20], the incompressibility of quan-\ntum Hall states [21], and a metal-insulator transition in\nrelatively low-mobility samples [22]. Our data reveal an\nunexpected non-monotonic behavior for the screening ef-\nficiency of the 2DES as it makes a transition from a\npinned WC state at low Tto an interacting electron liq-\nuid at high T. Most remarkably, the 2DES appears to\nbe particularly good in screening at a Twhich is close\nto the expected melting temperature of the WC. This\nnon-monotonic behavior is qualitatively different from\nthe monotonic behaviors we observe at other νwhere\nthe ground state of the 2DES is not a WC. Associat-\ning the temperature at which the 2DES shows maximum\nscreening with the melting temperature of the WC, we\ndetermine a Tvsνphase diagram which is tantalizingly\nsimilar to those expected and reported for the WC.\nOur sample is a modulation-doped, 70-nm-wide, GaAs\nquantum well (QW) grown via molecular beam epitaxy.\nThe 2DES has electron density n= 4.2×1010cm−2with\n≃8.5×106cm2/Vs low-temperature mobility. The sam-\nple has a van der Pauw ( ≃4×4 mm2) geometry, with\nsix, alloyedIn-Sn ohmic contactsmade to the 2DES: four\non the corners of the sample, and two in the middle of\ntwo opposite edges. The top and bottom gates are made2\nfrom Ti-Au and In, respectively. The distance between\nthe QW and the top (bottom) gate is ≃720 nm ( ≃0.45\nmm). The sample is cooled in a dilution refrigeratorwith\na base temperature of ≃37 mK. For in-plane, longitu-\ndinal (Rxx) and Hall ( Rxy) transport measurements, we\nuse low-frequency lock-in technique at 7 Hz, while keep-\ning the top and bottom gates grounded. For measure-\nments of the screeningefficiency, the configurationshown\nin Fig. 1 inset is used. We apply a small (10 mV) AC\nvoltage ( VAC) at 19 kHz to the bottom gate and measure\nthe current IPthat penetrates to the top gate through\nthe 2DES via a lock-in amplifier, while all the contacts\nto the 2DES are grounded [23]. Large IPindicates low\nscreening efficiency, and vice versa. Note that, because\nof the small VACamplitude and large distance between\nthe QW and bottom gate, the modulation of the 2DES\ndensity is negligible ( ≃105cm−2).\nFigure 1 provides an overview of our experimental re-\nsults. The in-plane transport traces, RxxandRxy, show\nthe features expected for a high-mobility 2DES, namely\ninteger and fractional quantum Hall states (IQHS and\nFQHS):Rxyexhibits plateaus with expected values, and\nRxxshows corresponding strong minima at integer and\n~\nVAC AIP\nE0EP\nFIG. 1. Overview of transport and penetration current re-\nsults. The inset shows a schematic of measurement configura-\ntion. The yellow layers are the top and bottom gates, and the\nblue layer is the 2DES. An AC voltage ( VAC) applied to the\nbottom gate generates the source electric field E0between\nthis gate and the 2DES. The penetration electric field EP\nreaches the top gate and induces IPwhich is measured by a\nlock-in amplifier in ammeter mode (A). The black trace is the\nlongitudinal magneto-resistance ( Rxx); the gray trace shows\nRxxreduced by a factor of 30 at high fields. The blue trace\nis the Hall resistance ( Rxy). The red trace is the penetration\ncurrent ( IP). TheIPmaximum observed at the filling factor\nbetween 2/9 and 1/5, where the reentrant WC is present, is\nmarked by a green arrow. The minima between the high-field\nWC and the FQHSs at ν= 2/9 and 1/5 are indicated by the\nred triangles.fractional fillings as marked on the top axis. Moreover,\non the flanks of the well-developed ν= 1/5 FQHS, Rxx\nshows highly resistive (insulating) states which are at-\ntributed to the WC pinned by disorder [6–13]. For con-\nvenience, we denote the insulating state on the lower- B\nside ofν= 1/5 as the reentrant WC (RWC).\nThe measured IP(red trace in Fig. 1) also reflects the\nrich phases of the 2DES as a function of B. At zero mag-\nnetic field, the highly conductive 2DES strongly screens\nthe source electric field E0from the bottom gate, re-\nsulting in a small IP. With increasing B,IPoverall\nincreases to a high level, which is consistent with the\ngeneral evolution of 2DES’ bulk resistance [24]. When\nthe 2DES is in an IQHS or FQHS, IPshows a peak be-\ncausethe 2DES bulk is in agapped, incompressiblestate,\nand therefore its screening efficiency is low. In particu-\nlar, for sufficiently strong QHSs, the screening efficiency\nof the incompressible 2DES is negligible so that the pen-\netration electric field EPis essentially as strong as E0,\nleading to IPmaxima at similar high values. Indeed, at\nν= 3,2,1,2/3 and 1/3, the measured IPhas peaks with\nessentiallythe sameheight (red, dashed horizontalline in\nFig. 1). Ideally, the IPmaximum value is simply deter-\nmined by the geometric capacitance between the top and\nbottom gates as if the 2DES were not present. The ex-\nperimental value of IPmaximum, marked by the dashed\nred line in Fig. 1, indeed agrees with our estimate (7 ±2\nnA) based on the sample geometry, namely the sample\narea and the distance between the top and bottom gates.\nWhen the 2DES is in a compressible, liquid state, e.g.,\natν= 1/2 and at νbetween adjacent FQHSs, IPis\nrelatively low because now the 2DES bulk is compress-\nible and conducting, and therefore the 2DES screening\nefficiency is relatively high.\nMost relevant to our study is the behavior of IPat\nvery high magnetic fields where the 2DES hosts the WC\nstate:IPhas high value and tends to reach the same\nlimit as seen for the strong IQHS/FQHSs. This is con-\nsistent with the insulating behavior of the pinned WC\nwhich should result in a low screening efficiency. There\nis also a maximum in IPat a filling between 1/5and 2/9,\n(green triangle in Fig. 1), corresponding to the position\nof the RWC, and there are maxima at ν= 1/5 and 2/9\nwhere incompressible FQHSs are present (the maximum\natν= 2/9 is very weak because of the weakness of the\n2/9 FQHS). Between these maxima, there are three clear\nminima marked by red triangles in Fig. 1, indicating\nthe higher screening-efficiency states separating the WC,\nRWC states and ν= 2/9, 1/5 FQHSs. In the remainder\nof the manuscript, we carefully monitor IPas a function\nofTandνtoelucidatetheevolutionofthevariousphases\nof the 2DES.\nIn Fig. 2(a) we present a color-density plot of IPas\na function of BandT. This plot summarizes many IP\nvsBtraces taken at different T; for typical traces in the\nhigh-Brange, see Fig. S1 in Supplemental Material [25].\nThe high- IPregimes (lighter color) locate the positions\nof FQHSs and WC/RWC, while low- IPvalleys (darker3\nFIG. 2. (a) Evolution of IPwith magnetic field ( B)\nand temperature ( T). (b)-(e) T-dependence of IPfor var-\nious phases of the 2DES: (b) compressible, electron liquid\nstates at ν= 1/2,1/4 and 3/10; (c) incompressible FQHSs at\nν= 1/3 and 1/5; (d) reentrant Wigner crystal (RWC) phase\natν= 0.21; and (e) WC phase at ν= 0.17. In (d) and (e),\ntheT-dependence of Rxx(gray trace) at the same νis also\nplotted. (b) and (c) share the same IPscale on the left. (d)\nand (e) have the same IPscale on the left, and Rxxscale on\nthe right (in gray). The arrows on top of each panel indicate\ntheBpositions for different phases.\ncolor) highlight the states separating these. At the low-\nestT, there are three dark regions seen above B= 8\nT (ν <2/9); these reflect the three IPminima marked\nby red triangles in Fig. 1. In our experiments, we also\nindependently measure IPvsTat fixed Bfor different\n2DES phases, and present the results in Figs. 2(b)-(e).\nFor the (compressible) Fermi liquid phases such as those\natν= 1/2,3/10 and 1 /4,IPshows only a weak Tde-\npendence and stays at a relatively low level [Fig. 2(b)].\nThis is consistent with the fact that these are metallic,conducting phases in the Trange of Fig. 2. At FQHSs\nsuch asν= 1/3 and 1/5, as seen in Fig. 2(c), IPde-\ncreases monotonically from the high level at low T; this\nis also expected as the quasi-particle excitations of these\nstates that are generated at higher Tare conducting and\nlead to screening.\nHowever,the Tdependenceof IPfortheRWCandWC\nphases, shown in Figs. 2(d) and (e), is qualitatively dif-\nferent from the monotonic behavior seen for other phases\nof the 2DES. With increasing temperature, IPdecreases\nfirst, reaches a minimum at a critical temperature ( TC)\nwhich depends on ν, and then increases and saturates at\na value which is lower than IPat base temperature. This\nsaturated IPvalue isalmostthe samein alargehigh-field\nrange at high T, implying the screening efficiency of the\n2DES at high Tis nearly independent of filling and T\n(also, see Fig. S2 in Supplemental Material).\nThe non-monotonic behavior of IPvsTfor the\nWC/RWC is surprising. To ensure that it is not an ar-\ntifact of our measurement circuit, we repeated the mea-\nsurements at multiple VACfrequencies, coveringovertwo\norders of magnitude (102-104Hz). At different frequen-\ncies,IPvsBtraces show qualitatively similar behavior,\ni.e.,IPminima separate the FQHSs and the WC/RWC,\nandT-dependence measurements show a similar non-\nmonotonic behavior, with IPminima at essentially the\nsameTC(see Fig. S3 in Supplemental Material). Also,\nthe origin of IP’s non-monotonicbehaviorcannot be sim-\nply attributed to the changes in the 2DES bulk resis-\ntance. In Figs. 2(d) and (e), we also plot the Tdepen-\ndence of Rxx(gray traces). At both ν= 0.21 [Fig. 2(d)]\nandν= 0.17 [Fig. 2(e)], Rxxshows a monotonic depen-\ndence on Tas expected [6], which is hard to link with the\nnon-monotonic behavior of IP. Moreover, at different ν,\nwhenIPreaches the minimum at TC,Rxxhas signifi-\ncantly different values [ ∼4 and 40 kΩ in Figs. 2(d) and\n(e), respectively], indicating that the IPminimum is not\nassociated with a certain value of Rxx.\nWe suspect that the IPminimum, i.e., the max screen-\ningefficiency, signalsaphasetransitioninthe WC/RWC.\nTheory suggests that an interacting 2DES is more com-\npressible when it arranges itself close to an ordered array\nwhich has strong positional correlation [26]. Indeed, in\nprevious studies of negative compressibility in a bilayer\n2DES [20], the 2DESshowsenhanced screeningefficiency\n(even over-screening) when νdecreases (by lowering the\n2DES density at fixed B), and suddenly loses its screen-\ning ability at very small ν(/lessorsimilar0.05). In Ref. [20], this\nsudden transition in screening efficiency was attributed\nto the localization of electrons in the random disorder\npotential.\nThis interpretation might be reasonable for the sam-\nples of Ref. [20], which had relatively lower quality (mo-\nbility of ≃2.1×106cm2/Vs atn= 7.5×1010cm−2),\nand consideringthat the transition in screeningefficiency\nhappened when the 2DES was depleted to a very low\nn(<1×1010cm−2). In contrast, nis much higher\n(4.2×1010cm−2) in our entire measurement range, and4\nFIG. 3. The measured critical temperature ( TC) vsν. The\nred and blue circles are the data measured at n= 4.2 and\n6.0×1010cm−2, respectively. The data represented by solid\ncircles are measured by sweeping Bat fixed T[Fig. 2(a)],\nwhile the empty circles are measured by sweeping Tat fixed\nB[Figs. 2(d)-(e)]. The gray zones indicate the regime of\nnearby FQHSs. The curves connecting data points serve as\nguides to the eye.\nthe sample’s high quality is evinced by the very largemo-\nbility (≃8.5×106cm2/Vs), as well as the rich sequence\nof FQHSs, and especially the well-developed ν= 1/5\nFQHS. These facts strongly suggest that the high- IP,\nlow-screening states at low fillings in our sample reflect\nthe formation of (pinned) collective WC and RWC states\nrather than the onset of single-electron localization by\nstrong disorder. We therefore surmise that the positions\nofIPminima in Fig. 2(a) might imply a phase transition\nin the WC/RWC.\nFrom Fig. 2(a), we extract the positions of IPmin-\nima and plot the data in Fig. 3 as solid red circles; the\nopen red circles are from the IPvsTdata of Figs. 2(d)\nand 2(e). The RWC ”dome” is flanked by ν= 1/5 and\n2/9 FQHSs (gray zones), and the ν= 1/5 FQHS sepa-\nrates the RWC and WC. The νdependence of TCin Fig.\n3 is qualitatively consistent with the WC-liquid phase\ndiagrams reported previously [7, 9, 11, 16]. Associat-\ningTCwith the melting temperature of the WC/RWC,\nthe well-pronounced and relatively sharp IPminima in\nour measurements allow us to unveil rich details of the\nmelting, especially the clear ”dome”-shaped boundary\nof the RWC. We note that our data imply somewhat\nlower melting temperatures compared to those reported\nin Ref. [16] for 2DES samples with quality comparableto ours. A possible reason for this discrepancy might\nbe that, unlike the GaAs/AlGaAs hetero-structures and\nnarrow QWs used in Ref. [16], the 2DES in our wide\n(70 nm) QW sample has a larger layer thickness. This\nlarger thickness can soften the Coulomb interaction be-\ntween electrons and lower the WC melting temperature.\nWe alsorepeated similar measurementsand analysisof\nIPat a higher 2DES density of n= 6.0×1010cm−2, at-\ntained by applying DC voltage biases to the top and bot-\ntom gates. The values of gate voltages were set carefully\nto keep the charge distribution in the QW symmetric.\nTheTCvsνdata for the higher density are plotted as\nblue circles in Fig. 3. The data are qualitatively similar\nto the data measured at lower n, but exhibit a larger TC\nin general. The overall higher TCis consistent with the\nstronger electron-electron interaction at higher nwhich\nshould help stabilize the WC and therefore increase the\nmelting temperature at a given filling.\nIn conclusion, via measurements of the penetrating\ncurrentIPthrough a 2DES, we probe its screening ef-\nficiency. The data show very different behaviors at dif-\nferent filling factors as the 2DES goes through its many-\nbody states. In particular, the WC and RWC states ex-\nhibit a high IP, revealing low screening efficiency. Sur-\nprisingly, IPshows well-defined minima as a function of\neither filling ortemperature for the WC and RWC states,\nand the positions of the minima are consistent with the\nmelting of these states. If the TCofIPminima could\nindeed be associated with the WC melting temperature,\nourdatademonstratethatthemeasurementsofscreening\nefficiencyprovideaprimetechniquetomapoutthe phase\ndiagram of the magnetic-field-induced WC precisely. Re-\ngardless of its possible association with the WC melting,\nthe non-monotonic behavior for the screening efficiency\nwe observe is novel, and begs theoretical explanation.\nACKNOWLEDGMENTS\nWe thank R. N. Bhatt, M. Dykman, D. Huse, J. K.\nJain, S. Kivelson and S. Sondhi for helpful discussions.\nWe acknowledgethe National Science Foundation (Grant\nDMR 1709076) for measurements, and the Gordon and\nBetty Moore Foundation (Grant GBMF4420), the De-\npartment of Energy Basic Energy Sciences (Grant DE-\nFG02-00-ER45841), and the National Science Founda-\ntion (Grants MRSEC DMR 1420541and ECCS 1508925)\nfor sample fabrication. L. W. E. is supported by the De-\npartment of Energy (Grant DE-FG02-05-ER46212).\n[1] E. Wigner, On the interaction of electrons in metals,\nPhys. Rev. 46, 1002 (1934).\n[2] Y. E. Lozovik and V. I. Yudson, Crystallization of a two-\ndimensional electron gas in a magnetic field, JETP Lett.\n22, 11 (1975).[3] P. K. Lam and S. M. Girvin, Liquid-solid transition and\nthe fractional quantum-Hall effect, Phys. Rev. B 30, 473\n(1984).\n[4] D. Levesque, J. J. Weis, and A. H. MacDonald, Crys-\ntallization of the incompressible quantum-fluid state of a5\ntwo-dimensional electron gas in a strong magnetic field,\nPhys. Rev. B 30, 1056 (1984).\n[5] E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams,\nE. Paris, and B. Etienne, Observation of a magnetically\ninduced Wigner solid, Phys. Rev. Lett. 60, 2765 (1988).\n[6] H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui,\nL. N. Pfeiffer, and K. W. West, Quantum liquid versus\nelectron solid around ν=1/5 Landau-level filling, Phys.\nRev. Lett. 65, 633 (1990).\n[7] V. J. Goldman, M. Santos, M. Shayegan, and J. E. Cun-\nningham, Evidence for two-dimentional quantumWigner\ncrystal, Phys. Rev. Lett. 65, 2189 (1990).\n[8] H. W. Jiang, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer,\nand K. W. West, Magnetotransport studies of the in-\nsulating phase around ν=1/5 Landau-level filling, Phys.\nRev. B44, 8107 (1991).\n[9] F. I. B. Williams, P. A. Wright, R. G. Clark, E. Y. An-\ndrei, G. Deville, D. C. Glattli, O. Probst, B. Etienne,\nC. Dorin, C. T. Foxon, and J. J. Harris, Conduction\nthreshold and pinning frequency of magnetically induced\nWigner solid, Phys. Rev. Lett. 66, 3285 (1991).\n[10] Y. P. Li, T. Sajoto, L. W. Engel, D. C. Tsui, and\nM. Shayegan, Low-frequency noise in the reentrant in-\nsulating phase around the 1/5 fractional quantum Hall\nliquid, Phys. Rev. Lett. 67, 1630 (1991).\n[11] M. A. Paalanen, R. L. Willett, R. R. Ruel, P. B. Little-\nwood, K. W. West, and L. N. Pfeiffer, Electrical con-\nductivity and Wigner crystallization, Phys. Rev. B 45,\n13784 (1992).\n[12] Y. Li, D. Tsui, T. Sajoto, L. Engel, M. Santos, and\nM. Shayegan, Observation of a giant dielectric constant\nin the re-entrant insulating phase of two-dimensional\nelectrons, Solid State Communications 95, 619 (1995).\n[13] For an early review, see M. Shayegan, ”Case for the\nmagnetic-field-induced two-dimensional Wigner crystal”,\ninPerspectives in quantum Hall effects , edited by S.D.\nSarma and A. Pinczuk (Wiley, New York, 1997), pp. 343-\n383.\n[14] W. Pan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W.\nBaldwin, and K. W. West, Transition from an electron\nsolid to the sequence of fractional quantum Hall states\nat very low Landau level filling factor, Phys. Rev. Lett.\n88, 176802 (2002).\n[15] P. D. Ye, L. W. Engel, D. C. Tsui, R. M. Lewis, L. N.\nPfeiffer, and K. West, Correlation lengths of the Wigner-\ncrystalorderinatwo-dimensionalelectronsystemathighmagnetic fields, Phys. Rev. Lett. 89, 176802 (2002).\n[16] Y. P. Chen, G. Sambandamurthy, Z. Wang, R. Lewis,\nL.Engel, D.Tsui, P.Ye,L.Pfeiffer, andK.West, Melting\nof a 2D quantum electron solid in high magnetic field,\nNat. Phys. 2, 452 (2006).\n[17] L. Tiemann, T. D. Rhone, N. Shibata, and K. Muraki,\nNMR profiling of quantum electron solids in high mag-\nnetic fields, Nat. Phys. 10, 648 (2014).\n[18] H. Deng, Y. Liu, I. Jo, L. N. Pfeiffer, K. W. West, K. W.\nBaldwin, and M. Shayegan, Commensurability oscilla-\ntions of composite fermions induced by the periodic po-\ntential of a Wigner crystal, Phys. Rev. Lett. 117, 096601\n(2016).\n[19] H. Deng, L. W. Engel, L. N. Pfeiffer, K. W. West, K. W.\nBaldwin, and M. Shayegan, Critical filling factor for the\nformation of a quantum Wigner crystal screened by a\nnearby layer, Phys. Rev. B 98, 081111 (2018).\n[20] J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Com-\npressibilityofthetwo-dimensional electrongas: Measure -\nments of the zero-field exchange energy and fractional\nquantum Hall gap, Phys. Rev. B 50, 1760 (1994).\n[21] A. A. Zibrov, E. M. Spanton, H. Zhou, C. Kometter,\nT. Taniguchi, K. Watanabe, and A. F. Young, Even-\ndenominator fractional quantum Hall states at an isospin\ntransition in monolayer graphene, Nat. Phys. 14, 930\n(2018).\n[22] S. C. Dultz and H. W. Jiang, Thermodynamic signature\nof a two-dimensional metal-insulator transition, Phys.\nRev. Lett. 84, 4689 (2000).\n[23] In all the plots presented in our manuscript, we show\nthe magnitude of the measured current IP. The out-of-\nphase component of IPis typically much larger than its\nin-phase component, and both components show quali-\ntatively similar behaviors as a function of filling and/or\ntemperature.\n[24] J. S. Moon, J. A. Simmons, J. L. Reno, and B. L. John-\nson, Multiterminal capacitance tensor elements of com-\nposite fermions and anomalous capacitance peaks at even\ndenominator fillings, Phys. Rev. Lett. 79, 4457 (1997).\n[25] See Supplementary Materials at this link for additiona l\ndata, including the dependence of IPon frequency.\n[26] B. Skinner, G. L. Yu, A. V. Kretinin, A. K. Geim, K. S.\nNovoselov, and B. I. Shklovskii, Effect of dielectric re-\nsponse on the quantum capacitance of graphene in a\nstrong magnetic field, Phys. Rev. B 88, 155417 (2013)."    },    {        "title": "1101.4572v4.Possible_Superconducting_Symmetry_and_Magnetic_Correlations_in_K0_8Fe2Se2__A_77Se_NMR_Study.pdf",        "content": "arXiv:1101.4572v4  [cond-mat.supr-con]  12 Apr 2011Journal of the Physical Society of Japan LETTERS\nPossible Superconducting Symmetry and Magnetic Correlati ons in\nK0.8Fe2Se2:\nA77Se–NMR Study\nHisashiKotegawa1,3∗, YudaiHara1, HirokiNohara1, Hideki Tou1,3,\nYoshikazu Mizuguchi2,3, Hiroyuki Takeya2,3, and Yoshihiko Takano2,3\n1Department of Physics, Kobe University, Kobe 658-8530, Jap an\n2National Institute for Materials Science, 1-2-1, Sengen, T sukuba 305-0047, Japan\n3JST, Transformative Research-Project on Iron Pnictides (T RIP), Chiyoda, Tokyo 102-0075, Japan\nWe report77Se-NMR measurements in the recent discovered K 0.8Fe2Se2with the superconducting tran-\nsition temperature Tc= 32 K. The characteristic point of this system is that the hig h density of state is\nlocated in the vicinity of the Fermi level. The comparison be tween Knight shift and nuclear spin lattice\nrelaxation rate 1 /T1shows that the antiferromagnetic spin correlation is not st rong in this system, but it is\nreasonable to consider that its development toward low temp eratures occurs. 1 /T1in the superconducting\nstate can be reproduced well by an s±-wave model, but a d-wave model also roughly reproduces the data.\nNon exponential behavior in 1 /T1at low temperatures disagrees with a single isotropic gap.\nKEYWORDS: K 0.8Fe2Se2, superconductivity, NMR\nThe recent discovery of superconductivity in\nK0.8Fe2Se2is accelerating the research on related\nmaterials.1,2)The research will spread widely because\nof the relatively high transition temperature Tcof\n∼32 K and absence of arsenic. Another interesting\npoint is whether superconductivity occurs in the same\nframework as that of other Fe-based superconductors.\nMost Fe-based superconductors possess both hole-like\nbands and electron-like bands. The nesting between\nthose bands is considered to yield multigap super-\nconductivity characterized by s±symmetry.3)The\nband calculations suggest that the electronic state\nof stoichiometric KFe 2Se2is far from that of other\nFe-based superconductors, and the appearance of the\ncommon band topology to other systems depends on\nthe amount of hole doping, which is probably induced\nby the deficiency of K and Fe.4–6)Recent angle-resolved\nphotoemission spectroscopy (ARPES) studies suggest\nthat K 0.8Fe2Se2is a heavily electron-doped system\ncompared with other Fe-based superconductors, and the\nhole-like Fermi surface disappears.7,8)If superconduc-\ntivity is realized under such a situation, s±symmetry\nshould be excluded.\nIn this letter, we report77Se-NMR results in single-\ncrystallineK 0.8Fe2Se2toinvestigatethesuperconducting\n(SC) symmetry and magnetic correlations.\nA single-crystalline sample with Tc= 32 K was pre-\npared as described elsewhere.9) 77Se-NMR measurement\nusing a standard spin-echo method was performed un-\nder a magnetic field of ∼8.995 T along the abplane\nin both the normal state and the SC state.77Se pos-\nsesses the nuclear spin of I= 1/2, which corresponds\nto one NMR transition. Tc= 31 K was estimated un-\nder∼8.995 T from the onset of diamagnetism. The\nKnight shift was obtained using the gyromagnetic ratio\nofγn= 8.13 MHz/T. The nuclear spin-lattice relaxation\n∗E-mail address: kotegawa@crystal.kobe-u.ac.jprate 1/T1was obtained by a nice fitting of the recovery\ncurve to the single exponential function in the normal\nstate. In the SC state, we omitted a small amount of fast\nrelaxation arising from the vortex core in the fitting.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s48/s46/s49/s52\n/s55/s50/s46/s56 /s55/s50/s46/s57 /s55/s51/s46/s48\n/s32/s32/s75/s110/s105/s103/s104/s116/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s84 /s32/s40/s75/s41/s84\n/s99\n/s40/s98/s41/s32/s32\n/s84 /s32/s40/s75/s41/s84\n/s99\n/s49/s50/s32/s75/s53/s48/s32/s75\n/s51/s50/s32/s75\n/s50/s49/s32/s75/s32\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s77/s72/s122/s41/s50/s57/s32/s75/s40/s97/s41/s55/s55\n/s83/s101/s45/s78/s77/s82/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s83/s67\nFig. 1. (a)77Se-NMR spectrum under ∼8.955 T at several tem-\nperatures. The spectrum shifts both below and above Tc. (b) Tem-\nperature dependence of Knight shift. The strong temperatur e de-\npendence at high temperatures originates in the band struct ure\neffect. Knight shift drastically decreases below Tc.\nFigure 1(a) shows the77Se-NMR spectra measured by\napplying ∼8.955 T along the abplane. The line width\nwas 25 kHz at 32 K. The spectral shape at high tem-\nperatures is not the Lorenz type but rather close to a\nrectangle, suggesting that slight inhomogeneity exists in\nthe sample. However, we confirmed that the electronic\nstate is homogeneous, within the accuracy of 1 /T1, from\nthe frequency dependence of 1 /T1. Knight shift was es-\ntimated from a central position of the spectrum in the\nnormal state. The spectrum shows a slight and asym-\nmetric broadening owing to the existence of a vortex in\nthe SC state, where Knight shift was obtained from the\npeak of the spectrum. The spectrum at 29 K just below\n1J. Phys. Soc. Jpn. LETTERS\nTcshifts from that at 32 K without obvious broadening,\nensuring that Tcis almost homogenous in the sample.\nFigure1(b)showsthe temperaturedependence ofKnight\nshift. In the normal state, Knight shift shows a strong\ntemperature dependence. This reduction in Knight shift\ntoward low temperatures originates in the high density\nof states (DOS) near the Fermi level,10)which is often\nseen in the electron-doped systems among Fe-based su-\nperconductors.11–14)In K0.8Fe2Se2, Knight shift contin-\nues to decrease down to Tcwithout the Fermi liquid re-\ngion, as shown in the inset. This indicates that the high\nDOS is located in the vicinity of the Fermi level, and the\nspin susceptibility of q= 0 is suppressed with decreas-\ning temperature. Knight shift markedly decreases below\nTc= 31 K and approaches ∼0.02% at zero tempera-\nture. Generally, Knight shift consists ofthe temperature-\ndependent spin part Ksand temperature-independent\norbital part (or chemical shift) Korb. This suggests that\nthe spin part of superconducting symmetry is a singlet\nandKorb∼0.02%, because field-induced DOS is not\nexpected at the present magnetic field sufficiently lower\nthanHc2.9)\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48\n/s40/s70/s101\n/s48/s46/s57/s67/s111\n/s48/s46/s49/s41/s83/s101\n/s32/s32/s49/s47 /s84\n/s49/s84 /s32/s40/s115/s45/s49\n/s75/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s70/s101/s83/s101/s75\n/s48/s46/s56/s70/s101\n/s50/s83/s101\n/s50\nFig. 2. (color online) Temperature dependences of 1 /T1Tfor\nK0.8Fe2Se2, FeSe, and non superconducting (Fe 0.9Co0.1)Se. 1/T1T\nalso shows a strong temperature dependence at high temperat ures,\nbut it is almost constant below ∼60 K. The large difference from\nthe related systems suggests that the high DOS near the Fermi\nlevel is characteristic feature of K 0.8Fe2Se2. In FeSe, the differ-\nence in the symbols indicates the difference in the sample. Lo w-\ntemperature data were obtained using a high-quality sample , and\nthere is no sample dependence at high temperatures.\nFigure 2 shows temperature dependences of 1 /T1T\nfor K 0.8Fe2Se2, FeSe, and non superconducting\n(Fe0.9Co0.1)Se. In K 0.8Fe2Se2, 1/T1Talso decreases\nwith decreasing temperature owing to the band struc-\nture effect. The marked difference from Knight shift\nis that 1 /T1Tis almost constant below ∼60 K, while\nKnight shift continues to decrease toward low tem-\nperatures. This disagreement suggests that the spin\nfluctuation of q/negationslash= 0, which is most likely an antiferro-\nmagnetic (AF) one, develops at least in this temperature\nregion, even though it is not so strong. A strong tem-\nperature dependence of DOS toward low temperatureslikely masks the development of spin fluctuations. We\ncompare 1 /T1Twith those in FeSe and electron-doped\n(Fe0.9Co0.1)Se. 1/T1Tin FeSe was measured using\na high-quality sample made with high-temperature\nannealing,15)and the NMR line width of which is almost\nthe same as that of stoichiometric FeSe.14)Even in this\nsample, 1 /T1Tseparated into two components below 40\nK, so that we plotted the short component of ∼75% in\nthe volume fraction, which is considered to be intrinsic.\nIn FeSe, an obvious increase in 1 /T1Tis seen below ∼40\nK owing to the development of a low-energy part of AF\nspin fluctuations. The development of1 /T1Tis enhanced\nunder pressure with a close relationship with Tc.14,16)\nIn the high-temperature region, the temperature de-\npendence of 1 /T1Tin K0.8Fe2Se2is markedly stronger\nthan that in FeSe, and it is weak in (Fe 0.9Co0.1)Se.\nThe reduction of 1 /T1Tin (Fe0.9Co0.1)Se upon electron\ndoping indicates that the high DOS, which is located\nbelow the Fermi level in FeSe, is removed from the Fermi\nlevel by electron doping. This doping suppresses the AF\nspin fluctuations and superconductivity. By contrast,\nstrong temperature dependence in 1 /T1Tabove∼60 K\nin K0.8Fe2Se2suggests that high DOS is located closer\nto the Fermi level than in the case of FeSe. However, the\ndevelopment ofAF spin fluctuations at low temperatures\nis not drastically enhanced in K 0.8Fe2Se2. This indicates\nthat the character of the band near the Fermi level is\ndifferent between FeSe and K 0.8Fe2Se2.\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s50/s51\n/s32/s32/s40/s49/s47 /s84\n/s49/s84 /s41/s49/s47/s50\n/s32/s40/s115/s101/s99/s45/s49/s47/s50\n/s75/s45/s49/s47/s50\n/s41\n/s75/s110/s105/s103/s104/s116/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41/s75/s32 /s40/s84 /s126/s48/s41/s75\n/s48/s46/s56/s70/s101\n/s50/s83/s101\n/s50\n/s32/s32/s75\n/s84 /s32/s40/s75/s41\nFig. 3. Relationship between Knight shift and (1 /T1T)1/2from\n32 to 260 K. The non linear relationship indicates the collap se of\nthe Korringa relation. The solid curve is a guide for the eye. The\narrow indicates Knight shift at the lowest temperature, whi ch is\nconsidered to correspond to Korb. The inset shows the temperature\ndependence of Korringa ratio Kestimated using Korb= 0.02%.\nFigure 3 shows Knight shift vs (1 /T1T)1/2in the nor-\nmal state. The Korringa ratio Kis given as follows.\nK=1\nT1TK2s/planckover2pi1\n4πkBγ2\ne\nγ2n(1)\nHere,γeis the electron gyromagnetic ratio. Kcorre-\n2J. Phys. Soc. Jpn. LETTERS\nsponds to the character of spin correlations, for exam-\nple, K<<1 corresponds to ferromagnetic correlations\nand K>>1 corresponds to AF correlations. The ob-\nserved strong temperature dependences of Knight shift\nand 1/T1Tare mainly attributed to the band structure\neffect, but Knight shift vs (1 /T1T)1/2does not show a\nlinear relationship, indicating that the Korringa relation\nofK=const.is not realized. The relationship of upward\nconvex suggests that Kincreases with decreasing tem-\nperature, because the slope in the figure is proportional\ntoK1/2. If we assume Korb= 0.02% asKs(T= 0) = 0,\nthe temperature dependence of Kis as indicated in the\ninset. Kis not very far from 1, indicating that spin cor-\nrelations are not strong in K 0.8Fe2Se2, but the obvious\nincrease in Ktoward low temperatures suggests the de-\nvelopment of AF spin correlations.\n/s49/s48 /s49/s48/s48/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s49/s48/s48\n/s126/s32 /s84\n/s32/s32\n/s75\n/s48/s46/s56/s70/s101\n/s50/s83/s101\n/s50/s49/s47 /s84\n/s49/s32/s40/s115/s101/s99/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s72 /s32/s126/s32/s56/s46/s57/s53/s53/s32/s84/s32/s40/s47/s47/s32/s97/s98/s41\n/s84\n/s99\n/s32/s115/s43/s45\n/s32/s45/s32/s119/s97/s118/s101/s32/s109/s111/s100/s101/s108\n/s32/s100 /s32/s45/s32/s119/s97/s118/s101/s32/s109/s111/s100/s101/s108\n/s32/s115/s105/s110/s103/s108/s101/s32/s105/s115/s111/s116/s114/s111/s112/s105/s99/s32/s103/s97/s112\nFig. 4. (color online) Temperature dependence of 1 /T1for\nK0.8Fe2Se2. 1/T1does not show a coherence peak just below Tc.\nNon exponential behavior excludes a single isotropic gap. T he red\nsolid curve are the best-fitting result obtained using the s±-wave\nmodel. The blue dotted curve (the green dashed curve) was ob-\ntained using the d-wave model (a single isotropic model).\nNext we focus on the symmetry of the superconduct-\ning gap in K 0.8Fe2Se2. Figure 4 shows the tempera-\nture dependence of 1 /T1. The steep decrease in 1 /T1\nwas observed below Tcwithout any sign of a coherence\npeak. The power is T6-like just below Tc, approaches\nT3-like below ∼Tc/2, and then starts to deviate from\nT3below∼Tc/5. Such power-law behavior in 1 /T1is\noften seen in most Fe-based superconductors. The ab-\nsence of the coherence peak and non exponential tem-\nperature dependence of 1 /T1clearly excludes the pos-\nsibility of a single isotropic gap pointed out from the\nARPES measurement.7)The green dashed curve is a cal-\nculation using a single isotropic gap of ∆ 0= 4kBTcig-\nnoring the coherence effect. This curve completely dis-\nagrees with the data for the low-temperature region.Non exponential behavior suggests that the multigap is\nrealized with different magnitudes of gap size or that\nthe anisotropic superconductivity is realized with a fi-\nnite DOS in the SC gap. The red solid curve is the re-\nsults of a calculation based on the s±isotropic gap.17)\nThis model has been adapted for other Fe-based sys-\ntems such as LaFeAsO, (Ba,K)Fe 2As2, and LiFeAs.18–20)\nWe can reproduce the data well using ∆ 1= 4.2kBTc,\n∆2= 1.7kBTc,N1:N2= 0.7 : 0.3, andη= 0.048∆1.\nHere, ∆ 1and ∆ 2are the magnitudes of the SC gap in\nthe respective bands, and N1andN2are respective DOS\nat the Fermi level. ηis the smearing factor due to the im-\npurity effect, and induces the residual DOS at the Fermi\nlevel. ∆ 1= 4.2kBTcindicates the superconductivity in\nthe strong-coupling regime, and is comparable to that\nof (Ba 0.6K0.4)Fe2As2.19)The value of ηis also compa-\nrable to those of other Fe-based superconductors with\ns±symmetry.18–20)The multigap feature is not obvious\nin the Knight shift probably due to the relatively large\nN1, which is similar to the case of (Ba 0.6K0.4)Fe2As2.19)\nThe third model is a two-dimensional d-wave model\n(∆(θ) = ∆0cos(2θ)) with line nodes that produce T3be-\nhavior at low temperatures. As shown by the blue dotted\ncurve,thismodel alsocanreproducethe dataroughlyus-\ning ∆0= 5kBTcin the strong-coupling regime and the\nresidual DOS of 3%, which is generally induced by im-\npurity scattering. The model of d-wave symmetry is not\nsufficiently far from the data to exclude the possibility\nthatd-wavesymmetryisrealizedinthis system.Wetried\nthe two-gap d-wavemodel, but it did not improve the fit-\nted result. The 1 /T1at the lowest temperature gives the\nupperlimitoftheresidualDOSof ∼7%.Thesmallresid-\nual DOS of ∼3−7%indicates that impurity scatteringis\nsmall in spite of the system with a deficiency of ions. The\ns±symmetry is more plausible, but for its identification,\nmeasurement at lower temperatures will be required, as\nwell as under lower field to avoid the contribution in the\nrelaxation process from the vortex core.\nThe ARPES measurement suggests the isotropic gap\nopens only in the electron pocket.7)The interpretation\nbased on our data seems to be inconsistent with this\nobservation. Another ARPES measurement also shows\nthat this system possesses only an electron pocket.8)If\nthe nesting between the electron pockets plays a cru-\ncial role in superconductivity, the d-wave symmetry is\nlikely to be realized. The present NMR result cannot ex-\nclude this possibility. From the results of our study, we\ncan claim that the single isotropic gap is excluded, and\nthes±-wave or d-wave with a strong-coupling regime are\npossible candidates.\nFinally, we mention earlier NMR reports presented\nby Yu and co-workers quite recently.21,22)The obtained\ndatainthisstudyaresimilartotheirdata.Thedifference\nin the absolute value of Knight shift seems to originate\nin the different value of γn. Yu and co-workersclaim that\nAF spin correlations are absent in this system from the\nKorringa ratio. Note that we defined the reciprocal of\ntheir Korringa ratio as K. The Kis sensitive to the esti-\nmationof Korb,andtheyestimated Korbconsideringthat\nKorringaratioisconstantagainsttemperature.(theyuse\n3J. Phys. Soc. Jpn. LETTERS\nthenotationof Kc).However,ourmeasurementathigher\ntemperatures shows that the Korringa ratio depends on\ntemperature. If we can obtain the susceptibility data re-\nmoving the contribution from impurity, it will help us to\nestimate the exact Korb. As for the symmetry of the SC\ngap, they tried the fitting of 1 /T1with an isotropic gap\nand discussed the possibility of the multigap. We clearly\nexcluded a single isotropic gap and suggest the multigap\nor the anisotropic gap from the fitting of 1 /T1.\nIn summary, we performed NMR measurements in\nK0.8Fe2Se2using a single-crystalline sample. The strong\ntemperature dependences in Knight shift and 1 /T1Tat\nhigh temperatures suggest that the high DOS is located\nin the vicinity of the Fermi level. This feature is much\nstronger than that in FeSe. The breakdown of the Kor-\nringa relation suggests the AF spin correlations develop\ntowardlowtemperatures,althoughit isnot obviouscom-\npared with FeSe. The temperature dependence of 1 /T1\nin the SC state can exclude the single isotropic gap. It\ncan be reproduced well by the s±-wave model, but the\npossibility of d-wave symmetry still exists.\nThis work has been partly supported by Grants-in-\nAid for Scientific Research (Nos. 19105006, 20740197,\n20102005, 22013011, and 22710231) from the Ministry\nof Education, Culture, Sports, Science, and Technology\n(MEXT) of Japan.\n1) J.Guo, S.Jin, G.Wang, S.Wang, K.Zhu, T.Zhou, M.He, and\nX. Chen: Phys. Rev. B 82(2010)180520(R).\n2) A.Krzton-Maziopa, Z.Shermadini, E.Pomjakushina, V.Po m-\njakushin, M. Bendele, A. Amato, R. Khasanov, H. Luetkens,\nand K. Conder: J. Phys.: Condens. Matter 23(2011) 052203.\n3) I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du: Phys.Rev. Lett. 101(2008) 057003.\n4) I. R. Shein and A. L. Ivanovskii: arXiv:1012.5164 (2010).\n5) C. Cao and J. Dai: arXiv:1012.5621 (2010).\n6) I. A. Nekrasov and M. V. Sadovskii: Pisma ZhETF 93(2011)\n182.\n7) Y. Zhang, L. X. Yang, M. Xu, Z. R. Ye, F. Chen, C. He, J.\nJiang, B. P. Xie, J. J. Ying, X. F. Wang, X. H.Chen, J. P. Hu,\nand D. L. Feng: arXiv:1012.5980 (2010).\n8) T. Qian, X.-P. Wang, W.-C. Jin, P. Zhang, P. Richard, G.\nXu, X. Dai, Z. Fang, J.-G. Guo, X.-L. Chen, and H. Ding:\narXiv:1012.6017 (2010).\n9) Y. Mizuguchi, H. Takeya, Y. Kawasaki, T. Ozaki, S. Tsuda,\nT. Yamaguchi and Y. Takano: Appl. Phys. Lett. 98(2011)\n042511.\n10) H. Ikeda: J. Phys. Soc. Jpn. 77(2008) 123707.\n11) Y. Nakai, K. Ishida, Y. Kamihara, M. Hirano, and H. Hosono :\nJ. Phys. Soc. Jpn. 77(2008) 073701.\n12) H.-J.Grafe, D.Paar, G.Lang, N.J.Curro, G.Behr, J.Wern er,\nJ. Hamann-Borrero, C. Hess, N. Leps, R. Klingeler, and B.\nB¨ uchner: Phys. Rev. Lett. 101(2008) 047003.\n13) F.Ning, K.Ahilan, T.Imai, A.S.Sefat, R.Jin, M.A.Mcgui re,\nB. C. Sales, D. Mandrus: J. Phys. Soc. Jpn. 77(2008) 103705.\n14) T. Imai, K. Ahilan, F. L. Ning, T. M. McQueen, and R. J.\nCava: Phys. Rev. Lett. 102(2009) 177005.\n15) Y. Mizuguchi and Y. Takano. J. Phys. Soc. Jpn. 79(2010)\n102001.\n16) S. Masaki, H. Kotegawa, Y. Hara, H. Tou, K. Murata, Y.\nMizuguchi,and Y.Takano: J.Phys.Soc.Jpn. 78(2009) 063704.\n17) Y. Nagai, N. Hayashi, N. Nakai, H. Nakamura, M. Okumura,\nand M. Machida: New J. Phys. 10(2008) 103026\n18) K. Matano, Z. Li, G. L. Sun, D. L. Sun, C. T. Lin, M. Ichioka,\nand G. -q. Zheng: EPL 87(2009) 27012.\n19) M. Yashima, H. Nishimura, H. Mukuda, Y. Kitaoka, K.\nMiyazawa, P. M. Shirage, K. Kihou, H. Kito, H. Eisaki, and\nA. Iyo; J. Phys. Soc. Jpn. 78(2009) 103702.\n20) Z. Li, Y. Ooe, X.-C. Wang, Q.-Q. Liu, C.-Q. Jin, M. Ichioka ,\nand G.-q. Zheng: J. Phys. Soc. Jpn. 79(2010) 083702.\n21) W. Yu, L. Ma, J. B. He, D. M. Wang, T.-L. Xia, and G. F.\nChen: arXiv:1101.1017 (2011).\n22) L. Ma, J. B. He, D. M. Wang, G. F. Chen, and W. Yu:\narXiv:1101.3687 (2011).\n4"    },    {        "title": "1612.04989v1.Electronic_correlation_effects_and_Coulomb_gap_in_the_Si_111_____sqrt_3__times_sqrt_3____Sn_surface.pdf",        "content": "Electronic correlation e\u000bects and Coulomb gap in the Si(111)- (p\n3\u0002p\n3)-Sn surface\nA.B. Odobescu,\u0003A.A. Maizlakh, N.I. Fedotov, and S.V. Zaitsev-Zotov\nKotel'nikov IRE RAS, Mokhovaya 11, bld.7, 125009 Moscow, Russia\n(Dated: February 9, 2020)\nElectronic transport properties of the Si(111)-(p\n3\u0002p\n3)-Sn surface formed on low doped Si\nsubstrates are studied using two-probe conductivity measurements and tunnelling spectroscopy.\nWe demonstrate that the ground state corresponds to Mott-Hubbard insulator with a band gap\n2\u0001 = 70 meV, which vanishes quickly upon temperature increase. The temperature dependence\nof the surface conductivity above T > 50 K corresponds to the Efros-Shklovskii hopping conduc-\ntion law. The energy gap at the Fermi level observed in tunnelling spectroscopy measurements at\nhigher temperatures could be described in terms of dynamic Coulomb blockade approximation. The\nobtained localization length of electron is \u0018= 7\u0017A.\nPACS numbers: 73.25.+i, 73.20.At, 71.30.+h, 72.40.+w\nINTRODUCTION\nTwo-dimensional electron gas formed on reconstructed\nsemiconductor surfaces where Coulomb interaction en-\nergy is comparable with charge carrier kinetic energy is\nvery promising systems for experimental study the e\u000bect\nof electron-electron interactions on the ground state and\ntemperature-dependent transport properties. Of great\ninterest for the study are \u000b-phase surfaces obtained by\ncovering an ideal (111) Si or Ge surface with the 1/3 ML\nof group I, IV adatoms forming ap\n3\u0002p\n3\u0000R30\u000ere-\nconstruction with \u00187\u0017A inter-adatom distance. These\nsystems have a very narrow half-\flled surface state band,\nwith an odd number of electrons per unit cell, which is\nmetallic but a\u000bected by various scenarios of structural\nand electronic instabilities at low temperatures: charge-\ndensity wave-like reversible 3 \u00023 periodic surface struc-\nture distortion in Pb/Ge(111) and Sn/Ge(111) [1, 2];\nundistortedp\n3\u0002p\n3 Mott-Hubbard insulator (MI) state\nin Si/SiC(0001)[3] and K:Si(111) \u0000(p\n3\u0002p\n3)R30\u000e-B [4].\nAn intermediate case is the Si(111)-(p\n3\u0002p\n3)R30\u000e-Sn\nsurface. For this structure a density functional calcula-\ntion in the local density approximation (LDA) predicts\nan undistorted ground metal state [5]. The experimental\nstudy of the electronic surface structure by photoemis-\nsion shows a metallic band structure as expected from the\nsimple electron counting but with clear indication of 3 \u00023\nperiodicity [6, 7]. Structural tools like STM and photo-\nelectron di\u000braction showing onlyp\n3 periodicity provide\nno \fngerprints of the 3 \u00023 structure [7, 8]. More recent\nlocal spin density (LSDA) + Hubbard U approximation\n[9] predicts a stable magnetic (Mott-Hubbard-like) insu-\nlatingp\n3\u0002p\n3 phase in Sn/Si(111) which forms two\nseparate 3\u00023 unit cells where one is fully ferromag-\nnetic and the other ferrimagnetic (two atoms spins up\nand one spin down). The calculation explains the \fnger-\nprints in photoemision intensity near EFat \u0000 observed\nin Sn/Si(111) [6] due to 3 \u00023 anti-ferromagnetic order.\nA scanning tunnelling spectroscopy (STS) demonstrated\nan insulating gap in the local density of states (LDOS)at the Fermi level [10] with a narrow band gap (2\u0001 = 40\nmeV), which disappears above 70K but the deep min-\nima in LDOS still exists up to the room temperature.\nThe answer on possible MI transition could be obtained\nfrom electronic transport measurements. Temperature\ndependence of conductivity of the Sn/Si(111)p\n3\u0002p\n3\nsurface shows an insulator behaviour in a wide tempera-\nture range 20{300 K with possible variable-range hopping\ntransport mechanism due to the defects or activation be-\nhaviour with very small activation energy 2\u0001 \u001910 meV,\nwhich was not uniquely determined [11].\nSimilar behaviour and discrepancy in various experi-\nmental results was observed in metallic Si(111)7 \u00027 sur-\nface [12{14]. The recent explanation [15] considers the\nnarrow band gap observed in the surface states at the\nFermi level as a result of the dynamic Coulomb blocade\nwhich appears due to STS on surfaces with the sheet con-\nductivity much less than the minimum metallic conduc-\ntivity 1=Rh=e2=h= 39\u0016S[16]. The deep band gap at\nLDOS in Sn/Si(111)p\n3\u0002p\n3 surface behaves very similar\nto the dynamic Coulomb gap in Si(111)-7 \u00027 and may\nfollow the same scenario.\nHere we report the results of detailed study of Si(111)-\n(p\n3\u0002p\n3)-Sn surface. We show by comparing the tem-\nperature dependence of LDOS measured by tunnelling\nspectroscopy with temperature dependence of the sur-\nface conductivity, that the surface ground state corre-\nsponds to insulator-like state with the surface band gap\n2\u0001 = 70 meV, in accordance with LSDA+U calcula-\ntion [9]. With increasing the temperature the MI spin-\nordered state vanishes due to thermal \ructuation and the\nsurface turns into a bad metal. The transport measure-\nments indicate the Efros-Shlklovskii hoping regime with\nthe charge carrier localization length \u0018\u00197\u0017A, right to\nthe interatomic distance of Si(111)-(p\n3\u0002p\n3)-Sn unit\ncell, and the energy gap in tunnelling spectroscopy at\nhigher temperatures is well described with temperature\ndependent dynamic Coulomb gap approximation [16, 17].arXiv:1612.04989v1  [cond-mat.str-el]  15 Dec 20162\nEXPERIMENTAL\nThe tunnelling spectroscopy measurements were per-\nformed in commercial UHV LT STM Omicron. We used\nlow doped n-andp-type Si crystals with \u001a= 1 \n\u0001cm\nas the substrate in order to eliminate the contribution\nof the bulk conductivity at low temperatures in electron\ntransport measurements and the e\u000bect of charge carrier\nconcentration change due to the band bending near the\nsurface [18]. Such crystals are insulating in the low-\ntemperature region and cannot be studied by the STM\ntechnique. Therefore, for STS study we used the exter-\nnal illumination to produce necessary bulk conduction at\nlow temperatures (see details below).\nThe Si(111)-7\u00027 clean surface was prepared by di-\nrect current heating up to 1250\u000eC for 30 s and com-\nputer controlled cooling. Then the 1/3 ML of Sn was\ndeposited at RT on the surface and annealed at 600\u000eC\nfor 5 min to create the Si(111)-(p\n3\u0002p\n3)-Sn reconstruc-\ntions. To eliminate the (p\n7\u0002p\n3)-Sn phase growth near\nthe step edges (Fig. 1a) the necessary quantity of de-\nposited Sn was \ftted by reducing the evaporation time\nuntil the 7\u00023 - Sn phase disappears completely. All\nSTM and STS measurements were performed with plat-\ninum STM tips, preliminary tested on Au foil to verify\nspectroscopy and microscopy capability. The I-Vcurves\nwere collected at \fxed temperature over the adatom of\nthe Si(111)-(p\n3\u0002p\n3)-Sn unit cell far away from defects\nand averaged over series of measurements consisting of\ntens individual cycles. The data collected over the indi-\nvidual defects are the same. The dI/dV were measured\nwith lock-in technique with a sinusoidal modulation of\nthe sample bias voltage with amplitude 30-50 mV at fre-\nquency f= 778 Hz, and veri\fed by comparison with cal-\nculated numerically dI/dV from averaged I-Vcurves.\nThe temperature dependent transport measurements\nwere performed with two-probe technique using Omicron\nLT SPM with modi\fed standard tip holder, equipped\nwith two\u000b30\u0016m platinum wire contacts. The distance\nbetween point contacts was l\u0019400\u0016m. The I-Vcurves\nwere measured using picoammeter Keithley 6487 with\nintegrated voltage source at \fxed sample temperatures\ncontrolled by LakeShore 325 temperature controller. In\napproximation of low conductive bulk the total conduc-\ntivity accounts only 2D geometrical spreading resistance\nof the surface:\n\u001b2p=I\n\u0019Vln(l\nd)\nwheredis the contact spot size. We take ln(l=d)\u00194:4\nwhich corresponds to d= 1\u000030\u0016m within 30% accuracy.\nThe contribution of the bulk conductivity into the mea-\nsured data for used low doped samples was found to be\nnegligibly small at low temperatures: a control measure-\nment obtained for the surface degraded at vacuum con-\ndition during 7 days shown at Fig. 2. The conductivity(a)\n(b)\nFigure 1: (a) - the STM image of the Si(111)-(p\n3\u0002p\n3)-Sn\nsurface with di\u000berent exposition time during Sn evaporation.\n(a): The extra Sn deposited adatoms forms triangle islands\nof the (p\n7\u0002p\n3)-Sn phase near terraces edges, T= 78K;\n(b) - the clean Si(111)-(p\n3\u0002p\n3)-Sn structure with average\ndefects value less than 2% is done at T= 5K under external\nillumination of the surface. Substrate Si n-type,\u001a= 1 \ncm,\nUT=\u00002 V,IT= 50 pA.\nof the bulk plus contribution through degraded surface\n\u001bis more than 2 orders of magnitude smaller than the\nclean surface conductivity below T= 100 K . This gives\nthe bulk contribution to be below 1%. The choice of two-\nprobe technique let us to extend the measurements over\n7 orders of surface conduction variation into a region of\nvery low conductivity not available with usual 4-probe\ntechnique.\nTo perform spectroscopy measurements at helium tem-\nperatures we used external illumination from typical LED\nsource with light intensity up to 10\u00004W\u0001cm2at the\nsample position. This provides necessary conduction of\nthe bulk required for STS and STM study and also re-\nmoves bands bending near the surface and restores initial\ncharge carrier concentration in the surface bands [19, 20].\nIllumination-induced shift of the surface bands energy\npositions by photovoltage (Fig. 3) was taken into ac-\ncount.\nRESULTS\nFig. 2 shows the temperature dependence of the sur-\nface conductivity of the Si(111)-(p\n3\u0002p\n3)-Sn surface\nmeasured on n-andp-type samples with various 2-point\nprobes in the temperature range of 25 \u0000260 K. At\n80< T < 20 K the conductivity corresponds to ac-\ntivation law \u001b/exp(\u0000\u0001\nkT) with the activation energy\n\u0001 = 34 meV, which is the same for both n-andp-type\nsamples. At lower temperatures ( T <25 K) the conduc-\ntivity of the surface drops below the measurement limit3\n0.010.020.030.040.0510−1310−1010−710−4\nT−1 (K−1)σ (Ω−1/square)\n  \nBulkSn/Si(111)-√\n3x√\n3\n∆ = 34 meVp−type\nn−type100 50 20T (K)\nFigure 2: Temperature variation of the surface conductivity\nfor the clean Si(111)-(p\n3\u0002p\n3)-Sn structure obtained with\nvarious 2 micro probes on n-type (crosses) and p-type (circles)\nSi substrates with \u001a= 1 \n\u0001cm. Di\u000berent coloured markers\ncorrespond to the individual measurements done with various\n2 micro probes on each time fresh prepared surface. The\nblack squares are the 2-point measurements of the Si(111)-\n(p\n3\u0002p\n3)-Sn surface degraded in the vacuum chamber for\n7 days, indicates the contribution of the bulk and subsurface\nlayer into the measured conductivity.\nof the used picoampermeter. The 25 K also was the limit\nfor STM measurements on our samples without using of\nexternal illumination of our samples.\nFig. 3 shows the STS data sets measured at T= 30K\nwith and without external illumination. The energy gap\nat zero voltage is clearly seen on both data sets collected\nin the dark. When the samples are illuminated, the en-\nergy gap is shifted by the value of surface photovoltage\nVph=\u00000:5 V andVph= 0:4 V for n-andp-samples\nrespectively. The values of photoshifts indicate that the\nShotky barrier near the surface almost completely disap-\npears. We observed no noticeable deviation in the shape\nand value of the energy gap measured with and without\nexternal illumination (Fig. 3c-d). The value of the gap\natT= 30 K is 2\u0001\u001940 meV similar with reported in\n[10] obtained on heavily doped Si samples at T= 5 K.\nAt lower temperatures it was impossible to detect any\nsurface states inside the bulk gap. The typical charac-\nteristicdI=dV spectra of the bulk are plotted in Fig. 4c.\nThe explanation for this is very low surface conductiv-\nity at helium temperatures and possible high resistance\nat contact sample-holder Rbulk\u0000hold, comparable with\ntunnelling resistance Rtun. In this case the bias volt-\nageVTis distributed partially between the vacuum tun-\nnelling junction and (bulk Si)-(metallic sample holder)\njunction (Fig 4a). By altering the distance between the\ntip and surface the Vbulk\u0000holdpart could be extracted\nand used for correction of dI=dV data. The method is\ndescribed in details in [13]. Namely, we measured I(V)\nspectra at distances shifted from set point at IT= 50 pA,\nUT= 2 V,RT= 4\u00011010\n at \u0001Z=\u00001\u0017A and\n−1 −0.5 0 0.510−210−1100\nVT (V)(dI/dV)RT\n  \nVph= −0.5Vn−type\nT = 30KLight On\nLight Off(a)\n−0.5 0 0.5 110−210−1100\nVT (V)(dI/dV)RT\n  \nVph = + 0.4 Vp−type\nT = 30KLight On\nLight Off(b)\n−0.2−0.1 00.10.200.050.1\nVT − Vph (V)(dI/dV)RT\n  \nLight ON\nT = 30K\n2∆ = 40 meVn−type\np−type(c)\n−0.2−0.1 00.10.200.050.1\nVT (V)(dI/dV)RT\n  \nLight Off\nT = 30K\n2∆ = 40 meVn−type, 30 K\np−type, 30 K(d)Figure 3: (a-b): the STS spectra ( dI=dV )RTof the Si(111)-\n(p\n3\u0002p\n3)-Sn surface done at T= 30K on illuminated (blue\nlines) n-andp-type samples, and the same point with no ex-\nternal illumination (black lines). (c-d): the value of the energy\ngap at Fermi level at Si(111)-(p\n3\u0002p\n3)-Sn surface for n-and\np-type samples depending on the illumination On or O\u000b. Set\npoint parameters are UT=\u00002 V,IT= 50 pA for n-type, and\nUT= 1:8 V,IT= 50 pA for p-type.\n\u0001Z= +1 \u0017A (Fig 4b). Then I(Vtip\u0000surface ) is calculated\nthe by subtraction I(Vbulk\u0000hold) fromI(VT) measured at\nhigher tunnelling distances (Fig 4d). Then the energy\ngap value varying from 70 to 100 meV for both n-andp-\ntype substrates was obtained from numerically calculated\n(dI=dVtip\u0000surface )RTspectra.\nAnother method is to decrease the bulk-holder resis-\ntance by preliminary evaporation of Ta contact areas\non the studied sample surface, so that the holder plates\nmount to this this Ta areas. In this case the I(V) spectra\nwas found to be not a\u000bected by the tip-surface distance,\nand the surface states could be distinguished in the bulk\ngap region. The resulted normalized dI=dV spectra are\nshown on Fig. 5. The energy gap values obtained by this\nmethod 2\u0001\u001970 meV for p-type sample and 2\u0001 \u001990\nmeV for n-type sample.\nDISCUSSION\nThe measured value of the energy gap in LDOS for\nthe Si(111)-p\n3\u0002p\n3-Sn surface at helium temperature is\n2\u0001\u001970\u000490meV and agrees well to the activation energy\nobtained in the transport measurements \u0001 = 34 meV.\nSo the ground state corresponds to the calculated spin-\norder magnetic Mott-Hubbard insulator state [9]. With\ntemperature increase the spin ordering should disappear4\nHolder Si Bulk Surface\nstatesVacuum Tip\nEF(a)\nEFEc\nEa\nEveVpveVtip−surf\neVbulk−holdeVT\n0 5 10−0.500.511.52\nIT (pA)VT (V)\n  \nVtip−surf\nVbulk−hold\nVph∆Z=0.1nm\n∆Z=−0.1nm(b)(b)\n−2 0 200.20.40.6(dI/dV)RT\nVT (V)VphSi bulk gapT = 5Kp−type(c)\n−0.2 00.20.400.20.4\n2∆ = 70 meV T = 5 K\nVtip−surf (V)(dI/dVtip−surf)RT(d)\nFigure 4: (a) - schematic energy diagram of tip-surface-sample\nholder bias voltage distribution of the illuminated sample,\nwhen the resistance of the bulk-holder part is comparable\nwith vacuum junction resistance; (c) - normalized dI=dV data\ntaken at set point \u0001 Z= 0 \u0017A,IT= 50 pA,UT= 2 V; (b)\n- I/V data of the same point taken at di\u000berent tip-sample\ndistances shifted from set point at \u0001 Z. (d) - corrected nor-\nmalized conductance ( dI=dV tip\u0000surface )RTof the Sn/Si(111)\nsurface at T = 5K.\n−1 0 100.20.4\nVT (V)(dI/dV)RT\n  \nT = 5K\nVph = − 0.5 Vn−type(a)\n−0.5 0 100.20.4\nVT (V)(dI/dV)RT\n  \nT = 5K\nVph = +0.4Vp−type(b)\nFigure 5: The normalized ( dI=dV )RTdata of the illuminated\nSi(111)-(p\n3\u0002p\n3)-Sn surface with preliminary deposited Ta\ncontact areas, T = 5K. (a) - n-type, (b) - p-type Si substrate\nwith\u001a= 1 \n cm. UT=\u00002 V,IT= 50 pA for n-type, and\nUT= 1:8 V,IT= 50 pA for p-type.\ndue to thermal \ructuations. The intersite antiferromag-\nnetic (AF) exchange coupling energy is J\u00183 meV\n[9], so vanishing of AF state at T&J\u001940 K is ex-\npected. The next phase for Si(111)-(p\n3\u0002p\n3)-Sn sur-\nface is pseudo-gap metal. The conductivity is expected\nto be the variable-range hopping conduction a\u000bected by\nstrong Coulomb interaction. Indeed, our data indicates\nthe Efros-Shklovskii temperature dependent conductivity\u001b\u0018exp(\u0000(TES=T)1=2) aboveT > 50 K (Fig. 6a) with\nestimatedTES\u00190:9 eV in good agreement with our\nsuggestion. This corresponds to the localization length\nof the charge carrier \u0018=2:8e2\n\u0014TES= 0:7 nm [22], where\n\u0014= (11:8 + 1)=2 dielectric constant of the Si bulk plus\nvacuum. The localization length precisely matches the\ndistance between Sn atoms in unit cell, which is very\nsimilar to what was observed in Si(111)-7 \u00027 surface\n[15].\nWith increasing the temperature the deep valley at\nzero bias in tunnelling LDOS is not connected any more\nto the band gap of the MI ground state, but could be de-\nscribed in terms of dynamic Coulomb blockade approx-\nimation as a sequential single electron tunnelling into a\nlow conductive surface with sheet conduction \u001b\u001ce2=h.\nIn this approximation the surface is presented as the\nresistanceRsurf and the capacitance Csurf (for more\ndetails see [16, 17]). The STS data at T= 40 K \ft-\nted within dynamic Coulomb approximation using the\nonly one \ftting parameter Csurf is presented in Fig.6b.\nFor higher temperatures the data also \fts well within\nCsurf = 1:1\u000110\u000018\u00041:2\u000110\u000018F. The relevant local-\nization length aof the tunnelled electron could be esti-\nmated asa\u0018C=8\u0019\u0014\"0\u00190:8 nm [23]. The value agrees\nperfectly with the localization length obtained from the\ntransport measurements.\nCONCLUSION\nIn conclusion, through analysis of transport conduc-\ntance measurements and tunnelling spectroscopy data of\nthe Si(111)-(p\n3\u0002p\n3)-Sn surface formed on low doped Si\nsamples, we found that the ground state of this surface\ncorresponds to the Mott-Hubbard insulator as was pro-\nposed before, with the surface band gap 2\u0001 = 70 meV.\nWith increasing the temperature the magnetic insulator\nstate vanishes due to thermal \ructuation and the surface\nturns into \"bad\" metal with a Coulomb gap at Fermi\nlevel. The temperature dependence of the surface con-\nductance corresponds to the Efros-Shklovskii conduction\natT > 50 K and the deep valley in STS data at higher\ntemperatures is described in terms of dynamic Coulomb\napproximation with measured localisation length per-\nfectly agrees with interatomic Sn atoms distance.\nACKNOWLEDGMENTS\nThe work was supported by the Russian Foundation\nfor Basic Research under contract 16-32-60156.\n\u0003arty@cplire.ru5\n0.06 0.08 0.10.12 0.1410−910−810−710−610−5\nT−1/2 (K−1/2)σ (Ω−1/square)\n  \nTES = 0.9 eVσ ~ exp( − (TES/T)−1/2)p−type\nn−type250 100 50T (K)\n(a)\n−0.4 −0.2 00.2 0.410−210−1100\nVT, (V)(dI/dV)RT\n  \nT = 40K\nC = 1.2e−18F\nDCB approx\nSTS data(b)\nFigure 6: (a) -the temperature variation of the Si(111)-p\n3\u0002p\n3-Sn surface conductivity presented in T\u00001=2coor-\ndinates. The data corresponds to the Efros-Sklovskii hopping\nconduction; (b) - the deep valley at surface Fermi level in nor-\nmalized (dI=dV )RTspectra of the Si(111)-p\n3\u0002p\n3-Sn surface\nat T = 40 K (black line) \ftted within dynamic Coulomb ap-\nproximation with Csurf= 1:2\u000110\u000018F,Rsurf= 4\u0001104\u0001Rh,\nT= 40 K.UT=\u00001:8 V,IT= 50 pA, sample n-type Si.\n[1] J.M. Carpinelli, H.H. Weitering, M. Bartkowiak, R.\nStumpf, and E.W. Plummer, Phys. Reb. Lett. 79, 2859\n(1997).\n[2] L. Petersen, Ismail, and E.W. Plummer, Prog. Surf. Sci.,\n711 (2002).\n[3] H.H. Weitering et. al., Phys. Rev. Lett. 78, 1331 (1997).[4] L.I. Johansson et. al., Surf. Sci. 360L478 (1996).\n[5] G.Profeta et. al., Phys. Rev. B. 62, 1556 (2000).\n[6] R.I.G. Uhrberg, H.M. Zhang, T. Balasubramanian, S.T.\nJemander, N. Lin, and G. V. Hansson, Phys. Rev . B.\n62, 8082 (2000).\n[7] J. Lobo, A. Tejeda, A. Mugarza, and E. G. Michel, Phys.\nRev. B. 68, 235332 (2003).\n[8] H. Morikawa, I. Matsuda, and S. Hasegawa, Phys. Rev.\nB.65, 201308 (2002).\n[9] G. Profeta and E. Tosatti, Phys. Rev. Lett. 98, 086401\n(2007).\n[10] S. Modesti, L. Petaccia, G. Ceballos, I. Vobornik, G.\nPanaccione, G. Rossi, L. Ottaviano, R. Larciprete, S.\nLizzit, and A. Goldoni, Phys. Rev. Lett. 98,126401\n(2007).\n[11] T. Hirahara, T. Komorida, Y. Gu, F. Nakamura, H.\nIdzuchi, H. Morikawa, and S. Hasegawa, Phys. Rev. B.\n80, 235419 (2009).\n[12] R. Losio, K. N. Altmann, and F. J. Himpsel, Phys. Rev.\nB61, 10845 (2000).\n[13] S. Modesti, H. Gutzmann, J. Wiebe, and R. Wiesendan-\nger, Phys. Rev. B 80, 125326 (2009).\n[14] T. Tanikawa, K. Yoo, I. Matsuda, S. Hasegawa, and Y.\nHasegawa, Phys. Rev. B 68, 113303 (2003).\n[15] A.B. Odobescu, A. A. Maizlakh and S. V. Zaitsev-Zotov,\nPhys. Rev. B 92, 165313 (2015).\n[16] P. Joyez and D. Esteve, Phys. Rev. B 561848 (1997).\n[17] C.Brun, K.H. Muler, I. Hong, F. Patthey, C. Flindt, W.-\nD. Shneider, Phys. Rev. Lett. 108, 126802 (2012).\n[18] Current carrier concentration change \u0001 Q=d\u0001Na, where\nd=r\n2\u000f\"\u0001E\ne2Na- is the depletion depth, and Na- doppant\nconcetration in the substrate, \u0001 E\u00190:4 eV - band bend-\ning value. For heavy doped Si with \u001a= 0:001\n\u0001cm the\ncarrier concentration change inp\n3\u0002p\n3-Sn surface is\nestimated about 10%.\n[19] A.B. Odobescu and S.V. Zaitsev-Zotov, J. Phys.: Con-\ndens. Matter 24, 395003 (2012).\n[20] S. Grafstr om, J. Appl. Phys. 91, 1717 2002.\n[21] J. Myslive\u0014 cek, A. Str\u0013 o_ zecka, J. Ste\u000f, P. Sobot\u0013 \u0010k, I.\nO\u0014 st'\u0013 adal, and B. Voigtl ander, Phys. Rev. B 73, 161302\n(2006).\n[22] B.I. Shklovskii, A.L. Efros. Electronic properties of doped\nsemiconductors. Springer-Verlag (1984).\n[23] The capacitance of tiny disk with characteristic dimen-\nsion adescribe spreading area of a tunnelled electron\nbefore the next tunnelling event is estimated as C=\n8a\u0019\u0014\" 0."    },    {        "title": "2303.07698v4.Higher_order_topological_superconductors_characterized_by_Fermi_level_crossings.pdf",        "content": "Higher-order topological superconductors characterized by Fermi level crossings\nHong Wang and Xiaoyu Zhu∗\nSchool of Physics, MOE Key Laboratory for Non-equilibrium Synthesis and Modulation of Condensed Matter,\nXi’an Jiaotong University, Xi’an, Shaanxi 710049, China\n(Dated: September 22, 2023)\nWe demonstrate that level crossings at the Fermi energy serve as robust indicators for higher-\norder topology in two-dimensional superconductors of symmetry class D. These crossings occur when\nthe boundary condition in one direction is continuously varied from periodic to open, revealing the\ntopological distinction between opposite edges. The associated Majorana numbers acquire nontrivial\nvalues whenever the system supports two Majorana zero modes distributed at its corners. Owing to\ntheir immunity to perturbations that break crystalline symmetries, Fermi level crossings are able to\ncharacterize a wide range of higher-order topological superconductors. By directly identifying the\nlevel-crossing points from the bulk Hamiltonian, we establish the correspondence between gapped\nbulk and Majorana corner states in higher-order phases. In the end, we illustrate this correspondence\nusing two toy models. Our findings suggest that Fermi level crossings offer a possible avenue for\ncharacterizing higher-order topological superconductors in a unifying framework.\nI. INTRODUCTION\nTopological states of matter are usually endowed with\na bulk-boundary correspondence, which facilitates the\nidentifications of topologically protected gapless bound-\nary modes without going into the details of the energy\nspectrum at open boundaries [1–3]. Recent advance-\nments in higher-order topological systems have extended\nthis correspondence to include gapped boundaries [4–13],\nwith gapless corner (hinge) modes appearing at the in-\ntersections of adjacent edges (surfaces). Tremendous ef-\nforts have been devoted to classifying and characterizing\nthese topological states, mostly in crystalline-symmetry\nprotected systems [14–34]. However, it is well known\nthat gapless corner or hinge states persist when crys-\ntalline symmetries are broken. This is especially evi-\ndent in higher-order topological superconductors [35–40],\nwhere Majorana zero modes [41–46] remain stable as mid\ngap states unless the bulk or boundary gap closes. Hence\nit would be desirable to characterize higher-order states\nregardless of whether crystalline symmetries are present.\nHigher-order topology can be understood from a\nboundary perspective, as different parts of the whole\nboundary, such as the four edges of a square lattice, may\nexhibit a distinct topology in higher-order phases. For\nintrinsic higher-order states, the relevant crystalline sym-\nmetry requires symmetry related edges or surfaces to be\ntopologically inequivalent [14–16]. A topology change is\nonly possible through bulk-gap closing. Consequently,\nbulk invariants, such as symmetry indicators related to\nthe crystalline symmetry, can be defined [17–20]. This\nstands in contrast with boundary-obstructed topological\nstates, which fall within an extrinsic higher-order classi-\nfication [11, 40]. Without the protections of crystalline\nsymmetries, the boundary topology in these states could\nchange while the bulk gap remains open. One may char-\n∗xiaoyu.zhu@xjtu.edu.cnacterize the topology by Wilson loop eigenvalues of Wan-\nnier bands that are obtained from Wilson loops of energy\nbands, the so-called nested Wilson loop approach [4, 5].\nHowever, the quantization of such topological invariants\nstill requires the presence of crystalline symmetries, such\nas mirror symmetry [11, 38]. Establishing bulk-boundary\ncorrespondence under broken crystalline symmetries re-\nmains an open question. Considering that boundary\ntopology is ultimately determined by the bulk proper-\nties for both intrinsic and boundary-obstructed phases,\nit should be possible to associate a topological invariant\nwith it based on bulk information, which applies in both\nphases.\nIn this paper, we focus on two-dimensional (2D) su-\nperconductors of symmetry class D [3] and higher-order\nphases featuring two Majorana corner states. The higher-\norder topology can be characterized by a pair of Majo-\nrana numbers, which are intimately related to Fermi level\ncrossings that emerge during the continuous variation of\nthe boundary condition along one direction, as illustrated\nin Fig. 1(a). We further introduce a generic method for\nlocating these crossings from the bulk Hamiltonian. As\na result, bulk-boundary correspondence is established in\nboth higher-order phases discussed earlier, due to the ro-\nbustness of the Fermi level crossings against crystalline-\nsymmetry-breaking perturbations.\nII. GENERAL THEORY\nTo demonstrate how Fermi level crossings determine\nthe higher-order topology of D-class superconductors, we\nstart from a 2D periodic lattice and modulate its bound-\nary condition in one direction. The resulting Bogoliubov-\nde Gennes (BdG) Hamiltonian can be expressed as\n˜Hλa=X\nk¯aHλa(k¯a) =˜H1−(1−λa)X\nk¯aBa(k¯a),(1)\nwhere ¯ a=y(x) when a=x(y), and the real parame-\nterλacontrols the boundary condition in the adirec-arXiv:2303.07698v4  [cond-mat.mes-hall]  21 Sep 20232\n(a)\nCylinderTorus\nFermi Level Crossing\nMajorana \nZero Mode\n(b)\n(c)\nOBC APBC PBC OBC APBC PBCbulk gapboundary gap\nFIG. 1. (a) Higher-order topology characterized by Fermi\nlevel crossings in the D symmetry class. Crossings occur\nat high-symmetry momenta K= 0, π, when the system on\na torus ( λa= 1) is continuously deformed into a cylinder\n(λa= 0). (b) Topology of edges determined by Majorana\nnumbers. When Mx(y)=−1, two opposite edges along the\ny(x) direction exhibit a distinct topology, indicated by dif-\nferent colors. (c) Schematic plots of the BdG spectrum at\nK= 0(π) with or without level crossings. The fermion parity\nof the ground state (“+” for even, “ −” for odd) switches at\neach crossing. The crossing may disappear when the bulk or\nboundary gap closes.\ntion, with λa= 1,−1,0 corresponding to the periodic\n(PBC), anti periodic (APBC), and open boundary condi-\ntion (OBC), respectively. In Eq. (1), Hλa(k¯a) represents\nthe 1D boundary-modulated Hamiltonian at wave vector\nk¯a, and Ba(k¯a) involves all terms that cross its bound-\nary. The lattice terminations we consider are compatible\nwith unit cells, thus allowing the specific form of Bto\nbe directly read off from the bulk Hamiltonian ˜H1. The\nprocess of varying λafrom 1 to 0 is akin to gradually\ncutting a torus along the ¯ adirection until it eventually\nbecomes a cylinder, as illustrated in Fig. 1(b) for the\ncase of a=x.\nHere, we consider a gapped bulk with trivial first-order\ntopology, which means the cylindrical system described\nby˜Hλa=0is fully gapped. Treating it as a quasi-1D\nsystem along the ¯ a-direction, we may characterize the\nhigher-order topology with the Majorana number [47–49]\nMa= sgnY\nKPf[−iHλa=0(K)], (2)\nwhere “Pf” is shorthand for Pfaffian, K= 0, πrepresents\nthe high-symmetry momentum, and Hrefers to the ma-\ntrix representation of Hin the Majorana basis. In 1D,\nthe Majorana number being −1 implies the presence of\na single Majorana zero mode at each end. If either Mx\norMy, or both of them, take the value of −1, we willinstead have two Majorana zero modes at the corners of\na 2D sheet. To elaborate this let us consider the cylin-\ndrical system in the lower left-hand panel of Fig. 1(b)\nwithMx=−1. If we cut it along the axis, the result-\ning two edges along the xdirection will each harbor one\nMajorana mode. Due to the trivial first-order topology,\nthese localized modes cannot propagate along the edges\nand must be confined to their respective ends, i.e., the\ncorners. If, in addition My=−1, the two modes would\nalso appear at the two edges in the ydirection. As a re-\nsult, they can only reside at opposite corners, as depicted\nin the upper right-hand panel of Fig. 1(b). If My= 1,\nhowever, they would appear at adjacent corners along\ntheydirection, as shown in the lower right-hand panel\nof Fig. 1(b).\nThe Majorana number defined in Eq. (2) is closely\nrelated to level crossings at the Fermi energy ϵ= 0 that\nappear while λavaries in the range [0 ,1]. Notably, Eq.\n(2) only involves the 1D Hamiltonian at high-symmetry\nmomenta K. Therefore, we only need to consider Fermi\nlevel crossings in these subsystems, as shown in Fig. 1(a).\nAt each crossing, the fermion parity of the ground state\nswitches, indicated by the sign change of Pf[ −iHλa(K)].\nWe can then characterize the fermion-parity difference\nbetween PBC and OBC by the number of crossings in\nbetween, denoted by ηa,K, as Fig. 1(c) demonstrates.\nThis is formally expressed as\n(−1)ηa,K=sgn Pf[ −iHλa=0(K)]\nsgn Pf[ −iHλa=1(K)]. (3)\nWe may also define a Majorana number for the toroidal\nsystem ˜Hλa=1(˜H1) similar to Eq.(2), which due to trivial\nfirst-order topology must be positive, i.e.,\nsgnY\nKPf[−iHλa=1(K)] = 1 . (4)\nCombining Eqs. (2)-(4), we arrive at\nMa=Y\nK(−1)ηa,K= (−1)ηa, (5)\nwhere ηadenotes the total number of crossings at K=\n0, π. An odd value of ηxorηyimplies the system re-\nsides in a higher-order phase. Fermi level crossings are\nprotected by fermion-parity conservation and particle-\nhole symmetry, making them immune to crystalline-\nsymmetry-breaking perturbations [50].\nIntuitively, we may understand the relation between\nFermi level crossings and higher-order topology from the\nviewpoint of boundary topology. As shown in Fig. 1(b),\nan odd value of ηa(Ma=−1) reveals that opposite edges\nalong ¯ aare topologically inequivalent (shown in different\ncolors). This explains the possible locations of Majorana\nzero modes, which appear at the intersections of topo-\nlogically distinct edges. In some simple models, as we\ndemonstrate later, the edge topology can be character-\nized by the sign of the mass gap in the edge Hamiltonian,\nallowing us to validate this argument.3\nTo establish the bulk-boundary correspondence, we\nwill demonstrate how the Fermi level crossings of the\n1D subsystems are identified from the bulk Hamiltonian.\nFor brevity, we use Hλto replace Hλa(K), where\nHλ=H1−(1−λ)B (6)\nrepresents a generic 1D Hamiltonian of D class. Follow-\ning the prescription given by Ref.[51], we first define a\nretarded Green’s function\nGλ(ϵ) = (ϵ− H λ+i˜δ)−1=A−1\nλ(ϵ)G1(ϵ), (7)\nwhere ˜δis a positive infinitesimal, G1(ϵ) is the Green’s\nfunction corresponding to the bulk Hamiltonian H1, and\nAλ(ϵ) = 1 + (1 −λ)G1(ϵ)B. Since we focus on the pa-\nrameter regime in which the bulk is fully gapped, in-gap\nstates of Hλare solely determined by the poles of A−1\nλ(ϵ).\nConsequently, level-crossing points are identified as roots\nof\ndet[Aλ(ϵ= 0)] = 0 , (8)\nwhere Aλis the matrix representation of Aλ. AsBonly\nincludes intra-cell terms crossing the boundary, we then\nhave [ Aλ]IJ=δIJifJdoes not appear in these terms.\nThis enables us to calculate det( A) using a much smaller\nmatrix D, which is obtained by projecting Ainto the\neigenspace of Band satisfies det( D) = det( A). The en-\ntries of Dare given by\n[Dλ(ϵ)]ij=δij+ (1−λ)X\nn,k⟨i|n, k⟩⟨n, k|B|j⟩\nϵ−ϵn,k,(9)\nwhere ϵn,kdenotes the energy spectrum of the bulk\nHamiltonian H1in the Brillouin zone, with |n, k⟩be-\ning the corresponding eigenstate, and |i⟩,|j⟩represent the\neigenvectors of B. The dimension of Dλis equal to the\nrank of B, denoted by Nb. We then obtain the charac-\nteristic equation\ndet[Dλ(ϵ= 0)] = 0 , (10)\nwhich has Nbroots in total. The number of Fermi level\ncrossings ηis half the number of real roots in the in-\nterval [0 ,1], from which we can readily obtain Majorana\nnumbers according to Eq. (5).\nCompared to Eq.(2), where Majorana numbers are de-\ntermined by calculating the Pfaffian of finite systems with\nopen boundaries [52], i.e., Pf[ −iHλa=0(K)], and the ac-\ncuracy crucially depends on system size, identifying the\nFermi level crossings is computationally more accurate\nand efficient for a translation-invariant system. It does\nnot suffer from finite-size effects, and the computational\ncost is similar to Wilson loop calculations. Moreover, it\nprovides a potential path to characterizing higher-order\ntopological superconductors in other symmetry classes\nsuch as the DIII or BDI classes, where Fermi level cross-\nings might be protected by their topological charges. Ad-\nditionally, by pinpointing the crossings directly from the\nN 1\n(a)\n(b)\nOBC\n APBC\n PBC\nFIG. 2. (a) Geometry of the two-leg Kitaev ladder. λcontrols\nthe boundary condition. (b) The fermion-parity switch in a\ndimerized lattice ( t1= 0). For the case of |m|>|δm|, the\nground state switches from the even-parity sector ( Ee,1, solid\nlines) to the odd-parity sector ( Eo,1, dashed lines) while the\nboundary condition varies from PBC (APBC) to OBC. Level\ncrossings are indicated by black arrows.\nbulk Hamiltonian, we establish the correspondence be-\ntween gapped bulk and gapless corner states in higher-\norder phases. In the following, we shall illustrate this in\nspecific models.\nIII. TOY MODELS\nFirst we consider a two-leg Kitaev ladder [53–56] as\nschematically shown in Fig. 2(a), and demonstrate how\nlevel crossings are identified from the bulk Hamiltonian.\nEach unit cell contains four Majorana fermions denoted\nbyαs,jandβs,j, with s= 1,2 being the chain in-\ndex and jreferring to the cell index. The boundary-\nmodulated Hamiltonian with Nunit cells has the form\nHλ= ΓTHλΓ in the Majorana basis Γ =LN\nj=1Γj,\nwhere Γ j={α1,j, α2,j, β1,j, β2,j}T/√\n2 and the Hamil-\ntonian matrix is given by\nHλ=X\nr=0,±1Tr⊗hr+λ(TN−1⊗h†\n1+ H.c.) .(11)\nHere, Tdenotes the translation operator that moves each\ncell by one site to the left, with T|j⟩=|j−1⟩andT|j=\n1⟩= 0 [57]. Hamiltonian (11) includes the intra cell\nterm h0=−t1τy−mσy−δmτ zσy, and inter cell term\nh1=h†\n−1=t2(τy+iτx)/2, with τandσbeing Pauli\nmatrices that act in the chain and rung space separately.\nt1andt2represent couplings of Majorana fermions along\nthe chain, while mandδmare those along the rung. For\nbrevity, we assume t1andt2to be non-negative.\nIn this model, mandδmdetermine whether level cross-\nings occur when λvaries in the range [0 ,1]. This is readily\nseen in a perfectly dimerized lattice ( t1= 0), in which\ncase only the boundary block shown in Fig. 2(b) depends4\n(a) (b)\n(c) (d)-11\n-1 0 1\n-1 0 1-11\n0\n-11\n0\n-11\n-1 -0.5 1 0 0.5-101\n-1 0 1-11\n0\n-11\nFIG. 3. (a)-(c) Fermi level crossings in a BdG spectrum.\nThe fermion parity switches at each crossing. The crossing\nappears when |m|>|δm|. Blue dashed lines indicate signs\nof the Pfaffian of a 1D Hamiltonian. (d) The variation of\ncrossing points λwith m.δm= 0.5,t1= 1, and t2= 2.\nonλ, and its Hamiltonian has the form\nHb=−2λt2(ψ†\n1ψ1+ψ†\n2ψ2−1)\n+ 2i(mψ†\n1ψ2+δmψ†\n1ψ†\n2−H.c.), (12)\nwhere ψs= (αs,1+iβs,N)/2 are fermionic operators.\nThe conservation of fermion number parity enables us\nto study the lowest energy levels in the even- and odd-\nparity sectors separately, with Ee,1=−2p\nλ2t2\n2+δm2\nand Eo,1=−2|m|. While the boundary condition\ngoes from PBC to OBC, the two levels would cross if\n0< m2−δm2< t2\n2, signaling a switch in the ground-\nstate fermion parity, as demonstrated in Fig. 2(b). This\nparity switch could be observed from the zero-bias peak\nin an experimental setup that consists of two quantum\ndots coupled by a nanowire-superconductor heterojunc-\ntion [58, 59]. The parameters mandδmare related to\nthe electrochemical potential of quantum dots, and t2or\nλis controlled by tuning the cross Andreev reflection and\nelastic cotunnelling.\nFor generic t1, we have H1=P\nkΓT\n−kHkΓkinkspace,\nwith the basis Γ k={α1,k, α2,k, β1,k, β2,k}T/√\n2, and the\nBloch Hamiltonian\nHk= (−t1+t2cosk)τy−t2sinkτx−mσy−δmτ zσy.(13)\nThe energy spectrum is given by\nϵn,k=±\u0012q\nt2\n1+t2\n2−2t1t2cosk+δm2±m\u0013\n,(14)\nwith nbeing the band index. Substituting ϵn,kandB=\nΓT\nNh1Γ1+ H.c. into Eqs. (9) and (10), we obtain\nλ2= 1−2Λ\nΛ−a+ 2t2\n2, (15)\nwith Λ =√\na2−b2,a=t2\n1+t2\n2+δm2−m2, and b= 2t1t2.\nFrom Eq. (15), we find that the number of crossings\n(a)\n0 �/200.4\n-0.4\n�(-1,-1)(1,-1) (-1,1)(1,1)\n(-1,-1) (-1,1) (1,-1)\n00.5\n00.5(b)\n(c)\nFIG. 4. (a) Phase diagram of the 2D model in ( θ, δm ) space.\nFour different phases are characterized by Majorana numbers\n(Mx,My). (b) Evolutions of crossing points with θ. No\ncrossings appear at K=πin this case and hence only those\natK= 0 are displayed. (c) Distributions of Majorana zero\nmodes in three nontrivial phases that are separated by bound-\nary phase transitions. Majorana zero modes appear either\nat two adjacent corners ( θ= 0, π/2), or at opposite corners\n(θ=π/4). The symbols “+” and “ −” indicate the signs of\nthe edge gaps. t1= 0.5,t2= 1, and m= 0.4.\nη= 1 when 0 < m2−δm2<(t2−t1)2andt1< t2,\nas shown in Fig. 3. This indicates that the boundary\nphase transition occurs at |m|=|δm|as in the dimerized\ncase, which is verified by the exact boundary spectrum\n(see Supplemental Material [60] and Ref. [61] therein).\nIn the special case where δm= 0, Hamiltonian (13) is in-\nvariant under inversion, with the corresponding operator\nbeing τy, up to a gauge factor. The inversion symme-\ntry facilitates the direct determination of the Fermi level\ncrossings from the differences of the ground-state inver-\nsion eigenvalues between PBC and APBC [60]. With the\nknowledge of ηin a 1D system, we can proceed to deter-\nmine the higher-order topology in a 2D system, according\nto Eq. (5).\nThe 2D Hamiltonian we consider takes the form\nH2D\nk= [t2(coskx+ cos ky)−t1−t2]τy−δmτ zσy(16)\n−t2(sinkxτx+ sin kyτzσz)−m(cosθσy+ sin θτyσx),\nwhen written in the Majorana basis as in Eq.(13), and\nreduces to the 1D Hamiltonian at ky= 0, θ= 0. This\nmodel is equivalent to the p±ipsuperconductor under\nan in-plane Zeeman field [62, 63]. According to Eq. (5),\nMajorana numbers ( Mx,My) are determined by Fermi\nlevel crossings of four 1D Hamiltonians, Hλa(K). In Fig.5\n4(a), we draw the ( θ, δm ) phase diagram. Here the cross-\nings only occur at K= 0 as Fig. 4(b) shows, although\nit is possible they emerge at K=πfort1andt2taking\nother values. Two Majorana corner states emerge when\nat least one Majorana number takes −1, as illustrated in\nFig. 4(c).\nTo corroborate previous arguments concerning the re-\nlation between level crossings and boundary topology, we\nobtain the mass gap for an arbitrary edge [60], given by\n∆(ϕ) =δm−mcos(ϕ−θ), (17)\nwhere ϕindicates the normal direction of the edge ( ϕ=\n0, π/2 for right and top edges respectively). The topology\nof the edges in D-class systems can be characterized by\nthe sign of the mass gap. As seen from the three repre-\nsentative cases in Fig. 4(c), gaps of opposite edges along\ny(x) indeed take different signs when ηx(y)is an odd num-\nber, or equivalently, Mx(y)=−1. This can be guaran-\nteed when inversion symmetry is enforced, by noting that\n∆(ϕ) =−∆(ϕ+π) in the absence of δm. In this intrinsic\nhigher-order phase, we always have Mx=My=−1.\nTurning on δmbreaks inversion symmetry and drives\nthe system into a boundary-obstructed phase, in which\nprocess the gap signs do not change immediately, so is\nthe number of Fermi level crossings. We can therefore\nuse Fermi level crossings to characterize the higher-order\ntopology in both phases.\nThe robustness of Fermi level crossings is also reflected\nin their persistence under weak disorder or boundary im-\npurities [60]. While Eqs. (5) and (10) may not be directlyapplicable due to potential broken translation symmetry,\nthe number of Fermi level crossings remains unchanged.\nThis reinforces their role as a reliable tool to characterize\nhigher-order topological superconductors.\nIV. CONCLUSION\nIn conclusion, Fermi level crossings can serve as use-\nful indicators for higher-order topology in the D symme-\ntry class when the nontrivial phase accommodates two\nMajorana corner states. The applicability of this ap-\nproach extends beyond the toy models introduced above,\nas demonstrated in the Supplemental Material [60] for a\nRashba bilayer system. The level crossings we consider\nemerge while the boundary condition continuously varies\nfrom PBC to OBC, during which two opposite edges\ngradually decouple. An odd number of crossings signals\na topological distinction between the two edges. From\nthis point of view, one may consider Fermi level crossings\nemerging under variations of other twisted boundary con-\nditions [64] when dealing with higher-order phases with\nfour or more Majorana corner states, where one needs to\nassociate the crossings with topological distinctions be-\ntween neighboring edges.\nACKNOWLEDGEMENTS\nThis work was supported by National Science Founda-\ntion of China (NSFC) under Grant No. 11704305, and\nthe Innovation Program for Quantum Science and Tech-\nnology (2021ZD0302400).\n[1] M. Z. Hasan and C. L. Kane, Colloquium : Topological\ninsulators, Rev. Mod. Phys. 82, 3045 (2010).\n[2] X.-L. Qi and S.-C. Zhang, Topological insulators and su-\nperconductors, Rev. Mod. Phys. 83, 1057 (2011).\n[3] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,\nClassification of topological quantum matter with sym-\nmetries, Rev. Mod. Phys. 88, 035005 (2016).\n[4] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes,\nQuantized electric multipole insulators, Science 357, 61\n(2017).\n[5] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes,\nElectric multipole moments, topological multipole mo-\nment pumping, and chiral hinge states in crystalline in-\nsulators, Phys. Rev. B 96, 245115 (2017).\n[6] J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen,\nand P. W. Brouwer, Reflection-Symmetric Second-Order\nTopological Insulators and Superconductors, Phys. Rev.\nLett. 119, 246401 (2017).\n[7] Z. Song, Z. Fang, and C. Fang, ( d - 2 ) -Dimensional\nEdge States of Rotation Symmetry Protected Topological\nStates, Phys. Rev. Lett. 119, 246402 (2017).\n[8] M. Ezawa, Higher-Order Topological Insulators and\nSemimetals on the Breathing Kagome and Pyrochlore\nLattices, Phys. Rev. Lett. 120, 026801 (2018).[9] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang,\nS. S. P. Parkin, B. A. Bernevig, and T. Neupert, Higher-\norder topological insulators, Sci. Adv. 4, eaat0346 (2018).\n[10] F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook,\nA. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock,\nS. Jeon, I. Drozdov, H. Bouchiat, S. Gu´ eron, A. Yazdani,\nB. A. Bernevig, and T. Neupert, Higher-order topology\nin bismuth, Nature Phys 14, 918 (2018).\n[11] E. Khalaf, W. A. Benalcazar, T. L. Hughes, and\nR. Queiroz, Boundary-obstructed topological phases,\nPhys. Rev. Research 3, 013239 (2021).\n[12] Q. Wang, C.-C. Liu, Y.-M. Lu, and F. Zhang, High-\nTemperature Majorana Corner States, Phys. Rev. Lett.\n121, 186801 (2018).\n[13] R.-X. Zhang, W. S. Cole, and S. Das Sarma, Helical\nHinge Majorana Modes in Iron-Based Superconductors,\nPhys. Rev. Lett. 122, 187001 (2019).\n[14] E. Khalaf, Higher-order topological insulators and super-\nconductors protected by inversion symmetry, Phys. Rev.\nB97, 205136 (2018).\n[15] M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer,\nSecond-order topological insulators and superconductors\nwith an order-two crystalline symmetry, Phys. Rev. B\n97, 205135 (2018).6\n[16] L. Trifunovic and P. W. Brouwer, Higher-Order Bulk-\nBoundary Correspondence for Topological Crystalline\nPhases, Phys. Rev. X 9, 011012 (2019).\n[17] A. Skurativska, T. Neupert, and M. H. Fischer, Atomic\nlimit and inversion-symmetry indicators for topological\nsuperconductors, Phys. Rev. Research 2, 013064 (2020).\n[18] S. Ono, H. C. Po, and H. Watanabe, Refined symme-\ntry indicators for topological superconductors in all space\ngroups, Sci. Adv. 6, eaaz8367 (2020).\n[19] R. Takahashi, Y. Tanaka, and S. Murakami, Bulk-edge\nand bulk-hinge correspondence in inversion-symmetric\ninsulators, Phys. Rev. Research 2, 013300 (2020).\n[20] Y.-T. Hsu, W. S. Cole, R.-X. Zhang, and J. D. Sau,\nInversion-protected Higher-order Topological Supercon-\nductivity in Monolayer WTe 2, Phys. Rev. Lett. 125,\n097001 (2020).\n[21] F. Tang, S. Ono, X. Wan, and H. Watanabe, High-\nThroughput Investigations of Topological and Nodal Su-\nperconductors, Phys. Rev. Lett. 129, 027001 (2022).\n[22] Z. Yan, Higher-Order Topological Odd-Parity Supercon-\nductors, Phys. Rev. Lett. 123, 177001 (2019).\n[23] Z. Zhang, J. Ren, Y. Qi, and C. Fang, Topological clas-\nsification of intrinsic three-dimensional superconductors\nusing anomalous surface construction, Phys. Rev. B 106,\nL121108 (2022).\n[24] A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Wil-\nson loop approach to fragile topology of split elemen-\ntary band representations and topological crystalline in-\nsulators with time-reversal symmetry, Phys. Rev. B 100,\n195135 (2019).\n[25] Y. Hwang, J. Ahn, and B.-J. Yang, Fragile topology pro-\ntected by inversion symmetry: Diagnosis, bulk-boundary\ncorrespondence, and Wilson loop, Phys. Rev. B 100,\n205126 (2019).\n[26] J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and\nR.-J. Slager, Topological Classification of Crystalline In-\nsulators through Band Structure Combinatorics, Phys.\nRev. X 7, 041069 (2017).\n[27] F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Compre-\nhensive search for topological materials using symmetry\nindicators, Nature 566, 486 (2019).\n[28] T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang,\nH. Weng, and C. Fang, Catalogue of topological elec-\ntronic materials, Nature 566, 475 (2019).\n[29] M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A.\nBernevig, and Z. Wang, A complete catalogue of high-\nquality topological materials, Nature 566, 480 (2019).\n[30] R.-X. Zhang, Bulk-Vortex Correspondence of\nHigher-Order Topological Superconductors (2022),\narXiv:2208.01652.\n[31] M. Jung, Y. Yu, and G. Shvets, Exact higher-order\nbulk-boundary correspondence of corner-localized states,\nPhys. Rev. B 104, 195437 (2021).\n[32] E. Roberts, J. Behrends, and B. B´ eri, Second-order\nbulk-boundary correspondence in rotationally symmetric\ntopological superconductors from stacked Dirac Hamilto-\nnians, Phys. Rev. B 101, 155133 (2020).\n[33] S. Kooi, G. van Miert, and C. Ortix, The bulk-corner cor-\nrespondence of time-reversal symmetric insulators, npj\nQuantum Mater. 6, 1 (2021).\n[34] S.-J. Huang and Y.-T. Hsu, Faithful derivation of symme-\ntry indicators: A case study for topological superconduc-\ntors with time-reversal and inversion symmetries, Phys.\nRev. Research 3, 013243 (2021).[35] X. Zhu, Second-Order Topological Superconductors with\nMixed Pairing, Phys. Rev. Lett. 122, 236401 (2019).\n[36] Z. Yan, F. Song, and Z. Wang, Majorana Corner Modes\nin a High-Temperature Platform, Phys. Rev. Lett. 121,\n096803 (2018).\n[37] T. Liu, J. J. He, and F. Nori, Majorana corner states\nin a two-dimensional magnetic topological insulator on\na high-temperature superconductor, Phys. Rev. B 98,\n245413 (2018).\n[38] A. Tiwari, A. Jahin, and Y. Wang, Chiral Dirac su-\nperconductors: Second-order and boundary-obstructed\ntopology, Phys. Rev. Research 2, 043300 (2020).\n[39] Y. Volpez, D. Loss, and J. Klinovaja, Second-Order\nTopological Superconductivity in π-Junction Rashba\nLayers, Phys. Rev. Lett. 122, 126402 (2019).\n[40] X. Wu, W. A. Benalcazar, Y. Li, R. Thomale, C.-X. Liu,\nand J. Hu, Boundary-Obstructed Topological High- T c\nSuperconductivity in Iron Pnictides, Phys. Rev. X 10,\n041014 (2020).\n[41] N. Read and D. Green, Paired states of fermions in\ntwo dimensions with breaking of parity and time-reversal\nsymmetries and the fractional quantum Hall effect, Phys.\nRev. B 61, 10267 (2000).\n[42] F. Wilczek, Majorana returns, Nature Phys 5, 614\n(2009).\n[43] J. Alicea, New directions in the pursuit of Majorana\nfermions in solid state systems, Rep. Prog. Phys. 75,\n076501 (2012).\n[44] T. D. Stanescu and S. Tewari, Majorana fermions in semi-\nconductor nanowires: Fundamentals, modeling, and ex-\nperiment, J. Phys.: Condens. Matter 25, 233201 (2013).\n[45] S. R. Elliott and M. Franz, Colloquium : Majorana\nfermions in nuclear, particle, and solid-state physics, Rev.\nMod. Phys. 87, 137 (2015).\n[46] R. Aguado, Majorana quasiparticles in condensed mat-\nter, Riv. Nuovo Cimento 40, 523 (2017).\n[47] A. Y. Kitaev, Unpaired Majorana fermions in quantum\nwires, Phys.-Usp. 44, 131 (2001).\n[48] M. Kheirkhah, Z. Yan, and F. Marsiglio, Vortex-line\ntopology in iron-based superconductors with and with-\nout second-order topology, Phys. Rev. B 103, L140502\n(2021).\n[49] P. P. Poduval, T. L. Schmidt, and A. Haller, Perfectly\nlocalized Majorana corner modes in fermionic lattices\n(2023), arXiv:2303.01535.\n[50] C. W. J. Beenakker, J. M. Edge, J. P. Dahlhaus, D. I.\nPikulin, S. Mi, and M. Wimmer, Wigner-Poisson Statis-\ntics of Topological Transitions in a Josephson Junction,\nPhys. Rev. Lett. 111, 037001 (2013).\n[51] J.-W. Rhim, J. H. Bardarson, and R.-J. Slager, Unified\nbulk-boundary correspondence for band insulators, Phys.\nRev. B 97, 115143 (2018).\n[52] M. Wimmer, Algorithm 923: Efficient Numerical Com-\nputation of the Pfaffian for Dense and Banded Skew-\nSymmetric Matrices, ACM Trans. Math. Softw. 38, 1\n(2012).\n[53] N. Wu, Topological phases of the two-leg Kitaev ladder,\nPhysics Letters A 376, 3530 (2012).\n[54] G. Y. Chitov, Local and nonlocal order parameters in the\nKitaev chain, Phys. Rev. B 97, 085131 (2018).\n[55] R. Wakatsuki, M. Ezawa, and N. Nagaosa, Majorana\nfermions and multiple topological phase transition in Ki-\ntaev ladder topological superconductors, Phys. Rev. B\n89, 174514 (2014).7\n[56] Y. Yan, L. Qi, D.-Y. Wang, Y. Xing, H.-F. Wang, and\nS. Zhang, Topological Phase Transition and Phase Dia-\ngrams in a Two-Leg Kitaev Ladder System, Ann. Phys.\n532, 1900479 (2020).\n[57] A. Alase, E. Cobanera, G. Ortiz, and L. Viola, Exact\nSolution of Quadratic Fermionic Hamiltonians for Arbi-\ntrary Boundary Conditions, Phys. Rev. Lett. 117, 076804\n(2016).\n[58] T. Dvir, G. Wang, N. van Loo, C.-X. Liu, G. P. Mazur,\nA. Bordin, S. L. D. ten Haaf, J.-Y. Wang, D. van Driel,\nF. Zatelli, X. Li, F. K. Malinowski, S. Gazibegovic,\nG. Badawy, E. P. A. M. Bakkers, M. Wimmer, and L. P.\nKouwenhoven, Realization of a minimal Kitaev chain in\ncoupled quantum dots, Nature 614, 445 (2023).\n[59] A. Bordin, X. Li, D. van Driel, J. C. Wolff, Q. Wang,\nS. L. D. ten Haaf, G. Wang, N. van Loo, L. P. Kouwen-\nhoven, and T. Dvir, Crossed Andreev reflection and elas-\ntic co-tunneling in a three-site Kitaev chain nanowire de-\nvice (2023), arXiv:2306.07696.[60] See Supplemental Material at [**] for detailed derivations\nof the boundary spectrum of the two-leg Kitaev ladder;\nthe role of inversion symmetry in determining Fermi-\nlevel crossings; the effective Hamiltonian for an arbitrary\nedge in the 2D model; the influence of bulk disorder and\nboundary impurities on Fermi-level crossings; Fermi-level\ncrossings in a Rashba bilayer superconducting system.\n[61] S. S. Pershoguba and V. M. Yakovenko, Shockley model\ndescription of surface states in topological insulators,\nPhys. Rev. B 86, 075304 (2012).\n[62] V. T. Phong, N. R. Walet, and F. Guinea, Majorana\nzero modes in a two-dimensional p -wave superconductor,\nPhys. Rev. B 96, 060505(R) (2017).\n[63] X. Zhu, Tunable Majorana corner states in a two-\ndimensional second-order topological superconductor in-\nduced by magnetic fields, Phys. Rev. B 97, 205134\n(2018).\n[64] Z.-D. Song, L. Elcoro, and B. A. Bernevig, Twisted\nbulk-boundary correspondence of fragile topology, Sci-\nence367, 794 (2020).1\nSupplemental Material for “Higher-order topological superconductors\ncharacterized by Fermi level crossings”\nIn this Supplemental Material, we provide detailed\nderivations of the boundary spectrum for the two-leg\nKitaev ladder, explore the role of inversion symmetry\nin determining Fermi level crossings, derive the effec-\ntive Hamiltonian for an arbitrary edge in the 2D model,\nand investigate the robustness of Fermi level crossings\nagainst bulk disorder and boundary impurities. We also\ndemonstrate in a Rashba bilayer superconducting system\nhow Fermi level crossings effectively identify higher-order\ntopological phases.\nAppendix A: A. Boundary spectrum\nWe consider a semi-infinite system with boundary at\nj= 1 in the two-leg Kitaev ladder. In the Hamiltonian\nHλ=0,σyis a good quantum number, and we can work\nin its eigenspace, where Hλ=0is block diagonal. Con-\nsequently, we can set σyto be ±1 in the two blocks,\nrespectively. In this toy model, each block with σy= 1\n(σy=−1) can be viewed as a particle (hole) version of\nSu-Schrieffer-Heeger (SSH) model, with the parameter m\nacting as a chemical potential term that shifts the energy\nspectrum in corresponding block. The two blocks do not\ncouple due to the conservation of σyin this simple model.\nIt is possible to introduce additional terms that couple\nthe two blocks and make the model more complicated.\nHowever, the main results do not change as we only re-\nquire particle-hole symmetry. The simplicity of this toy\nmodel allows us to obtain analytical results in a straight-\nforward manner, as we demonstrate in the following.\nWe will now derive the condition for the appearance\nof gapped boundary modes in the two-leg Kitaev ladder,\nas well as the boundary spectrum. Let’s first consider\nthe block with σy= 1, and the case with σy=−1 can\nbe obtained by sending m→ −mandδm→ −δm. The\nHamiltonian matrix with σy= +1 is given by\nH=X\nr=0,±1Tr⊗˜hr, (A1)\nwhere ˜h0=−t1τy−m−δmτ z,˜h1=˜h†\n−1=t2(τy+iτx)/2\nandTis the translation operator that moves each unit\ncell by one site to the left, with T|j⟩=|j−1⟩and\nT|j= 1⟩= 0. The wavefunction Ψ =L\nj=1ϕj=L\nj=1{uj, vj}Tsatisfies the Schr¨ odinger equation HΨ =\nEΨ, which has the form\n(˜h0−E)ϕ1+˜h1ϕ2= 0 (A2)\nand\n˜h†\n1ϕj−1+ (˜h0−E)ϕj+˜h1ϕj+1= 0 (A3)forj >1. Multiplying Eq.(A3) by qj−1and summing up\nall the equations, we obtain\n[q˜h†\n1+(˜h0−E)+q−1˜h1]G(q) = (˜h0−E+˜h1q−1)ϕ1+˜h1ϕ2\n(A4)\nwhere\nG(q) =+∞X\nj=1qj−1ϕj, (A5)\nandqis a complex number. Utilizing Eq.(A2) and (A4),\nwe can express G(q) as\nG(q) = [q2˜h†\n1+q(˜h0−E) +˜h1]−1˜h1ϕ1. (A6)\nFor Ψ to be a localized state at j= 1, all the poles of\nG(q) must satisfy |qp|>1 [61]. Substituting the specific\nforms of ˜h0and˜h1into Eq.(A6), we have\nG(q) =u1t2\nD\u001a\nt1−qt2,−i(m+δm+E)\u001bT\n,(A7)\nwith the denominator\nD= (t1−qt2)(t2−qt1) +q[(E+m)2−δm2].(A8)\nThe poles are decided from the two roots of D= 0, which\nhave the relation q1q2= 1, and therefore cannot both\nhave absolute values greater than one. So only one of\nthe two roots can be the pole and the other has to be\neliminated from numerator. By requiring ( E+m)2−\nδm2= 0, we could eliminate t1−qt2term in the first\nentry of Eq.(A7). However, only when E=−m−δm\ncan this term be eliminated in both entries, which leads\nto\nG(q) =u1t2\nt2−qt1{1,0}T. (A9)\nThe pole qp=t2/t1, satisfies |qp|>1 when t1< t2. A\nseries expansion of G(q) takes the form\nG(q) =+∞X\nj=1qj−1\u0012t1\nt2\u0013j−1\n{u1,0}T. (A10)\nComparing this equation with the definition of G(q), we\nimmediately find that\nϕj=\u0012t1\nt2\u0013j−1\n{u1,0}T, (A11)\nwhich is clearly localized at j= 1 if t1< t2. For the\nblock with σy=−1, the energy of bound state is given\nbyE=m+δm. Hence the boundary mode at j= 12\nappears when t1< t2. The boundary states for Hλ=0at\nj= 1 can be written in original Majorana basis, with\nEb1,+=m+δm,\nΨb1,+=u1√\n2M\nj=1\u0012t1\nt2\u0013j−1\n{1,−i,0,0}T(A12)\nEb1,−=−(m+δm),\nΨb1,−=u1√\n2M\nj=1\u0012t1\nt2\u0013j−1\n{1, i,0,0}T(A13)\nwhere u1is decided from normalization condition.\nBoundary modes at the other end can be obtained in\na similar way. We consider a semi-infinite system with\nboundary at j=N. Schr¨ odinger equation for σy= 1\nblock is given by\n˜h†\n1ϕN−1+˜h0ϕN=EϕN (A14)\nand\n˜h†\n1ϕj−1+ (˜h0−E)ϕj+˜h1ϕj+1= 0 (A15)\nforj∈(−∞, N−1]. Multiplying Eq.(A15) by qN−jand\nsumming up all the equations, we have\nG(q) =−∞X\nj=NqN−jϕj= [˜h†\n1+q(˜h0−E) +q2˜h1]−1˜h†\n1ϕN\n(A16)\nFollowing the same analysis in deriving boundary modes\natj= 1, we obtain the boundary states at j=N, with\nEbN,+=m−δm,\nΨbN,+=vN√\n2−∞M\nj=N\u0012t1\nt2\u0013N−j\n{0,0,1,−i}T(A17)\nEbN,−=−(m−δm),\nΨbN,−=vN√\n2−∞M\nj=N\u0012t1\nt2\u0013N−j\n{0,0,1, i}T. (A18)\nSo we have established that gapped boundary modes\nappear when t1< t2, and the boundary spectrum is in-\ndependent of t1, being ±(m+δm) at the end j= 1,\nand±(m−δm) at j=N. Note that, these bound-\nary modes may appear in the bulk continuum if bulk\ngap vanishes or is smaller than the boundary gap. For\na gapped bulk, boundary phase transition occurs when\n|m|=|δm|. In this context, we can treat each boundary\nas a zero-dimensional gapped system that belongs to D\nclass, which is again characterized by a Z2invariant [3].\nOne can in principle assign a different invariant at two\nsides of the phase transition point based on the number\nof Fermi level crossings. However, this is not related to\ngapless modes.\n-1-11\n0\n0 1\nAPBC\n PBC\n OBC\n-1 0 1\n(a)\n(b) (c)FIG. 1. (a) Fermion-parity switch with inversion symmetry\nenforced ( δm= 0). Although the ground state falls in even-\nparity sector for both PBC and APBC, they have distinct in-\nversion eigenvalues (shown in the parenthesis). (b) For t1< t 2\nthe spectrum exhibits spectral flows with some states moving\nfrom occupied (negative energies) to unoccupied (positive en-\nergies) bands. (c) Due to inversion symmetry, level crossings\ncan only disappear when bulk gap closes and reopen.\nAppendix B: B. Inversion symmetry\nIn this section, we demonstrate that with inversion\nsymmetry enforced, level crossings can be inferred from\nground-state difference of inversion eigenvalues for sys-\ntem under PBC and APBC.\nIn the absence of δm, the 1D Kitaev ladder is invariant\nunder inversion, which transforms Majorana fermions as\nIαs,jI−1=−βs,N+1−j,Iβs,jI−1=αs,N+1−j.(B1)\nAccordingly, fermionic operators ψsfollow transforma-\ntionIψsI−1=iψs. Although fermion parity is still\nthe same for PBC and APBC, we could discriminate the\ntwo ground states by inversion eigenvalues. Specifically,\nfort1= 0, provided mis finite, the ground state would\nevolve from |11⟩=ψ†\n1ψ†\n2|00⟩under PBC to |00⟩in APBC.\nThe inversion eigenvalues of the two states are −1 and 1\nrespectively. This distinction leads to level crossings, as\nshown in Fig. 1(a).\nAs Fig. 1(b) shows, the level crossings persist for finite\nt1, with two states from occupied bands (negative energy)\nmoving straight into unoccupied bands (positive energy)\nasλvaries. This suggests that ground state under APBC\nshould be different from that of PBC in some aspect.\nAs we illustrate in Fig. 1(a) for t1= 0, the difference\nlies in their inversion eigenvalues. To investigate this\nfor general t1, we may write down the mean-field ground\nstate explicitly. Define fermionic operators cs,k= (αs,k+\niβs,k)/2, and we could express the 1D Hamiltonian in\nparticle-hole basis {c1,k, c2,k, c†\n1,−k, c†\n2,−k}T, which takes\nthe form\n˜Hk= (t1−t2cosk)˜τz+t2sink˜τy−m˜σy, (B2)3\nwith ˜ τand ˜σbeing Pauli matrices acting in Nambu space\nand chain space. Ground state of this Bogoliubov-de\nGennes (BdG) Hamiltonian is given by [41]\n|G⟩=Nexp\nX\nk>0,sgkc†\ns,kc†\ns,−k\nY\nn∈occ,K(ϕ†\nn,K)νn,K|0⟩,\n(B3)\nwhere Nis a normalization factor, gk=it2sink/[ϵ0+\n(t1−t2cosk)] with ϵ0=p\nt2\n1+t2\n2−2t1t2cosk, and ϕ†\nn,K\nis the creation operator of Bogoliubov quasiparticle in\noccupied bands at high symmetry momenta. Occupation\nnumber νn,Kis determined by the eigenstate of ϕn,K, de-\nnoted by |gn, K⟩={un1,K, un2,K, vn1,K, vn2,K}T, where\nuns,K andvns,K are particle and hole components re-\nspectively. We would then have ϕ†\nn,K=P\nsuns,Kc†\ns,K+\nvns,Kcs,K. AtK, there is no pairing term. Only the state\nthat is of particle type ( vns,K = 0) will be occupied in\nthe ground state, with νn,K= 1.\nAccording to Eq.(B1), Ics,kI−1=ics,−ke−ik(N+1),\nwhich is differing by a sign for PBC and APBC. Inver-\nsion symmetry requires BdG Hamiltonian (B2) to obey\nU†\nI(−k)˜H(k)UI(−k) = ˜H(−k), with UI(k) = ieik˜τz.\nEach state at Kwould be an eigenstate of ˜ τz, of which\nparticle states satisfying ˜ τz|gn, K⟩=|gn, K⟩. Therefore,\nνn,Kis identified as the number of occupied states |gn, K⟩\nwith ˜ τz= 1. Under inversion transformations, pairing\nterms in Eq.(B3) remain the same while ϕ†\nn,Kacquires a\nfactor of −ieiK(under PBC). The difference of ground-\nstate inversion eigenvalues between PBC and APBC can\nthen be given by\nP=Y\nn∈occ(−i)νn,0−νn,π, (B4)\nwhich is valid regardless of Nbeing even or odd. Pcan\ntake four different values and therefore serves as a Z4in-\nvariant. Ground-state fermion-parity difference between\nPBC and APBC may also be expressed with the occupa-\ntion numbers, given by\nnF=Y\nn∈occ,K(−1)νn,K. (B5)\nForP ̸= 1, the spectrum exhibits spectral flows while\nλvaries, with some states moving from occupied bands\nto unoccupied bands, and therefore level crossing is in-\nevitable. When P=±i, there would be an odd num-\nber of level crossings while λvaries between PBC and\nAPBC due to the fermion-parity difference ( nF=−1),\nthus leaving an unpaired Majorana zero mode at each\nopen boundary. The gapped boundaries with nontrivial\ntopology is characterized by P=−1.\nAppendix C: C. Effective Edge Hamiltonian\nIn this section, we derive the effective boundary Hamil-\ntonian of the 2D model for an arbitrary edge.\nFIG. 2. A semi-infinite 2D system with boundary (blue line)\nalong k2direction. Two sets of coordinate systems are related\nby an in-plane rotation. ˆ nrepresents the unit vector pointing\nalong normal direction of the edge.\nIn the absence of mandδmterms, the bulk gap closes\nat Γ point when t1=t2. Near this critical point, we\nwrite down the continuum model by expanding the bulk\nHamiltonian at Γ point up to second order in k, which\nreads\nH(kx, ky) = [( t2−t1)−t2\n2(k2\nx+k2\ny)]τy−t2(kxτx+kyτzσz).\n(C1)\nTo derive the effective Hamiltonian of an arbitrary edge,\nwe consider another coordinate system k1k2that is ob-\ntained by rotating kxkysystem counterclockwise by an\nangle ϕ, as illustrated in Fig. 2. Coordinates in the two\nsystems are related by\nkx=k1cosϕ−k2sinϕ, k y=k1sinϕ+k2cosϕ.(C2)\nSubstituting Eq.(C2) into the continuum Hamiltonian,\nwe have\nH′(k1, k2) = [( t2−t1)−t2\n2(k2\n1+k2\n2)]τy (C3)\n−t2[(k1cosϕ−k2sinϕ)τx+ (k1sinϕ+k2cosϕ)τzσz].\nWe further rotate the inner basis with Uϕ=e−iϕ/2τyσz,\nand the resulting Hamiltonian for finite mandδmtakes\nthe form\n˜H(k1, k2) =UH1(k1, k2)U†(C4)\n= [(t2−t1)−t2\n2(k2\n1+k2\n2)]τy\n−t2(k1τx+k2τzσz)\n−δmτ zσy−mσyei(ϕ−θ)τyσz\nTo derive boundary Hamiltonian, we consider a semi-\ninfinite system with edges along k2direction (blue lines\nin Fig. 2). Hamiltonian for this semi-infinite 2D system4\n10-110-4\n10-110-3\n(a) (b)\n(c)(d) (e)\n(f)\nFIG. 3. (a) The variation of lowest energy level (nonnegative) with λand disorder strength σt. Level crossings are obvious\nat weak disorder and become obscured when the disorder is strong enough. Comparing energy spectrum at σt= 0.3 in (b)\nandσt= 0.8 in (c), we find that disorder in t1term mainly influences the bulk gap. (d) The variation of lowest energy level\n(nonnegative) with λand impurity strength Vm. The impurity is added solely at site j= 1. The level crossings move under\nthe variation of impurity strength and disappear when boundary gap closes. Energy spectrum at two representative impurity\nstrength, Vm= 0.7 and Vm= 1.2, are plotted in (e) and (f) respectively. In all the figures, ¯t1= 1,t2= 2,m= 0.5,δm= 0.2.\nis obtained by making a substitution k1→ − i∂xwhile\nkeeping k2intact, which leads to\n˜H(−i∂x, k2) = [( t2−t1) +t2\n2∂2\nx]τy+it2∂xτx\n−t2\n2k2\n2τy−t2k2τzσz\n−δmτ zσy−mσyei(ϕ−θ)τyσz(C5)\nIn the absence of mandδmterms, the model supports\ntwo Majorana zero modes at k2= 0 in topological phase,\nwhich are localized on the edge. Eigenstates for these\ntwo modes are obtained by solving Schr¨ odinger equation\nform=δm= 0, i.e.,\n˜H(−i∂x, k2= 0)|ψ⟩= 0. (C6)\nThe direction of an arbitrary edge is represented by a unit\nvector ˆ nϕpointing along its normal direction outwards,\nas shown in Fig. 2. In the following, we simply refer it\nto edge ˆ nϕ. Eigenstates of the two zero modes at edge\nˆnϕis given by\n|ψn⟩=c(eη1x−eη2x)|φn⟩, n= 1,2 (C7)\n|φ1⟩= (1 0)T⊗(1 0)T,|φ2⟩= (0 1)T⊗(0 1)T,\nwhere η1/2are the roots oft2\n2η2−t2η+ (t2−t1) = 0. We\nthen obtain the effective edge Hamiltonian by projecting\nthe bulk Hamiltonian in Eq.(C5) into eigenspace spanned\nby basis {|ψ1⟩,|ψ2⟩}T, which reads\nHE(ϕ) =t2k2sz+δms y−mcos(ϕ−θ)sy, (C8)\nwith sbeing Pauli matrices acting in the zero mode basis.\nFrom the effective edge Hamiltonian, we immediately ob-\ntain the edge gap ∆( ϕ) =δm−mcos(ϕ−θ), whose size as\nwell as sign depends on edge orientation. Majorana cor-\nner states appear whenever gaps of adjacent edges take\nopposite signs.Appendix D: D. Effects of disorder and impurity on\nlevel crossings\nIn this section, we investigate the stability of level-\ncrossing points against bulk disorder and boundary im-\npurities, and study their influences on Majorana corner\nstates in 2D.\nAs we pointed out in the main text, level crossings\nare protected by fermion parity conservation. A single\ncrossing cannot disappear unless bulk or boundary gap\nis closed. Therefore, if the disorder or impurity doesn’t\nclose the two gaps, we can expect the level crossings to\npersist.\nFirst, let us consider bulk disorder of t1term in the 1D\nmodel, which follows Gaussian distribution with mean\nvalue ¯t1and standard deviation σt. We plotted the low-\nest energy level (nonnegative) in ( σt, λ) parameter space,\nas shown in Fig. 3(a). For weak disorder (small σt), the\nlevel crossing points remain stable, as verified by energy\nspectrum shown in Fig. 3(b). With the increase of bulk\ndisorder, more and more states move close to zero energy,\nindicating the closure of bulk gap, as Fig. 3(c) demon-\nstrates. In the latter case, there is no longer any level\ncrossing. Therefore, the disorder in t1term mainly influ-\nences the bulk gap and is expected to close the gap when\nit is strong enough.\nIn contrast to t1term, morδmterm would influence\nthe boundary gap. We consider impurities of strength\nVmat boundary site j= 1, which has the the same form\nasmterm, i.e., VmΓT\n1σyΓ1. In Fig. 3(d), we find that\nwith the increase of Vm, the two level-crossing points at\n±λmove towards λ= 0 and annihilate with each other\nwhere boundary gap closes. From Fig. 3(e) and (f), we\nfind that the bulk gap doesn’t change in this process,\nbut the boundary gap closes and reopens, accompanied\nby the disappearance of level crossings.5\nFIG. 4. Fermi level crossings in a 2D system with disorder (upper panel) and boundary impurities (lower panel). Boundary\nimpurities are added uniformly on the left edge. The system size is 40 ×40, and ¯t1= 1,t2= 2,m= 0.7,δm= 0.2,θ=π/4.\n-0.40.4\n0\n-0.40.4\n0\n0 1.0 2.0 0 1.5 3.0(d) (e)\n(f)(a) (b)\n(c)\nFIG. 5. (a) Evolution of energy spectrum for a finite 2D system with bulk disorder σt. (b) and (c) Probability distributions\nof energy states closest to zero energy for σt= 0.9 and σt= 2.4. Majorana corner states remain stable at weak disorder, and\ncouple with bulk states when disorder is strong enough. (d) Evolution of energy spectrum with impurity strength Vm. The\nimpurities are added uniformly on the left edge. Bulk gap is not influenced. Boundary gap of the left edge is expected to\nclose and reopen with the variation of Vm. Due to finite size effect, the edge gap doesn’t really close, but the topology of left\nedge may change when impurity strength is strong enough. Consequently, one of the Majorana corner states is transferred to\nneighboring corner, as can be seen by comparing the cases in (e) with Vm= 0.7 and (f) with Vm= 1.2. In all the figures,\n¯t1= 1,t2= 2,m= 0.7,δm= 0.2,θ=π/4.\nTurning to 2D system, the topological invariant intro-\nduced in the main text is determined from level crossings\nat high symmetry momenta, and hence relies on transla-\ntion symmetry. When disorder or impurities break trans-\nlation symmetry, we can no longer say that the crossing\nappears at 1D subsystem with K= 0 or π, but have to\nlook at the 2D spectrum instead. We should emphasize\nthatηais the sum of the crossings at K= 0 and K=π,\nand doesn’t necessarily equal the number of crossings\nthat appear while a toroidal system is deformed into a\ncylindrical one. This is because K=πis not allowed\nwhen a periodic system has an odd number of unit cells\nbut is allowed under anti-periodic boundary condition.\nWhen the numbers of unit cells along both directions are\neven, we could safely say that the two numbers are equal\nto each other. In Fig. 4, we show the energy spectrum of\na 40×40 lattice, and Fermi level crossings indeed surviveunder weak bulk disorder and boundary impurities that\nare added uniformly on the left edge.\nConsidering that the higher-order topology is inti-\nmately related to Fermi level crossings, we could expect\nMajorana corner states to be robust under these pertur-\nbations. In Fig. 5(a) and (d) we plotted the variations of\nenergy spectrum at open boundaries with bulk disorder\noft1term, as well as boundary impurities Vmthat are\nuniformly distributed on the left edge. Indeed, Majorana\ncorner states survive under weak disorder, and disappear\nwhen disorder becomes so strong that the bulk gap closes,\nas shown in Fig. 5(b) and (c). Adding impurities on one\nedge only influences the boundary spectrum. The topol-\nogy of this particular edge would change when the impu-\nrities are strong enough, so is the topological difference\nbetween it and neighboring edges. As a result, Majorana\ncorner states do not disappear but may hop from one6\n01\n01\n0 1 201\n01\n0 1 2\nTrivialWeyl SCHOTS\nTrivialWeyl SCHOTS(a)\n(b)\nFIG. 6. Fermi Level crossings in trivial, Weyl superconductor\n(SC) and higher-order topological superconductor (HOTS)\nphases. (a) ϕ=π. Both Majorana numbers take −1 in\nhigher-order phase, suggesting the two Majorana corner states\n(shown in the inset) sit at opposite corners. (b) ϕ= 3π/4.\nOnlyMx=−1. The two Majorana corner states sit at ad-\njacent corners. Note that level-crossings only occur at K= 0\nwith given parameters. In all the figures, t=α= ∆ = 2∆ Z=\n1,µ= 0,θ=π/3.\ncorner to an adjacent one, as shown in Fig. 5(e) and (f).Appendix E: E. Rashba Bilayer\nIn this section, we apply the topological invariant pro-\nposed in the main text to a Rashba bilayer supercon-\nducting system that is known to support higher-order\nphases [39]. There are three key ingredients that make\nit a higher-order superconductor: Rashba spin-orbit cou-\npling, in-plane Zeeman field and phase difference of s-\nwave pairing between the two layers. The model Hamil-\ntonian in k-space is given by\nH(k) =[µ+ 4t−2t(coskx+ cos ky)]τz (E1)\n+α(sinkysx−sinkxτzsy)σz\n+∆\n2τysy[(σ0+σz) +eiϕτz(σ0−σz)]\n+Γτzσx+ ∆ Z(cosθτzsx+ sin θsy),\nwhere τ,sandσare Pauli matrices that act in Nambu,\nspin and layer space respectively. In this model, µrep-\nresents chemical potential, tis amplitude of nearest-\nneighboring hopping, αrepresent the strength of Rashba\nspin-orbit coupling, ∆ is the pairing of upper layer, ϕis\nphase difference between the two layers, Γ denotes the\ncoupling between two layers, and ∆ Zis in-plane Zeeman\nfield with θbeing its direction.\nForϕ=πand ∆ Z<∆, the model realizes higher-\norder phase with two Majorana corner states at oppo-\nsite corner when Γ >|∆ + ∆ Z|and is in trivial phase\nwhen Γ <|∆−∆Z|. In between the two phases, the\nsystem becomes a Weyl superconductor. Indeed, Fermi\nlevel crossings (real root λ∈(0,1)) appear in the higher-\norder phase, as can be seen in Fig. 6(a). In the Weyl\nsuperconductor phase, level crossings appear exactly at\nλ= 0. It should be noted that the higher-order phase\npersists for a wide range of ϕ, in which case Majorana\ncorner states may reside at neighboring corners instead\nof opposite corners, as shown in the inset of Fig. 6(b)."    },    {        "title": "1402.0403v1.The_Fermi_Bubbles_Revisited.pdf",        "content": "arXiv:1402.0403v1  [astro-ph.HE]  3 Feb 2014Astronomy & Astrophysics manuscript no. slicev14 c/circlecopyrtESO 2021\nJuly 13, 2021\nThe Fermi Bubbles Revisited 1\nRui-zhi Yang1,2, Felix Aharonian1,3, and Roland Crocker4,12\n1Max-Planck-Institut f¨ ur Kernphysik, P.O. Box 103980, 690 29 Heidelberg, Germany 3\n2Key Laboratory of Dark Matter and Space Astronomy, Purple Mo untain Observatory, Chinese Academy of Sciences, 4\nNanjing, 210008, China 5\n3Dublin Institute for Advanced Studies, 31 Fitzwilliam Plac e, Dublin 2, Ireland 6\n4Research School of Astronomy and Astrophysics, Australian National University, Canberra, Australia 7\nPreprint online version: July 13, 2021 8\nABSTRACT 9\nWe analyze 60 months of all sky data from the Fermi-LAT. The Fermi Bubble structures discovered previously ar e 10\nclearly revealed by our analysis. With more data and, conseq uently, better statistics we can now divide each bubble into 11\nconstant longitude slices to investigate their gross γ-ray spectral morphology. While the detailed spectral beha viour of 12\neach slice derived within our analysis is somewhat dependen t on the assumed background model, we find, robustly, a 13\nrelative deficit of the flux at low energies (i.e., hardening) towards the top of the South Bubble. In neither Bubble does 14\nthe spectrum soften with longitude. The morphology of the Fe rmi Bubbles is also revealed to be energy dependent: at 15\nhigh energies they are more extended. We conclude from the ga mma-ray spectrum at high latitudes that a low energy 16\nbreak in the parent cosmic ray population is required in both leptonic and hadronic models. We briefly discuss possible 17\nleptonic and hadronic interpretation of this phenomenolog y. 18\nKey words. Gamma rays: diffuse background, ISM: cosmic rays\n1. Introduction19\nTwohuge,bubble-likestructureshavebeenreportedbySu et al.( 2010),Dobler et al.(2010),andSu & Finkbeiner(2012) 20\ninFermi-LAT gamma ray data to extend ∼50◦above and below the Galactic center. The gamma ray emission from th ese 21\nstructures, dubbed the ‘Fermi Bubbles’ (FBs), exhibits a E−2type spectrum, significantly harder than the spectrum of 22\nthe diffuse gamma-ray emission from the Galactic disk. Remarkably, s tructures coincident or similar to the FBs can be 23\nseen at other wavelengths, including the (total intensity) microwa ve haze found in WMAP (Finkbeiner 2004) and, most 24\nrecently, in Planck data (Planck Collaboration et al. 2013); the polar ized microwave structures reported by Jones et al. 25\n(2012);thelarge-scale,biconicalstructuresfound(Bland-Haw thorn & Cohen2003)inROSATX-raydata(Snowden et al. 26\n1997); and the giant polarised radio lobes recently found at 2.3 GHz ( Carretti et al. 2013). Several models have been 27\nproposed to explain both the morphology and spectral properties of the FBs (Cheng et al. 2011; Mertsch & Sarkar 28\n2011; Crocker & Aharonian 2011; Crocker 2012; Zubovas et al. 20 11; Zubovas & Nayakshin 2012; Guo & Mathews 2012; 29\nGuo et al. 2012; Yang et al. 2012; Crocker et al. 2013). These mode ls predict different energy dependent morphologies. 30\nFor example, in the simplest IC model (e.g., discussed by Su et al. 2010 ) with a spatially-independent electron spectrum 31\ndescribed by a power law electron population with a low energy cutoff a t 500 GeV, one expects a tendency for softening 32\nof the gamma-ray spectra at high latitudes. This is due to the gradu al reduction of the IC components produced via 33\nupscattering of the optical and UV photons - the main target fields contributing to the production of gamma-rays 34\nabove 10 GeV that also decline towards high latitudes. On the other h and, in a simple one-zone hadronic model (e.g., 35\nCrocker & Aharonian 2011) the protons’ steady state distributio n should produce a similar spectrum of gamma rays at 36\nall latitudes. Thus, studies of the energy-dependent morphology of the FBs may shed light on the nature of the gamma 37\nray emission mechanism(s). This is the basic motivation for the curre nt study. 38\nHerewepresentananalysisbasedon60months’ Fermi-LATdata.Wefind asimilaroverallmorphologyandspectrum 39\nfor the FBs to those obtained by Su et al. (2010). The FBs exhibit a r ather homogeneous surfacebrightness. Importantly, 40\nthis implies non-homogeneous volumetric emissivities considering projection effects (Su et al. 2010). The FBs , however, 41\nare not completely uniform: they exhibit some ‘hot spots’ (referre d to by Su et al. (2010); Su & Finkbeiner (2012) as the 42\n‘jet’, ‘donut’ and ‘cocoon’).43\nIn this paper the significantly larger photon statistics and the availa bility of the recently updated Fermi science 44\nsoftware tools (Ackermann et al. 2012) allow us to investigate the g ross spectral morphology of the FBs. A somewhat 45\nsimilarstudy hasrecentlybeen conducted byHooper & Slatyer(201 3). However,they concentratespectralfeaturesin low 46\nlatitudes while we focus on the high latitude, especially the top of the b ubble. We proceed by dividing each bubble into 47\nseveral slices to investigate possible gamma-ray spectral change with latitude. Employing different background models, 48\nwe find, robustly, a spectral hardening, or more specifically, a defi cit of low energy flux towards the top of the SFB. We 49\nare also able to extend the spectrum of the FBs to lower energy tha n previously attempted because of the improvement of 50\nthe analysis software and the instruments response functions. T he paper is structured as follows: in section 2 we present 51\n1Rui-zhi Yang et al.: The Fermi Bubbles Revisited\nthe results of our data analysis, in section 3 we discuss the fitting of leptonic and hadronic models to the gamma ray 52\ndata, and in section 4 we set out our conclusions.53\n2. Data Analysis54\nWe use the publicly-available data obtained by the Fermi-LAT in the gamma-ray energy interval from 100 MeV to 300 55\nGeV over the period of 4 Aug 2008 and 17 July 2013 (MET 239557417 - MET 395797613). To avoid contamination from 56\ncharged particles we use the ULTRACLEAN data set in the analysis. We adopt the instrument response functio n version 57\nP7V6 (Ackermann et al. 2012). All events are binned in the all-sky ma p in HEALPIX format with NSIDE=256 . The 58\nsources in the 2ndFermi catalog (Abdo et al. 2012) are subtracted from the counts map using the flux in the catalog. 59\nTo take into account the energy dependent point spread function (PSF) of the Fermi-LAT, we use the functional form 60\ndescribed in the LAT homepage1when masking sources in the catalog. 61\n62\nWe use a likelihood method for determining the FBs’ apparent morpho logy and spectral features for different diffuse 63\nemission templates. The likelihood function has the form log(L) =/summationtext\nkilog(µi)−µi−log(ki!), where kis the number 64\nof photons in the i-th bin in the counts map and µiis the predicted number of photons within a particular linear 65\ncombination of the templates. The sum is over all the spatial bins in th e map. The likelihood function is determined in 66\ndifferent energy bins to derive the energy spectrum. For fitting diff erent energy bins, rather than smoothing the map 67\nto an universal FWHM (full width at half maximum) as used in Su et al. (2 010), we only smooth the diffuse templates 68\nwith the Fermi PSF and then fit the counts map as a linear combination of these smoothed diffuse templates. The 69\nnormalization of each diffuse template is left free in the likelihood fitting . We do not assume a priorithe existence of a 70\npairofbubble-likestructures.Weonlyusethespatialtemplatesfo rπ0decayandICgamma-raysgeneratedbyGALPROP 71\n(Vladimirov et al. 2011) in addition to the isotropic template related to the extragalactic gamma ray background and 72\ncosmic ray contamination. Finally, we take into account the large diffu se structure high above the plane to Galactic north 73\nwhich may be connected with the nearby ISM feature Loop 1 (Su et a l. 2010). In generating the diffuse templates with 74\nGALPROP, we adopt the default Galdef webrun setting, the 2D plane diffusion model that was tuned to fit the ACE 75\ndata.76\nAfter subtracting the best-fit linear combination of the diffuse tem plates, we find residual maps. By summing over 77\nall energy bins above 1 GeV we obtain the image in Fig. 1 in which two bubb le-like structures are clearly seen. Next we 78\ngenerate spatial templates for the FBs from the residual map. In the second step of our analysis these Bubble templates 79\nare included. We employ the likelihood method mentioned above once ag ain and obtain the spectrum of all the diffuse 80\ntemplates.81\nTo derive the spectrum in different parts of the FBs, we divide the SF B (−55◦< b <−25◦) into four slices and the 82\nNorth Fermi Bubble (25◦< b <50◦; ‘NFB’) into three. To avoid contamination from the Galactic plane, in t he fitting 83\nwe mask out the inner ±25◦region. The position of each analysis slice can be found in Fig. 3. The SE D of each slice 84\nis shown in Fig. 2 where the numbers 1 to 4 run from lowest to highest la titude. From the SEDs it is evident that the 85\nspectrum of the highest southern slice is in deficit at low energies rela tive to the other slices. It is important to note that 86\nthis same slice suffers little from geometric projection effects and is t hus a true reflection of the spectrum at the top of 87\nthe SFB. The spectrum at the top of the SFB is, therefore, signific antly different from that in the interior. Fig. 3 reveals 88\nthe different morphologies of the FBs in different energy bins; at high energies the SFB is clearly more extended than at 89\nlow energies. This is highlighted in Fig. 4. It should also be noted that in t he top slice of the SFB (South 4) there are no 90\nknown point sources from the 2ndFermi catalog, making our conclusion about the spectral variation quite robust. 91\nRegarding the energy-dependent morphology of the FBs, we also fl ag the following point to be dealt with in further 92\nwork: as evident from figs. 3 and 4, the SFB is also relatively more ext ended to Galactic westat high energies than low 93\nenergies implying a spectral hardening going from east to west. As f ar as we are aware, no model for the FBs currently 94\naccounts for this effect. It is interesting that, because of this ex tension to the west, at high energies, the FBs come to 95\nmore closely resemble the polarised radio lobes detected at 2.3 GHz (C arretti et al. 2013)2. 96\nAccurate determination of the morphologyand spectral feature s of the Bubbles strongly depends on reliable modeling 97\nof the diffuse background. Unfortunately, because of uncertain ties in the distributions of cosmic rays and interstellar 98\ngas, our knowledge about the diffuse gamma-ray background is impe rfect. In such circumstances, one can only introduce 99\ndifferent templates that instantiate different estimates of the backgroundand try to derive the true Bubble flux with them 100\nusing a likelihood method. Unfortunately, different choices of backg round templates alter the final result significantly 101\nand this may lead to significant systematic errors. To study such po ssible errors, we investigate the 128 GALPROP 102\nmodels listed by Ackermann et al. (2012). The aim of Ackermann et al. (2012) was to study the origin and propagation 103\nof cosmic-rays and the distribution of the interstellar medium by simu ltaneously fitting diffuse gamma ray emission and 104\ncosmic ray data. Although all their models underestimate the GeV em ission at low latitudes (Ackermann et al. 2012), 105\nthis has little influence on our results given our masking of the inner ±25◦when deriving spectra. The 128 models 106\nare different in cosmic ray (CR) source distribution, CR halo size, HI s pin temperature and E(B−V) magnitude cut. 107\nFollowing perusal of the the online material for Ackermann et al. (20 12), we found that 64 models with z= 8 kpc and 108\nz= 10 kpc do not fit the9Be/10Bedata well (the derived curves fall outside of the error bars of nea rly all the data 109\n1http://fermi.gsfc.nasa.gov/ssc/data/analysis/docume ntation/\nCicerone/Cicerone LATIRFs/IRF PSF.html\n2RMC thanks Ettore Carretti for raising this point in convers ation.\n2Rui-zhi Yang et al.: The Fermi Bubbles Revisited\nFig.1.The residual map above 2 GeV. Background subtraction is describe d in the text. Two bubble-like structures can\nbe seen. To render the picture clearer we mask the bright Galactic p lane|b|<5◦\n.\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \nFig.2.SEDs of different slices of the North and South FBs. The numbers 1 t o 4 run from low to high latitudes. Left\npanel: North Bubble (NFB), Right panel: south Bubble (SFB).\npoints), where zis the height of the CR halo. Therefore we abandon these, and inves tigate the other 64 models as diffuse 110\nemission templates.111\nWith the same procedure as described above, we find the SEDs in diffe rent slices with all chosen templates; the results 112\nare summarized in Fig. 5. Because of the large systematic errors, w e may not claim a deficit at low energies and high 113\nlatitude in the NFB. However, in the SFB the low energy deficit of the t op slice is significant for every template. It should 114\nbe noted that, due to the slight differences in overall flux normalisat ion for the different templates choices, differences of 115\nspectral shape are smeared out if we plot them all together as in Fig . 5 which, therefore, suggests a smaller difference 116\nbetween the top and bottom slices than exists in reality. In Fig. 6 we s how 8 examples of SEDs for the slices in the SFB 117\nwhich give the extreme cases of the gamma ray flux. From these indiv idual examples the low energy deficit in the top 118\nslice is much more evident. Results for the northern slices are also sh own in Fig. 7 which reveal the low energy part of 119\nthe top slice has very large uncertainties. The large systematics in t he NFB are likely due to the fact that it is partially 120\ncoincident with the extended Loop 1 structure whose physical orig in and exact morphology and spectral features are still 121\nuncertain.122\n3. Discussion123\nThe gamma-ray residual maps obtained in the current study – base d on six years’ Fermi-LAT data – confirm that the 124\nsurface brightness of each bubble is homogeneous at a gross level. This implies that the volumetric emissivity is not 125\nhomogeneous, otherwise we would detect a higher brightness in the center of each bubble due to projection (Su et al. 126\n2010). One possible explanation of this may be higher turbulence in th e bubbles’ edges generating more efficient particle 127\nacceleration or stronger confinement of high energy particles the re with a resulting increase in the local gamma ray 128\n3Rui-zhi Yang et al.: The Fermi Bubbles Revisited\n00.71.42.12.83.54.24.95.66.37\n320:00:00.0 340:00:00.0 0:00:00.0 20:00:00.0\n-50:00:00.0 -30:00:00.0 10:00:00.0 30:00:00.0 50:00:00.0South 4South 3South 2South 1North 1North 2North 3\n00.10.20.30.40.50.60.70.80.91\n320:00:00.0 340:00:00.0 0:00:00.0 20:00:00.0\n-50:00:00.0 -30:00:00.0 10:00:00.0 30:00:00.0 50:00:00.0South 4South 3South 2South 1North 1North 2North 3\nFig.3.Residual maps for different energy bins. The left and right panels co rrespond to energy intervals 1 −2 GeV and\n10−30 GeV, respectively. The inner disk of the Galaxy ( |b|<2◦) is masked. The boxes laid over the residual maps show\nthe position of each slice.\n0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7\n330:00:00.0 340:00:00.0 350:00:00.0 0:00:00.0 10:00:00.0 20:00:00.0\n-50:00:00.0 -40:00:00.0 -30:00:00.0South 4South 3South 2South 1\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n330:00:00.0 340:00:00.0 350:00:00.0 0:00:00.0 10:00:00.0 20:00:00.0\n-50:00:00.0 -40:00:00.0 -30:00:00.0South 4South 3South 2South 1\nFig.4.Zoom in of Fig. 3 for the SFB region. Left panel 1 −2 GeV and right panel 10 −30 GeV.\n10-710-6\n 1  10  100φ(E)(GeV s-1 m-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 10-710-6\n 1  10  100φ(E)(GeV s-1 m-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \nFig.5.The SEDs for the seven slices with all 64 templates we use in our analys is. The shaded areas span all derived\nSEDs. The left panel is for the NFB while the right is for the SFB.\nemissivity; alternatively, in a non-saturation hadronic model, higher gas densities towards the edges may have the same 129\neffect (Crocker et al. 2013).130\nBelow we assume the low energy deficit of the gamma ray spectrum at high latitudes found above is a real effect and 131\ndiscuss its possible implications.132\n133\nTo simplify our modeling, we assume each bubble is a sphere with a radius 3.4 kpc. The 2-d projection of each 134\nbubble can be approximated as a half sphere in the region b >30◦for the NFB and b <−30◦for the SFB. The 135\n4Rui-zhi Yang et al.: The Fermi Bubbles Revisited\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \n(a) Lorimer distribution\n(Lorimer et al. 2006), zh= 4,Rh=\n20,TS= 150, and E(B−V)cut= 510-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \n(b) Lorimer distribution, zh= 4,Rh=\n30,TS= 150, and E(B−V)cut= 2\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \n(c) SNR distribution\n(Case & Bhattacharya 1998),\nzh= 6,Rh= 20,TS= 150, and\nE(B−V)cut= 210-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \n(d) SNR distribution, zh= 4,Rh=\n20,TS= 100000, and E(B−V)cut= 2\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \n(e) Yusifov distribution\n(Yusifov & K¨ u¸ c¨ uk 2004), zh= 6,Rh=\n30,TS= 150, and E(B−V)cut= 510-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1 \nSouth 2 \nSouth 3 \nSouth 4 \n(f) OBstars distribution\n(Bronfman et al. 2000), zh= 4,Rh=\n20,TS= 150, and E(B−V)cut= 2\nFig.6.Six examples of SEDs for slices in the SFB.\ncenter of each bubble is located at (0,0, ±5kpc) in Galactic coordinates. Assuming an intrinsic north-south symmetry, 136\nwe only analyse the SFB and, to reveal most starkly the spectral c hange with latitude, we consider only slice South 1 137\n(−33◦< b <−25◦) and South 4 ( −55◦< b <−47◦) in detail. Slice South 1 covers the center of the SFB. We use the 138\nISRF value in Galprop at (0,0,5 kpc) for the average value of the slice S outh 1 in the calculation, wIR= 0.18 eV/cm3139\nandwopt= 0.9 eV/cm3. The height of South 4 is about 7 kpc, thus we adopt the ISRF energ y density from Galprop 140\nat (0,0,7 kpc), 0 .7 eV/cm3for the optical component and 0 .15 eV/cm3for the IR component. The IR and optical 141\nphoton fields are modeled as diluted blackbody (gray body) spectra with temperatures of 100 K and 5000 K, respectively. 142\n143\nIn the hadronic scenario, motivated by the relative low energy defic it of the gamma ray flux in the top slice, we fit the 144\nSED in South 1 with a pure power law proton spectrum while for South 4 we use a power law with a low energy break 145\nthat we find needs to be at 20 GeV to fit the data. We assume the pro ton flux below the break is zero for simplicity (see 146\nFig. 8). Our results are shown in Fig. 9. The required low energy brea k may arise naturally due to energy-dependent 147\ndiffusion effects in a non-steady-state scenario: given South 4 is fa r from the (assumed) injection source in the plane, 148\nit may be that only high energy particles (which diffuse faster) have h ad time to travel there since a previous injection 149\nevent or given the age of the structure. The formalism describing t hese energy-dependent diffusion effects can be found in 150\nAharonian & Atoyan (1996) where both impulsive and continuous inje ction cases are described. The position of the low 151\nenergy break can be estimated by equating the age of the bubble (o r the time since a previous burst) with the diffusion 152\n5Rui-zhi Yang et al.: The Fermi Bubbles Revisited\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 \n(a) Lorimer distribution, zh= 4,Rh=\n20,TS= 150, and E(B−V)cut= 510-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 \n(b) Lorimer distribution, zh= 4,Rh=\n30,TS= 150, and E(B−V)cut= 2\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 \n(c) SNR distribution, zh= 6,Rh=\n20,TS= 150, and E(B−V)cut= 210-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 \n(d) SNR distribution, zh= 4,Rh=\n20,TS= 100000, and E(B−V)cut= 2\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 \n(e) Yusifov distribution, zh= 6,Rh=\n30,TS= 150, and E(B−V)cut= 510-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)North 1 \nNorth 2 \nNorth 3 \n(f) OBstars distribution, zh= 4,Rh=\n20,TS= 150, and E(B−V)cut= 2\nFig.7.Six examples of SEDs for slices in the NFB.\ntime, i.e.,the beak energy is Ebk≃(d2/(D0t))1\nδ, wheredis the distance of the top slice from the injection source, tis 153\nthe age of the bubble (or burst) and D(E) =D0Eδis the energy dependent diffusion coefficient. If we assume protons 154\nare injected at the Galactic center, dis about 7 kpc. Notice that in the kinetic equations which govern prot on diffusion 155\nonly the combination D0tappears. Thus any timescale can be obtained if we tune the diffusion c oefficient D0. IfD(E) 156\ntakes a value similar to that in the Galactic plane, say 4 ×1028cm2s−1at 1 GeV and δ= 0.3, the age of the FBs should 157\nbet∼108yr (cf. Crocker et al. 2013). Alternatively, if there is continuous p roton injection into the FBs over 1010yrs 158\nwe need a rather small diffusion coefficient, D0∼1026cm2s−1at 1 GeV, which is about 1/100 of that in the Galactic 159\nplane.160\nIf we assume the injection is impulsive and the ISM density is 0 .005 cm−3the total energy in protons needed to light 161\nthe Bubbles is of the order 1056erg. Note that if we assume the simple geometry mentioned above an d a magnetic field 162\nB∼10µG as suggested by Carretti et al. (2013), then the energy densit ies of cosmic rays and magnetic fields are close 163\nto equipartition. A burst-like injection event requires that the dur ation of the injection is much smaller than the age of 164\nthe structure which is t∼108yr (forD0= 4×1028cm2s−1at 1 GeV), i.e., the duration of the injection event would 165\nhave to be of order 107yr or less in this case. Then the injection rate is of the order 1041erg/s, two orders of magnitude 166\nlarger than the the X-ray luminosity of a X-ray reflection nebulae ne ar the Galactic center (e.g., Gando Ryu et al. 2012), 167\nbut still only one thousandth of the Eddington Luminosity of Sgr A∗,∼5×1044erg/s. 168\nA possible concern for an energy-dependent diffusion scenario is th at, in the high energy range for which the diffusion 169\nlength is much larger than the distance to the source, the predicte d proton density should be the same at all latitudes 170\nfor the impulsive case, or scale as 1 /R(whereRis the distance to the injection source) for the continuous case. T hus, 171\ngiven the (expected) relatively smaller line of sight through the top o f the SFB, one naively anticipates that the gamma 172\nray flux in South 4 is smaller than that in South 1. The observed data r eveal a similar total flux in both slices at high 173\n6Rui-zhi Yang et al.: The Fermi Bubbles Revisited\n10-1410-1310-12\n 1  10  100  1000  10000φ(E)(GeV cm-3 )\nEnergy(GeV)South 1\nSouth 410-1610-1510-14\n 1  10  100  1000  10000φ(E)(GeV cm-3 )\nEnergy(GeV)broken power law\nMaxwellian\nFig.8.Assumed CR proton (left) and electron (right) distributions in Slices 1 and 4 of the SFB\n.\nenergy, however. One possible resolution to this tension may be tha t the target gas density is higher at high latitudes. 174\nIt is also likely that the geometry of the FBs is different from our simple assumption of sphericity thereby entailing 175\nnon-trivial projection effects.176\n177\nIn a leptonic scenario the low energy deficit of gamma ray flux in the to p slice also requires a corresponding low 178\nenergy break in the electron distribution (see Fig. 8 right). This can , again, be produced by several mechanisms. Firstly, 179\nthe energy-dependent diffusion effects mentioned above also work for electrons in principle (cf. McQuinn & Zaldarriaga 180\n2011). However the much faster cooling of electrons relative to pr otons tends to prevent such effects playing an important 181\nrole. To fit the high energy gamma ray data we need TeV electrons, w hose cooling time scale is about 105yrs in the 182\ncircumstances of the FBs. Thus, even if we assume very fast diffus ion with the diffusion coefficient of 4 ×1029cm2/s at 183\n1 GeV (and an index of 0.6), the diffusion distance of TeV electrons in 1 05yrs is only 2 kpc, significantly less than the 184\ndistance of South 4 from the plane (which is more than 5 kpc). This imp lies high energy electrons accelerated at low 185\nlatitudes can never reach South 4 and, as a result, if we want to exp lain the radiation of the FBs in a leptonic scenario, 186\nthe electrons should be accelerated in situ. 187\n188\nThe second problem related to electrons concerns the interpreta tion of the very hard energy spectrum of radiation 189\nbelow 5 GeV. Such a hard spectrum formally can be explained by assum ing a low energy break in the electron spectrum. 190\nHowever, if the electrons are cooled, no matter how hard the injec tion spectrum is, because of the radiative ( dE/dt∼E2191\ntype) losses the resulted steady-state distribution below the break in the injection spectrum, will have have a standard, 192\nE−2power law-spectrum. The latter will result in IC gamma-ray spectru m with a photon index 1.5 which is still not 193\nsufficient to explain the observed gamma-ray spectrum of South 4; below 5 GeV the photon index of the latter is as small 194\nas 1. One can formally overcome this problem assuming that the elect rons stay uncooled in FBs. However in order to 195\nreproduce the break in the gamma-ray spectrum, one would need a corresponding break at 100 GeV. The corresponding 196\nsynchrotron cooling time at this energy, t∼8×105(100 GeV\nE)(B\n10µG)−2yr, appears significantly shorter than the age 197\nof FBs within any reasonable model of the latter. Thus we need very efficient continuous acceleration acceleration of 198\nelectrons which would dominate over the rate of energy losses and in that way keep very hard the steady-state spectrum. 199\nThis in principle can be realized with stochastic acceleration which can p roduce Maxwellian type steady-state energy 200\ndistribution of electrons. Indeed adopting an electron distribution likeN(E)∼E2exp(−E/E0) withE0= 300 GeV, 201\nwe can have a reasonable fit to the detected gamma-ray spectrum in South 4, as it is shown in Fig. 9. For comparison, 202\nwe also show IC emission at the position of South 1 and South 4 by adop ting a broken power law electron distribution 203\nwith an index 2 below the break at 200 GeV and 3 above. The results fit the SED of South 1 well, but fail to fit that of 204\nSouth 4, which are also shown in Fig. 9.205\nAnisotropy effects may, in principle, increase the IC gamma ray flux a t high latitudes since most of seed photons 206\ncome from regions in the Galactic plane; thus gamma-rays we detect at high latitudes are a result of (almost) head-on 207\ncollisions. To probe this effect we use the formalism described in Moska lenko & Strong (2000). The enhancement factor 208\nis defined by the flux ratio of the anisotropic IC to the isotropic IC as suming the the same ISRF energy density. Results 209\nfor the positions of South 1 and South 4 are shown in Fig. 9. We can se e that although the anisotropic IC effect may 210\ncause a spectral a hardening, however the difference between So uth 1 and South 4 is rather small, less than 10%. It 211\nappears not sufficient to explain the very hard spectrum of South 4 . 212\n4. Summary213\nIn this paper we re-analyze the Fermi-LAT data covering the FB region. With an improved instrumental re sponse 214\nfunction and more data, we are able to extend the previously obtain ed spectrum to lower energies. Furthermore we 215\ndivide each bubble into slices to investigate possible variation of the sp ectrum with latitude. Given the improved data, 216\nfor the first time, we can determined robustly that the spectrum o f the top of South Bubble drops appreciably at low 217\n7Rui-zhi Yang et al.: The Fermi Bubbles Revisited\n10-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 1\nSouth 410-710-6\n 1  10  100φ(E)(GeV s-1 cm-2 sr-1)\nEnergy(GeV)South 4 broken power law\nSouth 1 broken power law\nMaxwellian type 0.6 0.8 1 1.2 1.4 1.6 1.8 2\n 0.1  1  10  100  1000Enhancement Factor\nEnergy(GeV)100K 7kpc\n100K 5kpc\n5000K 7kpc\n5000K 5kpc\nFig.9. Left and Middle panel: Fitting the FBs’ SED with the hadronic model (le ft panel) and the leptonic model\n(middle panel) described in the text. The red curve shows the fitted spectrum and the red hatched region spans all\nspectra resulting with from different templates for South 1; blue cu rve and hatched region are for South 4. For leptonic\nmodels, for South 1 we assume the broken power law described in the text and the ISRF value given by GALPROP.\nFor South 4, the black curve is for the same broken power law with th e ISRF floating while for the gray curve we use\nthe ISRF in GALPROP at the position of South 4 and a Maxwellian type ele ctron distribution with Ebk= 300 GeV.\nIn our hadronic scenario energy dependent diffusion effects are co nsidered as described in the text. Right panel: The\nenhancement factor due to anisotropic IC. The IR and Opt/UV com ponents of the ISRF have effective temperatures of\n100 K and 5000 K, respectively. Results for z= 5 kpc and z= 7 kpc are shown (approximate heights of South 1 and\nSouth 4 above the Galactic plane).\nenergies relative to the spectrum determined at lower latitudes. We also show that the morphology of the South Bubble 218\nis energy dependent; at high energy the structure is relatively mor e extended to both Galactic south and west. 219\nWe have also investigated the influence of different choices of backg round model on the results of our analysis. We 220\nfound that background models may significantly alter the apparent gamma-ray signal from the FBs. Nevertheless, the 221\nspectral hardening with latitude in the South Bubble remains a robus t result. In neither Bubble do we find a spectral 222\nsteepening with latitude. 223\nThe relative suppression of the low energy gamma ray spectrum may be explained within a hadronic model for the 224\nFBs wherein energy-dependent diffusion leads to a relative deficit of low energy protons at high latitudes. Specifically, 225\nif protons are injected in the plane, the finite age of the FBs may only allow high energy protons to propagate to high 226\nlatitude thereby predicting a gradual hardening of the gamma ray s pectrum with latitude. 227\nIn leptonic models the fast coolingof electronsmeans they cannot m ove too far from their acceleratorsand distributed 228\nacceleration inside the FBs seems to be favored. For the case of st ochastic acceleration the maximal electron energy 229\nmight be expected to be 1 TeV, implying the optical/UV and IR compone nt of the ISRF may play an important role 230\nin gamma ray production inside the FBs. However, the attenuation o f these components of the ISRF at high latitudes 231\nis, at least naively, in conflict with the observed hardening of the gam ma ray spectrum in the highest slice of the South 232\nBubble unless a very specific electron spectrum evolution is realized.233\n234\nReferences235\nAbdo et al. 2012, Astrophys.J.Suppl., 199, 31236\nAckermann, M., Ajello, M., Atwood, W. B., et al. 2012, ApJ, 75 0, 3 237\nAckermann, M. et al. 2012, Astrophys.J.Suppl., 203, 4238\nAharonian, F. A. & Atoyan, A. M. 1996, A&A, 309, 917239\nBland-Hawthorn, J. & Cohen, M. 2003, ApJ, 582, 246240\nBronfman, L., Casassus, S., May, J., & Nyman, L.- ˚A. 2000, A&A, 358, 521 241\nCarretti, E., Crocker, R. M., Staveley-Smith, L., et al. 201 3, Nature, 493, 66 242\nCase, G. L. & Bhattacharya, D. 1998, The Astrophysical Journ al, 504, 761 243\nCheng, K.-S., Chernyshov, D. O., Dogiel, V. A., Ko, C.-M., & I p, W.-H. 2011, ApJ, 731, L17 244\nCrocker, R. M. 2012, MNRAS, 423, 3512245\nCrocker, R. M. & Aharonian, F. 2011, Physical Review Letters , 106, 101102 246\nCrocker, R. M., Bicknell, G. V., Carretti, E., Hill, A. S., & S utherland, R. S. 2013, ArXiv e-prints 247\nDobler, G., Finkbeiner, D. P., Cholis, I., Slatyer, T., & Wei ner, N. 2010, ApJ, 717, 825 248\nFinkbeiner, D. P. 2004, ApJ, 614, 186249\nGando Ryu, S., Nobukawa, M., Nakashima, S., et al. 2012, ArXi v e-prints 250\nGuo, F. & Mathews, W. G. 2012, ApJ, 756, 181251\nGuo, F., Mathews, W. G., Dobler, G., & Oh, S. P. 2012, ApJ, 756, 182 252\nHooper, D. & Slatyer, T. R. 2013, Physics of the Dark Universe , 2, 118 253\nJones, D. I., Crocker, R. M., Reich, W., Ott, J., & Aharonian, F. A. 2012, ApJ, 747, L12 254\nLorimer, D. R., Faulkner, A. J., Lyne, A. G., et al. 2006, MNRA S, 372, 777 255\nMcQuinn, M. & Zaldarriaga, M. 2011, MNRAS, 414, 3577256\nMertsch, P. & Sarkar, S. 2011, Mem. Soc. Astron. Italiana, 82 , 876 257\nMoskalenko, I. V. & Strong, A. W. 2000, ApJ, 528, 357258\nPlanck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2013 , A&A, 554, A139 259\nSnowden, S. L., Egger, R., Freyberg, M. J., et al. 1997, ApJ, 4 85, 125 260\nSu, M. & Finkbeiner, D. P. 2012, ApJ, 753, 61261\nSu, M., Slatyer, T. R., & Finkbeiner, D. P. 2010, ApJ, 724, 104 4 262\n8Rui-zhi Yang et al.: The Fermi Bubbles Revisited\nVladimirov, A. E., Digel, S. W., J´ ohannesson, G., et al. 201 1, Computer Physics Communications, 182, 1156 263\nYang, H.-Y. K., Ruszkowski, M., Ricker, P. M., Zweibel, E., & Lee, D. 2012, ApJ, 761, 185 264\nYusifov, I. & K¨ u¸ c¨ uk, I. 2004, A&A, 422, 545265\nZubovas, K., King, A. R., & Nayakshin, S. 2011, MNRAS, 415, L2 1 266\nZubovas, K. & Nayakshin, S. 2012, MNRAS, 424, 666267\n9"    },    {        "title": "1009.3409v2.Fermi_s_golden_rule_applied_to_the_gamma_decay_in_the_quasicontinuum_of_46Ti.pdf",        "content": "Fermi’s golden rule applied to the gdecay in the quasicontinuum of46Ti\nM. Guttormsen,1,\u0003A. C. Larsen,1A. B ¨urger,1A. G ¨orgen,1S. Harissopulos,2M. Kmiecik,3T. Konstantinopoulos,2\nM. Krti ˘cka,4A. Lagoyannis,2T. L¨onnroth,5K. Mazurek,3M. Norrby,5H.T. Nyhus,1G. Perdikakis†,2\nA. Schiller,6S. Siem,1A. Spyrou†,2N.U.H. Syed,1H.K. Toft,1G.M. Tveten,1and A. V oinov6\n1Department of Physics, University of Oslo, N-0316 Oslo, Norway\n2Institute of Nuclear Physics, NCSR ”Demokritos”, 153.10 Aghia Paraskevi, Athens, Greece\n3Institute of Nuclear Physics PAN, Krak ´ow, Poland\n4Institute of Particle and Nuclear Physics, Charles University, Prague, Czech Republic\n5Department of Physics, ˚Abo Akademi University, FIN-20500 ˚Abo, Finland\n6Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA\n(Dated: November 6, 2018)\nParticle- gcoincidences from the46Ti(p;p0g)46Ti inelastic scattering reaction with 15-MeV protons are uti-\nlized to obtain g-ray spectra as a function of excitation energy. The rich data set allows analysis of the coinci-\ndence data with various gates on excitation energy. For many independent data sets, this enables a simultaneous\nextraction of level density and radiative strength function (RSF). The results are consistent with one common\nlevel density. The data seem to exhibit a universal RSF as the deduced RSFs from different excitation energies\nshow only small fluctuations provided that only excitation energies above 3 MeV are taken into account. If\ntransitions to well-separated low-energy levels are included, the deduced RSF may change by a factor of 2 \u00003,\nwhich might be expected because the involved Porter-Thomas fluctuations.\nPACS numbers: 21.10.Ma, 25.20.Lj, 27.40.+z, 25.40.Hs\nI. INTRODUCTION\nFermi’s golden rule predicts the transition rate from one\nstate to a set of final states in a quantum system. The the-\noretical foundation, which has been successfully applied in\nmany disciplines of physics, was first established by Dirac in\n1927 [1] and emphasized by Fermi in his book in 1950 [2].\nThe rule is based on first-order perturbation theory, where\nthe transition matrix element is assumed to be small. In nu-\nclear physics, this assumption is well fulfilled for bandgde-\ncay. Thus, we take the validity of the Fermi’s golden rule as\ngranted, rather than testing this rule. In this work, we study\nthegdecay between states in the quasi-continuum of46Ti and\napply Fermi’s golden rule to disentangle the gstrength and\nlevel density.\nThe Oslo nuclear physics group has developed a method to\ndetermine simultaneously the level density and the radiative\nstrength function (RSF) from particle- gcoincidences. These\nquantities provide information on the average properties of ex-\ncited nuclei and are indispensible in nuclear reaction theories\nas they are the only quantities needed for complete description\nof the gdecay at higher excitation energies.\nThe nuclear level density can be determined reliably up to a\nfew MeV of excitation energy from the counting of low-lying\ndiscrete known levels [3]. There is also reliable level-density\ninformation from neutron resonances at higher excitation en-\nergies; however, this information is restricted in energy as well\nas spin range.\nThe most rich experimental information on the RSF was ob-\n†Current address: National Superconducting Cyclotron Laboratory, Michigan\nState University, East Lansing, MI 48824-1321, USA.\n\u0003Electronic address: magne.guttormsen@fys.uio.notained from the study of photonuclear cross-sections [4], and\nthus limited only to energies above the particle separation en-\nergy. It was established that the giant electric dipole resonance\n(GEDR) dominates the RSF in all nuclei. The information on\nRSF below particle separation energy is significantly less. It\nhas been obtained mainly from the Oslo method and (n, g) and\n(g,g0) reactions.\nThe Oslo method, which is applicable for excitation ener-\ngies below the particle separation, is described in detail in\nRef. [5]. In this work, we report on results obtained for\nthe46Ti nucleus, which has two protons and four neutrons\noutside the doubly-magic40Ca core. The advantages of the\n(p;p0)reaction compared to the commonly used (3He,3He0)\nand (3He,4He) reactions are much higher cross-sections and\nbetter particle resolution. This allows us to make a detailed\nstudy of g-decay as a function of excitation energy.\nIt can be discussed if the concept of one unique RSF is\nvalid for light nuclei and in particular at low excitation ener-\ngies. This question together with the applicability of the Oslo\nmethod for light systems as titanium is the main subject of this\nwork.\nIn Sec. II the experimental results are described. The nu-\nclear level density and RSF are extracted in Sec. III, and in\nSec. IV , the application of Fermi’s golden rule and the Brink\nhypothesis is discussed. Summary and conclusions are given\nin Sec. V .\nII. EXPERIMENTAL RESULTS\nThe experiment was conducted at the Oslo Cyclotron Labo-\nratory (OCL) with a 15-MeV proton beam bombarding a self-\nsupporting target of46Ti with thickness of 1.8 mg/cm2. The\ntarget was enriched to 86%46Ti with48Ti (11%) as the main\nimpurity. This small admixture of48Ti is not expected to playarXiv:1009.3409v2  [nucl-ex]  20 Jan 20112\nCounts / 39.3 keV\n05001000150020002500310×\nTi46(a) Singles+0+2\n+4\n+0\nProton energy E (MeV)4 6 810 12 14Counts / 39.3 keV\n0200400600800100012001400310×\n(b) Coincidences+2\n+4\n+0\nFIG. 1: (Color online) Singles (a) and coincidence (b) proton spectra\nrecorded with 15-MeV protons on46Ti.\na major role. This is supported by the fact that at low exci-\ntation energies where the level density is small, we could not\nidentify any important contributions of48Ti into the g-spectra.\nIn addition, both titaniums are expected to behave similarly.\nParticle- gcoincidences were measured with eight Si DE\u0000\nEparticle telescopes and the CACTUS multidetector sys-\ntem [6]. The Si detectors were placed in forward direction,\n45\u000erelative to the beam axis. The front ( DE) and end ( E) de-\ntectors had a thickness of 140 mm and 1500 mm, respectively.\nThe CACTUS array consists of 28 collimated 5” \u00025” NaI(Tl)\ngdetectors with a total efficiency of 15 :2% at Eg=1:33 MeV .\nThe singles-proton spectrum and protons in coincidence with\ng-rays are shown in Fig. 1.\nIn total, 110 million coincidence events were collected in\none week with a beam current of 1.5 nA. Using reaction kine-\nmatics, the measured proton energy was transformed into ex-\ncitation energy of the residual nucleus. In this way, a set of\ng-ray spectra is assigned to a specific initial excitation energy\nEiin46Ti. Furthermore, the g-ray spectra are corrected for\nthe known response functions of the CACTUS array following\nthe procedure described in Ref. [7]. The unfolded coincidence\nmatrix (Ei;Eg)of46Ti is shown in Fig. 2.\nThe coincidence matrix displays vertical lines that represent\nyrast transitions from the last steps in the g-cascades. How-\never, there are also clear diagonal lines where the genergy\nmatches the direct decay to the ground state ( Ei=Eg) and to\nthe first and second excited states at 889 keV (2+) and 2010\nkeV (4+), respectively. These g-rays are primary transitions\nin the g-cascades. By studying the energy distribution of all\nprimary g-rays originating from various excitation energies,\n (keV)γE200040006000800010000 (keV) iE\n0200040006000800010000\n210310410\n210310410FIG. 2: (Color online) The particle- gcoincidence matrix for46Ti.\nTheg-ray spectra have been unfolded with the NaI response func-\ntions.\ninformation on the level density and RSF can be extracted si-\nmultaneously.\nAn iterative subtraction technique has been developed to\nseparate out the first-generation (primary) g-transitions from\nthe total cascade [8]. The subtraction technique is based on\nthe assumption that the decay pattern is the same whether the\nlevels were initiated directly by the nuclear reaction or by g\ndecay from higher-lying states. This assumption is automat-\nically fulfilled when states have the same relative probability\nto be populated by the two processes, since g-branching ra-\ntios are properties of the levels themselves. If the excitation\nbins contain many levels, it is likely to find the same g-energy\ndistribution from this set of levels independent of the type of\npopulation. However, the assumption is more problematic if\nthe decay involves only a few (but not one) levels within the\nenergy bin.\nFermi’s golden rule predicts that the decay probability may\nbe factorized into the transition matrix element between the\ninitial and final states, and the density at the final state [2]:\nli!f=2p\n¯hjhfjH0jiij2rf: (1)\nRealizing that the first generation g-ray spectra P(Ei;Eg)\nare proportional to the decay probability from Ei!Ef=\nEi\u0000Eg, i.e. li!f, we may write the equivalent expression of\nEq. (1) as:\nP(Ei;Eg)µTi!frf; (2)\nwhereTi!fis the g-ray transmission coefficient, and rf=\nr(Ei\u0000Eg)is the level density at the excitation energy Efaf-3\nter the primary g-ray emission. This expression does not allow\nus to extract simultaneously Ti!fandrfrom the experimen-\ntalP(Ei;Eg)matrix. To do that, either one of the factorial\nfunctions must be known, or some restrictions have to be in-\ntroduced. According to the Brink hypothesis [9], the g-ray\ntransmission coefficient is independent of excitation energy;\nonly the transitional energy Egplays a role. Thus, we replace\nTi!fwithT(Eg), giving\nP(Ei;Eg)µT(Eg)rf; (3)\nwhich permits a simultaneous extraction of the two multi-\nplicative functions. We then fit about N2=2 data points of the\nPmatrix with 2 Nfree parameters. A least c2fit is then pos-\nsible, since it is many more data points than fit parameters; in\nthe present case we have 150 free parameters to fit 2240 data\npoints.\nAt low excitation energy, the gdecay is highly dependent\non the individual initial and final state; therefore, we have ex-\ncluded g-ray spectra originating from excitation energy bins\nbelow Ei=5:5 MeV .\nIt is well known that the Brink hypothesis is violated when\nhigh temperatures and/or spins are involved, see Ref. [10] and\nreferences therein. However, in the present Oslo experiment,\nthe temperature reached is low ( T\u00181:5 MeV) and is assumed\nto be rather constant. The dependency on spin is of minor im-\nportance. The measured ratios of the gfeeding into the ground\nband indicate a low spin window of I\u00180\u00006¯h. At excita-\ntion energy E\u00188 MeV the ratios are 57:100:9 for the 2+, 4+\nand 6+states, and at E\u001810 MeV they are 55:100:10, which\nare equal within the error bars for extracting these ratios. Of\ncourse, at low excitation energy the spin distribution fluctuates\ndue to a few levels (and spins) present within each 118 keV\nexcitation bin.\nIn principle, the Brink hypothesis assumed in expression (3)\ncould bias the analysis, so that the Oslo method in the next\nturn validates the Brink hypothesis. This issue will be ad-\ndressed in Sec. IV , where we find one common level density\n(according to Fermi’s golden rule), which in turn results in\none universal RSF in the quasi-continuum of46Ti.\nIII. LEVEL DENSITY AND RADIATIVE STRENGTH\nFUNCTION\nIn our first investigations, we rely on the Brink hypothe-\nsis from the expression (3) and factorize the first-generation\ng-matrix P(Ei;Eg)into one transmission coefficient T(Eg)\nand one level density r(E). Since the decay between states at\nlow excitation energy cannot be treated within a statistical ap-\nproach, a cut of the matrix with Eg>1:8 MeV and 5 :5 MeV\n<Ei<10:0 MeV was used to exclude clear non-statistical\ndecay routes1. The least c2fitting of the two multiplicative\nfunctions follows the iterative procedure of Ref. [5].\n1The lower excitation cut concerns only the initial excitation energy Ei; one\nstill needs g-spectra originating from excitation regions down to the ground\nstate in order to subtract higher-order generations of g-rays.To demonstrate how well the procedure works, we compare\nin Fig. 3 for some initial excitation energies Eithe experimen-\ntal first-generation spectra Pwith the ones obtained by multi-\nplying the extracted Tandrfunctions. The agreement be-\ntween calculated and experimental first-generation spectra is\nexcellent for decay from higher excitations; however there are\nlocally strong deviations where the calculated spectra fall out-\nside of the experimental error bars for populations of lower ex-\ncitations energies, as seen in the Ei=5:6 and 6 :4 MeV gates.\nThese deviations could be the consequence of Porter-Thomas\nfluctuations [11], which will be further discussed in Sec. IV .\nThe general good agreement holds also for all the other 40\nspectra (not shown) included in the global fit with the same\nT(Eg)andr(E)functions.\nThe experimental statistical errors are very small as seen\nin Fig. 3. Thus, the deviations are due to other sources of\nerrors. For example for Ei=6:4 MeV , the T\u0001rprediction\noverestimates around Eg=3 MeV and underestimates around\nEg=5:5 MeV with several standard deviations. Thus, there\nare indications that a common Tandrfunction that simulta-\nneously fits the P(Ei;Eg)matrix, could not be found. The sys-\ntematic errors behind these deviations could be due to several\nfactors as experimental shortcomings, assumptions behind the\nOslo method, the Brink hypothesis [9] and, most probable,\nPorter-Thomas fluctuations. Keeping all these possibilities in\nmind, the results of Fig. 3 are very gratifying.\nThere exist infinitely many Tandrfunctions making\nidentical fits to the data [5] as the examples shown in Fig. 3.\nThese functions can be generated by the transformations\n˜r(Ei\u0000Eg) = Aexp[a(Ei\u0000Eg)]r(Ei\u0000Eg); (4)\n˜T(Eg) = Bexp(aEg)T(Eg): (5)\nIn the following, we will try to determine the parameters A,a,\nandB. This information is not available from our experiment\nand has to be determined from other experimental results.\nFirst, the level density at high excitation energy is nor-\nmalized to the level density at the neutron separation energy\nr(Sn). This data point is calculated from neutron resonance\nspacings D0(see, e.g., Ref. [12]) with a spin distribution given\nby [13]\ng(E;I)'2I+1\n2s2exp\u0002\n\u0000(I+1=2)2=2s2\u0003\n; (6)\nwhere Eis excitation energy and Iis spin.\nThere exists no neutron resonance data for46Ti. We there-\nfore estimate r(Sn)from the parameterizations of von Egidy\nand Bucurescu [14] using the back-shifted Fermi gas (BSFG)\nmodel, which reads\nrBSFG(E) =hexp(2p\naU)\n12p\n2a1=4U5=4s; (7)\nwhere ais the level density parameter, U=E\u0000E1 is the in-\ntrinsic excitation energy, and E1 is the back-shift energy pa-\nrameter. The spin cut-off parameter sis given by [14]\ns2=0:0146 A5=31+p1+4aU\n2a; (8)4\n (MeV)γE0246810Probability / 118 keV0.020.040.060.080.1\n = 5.6 MeViE\n (MeV)γE0246810Probability / 118 keV0.020.040.060.080.1\n = 8.0 MeViE (MeV)γE0246810Probability / 118 keV = 6.4 MeViE\n (MeV)γE0246810Probability / 118 keV = 8.9 MeViE (MeV)γE0246810Probability / 118 keV = 7.2 MeViE\n (MeV)γE0246810Probability / 118 keV = 9.7 MeViE\nFIG. 3: (Color online) Comparison of experimental first-generation spectra (squares) and the ones obtained from multiplying the extracted T\nandrfunctions (red line). The initial excitation energy bins Eiare 118 keV broad. The error bars represent the experimental statistical errors.\nAbeing the nuclear mass number.\nFigure 4 shows the extracted total level densities for tita-\nnium isotopes with known resonance spacings D0atSn(filled\ntriangles). The resonance spacings for `=0 neutrons only\ngive the level densities for one or two spins and only one par-\nity. In order to extract the level density for all spin and parities,\nwe use Eq. (6) and assume equally many positive and negative\nparity states at Sn. This is supported by combinatorial quasi-\nparticle models [15–17], where the numbers of positive and\nnegative parity states at Snare predicted to be the same.\nThe points connected with lines are based on the semi-\nempirical estimate of von Egidy and Bucurescu [14] with a\ncommon scaling of h=0:5 in order to match qualitatively\nthe experimental r(Sn)points. The estimated value for46Ti is\nr= (4650\u00061000)MeV\u00001atSn=13:189 MeV (see Fig. 4).\nThe error bar chosen reflects roughly the general deviation\nbetween the global estimates and the points derived from neu-\ntron resonance data.\nNow the scaling ( A) and slope ( a) parameters of the level\ndensity can be determined as shown in Fig. 5. The normal-\nization of the level density is fitted to the known level density\naround 3.5 MeV of excitation energy, and to the extrapola-\ntion from r(Sn)using the BSFG model with parameters sum-\nmarized in Table I. By choosing other r(Sn)values (within\nthe assumed uncertainty of \u00061000 MeV\u00001), the logarithmi-\ncal slope of the randTwill change accordingly, as the a-\nparameter of Eqs. (4, 5) has to be adjusted. The rcurve is\nwell fixed at\u00183:5 MeV and will rotate around this point with\ndifferent choices of a. Level densities in the 14 \u000019 MeV\nexcitation region of46Ti have been measured from Ericson\nfluctuations, see Ref. [18] and references therein. However,\nthe values reported by the various experimental groups differ\nwithin a factor of ten. The measurement of the Ohio group[18] is shown in Fig. 5 for comparison (see triangle point at\n15.5 MeV).\nOne should stress that the level density found from the ex-\nperiment is based on the spin and parity levels populated in the\n(p;p0)reaction. Thus, the normalization to the total level den-\nsity described above, rests on the assumption that the struc-\nture of the level density remains approximately the same if all\nspins and parities are included. This assumption is reasonable\nfulfilled according to Ref. [16], where the level density for\nspin windows of 2 \u00006¯hand 0\u000030¯hhave been calculated for\n46Ti within a combinatorial quasiparticle model [17]. From\nthese estimates, our measurement includes 70 \u000080% of the\ntotal level density and the level density fine structures for the\ntwo spin windows are very similar.\nThe measured level density describes nicely the known\nlevel density up to around 4 MeV of excitation energy. The ex-\nperimental resolution of ris about 0 :3 MeV at low excitation\nenergy, as seen for the 2+state at 889 keV . At around 3 MeV ,\nwe see an abrupt increase in level density due to the breaking\nof proton and/or neutron pairs coupled in time-reversed or-\nbitals (Cooper pairs [19]). This results in \u001810 times more lev-\nels, making it difficult to determine the level density at higher\nexcitation energies with conventional spectroscopic methods.\nIt remains to determine the scaling parameter Bof the trans-\nmission coefficient T(Eg). Here we use the radiative width\nhGgiatSnassuming that the g-decay is dominated by dipole\ntransitions. For initial spin Iand parity p, the width is given\nby [20]\nhGgi=1\n2pr(Sn;I;p)å\nIfZSn\n0dEgBT(Eg)\n\u0002r(Sn\u0000Eg;If); (9)5\n (MeV)nNeutron separation energy S67891011121314)-1 (MeVnLevel density at S\n210310410Calculated level density\nLevel density from neutron res. data\nEstimated level density\nTi45\nTi47\nTi49\nTi51\nTi53Ti46\nTi48\nTi50\nTi52\nFIG. 4: Deduced total r(Sn)from neutron resonance spacings (trian-\ngles). The data point for46Ti (square) is estimated by extrapolations\nfrom the BSFG model with global parameterization of von Egidy and\nBucurescu [14] (cirles with lines to guide the eye).\nwhere the summation and integration run over all final levels\nwith spin Ifthat are accessible by gradiation with energy Eg\nand multipolarity E1 orM1. Further details on the normaliza-\ntion procedure are found in Refs. [5, 21].\nSince there exists no neutron resonance data, we again have\nto rely on systematics from other isotopes. From Fig. 6, we\nestimate an average g-width ofhGgi= (1200\u0006500)meV at\nSn. The large uncertainty in hGgigives an absolute normal-\nization of T(Eg)and the RSF within about \u000650%. However,\nthis uncertainty does not influence the energy dependence.\nThe deduced RSF for dipole radiation can be calculated\nfrom the normalized transmission coefficient T(Eg)by [22]\nf(Eg) =1\n2pT(Eg)\nE3g: (10)\nThe normalized RSF is shown in Fig. 7. For comparison, the\nGEDR data from Ref. [23] are also shown, which have been\ntranslated from photo neutron cross section sto RSF by [22]\nf(Eg) =1\n3p2¯h2c2s(Eg)\nEg: (11)\nUnfortunately, there is a large energy gap between our data\nending at a Eg=10 MeV and the GEDR data that start at\n14 MeV .\nOur data on the RSF display a minimum near 4 \u00006 MeV\nand some structures at lower g-ray energies. The uncertain-\nExcitation energy E (MeV)02468101214)-1 (E) (MeVρ Level density \n110210310410Present experiment\nKnown levels\nBack-shifted Fermi gas\nFrom neutron res. data\nFrom Ericson fluctuation dataFIG. 5: Normalization of the nuclear level density (filled squares) of\n46Ti. At low excitation energies, the data are normalized (between\nthe arrows) to known discrete levels (solid line). At higher excitation\nenergies, the data are normalized to the BSFG level density (dashed\nline) going through the point r(Sn)(open square), which is estimated\nfrom the systematics of Fig. 4. For comparison a data point [18]\nobtained from Ericson fluctuations are shown (black triangle).\nties in the value of ratSnwill change the slope of the RSF,\nand thus also the degree of low-energy enhancement. How-\never, the lines of Fig. 7 show that the enhancement is not very\nsensitive to reasonable choices of the value r(Sn).\nSuch or similar enhancement has been seen in several\nlight nuclei with mass A<100, see Ref. [24] and references\ntherein. There is still no theoretical explanation for this very\ninteresting phenomenon.\nTABLE I: Parameters used for the extraction of level density and\nradiative strength function in46Ti.\nSn a E 1s r(Sn)hGg(Sn)i\n(MeV) (MeV\u00001) (MeV) (MeV\u00001) (meV)\n13.189 4.7 -1.0 4.0 4650(1000) 1200(500)\nIV . FERMI’S GOLDEN RULE AND BRINK HYPOTHESIS\nAccording to Fermi’s golden rule, it is possible to factor-\nize out the level density from the g-decay probability. Since\nthe level density is a unique property of the nucleus, the same6\n (MeV)nNeutron separation energy S67891011121314 (meV)〉γΓ〈 \n0500100015002000250030003500Neutron resonance data\nEstimated\nTi47Ti49\nTi51Ti46Ti48\nTi50\nFIG. 6: Experimental average g-widthhGgifrom neutron resonance\ndata (triangles) [22]. Only a rough estimate for46Ti can be achieved\n(square).\nextracted level density is expected from the decay probability\ndeduced at different excitation energy regions within the ex-\nperimental restrictions, in particular the energy resolution of\nCACTUS. In order to see if the Oslo method gives a unique\nlevel density, we have divided the data set into three statis-\ntically independent initial excitation-energy regions, namely\nEi=5:5\u00007:0 MeV , 7 :0\u00008:5 MeV and 8 :5\u000010:0 MeV , and\napplied the same methodology as described in Sec. III. The\nresults shown in Fig. 8 are very satisfactory. The different\ndata sets give approximately the same level density. Thus, the\ndisentanglement of level density and transmission coefficient\nfrom g-particle coincidences seems to work very well accord-\ning to Eq. (2).\nThe fact that we measure approximately the same level den-\nsity function for primary g-ray spectra taken at different ex-\ncitation regions indicates that the RSF depends only on g-\nray energy and not on excitation energy. This seems to in-\ndicate the validity of Brink hypothesis, which will be tested\nmore thoroughly in the following. In principle, the test could\nbe performed by dividing the data set into even more initial\nexcitation-energy regions. However, the statistics of the ex-\nperiment does not permit such an approach, and a different\napproach has been used.\nBy accepting the level density obtained with the global fit\nof all relevant data (see lowest curve in Fig. 8), we may further\ninvestigate the transmission coefficient in detail, and thus the\nvalidity of the Brink hypothesis. We first adopt the solutions\nTandrfrom Sec. III and write\nN(Ei)P(Ei;Eg)\u0019T(Eg)r(Ei\u0000Eg): (12)\nThe normalization factor for each initial excitation bin is de-\n (MeV)γ-ray energy Eγ02468101214161820)-3Radiative strength function (MeV-810-710Present experiment\n (Sn)ρHigh \n (Sn)ρLow \n,abs) resonance dataγ(FIG. 7: (Color online) Experimental radiative strength function for\n46Ti (squares). The RSFs assuming high (5650 MeV\u00001) and low\n(3650 MeV\u00001) level density at Snare also shown. For comparison\nthe GEDR data from the ( g;abs) reaction [23] are shown (triangles).\nfined by\nN(Ei) =REi\n0dEgT(Eg)r(Ei\u0000Eg)\nREi\n0dEgP(Ei;Eg): (13)\nThe degree of correctness of the approximation (12) is typi-\ncally illustrated by the fits in Fig. 3.\nIn the following, we will investigate the dependency of the\ndeduced RSF on initial and final excitation energy. If the\nBrink hypothesis is valid, there exists the same RSF for all ex-\ncitation energies in this nucleus. We will call it the universal\nRSF. In reality the Porter-Thomas fluctuations involved influ-\nence the RSF obtained from different excitation regions. We\nwill use the term deduced RSF in the following for the quan-\ntity obtained from experimental data. The deduced RSFs are\nexpected to fluctuate around the universal RSF and the fluc-\ntuations are expected to be stronger if the number of transi-\ntions used in the determination of the deduced RSF is smaller.\nThus, the deduced RSFs extracted from data sets involving\ninitial and final regions of high level density should be much\ncloser to the universal RSF than the RSFs deduced from re-\ngions of low level density.\nSince there exists only one unique level density, we con-\nstruct the counterpart to Eq. (12) in the case that the transmis-\nsion coefficient depends on the initial excitation energy:\nN0(Ei)P(Ei;Eg)\u0019T(Eg;Ei)r(Ei\u0000Eg); (14)7\nExcitation energy E (MeV)0246810)-1 (MeVρ  Level density \n110210310410\nx15.5-10.0 MeV\nx38.5-10.0 MeV\nx97.0-8.5 MeV\nx275.5-7.0 MeV\nFIG. 8: Level density extracted from statistically independent data\nsets, taken from various initial excitation energy bins Ei(the three\nupper curves). The lower curve is the result for the whole energy\nregion and is identical to the one of Fig. 5.\nwhereN0is determined analogously to Eq. (13). We expect\nthatT(Ei;Eg)fluctuates on the average around T(Eg). Thus,\nit is reasonable to expect that N0\u0019N, which gives a trans-\nmission coefficient of:\nT(Ei;Eg)\u0019N(Ei)P(Ei;Eg)\nr(Ei\u0000Eg): (15)\nSimilarly, the transmission coefficient as a function of the final\nexcitation energy Ef=Ei\u0000Egis given by\nT(Ef;Eg)\u0019N(Ef+Eg)P(Ef+Eg;Eg)\nr(Ef): (16)\nThe validity of the approximations (14) and (15) is demon-\nstrated in Fig. 9 by comparing the deduced RSF from this\napproximation (lines) with the independent fits of Tandr\n(data points). The RSFs determined for the whole energy re-\ngion 5 :5\u000010 MeV are very similar, especially for low Eg,\nas shown in the lower panels. Also the similarity in the de-\ntailed structures of the two methods is recognized, although\nsome differences are present. The deviations are largest for\nthe high g-energy part of the RSFs populating the lowest 0+,\n2+, and 4+states, where large Porter-Thomas fluctuations\nare expected (these fluctuations are not included in the error\nbars). However, the overall good resemblance encourages us\nto study the detailed evolution of the RSFs as a function of\ninitial excitation energies using the approximations (14) and\n0246810)-3RSF (MeV\n-910-810 = 5.5-7.0 MeViE T× ρFit \nρP/\n0246810)-3RSF (MeV\n-910-810 = 7.0-8.5 MeViE\n0246810)-3RSF (MeV\n-910-810 = 8.5-10.0 MeViE\n (MeV)γE0 2 4 6 810)-3RSF (MeV\n-910-810 = 5.5-10.0 MeViEFIG. 9: (Color online) Gamma-ray strength functions extracted from\nvarious initial excitation bins Ei. The data points of the three\nupper deduced RSFs are from statistically independent data sets.\nThe data points in the lower panel are identical with the RSF of\nFig. 7. The RSFs displayed as red lines, are evaluated from the ra-\ntioP(Ei;Eg)=r(Ef)(see text). The resemblance between the two\nmethods confirms that the level density is common for the various\nexcitation regions, in accordance with Fermi’s golden rule.\n(15). Since the approximation (16) is a simple transformation\nusing Ef=Ei\u0000Eg, we may also investigate the dependencies\nof the RSF on final excitation energy Ef.\nIn Figs. 10 and 11, the RSFs f(Ei;Eg)and f(Ef;Eg)are\nshown for eight excitation regions. For all these deduced\nRSFs, we use one common level density in the evaluation\nof the approximations (15) and (16). The data points of the\nexperimental Pmatrix cover only a certain region in Eiand\nEg, making restrictions on the deduced RSFs. Thus, f(Ei;Eg)\nis limited to 5.5 MeV <Ei<Snand 1.8 MeV <Eg<Ei.\nThe limits for f(Ef;Eg)are 0 <Ef<Sn\u00001:8 MeV and\n1.8 MeV <Eg<Sn\u0000Ef. For a consistency check, we have\ntested that the average RSF for all Efenergies equals the one\nfor all Eienergies (not shown here).\nFigure 10 shows that the deduced RSF changes as a func-\ntion of Ei, when low excitation energy is populated after the g-\nemission. The lower, left panel, where two consecutive RSFs\nare plotted together, illustrates this. Here, the bumps seen at\nhigh g-ray energies are the artifacts of the decay to specific8\n (MeV)γE0 2 4 6 8 10 -810 = 5.4-5.9 MeViE\n (MeV)γE0 2 4 6 8 10 -810 = 6.6-7.1 MeViE\n (MeV)γE0 2 4 6 8 10)-3RSF (MeV-810 = 7.8-8.3 MeViE\n (MeV)γE0 2 4 6 8 10 -810 = 9.0-9.4 MeViE\n (MeV)γE0246810-810Two last RSFs (MeV)γE0 2 4 6 8 10 -810 = 6.0-6.5 MeViE\n (MeV)γE0 2 4 6 8 10 -810 = 7.2-7.7 MeViE\n (MeV)γE0 2 4 6 8 10 -810 = 8.4-8.9 MeViE\n (MeV)γE0 2 4 6 8 10 -810 = 9.6-10.0 MeViE\n (MeV)γE0246810-810All RSFs\nFIG. 10: (Color online) Deduced RSFs from various initial excitation bins Ei. The RSFs are evaluated from the ratio P(Ei;Eg)=r(Ef)as\ndescribed in the text.\nisolated levels below 3 MeV of excitation. This is also the\nreason why apparently the fluctuations are typically a factor\nof 2\u00003 in the panel where all deduced RSFs are plotted to-\ngether (lower, right panel). In the plot of all deduced RSFs,\nwe see a minimum at about 4 \u00006 MeV and some interest-\ning structures at low g-ray energies. These structures and the\nminimum are independent of the uncertainty in the log-slope\nof the level density, as shown in Fig. 7.\nThe various f(Ef;Eg)plots in Fig. 11 are difficult to com-\npare due to different limits appearing at both low and high\ng-energies for the various Efregions. For example, the first\nthree spectra do not reveal the region of low g-energy en-\nhancement because Eg>5:5 MeV\u0000Ef>3:7 MeV . These\nspectra represent the decay to the 0+, 2+and 4+ground band\nstates, respectively, where the experimental energy resolution\nmakes some overlap between these states. In general the var-\nious spectra show strong fluctuations in the deduced RSFs\nwhen the gemission ends up at low excitation energy, typi-\ncally Ef<3 MeV . The panel with all deduced RSFs plotted\ntogether (lower, right panel) shows approximately the same\nscattering of data points as in Fig. 10.\nIt is thus clear that the experimentally deduced RSFs for\nwhich states below 3 MeV of excitation energy are involved\nin the gdecay are very different. Figure 5 shows that this\nexcitation region coincides with a region of few and well sep-\narated low-lying levels with specific structures. The bumps in\nthe deduced RSFs are specific to the low energy level scheme,and the changes between the various deduced RSFs are large\ndue to Porter-Thomas fluctuations.\nIn order to show the similarity of the deduced RSFs in the\ncase of strongly suppressed Porter-Thomas fluctuations, we\nhave compared the two uppermost excitation energy gates Ei\nandEfin the lower, left panels of Figs. 10 and 11, respec-\ntively. The deduced RSFs are here extracted for gdecay be-\ntween states in quasi-continuum, except for the data points\nwith Eg>7 MeV in Fig. 10 where the final excitation en-\nergy is <3 MeV . The deduced RSFs extracted from the quasi-\ncontinuum region behave similar to the deduced RSF from the\nall-over fit displayed in Fig. 7.\nThe good agreement between the deduced RSFs at higher\nenergies is consistent with the expectation of suppressed\nPorter-Thomas fluctuations due to more initial and final lev-\nels in the evaluation of the gstrength. These results strongly\nindicate that the concept of an RSF, which is independent on\nexcitation energy, is valid already at relatively low excitation\nenergies in46Ti. At the same time, the results give us con-\nfidence that Oslo method works reasonably. The differences\nin deduced RSFs from lower energies indicate that the Porter-\nThomas fluctuations are so poorly suppressed that it is difficult\nto predict the shape of the universal RSF. This is not very sur-\nprising and the differences in the deduced RSFs seem to be\nconsistent with the expected fluctuations.\nIn order to display the small differences in the deduced\nRSFs obtained from regions of higher level density, we have9\n (MeV)γE0 2 4 6 8 10 -810 = 0.0-0.6 MeVfE\n (MeV)γE0 2 4 6 8 10 -810 = 1.2-1.8 MeVfE\n (MeV)γE0 2 4 6 8 10)-3RSF (MeV-810 = 2.4-3.0 MeVfE\n (MeV)γE0 2 4 6 8 10 -810 = 3.6-4.2 MeVfE\n (MeV)γE0246810-810Two last RSFs (MeV)γE0 2 4 6 8 10 -810 = 0.6-1.2 MeVfE\n (MeV)γE0 2 4 6 8 10 -810 = 1.8-2.4 MeVfE\n (MeV)γE0 2 4 6 8 10 -810 = 3.0-3.6 MeVfE\n (MeV)γE0 2 4 6 8 10 -810 = 4.2-4.8 MeVfE\n (MeV)γE0246810-810All RSFs\nFIG. 11: (Color online) Deduced RSFs populating various final excitation bins Ef. The RSFs are evaluated from the ratio P(Ef+\nEg;Eg)=r(Ef)as described in the text. It should be noted that there are no final states at Ef=1:2\u00001:8 MeV , however the experimental\nresolution (see Fig. 5) is responsible for including the 2+and 4+ground band states in this gate.\ntaken the four highest gates shown in Fig. 10 and only con-\nsidered genergies up to 5.1 MeV . Thus, these statistically in-\ndependent RSFs are evaluated in quasi-continuum with initial\nand final excitation energies of roughly Ei=8\u000010 MeV and\nEf=3\u00007 MeV , respectively. The deduced RSFs are pre-\nsented in the upper panel of Fig. 12.\nFor this data set we evaluated the relative deviations by\nri j=fi j\u0000hfii\nhfii; (17)\nwhere the index irepresents the genergy and jis the initial\nexcitation energy. The average strength at each genergy is\nestimated from the four individual RFSs\nhfii=1\nnjå\njfi j: (18)\nIn the lower panel of Fig. 12 the ri jvalues are plotted showing\nthe relative fluctuations from the mean value at each genergy.\nThe average ratio for all data points ( ni=28 and nj=4) is\ntaken as\nr=1\nninjå\ni jjri jj; (19)\ngiving r\u00186%. The differences in the deduced RSFs can eas-\nily be interpreted as remnants of the Porter-Thomas fluctua-\ntions. However, a quantitative estimate of the fluctuations isdifficult to determine since the experimental statistical errors\n(see error bars) are also of the same order.\nV . CONCLUSIONS\nThe level density and radiative strength function for46Ti\nhave been determined using the Oslo method. Similar level\ndensity functions have been extracted from statistically inde-\npendent data sets covering different excitation energies. This\ngives confidence to the Oslo method, since the disentangle-\nment of the level density by Fermi’s golden rule predicts one\nand only one unique level density, independent of the data set.\nThe deduced RSF displays an enhancement at low g-ray en-\nergy where we see a bump around 3 MeV and another struc-\nture at energies near 2 MeV . A similar enhancement has been\nseen in several other light mass nuclei and is still not ac-\ncounted for by present theories.\nA method to study the evolution of the deduced RSFs as a\nfunction of initial and final energy regions has been described.\nThe deduced RSFs are found to display strong variations for\ndifferent initial and final excitation energies if transitions to\nthe lowest excitations are involved. The reason for the violent\nfluctuations of a factor of 2 \u00003 is that only a few isolated lev-\nels are present at low excitation energies with Ef<3 MeV .\nThe differences in the RSFs obtained from a few number of10\n (MeV)γE\n22.533.544.55)-3 MeV-9RSF (10\n0246810121416-910×\n = 7.8-8.3 MeViE\n = 8.4-8.9 MeViE\n = 9.0-9.4 MeViE\n = 9.6-10.0 MeViE(a)\n (MeV)γE22.533.544.55Difference in RSF (%) -20-1001020(b)\nFIG. 12: (Color online)(a) Radiative strength functions for gtransi-\ntions between states in quasi-continuum. Data from the four highest\nexcitation energy gates of Figs. 10 have been chosen. (b) Ratios of\nthe deviation from the average RSF at each genergy, see text.transitions, that can be explained as a consequence of Porter-\nThomas fluctuations of individual intensities, show that this\nenergy region cannot be used for determination of the univer-\nsal RSF. Even though, the deduced RSFs based on a restricted\nnumber of transitions still indicate that the decay is governed\nby a universal RSF.\nHowever, the present work shows that it is possible to get\nmore precise experimental information on the universal RSF.\nBy imposing restrictions on the initial and final excitation\nenergies, the RSFs for the decay between states in quasi-\ncontinuum can be extracted (i.e. for Ef>\u00183 MeV). The results\nfrom this selected data set show that the decay is consistent\nwith an RSF which is idependent of excitation energy within\nless than\u00186 % already at these relatively low excitations.\nIn summary, the observation of almost identical level den-\nsities and RSFs extracted from different data sets in quasi-\ncontinuum, indicates that the Oslo method works well. Pro-\nvided that we use data from the quasi-continuum, a universal\nRSF in the light mass region of the46Ti nucleus can be ex-\ntracted.\nAcknowledgments\nThe authors wish to thank E.A. Olsen and J. Wikne for ex-\ncellent experimental conditions. This work was supported by\nthe Research Council of Norway (NFR).\n[1] P.A.M. Dirac, Proc. R. Soc. London A 114, 243-265 (1927).\n[2] E. Fermi, Nuclear Physics (University of Chicago Press,\nChicago, 1950).\n[3] Data extracted using the NNDC On-Line Data Service from the\nENSDF database. [http://www.nndcnbnl.gov./ensdf/]\n[4] S.S. Dietrich and B.L. Berman, At. Data Nucl. Data Tables 38,\n199 (1988).\n[5] A. Schiller, L. Bergholt, M. Guttormsen, E. Melby, J. Rekstad,\nS. Siem, Nucl. Instrum. Methods Phys. Res. A 447, 498 (2000).\n[6] M. Guttormsen, A. Atac, G. Løvhøiden, S. Messelt, T. Ramsøy,\nJ. Rekstad, T.F. Thorsteinsen, T.S. Tveter, and Z. Zelazny, Phys.\nScr.T 32 , 54 (1990).\n[7] M. Guttormsen, T. S. Tveter, L. Bergholt, F. Ingebretsen, and\nJ. Rekstad, Nucl. Instrum. Methods Phys. Res. A 374, 371\n(1996).\n[8] M. Guttormsen, T. Ramsøy, and J. Rekstad, Nucl. Instrum.\nMethods Phys. Res. A 255, 518 (1987).\n[9] D. M. Brink, Ph.D. thesis, Oxford University, 1955.\n[10] A. Schiller and M. Thoennessen, At. Data Nucl. Data Tables\n93, 549 (2007).\n[11] C.E. Porter and R.G. Thomas, Phys. Rev. 104, 483 (1956).\n[12] N.U.H. Syed, M. Guttormsen, F. Ingebretsen, A. C. Larsen,\nT. L¨onnroth, J. Rekstad, A. Schiller, S. Siem, and A. V oinov,Phys. Rev. C 79, 024316 (2009).\n[13] A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43, 1446\n(1965).\n[14] T. von Egidy and D. Bucurescu, Phys. Rev. C 72, 044311\n(2005); Phys. Rev. C 73, 049901(E) (2006).\n[15] S. Goriely, S. Hilaire, and A.J. Koning,, Phys. Rev. C 78,\n064307 (2008).\n[16] Kristine Wikan, Master thesis in Physics, De-\npartment of Physics, University of Oslo, 2010,\nhttp://www.duo.uio.no/publ/fysikk/2010/105030/Master.pdf\n[17] N.U.H. Syed, A. C. Larsen, A. ¨urger, M. Guttormsen, S. Haris-\nsopulos, M. Kmiecik, T. Konstantinopoulos, M. Krti ˘cka, A.\nLagoyannis, T. L ¨onnroth, K. Mazurek4, M. Norby, H. T. Ny-\nhus, G. Perdikakis, S. Siem, and A. Spyrou, Phys. Rev. C 80,\n044309 (2009).\n[18] Merico Salas-Bacci, Steven M. Grimes, Thomas N. Massey,\nYannis Parpottas, Raymond T. Wheller, and James E. Olden-\ndick, Phys. Rev. C 70, 024311 (2004).\n[19] J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 108,\n1175 (1957).\n[20] J. Kopecky and M. Uhl, Phys. Rev. C 411941 (1990).\n[21] A. V oinov, M. Guttormsen, E. Melby, J. Rekstad, A. Schiller,\nand S. Siem, Phys. Rev. C 63, 044313 (2001).11\n[22] RIPL-3 Handbook for calculation of nuclear reaction, (2009);\navailable at http://www-nds.iaea.org/RIPL-3/\n[23] B.S. Ishkhanov, I.M. Kapitonov, E.I. Lileeva, E.V . Shirokov,\nV .A. Erokhova, M.A. Elkin, A.V . Izotova, Moscow State Univ.\nInst. of Nucl. Phys., Reports No.2002, p.27/711 (2002).[24] A.C. Larsen, M. Guttormsen, R. Chankova, F. Ingebretsen,\nT. L ¨onnroth, S. Messelt, J. Rekstad, A. Schiller, S. Siem,\nN.U.H. Syed, and A. V oinov, Phys. Rev. C 76, 044303 (2007)."    },    {        "title": "0802.2455v1.Pacifying_the_Fermi_liquid__battling_the_devious_fermion_signs.pdf",        "content": "arXiv:0802.2455v1  [cond-mat.other]  18 Feb 2008Pacifying the Fermi-liquid: battling the devious fermion s igns.\nJ. Zaanen,1,∗F. Krueger,1,†J.-H. She,1,‡D. Sadri,1,§and S.I. Mukhin2,¶\n1Instituut Lorentz voor de theoretische natuurkunde, Unive rsiteit Leiden,\nP.O. Box 9506, NL-2300 RA Leiden, The Netherlands\n2Moscow Institute for Steel and alloys,Leninski ave. 4, 1190 49 Moscow, Russia\n(Dated: November 11, 2018)\nThe fermion sign problem is studied in the path integral form alism. The standard picture of Fermi\nliquids is first critically analyzed, pointing out some of it s rather peculiar properties. The insightful\nwork of Ceperley in constructing fermionic path integrals i n terms of constrained world-lines is\nthen reviewed. In this representation, the minus signs asso ciated with Fermi-Dirac statistics are\nself consistently translated into a geometrical constrain t structure (the nodal hypersurface ) acting\non an effective bosonic dynamics. As an illustrative example we use this formalism to study 1+1-\ndimensionalsystems, wherestatistics areirrelevant, and hencethesignproblemcanbecircumvented.\nIn this low-dimensional example, the structure of the nodal constraints leads to a lucid picture of\nthe entropic interaction essential to one-dimensional phy sics. Working with the path integral in\nmomentum space, we then show that the Fermi gas can be underst ood by analogy to a Mott\ninsulator in a harmonic trap. Going back to real space, we dis cuss the topological properties of the\nnodal cells, and suggest a new holographic conjecture relat ing Fermi liquids in higher dimensions\nto soft-core bosons in one dimension. We also discuss some po ssible connections between mixed\nBose/Fermi systems and supersymmetry.\nI. INTRODUCTION\nSince the last twenty years or so a serious intel-\nlectual crisis has developed in the condensed matter\nphysics enterprise dealing with strongly interacting elec-\ntrons in solids. An intellectual crisis in science is ac-\ntually the best one can hope for – it should not be\nconfused with an economic- or sociological crisis. This\nfield is flourishing right now1and there is a general\nperception that after a slump in the 1990’s the field\nhas reinvented itself. What is this intellectual crisis\nabout? Substantial progressis made on the experimental\nside, both with regard to the discovery of electron sys-\ntems in solids that behave in very interesting and puz-\nzling ways (high- Tcsuperconductors2and other oxides3,\n’heavy fermion’ intermetallics4, organics5, 2DEG’s in\nsemiconductors6), and in the rapid progress of new in-\nstruments that make possible to probe deeper and far-\nther in these mysterious electron worlds (scanning tun-\nneling spectroscopy7, photoemission8, neutron-9and res-\nonant X-ray scattering10). On the theoretical side there\nis also much action. This is energized by the ’quantum\nfield theory’11revolution that started in the 1970’s in\nhigh energy physics, and is still in the process of unfold-\ning its full potential in the low energy realms, as exem-\nplified by topological quantum computation12, quantum\ncriticality13and so forth.\nBut the intellectual crisis manifests itself through the\nfact that the experimental- and theoretical communities\nare increasingly drifting apart despite all the pressures to\nstay together. This is not because the people are bad sci-\nentists but instead it is caused by the dynamics of science\nitself. The theorists deduce from their powerful field the-\nory very interesting suggestions for experiment but these\nare either impossible to realize in the laboratory or they\nare positioned on the fringe of the experimental mainstream. The attitude of the experimentalists is deter-\nmined by the expectation that theorists are just there to\nexplain why the data do not resemble anything that is\nfound in the books.\nThe cause is obvious. The experimentalists mea-\nsure systems formed from electrons and electrons are\nfermions. The theorists are playing with the mathemat-\nical marvel called quantum field theory. But the latter\nworks so well because via the euclidean path integral it\nboils down to exercises in equilibrium statistical physics.\nIt is about computing probabilistic partition sums in eu-\nclidean space-time following the recipe of Boltzmann and\nthis seems to have no secrets for humanity. The origin\nof the crisis is that this Boltzmannian path integral logic\ndoesnot workat all when onewantsto describeproblems\ncharacterized by a finite density of fermionic particles.\nThe culprit is that the path integral is suffering from the\nfermion sign problem. The Boltzmannian computation is\nspoiledby’negativeprobabilities’renderingtheapproach\nto be mathematically ill defined. In fact, the mathemat-\nics is as bad as it can be: Troyer and Wiese14showed\nrecently that the sign problem falls in the mathematical\ncomplexity class ”NP hard”, and the Clay Mathematics\nInstitute has put one of its 7 one million dollar prizes on\nthe proof that such problems cannot be solved in poly-\nnomial time.\nAlthough not always appreciated, the fermion sign\nproblem is quite consequential for the understanding\nof the physical world. Understanding matter revolves\naround the understanding of the emergence principles\nprescribing how large number of simple constituents (like\nelementary particles) manage to acquire very different\nproperties when they form a wholeness. The path inte-\ngral is telling us that in the absence of the signs these\nprinciples are the same for quantum matter as they\nare for classical matter. But these classical emergence2\nprinciples are in turn resting on Bolzmannian statistical\nphysics. When this fails because of the fermion signs, we\ncan no longer be confident regarding our understanding\nofemergence. To put it positively, dealing with fermionic\nquantum matter there is room for surprises that can be\nvery different from anything we know from the classical\nrealms that shape our intuition. In fact, we have only\ncomprehended one such form of fermionic matter: the\nFermi-gas, and its ”derivative” the Fermi-liquid. The\nembarrassment is that we are completely in the dark re-\ngardingthe nature of other forms of fermionic matter, al-\nthough we know that they exist because the experiments\nare telling us so.\nThe ’quantum weirdness’ of the Fermi-gas is obvi-\nous: how to understand the Fermi-surface, the Fermi-\nenergy and so forth, just knowing about classical sta-\ntistical physics? The interacting Fermi-liquid is a bit\nmore than the Fermi-gas, but focusing on the emergence\nprinciples it is deep inside the same thing. As Lan-\ndau pointed out, the Fermi-liquid is connected by adi-\nabatic continuation to the Fermi-gas meaning that the\ntwo are qualitatively indistinguishable at the long times\nand distances where emergence is in full effect. The great\nframework of diagrammatic perturbation theory devel-\noped in the 1950’s15does allow to arrive at quite non\ntrivial statements associated with the presence of the in-\nteractions but it only works under the condition that\nthe Fermi-liquid is adiabatically connected to the Fermi\ngas. But conventional Feynman diagrams are impotent\nwith regard to revealing the nature of ’non Fermi liq-\nuids’. To complete the ’fermionic’ repertoire of theo-\nretical physics, Bardeen, Cooper and Schrieffer discov-\nered the ”Hartree-Fock” mechanism16, showing how the\nFermi-gas can become unstable towards a bosonic state,\nlike the superfluids- and conductors, charge- and spin\ndensity wave states and so forth. Despite fermionic pe-\nculiarities (like the gap function), this is eventually a\nrecipe telling us how the fermi-gas can turn into bosonic\nmatter that is in turn ruled by the Ginzburg-Landau-\nWilson classical emergence rules. In this regard, the\nother theoretical main streams in correlated electron\nphysics rest on the same bosonization moral: the one di-\nmensional electron systems17, as discussed in section VI;\nthe Kondo-typeimpurity problemsbeingbosonproblems\nin disguise18, which in turn form the fundament for the\npopular dynamical mean-field theory19, and so forth.\nSummarizing, given the present repertoire of theoreti-\ncal physics, all we know to do with fermionic matter is to\nhope that it is a Fermi gas or bound in bosons. But we\nare facing a zoo of ’non-Fermi-liquid’ states of electrons\ncoming out of the experimental laboratories and the the-\norists are standing empty handed because the fermion\nsigns render all the fancy theoretical technologies to be\nuseless. The NP hardness of the sign problem tells us\nthat there is no mathematically exact solution but how\nmany features of the physical world we understand well\nare actually based on exact mathematics? Nearly all of\nit is based on an effective description, mathematics thatis tractable while it does describe accuratelywhat nature\nis doing although it is not derived with exact mathemat-\nics from the first principles. Is there a way to handle\nnon-Fermi-liquid matter on this phenomenological level?\nThe remainder of this paper is dedicated to the case\nthat there is reason to be optimistic. This optimism is\nbased on a brilliant discovery some fifteen years ago of\nan alternative path-integral description of the fermion\nproblem by David Ceperley20,21. This ’constrained’ or\n’Ceperley’ path integral has a Boltzmannian structure\n(i.e., only positive probabilities) but the signs are traded\nin for another unfamiliar structure: a structure of con-\nstraints acting on a ’bosonic’ configuration space that\nis coding for all the effects of Fermi-Dirac statistics.\nThis is called the reach and it amounts to the require-\nment that for all imaginary times τbetween zero and\n/planckover2pi1β(β= 1/(kBT)) the worldline configurations should\nnot cross the hypersurface determined by the zero’s of\nthe fullN-particle, imaginary time density matrix. Al-\nthough the constrained path integral suffers from a self-\nconsistency problem since the exact constrain structure\nis not known except for the non-interacting Fermi-gas, it\nappears that this path integral is quite powerful for the\nconstruction of phenomenological effective theories. The\ninformation carried by the reach lives ’inside’ the func-\ntional integral and should therefore be averaged. This\nimplies that only global- and averaged properties of this\nreach should matter for the physics in the scaling limit.\nThe reach is in essence a high dimensional geometrical\nobject, closely related to the more familiar ’nodal hyper-\nsurface’ associated with the sign changes of ground state\nwave function. The theoretical program is to classify\nthe geometrical and topologicalproperties of the reach in\ngeneral terms, to find out how this information is aver-\naged over in the path integral, with the potential to yield\neventually a systematic classification of phenomenologi-\ncal theories of fermionic matter.\nGiven that Ceperley derived his path integral already\nquite some time ago, why is it not famous affair? These\npath integral are not so easy to handle. Although vari-\nous interesting results were obtained22, even the attempt\nto reconstructthe Fermi-liquid in this languagestalled23.\nBut these efforts were limited to a very small community,\nwith a focus on large scale numerical calculations. The\npotential of the Ceperley path integral to address mat-\nters ofprinciple appearsto be overlookedin the past. We\ndiscovered the Ceperley path integral in an attempt to\nunderstandthescaleinvariantfermionicquantumcritical\nstates as found in the heavy fermion intermetallics. We\nstarted out on the more primitive level of wave function\nnodal structure, discovering by accident the much more\npowerful Ceperley path integral approach. We believe\nthat we have delivered proof of principle24that this lan-\nguagegives penetrating insights in the nature of a promi-\nnent non-Fermi liquid state: the fermionic quantum crit-\nical states realized in the heavy fermion intermetallics.\nSince this work is still under review we will not address\nit in any detail. However, to make further progress, we3\nwere confronted with the need to better understand the\ndetailed workings of the Ceperley path integral and we\ndecided to revisit the description of the Fermi gas and\nthe Fermi liquid. The outcomes of this pursuit are sum-\nmarized in this paper. This paper contains some new\nresults: the supersymmetric quantum gas as implied by\neven permutations (IV), and especially the closed solu-\ntion of the Ceperley path integral of the Fermi gas in\nmomentum space (section VII). But there are also many\nloose ends and this paper is in first instance intended as\nan easy to read tutorial on the Ceperley path integral.\nWe hope that it will infect others to take up this fresh-\nand wide open subject, where much terrain can be con-\nquered.\nThis tutorial is organized as follows. We start out\nin chapter II with a somewhat unconventional discus-\nsion of the Fermi-liquid. To get the problem sharply in\nfocus, we step back from the usual textbook viewpoint\nand instead consider the Fermi-liquid from the perspec-\ntive of the emergence principles governing classical- and\nbosonic matter. In the present context, this perspective\nhasaspecialrelevance: Ceperley’spathintegraltellsthat\nfermionic matter is also subjected to the rules of statis-\ntical physics and viewed in this light the Fermi-liquid\nturns into an outrageous, confusing entity. We will make\nthe case that the Fermi-liquid is holographic in the same\nsense of the holographic principle associated with black\nholes and string theory25. We claim that regardless the\nnumber of space dimensions the physics of the Fermi-\nliquid at low temperature is in one-to-one relation with\nthe physics of a system of soft core interacting bosons\nin one dimension. This might sound absurd but it will\nturn out to become morereasonabledealing with the real\nspace representation of the Ceperley path integral in the\nfinal chapter.\nWe continue in chapter III reviewing the only sign-\nful fermion path integral that can be solved: the classic\nFeynman path integral for the Fermi-gas. This is just a\nsummary of the beautiful treatment found in Kleinert’s\npath integral book26. This story appears to be not as\nwidely known as it should. It shows that the Fermi-gas\nis quite like the Bose gas, where the hard work is done\nby worldlinesthat at low temperature wrapinfinite times\naround the imaginary time circle. However, the negative\nprobabilities interfere, turning into alternating sums over\nwinding numbers that eventually take the shape of the\nFermi-Diracdistributionfunction. ChaptersIVisdealing\nwith side lines inspired on section III but we found both\ninstructive to an extent that they should be included.\nChapter IV is actually of some relevance for the Ceper-\nley path integral but it might well have a broader signifi-\ncance. We were puzzled by the issue of how to deal with\nthe prescription that only even permutations should be\nsummed over in the Ceperley path integrals. It is well\nunderstood that the braiding properties of worldlines un-\nderpin the workings of quantum statistics, referring for\ninstance to the understanding of anyons and topological\nquantum computation in two space dimension. What isthen the meaning of the even permutations? As we will\nshow in section IV, it renders the free quantum gas to\nbecome supersymmetric!\nIn Chapter V the core business starts: we introduce\nthe Ceperley path integral, reviewing it’s derivation as\nwell as various other technical issues. As a first example\nof the workings of this path integral we will discuss in\nsection VI the one dimensional Fermi-gas. In a way it is\nnothingnew,butwewillmakethecasethatthebosoniza-\ntion step becomes particularly transparent in the Ceper-\nleylanguage: the reachbecomes the ’Paulihypersurface’,\nmeaning that the fermion statistics just takes the form\nof hard core interactions between the particles. We sub-\nsequently highlight a ’maximally’ statistical physics view\non the one dimensional electron systems that one of us\ndeveloped some time ago27,28but which is not particu-\nlarly well known in the community. This emphasizes the\naspect that the typical ’fermionic’ aspects of physics in\none dimensions are actually coding for the rather intri-\ncate effects of entropic interactions, using tricks from soft\nmatter physics to reconstruct the Fermi-gas.\nChapter VII is intended to be the highlight of this pa-\nper. We present a quite simple solution of the Ceperley\npath integral for the Fermi-gas: the Fermi-gas turns out\nto be in one-to-one correspondence with a system of cold\natoms in an harmonic trap, subjected to a deep opti-\ncal lattice potential such that the atoms form a perfect\nBose Mott-insulator! This can be taken as completely\nliteral, except that this atom trap lives in single particle\nmomentum space instead of the real space of the atoms.\nYou might have already figured out that this is a cor-\nrect statement, and you might wonder why this is not in\nthe undergraduate books. The reason is that one needs\nthe Ceperley path integral to forget once and for all that\nfermions are incomprehensible.\nFinally in Chapter VIII we turn to the real space de-\nscription of the Fermi-gas. This was the alley tried by\nCeperley and coworkers where they got stuck, and we\nhave not managed either to get it fully under control.\nTheperspectiveonthis issuedoeschange, knowingabout\nthe momentum space solution of chapter VII. What is at\nstake is the structure of duality transformations in the\nCeperley formalism: real- and momentum space dynam-\nics are dual to each other. In order to learn how to ad-\ndress the problems at arbitrary couplings it is important\nto understand the duality structure. On the real space\nside one gets a better view on the richness of the Ceper-\nley path integral. A highlight is the understanding of the\ntopologyofthereach,basedonaconjecturebyCeperley20\nthat was recently proven by Mitas29. The outcome is\nthat there is no topological restriction on the windings\nof the ’Ceperley particle’ worldlines as long as these are\nconstructed from triple particle exchanges. We will ar-\ngue that the low temperature thermodynamics should be\ngovernedby the winding sector and the zero temperature\nFermi gas can be viewed as a Bose condensate of Ceper-\nley particles. However, the presence of the reach changes\nradically the winding statistics as compared to the boson4\ncase and it appears that the windings of the Ceperley\nparticles in anyhigher dimension are counted as if they\nare the windings associated with soft core bosons living\nin one space dimension – the literal interpretation of the\n’Fermi-liquid holography’ introduced in section II.\nII. KNOW THE ENEMY: THE STRANGENESS\nOF THE FERMI-LIQUID.\nThe only exactly solvable many Fermion problem is\nthe non-interacting Fermi-gas. Surely, every student in\nphysics knows the canonical solution. Introduce creation\nand annihilation operators that anti-commute,\n{c†\n/vectork,c/vectork′}=δ/vectork,/vectork′, (1a)\n{c†\n/vectork,c†\n/vectork′}={c/vectork,c/vectork′}= 0, (1b)\nand the Hamiltonian is\nH0=/summationdisplay\n/vectorkεkc†\n/vectorkc/vectork, (2)\nwhere/vectorkis some set of single particle quantum numbers;\na representativeexample is the spinless gas in the contin-\nuum where /vectorkrepresents single particle momentum and\nεk=/planckover2pi12k2/2m. It follows from standard manipulations\nthat its grand canonical free energy is\nFG=−1\nβ/summationdisplay\n/vectorkln/parenleftig\n1+e−β(ε/vectork−µ)/parenrightig\n, (3)\nwhereβ= 1/(kBT) andµthe chemical potential, tend-\ning to the Fermi-energy EFwhenT→0. The particle\nnumber is\nN=/summationdisplay\n/vectorkn/vectork, (4a)\nn/vectork=1\neβ(ε/vectork−µ)−1, (4b)\nwheren/vectorkis recognized as the momentum distribution\nfunction. At zero temperature this momentum distri-\nbution function turns into a step function: n/vectork= 1 for\n|/vectork| ≤kFand zero otherwise where the Fermi-momentum\nkF=/radicalbig\n2mEF//planckover2pi12. The step smears at finite tempera-\nture, and this is another way of stating the fact that only\nat zero temperature one is dealing with a Fermi-surface\nwith a precise locus in single particle momentum space\nseparating occupied- and unoccupied states.\nThe simplicity of the Fermi-gas is deceptive. This can\nbe highlighted by a less familiar but illuminating argu-\nment. As Landau guessed correctly15,16, the Fermi-gas\ncan be adiabatically continued to the interacting Fermi-\nliquid. The meaning of this statement is that when oneconsiders the system at sufficiently large times and dis-\ntancesandsufficientlysmalltemperatures(’scalinglimit’)\nastateofinteractingfermionicmatter existsthat isphys-\nically indistinguishable from the Fermi-gas. It is charac-\nterized by a sharp Fermi surface and a Fermi energy but\nnowthese areformed from a gasofnon-interactingquasi-\nparticles that have still a finite overlap (’pole strength’\nZ/vectork) with the bare fermions, because the former are just\nperturbatively dressed versions of the latter, differing\nfrom each other only on microscopic scales15. This is\nthe standard lore, but let us now consider these matters\nwith a bit more rigor. The term describing the inter-\nactions between the bare fermions will have the general\nform,\nH1=/summationdisplay\n/vectork,/vectork′/vector qV(/vectork,/vectork′,/vector q)c†\n/vectork+/vector qc/vectorkc†\n/vectork′−/vector qc/vectork′.(5)\nIt is obvious that single particle momentum does not\ncommute with the interaction term,\n/bracketleftig\nc†\n/vectorkc/vectork,H1/bracketrightig\n/ne}ationslash= 0, (6)\nhenceforth, single particle momentum is in the presence\nof interactions no longer a quantum number and sin-\ngle particle momentum space becomes therefore a fuzzy,\nquantum fluctuating entity. But according to Landau we\ncanstillpoint atasurfacewith asharplocusinthis space\nalthough this space does not exist in a rigorous manner\nin the presence of interactions!\nIn the textbook treatments of the Fermi-liquid this ob-\nvious difficulty is worked under the rug. Since the above\nargumentisrigorous,ithastobe the casethat the Fermi-\nsurface does not exist when one is dealing with any finite\nnumber of particles! Since we know empirically that the\nFermi-liquid exists in the precise sense that interacting\nFermi-systems are characterized by a Fermi-surface that\nis precisely localized in momentum space in the ther-\nmodynamic limit it has to be that this system profits\nfrom the singular nature of the thermodynamic limit, in\nanalogy with the mechanism of spontaneously symmetry\nbreaking that rules bosonic matter.\nWe refer to the peculiarity of bosonic- and classi-\ncal systems that (quantum) phases of matter acquire a\nsharp identity only when they are formed from an infin-\nity of constituents16,30. Consider for instance the quan-\ntum crystal, breaking spatial translations and rotations.\nSurely, one can employ a STM needle to find out that\nthe atoms making up the crystal take definite positions\nin space but this is manifestly violating the quantum me-\nchanical requirement that ’true’ quantum objects should\ndelocalize over all of space when it is homogeneous and\nisotropic. The resolution of this apparent paradox is well\nknown. One should add to the Hamiltonian an ’order\nparameter’ potential V(R) whereRrefers to the dNdi-\nmensional configuration space of Natoms inddimen-\nsional space, having little potential valleys at the real5\nspace positions of the atoms in the crystal. It is then a\nmatter of order of limits,\nlim\nN→∞lim\nV→0/an}bracketle{t/summationdisplay\niδ(/vector ri−/vector r0\ni)/an}bracketri}ht= 0, (7a)\nlim\nV→0lim\nN→∞/an}bracketle{t/summationdisplay\niδ(/vector ri−/vector r0\ni)/an}bracketri}ht /ne}ationslash= 0, (7b)\nwhere/vector riand/vector r0\niare the position operator and the equi-\nlibrium position of the i-th atom forming the crystal.\nHenceforth, the precise positions of the atoms in the\nsolid, violating the demands of quantum mechanical in-\nvariance, emerge in the thermodynamic limit – we know\nthat a small number of atoms cannot form a crystal in a\nrigorous sense.\nReturning to the Fermi-liquid, the commonality with\nconventionalsymmetrybreakingisthatinbothcasesnon\nexistent quantum numbers (position of atoms in a crys-\ntal, single particle momentum in the Fermi-liquid) come\ninto existence via an ’asymptotic’ emergence mechanism\nrequiring an infinite number of constituents, at least in\nprinciple. But this is as far the analogy goes. In every\nother regard, the Fermi-liquid has no dealings with the\nclassical emergence principles, that also govern bosonic\nmatter.\nAlthough it is unavoidable that the Fermi-liquid needs\nthe thermodynamic limit it is not at all clear what to\ntake for the order parameter potential V. In this re-\ngard, the Fermi-liquid is plainly mysterious. The text-\nbook treatises of the Fermi-liquid, including the quite\nsophisticated ’existence proofs’, share a very perturba-\ntive attitude. The best treatments on the market are the\nones based on functional renormalization and the closely\nrelated constructive field theory31,32. Their essence is as\nfollows: start out with a Fermi gas and add an infinites-\nimal interaction, follow the (functional) renormalization\nflow from the UV to the IR to find out that all interac-\ntions are irrelevant operators. Undoubtedly, the conclu-\nsions from these tedious calculations that the Fermi-gas\nis in a renormalization group sense stable against small\nperturbations are correct. The problem is that these per-\nturbative treatments lack the mighty general emergence\nprinciples that we worship when dealing with classical\nand bosonic matter.\nTo stress this further, let us consider a rather classic\nproblem that seems to be more or less forgotten although\nit was quite famous a long time ago: the puzzle of the\n3He Fermi-liquid16. The3He liquid at temperatures in\nthe Kelvin range is not yet cohering and it is well under-\nstood that it forms a dense van der Waals liquid. Such\nliquidshaveabadreputation; allmotionsinsuchaclassi-\ncal liquid are highly cooperative to an extent that all one\ncandois toput them into acomputerandsolvethe equa-\ntionsofmotionsbybruteforceusingmoleculardynamics.\nWhen one cools this to the millikelvin range, quantum\ncoherence sets in and eventually one finds the impecca-\nble textbookversionofthe Fermi-liquid: the macroscopicproperties arise from dressed helium atoms that have be-\ncome completely transparent to each other, except that\nthey communicate via the Pauli principle, while they are\nroughly ten times as heavy as real3He atoms. When\none now measures the liquid structure factor using neu-\ntron scattering one finds out that on microscopic scales\nthis Helium Fermi-liquid is more or less indistinguishable\nfrom the classical van der Waals fluid! Hence, at micro-\nscopic scales one is dealing with the same ’crowded disco’\ndynamics as in the classical liquid except that now the\natomsarekeptgoingbythe quantum zero-pointmotions.\nOn the microscopic scale there is of course no such thing\nas a Fermi surface. For sure, the idea of renormaliza-\ntion flow should still apply, and since one knows what is\ngoing on in the UV and IR one can guess the workings\nof the renormalization flow in the3He case: one starts\nout with a messy van der Waals ultraviolet, and when\none renormalizes by integrating out short distance de-\ngrees of freedom one meets a ’relevant operator creating\nthe Fermi-surface’. At a time scale that is supposedly co-\nincident with the inverse renormalized Fermi-energy this\nrelevant operator takes over and drags the system to the\nstable Fermi-liquid fixed point. How to construct such\na ’Fermi-surface creation operator’? Nobody seems to\nhave a clue!\nAlthough the microscopic details are quite different,\nthe situation one encounters in interesting electron sys-\ntem like the ones realized in manganites3,33, heavy\nfermion intermetallics4and cuprate superconductors2is\nin gross outlines very similar as in3He. In various guises\none finds coherent quasiparticles (or variations on the\ntheme, like the Bogoliubons in the cuprates) only at very\nlow energiesand low temperatures. Undoubtedly the UV\nin these systems has much more to do with the van der\nWaals quantum liquid than with a free Fermi-gas. Still,\nthe only activity the theorists seem capable off is to de-\nclare the UV to be a Fermi-gas that is hit by small in-\nteractions. It is not because these theorists are incompe-\ntent: humanity is facing the proverbial brick wall called\nthe fermion sign problem that frustrates any attempt to\ndo better.\nArrived at this point we hope that we have convinced\nthe reader that even the ’simple’ Fermi-liquid is plainly\nmysterious. This mystery is of course rooted in the fact\nthatthe fermionsignsdisconnectthemanyfermionprob-\nlem from the powerful principles of statistical physics\nthat allow us to fully comprehend the emergence logic\nof classical- and bosonic matter. We already stressed\nthat nobody has a clue how to construct a mathemat-\nical definition of a ’Fermi-surface generating renormal-\nization group operator’ and the closely related issue of\nthe ’Fermi-surface stabilizing order parameter potential’.\nBut there are a couple of other features that are discon-\nnected from anything we know in statistical physics.\nThe relationshipbetween thermal andquantum fluctu-\nations is plainly weird in Fermi-liquids and Fermi-gasses.\nIn sign free, i.e. bosonic or ’Bolzmannion’, quantum sys-\ntems one has a simple rule telling how these fluctuations6\nrelate, which is rooted in the postulates. The thermal\npath-integral can be taken as basic postulate, being both\napplicable to fermionic and sign-free quantum matter13.\nIt states that everything takes place in Euclidean space-\ntime, being spanned by the space dimensions and imagi-\nnary timeτ. In this formalism, temperature determines\nthe ’maximumdurationofimaginarytime’: foropenspa-\ntial boundaries, euclidean space-time has the topology\nof a cylinder where imaginary time is the compact di-\nrection with a compactification radius Rτ=/planckover2pi1/(kBT).\nHenceforth, at zero temperature everything takes place\nin a (d+ 1)-dimensional space (ignoring complications\nlike an external heat bath) where dis the number of\nspace dimensions. Addressing general scaling limit is-\nsues, like the existence (or not) of order, one is at finite\ntemperatures interested in times long compared to Rτ.\nIt follows that thermal fluctuations are ’one dimension\nmore important’ than quantum fluctuations, at least as\nlong one can get away with the well understood role of\ntarget space dimensionality in statistical physics. Hence-\nforth, the well known Mermin-Wagner rules34imply that\nat zero temperature one can have algebraic long range\norder ind= 1, while at any finite temperature one needs\nat leastd= 2; one can truly break a continuous sym-\nmetry ind= 2 atT= 0, but at any finite temperature\none needsd= 3, and so forth. This ’space-time geogra-\nphy’ applies as well to fermionic problems but the com-\nplication is of course that the connection with statistical\nphysics is shattered! This has avery strangeconsequence\nthat can be easily overlooked. We argued already that\nin some weird fermionic sense, the Fermi-liquid ’breaks\nsymmetry’. But from the canonical side we know some\nanswers: from the discontinuity in the momentum distri-\nbutions we learn that the Fermi-surface acquires a pre-\ncise locus in momentum space only at zero temperature\n(omitting the non-generic d= 1 case that follows the bo-\nson rules17). The Fermi-Dirac distribution teaches us in\nturn that the Fermi-surface ’smears’ in momentum space\nat any finite temperature, regardless the dimensionality\nof target space. Henceforth, one has zero temperature\norder, and finite temperature disorder, regardless dimen-\nsionality to the extent that it is even true in d=∞. This\nis quite hard to comprehend when you would only know\nstatistical physics!\nWe can actually push this further by considering the\nthermodynamics of the Fermi-liquid in more detail, just\nforgetting for the moment how we got there, and insist-\ning that there exist eventually a bosonic/Boltzmannian\ndescription. Theargumentisano-brainerbuttheconclu-\nsion is quite spectacular: the Fermi-liquid demonstrates\nanextremeformofthe holographic principle thatwasdis-\ncovered in the context of the quantum physics of black\nholes25. The precise statement is: the low energy physics\nof a Fermi-liquid in arbitrary dimensions is in precise\ncorrespondence with an interacting Boltzmannian system\nin 1+1 dimensions . This is surely consistent with the\nobservation that the Fermi-liquid shows a zero tempera-\nture (algebraic) order, while it is disordered at any finitetemperature. This is the typical trait of one dimensional\nbosonic physics but the Fermi-liquid weirdness is that\nit is doing this job in all dimensions. Let us make this\nclaim more precise, by considering the grand canonical\nfree energy of the Sommerfeld gas in arbitrary space di-\nmensiond >2. This can be regarded as representative\nfor the Fermi-liquid in the scaling limit, i.e. modulo the\nrenormalizationofthe Fermi-energyand at temperatures\nsufficiently small compared to the Fermi-energy,\nFG=−2\nd+2NEF/bracketleftigg\n1+π2\n6/parenleftbiggd\n2+1/parenrightbigg/parenleftbiggkBT\nEF/parenrightbigg2\n+\nO/parenleftigg/parenleftbiggkBT\nEF/parenrightbigg4/parenrightigg/bracketrightigg\n. (8)\nFrom this free energy follows the temperature depen-\ndence of the specific heat,\nCV=dNkB/bracketleftigg\nπ2\n6kBT\nEF+O/parenleftigg/parenleftbiggkBT\nEF/parenrightbigg3/parenrightigg/bracketrightigg\n,(9)\nand the chemical potential,\nµ=EF/bracketleftigg\n1−π2\n12(d−2)/parenleftbiggkBT\nEF/parenrightbigg2\n+O/parenleftigg/parenleftbiggkBT\nEF/parenrightbigg4/parenrightigg/bracketrightigg\n. (10)\nThe Sommerfeld expansion breaks down in d= 2 where\nthe abovethermodynamicfunctions becomenon-analytic\nfunctions at T= 0 and cannot be expanded in powers of\nkBT/EF. For example, for the chemical potential in d=\n2 one obtains µ=kBTln[exp(EF/kBT)−1]. However,\nthe above expressions strictly hold in d= 2+ǫ.\nLet us now consider an arbitrary interacting massless\nbosonic system. In any space dimension d≥1 such a\nsystem cannot avoid (algebraic) long range order and the\nthermodynamics is set by the massless Goldstone bosons\ncharacterized by a dispersion ǫ(k) =c/planckover2pi1k. Assuming that\nthe order survives at finite but small temperatures the\nFree energy becomes,\nF=−ΓdVkBT/parenleftbiggkBT\n/planckover2pi1c/parenrightbiggd\n, (11)\nwith Γ da dimensionless prefactor and it follows that\nCV=d(d+ 1)Γ dkBV/parenleftbigkBT\n/planckover2pi1c/parenrightbigd, while for the chemical\npotential it is interesting to consider a superfluid where\nµ∼ −T2for 1+1D Boson systems.\nAswelearnedinfreshmencourses,thetemperaturede-\npendence of thermodynamic quantities of Boltzmannions\nstrongly depends on temperature, like the Debye specific7\nheatCV∼Td, reflecting that for increasing dimension-\nality more collective degrees of freedom become available\nwith the effect that entropy increases more rapidly for\nincreasing temperature. On the other hand, the number\nof degrees of freedom counted by the increase of entropy\nfor increasing temperature of the Fermi-liquid is entirely\nindependent of dimensionality , being actually coincident\nwith the number ofdegrees offreedom ofa 1+1Dbosonic\nsystem. We learned to comprehend this on the signfull\nside by arguing that the microscopic degrees of freedom\nare locked up in the Fermi sea, while for rising tempera-\nture only degrees of freedom are released in a thin shell\n∼kBT/EFaround the Fermi-surface. This is of course a\nfine explanation but to make it work we need the fermion\nsigns. But we have now learned that a Boltzmannian de-\nscription of the Fermi-liquid exists, in the form of the\nCeperley path integral. Although the constrained struc-\nture is quite non-trivial, it cannot cause miracles and\nbecause it is a Boltzmannian machine it has to give in\neventually to the iron ’Mermin-Wagner’ order parameter\nrules. Henceforth, it has to be that in the Ceperley for-\nmalism we are dealing with an effective 1+1 dimensional\norder parameter theory.\nThelast’anomaly’ofthe Fermi-liquidappearsagainas\nratherinnocentwhenonehasjustworkedoneselfthrough\na fermiology textbook. However, giving this a further\nthought, it is actually the most remarkable and most\nmysterious feature of the Fermi-liquid. Without exag-\ngeration, one can call it a ’UV-IR connection’, indicat-\ning the rather unreasonable way in which microscopic in-\nformation is remembered in the scaling limit. It refers\nto the well known fermiology fact that by measuring\nmagneto-oscillationsin theelectricaltransport(Haasvan\nAlphen-, and Shubnikov de Haas effects) one can deter-\nmine directly the average distance between the micro-\nscopic fermions by executing measurements on a macro-\nscopic scale35. This is as a rule fundamentally impossible\nin strongly interacting classical- and sign free quantum\nmatter. Surely, this is possible in a weakly interacting\nand dilute classical gas, as used with great effect by van\nder Waals in the 19-th century to proof the existence of\nmolecules. But the trick does not work in dense, strongly\ninteracting classical fluids: from the hydrodynamics of\nwater one cannot extract any data regarding the prop-\nerties of water molecules. Surely, the weakly interacting\nFermi-gas is similar to the van der Waals gas but a more\nrelevantexampleisthestronglyinteracting3He, oreither\nthe heavy fermion Fermi-liquid. At microscopic scales it\nis of course trivial to measure the inter-particle distances\nand the liquid structure factor of3He will directly re-\nveal that the helium atoms are apart by 4 angstroms or\nso. But we already convinced the reader that there is no\nsuch thing as a Fermi surface on these scales. Descend-\ning to the scaling limit, a Fermi-surface emerges and it\nencloses a volume that is protected by the famous Lut-\ntinger theorem36,37:it has to enclose the same volume\nas the non-interacting Fermi gas at the same density!\nUsing macroscopic magnetic fields, macroscopic samplesand macroscopic distances between the electrical con-\ntactsonecannowmeasureviade Haasvan Alphen effect,\netcetera, what kFis and the Fermi momentum is just the\ninverseoftheinter-particledistancemodulofactorsof2 π.\nThis is strictly unreasonable. We repeat, on microscopic\nscales the system has knowledge about the inter-particle\ndistance but there is no Fermi-surface; the Fermi sur-\nface emerges on a scale that is supposedly in some heavy\nfermion systems a factor 100or even 1000largerthan the\nmicroscopic scale. But this emerging Fermi-surface still\ngets its information from somewhere, so that it knows\nto fix its volume satisfying Luttinger’s rule! In a later\nsection we hope to shed some light on the ’mysteries’ ad-\ndressed in this section using Ceperley’s path integral but\nwe are still completely in the dark regarding this partic-\nular issue. It might well be that there are even much\ndeeper meanings involved; we believe that it has dealings\nwith the famous anomalies in quantum field theories38.\nThese are tied to Dirac fermions and the bottom line is\nthat these process in rather mysterious ways ultraviolet\n(Planckscale) information to the infrared, with the effect\nthat a gauge symmetry that is manifest on the classical\nlevel is destroyed by this ’quantum effect’.\nTo summarize, in this section we have discussed the\nfeatures of the Fermi-liquid that appear to be utterly\nmysterious to a physicist believing that any true under-\nstanding of physics has to rest on Boltzmannian princi-\nple:\n(i) What is the order parameter and order parameter po-\ntential of the zero temperature Fermi-liquid?\n(ii) How to construct a ’Fermi-surface creation operator’,\nwhich is supposed to be the relevant operator associated\nwith the IR stability in the renormalization group flow?\n(iii) Why is there ’Fermi-liquid order’ at zero tempera-\nture in any d≥2, while it gets destroyed by any finite\ntemperature regardless dimensions? More precisely, why\nis the Fermi-liquid holographic and what are the degrees\nof freedom of the ’holographic screen’ populated by the\neffective bosons?\n(iv)Whyisitpossibletoretrievemicroscopicinformation\nvia the Luttinger sum rule by performing macroscopic\nmagneto-transport measurements, even in the asymptot-\nically strongly interacting Fermi-liquid?\nThe bottom line of this paper will be, although we\nknow for sure that there are ’Bolzmannian’ answers to\nthese questions, and although we know quite well where\nto look for them, we have no conclusive answers in the\noffering right now. But the remainder will make clear\nwhy these questions are so interesting.\nIII. THE SIGN-FULL WORLDLINE PATH\nINTEGRAL.\nThere is just one sign-full path integral problem that\ncan be completely solved: the non-interacting Fermi gas\nin worldline representation, in any dimension. It is the\nusual business, when one can solve a problem exactly in8\none representation (i.e. canonical) it can also be solved\nin any other representation. In fact, the Fermi-gas path\nintegral is a textbook problem, although we are aware\nof only one textbook where it is worked out in detail:\nKleinert’s Path integral book26. Let us first summarize\nthe ’mechanics’ of this path-integral, referring the reader\nto Kleinert’s book for the details, to subsequently use\nthis solvable case as an example to highlight the rather\nawkward and counterintuitive workings of the ’negative\nprobabilities’. All along it is interesting to compare it\nwith the free boson path integral which works the same\nway except that it corresponds with a well behaved Bolz-\nmannian problem.\nConsiderthepartitionfunctionforBosonsorFermions;\nthiscanbewrittenasanintegraloverconfigurationspace\nR= (r1,...,rN)∈RNdof the diagonal density matrix\nevaluated at an imaginary /planckover2pi1β,\nZ= Tre−βH=/integraldisplay\ndRρ(R,R;β).(12)\nThe path integral formulation of the partition function\nrests on a formal analogy between the quantum mechan-\nical time evolution operator in real time e−iˆHt//planckover2pi1and the\nfinite temperature quantum statistical density operator\nˆρ=e−βˆH, wherethe inversetemperature β= 1/kBThas\nto be identified with the imaginary time it//planckover2pi1. The par-\ntition function defined as the trace of this operator and\nexpression (12) simply evaluates this trace in position\nspace. More formally this can viewed as a Wick rotation\nof the quantum mechanical path integral, and requires\na proper analytic continuation to complex times. This\nrotation tells us that the path integral defining the par-\ntition function lives in D-dimensional Euclidean space,\nwithD=d+1 anddthe spatial dimension of the equi-\nlibrium system. This analogy tells us that to study the\nequilibrium statistical mechanics of a quantum system in\nindspace dimensions, we can study the quantum sys-\ntem in a Euclidean space of dimension d+1, where the\nextra dimension is now identified as a ”thermal” circle\nof extentβ. At finite temperature this circle is compact\nand world-lines of particles in the many-body path inte-\ngral (12) then wrap around the circle, with appropriate\nboundaryconditionsforbosonsorfermions. The discrete\nMatsubarafrequencies that arisefromFouriertransform-\ning modes on this circle carry the idea of Kaluza-Klein\ncompactification to statistical mechanics. We will come\nback to a careful consideration of the evaluation of the\npartition function (12) in section IV when we discuss the\nconnection between winding vs. cycle decomposition in\npreparation for some observations regarding supersym-\nmetry.\nFor distinguishable particles interactingvia a potential\nVthe density matrix can be written in a worldline pathintegral form as,\nρD(R,R′;β) =/integraldisplay\nR→R′DRexp(−S[R]//planckover2pi1),(13a)\nS[R] =/integraldisplay/planckover2pi1β\n0dτ/parenleftigm\n2˙R2(τ)+V(R(τ))/parenrightig\n,(13b)\nbut for indistinguishable bosons or fermions one has also\nto sum over all N! permutations Pof the particle coor-\ndinates,\nρB/F(R,R;β) =1\nN!/summationdisplay\nP(±1)pρD(R,PR;β),(14)\nwherepis he parity of the permutation. For the bosons\none gets awaywith the positive sign, but for fermions the\ncontribution of a permutation with uneven parity to the\npartition sum is a ’negative probability’, as required by\nthe anti-symmetry of the fermionic density matrix. This\nis the origin of the sign problem.\nThe partition sum describes worldlines that ’lasso’ the\ncircle in the time direction. Every permutation in the\nsum is composed out of so called permutation cycles. For\ninstance, consider three particles. One particular contri-\nbution is given by a cyclic exchange of the three particles\ncorresponding with a single worldline that winds three\ntimes around the time direction with winding number\nw= 3 (see Fig. 1), a next class of contributions cor-\nrespond with a ’one cycle’ with w= 1 and a two-cycle\nwithw= 2 (one particle returns to itself while the other\ntwo particles are exchanged), and finally one can have\nthree one cycles (all particles return to their initial posi-\ntions). For bosons this is just equivalent to a problem of\ninteracting ring polymers and this can be solved to any\nrequired accuracy using quantum Monte-Carlo – see e.g.\nthe impressive work by Ceperley on4He,39,40making the\ncase that this strongly interacting boson problem has no\nsecrets left. But for fermions one can only handle the\nnon-interacting limit, because of the fermion signs.\nFor the non-interactingBose and Fermi gasthe evalua-\ntion of the path integral reduces to a combinatorial exer-\ncise. Let us first illustrate these matters for the example\nofN= 3 particles. It is straightforward to demonstrate,\nthattheidentitypermutationgivesacontribution Z0(β)3\nto the partition function (here Z0(β) denotes the parti-\ntion function of a single particle), whereas an exchange\nof all three particles contribute as Z0(3β). The meaning\nis simple: in the absence of interactions the 3-cycle can\nbe identified with a single particle worldline returning to\nits initial position at an effective inverse temperature 3 β\n(see Fig. 1). Further on, a permutation consisting of a\nw= 1 and aw= 2 cycle contributes with Z0(β)Z0(2β).\nTo write down the canonical partition function for N= 3\nnon-interacting bosons or fermions we only have to know\nthe combinatorial factors (e.g. there are 3 permutations\nmade out of a w= 1 and aw= 2 cycle) and the parity\nof the permutation to obtain9\nFIG. 1: Worldline configuration corresponding to a cyclic ex -\nchange of three particles, 1 →2, 2→3, and 3 →1, or in short\nnotation (123) (upper left). On a cylinder (upper right), th e\nworldlines form a closed loop winding w= 3 times around the\ncylinder. Intheextendedzonescheme(bottom), theexchang e\nprocess of three particles can be identified with a worldline of\na single particle at an effective inverse temperature 3 β.\nZ(N=3)\nB/F(β) =1\n3![Z0(β)3±3Z0(β)Z0(2β)\n+2Z0(3β)]. (15)\nThis result can easily be generalized to Nparticles. We\ndenote the number of 1-cycles, 2-cycles, 3-cycles, ...N-\ncycles the permutation is build of with C1,C2,C3,...,\nCNand denote the combinatorial factors counting the\nnumbers of permutations with the same cycle decompo-\nsitionC1,...CNwithM(C1,...CN). ForNparticles we\nhave to respect the overall constraint N=/summationtext\nwCwand\nobtain\nZ(N)\nB/F(β) =1\nN!N=P\nwCw/summationdisplay\nC1,...CNM(C1,...CN)(±1)P\nw(w−1)Cw\n×N/productdisplay\nw=1[Z0(wβ)]Cw. (16)\nAlthough the combinatorial factors can be written\ndown in closed form,\nM(C1,...CN) =N!/producttext\nwCw!wCw, (17)\nthe canonical partition function (16) is very clumsy to\nwork with because of the constraint acting on the sumover cycle decompositions. However, it is possible to de-\nrive a recursion relation for the canonical partition func-\ntion or examine it in terms of so called loop decompo-\nsitions. For details we refer the reader to the appendix.\nThe constraint problem can be circumvented by going\nto the grand-canonical ensemble. After simple algebraic\nmanipulations we arrive at the grand-canonical partition\nfunction\nZG(β,µ) =∞/summationdisplay\nN=0Z(N)\nB/F(β)eβµN\n= exp/parenleftigg∞/summationdisplay\nw=1(±1)w−1Z0(wβ)\nwewβµ/parenrightigg\n,(18)\ncorresponding to a grand-canonical free energy\nFG(β) =−1\nβlnZG(β,µ)\n=−1\nβ∞/summationdisplay\nw=1(±1)w−1Z0(wβ)\nweβwµ,(19)\nwith the ±inside the sum referring to bosons (+) and\nfermions ( −), respectively. This is a quite elegant re-\nsult: in the grand-canonical ensemble one can just sum\nover worldlines that wind wtimes around the time axis;\nthe cycle combinatorics just adds a factor 1 /wwhile\nZ0(wβ)exp(βwµ) refers to the return probability of a\nsingle worldline of overall length wβ. In the case of zero\nexternal potential we can further simplify\nZ0(wβ) =Vd\n/radicalbig\n2π/planckover2pi12wβ/Md\n=Z0(β)1\nwd/2, (20)\nto obtain for the free energy and averageparticle number\nNG, respectively,\nFG=−Z0(β)\nβ∞/summationdisplay\nw=1(±1)w−1eβwµ\nwd/2+1,(21a)\nNG=−∂FG\n∂µ=Z0(β)∞/summationdisplay\nw=1(±1)w−1eβwµ\nwd/2.(21b)\nTo establish contact with the textbook results for the\nBose- and Fermi-gas one just needs that the sums over\nwindings can be written in an integral representation as,\n∞/summationdisplay\nw=1(±1)w−1eβwµ\nwν=1\nΓ(ν)/integraldisplay∞\n0dεεν−1\neβ(ε−µ)∓1,(22)\nand one recognizes the usual expressions involving an in-\ntegral of the density of states ( N(ε)∼εd/2indspace10\ndimensions) weighted by Bose-Einstein or Fermi-Dirac\nfactors.\nThe bottom line is that at least for bosons Eq. (21)\nhas the structure of a Bolzmannian partition sum and\none can rest on the powerful conceptual machinery of\nstatistical physics. Surely, the canonical route is shorter\nbut it has the ’transfer-matrix attitude’: very powerful\nwhen it solves the problem exactly but it is ’overly alge-\nbraic’, not adding much to the ’no-nonsense’ conceptual\nstructure, characteristic for statistical physics.\nFor example, Einstein just used Eq.’s (21,22) to de-\nduce that for d≥3 a true phase transition occurs in\nthe Bose gas from a high temperature classical gas to\nthe Bose-Einstein condensate. Given that there are no\ninteractions this is peculiar and just knowing about the\ncanonical side one can only take Bose-Einstein conden-\nsation as a mysterious quantum phenomenon that drops\nout from the algebra. But knowing the path-integral side\none cannot afford mysteries: it is just an equilibrium\nring polymer problem, and plastic cannot have secrets!\nIndeed, in the winding representation one meets a meat-\nand-potato thermodynamic singularity. At the transi-\ntionµ→0 and one directly infers from Eq. (21) that\nvery long worldlines corresponding with winding num-\nbersw∼Nare no longer penalized, while there are\nmany more long winding- than short winding contribu-\ntions in the sum. It is straightforwardto showthat in the\nthermodynamic limit worldlineswith wbetween√\nNand\nNhave a vanishing weight above the BEC temperature,\nwhile these infinite long lines dominate the partition sum\nin the condensate41.\nA related issue is the well known fact that the non-\ninteracting Bose-Einstein condensate and the superfluid\nthat occurs in the presence of finite repulsions are adi-\nabatically connected: when one switches on interactions\nthe free condensate just turns smoothly into the super-\nfluid and there is no sign of a phase transition. This can\nbe seen easily from the canonical Bogoliubov theory42,43.\nAgain, although the algebrais fine matters are a bit mys-\nterious. The superfluid breaks spontaneous U(1) symme-\ntry, thereby carrying rigidity as examplified by the fact\nthat it carries a Goldstone sound mode while it expels\nvorticity. The free condensate is a non-rigid state, that\ndoes not break symmetry manifestly, so why are they\nadiabatically connected? The answer is obvious in the\npath-integral representation39,40: although interactions\nwill hinder the free meandering of the polymers, a lot of\nthishindranceisrequiredtomakeitimpossibleforworld-\nlines to become infinitely long below some temperature.\nThe fraction of infinitely long worldlines is just the con-\ndensate fraction and even in the very strongly coupled\n4He superfluid these still make up for roughly 30% of all\nworldlines! The only way one can get rid of the infinite\nwindings in the interacting system is to turn it into a\nstatic array of one cycles - the4He crystal. This simple\nargument underlies the widespread believe that ’simple’\nbosons can only form superfluids or crystals, while for\ninstance a non-superfluid ’Bose metal’ cannot possiblyexist44. How to avoid the windings when the worldlines\ncan meander over infinite distances?\nThis preceding paragraph illustrates the reasons to\nworship path-integrals when one has learned the lan-\nguage. Are they helpful dealing with fermions? Let us\nattempt to address the ’mysteries’ (i) and (iii) of Section\nII: the Fermi-gas represents a form of order (the locus of\nthe Fermi surface, the jump in nk) but in all dimensions\nd≥2 this ’order’ disappears at any finite temperature.\nThinking in a statistical physics language it appears at\nfirst sight that the only source of this thermodynamic\nsingularity can reside in the ’infinite windings’. Let us\nfirst consider the zero temperature case,where the Fermi\ngas is described by the wave function\nΨ =1√\nN!det\neik1x1···eikNx1\n......\neik1xN···eikNxN\n,(23)\nHere the momenta k1,···,kNfill the Fermi sea. The\nslater determinant is a signful summation over N! differ-\nentpermutationpatterns, eachcharacterizedbyacertain\ncycle decomposition as explained previously,\nΨ =1√\nN!/summationdisplay\nP(−1)peik1xp(1)···eikNxp(N)\n=1√\nN!/summationdisplay\nPψP. (24)\nNow we want to ask the question, what is the proba-\nbility for large cycles to occur? This probability can be\nwritten as an incoherent sum\nPL=1\nN!/summationdisplay\nlarge−cycles|ψP|2, (25)\nwhich is insensitive to the fermion sign, thus leading to\nthe sameresultasforbosons. PLcanbe computed byex-\namining the cycle structure of the random permutations\nofNobjects. This is already done by the mathemati-\ncians Golomb and Gaal45. According to their result, the\nprobabilitythat the greatestcycle length is ksatisfies the\nrecursion relation\nP(k,N) =[N/k]/summationdisplay\nj=11\nj!kjN!\n(N−kj)!\n×min(k−1,N−kj)/summationdisplay\nt=1P(t,N−kj),(26)\nwhere [a] represents the largest integer not greater than\na. ForN/2< k≤N, the probability has the simple\nformP(k,N) = 1/k. For large N, the probability that\nthe length of greatest cycle lies between N/2 andNis11\nN/summationdisplay\nk=N/2P(k,N)≈ln2≈0.693. (27)\nForN/3≤k < N/ 2, the total probability is/summationtextN/2\nk=N/3P(k,N)≈0.258, while for N/4≤k < N/3,\n/summationtextN/3\nk=N/4P(k,N)≈0.044. Thus there is 99 .5 per cent\npossibility that the greatest cycle has length larger than\nN/4. Our conjecture is that in the large Nlimit, the\nprobability to have infinite winding is unity\nlim\nN→∞N/summationdisplay\nk=NcP(k,N) = 1, (28)\nwhere the lower bound Nc→ ∞asN→ ∞, and\nlim\nN→∞Nc/N→0. For example, Nccan be chosen as\n√\nNand one indeed finds that Eq. (28) is satisfied.\nSo we have rigorously proved that the zero tempera-\nture Fermi gas is characterized by infinite windings. As\nfor bosons, does this mean that at any finite tempera-\nture a sudden change has occurred to short windings?\nOne can easily deduce the answer from Eq. (21). At a\ntemperature of order of the Fermi temperature TFthe\nchemical potential of the Fermi-gas switches from nega-\ntive to positive and this means that in any term of the\nwinding sum a worldline which has a winding number\nthat is larger by one unit than another is more impor-\ntant for the free energy by a factor eβµ, a very large\nfactor indeed for µ≃EFandβ→ ∞! Henceforth, when\nthe effects of Fermi-Dirac statistics become noticeable at\ntemperatures T≤TFthe sum is by default dominated\nby infinite long windings that however cancel each other\nnearly completely because of the alternating sign in Eq.\n(21)!\nThe resolution of this ’paradox’ of course lies in the\nfact that when one is dealing with an alternating sum\nwhich is not absolutely convergent every individual term\nin the sum is meaningless, while one has to consider the\nsum as a whole. To cite a well known example,\nh= 1−2+4−8+16−···\n= 1−2h,\nand it follows that h= 1/3. This is of course the same\nthing as a Fermi-Dirac factor, and after performing the\ntransformation to the integral representation, Eq. (22),\none just discovers that the sum over winding boils down\nto the usual result that the Fermi-Dirac distributions\nturns into a step function only when β→ ∞. The\nfermion signs translate in the Fermi-Dirac alternating\nwinding sums in the case of the free Fermi gas and that\nis the end of the story. One can only handle these sums\nwhen the problem is exactly solvable, and the signs havethe net effect of destroying the connection with the pow-\nerful conceptual structure of statistical physics.\nThe Fermi-gas is to an extent misleading: it is actually\nthe only example of a fermion system that is completely\nsolved. No closed form, exact solution exists for finite\ninteractions. Surely, there is no doubt that the Fermi-\nliquid exists but there is no general mathematical proof\nstating that it has to exist given specific microscopic cir-\ncumstances. The claims in this direction are all based on\nspecial,orevenpathologicallimits, liketheweakcoupling\nand low density limits. The Fermi-liquid is at the same\ntime a monument of scaling limit phenomenology31,32:\nconsidering long times and large distances, physics sim-\nplifies and emergence principles take over that can be\ncaught in simple but powerful mathematical structures.\nTofind suchprinciplesforfermionicmatteristheoutlook\nof this paper.\nTo conclude this exposition of the sign problem, the\nbest way to highlight its severity is by employing the rig-\norous language of mathematical complexity theory. One\ncan classify mathematical problems according to the cri-\nteriumofhowthecomputationtimeofsomehypothetical\ncomputer will scale with the number of degrees of free-\ndomN. When this time is polynomial in Nthe problem\nis considered as solvable in principle. This is different\nfor problems in the ’nondeterministic polynomial’ (NP)\nclass, that have a time that grows exponentially with N\non a classical, deterministic computer, and a special sub-\nclassofsuchproblemsarecalled’NPhard’ifanyproblem\ninNPcanbemappedontoitwithpolynomialcomplexity.\nThe bottom line is that when one can solve one partic-\nular NP-hard problem, one has solved all NP problems.\nTo illustrate the gravity of this affair: the traveling sales-\nman and the spin glass problem are NP hard. Recently,\nTroyerandWiese14provedthat the fermionsignproblem\nis NP hard. Hence, when you manage to demonstrate a\npolynomial time solution for the sign problem you can\ncash a million dollars at the Clay Mathematics Institute!\nThis proof is actually remarkably simple. The classical\nIsing spin glass in three dimensions,\nH=−/summationdisplay\n<j,k>Jjkσz\njσz\nk (29)\nwhere the spins σjtake the values ±1 while the couplings\nJarerandomlychosenfrom0 ,±1, isthe complexityclass\nNP hard. By a trivial rotation of the quantization axis\none can write this as well as a signful quantum problem,\nH=−/summationdisplay\n<j,k>Jjkσx\njσx\nk (30)\nthe Hamiltonian has only off diagonal matrix elements in\na basis spanned by eigenstates of σz. When the exchange\ncouplingsJwould be all positive definite all matrix ele-\nments would be negativeand the groundstate wavefunc-\ntion would be nodeless and therefore bosonic. This prob-\nlem is easy to solve in polynomial time. However, when\nthe signs of the J’s can be positive and negative the signs12\nappear, while at the same time the frustrations switch\non causing the NP-hardness of the equivalent spin-glass\nproblem: Q.E.D.\nIV. SUPERSYMMETRY\nIn this section we will consider the meaning of su-\npersymmetry in the world-line path integral formal-\nism. It is a new symmetry introduced to related the\nphysics of bosons (which mediate forces) and fermions\n(the constituents of matter)46, and leads to many beau-\ntiful mathematical properties47,48. Supersymmetry is\nan idea that has a long and illustrious history in the\nhigh-energy physics community and has made appear-\nances even within condensed matter. It is presently the\nmost promisingidea fornew physicsbeyondthe standard\nmodel and there is much hope that the first glimpses of\nit will be gleaned at the Large Hadron Collider in the\nnext few years. This hope is predicated on the ability of\nsupersymmetry to solve one of the principal open issues\nfacing high-energy physics (aside from gravity), that of\nthe hierarchy problem associated with the mass of the\nHiggs boson, which is believed to drive the electroweak\nsymmetry-breaking phase transition. The solution de-\nrives from a new class of non-renormalization theorems\ncontrolling certain quantum corrections in manifestly su-\npersymmetric systems. In the context of gravitational\nphysics, supersymmetry has been instrumental in formu-\nlating a consistent theory of quantum gravity within the\nframework of string theory, and has led to the discovery\nof deep and profound dualities relating seemingly incon-\ngruous theories.\nIt is standard practice when teaching quantum field\ntheory to high-energy physicists these days to focus on\nthe second quantization of theories at zero temperature\nandchemicalpotential, thisbeingbothaconvenientlimit\nand a good starting point for those interested in the kind\nof collider physics that is the experimental underpinning\nof high-energy particle physics today. There are, though,\na new generation of experiments, for example the Rela-\ntivistic Heavy Ion Collider (RHIC)54,55, which mean to\nprobe our understanding of Quantum Chromodynamics\n(QCD, the theory of quarks) at ever higher temperatures\nand densities, with an eye to understanding the phase\ntransition in the early universe that is conjectured to\nhave transformed a plasma of quarks and gluons into the\nHadrons(protons, neutronsandsoon)that dominatethe\nlow energy world we see.\nYet, given the prominent role of phase transitions and\nsymmetry-breakingin our understanding of the standard\nmodel and extensions thereof, together with the postu-\nlated role of supersymmetry, a natural question to pon-\nder is the meaning of supersymmetry at finite temper-\nature and density. It is often stated that supersymme-\ntry is broken at any finite non-zero temperature, though\nthe precise meaning of this statement and the conse-\nquent implications are not immediately obvious. Thisbelief relies on how supersymmetry relates bosonic and\nfermionic degrees of freedom and the fact that at fi-\nnite temperatures they follow different statistical distri-\nbutions. Studies of this question49,50,51,52,53leave open\nsome unresolved questions, in particular the influence\nof non-renormalization theorems when zero-temperature\nsupersymmetric systems are raised to a finite tempera-\nture.\nWe will not address these issues in full generality here,\nbutwillpointoutaninterestingobservation(hithertoun-\nknown) about free supersymmetric systems when formu-\nlated in the language of first-quantized world-line path\nintegrals, and suggest ways to press into the regime of\ninteracting systems. To set the stage for this discussion,\nwe will first take a detour to study the partition func-\ntion of a free system in the world-line language, making\nexplicit the sum over windings. We then demonstrate\nthat the combinatoric sums can be reorganizedinto sums\nover numbers of cycles. This will be our goal for section\nVIIIB. With this tool kit at our disposal, in section IV\nwe then show how supersymmetry can be understood as\na restriction on the types of cycles we must sum over\nwhen constructing partition functions.\nWewillnowpresentsomethoughtsonthenatureofsu-\npersymmetryinthe languageofworld-linepathintegrals.\nThough we only consider the case of free particles explic-\nitly, we feel that this way of looking at supersymmetry\nmight suggest new insights into the underlying meaning\nof supersymmetry in a way that allows one to move to\nstudy its properties and implications at finite tempera-\nture and chemical potential.\nWe start by considering the physics of a gas of bosons\nintermingled with a gasof fermions. The grandcanonical\nfree energy for a gas of free bosons, written as a sum over\nwindings, is\nFb\nG=−1\nβZ0(β)∞/summationdisplay\nw=1ewβµ\nwd\n2+1, (31)\nwithZ0(β) the partition function of a single particle at\ninverse temperature β, andDis the spatial dimensional-\nity of the system. For free fermions\nFf\nG=−1\nβZ0(β)∞/summationdisplay\nw=1(−1)(w−1)ewβµ′\nwd\n2+1,(32)\nwhere the chemical potentials are in general unrelated,\nand consistency requiring that the chemical potential of\nfree bosons be negative semi-definite. Since no such con-\nstraint exists for free fermions, we are free to choose the\nchemical potential of the fermions such that µ′=µ≤0.\nThe free energy of the full system then becomes\nFG=−2\nβZ0(β)/summationdisplay\nw oddewβµ\nwd\n2+1(33)\nHere we notice an intriguing property of a gas of free\nbosons and fermions with equal chemical potential: it is13\nequivalent(due tocancellations)toasystemcomposedof\na new type of particle, with the same chemical potential,\nbut the strange property that it can only wind an odd\nnumber of times around the thermal circle. Going back\nto the original picture in terms of a gas of free fermions\nand bosons, we note that, in the zero temperature limit,\nit is now easy to check the expected value of the internal\nenergy vanishes. The vanishing of the energy of the sys-\ntem is precisely the order parameter for unbroken super-\nsymmetry. Thus a system of free particles which are only\nallowed to wind an odd number of times is supersymmet-\nric! This is true even for finite chemical potential. The\nvanishing of the energy at zero temperature is of course\nalso a feature of the normal Bose-Einstein condensate.\nThis new class of particle which only experiences odd\nwindings also undergoes a condensation in dimensions\ngreater than two, but with a critical temperature that is\nshifted higher relative to the normal boson case. At tem-\nperaturesbelow this critical value, the chemicalpotential\nof the system vanishes (for d>2), which implies that the\nFermi surface for the original fermions also degenerates\nand the fermion occupation vanishes since the fermions\ncan not condense in the zero-momentum state. As we\ncross above the critical temperature the chemical poten-\ntial becomes finite, a Fermi surface appears, but simul-\ntaneously we have the situation that the occupation of\nthe zero momentum state becomes non-macroscopic. So\nmagically the appearance of a finite density of fermions\nis associated with the change from a macroscopically oc-\ncupied zero-momentum state to non-macroscopic occu-\npation. The fermions kick out the bosons!\nWe have arrived at this conclusion without any\nmention of an underlying algebraic system describing\nfermionic symmetries of the system, which is how su-\npersymmetry is usually discussed in the context of both\nclassical and quantum field theories, though this struc-\ntureisimplicitinthewayweconstructedthesystemfrom\na gas of bosons and fermions. We can take this winding\nrule as a new definition characterizing supersymmetric\nsystems (at least free ones), even at finite temperature.\nSome open questions to ponder are: (1) Is it possi-\nble to relate the sum over odd windings to a symme-\ntry algebra? It is evident from our construction that\nthere is a symmetry relating bosons to fermions. (2)\nHow do we include interactions into this picture? Af-\nter all, the power of supersymmetry lies in its ability to\nhelp us understand complicated interacting systems; free\nsystems are too easy. Here perturbation theory, perhaps\nin the relativistic notation we introduced by Feynman56\nshould be analyzed. (3) Taking this new view, what do\nwe learn about supersymmetric systems at finite temper-\nature? A well known property of supersymmetric sys-\ntems is the non-renormalization theorems which protect\ncertain quantities against quantum corrections. These\ntheorems are usually presented in the context of zero\ntemperature quantum field theory. Since, in the path\nintegral formulation of the quantum statistical partition\nfunction, quantum fluctuations are associatedwith imag-inarytime dynamics, we conjecture that the contribution\nmade to any thermodynamic quantity by these fluctua-\ntions cancel, at all temperatures, though at finite tem-\nperature, corrections arising from thermal fluctuations\nsurvive, a result of the fact that away from T= 0 the\nBose and Fermi distributions differ from each other. It\nis usually claimed that supersymmetry is broken at finite\ntemperature as a result of this difference. For example,\nthe mass renormalizations for bosons and fermions will\nbe different. Our argument suggests that their quan-\ntum renormalization still cancel, and the breaking of the\nBose/Fermi degeneracy is strictly a thermal effect.\nV. THE ENLIGHTENMENT: CEPERLEY’S\nCONSTRAINED PATH INTEGRAL.\nAfter these preliminaries, we have arrived at the core\nofthispaper: Ceperley’s1991discoveryofapathintegral\nrepresentation for arbitrary fermion problems that is not\nsuffering from the ’negative probabilities’ of the standard\nformulation20. Surely, one cannotnegotiatewith the NP-\nhardness of the fermion problem and Ceperley’s path\nintegral is not solving this problem in a mathematical\nsense. However, the negative signs are transformed away\nat the expense of a structure of constraints limiting the\nBoltzmannian sum over world-line configurations. These\nconstraints in turn can be related to a geometrical mani-\nfold embedded in configuration space: the ’reach’, which\nis a generalization of the nodal hypersurface characteriz-\ning wave functions to the fermion density matrix. This\nreach should be computed self-consistently: it is gov-\nerned by the constrained path integral that needs itself\nthe reach to be computed. This is again a NP-hard prob-\nlem and Ceperley’s path integral is therefore not solving\nthe sign problem. However, the reach contains all the\ndata associated with the differences between bosonic and\nfermionic matter, and only its averageand globalproper-\ntiesshouldmatterforthephysicsinthescalinglimitsince\nitactsonworldlineconfigurationsthatthemselvesareav-\neraged. Henceforth, it should be possible in principle to\nclassify all forms of fermionic matter in a phenomeno-\nlogical way by classifying the average geometrical- and\ntopological properties of the reach, to subsequently use\nthis data as an input to solve the resulting bosonic path\nintegral problem. This procedure is supposedly a unique\nextension of the Ginzburg-Landau-Wilson paradigm for\nbosonic matter to fermionic matter. We do not have a\nmathematical proof that this procedure will yield a com-\nplete classification offermionic matter, but we have some\nvery strong circumferential evidences in the offering that\nit will work. The status of our claim is conjectural in the\nmathematical sense.\nLet us start out presenting the answer. Ceperley\nproved in 1991 that the following path integral is strictly\nequivalent to the standard fermion path integral Eq.\n(13,14),14\nρF(R,R;β) =1\nN!/summationdisplay\nP,even/integraldisplayγ∈Γβ(R)\nγ:R→PRDRe−S[R]//planckover2pi1.(34)\nThis is quite like the standard path integral, except\nthat one should only sum over evenpermutations (the\nreason to address this in section IV), while the allowed\nworldline configurations γare constrained to lie ’within\nthe reach Γ’. This reach is defined as,\nΓβ(R) ={γ:R→R′|ρF(R,R(τ);τ)/ne}ationslash= 0}(35)\nfor all imaginary times 0 <τ </planckover2pi1β. In words, only those\nwordline configurations should be taken into account in\nEq. (34) that do not cause a sign change of the full den-\nsitymatrixateveryintermediateimaginarytimebetween\n0 and/planckover2pi1β. In outline, the proof of this result is as follows.\nThe fermion density matrix is defined as a solution to the\nBloch equation\ndρF(R0,R;β)\ndβ=−HρF(R0,R;β) (36)\nwith initial conditions\nρF(R0,R;β= 0) =1\nN!/summationdisplay\nP(−1)pδ(R0−PR).(37)\nIn the following we fix the reference point R0and de-\nfine the reach Γ( R0,τ) as before as the set of points\n{Rτ}forwhich there exists a continuous space-timepath\nwithρF(R0,Rτ′;τ′)>0 for 0≤τ′< τ. Suppose that\nthe reach is known in advance. It is a simple matter to\nshow that the problematical initial condition, Eq. (37),\nimposing the anti-symmetry can be replaced by a zero\nboundary condition on the surface of the reach. It fol-\nlows because the fermion density matrix is a unique so-\nlution to the Bloch equation (36) with the zero bound-\nary condition. One can now find a path integral so-\nlution without the minus signs. One simply restricts\nthe paths to lie in the reach Γ( R0,τ) imposing the zero\nboundary condition on the surface of the reach. The\nodd permutations fall for sure out of the reach since\nρF(R0,PoddR0) =−ρF(R0,R0).\nThe Ceperley path integral revolves around the reach.\nHow to think about this object? The way the path inte-\ngral is constructed seems to break imaginary time trans-\nlations. One has to first pick some ’reference point’ Rin\nconfiguration space at imaginary time 0 or /planckover2pi1β. Starting\nfrom this set of particle coordinates, one has to spread\nthem out in the form of wordline configurations to check\nat every time slice that the density matrix does not\nchange sign. The dimensionality of the density matrix\nis 2dN+ 1 (twice configuration space plus a time axis)\nand the dimensionality of the reach is therefore 2 dN(oneoverall constraint). However, when we first pick a refer-\nencepoint Randwefocusonaparticularimaginarytime\nthe dimensionality of this restricted reach is dN−1. In\nthe limitτ→ ∞this restricted reach turns into a more\nfamiliar object: the nodal hypersurface associated with\nthe ground state wave function. The density matrix be-\ncomes for a given Rin this limit,\nρ(R,R′;β=∞) = Ψ∗(R)Ψ(R′) (38)\nand the zero’s of the density matrix are just coinci-\ndent with the nodes of the ground-state wave function,\nΨ(R) = 0, where we have assumed that the ground state\nis non-degenerate. The wave function is anti-symmetric\nin terms of the fermion coordinates,\nΨ(···,ri,···,rj,···) =−Ψ(···,rj,···,ri,···),(39)\nand therefore the nodal hypersurface\nΩ ={R∈RNd|Ψ(R) = 0} (40)\nis a manifold of dimensionality dimΩ = Nd−1 embed-\nded inNd-dimensional configuration space. This nodal\nsurface Ω is surely an object that is simpler than the\nfull reach Γ and it is rather natural to train the intu-\nition using the former. According to Ceperley’s numer-\nical results20, it appears that at least for the Fermi gas\nthe main features of the reach are already encoded in Ω.\nIn a way, the dependence on imaginary time is remark-\nably smooth and unspectacular. A greater concern is\nthe role of the reference point, or either the fact that the\nreachdepends ontwoconfigurationspacecoordinates. In\nthe long imaginary time limit, the reach factorizes in the\nnodal surfaces (Eq. (38)), which means that one can get\naway just considering the nodal surface of the ground\nstate wave function, but this is not the case at finite\nimaginary times. It is not at all that clear what role the\n’relative distance’ R−R′plays, although there is some\nevidence that it can be quite important as we will discuss\nin Section IX. Notice that the conventional ’fixed-node’\nquantum Monte-Carlo methods aim at a description of\nthe ground state, using typically diffusion Monte-Carlo\nmethods. As input for the ’fermionic-side’, these only re-\nquire the wave function nodal structure. The difference\nbetween the reach and this nodal structure is telling us\neventually about the special nature of the excitations in\nthe fermion systems since the Ceperley path integral can\nbe used to calculate dynamics, either in the form of fi-\nnite temperature thermodynamics or, by Wick rotation\nto real time, about dynamical linear response. At this\nmoment in time it is not well understood what the pre-\ncise meaning is of these ’dynamical signs’ encoded in the\nnon-local nature of the reach.\nAnother useful geometrical object associated with\nFermi-Dirac statistics is the Pauli surface, correspond-\ning with the hypersurface in configuration space where15\nFIG. 2: Cut through the nodal hypersurface of the ground-\nstate wave function of N= 49 free, spinless fermions in a\ntwo-dimensional box with periodic boundary conditions. Th e\ncut is obtained by fixing N−1 fermions at random positions\n(black dots) and moving the remaining particle (white dot)\nover the system. The lines indicate the zeros of the wave func -\ntion (nodes). Note that the nodal surface cut has to connect\ntheN−1 fixed particles since the Pauli surface is a lower\ndimensional submanifold of dimension Nd−dincluded in the\nnodal hypersurface with dimension Nd−1.\nthe wave function vanishes because the fermions are co-\nincident in real space,\nP=/uniondisplay\ni/ne}ationslash=jPij\nPij={R∈RNd|ri=rj}. (41)\nObviously, the Pauli surface is a submanifold of the\nnodal hypersurface of dimension dim P=Nd−d. The\nspecialty of one dimension is that the Pauli- and nodal\nhypersurfaces are coincident. This property that the\nnodes are ’attached’ to the particles is the key to the\nspecial status of one dimensional physics as we will ex-\nplain in detail in the next section.\nIn the next sections we will discuss in more detail the\nfew facts that are known about the reach and nodal hy-\npersurface geometry and topology. To complete the dis-\ncussion of the basic structure of the Ceperley Path In-\ntegral, let us once more emphasize that according to its\ndefinition Eq. (34) one still has to sum over evenpermu-\ntations in so far these do not violate the reach. As for\nthe signful path integral, this translates via the sum over\ncycles into a sum over winding numbers that are now as-\nsociated with triple exchanges of particles. We explained\nalready in detail in section IV that this has the pecu-\nliar consequence that it codes for supersymmetry when\none is dealing with the free quantum gas that just knows\nabout the even permutation requirement. Because of theFIG. 3: Nodal constraint structure in space-time seen by\none particular particle. In the constraint path integral on ly\nworld-line configurations {Rτ}are allowed that do not cross\nor touch a node of the density matrix on all time slices,\nρF(R0,Rτ,τ)/ne}ationslash= 0 for 0 ≤τ </planckover2pi1β. Therefore, a particular\nparticle (white circle) is constrained by the dynamical nod al\ntent (grey surface) spanned by the N−1 remaining particles\ntrajectories (black circles). In a Fermi liquid the nodal te nt\nhas a characteristic dimensions and particles feel the noda l\nconstraints at an average time scale τc. In later chapter we\nwill see that these scales are in one-to-one correspondence\nwith the Fermi degeneracy scale EF.\nconstraints, the ’particles’ of the Ceperley path integral\nare actually very strongly interacting and it is unclear to\nwhat extent this supersymmetry is of any relevance to\nthe final solution. In fact, we do know for the Fermi-gas\nthat the combined effect of the constraints and the triple\nexchangesistoeventuallygivebackafreegaswithFermi-\nDirac statistics. As we discussed in section IV, there is\na ’don’t worry theorem’ at work because the thermody-\nnamics of the supersymmetric gas is quite similar to the\nBose gas.\nIn conclusion, Ceperley has demonstrated that in prin-\nciple fermion problems can be formulated in a prob-\nabilistic, Boltzmannian mathematical language, paying\nthe prize of a far from trivial constraint structure that\nis a-priori not known while it cannot be exactly com-\nputed. Qualitatively, the reach is like the nodal struc-\nture of a wave function. It is obvious that the nodal\nstructure codes for physics but this connection is largely\nunexplored, while the remainder of this paper is dedi-\ncated to the case that it is actually quite easy to make\nprogress, at least with regard to the Fermi-liquid. One\nparticular property is so important that it should be al-\nready introduced here. Any wave function of a system\nof fermions has the anti-symmetry property Eq. (39)\nand naively one could interpret this as ’any physical sys-\ntem of fermions has its fermionic physics encoded in a16\nNd−1 dimensional nodal surface’. This is obviously not\nthe case. It is easy to identify a variety of fermionic sys-\ntems where many more nodes are present in the fermion\nwave function than are required to encode the physics.\nA first example are Mott-insulating antiferromagnets on\nbipartite lattices. Because the electrons are localized\nthey become effectively distinguishable. One can there-\nforetransformawayremnantsignsin the Heisenbergspin\nproblem by Marshall sign transformations: the bottom\nline is that such Mott-insulators can be handled by stan-\ndard bosonic quantum Monte Carlo methods. A next\nexample is physics in one dimensions, as we will discuss\nin the next section, where again the fermion signs can\nbe transformed away completely, in a way that can be\nneatly understood in terms of the topology of the nodal\nsurface. Nodal structure is therefore like a gauge field: it\ncarries redundant information that is inconsequential for\nthe physics. Nodal structure that is in this ’gauge vol-\nume’ we call reducible nodal structure, while the ’gauge\ninvariant’ (physical) part of the nodal structure we call\nirreducible , and as a first step one should always first\nisolate the true, irreducible signs.\nVI. THE CEPERLEY PATH INTEGRAL IN 1+1\nDIMENSIONS.\nThe physics of quantum matter in one space dimen-\nsion can be regarded as completely understood17. The\ndeep reason is that quantum statistics has no physical\nmeaning in 1+1D, and it is always possible to find a\nrepresentation where the sign structure drops out com-\npletely. All signs are reducible in the language of the\nprevious paragraph. Accordingly, the quantum problem\nis equivalent to a statistical physics problem in 2 classi-\ncal dimensions, and it appears that the problem solving\ncapacity of statistical physics has no limit in this dimen-\nsion. The reader might be familiar with the standard\nbosonization techniques. A most elementary example is\nthe Jordan-Wignertransformationwhichis usuallyintro-\nduced to demonstrate that S= 1/2 quantum spin chain\nproblems are equivalent to interacting spinless fermion\nproblems, with as special cases the transversal field Ising\nmodel (equivalent to 2D Ising) mapping onto free Majo-\nrana fermions57, and the XY spin chain being equivalent\nto just free Dirac fermions11. It is instructive to find\nout how this is processed by the Ceperley path integral.\nOn the one hand, although the canonical Jordan-Wigner\nand bosonization methods are of course correct, the way\ntheydealwiththe(anti)symmetryofthestatesinHilbert\nspace is somewhat implicit and in this regard a discus-\nsionintermsofCeperley’sreachismostinformative. The\nother side is about the powers of fermionization; the sim-\nple free spinless Fermi-gas becomes in the Ceperley path\nintegral representation a very serious statistical physics\nproblem. It is difficult to imagine a harder 2D statistical\nphysicsproblem: itisthe’Pokrovsky-Talapov’problem58\nof fluctuating polymers interacting via purely steric con-straints. These correspond with infinitely strong delta\nfunction potentials (real, finite range interactions sim-\nplify the problem!) and accordingly everything is about\nentropic interactions and order-out-of-disorder physics.\nRemarkably, this problem can be solved in a few lines\nusing canonical fermions. Although the Ceperley path\nintegral has a much richer structure in higher dimensions\nit is surely the case that the higher dimensional Fermi-\nliquids have to know in one or the other way about this\n’entropic dynamics’. To highlight this aspect we will re-\nview here the one dimensional fermion story in a less\nfamiliar, radically statistical physics way27,28.\nLet us first focus on the workingsof quantum statistics\nin 1+1D, using the Jordan-Wigner transformation as a\ntemplate. Consider a chain of interacting s= 1/2 spins,\ndescribed by SU(2) operators,/bracketleftbig\nSα,Sβ/bracketrightbig\n=iεαβγSγ.\nSpins live in simple tensor product space. In condensed\nmatter physics they describe electrons that through a\nMott condition got localized and localized electrons are\n’Boltzmannions’, i.e. distinguishable particles. The stan-\ndard construction continues claiming that on every site\nthere are two available states (spin-up and -down) and\nthis is no different from spinless fermions leaving a site\nunoccupied or singly occupied. But the difference is\nclearly in the antisymmetry of the fermion-Hilbert space,\nas encoded in the anti-commutation property of the\nfermion operators. This problem can be dealt with by\nthe Jordan-Wigner sign string that works by the virtue\nthat in one space dimension a string can see all the par-\nticles covered by itself between its two end points,\nc(n) =\n/productdisplay\nj<n[−σz(j)]\nσ−(n),(42a)\nc†(n) =σ+(n)\n/productdisplay\nj<n[−σz(j)]\n.(42b)\nHerec(n) andc†(n) denote fermionic annihilation and\ncreationoperatorsonsite n, respectively,and σ±= (σx±\nσy)/2 withσα=2\n/planckover2pi1Sαthe conventional spin-1/2 Pauli\noperators.\nHavingtheseoperatoridentitiesit becomesthen trivial\nto rewrite the spin-Hamiltonian in terms of the fermion\noperators and one finds out in no time that the sign\nstrings cancel out, and one obtains a problem that is\nlocal in the fermions.\nA little miracle has happened: we started out with\nBoltmannions and by the magic of the above operator\nidentities wefind out that wemightaswell considerthese\ndistinguishable particles as fermions. In fact, one has\nthe free choice to invoke hard core bosons as well in the\nintermediate stage since these share the property with\nspinss= 1/2 and spinless fermions that one has two\navailable states per site.\nThis is surely correct but in the canonical language\nit just appears as a mathematical fact associated with17\noperator identities. The Ceperley path integral is in this\nregard more transparent. Let us consider the meaning\nof the reach of the one dimensional Fermi-gas. For a\ngiven reference point R0and imaginary time τone can\nassociate with the fermion density matrix ρF(R0,R;τ)\na (Nd−1)-dimensional nodal hypersurface. However,\nwe know that the ( Nd−d)-dimensional Pauli surface is\na submanifold of the nodal hypersurface and for d= 1\nthe Pauli- and the nodal hypersurfaces have the same\ndimensionality and they are therefore the same! This is\nnothing else than the well established wisdom that in one\ndimension the nodes of the wave functions are attached\nto the particle positions, a fact that is at the heart of\nJordan-Wigner and all other bosonization constructions.\nIn dimensions larger than one ’signs can have a life of\ntheir own’ because the nodal hypersurface has a larger\ndimensions than the Pauli surface. This is the simple but\ndeep reason for the complete failure of all attempts to\nconstruct Jordan-Wigner style bosonization procedures\nin higher dimensions.\nGiven that the Pauli-and nodalhypersurfacescoincide\nit becomes quite easy to read the reach. Start out with a\nreference point R0= (x1,x2,···,xN) ordering the par-\nticles for instance like x1< x2<···< xN. ’Spread\nout’ this configuration in terms of world lines meander-\ning along the time direction and the Pauli-hypersurface\nreach tells that only configurations are allowed where\nthese worldlines never cross each other at any imagi-\nnary time. This is just the problem of an ensemble of\npolymers with only steric, hard core interactions in 2 di-\nmensions! What is the fate of the quantum statistics?\nLet us permute two coordinates in the reference point\nR′0= (x2,x1,···,xN). Because the particles one and\ntwo cannot pass each other these two starting configu-\nrations are disconnected: they belong to two different\nnodal cells. Since this is true for any of the N! permuta-\ntions, in one dimensions one finds N! nodal cells, instead\nof the two nodal cells of the higher dimensional Fermi-\ngas, as discussed in the next section. The full partition\nsum consists of N! copies of the same one-cycle ’Boltz-\nmannion’ partition function starting from some particle\nsequence that is just divided by N!. The bottom line is\nthat bosonic symmetry or fermionic anti-symmetry turns\nin the presence of the Pauli-hypersurface reach into a\nmere redundancy of the description. It has the status of\na gauge volume and gauge invariant reality is caught in\nterms of a Boltzmannion ’gauge fix’. Surely these wis-\ndoms are well known from general considerations invok-\ning the braid group, but the merit of the Ceperley path\nintegral is that it incorporates these considerations in a\nmost explicit way.\nWe now have landed on the statistical physics side:\nthe Fermi-gas in 1+1D just corresponds with a problem\nof ’one cycle’ ring polymers interacting merely through\nsteric constraints58. Like the van der Waals liquid, prob-\nlems with just steric interactions have a bad reputation.\nThisisobviouslyaboutstronginteractionswith theextra\ndifficulty that the potentials are singular: a crossing ofFIG. 4: Allowed world line configurations in Ceperley path\nintegral for one-dimensional fermions. In 1D the nodal hype r-\nsurface coincides with the Pauli surface and the constraint s\nturn into hard-core repulsions between the particles. Due t o\nthe hard-core constraints particle exchanges are strictly for-\nbidden and the particles become distinguishable. When the\nmean square displacement of the particles becomes of the or-\nder of the average inter-particle spacing, collisions betw een\nthe particles start to happen. This characteristic time sca le\nτcis directly related to the Fermi energy, EF=/planckover2pi1/τc. From\nthe statistical physics point of view the Fermi-gas in 1+1D\njust corresponds with a problem of ’one-cycle’ ring polymer s\ninteracting merely through steric constraints.\nworldlines costs an infinite potential energy. The ramifi-\ncation is that all the physics is driven by entropy. Surely,\nthe easy way to solve this problem is to fermionize it and\neverything follows from the simple Fermi-gas solution.\nBut how to read these Fermi-gas wisdoms in the physical\nproblem of the ring polymers? In fact, one of us27was\nfacing some time ago a problem which is quite similar to\nthe 2D ring polymers but where fermionization technol-\nogy fails: the gas of hard core directed quantum strings\nin 2+1 dimensions, which is equivalent to the problem\nof directed elastic membranes in 3D interacting through\nsteric hindrance. This problem came up as a toy exercise\nin the investigation of quantum stripes in cuprates and it\nis obviously the direct generalization of the 1+1D Fermi-\ngas to 2 space dimensions: just attach an extra space\ndimension and the world-lines spread out in the string\nworldsheets. As it turns out, both the 1+1D Fermi gas\nand the 2+1D string gas can be addressed using a ’self-\nconsistent phonon’ method discovered by Helfrich59to\ndeal with the entropic interactions associated with bi-\nological (extrinsic curvature) membranes. Let us just\nsketch the derivation.\nIn the polymer language, the partition sum of the\nFermi-gas can be written as,\nZ=N/productdisplay\ni=1/productdisplay\nτ/integraldisplay\ndφi(τ)e1\n/planckover2pi1S,\nS=/integraldisplay\ndτ/summationdisplay\niM\n2(∂τφi)2, (43)18\nwhereφiis the spatial displacement field of the i-th par-\nticle andthe wordlineisa1Delasticmanifoldwith spring\nconstant set by the mass Mof the particles. This is sup-\nplemented by the avoidance condition,\nφ1<φ2<···<φN. (44)\nDespite its simple formulation this is a rich problem\ncharacterized by various scales. Let us first inspect the\nultraviolet ofthe problem. The averagedistance between\nworldlines is rs=L/N= 1/nwhereLis the length of\nthe system and nthe particle density. The worldlines\nonly know about each others existence when they collide\nbecause of the steric nature of the interactions. Hence-\nforth, at sufficiently short times the worldlines will have\nmeandered over distances that are small compared to rs\nand they cannothave knowledgeof eachothers existence.\nThe problem becomes cooperative at the characteristic\ntime scale for collisions to occur. This is straightforward\nto estimate. By detailed balance the mean square fluc-\ntuation of the elastic line grows with τas,\nl2\nc(τ) =/an}bracketle{t(φi(τ)−φi(0))2/an}bracketri}ht=/planckover2pi1\nMτ. (45)\nWhenlc(τ) becomes similar to rsa collisions will occur\nand this will take a typical time\nτc≃M\n/planckover2pi1n−2, (46)\nwhich in the quantum interpretation this time is associ-\nated with an energy,\nEF=/planckover2pi1\nτc=/planckover2pi12\n2mn2, (47)\nand we recognize the Fermi energy27! This makes per-\nfect sense. The Fermi energy is the characteristic energy\nwhere the effects of the quantum statistics on the oth-\nerwise free system becomes noticeable. In the Ceperley\npath integral this translates into the time scale where for\nthe first time the particles become aware of the presence\nof the constraints. In the next section we will find that\nthe above argument can be trivially extended to higher\ndimensions.\nThe Fermi energy is easy but now the trouble starts.\nWhen the collisionsstartto happen the problembecomes\nhighly cooperative and in principle hard to deal with in\nterms of exact methods. The key is entropic repulsions,\nthe same effect causing a rubber band to stiffen up when\nyou heat it. The qualitative argument runs as follows27.\nTake the gas of non-interacting polymers as a reference\npoint. In this free gas there are two possibilities when\ntwo polymers meet each other: either they cross or do\nnot cross. But in the steric avoidance gas, there is only\none possibility (do not cross) and therefore the systemhas to pay an entropy cost of log(2) at every collision\nrelative to the non-interacting system. This adds a posi-\ntive (repulsive) term nc(T)kBTlog(2) to the free energy\nwherenc(T) is the density of collisions at temperature\nT. This will have the effect that upon coarse graining\nthe polymers start to repel each other: the collisions at\nshort distances caused by the strong microscopic fluctua-\ntionsrenormalizeatlargedistancesinto’entropicsprings’\nkeeping the polymers apart!\nTo make this more quantitative27we need the inge-\nnious trick devised by Helfrich59. Quite generally, the\neffect of the entropic repulsions will be to build up crys-\ntalline correlations and in the scaling limit one has to\nfind a crystal with algebraic long range order (2D, fi-\nnite temperature or 1+1D at finite coupling constant).\nThis worldline crystal does not carry shear rigidity be-\ncause the worldlines are incompressible in the time direc-\ntion. Henceforth, the crystal is characterized by a space-\ndirection compressionalmodulus B0, besides the time di-\nrectionmass’springconstant’. The coarse-grainedaction\ncan be written in terms of the effective elastic fields ψas\nSeff=1\n2/integraldisplay\ndτ/integraldisplay\ndx/bracketleftbig\nρ(∂τψ)2+B0(∂xψ)2/bracketrightbig\n,(48)\nwhereρ=nMis the mass density. Now Helfrich’s trick\ncomes: for finite B0fluctuations are suppressed relative\nto the case that B0vanishes and this entropy cost raises\nthefreeenergybyanamount∆ F(B0) =F(B0)−F(B0=\n0). But by general principle it has to be that the ’true’\nlong wavelength modulus Bshould satisfy\nB=r2\ns∂2\n∂r2s/parenleftbigg∆F(B0)\nL/parenrightbigg\n, (49)\nand in the case of steric interactions, the only source of\nlong wavelength rigidity is the fluctuation contribution\nto ∆F. This implies that B0=Band Eq. (49) turns\ninto a self-consistency condition for this ’true’ entropic\nmodulusB! It is an easy exercise to work this out and\none finds\nB=9π2\nη4/planckover2pi12\nMr3s, (50)\nwhereηis a fudge factor associated with the ultraviolet\ncut-off; the self-consistent phonon theory should set in\nat a length xminwhere crystalline correlations become\nnoticeable and because this requires some number of col-\nlisionsxmin=ηrswhereη>1.\nTo identify this ’order out of disorder’ physics\nwith our canonical Fermi-gas we have to invoke some\nmore bosonization wisdoms. The bottom line of the\nTomonaga-Luttinger liquid (Luttinger liquids are con-\nnected adiabatically to the Fermi-gas) is that he elec-\ntron system is nothing else than a 1+1D ’floating’ (alge-\nbraically ordered) crystal, characterized by a spectrum19\nof compressional phonons, the bosonization modes. This\nspectrum is characterized by a single dimension: the\nsound velocity that is coincident with the Fermi veloc-\nity in case of the free Fermi-gas. This velocity is in the\n’entropic world’ Eq. (48) given by vF=/radicalbig\nB/ρand the\ncorrect answer follows from Eq. (50) for a reasonable\nη=√\n6.\nTo complete this story, one can treat the 2+1D di-\nrected string gas with the same methodology27to find\nout the peculiar result that now both the Fermi energy\nand the Fermi-velocitybecome exponentially smallin the\ndensity, like vF∼exp(−const/µ1/3) whereµ=/planckover2pi1/(ρcd2)\nwithcthe worldsheet velocity and dthe interstring dis-\ntance. This reflects the fact that strings fluctuate a lot\nless than particles, and the entropic interactions are sup-\npressed. But in any other regard this string gas is quite\nlike the 1+1D Fermi gas!\nWehopethatthereaderhasappreciatedthisstory. We\nperceive it to an extent as demystifying. The bottom line\nis that the rather abstract mathematical procedures of\none dimensional physics are just coding accurately for a\nphysical world that is dominated by entropic interactions\nand order-out-of-disorder physics. It hits home the case\nthat it is quite misleading to call the Fermi-gas a ’gas’.\nOne could argue naively like: ’the system only knows\nabout kinetic energy so how can it be anything else than\na gas’? But Fermi-Dirac statistics translates into a sta-\ntistical physics via Ceperley’s path integral characterized\nby a structure of constraints that is essentially steric in\nnature. This is obviously the case in 1+1D but in this\nbasic regard things are the same in the less well under-\nstood higher dimensional cases. These steric constraint\nproblems have as a generic feature that the microscopic\n’kinetic energy’ fluctuations and the macroscopic ’poten-\ntial energy’ scales associated with order are governed by\nthe same dimensions. Order-out-of-disorder is in these\nkind of worlds an ubiquitous, hard to avoid mechanism\nand the bottom line is that all fermion systems in 1+1D\nfall victim to the algebraic order. This is in turn the key\nto the ’universal success’ of bosonization17. Bosoniza-\ntion is just geared to deal accurately with the fluctu-\nations around the ordered state. Turning to the spin-\nfull systems there is more life than just crystallization\n(and umklapp pinning). It becomes possible for word-\nlines to ’come to an end’ and these correspond with dis-\nlocations in the space time worldline crystals that are\ndynamically indistinguishable from XY vortices. Hence-\nforth, one finds a quantum melting equivalent to the 2D\nKosterlitz-Thouless transition leading into the gapped\nLuther-Emery state60. Another fascinating feature is the\nhidden ’squeezed lattice’ geometrical order that lies at\nthe heart of spin-charge separation. This is accurately\nencodedinstandardbosonization,butonlyrecentlychar-\nacterized in full using a statistical physics style non-local\norder parameter structure61.\nThere is much more to tell about physics in one dimen-\nsions, but this is less fun because it is fully understood.\nThe take home message is a warning: even when onefully grasps the nodal structure (not true in higher di-\nmension), while the problem maps onto something that\nexists in classical nature (not true either), a seemingly\ntrivial fermion problem (the 1+1D Fermi gas) turns out\nto code for a remarkably complex and rich world devoid\nof fermion signs!\nVII. THE FERMI GAS AS A COLD ATOM\nMOTT-INSULATOR IN MOMENTUM SPACE.\nThe Fermi-gas of the canonical formalism is very easy\nto solve exactly, and one would expect that in one or the\nother way this should mean that the constrained path\nintegral is also easy to solve. This is not true at all in the\npositionrepresentation, aswewill discuss in the next sec-\ntion. However, considering the derivation of the Ceper-\nley path integral there is actually no preferred status of\nreal space. The construction is completely independent\nof the representation one chooses for the single particle\nstates. On the canonical side momentum space is the\nconvenient representation to start from in the galilean\ncontinuum, or either any other basis that diagonalizes\nthe single particle problem. As we will show in this sec-\ntion, also the Ceperley path integral of the Fermi-gas\nbecomes very easy indeed when one chooses to formulate\nit in momentum space. After a couple of straightforward\nmanipulations one finds a sign free, Boltzmannian path\nintegralshowingamostentertainingcorrespondence: the\nFermi-gas is in one-to-one correspondence with a sys-\ntem of classical atoms forming a Mott insulating state\nin the presence of a commensurate optical lattice of infi-\nnite strength, living in a harmonic potential trap of finite\nstrength (see Fig. 5a). This is literal and the only odd-\nity is that this trap lives in momentum space instead of\nreal space; the Fermi surface is just the boundary be-\ntween the occupied optical lattice sites and the empty\nones. This boundary is sharp at zero temperature but it\nsmears at finite temperature because of the entropy that\ncan be gained by exciting atoms out of the trap! When\nyou are quick, you should already have realized that this\ntrap interpretation is actually consistent with everything\nwe know about the Fermi-gas. Let us now proof it by\nconstructing the Ceperley path integral.\nThe central wheel of the Ceperley path integral is the\nfermion density matrix. One should first guess an ansatz,\nuse it to construct the path integral, to check if the same\ndensity matrix is produced by the path integral. Surely\nweknowthefullfermiondensitymatrixfortheFermigas,\nandinmomentumspacethisturnsouttobearemarkably\nsimple affair. The k-space density matrix can be written\nas the determinant formed from imaginary time single\nparticle propagators in the galilean continuum,\ng(k,k′;τ) = 2πδ(k−k′)e−|k|2τ\n2/planckover2pi1M. (51)\nSincewelivein thespaceofexactsinglequantumnum-\nbers these propagators are diagonal; in the galilean con-20\nV(r)= αr2\nxyµ\nkxky\nkxkyτ\n0hβ(a)\n(b)(c)\nFIG. 5: (a) a system of classical atoms forming a Mott in-\nsulating state in the presence of a commensurate optical lat-\ntice of infinite strength , living in a harmonic potential trap\nV(/vector r) =αr2of finite strength; (b) the trap in momentum\nspacekx,kyinstead of real space; the Fermi surface is just\nthe boundary between the occupied optical lattice sites and\nthe empty ones; (c) a grid of allowed momentum states\nk= (2π/L)(kx,ky,kz,....) where the ki’s are the usual in-\ntegers and any worldline just closes on itself along the imag -\ninary time τdirection 0 →β: single particle momentum\nconservations prohibit anything but the one cycles.\ntinuum this just means the conservation of momentum,\nbut when translational symmetry is broken one should\nuse here just the basis diagonalizing the single particle\nHamiltonian.\nConsider now the full momentum configuration space\nK= (k1,...,kN) imaginary time density matrix,\nρF(K,K′;τ) =1\nN!/summationdisplay\nP(−1)pN/productdisplay\ni=1g/parenleftbig\nkp(i),k′\ni;τ/parenrightbig\n.(52)\nWe find that the delta functions cause a great simpli-\nfication. Substituting the single-fermion expression Eq.\n(51) in this expression for the density matrix Eq. (52)\nwe obtain:\nρF(K,K′;τ) =1\nN!e−PN\ni=1|ki|2τ\n2/planckover2pi1M\n×/summationdisplay\nP(−1)pN/productdisplay\ni=12πδ(kp(i)−k′\ni).(53)\nSince the single particle propagators are eigenstates of\nthe Hamiltonian, the momentum world lines go ’straight\nup’ in the time direction until they arrive at the time τ\nwhere the reconnections can take place associated with\nthe permutations. But the δfunction enforces that the\npermuted momentum has to be the same as the non-\npermuted one, and the worldlines can therefore not wind\nexcept when the momenta of some pairs of fermions coin-\ncide. But now the sum of the permutations in Eq. (53) is\nzero due to the Pauli principle. Mathematically, this fol-\nlows from the fact that the expression on the right hand\nside of Eq. (53) is actually a Slater determinant formed\nfrom the delta-functions 2 πδ(kp(i)−k′\ni) as the matrixelements of the Nd×Ndmatrix, that are indexed by\nmomenta {kp(i),k′\ni}. Hence, when two of the momenta\ncoincide (e.g. ki=kj,i/ne}ationslash=j)there are two coinciding\nraws/columns in the matrix and the Slater determinant\nequals zero. The result is that Eq. (52) factorizes in\nN! relabeling copies, associated with N! nodal cells like\nin 1+1D, of the following simple density matrix describ-\ning distinguishable and localized particles in momentum\nspace,\nρF(K,K′;τ) =N/productdisplay\nk1/ne}ationslash=k2/ne}ationslash=···/ne}ationslash=kN2πδ(ki−k′\ni)e−|ki|2τ\n2/planckover2pi1M.(54)\nThishasthestructureofaBoltzmannianpartitionsum\nof a system subjected to steric constraints: it is actually\nthe solution of the Ceperley path integral for the Fermi\ngas in momentum space! Let us apply periodic bound-\nary conditions so that on every time slice of the Ceperley\npath integral we find a grid of allowed momentum states\nki= (2π/L)(ki,x,ki,y,ki,z,....) where the ki,α’s are the\nusual integers (see Fig. 5b). We learn directly from Eq.\n(54) that we can ascribe a distinguishable particle with\neverymomentum cell, with aworldlinethat just closeson\nitself along the time direction: single particle momentum\nconservation prohibits anything but the one cycles (see\nFig. 5c). Inaddition, wefindthatthereachjustcollapses\nto the Pauli hypersurface, just as in one dimensions: per\nmomentum space cell either zero or one worldline can be\npresent. These worldlines are given by Eq. (51): since\nwe are living in exact quantum number space these just\ngostraight up along the time direction, since there areno\nquantum fluctuations: these are actually classical parti-\ncles living in momentum space. We do have to remember\nthat these world ’rods’ carry a fugacity set by a potential\n|k|2τ\n/planckover2pi1M. Henceforth, we have a problem of an ensemble of\nclassical hard core particles that live on a lattice of ’cells’\nin momentum space where every cell can either contain\none or no particle, with an overall harmonic potential\nenvelope centered at k= 0: this is literally the problem\nof cold atoms living in a harmonic trap, subjected to an\ninfinite strong optical lattice potential, tuned such that\nthey form a Mott-insulating state. The ground state is\nsimple: occupy the cells starting at k= 0, while the par-\nticles are put into cells at increasing trap potential until\nthe trap is filled up with the available particles. At zero\ntemperature there are no fluctuations and when one ex-\nceeds the chemical potential the cells remain empty, and\nthere is a sharp ( d−1)-dimensional interface between the\noccupied- and unoccupied trap states. This is of course\nthe way we explain the Fermi-gas to our undergraduate\nstudents. It invokes an odd metaphor that howeverturns\nout to express an exact identification since we learned to\nhandle the Ceperley path integral!\nHaving a statistical physics interpretation, can we now\naddress the questions posed in section II? First, what is\nthe order parameter of the Fermi-liquid? The answer\nis: the same order parameter that governs the Mott-21\ninsulator. This order parameter is well understood11,\nalthough it is of an unconventional kind: it is the ’stay\nat home’ emergent U(1) gauge symmetry62, stating that\nat every site and at all times there is precisely one par-\nticle per site. The particle number is locally conserved\nand henceforthalocal U(1)symmetry emerges. The ’dis-\norder operators’ that govern the finite temperature fate\nof the order parameter are just substitutional-interstitial\ndefects: there is a finite thermal probability to excite a\nparticle out of the trap, and the presence of the vacancies\ndestroysthe U(1)gaugesymmetry. Sincethedisorderop-\nerators are zero-dimensional particles regardless the di-\nmensionality of momentum space, thermal melting of the\nMott-insulator occurs at any finite temperature regard-\nless dimensionality. In the next section we will discuss\nhow this might relate to the ”holography” introduced in\nsection II.\nWe repeat, this is just a rephrasing of the standard\nFermi gas wisdoms in a non-standard language. The\nstrange powers of the Ceperley path integral become\nmore obvious when interactions are switched on. In the\npresence of the interactions single-particle momentum is\nno longer conserved, and this means that the worldlines\nof the Ceperley particles in momentum states get quan-\ntized: it is analogousto making the optical potential bar-\nriers finite in the cold gas Mott-insulator with the effect\nthat the particles acquire a finite tunneling rate between\nthe potential wells. One gets directly a hint regarding\nthe stability of the Fermi-liquid: Mott-insulators are sta-\nble states that need a rather large tunneling rate to get\ndestroyed. But the story is quite a bit more interesting\nthan that, as can be easily argued from the knowledge on\nthe canonical side. Let’s consider first what would hap-\npen in a literal cold atom Mott insulator when we start\nto quantize the atoms. Deep inside the trap motions are\nonly possibly by doubly occupying the nodal cells and\ngiven that in the non-interacting limit the ’Hubbard U’\nis infinite (expressing the Pauli surface) such processes\nare strongly suppressed. In the bulk of the trap the Mott\nstate would be very robust. However, at the boundary\none can make cheap particle-hole excitations, and at any\nfinitetthe interface would no longer be infinitely sharp\nonthemicroscopicscale: thedensityprofilewouldchange\nsmoothly. Eventually one would meet the ’wedding cake’\nsituation where the bulk is still Mott-insulating while the\ninterfacewouldturnintoasuperfluid(weliveinabosonic\nworld). How different is the Fermi-liquid! We know how\nit behaves from the canonical side. The single-fermion\nself-energy tells us directly about the fate of the k-space\nMott insulator. We learn that the time required to loose\ninformation on single-particle momentum is just given\nby the imaginary part of the self-energy and that be-\nhaves as151/τk∼(k−kF)2, Henceforth, it diverges at\nthe interface while it get shorter moving into the bulk.\nIn the Ceperley bosonic language the Fermi-liquid is like\na grilled marshmallow: It has a ’crispy’, solid Mott insu-\nlating crust while it becomes increasingly fluid when one\nmoves inside!More precisely, the worldlines near the interface are\nfluctuating at short times, since we know that the mo-\nmentum distribution of the bare electrons do smear\naround the Fermi-momentum - they do ’spill out of the\ntrap’. However, the effect of integrating out these micro-\nscopic fluctuations is to renormalize the ’optical lattice\npotential’upwards. Thishastobethecasebecauseinthe\nscaling limit the renormalized worldlines represent the\nquasiparticles and since they produce a perfectly sharp\ninterface (i.e. unit jump in the quasiparticle nk), the\nMottness has to be perfect. This can only be caused by\ninfinitely high effective potential barriers. This physics\nis of course coming from the modifications happening in\nthe reach when interactions are turned on. The phase\nspace restrictions giving rise to Σ′′∼ω2are rooted in\nFermi-Dirac statistics and all the statistical effects are\ncoded in the reach when dealing with the Ceperley for-\nmalism. These aspects can be computed by controlled\nperturbation theory and in a future publication they will\nbe analyzed in detail.\nVIII. THE FERMI-LIQUID IN REAL SPACE:\nHOLOGRAPHIC DUALITY.\nWe showed in the previous section that at least for\nthe Fermi gas the momentum space Ceperley path inte-\ngral becomes a quite simple affair. Momentum space is a\nnatural place to be when one is dealing with a quantum\ngas or -liquid, but dealing with a bosonic- or statisti-\ncal physics systems one invariably runs into the general\nnotion of duality57,63. Dealing with conjugate degrees of\nfreedom, like momentum and position or phase and num-\nber, one can reformulate the manifestly local order on\none ’side’ into some non-local topological order parame-\nter on the dual side. An elementary example is the Bose-\nEinstein condensate. In the language of the previous sec-\ntion, one can either form a ’black hole’ in the momentum\nspace ’trap’, by putting all bosons in the k= 0 ’optical\nlattice cell’. But one can also view it in real space, to dis-\ncover the lively world of Section III where the local order\nin momentum space translates into a global, topologi-\ncal description revolving around the infinite windings of\nworldlines around the time direction. Such duality struc-\ntures are ubiquitous in Bolzmannian systems, and they\nare at the heart of our complete understanding of such\nsystems: when one has a complete duality ’map’ one un-\nderstands the system from all possible sides and there is\nno room for surprises. For instance, when one is dealing\nwith a strongly interacting system like4He one prefers\nthe real space side because it is much easier to track the\neffects of the interactions39. Also in the strongly inter-\nacting fermion systems one expects that one is better\noff on the real space side. In this concluding section we\nwill address the issue of the dual, real space description\nof the Fermi-liquid in the Ceperley path integral formal-\nism. This real space side is remarkably complex: de-\nspite an intense effort even Ceperley and coworkers23got22\nstuck to the degree that they even did not manage to get\nthings working by brute computer force. They ran into a\nrather mysterious ’reference point glassification’ problem\nin their quantum Monte Carlo simulations, likely related\ntoacontrived’energylandscape’problemassociatedwith\nthe workings of the reach.\nThis is a fascinating problem: there has to be a simple,\ndual real space description of the Fermi gas. The obvious\ndifficulty as compared to straightforward bosonic duality\nis the presence of the reach. One has to dualize not only\nthe ’life of the worldlines’ but also the constraints coding\nfor the Fermi-Dirac statistics. Topology is at the heart of\nduality constructions and in this regard Ceperley20, and\nmore recently Mitas29, have obtained some remarkably\ndeep results, which will be discussed at length in the first\nsubsection: the topology of the reach of the Fermi-liquid\nind≥2 is such that the reach is open for all cycles of\nCeperley worldlines based on even permutations or triple\nexchange. Henceforth, there is no topological principle\nthat prevents infinitely long worldlines to occur and in\nsubsection B we will argue that the zero temperature or-\nder ofthe Fermi-liquid has to be a Bosecondensateofthe\n’Ceperley particles’. This is conjectural but if it proves\nto be correct the Fermi-liquid holography we discussed\nin section II acquires a fascinating meaning: the scal-\ning limit thermodynamics of the Fermi-gas in any spatial\ndimension d >1is governed entirely by the statistical\nphysics associated with distributing the Ceperley world-\nlines over the cycles associated with even permutations,\nand this effective partition sum is indistinguishable from\nthe partition sum enumerating the cycles of a soft-core\nboson system in one space dimension.\nA. The topology of the Fermi-liquid nodal surface.\nTo decipher the structure of constraints as needed for\nthe real space Ceperley path integral one has to find out\nwhere the zero’s of the real space density matrix are.\nBy continuation, these should be in qualitative regards\nthe same in the Fermi-liquid as in the Fermi gas, and in\nthe latter case we have an expression of the full density\nmatrix in closed form,\nρF(R0,R;τ) = (4πλτ)−dN/2\n×detexp/bracketleftbigg\n−(ri−rj0)2\n4λτ/bracketrightbigg\n,(55)\nwhereλ=/planckover2pi12/(2M). Henceforth, one needs to find out\nthe zero’s of this quantity for all R0,Rin the imaginary\ntime interval 0 < τ < β . In real space, this is not an\neasy task. Part of the trouble is that at low temperature\nthe zero’s of the determinant depend on all coordinates\nat the same time. Only in the high temperature limit\n(τ→0) the nodal surface of the density matrix becomes\nextremely simple20. To see this, define first a permuta-\ntion cell ∆P(R0) as the set of points closer to PR0thanto any other P′R0. Obviously, the configuration space is\ndivided into N! permutation cells which are convex poly-\nhedra bounded by hyperplanes, R·(PR0−P′R0) = 0.\nThe density matrix is simply a sum over all permuta-\ntions and for R∈∆P(R0) and sufficiently high tem-\nperatures this sum is completely dominated by the term\n(−1)pexp[−(R−PR0)2/(4λτ)] since all the other terms\nareexponentiallydamped relativeto it. Therefore, in the\nhightemperaturelimit, ρF(R0,R;τ) willhavethesignof\nPinside of ∆ P(R0) and the nodal hypersurface is simply\ngiven by the common faces shared by permutation cells\nof different parities.\nThe reach acts both in a local way, much in the same\nway as we learned in the (1+1)-dimensional case as a\nspecial ’steric hindrance’ structure having to do with\nentropic interactions, etcetera. However, it also carries\nglobal, topological properties and these are now well un-\nderstood because of some remarkable results by Mitas29,\nwho managed to proof the ’two nodal cell’ (or ’nodal\ndomain’) property of the higher dimensional Fermi-gas\nreach20. The topology of the nodal surface is associated\nwith the structure of cycles as discussed in section III\nbut now for the Ceperley path integral. The latter can\nbe written as\nZ=/summationdisplay\nPe/integraldisplay\ndR˜ρD(R,PeR;β), (56)\nwherePerefers to even permutations, while ˜ ρDrefers to\nthe density matrix of distinguishable particles that are\nhowever still subjected to the reach constraints. As in\nthe case of the Feynman path integral, this sum over\neven permutations can be recasted in a sum over cy-\ncles associated with all possible ways one can reconnect\nthe worldlines at the temporal boundary, of course lim-\niting this sum to those cycles that are associated with\neven permutations. We learned in section IV that for\nfree wordlines even permutations translate into the su-\npersymmetric quantum gas. But the Ceperley particles\nare not at all free, and the topology of the nodal surface\ntells us about global restrictions on the cycles that can\ncontribute to Eq. (56).\nIt is immediately clear that the counting of cycles is\ngovernedbytopology: tofind outhowtoreconnectword-\nlines arriving at the temporal boundary from the imag-\ninary time past, to worldlines that depart to the imagi-\nnary time future one needs obviously globaldata. This\nglobal information residing in the reach is just the divi-\nsion of the reach in nodal cells we already encountered in\nthe (1+1)-dimensional context and the momentum space\nFermi gas. There we found that the space of all permuta-\ntions got divided in N! nodal cells, with the ramification\nthat the sum in Eq. (56) is actually reduced to one cy-\ncles. Mitashasdeliveredtheproofthatin d≥2thereach\ncarries a two nodal cell topology, implying that all cycles\nbased on even permutations lie within the reach. Since\nonly this topological property of the reach can impose\nthat certain cycles have to rigorously disappear from the23\ncycle sum, this does imply that all cycles based on even\npermutationscancontributetothepartitionsum, includ-\ning the cycles containing macroscopic winding numbers.\nHenceforth, the Ceperley worldlines can Bose condense\nin principle and it is now just matter of finding out what\nthe distributions of the winding numbers are as function\nof temperature. This is what really matters for the main\nline of this story. Finding out the the way that Mitas\ndetermined the two-cell property is quite interesting and\nwe will sketch it here for those who are interested. When\nyou just want to understand the big picture, you might\nwant to skip the remainder of this subsection.\nQuite recently Mitas29proved a conjecture due to\nCeperley20, stating that the reach of the higher dimen-\nsionalFermigasis’maximal’inthesensethat, foragiven\nR0andτ, the nodalsurfaceof ρF(R0,R;τ)separatesthe\nconfigurationspace in just two nodal cells, corresponding\nwithρFbeing positive- and negative respectively. This\nis a quite remarkable property: for every pair RandR′\nin the same domain (lets say ρF>0), one can change R\nintoR′without encountering a zero crossing of ρF.\nThe easy way to prove this property goes as follows29.\nFirst, it can be demonstrated20that once there are only\ntwo nodal cells at some initial τ0than this property has\nto hold for any τ > τ 0. This follows straightforwardly\nfrom the imaginary time Bloch equation for the density\nmatrix,\n−∂ρ(R,R′;τ)\n∂τ=Hρ(R,R′;τ) (57)\nwith initial condition,\nρ(R,R′;0) = det/bracketleftbig\nδ(ri−r′\nj)/bracketrightbig\n(58)\nand the Bloch equation is a linear equation. This is a\nvery powerful result because it gives away that the two-\ncell property ’descents for the ultraviolet’: one has just\nto prove it at an arbitrary short imaginary time which is\nthe same as arbitrary high temperature. Ignoring Planck\nscale uncertainties, etcetera, the form Eq. (55) has to\nbecome asymptotically exact for sufficiently small β, also\nin the presence of arbitrary interactions as long as they\nare not UV-singular! As we already noticed, this high\ntemperature limit is rather tractable.\nWe now need to realize that we still have to take into\naccount the ’remnant’ of quantum statistics in the form\nof even permutations. Every even permutation can be\nwritten as a succession of exchanges of three particles\ni,j,k→j,k,ibecause these amount to two particle ex-\nchanges. When such an exchange does not cross a node\n(i.e. it resides inside the reach) the three particles are\ncalled ’connected’. By successions of three particle ex-\nchanges one can build up clusters of connected particles.\nAll one has now to demonstrate is that a point Rtexists\nwhereallparticles are connected in a single cluster, be-\ncause this complete set of even permutations exhaust allpermutations for a cell of one sign, because the odd per-\nmutations necessarily change the sign. One now needs a\nsecond property called tiling stating that when the parti-\nclesareconnectedforthe special point Rtthis hasalsoto\nbe the case for all points in the cell. And tiling is proved\nby Ceperley for non-degenerate ground states and also\nfor finite temperature. Actually due to the linearity of\nthe Bloch equation, its fixed node solution is unique, and\nthe tiling propertyin the high temperature limit will lead\nto the same property at any lower temperature.\nBefore we prove that the above holds for the high tem-\nperature limit density matrix, let us just dwell for a sec-\nond on what this means for the winding properties of the\nconstrainedpathintegral. Theevenpermutationrequire-\nment means that, as for the standard worldline pathinte-\ngrals, we have to connect the worldlines with each other\nat the temporal boundary, but now we have to take care\nthat we single out those cycles corresponding with even\n(or three particle) exchanges. The ’maximal reach’ just\nmeans that cycles containing worldlines that wind an ar-\nbitrary large number of times around the time axis never\nencounter a node ! As noted before by Ceperley, this\nhas the peculiar implication that in some non-obvious\nway the Fermi-gas has to know about Bose condensa-\ntion. Since nodal constraints do allow for infinite wind-\nings there seems to be no ’force in the universe’ that\ncan forbid these infinite windings to happen and since\nthe Cepereley path integral is probabilistic, when these\ninfinite windings happen one has to accept it as Bose\ncondensation. We will come back to this theme in a mo-\nment.\nFollowing Mitas, one can now prove the two cell prop-\nerty of the high temperature limit using an inductive\nmethod. Assume that all Nparticles in the low βlimit\nat a fixed R0are connected in one cluster, to see what\nhappens when an additional N+1 particle is added. Sin-\ngle out two other particles N−1,Nand move these\nthree particles away from the rest without crossing a\nnode. Now we can profit from the fact that in the low β\nlimit the density becomes factorizable: the determinant\nfactors into a product of the determinant of the three\nspecial particles and the determinant of the rest. It is\neasy to show that the three particle determinant has the\ntwo cell property, proving that the N+1’s particle is in\nthe cluster of Nparticles. Since this is true for any N,\nthe starting assumption that all particles in the cluster\nis hereby proven.\nFor free fermions, Mitas also proved the two nodal cell\nproperty for non-degenerate ground states using a simi-\nlar induction procedure. The trick is to choose a special\npointRtin theconfigurationspace, atwhichonecaneas-\nily show how all the particles are connected into a single\ncluster. Once proven for this single point, tiling ensures\nthat the same is true for the entire nodal cell. Mitas\naligned the particles into lines and planes, thus forming\nsome square lattice in the real space. This way the num-\nber of arguments of the wave functions is reduced and\nmore importantly, the higher dimensional wave functions24\ncan be factorized into products of sine functions and the\none dimensional wave functions, which are much easier\nto deal with than their higher dimensional counterparts.\nOne distinct property of the 1 dimensional wave func-\ntions is that they are invariant under cyclic exchanges of\nodd numbers of particles, namely for Nodd,\nCx\n+1Ψ1D(1,···,N) = Ψ1D(1,···,N),(59)\nwhereCx\n+1represents the action to move every particle\nby one site in the + xdirection, with the last particle\nmoved to the position of the first one, that is 1 →2,2→\n3,···,N→1.\nConsider for example the non-degenerate ground state\nof 5 particles in 2 dimensions. For this state, it becomes\nstraightforward to show that each group of the 3 near\nneighbors living in the real space square lattice are con-\nnectedbyproductsoffourtripletexchanges,whichareall\nperformedalongthe1dimensionallines. Proventhis, one\ncan proceed as in the high temperature limit, by adding\nmore particles to the lattice. And these newly added\nparticles can be shown to be connected to the original\nparticles’ cluster by the similar method used for 5 parti-\ncles. Theonlydifferenceisthatnowoneneedstoconsider\nthe whole line of particles, on which the new particle is\nadded, and thus a sequence of four cyclic exchanges, in-\nstead of the special triplet exchanges are required. Since\nfor non-degenerate ground states, there are odd number\nof particles on each line, cyclic exchanges will not pro-\nduce extra minus signs, thus leading to the same result\nas triplet exchanges. This completes the proof for 2 di-\nmensions, and the high dimensional cases are essentially\nthe same.\nHowever, winding is a topologicalpropertythat should\nbe independent of representation. In the long time\nβ→ ∞limit the path integral contains the same in-\nformation as the ground state wave function, and for the\nFermi-gas we can actually easily determine the winding\nproperties inside one of the nodal cells using the random\npermutation theory. This demonstratesthat at zerotem-\nperature the Fermi-gas is indeed precisely equivalent to\nthe Bose gas, within the nodal cell.\nB. There is only room for winding at the bottom.\nThe conclusion of the previous subsection is that the\nCeperleywordlinescaninprinciplebecomeinfinitelylong\nbecause the topology of the reach allows them to become\nmacroscopic. Does this mean that the zero-temperature\norder parameter of the Fermi-liquid is just an algebraic\nbose condensate ofCeperley wordlinescharacterizedby a\ndomination of the partition sum by macroscopic cycles?\nThe two nodal cell topological property is a necessary\nbut insufficient condition for this to be true. However,\nthere are more reasons to believe that the Fermi-liquid\nhas to be of this kind.\nAs we discussed at length in section II, the zero- and\nfinite temperature Fermi-liquid are separated by a phasetransition and it appears that only the winding sector\nof the Ceperley path integral can be responsible for this\ntransition. The argumentis simple and general. With re-\ngard to ordering dynamics the real space Ceperley path\nintegral is governed by Boltzmannian principle and let\nus find out what ’substance’ is available to form an or-\nder parameter. The nodal surface in isolation cannot\nbe responsible, since it is an immaterial object that just\ngoverns the behavior of the ’Ceperley particles’ . Hence-\nforth, whatever its (singular) properties, these have to be\nreflected in the behavior of the matter. In principle one\ncan imagine subtle topological changes occurring in the\nnodalsurfacebutintheprevioussubsectionwefoundthis\nnot to be the case in the Fermi-gas. Henceforth, search-\ning for the thermodynamic singularity we should keep\nour eyes on the worldlines and these should be subjected\nto the generalities associated with bosonic matter. One\nsourceofthermodynamicsingularityisthatthe systemof\nbosons breaks the translational- and/or rotational sym-\nmetry of space, forming a crystal or some liquid crystal.\nAlthough the one dimensional Fermi-gas is such a crys-\ntal in disguise, it is impossible to hide a (partial) crys-\ntallization in higher dimensions: the higher dimensional\nFermi-liquid is undoubtedly a true liquid. The worldlines\nhaveto be delocalized, but dealingwith indistinguishable\nparticles, being bosons or the ’even permuting’ Ceperley\nparticles, one has to account for an extra set of degrees\nof freedom: the reconnections at the temporal boundary.\nFrom astatistical physics perspective, Bosecondensation\nappears as an order out of disorder phenomenon. Lower-\ning temperature has the net effect of increasing the ’con-\nfigurational entropy’ associated with all possible ways of\nreconnecting worldlines, or either the appearances of cy-\ncles characterized by different windings. Worldlines get\nlonger and thereby the length over which they can me-\nander increases, and this in turn increases effectively the\nfugacity of long cycles. The more cycles can contribute,\nthe larger the ’configurational entropy’ associated with\nthe cycles and this gain in space time ’configurational\nentropy’ (physically the decrease of quantum zero point\nenergy) causes eventually a flat distribution of the wind-\ning configurations, and in the Bose system this sets in at\na sudden phase transition. Since all particles ’are part\nof the same wordline’ the Bose condensate is macroscop-\nically coherent. We learned that the reach allows the\nCeperley particles to form infinite windings. We learn\nfrom the Bose condensate that at zero temperature only\ncrystallization can prohibit the ’reconnection entropy’ to\ntake over, because the thermal de Broglie wavelength di-\nverges. Henceforth, theredoesnotseemtobeanyfeature\nof the reach that can prohibit this to happen as well to\nthe Ceperley worldlines at zero temperature.\nThere is a quite direct argument to support this view\nwhich was put forward by Ceperley some time ago20,21.\nAs we already emphasized a number of times, on the\ncanonical side the Fermi-liquid order manifests itself\nthrough the jump in the momentum distribution. Let\nus now turn to the zero temperature single particle den-25\nsity matrix,\nn(r) =/integraldisplay\ndRρ(r1,r2,···rN;r1+r,r2,···rN;∞)\n=/integraldisplay\ndkeik·rnk. (60)\nIn the boson condensate nB(r)→constant revealing\nthe off-diagonal long-range order which is equivalent to\nthe domination of infinite cycles. In the Femi-liquid on\nthe other hand,\nnF(r)≃1\n(kFr)d/2Jd/2(kFr). (61)\nThe oscillations governed by the Bessel function\nJd/2(kFr) can be easily traced back to the size of the\nnodal pocket as discussed in a moment. However, the\nenvelope function ( kFr)−d/2just behaves like the one\nparticle density matrix of a Bose condensate showing off-\ndiagonal long range order, like in the interacting Bose\nsystem in 1+1D at zero temperature. Relating this to\nthe real space Ceperley path integral, this signals the\npresence of infinite cycles formed from Ceperley world\nlines.\nBelieving the arguments in the above, indicating that\nthe Fermi liquid ’order’ in the real space language is as-\nsociated with the statistics of the windings, we are still\nfacing the problem why this behaves as a system of soft\ncore, interacting bosons in 1+1D: as we discussed in sec-\ntion II, we learn from the canonical description that the\n’reconnection entropy’ of the Fermi-liquid in arbitrary\ndimensions at finite temperature is counted in the same\nway as that of a soft-core (i.e. winding) Bose system in\n1+1 dimensions. This is surely not due to some remnant\nof the ’supersymmetric’ quantum statistics coming from\nthe even windings because we learned in Section IV that\nthis is quite similar tonormalBosestatistics. We suspect\nthat this ”holography”hasits originin anon-topological,\nmicroscopic feature of the reach.\nIt seems impossible to explain such a ’dimensional re-\nduction’ In terms of local particle-particle interactions\nand in one or the other way the reach apparently acts\nlikely as a many particle interaction. An indication that\nthis mysterious ’interaction’ on microscopic time and\nlength scales follows from the thermodynamics of the\nSommerfeld gas. We observe that temperature enters\nthe Eq.’s (8)-(10) always in the ratio T/EF. Tempera-\nture has the role of limiting the length of the imaginary\ntime axis and this information enters the free energy via\nsome effective return probabilities. These in turn relate\nin a simple way with the Fermi-energy because otherwise\nthe free energy would not be a simple algebraic function\nofT/EF, and the meaning of the Fermi-energy is quite\nclear. In analogy with the one dimensional case, the\nFermi-energy is the quantum kinetic energy associated\nwith the fact that the free volume in which a Ceperley\nparticle moves is restricted by the reach to be of order ofthe inter-particle distance. The reach is for a given R0\nandτjust a (Nd−1)-dimensional manifold embedded\ninNddimensional space. It can be visualized by con-\nstructing d dimensional cuts through configuration space\nobtained by fixing N−1 particle coordinates and moving\none particle around, trackingthe sign changesofthe den-\nsity matrix, as in Fig. 2. For the Fermi-gas this is obvi-\nously a smooth, non-fractal manifold. We also know that\nthe Pauli hypersurface lies on the nodal surface. Given\nthe typical inter-particle distance rs, one expects that\nevery particle moves around in a free volume with a typ-\nical linear dimension ≃rs, enclosed by the impenetrable\nnodal surface. This is the well known ’nodal pocket’; fol-\nlowing Ceperley20,21, one can isolate it by analyzing the\nsign changes of the one particle density matrix. But one\nmight as well consider the meaning of the Fermi-energy.\nEFsets the ultraviolet scale of the higher dimensional\nFermi gas in much the same way as it does in 1+1D.\nThere we saw that it corresponds to the time where the\nconstraints become noticeable in the form of a confine-\nmenteffect. Thenodalpocketsinhigherdimensionshave\nthe same confining effect as the ’Pauli pockets’ in one di-\nmension: the nodal surface ’hangs’ over the Pauli surface\nforming ’cages’ in which the worldlines of individual par-\nticles are locked up (see Fig 3). The derivation Eq.’s\n(45-47) generalizes straightforwardly to any dimension\nafter adjusting volume factors. The inter-particle den-\nsityrs= (V/N)1/d=n1/dwherenis the density. It\nfollows for the mean square displacement l2\nc(τ), the col-\nlision time τcand the characteristic energy scale,\nl2\nc(τ) =/an}bracketle{t(φi(τ)−φi(0))2/an}bracketri}ht=d/planckover2pi1\nMτ,(62a)\nτc≃1\ndM\n/planckover2pi1n−2/d, (62b)\nEF=/planckover2pi1\nτc=d/planckover2pi12\n2mn2/d, (62c)\nand one recognizes immediately that the characteristic\nenergy scale coincides with the Fermi-energy.\nLetusnowreturntothequestionofwhatisresponsible\nfor the holography. The Fermi-energy emerges at micro-\nscopic times, and has a simple interpretation in terms of\nthemicroscopicgeometryofthereach(thenodalpocket).\nCombining this observation with the fact that the free\nenergy of the Fermi-gas at finite temperature is a sim-\nple algebraic function of the ratio T/EFsuggests that\nthe winding statistics responsible for the thermodynam-\nics is in some quite direct way influenced by the nodal\npocket property of the reach. It is intriguing that the\ndimensionality of the nodal pocket is the same as the di-\nmensionality of space, suggesting that it has in principle\nthe capacity to render the winding statistics to become\nindependent of dimensionality, but we have not managed\nyet to find out how this works in detail. The reader is\ncordially invited to give this fascinating problem a deep\nthought.26\nAPPENDIX\nWindings vs. Loops\nThe partition function in the canonical ensemble for\nsuch a system, given in terms of the density matrix as\nin (12) and (14), with V= 0, factorizes into a product\nof Gaussian convolution integrals64. The convolutions\nthen extend the path lengths in the imaginarytime direc-\ntion of paths where the end-points are permuted among\neach other, as seen in figure 1. In the extended zone\nscheme such paths satisfy periodic boundary conditions,\nregardlessofwhethertheparticlesarebosonsorfermions,\nand have path lengths which are integer multiples of the\nlength of the temporal direction (inverse temperature),\nw/planckover2pi1β, where the integer is interpreted as a winding num-\nber. The bose/ferminatureoftheparticlesisstillpresent\nin an overall permutation sign in summing over all per-\nmutations which generate the windings. The contribu-\ntion of each path of winding number wto the partition\nfunction of Nparticles is the same as that ofa single par-\nticle (which can only have winding number one) living at\nwtimes the inverse temperature\n∆Z(N),w\n0=Z0(wβ). (A-1)\nWe will now show that the partition function of a free\nsystem ofNparticles at inverse temperature βcan be\nwritten in terms of a sum over windings subject to a cer-\ntain constraint, with the partition function of a single\nparticle appearing in the product. There are altogether\nN! permutations of the the Nparticles. When decom-\nposing these into cycles we need to keep track of the\nnumber of cycles of length win a given permutation. Let\nus denote, for a given permutation, the number of cycles\nin that permutation of length w, byCw. Then with each\npermutation is associated a series of numbers Cw, with\nw= 1,...,N. Then, a sum over all permutations can be\nrewritten as a sum over all integers assigned to the var-\niousCw, subject of course to an overall constraint, this\nconstraint being that the total length of all cycles taken\ntogether must be N. We write this constraint as\nN/summationdisplay\nw=1wCw=N (A-2)\nsince the length of cycles in the class Cwisw. Among\ntheN! permutations, the number of which have a given\nset of values of Cwhas been given already in (17), which\nwe reproduce here\nM(C1,...,CN) =N!/producttextN\nw=1Cw!wCw.(A-3)\nIf we sum this quantity over all configurations of C′\nws\nsuch that/summationtextN\nw=1=L, for any 1 ≤L≤N, at fixedN,\n/summationdisplay\n{C1,···CN}PN\nw=1Cw=LM(C1,...,CN) =S1(N,L) (A-4)whereS1(N,L) is the Stirling number (unsigned) of the\nfirst kind, and counts the number of permutationsamong\nallN! permutations that consist of Lloops, for any num-\nber of loops 1 ≤L≤N. Since there are N! permuta-\ntions, summing this quantity over all allowed loop num-\nbers must reproduce\nN/summationdisplay\nL=1S1(N,L) =N!. (A-5)\nNaturally, summing over all cycles of various lengths, we\nhave the total number of cycles\nN/summationdisplay\nw=1Cw= Total number of cycles. (A-6)\nIn writing the partition function in the canonical en-\nsemble in terms of cycles we encounter a sum of products\nof the form\n/summationdisplay\nP(±1)PN/productdisplay\nw=1, (A-7)\nwithPdenoting a permutation. Note here that the first\nsum over all permutations fixes the values of the Cw,\nsubject to the constraint on total length/summationtext\nwwCw=N.\nThis sum of products can be rewritten as a sum over all\nvalues of the various Cw, again subject to the constraint\non total length\n/summationdisplay\n{C1···CN}P\nwwCw=Nǫw(C1,···,CN)N/productdisplay\nw=1,(A-8)\nwhereǫw(C1,···,CN) = (±1)P\nw(1+w)Cwis the parity\nof a given permutation. With these we can write the\ncanonical partition function as26\nZ(N)\n0(β) =1\nN!/summationdisplay\n{C1···CN}P\nwwCw=NMǫwN/productdisplay\nw=1[Z0(wβ)]Cw,\n(A-9)\n(dropping for clarity the C1,···,CNonMandǫw). Af-\nter some starightforward reorganization this becomes\nZ(N)\n0(β) =/summationdisplay\n{C1···CN}P\nwwCw=NN/productdisplay\nw=11\nCw!/bracketleftbigg\n(±1)w−1Z0(wβ)\nw/bracketrightbiggCw\n.\n(A-10)\nOne may readily check that the partition function above\nsatisfies the recursion relation\nZ(N)\n0(β) =1\nNN/summationdisplay\nn=1(±1)n−1Z(1)\n0(nβ)Z(N−n)\n0(β),(A-11)\nsubject to the boundary condition Z(0)\n0= 1, allowing one\nto generate the N-particle partition functions from the\nbottom up.27\nIn writing the partition function for an N-particle sys-\ntem as it appears in (A-9) and (A-10), we have taken\nthe viewpoint that we sum over all windings. Yet, it is\nevident in equations (A-4) to (A-6) that we might also\nchoose to view the same partition function as a sum over\nall possible numbers of loops. In other words, we may\nidentify the sums as follows\n/summationdisplay\n{C1···CN}P\nwwCw=N=N/summationdisplay\nR=1/summationdisplay\n{C1···CN}P\nwCw=RP\nwwCw=N(A-12)\nThe second sum on the right hand side is subject to two\nconstraints. But we will see shortly that when we con-\nstruct the grand canonical partition function, the con-\nstraint/summationtext\nwwCw=Nwill be removed, since we sum\nover all canonical partition functions for the various par-\nticle numbers, leaving only the one enforcing the number\nof loops/summationtext\nwCw=R.\nDefine the partition function for N-particles and R-\nloops as\n˜Z(N)\n0,R(β) =/summationdisplay\n{C1···CN}P\nwCw=RP\nwwCw=NMN/productdisplay\nw=1/bracketleftig\nZ(1)\n0(wβ)/bracketrightigCw.(A-13)\nNote that no sign appears here, and so this gives the\nsame result whether we are studying bosons or fermions.\nWe can now perform the winding to loop regrouping on\n(A-9), to see that\nZ(N)\n0(β) =(±1)N\nN!N/summationdisplay\nR=1(±1)R˜Z(N)\n0,R(β).(A-14)\nWe note that by going to this view, we have greatly re-\nduced the number of alternating signs in the sums that\nwe have to deal with, from N! signs in (A-9) to Nsings\nin (A-14). For those interested in numerical simulations\nthis from of the partition function might offer a substan-\ntial improvement in the rate of convergence.\nThe analogue of the recursion relation (A-11) is then\ncycle form is\n˜Z(N)\n0,R(β) =1\nNN−1/summationdisplay\nn=1Z0(nβ)˜Z(N−n)\n0,R−1(β) (R≥2),\n˜Z(N)\n0,1(β) =1\nNZ0(Nβ). (A-15)We are now going to use the canonical partition func-\ntion to construct the grand canonical one, where the par-\nticle number is no longer a fixed, but a fluctuating quan-\ntity. The grand canonical partition function is the sum\nover all canonical partition functions of different particle\nnumbers, each weighted by a Boltzmann for Nparticles,\nand at chemical potential µ,\nZG,0≡∞/summationdisplay\nN=0Z(N)\n0(β)eβµN,(A-16)\nwith the free partition function satisfying\nZG,0(wβ) =ZG,0(β)\nwd\n2. (A-17)\nUsing the form of the canonical partition function in\n(A-10), we have then for the grand canonical partition\nfunction, after a rearrangement, the following\nZG,0(β) =∞/productdisplay\nw=1∞/summationdisplay\nCw=01\nCw!/bracketleftbigg\n(±1)w−1Z0(wβ)ewβµ\nw/bracketrightbiggCw\n,\n(A-18)\nwhere because we sum overall partical numbers, the con-\nstraint in (A-10) has been removed (as promised), i.e.\n∞/summationdisplay\nN=0/summationdisplay\n{C1···CN}P\nwwCw=N=∞/productdisplay\nw=1∞/summationdisplay\nCw=0. (A-19)\nIn terms of cycles, the removal of the constraint means\nwe have\n∞/summationdisplay\nN=0/summationdisplay\n{C1···CN}P\nwwCw=N=∞/summationdisplay\nR=1/summationdisplay\n{C1···CN}P\nwCw=R(A-20)\nIX. ACKNOWLEDGMENTS\nWeacknowledgeinsightfuldiscussionswithD.I. Ceper-\nley. This research was supported by the ”Nederlandse\norganisatie voor Wetenschappelijk Onderzoek” (NWO)\nand by the ”Stichting voor Fundamenteel Onderzoek der\nMaterie” (FOM).\n∗Electronic address: jan@lorentz.leidenuniv.nl\n†Electronic address: Krueger@lorentz.leidenuniv.nl\n‡Electronic address: she@lorentz.leidenuniv.nl\n§Electronic address: sadri@lorentz.leidenuniv.nl\n¶Electronic address: sergeimoscow@online.ru1see special issue on ’Quantum matter’, Science (2008).\n2J. Zaanen, Nature Physics 2, 138 (2006) and ref.’s therein.\n3M. Imada, A. Fujimori and Y. Tokura, Rev. Mod. Phys.\n70, 1039 (1998).\n4H. von Lohneysen et al., Rev. Mod. Phys. 79, 1015 (2007).28\n5F. Kagawa, K. Miyagawa and K. Kanoda, Nature 436, 534\n(2005).\n6S.V. Kravchenko, Rep. Prog. Phys. 67, 1 (2004).\n7J. Zaanen, Science 315, 1372 (2007); ibid., Nature 422,\n569 (2003).\n8A. Damascelli Z. Hussain and Z.X. Shen, Rev. Mod. Phys.\n75, 473 (2003).\n9J. Zaanen, Nature 440, 1118 (2006); ibid.. Nature 404,\n714 (2000); S.A. Kivelson et al., Rev. Mod. Phys. 75, 1201\n(2003).\n10P. Abbamonte et al., Nature Physics 1, 155 (2005).\n11e.g. E. Fradkin, Field theories of condensed matter sys-\ntems, Westview Press (1991) ; X.-G. Wen, Quantum Field\nTheory of Many Body Systems , Oxford Univ. Press (2004).\n12A.Y. Kitaev, Ann. Phys. (NY) 303, 2 (2003); see also Tai-\nKai Ng, Yi Zhou and Lei-Han Lang, Topological order in\nclassical system , preprint.\n13S. Sachdev, Quantum Phase Transitions , Cambridge Univ.\nPress, New York (1999).\n14M.TroyerandU.Wiese, Phys.Rev.Lett. 94, 17201(2005).\n15A.A.Abrikosov, L.P.GorkovandI.E.Dzyaloshinski, Meth-\nods of quantum field theory in statistical physics , Dover\nPublications, Revised edition (1975).\n16P. W. Anderson, Basic Notions of Condensed Matter\nPhysics, Benjamin/Cummings, Menlo Park (1984).\n17T. Giamarchi, Quantum physics in one dimension , Oxford\nScience Publications, New York (2004); J. Voit, Rep. Prog.\nPhys.57, 977 (1994).\n18D.L. Cox and A. Zawadowski, Exotic Kondo effects in met-\nals, Taylor & Francis, London (1999).\n19A. Georges et al., Rev. Mod. Phys. 68, 13 (1996).\n20D.M. Ceperley, J. Stat. Phys. 63, 1237 (1991).\n21D.M. Ceperley, Path Integral Monte Carlo Methods for\nFermions , inMonte Carlo and Molecular Dynamics of\nCondensed Matter Systems , Ed. K. Binder and G. Ciccotti,\nEditrice Compositori, Bologna, Italy, 1996.\n22D.M. Ceperley, Phys.Rev.Lett. 69, 331(1992); M.Bonin-\nsegni and D.M. Ceperley, Phys. Rev. Lett. 72, 2288 (1995);\nC. Pierleoni and D.M. Ceperley and B. Bernu and W.R.\nMagro, Phys. Rev. Lett. 73, 2145 (1994); W.R. Magro\nand D.M. Ceperley and C. Pierleoni and B. Bernu, Phys.\nRev. Lett. 76, 1240 (1996); B. Militzer and D.M. Ceperley,\nPhys. Rev. Lett. 85, 1890 (2000).\n23D.M. Ceperley, private communication .\n24F. Krueger and J. Zaanen, unpublished .\n25J. Zaanen, Nature 448, 1000 (2007) and ref.’s therein.\n26H. Kleinert, Path integrals in quantum mechanics, statis-\ntics, polymer physics, and financial markets , World Scien-\ntific, Singapore (2006).\n27J. Zaanen , Phys. Rev. Lett. 84, 753 (2000).\n28S.I. Mukhin, W. van Saarloos and J. Zaanen, Phys. Rev.\nB64, 115105 (2001).\n29L. Mitas, Phys. Rev. Lett. 96, 240402 (2006); L. Mitas,\narXiv:cond-mat/0605550v1.\n30J. van Wezel, J. van den Brink and J. Zaanen, Phys. Rev.\nLett.94, 230401 (2005).\n31M. Salmhofer and C. Honerkampf, Prog. Theor. Phys. 105,\n1 (2001).\n32J. Feldman, H. Knoerrer and E. Trubowitz, Fermionic\nfunctional integrals and the renormalization group , CMR\nmonographs series 16, Am. Math. Soc., Providence (2002).\n33N. Mannella at al., Nature 438, 474 (2005).34A.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966); P.C. Hohenberg, Phys. Rev. 158, 383 (1967); A.\nGelfert and W. Nolting, J. Phys. Cond. Mat. 13, R505\n(2001).\n35N.W. Ashcroft and N. D. Mermin, Solid State Physics ,\nSaunders College Publishing, Nw York (1976).\n36J.M. Luttinger, Phys. Rev. 119, 1153 (1960).\n37M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000).\n38S. Weinberg, The Quantum Theory of Fields Vol. II , Cam-\nbridge Univ. Press, Cambridge (2005).\n39D.M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).\n40D.M. Ceperley and E.L. Pollock, Phys. Rev. Lett. 56, 351\n(1986); E.L. Pollock and D.M. Ceperley, Phys. Rev. B 36,\n8343 (1987).\n41S. Bund and A. M. J. Schakel, Mod. Phys. Lett. B 13, 349\n(1999).\n42N.N. Bogoliubov, J. Phys. USSR 5, 71 (1947).\n43A.L. Fetter and J.D. Walecka, Quantum theory of many\nparticle systems , Dover Pub. (2003); J.O. Andersen, Rev.\nMod. Phys. 76, 559 (2004).\n44P. Phillips and D. Dalidovich, Science 302, 5643 (2003).\n45S.W. Golomb and P. Gaal, Advances in Applied Mathe-\nmatics20, 98 (1998).\n46J. Wess and J. Bagger, Supersymmetry and supergravity ,\nPrinceton, USA: Univ. Pr. (1992).\n47J.Terning, Modern supersymmetry: Dynamics and duality ,\nOxford, UK: Clarendon (2006).\n48S. Weinberg, The quantum theory of fields. Vol. 3: Super-\nsymmetry , Cambridge, UK: Univ. Pr. (2000).\n49M. Moshe and J. Zinn-Justin, Nucl. Phys. B 648, 131\n(2003).\n50J. R. Hiller, Supersymmetric two-dimensional QCD at fi-\nnite temperature , arXiv:hep-th/0608039.\n51C. Lucchesi, Thermal supersymmetry in thermal super-\nspace, arXiv:hep-ph/9808435.\n52K. Kratzert, Annals Phys. 308, 285 (2003).\n53W. H. Kye, S. K. Kang and J. K. Kim, Phys. Rev. D 46,\n1835 (1992).\n54J. Y. Ollitrault, Pramana 67, 899 (2006).\n55K. Yagi, T. Hatsuda and Y. Miake, Quark-Gluon Plasma:\nFrom Big Bang To Little Bang , Cambridge, UK: Univ. Pr.\n(2005).\n56R. P. Feynman, The principle of least action in quantum\nmechanics .\n57J.B. Kogut, Rev. Mod. Phys. 51, 659 (1979).\n58V.L. Pokrovsky and A.L. Talapov, Phys. Rev. Lett. 42, 65\n(1979).\n59W. Helfrich, Z. Naturforsch. A 33, 305 (1978); W. Helfrich\nand R.M. Servuss, Nuove Cimento Soc. Ital. Fis. 3D, 137\n(1984).\n60A.LutherandV.J.Emery, Phys.Rev.Lett. 33, 589(1974).\n61H.V. Kruis et al., Phys. Rev. B 70, 075109 (2004); J. Za-\nanenet al., Phil. Mag. B 81, 1485 (2001).\n62G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580\n(1988).\n63V. Cvetkovic and J. Zaanen, Phys. Rev. B 74, 134504\n(2006);ibid., Phys. Rev. Lett. 97, 045701 (2006); J. Zaa-\nnen, Z. Nussinov and S.I. Mukhin, Ann. Phys. (NY) 310,\n181 (2004).\n64The details of this calculation can be found in chapter 7\nof26."    },    {        "title": "1407.5595v2.Dissipation_in_Josephson_tunneling_junctions_at_low_temperatures.pdf",        "content": "arXiv:1407.5595v2  [cond-mat.supr-con]  14 Aug 2014L.N.Bulaevskii andS.-Z.Lin, Phys. Rev. B 90, 054509 (2014).\nDissipationinJosephsontunneling junctions atlowtemper atures\nLev N. Bulaevskii and Shi-Zeng Lin\nTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: February 11, 2020)\nIt is important to know the decoherence mechanism of a qubit b ased on Josephson junctions. At low tem-\nperatures, as quasiparticle concentration becomes expone ntially small, one needs to consider energy transfer\nfromtunnelingelectronstootherdegreesoffreedomtofindd issipationinJosephsonjunctionsanddecoherence\nin qubits. Here we discuss the energy transfer to two-level s ystems, i.e. the transitions between two di fferent\nconfigurations of ions inside insulating layer separated by a potential barrier. We derive a general equation\nof motion for the phase di fference between two superconducting electrodes and we find a r etarded dissipation\nterm due to electromagnetic mechanism and also contributio n due to electron tunneling mechanism. Using the\nequationof motionwecalculate the decayof Rabioscillatio ns andfrequency shiftinqubits due tothepresence\nofthetwo-levelsystems. Inthelongtimelimitourresultsc oincide withthoseobtainedbyMartinis etal.[J.M.\nMartinisetal.. Phys. Rev. Lett. 95, 210503 (2005)] withinthe Fermi’sGolden rule approach up t oa numerical\nfactor.\nPACS numbers: 74.78.-w,74.50. +r, 85.25.Pb\nI. INTRODUCTION\nAJosephsontunnelingjunctioncanoperatee ffectivelyasa\nphase qubit if one can achieve a long coherence time by re-\nducing dissipation and the corresponding noise.1–4It is well\nestablished that dissipation in tunneling Josephson junct ions\nnear the critical temperature Tcand at intermediate tempera-\nture is caused by quasiparticles.5,6However,at verylow tem-\nperatures, T≪2∆g,thequasiparticlesarefrozenoutandtheir\ncontribution to dissipation is exponentially small being p ro-\nportional to exp(−2∆g/T), where∆gis the superconducting\ngap.7,8Here we have assumedthat the electrodesare madeof\ns-wave superconductors. Hence, another mechanism of irre-\nversible energy transfer from tunneling Cooper pairs to oth er\ndegrees of freedom should be at work as experimental data\nconfirmtheexistenceofdissipationandphasedecoherencei n\nqubitsatverylowtemperatures.1,2,9Weconsiderinthefollow-\ningonlysourcesofdissipationintrinsictoJosephsonjunc tions\nexcluding external sources such as photon induced tunnelin g\ndiscussedinRef. 10. Dissipationduetoexcitationofphonons\ndiscussed in Refs. 11–14is not effective at very low energies\nandwewill notaccountforithere.\nFollowing the discussions by Martinis et al.1we will con-\nsider the dissipation at low temperatures T≪∆goriginat-\ning from two-level systems, i.e. the transitions between tw o\ndifferent configurations of ions inside amorphous insulating\nlayer separated by a potential barrier. It was well establis hed\nexperimentallythattwo-levelsystemsareresponsiblefor spe-\ncific heatandac dielectriclosses at lowtemperaturesandfr e-\nquenciesbelow20GHz in amorphousdielectrics15whichare\ninevitablypresentinsideJosephsonjunctions.\nInRef.1,thedecayofRabioscillationswasobtainedusing\nthe Fermi’s Golden rule approach. The dynamics of Joseph-\nson junctions is governed by an equation of motion for the\ngauge invariant phase di fference. Such an equation account-\ning for the presence of the two-level systems is not availabl e\nto date. In the present work, we derive such an equation of\nmotion by accounting for both the electromagnetic and tun-\nneling mechanismfor the dissipation caused by the two-leve lsystems. Theresultinggeneralequationofmotioncanbeuse d\ntodescribethedynamicsofjunctionatarbitrarytimeafter the\njunctionisperturbedawayfromequilibrium.\nWe will derive a general form of the dissipation term in\nthe equation of motion for the phase di fference in tunneling\njunctions with amorphousdielectric layer. We model this di -\nelectric layer as an ensemble of two-level systems. Interac -\ntionofthephasedi fferencewiththetwo-levelsystemsiselec-\ntromagnetic in nature and results in retarded dissipation, i.e.\nthedissipativeterm inthe equationforthephase di fferenceis\nnonlocalintimeandthecorrespondingequationforthephas e\ndifferenceisanintegralonewithrespecttotime. Wealsocon-\nsider the tunneling mechanism for dissipation and show that\nit is less effectivethanthe electromagneticone. We then treat\na qubit as a two-levelsystem and use the Bloch equationsfor\na “spin” to describe the qubit and two-level systems in the\ninsulating layer. We show that in the long time limit the dis-\nsipation of the low-amplitudeRabi oscillation in qubit di ffers\nfromthe resultsof Martinis et al.1only by a numericalfactor\nanda weaklyfrequency-dependentlogarithmicfactor.\nII. ELECTROMAGNETICMECHANISM\nWe consider low energy excitations inherent to two-level\nsystems in amorphous dielectrics (i.e. SiO 2and SiN x) be-\ntween junction electrodes. Specifically, oxygen ions in SiO 2\nor nitrogen ions in SiN xmay occupy two close positions in\nSi matrix separated by a distance bαof the order of atomic\nlength. Inenergyspacethesetwoconfigurationsareseparat ed\nby a potential barrier. We denote these positions as L and R,\nwe take the energyofsuch states as ǫLL=∆αandǫRR=−∆α.\nIn the space of states |L/angb∇acket∇ightand|R/angb∇acket∇ighttunneling of ion between\nthese two configurationsresults in the o ff-diagonalmatrix el-\nementsǫLR=−ǫRL=∆0,αin the Hamiltonian describing the\ntwo-levelsystems. The valueof ∆0,αisrelated tothe distance\nbαbetweenthetwo ionsinthestates |L/angb∇acket∇ightand|R/angb∇acket∇ightas\n2∆0,α(bα)=ǫaexp(−bα/a), (1)2\nwhereǫaisoftheorderofatomicenergyand aisacharacteris-\nticatomiclength. Foragivenjunction,weassumethat ǫaand\naarefixedwhile bαvariesfromsitetositewithanindex α. In\nthepresenceofanelectricfield E=/planckover2pi1˙φ/2edbetweenjunction\nelectrodes, the diagonal energies admit additional contri bu-\ntions±e∗bαEcosηαforǫLLandǫRR,respectively. Here φisthe\nsuperconductingphasedi fferenceand ˙φisitsfirsttimederiva-\ntive.dis the junction thickness, e∗is the effective charge of\ntunneling ions and ηαis the angle between the vector bαand\nthedirectionofthe electricfield betweenelectrodes, z. These\ncontributions due to the electric field describe the interac tion\nHintbetween the two-level systems and the phase di fference\nφ(t). Theeigenstatesofthetwo-levelsystem are\n|−/angb∇acket∇ight=sin(θα/2)|L/angb∇acket∇ight−cos(θα/2)|R/angb∇acket∇ight, (2)\n|+/angb∇acket∇ight=sin(θα/2)|L/angb∇acket∇ight+cos(θα/2)|R/angb∇acket∇ight, (3)\nwiththeeigenvalues ±/planckover2pi1Ωα/2≡±(∆2\nα+∆2\n0,α)1/2. Heretanθα=\n∆0,α/∆α. The dipole moment of the two-level systems is\n(e∗bα)cosθα(|+/angb∇acket∇ight−|−/angb∇acket∇ight). The matrix element Hintbetween\nstates|±/angb∇acket∇ightis/angb∇acketleft−|Hint|+/angb∇acket∇ight=Ee∗bαsinθαcosηα. The distribu-\ntion of two-level system parameters ��αandbαis assumed to\nbeuniform,\nP(∆α,bα)d∆αdbα=Pd∆αdbα, (4)\nwhere the normalization parameter Pwill be defined later.\nSuch a tunneling model describes the experimental data on\nthe specific heat and electric losses in amorphous dielectri cs\nquite well up to frequencies about 20 GHz.15This frequency\nismuchhigherthantheJosephsonfrequencyandwetakethis\nfrequencyasacuto fffrequencyinourtheory. Inthe |±/angb∇acket∇ightrepre-\nsentationtheHamiltonianofinteractionbetweenthetwo-l evel\nsystemwith theindex αandthephasedifferenceφis\nHs=/planckover2pi1[ΩαˆSz+λα˙φ(cotθαˆSz+ˆSx)], (5)\nλα=(2e∗bα/ed)sinθαcosηα, (6)\nwhereˆSk=ˆσk/2 and ˆσkare Pauli matrices with k=x,y,z.\nThejunctionHamiltonianis\nHJ=J0A[1−cosφ+ω−2\nJ˙φ2/2−Iφ], (7)\nwhereJ0is the Josephson coupling density, Ais the junction\narea,ωJis the Josephson frequency and Iis the bias current\nIBviathejunctioninunitsoftheJosephsoncriticalcurrent Ic,\nI≡IB/Ic.\nUsing the Heisenberg equation, we obtain the Bloch equa-\ntionsfor“spin”variables Sα\n˙Sx,α=−(Ωα+λα˙φcotθα)Sy,α, (8)\n˙Sy,α=(Ωα+λα˙φcotθα)Sx,α−λα˙φSz,α.(9)\nThe solution in the case of weak coupling, i.e. small\nSx,α,Sy,α≪1andSz,α≈1whenλα˙φ≪Ωα,is\nSx,α=λα/integraldisplayt\n0dt′˙φ(t′)sin[Ωα(t−t′)], (10)\n˙Sx,α=λαΩα/integraldisplayt\n0dt′˙φ(t′)cos[Ωα(t−t′)],and the two-level system frequency is renormalized from Ωα\ntoΩα+˙φλαcotθα. WereplaceΩα+˙φλαcotθαbyΩαinthefol-\nlowing discussions. Initially the junction is assumed in eq ui-\nlibrium,˙φ(t=0)=0. In theequationforthephasedi fference\nweobtainthedissipationcontributionduetothetwo-level sys-\ntems:\nω−2\nJ¨φ+sinφ−/summationdisplay\nα(/planckover2pi1λα/J0A)˙Sx,α=I.(11)\nWe replace summation over αby integration over ∆andb\nwith a uniform distribution function Eq. ( 4). In the integral\nover∆andbwe replace variables ∆andbby new variables\n/planckover2pi1Ω=(∆2+∆2\n0)1/2andsinθ=∆0/(/planckover2pi1Ω):\nd∆db=−a\n∆0d∆d∆0=a/planckover2pi1\n∆0||W||dΩd(sinθ), (12)\n||W||=/planckover2pi1(∂∆Ω∂∆0sinθ−∂∆0Ω∂∆sinθ)=cosθ/(/planckover2pi1Ω),(13)\nThe last term in the left hand side of Eq. ( 11) gives the dissi-\npationterm\n2/planckover2pi1µσd\nJ0Ωm/integraldisplayΩm\n0dΩΩd(sinθ)dηtanθln2/parenleftbiggǫa\n/planckover2pi1Ωsinθ/parenrightbigg\ncos2η×\n/integraldisplayt\n0dt′˙φ(t′)cos[Ω(t−t′)]\n≈/integraldisplayΩm\n0dΩ\nΩmE(Ω)/integraldisplayt\n0dt′˙φ(t′)cos[Ω(t−t′)], (14)\nE(Ω)=π/planckover2pi1µσΩd\n2J0ln2/parenleftiggΩa\nΩ/parenrightigg\n=ΩΓ0\nω2\nJln2/parenleftiggΩa\nΩ/parenrightigg\n,\nΓ0=2π2µσd2e2\nǫd/planckover2pi1,Ωa=ǫa\n/planckover2pi1, (15)\nwhereµ=2e∗a/ed, while the cutofffrequency isΩm≈20\nGHz andσis the volume density of the two-level systems.\nHereǫdis the high frequency dielectric function. In obtain-\ning Eq. ( 14), we have neglected sin θinside logarithm. It\nonly results in a numerical factor of the order of unity insid e\nthe logarithm, ln2(ǫa//planckover2pi1Ω), which is omitted within logarith-\nmic accuracy. The logarithmic factor ln( Ωa/Ω) is valid only\nforΩ>ω0≈Ωaexp(−1/σ1/3a) as we have assumed that the\ntwo-level systems do not overlap. At lower frequencies, Ω\ninsidethelogarithmicfactorshouldbereplacedby ω0.\nIII. TUNNELINGMECHANISM\nDisplacements of ions described by the two-level system\nmodel induce the electric field inside the dielectric and thu s\nchange the electric potential Vbarrier for tunneling electron.\nTheelectrontunnelingintegral βisgivenas\nβ=ǫaexp/bracketleftigg\n−/planckover2pi1−1/integraldisplayd\n0dz(2me[eV(z)−ǫe])1/2/bracketrightigg\n,(16)\nwheremeistheelectronmassand ǫeisitsenergy. Weconsider\nthecasethatthefrequencies Ωoftwo-levelsystemsaremuch3\nlower than the inverse transversal time for tunneling, 1 /τT,\nwhereτTisgivenbythequasiclassicalformula\nτT=/integraldisplayd\n0dz/radicalbiggme\n2[eV(z)−ǫe]. (17)\nThechangeofβdueto iondisplacements, δβ=β−β0, is\nδβ\nβ0=meed\n/planckover2pi12ln(ǫa/β0)/integraldisplayd\n0dzδV(z)=ln(ǫa/β0)\nV0d/integraldisplayd\n0dzδV(z),\n(18)\nwhereV0≈e/a. The change of potential, δV(z), induced\nby the dipoles pα=e∗bαof two-level systems positioned at\nrα,zα,withrα=(xα,yα),is\nδV(z)=e∗\nǫd(b·∇)/bracketleftigg1\n[(r−rα)2+(z−zα)2]1/2/bracketrightigg\n.(19)\nThe change in the Josephson coupling density δJ0caused by\nthe change in electron tunneling δβisδJ0/J0≈2δβ/β0. For\nthe tunneling mechanism the Hamiltonian for the interactio n\nbetweenthetwo-levelsystemswithindex αandthephasedif-\nferenceis\nHT,α=−J0/integraldisplay\ndrln(ǫa/β0)\nV0d(1−cosφ)ˆSx/integraldisplayd\n0dzδV(r,z)=\nJ0Aln(ǫa/β0)\nV0d(1−cosφ)ˆSx/integraldisplayd\n0dzδV(r=0,z). (20)\nUsingtheBlochequationsfora “spins”wefind\nSx,α(t)=−ΩαBα/integraldisplayt\n0dt′[1−cosφ(t′)]sin[Ωα(t−t′)]=\n−Bα/integraldisplayt\n0dt′sinφ(t′)cos[Ωα(t−t′)], (21)\nBα=/integraldisplayd\n0dz[bz,α(z−zα)−(bx,αxα−by,αyα)]R−3(z),\nR2(z)=(z−zα)2+r2\nα. (22)\nThe coordinates rα,zαare defined up to the size of the two-\nlevelsystemdipole b. Intheequationforthephasedi fference,\nEq. (11),wehaveanadditionaltwo-levelsystemcontribution\nduetothetunnelingmechanism\n/bracketleftigge∗ln(ǫa/β0)\nǫdV0d/bracketrightigg2/summationdisplay\nαAαsinφ(t)/integraldisplayt\n0dt′sinφ(t′)cos[Ω(t−t′)],\nAα=/integraldisplayd\n0du/integraldisplayd\n0dvb2\nz,α(u−zα)(v−zα)+(bx,αxα+by,αyα)2\nR3(u)R3(v).\nWereplacethesummation/summationtext\nαbyintegrationovercoordinates\nrα,zαoftwo-levelsystems\nA=/summationdisplay\nαAα⇒σ/integraldisplay\ndrα/integraldisplayd\n0dzαA(rα,zα).(23)Integrationover rαandaveragingoverdirectionsoftwo-level\nsystemdipoles bαgives\nA=πb2σ\n2/integraldisplayd\n0dz/integraldisplayd−z\n−z/integraldisplayd−z\n−zdudv/bracketleftigg2uv\n|u||v|(|u|+|v|)2+\n√\n2(u4+v4)1/2\n(u2+v2)2+√\n2(u2+v2)(u4+v4)1/2≈\nπ2b2σdln/parenleftiggd\nb/parenrightigg\n. (24)\nWe estimatethecontributiontothedissipationduetothetu n-\nnelingmechanismintheequationforthephasedi fference,Eq.\n(11), as\n/integraldisplayΩm\n0dΩ\nΩmT(Ω)sinφ(t)/integraldisplayt\n0dt′˙φ(t′)sinφ(t′)cos[Ω(t−t′)].\nT(Ω)=/parenleftiggπe∗ln(ǫa/β0)\nǫde/parenrightigg2a4σln(d/a)\ndln2/parenleftiggΩa\nΩ/parenrightigg\n.(25)\nTheratio rofthetunnelingcontributiontothedissipationand\nthatoftheelectromagneticmechanismatagiven Ωandsmall\nφis\nr≈φ2π2ln(ǫα/β0)\nǫ2\nd/parenleftbigga\nd/parenrightbigg/planckover2pi1dω2\nJ\ne2Ωln/parenleftiggd\na/parenrightigg\n.(26)\nDue toa≪dandωJ≪e2/(/planckover2pi1d) the tunneling mechanism\ncontributionissmalleverywhereexceptatlow Ω,whereitre-\nmains nonzeroin the limit Ω→0, while the electromagnetic\ncontributionvanishes in this limit. The di fferent behavior for\nbothmechanismsinthislimitisbecausethetwo-levelsyste m\ninteracts directly with the phase di fference via cosφfor the\ntunnelingmechanismas shown in Eq. ( 20), while in the elec-\ntromagneticmechanismitinteractswith ˙φasshowninEq.( 5).\nWe note that the tunneling mechanism gives contribution to\nthe dissipation which is nonlinearin the phase di fferenceand\nthusit isnegligibleatsmall deviationsfromtheequilibri um.\nIV. DECAYOF RABIOSCILLATIONSINTHE ABSENCE\nOF BIASCURRENT\nWe consider dissipation for small-amplitudeoscillations at\nIB=0neglectingthetunnelingcontribution. Weneedtosolve\ntheequation\nω−2\nJ¨φ+φ+\n/integraldisplayΩm\n0dΩ\nΩmE(Ω)/integraldisplayt\n0dt′˙φ(t′)cos[Ω(t−t′)]=0.(27)\nNotethatifEisfrequencyindependentand Ωmisinfinite,we\nwouldobtainforthelasttermintheleft-handsidethestand ard\ndissipationterm ˜γ˙φ.\nNeglectingthelogarithmicfactorin E(Ω),weintegrateover\nΩ. Afterchangingvariables t−t′=u/Ωm, weget\nω−2\nJ¨φ+φ+Γ0\nω2\nJ/integraldisplayΩmt\n0du˙φ(t−u/Ωm)f(u)=0,(28)\nf(u)=sinu/u−(1−cosu)/u2.4\nTounderstandthelong-timebehaviorwetake φ(t)=exp(iωt)\ninEq.(28)andobtainfortheleft-handside\n−ω−2\nJω2φ+φ+iωΓ0\nω2\nJ/integraldisplayΩmt\n0duexp(−iωu/Ωm)f(u)exp(iωt).\nWe see thatφ(t)=exp(iωt) is indeed the solution of Eq. ( 28)\ninthelongtimelimit Ωmt≫1ifwetakethecomplexnumber\nωasω=ωJ+δω+iγwith\nγ≈πωJΓ0\n4Ωm, δω≈Γ0ωJ\n2Ωmln/parenleftiggΩm\nωJ/parenrightigg\n(29)\nforωJ≪Ωm. The parameterγis the dissipation rate and\nδωistheshiftofRabioscillationfrequencyduetointeractio n\nwithtwo-levelsystems. Accountingforthelogarithmicfac tor,\nwe obtain\nγ≈πωJΓ0\n4Ωmln2/parenleftiggΩa\nΩm/parenrightigg\n, (30)\nδω≈Γ0ωJ\n2Ωmln/parenleftiggΩm\nωJ/parenrightigg\nln2/parenleftiggΩa\nΩm/parenrightigg\n. (31)\nThustheretardatione ffectisnotimportantat Ωm≫ωJinthe\nstationarysolution whichis established in the long-timel imit\nΩmt≫1.\nV. DISSIPATIONOFRABI OSCILLATIONSIN A QUBIT\nWe consider now a Josephson junction in the presence of\na bias current. Then energylevels in the quantumregime be-\ncome non-equidistantand we considerthe operationonly be-\ntween the ground state and the first excited state by ignoring\ntheotherenergylevelsvalidforasu fficientlyhighbiascurrent\nIB. We replace ˙φby the momentum operator and the Hamil-\ntonianforaqubittakesthe form\nHJ=J0A(1−cosφ)−4Ec∂2/∂φ2−/planckover2pi1IBφ/(2e),(32)\nwhereEc=e2/2CandCisthejunctioncapacitance. Thebias\ncurrent consists of a dc and an ac component, IB=Iac+Idc.\nWe find the Hamiltonian of interactionbetween the oscillato r\nandtwo-levelsystemsbyreplacing ˙φbythemomentumoper-\natorˆP=i∂/∂φaccording to the relation ˙φ→8ˆPEc//planckover2pi1. Then\nthe qubit is described by a two-level system with the Pauli\nmatrices ˆQfor a qubit “spin”. The matrix element of the op-\neratorˆPbetweenthegroundstateandthefirstexcitedstateof\nthe oscillator is given by the expression i(/planckover2pi1ω10e2/2C)1/2and\nwe write thetotal Hamiltonianforthe qubitandthe two-leve l\nsystemsas\nH=/planckover2pi1ω10\n2ˆQz−IacˆQy/(2e)+/summationdisplay\nαΩα\n2ˆSz,α+Hint,(33)\nHint=1\n2/summationdisplay\nαPαˆQxˆSx,α,Pα=i4bz,α\nd/parenleftigg/planckover2pi1ω10ee∗\n2C/parenrightigg1/2\n,(34)\nwhereω10is the energy difference between the first excited\nstate and the ground state of the qubit. This Hamiltoniandescribes the transfer of energy from the qubit to the two-\nlevel systems, and then back from the two-level systems to\nthequbit. Experimentally,acoherentstateofqubitoscill ation\nbetween the ground and the first excited state is prepared at\ntimet=0andthenprobabilitytofindthe qubitinthe excited\nstate at time tis measured. Using the Heisenbergequationof\nmotion for “spin” operators, we obtain the Bloch equations\nforthe system\n˙Qx=−ω10Qy−IacQz/e,˙Qy=ω10Qx−/summationdisplay\nαPαSx,αQz//planckover2pi1,\n˙Sx,α=−ΩαSy,α,˙Sy,α=ΩαSx,α+PαSz,αQx//planckover2pi1.(35)\nThese equationsdescribe the dynamicsof a qubit in the pres-\nence ofIacand two-level systems in the insulating layer. For\nweak excitations, Qx,Qy≪1,Sx,α,Sy,α≪1 andQz≈1,\nSz,α≈1 atIac=0, after excludingthe operatorsfor the two-\nlevelsystems,we obtaintheequationforthe qubit\n¨Qx+ω2\n10Qx+/summationdisplay\nα|Pα|2\n/planckover2pi12/integraldisplayt\n0dt′Qx(t′)sin[Ωα(t−t′)]=0.\nAfter summation over all two-level systems, this equation\nwithinalogarithmicaccuracytakestheform\nω−2\n10¨Qx+Qx+\nΓ0ln2/parenleftiggΩa\nΩm/parenrightigg/integraldisplayt\n0dt′Qx(t′)1−cos[Ωm(t−t′)]\nΩm(t−t′)=0.(36)\nWe assume that the qubit is perturbedaway from equilibrium\nstate at time t=0, i.e.Qx(t=0)/nequal0. For time t<Ω−1\nmdis-\nsipation is absent due to its retarded nature. In the station ary\nstate attime t≫Ω−1\nm, theretardationbecomesine ffectiveand\nwetakethesolutionas Qx(t)∼exp(iωt)withω=ω10+δω+iγ\ndeterminedby\n1−ω2\nω2\n10+Γ0\nΩmln2/parenleftiggΩa\nΩm/parenrightigg/integraldisplay∞\n0duexp/bracketleftiggiωu\nΩm/bracketrightigg\ng(u)=0,(37)\ng(u)=(1−cosu)/u.\nWethenobtainforthedissipationrateandthefrequencyshi ft\nγ≈πω10Γ0\n2Ωmln2/parenleftiggΩa\nΩm/parenrightigg\n. (38)\nδω≈ω10Γ0\nΩmln/parenleftiggΩm\nω10/parenrightigg\nln2/parenleftiggΩa\nΩm/parenrightigg\n. (39)\nAt intermediatetime t∼Ω−1\nm, oneneedstosolveEq. ( 36).\nResult forγcoincides with that of Martinis et al.1,γ≈\nπω10Γ0/(6Ωm),exceptforanumericalfactorandthelogarith-\nmic factor ln2(Ωa/Ωm) missed in their treatment. In our ap-\nproach,aswellasthatofRef. 1,transferofenergytononinter-\nacting two-level systems was assumed, i.e. the dipole-dipo le\ninteraction of two-level systems was neglected. As shown in\nRef.16, this is possible if the amplitude of Rabi oscillations,\nQx(t=0), is small, i.e. (4 πσ//planckover2pi1Ωm)(e∗a)2√Qx(t=0)≪1.\nForσ//planckover2pi1Ωm≈3×1029(erg·cm3)−1,seeRef. 1,thiscondition\nisfulfilledverywell.\nExcitationofRabioscillationsbyanaccurrentisdescribe d\nby Eqs. (35). For short external current pulses with duration5\nτ, the retarded dissipation due to two-level systems may be\nneglectedatanycurrentif Ωmτ≪1. Otherwise,thecondition\nof ignoringdissipation is Idc/e≫γ. Forstationary solutions,\ntheretardednatureofdissipationtermisnotimportant.\nVI. CONCLUSIONS\nIn conclusion,we havederiveda generalformof two-level\nsystemsdrivendissipativetermsinequationofmotionfort he\nphase difference in tunneling junctions with an amorphous\ndielectric layer at very low temperatures when dissipation\ncaused by quasiparticlesis negligible. We account for the d i-\nrect electromagneticmechanismsof phase and two-level sys -\ntemsinteractionandalsoforthee ffectoftwo-levelsystemson\nelectrontunnelingin junctions. We show that theygiveterm s\nwhicharenonlinearandlinearinphase,respectively. We fin dthat the dissipation from the tunneling mechanism is less ef -\nfectivethanthe electromagneticone. Finallywe havederiv ed\nthe decay rate of the Rabi oscillations due to the presence of\ntwo-level systems in the insulating layer of Josephson junc -\ntions. Our results are consistent with those obtained in Ref .\n1up to a numericalfactor and a weakly frequency-dependent\nlogarithmicfactor.\nVII. ACKNOWLEDGMENTS\nWe acknowledge helpful discussions with V. B. Geshken-\nbein,G. Blatter,J. Martinis,L.Io ffe,A.Golubov,A.Ustinov,\nV. Ryazanov and D. Khmel’nitskii. L.N.B. thanks the Pauli\nCenetr, ETH, Zurich for hospitality. The work was carried\noutundertheauspicesoftheNNSAoftheUSDOEatLANL\nunderContractNo. DE-AC52-06NA25396.\n1J.M. Martinis, K.B. Cooper, R. McDermott, M. Ste ffen, M. Ans-\nmann, K.D. Osborn, K. Cicak, S. Oh, D.P. Pappas, R.W. Sim-\nmonds, andC.C.Yu,Phys. Rev. Lett. 95, 210503 (2005).\n2J.M.Martinis,S.Nam,J.Aumentado, K.M.Lang,andC.Urbina ,\nPhys.Rev. B 67, 094510 (2003).\n3Y.Makhlin, G.Sch¨ on,andA.Shnirman,Rev.Mod.Phys. 73,357\n(2001).\n4C.H. van der Wal, F.K. Wilhelm, C.J.P.M. Harmans, and J.E.\nMooij,, Eur.Phys.J. B 31, 111 (2003).\n5K.K. Likharev, Dynamics of Josephson Junctions and Circuits\n(Gordon and Breach, NewYork, 1996).\n6I.O.Kulik,JETPLett, 2, 84(1965).\n7U. Eckern, G. Sch¨ on, V. Ambegaokar, Phys. Rev. B 30, 6419\n(1984).\n8A.I.Larkin, Yu.N.Ovchinnikov, Phys. Rev. B 28, 6281 (1983).\n9I. Chioresku, Y. Nakamura, C.J.P.M. Harmans, and J.E. Mooij ,\nScience299, 1869 (2003).10J.P. Pekola, V.F. Maisi, S. Kafanov, N. Chekurov, A. Kemppin en,\nYu.A.Pashkin,O.-P.Saira,M.M¨ ott¨ onen,andJ.S.Tsai,Ph ys.Rev.\nLett.105, 026803 (2010).\n11Ch. Helm, Ch. Preis, F. Forsthofer, J. Keller, K. Schlenga, R .\nKleiner,P.M¨ uller, Phys.Rev. Lett. 79, 737 (1997).\n12L.B. Ioffe, V.B. Geshkenbein, Ch. Helm, G. Blatter, Phys. Rev.\nLett.93, 057001 (2004).\n13Yu.M. Ivanchenko, Yu.V. Medvedev, Sov. Phys.JETP 33, 1223\n(1971).\n14E.G.Maksimov,P.I.Arseyev,N.S.Maslova,SolidStateComm un.\n111, 391 (1999).\n15S. Hunklenger and A.K. Raychaudhury, Prog. Low Temp. Phys.\n9, 265 (1986); Amorphous Solids: Low-Temperature Properties ,\neditedby W.A.Philips (Springer,Berlin,1981).\n16A.L. Burin, Yu. Kagan, I.Ya. Polishchuk, Phys. Rev. Lett. 86,\n5616 (2001)."    },    {        "title": "2109.00004v2.Ultraviolet_Infrared_Mixing_in_Marginal_Fermi_Liquids.pdf",        "content": "UV/IR Mixing in Marginal Fermi Liquids\nWeicheng Ye,1, 2Sung-Sik Lee,1, 3and Liujun Zou1\n1Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5\n2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1\n3Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1\nWhen Fermi surfaces (FS) are subject to long-range interactions that are marginal in the\nrenormalization-group sense, Landau Fermi liquids are destroyed, but only barely. With the in-\nteraction further screened by particle-hole excitations through one-loop quantum corrections, it\nhas been believed that these marginal Fermi liquids (MFLs) are described by weakly coupled field\ntheories at low energies. In this paper, we point out a possibility in which higher-loop processes\nqualitatively change the picture through UV/IR mixing, in which the size of FS enters as a relevant\nscale. The UV/IR mixing effect enhances the coupling at low energies, such that the basin of attrac-\ntion for the weakly coupled fixed point of a (2+1)-dimemsional MFL shrinks to a measure-zero set in\nthe low-energy limit. This UV/IR mixing is caused by gapless virtual Cooper pairs that spread over\nthe entire FS through marginal long-range interactions. Our finding signals a possible breakdown of\nthe patch description for the MFL, and questions the validity of using the MFL as the base theory\nin a controlled scheme for non-Fermi liquids that arise from relevant long-range interactions.\nIntroduction. Non-Fermi liquids (nFLs) arise ubiqui-\ntously when Fermi surfaces (FS) are coupled to gapless\ncollective modes that mediate long-range interactions.\nThe physics of nFLs is central to the strange metal-\nlic behavior and/or unconventional superconductivity\nin various systems, including cuprates, heavy-fermion\ncompounds, half-filled Landau level and pnictides [1–\n3]. However, understanding nFLs has been a long-\nstanding challenge due to strong quantum fluctuations\namplified by abundant gapless modes near FS [4–9].\nUnder renormalization group (RG) flow, most the-\nories for (2+1)-dimensional nFLs flow to the strong-\ncoupling regime at low energies, and non-perturbative\nmethods are required to understand their universal\nlong-distance physics [10, 11]. However, there is a spe-\ncial class of nFLs, marginal Fermi liquids (MFLs)1,\nwhere interaction effects are relatively weak, i.e.,\nmarginal in the RG sense with logarithmic (log) per-\nturbative corrections. If the marginal interactions are\nfurther screened, the long-distance physics should be\ncaptured by weakly coupled theories. While the MFL\nwas first introduced for cuprates [12], it is relevant in\nrather broad contexts. First, metallic states realized at\nthe half-filled Landau level and exotic Mott transitions\nmay be related to MFLs [13, 14]. Second, MFLs have\nbeen used as a foothold to gain a controlled access to\nstrongly coupled nFLs in an expansion scheme, where\nthe exponent with which long-range interactions decay\nin space is used as a control parameter [15, 16].\nIn nFLs, fermionic quasiparticles are destroyed by\nscatterings that are singularly enhanced at small mo-\nmenta. If large-momentum scatterings are suppressed\n1In this paper, the terms “metal”, “nFL” and “MFL” may\nrefer to that of either electrons or some emergent fermions,\ndepending on the context.strongly enough, one can understand physical observ-\nables that are local in momentum space (e.g., the\nsingle-particle spectral function) within local patches\nof FS that include the momentum point of interest (see\nFig. 1). Although this patch description becomes ul-\ntimately invalid in the presence of a pairing instabil-\nity driven by short-range four-fermion couplings, which\nisnon-local in momentum space, one may hope that\nthe dominant physics in the normal state can be cap-\ntured within the patch theory without invoking the\nentire FS. So the patch theory [8, 9, 16–18] has been\nwidely used to describe a large class of nFLs (see Refs.\n[13, 14, 19, 20] for some prominent examples) .\nIn MFLs, however, the validity of the patch the-\nory is questionable even before taking into account\nthe short-range four-fermion couplings, because large-\nmomentum scatterings are only marginally suppressed.\nIf large-momentum scatterings create significant inter-\npatch couplings, the patch theory fails even for the\npurpose of describing observables local in momentum\nspace. In this case, the size of the FS, a UV paramter,\nqualitatively modifies the IR scaling behavior, show-\ncasing UV/IR mixing.\nWhile such UV/IR mixing does not show up at low\norders in the perturbative expansion [16, 18], a system-\natic understanding of higher-order effects is still lack-\ning. This issue is also pertinent to multiple experimen-\ntally relevant problems. First, in the context of quan-\ntum Hall physics, it is debated whether the Halperin-\nLee-Read theory [13] and the recently proposed Son’s\ntheory [21] describe the same universal physics of the\ncomposite Fermi liquid (CFL). Second, in the contin-\nuous Mott transitions reported in κ-(ET) 2Cu2(CN) 3\n[22, 23] and moir´ e materials [24, 25], the observed phe-\nnomena appear to be compatible with the predictions\nof the patch theory [14], but some specific critical prop-\nerties seem to disagree [24]. To resolve these issues,arXiv:2109.00004v2  [cond-mat.str-el]  15 Mar 20222\nit is crucial to understand the behaviors of the cor-\nresponding MFL theories. Moreover, understanding\nhigher-order effects in nFLs in general may provide new\ninsight on the nature of quantum phase transitions as-\nsociated with sudden jumps of the FS size [26, 27], and\ndeconfined metallic quantum criticality [28–31].\nIn this paper, we study the higher-order behaviors\nof a theory of Nflavors of two-dimensional FS coupled\nto a dynamical U(1) gauge field, whose kinetic energy\nscales ask1+/epsilon1\ny. For the marginal exponent ( /epsilon1= 0), we\nindeed find potential UV/IR mixing in four-loop pro-\ncesses that renormalize the gauge coupling (see Fig.\n2), with a strength logarithmically singular in the FS\nsize. This is caused by gapless virtual Cooper pairs\nthat manage to explore the entire FS, assisted by large-\nmomentum scatterings that are only marginally sup-\npressed.\nFIG. 1: In the patch theory, a FS is partitioned into\nmultiple patches ( ∼kF/Λyof them, with kFthe\nFermi momentum and Λ ythe patch size). Ignoring\nthe short-range four-fermion interactions, couplings\nbetween modes from different patches are weak unless\nthey have almost parallel Fermi velocities. In this\ncase, one can focus on a pair of antipodal patches\nthat have nearly collinear Fermi velocities.\nModel and regularization scheme. We denote the\nlow-energy fermion fields near a pair of antipodal\npatches by ψip, withp=±the patch index and\ni= 1,...,N the flavor index. arepresents the gauge\nfield (see Fig. 1). Due to the kinematic constraints, the\nmost important interactions occur between fermionic\nmodes and the gauge bosons with momenta nearly per-\npendicular to their Fermi velocity [4, 17]. The Eu-\nclidean action for the patch theory is [16]\nS=Sψ+Sint+Sa, (1)where\nSψ=/integraldisplay\n[dk]/summationdisplay\ni,pψ†\nip(k)(−ikτ+pkx+k2\ny)ψip(k),\nSint=/integraldisplay\n[dk1][dk2]/summationdisplay\ni,pλpa(k1)ψ†\nip(k1+k2)ψip(k2),\nSa=/integraldisplay\n[dk]N\n2e2|ky|1+/epsilon1a(−k)a(k),(2)\nwith [dk] =dkτdkxdky\n(2π)3andλ±=±1. The reason\nfor the opposite signs of λ±is because acouples to\nthe currents of the fermions, and fermions from the\ntwo patches have opposite Fermi velocities. By power\ncounting, the coupling e2is marginal (relevant) if /epsilon1= 0\n(/epsilon1 > 0). Physically, with decreasing /epsilon1, the fermion-\nboson coupling gets weaker at small momenta, but\nlarge-momentum scatterings become stronger, which\nincreases the “risk” of UV/IR mixing. Below we pri-\nmarily focus on the marginal case with /epsilon1= 0. Note\nthat a large Nis still useful in organizing the calcula-\ntions when /epsilon1= 0.\nOne can introduce two cutoffs. Λ denotes the energy\ncutoff2, and Λyis the cutoff of y-momentum. The for-\nmer is the usual UV cutoff, while the latter represents\nthe size of patch. We take Λ →∞ for simplicity, i.e.,\nthe theory is regularized by Λ yonly. Crucially, Λ yalso\nserves as IR data that measures the number of gapless\nmodes near the FS, and there is a priori no guarantee\nthat low-energy observables are insensitive to Λ y. If\nthe Λy-dependence cannot be removed in low-energy\nobservables by renormalization, the theory has UV/IR\nmixing, and the patch description fails.\nUV/IR mixing. We consider the photon self-energy,\nΠ(k), which is O(N1) to the leading order. To or-\nderN0, Π(k) is finite as Λ y→ ∞ [9, 16], due to\na kinematic constraint that is nevertheless absent at\nhigher orders (see Eq. (30) of the Supplemental Ma-\nterial [32] or [33]). At order N−1, by exact calcula-\ntions we find a double -log divergence in the diagram\nin Fig. 2a: Π 0(kτ= 0,ky) =1\nN|ky|\ne22α4\n3π2/parenleftBig\nln/parenleftBig\nΛy\nky/parenrightBig/parenrightBig2\n,\nwithα≡e2/(4π) (see Sec. III.C.3 in [32]). Other\ndiagrams are harder to compute explicitly. However,\nunder reasonable assumptions, we argue that the only\nother net contribution to the double-log divergence is\nfrom Fig. 2b, whose contribution is also Π 0atkτ= 0\n(see Sec. III.C.4 in [32]). Double-log divergences are\nusually from divergences in sub-diagrams, which can\nthen be cancelled by diagrams with counter terms.\nBut the present double-log divergences are not due to\n2UV cutoff associated to high-energy modes can be imposed\non thex-momentum relative to the FS as well, yet this cutoff\nviolates an emergent gauge invariance [16].3\n(a)\n(b)\nFIG. 2: Eq. (3) comes from these two diagrams,\ntogether with their cousins with both internal fermion\nloops flipped in direction, which are not shown here.\nNote that only when the two fermion loops in each\ndiagram run in the same direction do they contribute\nto double-log divergence. See Sec. III.C.2 in [32] for\nall diagrams at this order.\nthis, since the only divergent sub-diagram in Fig. 2 is\nthe three-loop vertex correction, but the corresponding\ncounter term does not contribute to the renormaliza-\ntion of the boson kinetic term to order N−1. Taken\nall diagrams together (including the ones with counter\nterms), the total double-log divergence to order N−1\nis\nΠ(kτ= 0,ky)∼1\nN|ky|\ne24α4\n3π2/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n.(3)\nTo better understand this result, first consider the\nusual renormalizable field theories without a FS, e.g.,\n3 + 1 dimensional φ4theory. In such theories, given a\nUV cutoff Λ, quantities like dΠ/dln Λ are analytic in\nthe external momentum k/lessmuchΛ, since this derivative\nmeasures the contribution of high-energy modes in the\nenergy window [Λ ,Λ +dΛ) (see Fig. 3). Consequently,\nthe non-analyticity in Π can at most take the form\nofk2lnΛ\nk, and the Λ-dependence of the results can\nthen be eliminated by local counter terms, allowing any\nobservable at a scale k1/lessmuchΛ to be expressed solely in\nterms of renormalized quantities measured at another\nscalek2/lessmuchΛ, and the IR physics is insensitive to the\nUV physics.\nHowever, the present theory has another short-\ndistance scale, Λ y, which measures the number of gap-\nlessmodes near the FS. Low-energy observables in gen-\n𝑑𝑑ΛΛ𝑑𝑑Λ𝑦𝑦\nΛ𝑦𝑦FIG. 3: The gray regions illustrate modes that are\nintegrated out ifwe tune Λ yin MFL (left) or Λ in a\nusual field theory without FS (right). The latter has\ngapless modes only at a single point in the\nmomentum space (shown in red), while the former\nhas gapless modes overlapping with the gray regions.\nIn MFL, Π( k) calculated at a fixed Λycan have IR\nsingularities stronger than ln(Λ y/k).\neral can depend on Λ yin a sensitive manner. Espe-\ncially,dΠ/dln Λydoes not have to be analytic in k(see\nFig. 3). Gapless modes can not only renormalize the\nexisting non-local term through |ky|ln(Λy/|ky|), but\nalso generate stronger non-analyticity in the quantum\neffective action, such as |ky|lnn(Λy/|ky|) withn > 1\n(n= 2 in Eq. (3)). In this case, the Λ y-dependence\ncannot be removed in low-energy observables through\nrenormalization of the existing terms (local or not) in\nthe action, signaling UV/IR mixing.\nUV/IR mixing is known to arise in metals3. First,\nthe FS size kF, a UV parameter, becomes relevant at\nlow energies when a critical boson is coupled with FS\nwhose dimension is greater than one, as a boson can\ndecay into particle-hole pairs along the “great circle”\nof FS whose tangent space includes the boson momen-\ntum [39, 40]. Second, the FS size is important in the\npresence of pairing instabilities driven by short-range\nfour-fermion interactions, via which Cooper pairs re-\nsiding on the FS with zero total momentum can be\nscattered throughout the entire FS without violating\nmomentum or energy conservation [18, 41, 42].\nThe origin of the UV/IR mixing we find here is re-\nlated to the second one but different. The contribution\nin Eq. (3) comes from virtual Cooper pairs (VCP),\nrepresented by the two fermion loops that come from\nopposite patches and run in the same direction in Fig.\n2. Via the marginal long-range interactions mediated\nby the gauge field, these VCP spread over the entire\nFS, which enjoys a large phase space for scattering and\n3We note that UV/IR mixing with different origins is also pro-\nposed in other setups [34–38].4\ncan have singular contributions (see Eq. (86) in [32]).\nIndeed, the double-log divergence disappears if either\nthe VCP are absent (e.g., by taking the fermion loops\nin Fig. 2 to be in the same patch and/or run in op-\nposite directions), or large-momentum scatterings are\nfurther suppressed (e.g., by taking /epsilon1 > 0)4. This is\nreminiscent to the enhanced quasiparticle decay rate\ndue to VCP in Fermi liquids [45].\nConsequences of UV/IR mixing. Eq. (3) forces us\nto view Λ yas another coupling constant of the theory\n[39]. In particular, ˜Λy≡Λy/µplays the role of a\nrelevant coupling as the size of FS blows up relative\nto the decreasing scale µ. The beta functions of the\ntheory are (see Sec. I and II in [32] for details)\nd˜Λy\ndlnµ=−˜Λy,dα\ndlnµ=2α2\nπN−8α5\n3π2N2ln˜Λy.(4)\nLet us analyze these beta functions in the weak-\ncoupling regime with α/lessorsimilar1 and low-energy limit\nwithµ/lessmuchΛy, together with a large but finite N.\nThe first term in dα/d lnµis the lowest-order term\nin 1/Nand˜Λy-independent. As the scale µis lowered,\nit makes the gauge coupling decrease logarithmically\nthrough screening. If the initial coupling α0defined\nat energy scale µ0satisfies (α3\n0/N)·ln(Λy/µ0)/lessorsimilar1,\nthis term dominates and the gauge coupling flows\nto zero at low energies. On the other hand, for\n(α3\n0/N)·ln(Λy/µ0)/greaterorsimilar1, the second term dominates,\nwhich tends to enhance the coupling at low ener-\ngies. In this case, we can ignore the first term to\nthe leading order. Then the gauge coupling grows\nasα=α0/bracketleftBig\n1−16\n3π2N2α4\n0/parenleftBig\nln2/parenleftBig\nµ\nΛy/parenrightBig\n−ln2/parenleftBig\nµ0\nΛy/parenrightBig/parenrightBig/bracketrightBig−1/4\n.\nThis solution shows a divergence of the gauge coupling\nwith decreasing µ, although it cannot be trusted in the\nstrong coupling regime. For theories defined at scale\nµ0, the basin of attraction for the α= 0 fixed point\nis given byBµ0≡{α0|α3\n0< cN/ ln(Λy/µ0)}, withc\nanO(1) constant (see the shaded region in Fig. 4).\nThe salient feature is that Bµ0shrinks to a measure-\nzero set in the low-energy limit (i.e., µ0/lessmuchΛy), due\nto the scale dependence in the beta function. The fact\nthat the beta function explicitly depends on Λ yis a\nhallmark of UV/IR mixing.\nIn the presence of the UV/IR mixing, the FS size\ncannot be dropped in low-energy physical observables.\nFor example, the single-fermion spectral function takes\nthe form ofA(ω,k,T) =ω∆f/parenleftBig\nω\nkz\n/bardbl,k/bardbl\nkz/prime\nF,ω\nT/parenrightBig\n, wherekFis\n4Our double-log divergence appears similar to the Sudakov dou-\nble pole [43, 44] in QED and effective theories of QCD. Both of\nthem originate from gapless degrees of freedom. However, the\nSudakov double-log originates from small momentum modes\nwhile our double-log divergence is from modes with large y-\nmomentum.\n𝛼𝛼\n𝜇𝜇0𝛼𝛼0\nB𝜇𝜇0\n𝜇𝜇FIG. 4: The flow of α=e2/(4π) with initial condition\nα=α0atµ=µ0. For eachµ0, there is a critical\nvalueα∗≈N/ln(Λy/µ0) (the dashed curve) : when\nα0<α∗the gauge coupling flows to zero at low\nenergies (green), while when α>α∗it flows to\ninfinity (orange).\nthe FS size, k/bardblis the distance of kaway from the FS, T\nis the temperature, ∆, zandz/primeare critical exponents\n(z/prime= 2 from Eq. (3)), and fis a universal function\n[39]. It is interesting to test this in CFLs at various\nfilling factors that can be realized in Chern bands [29].\nThe UV/IR mixing in MFLs also has implications for\nthe/epsilon1-expansion scheme [15, 16], which has been used\nto approach nFLs with /epsilon1= 1 from MFLs with /epsilon1= 0\nperturbatively in /epsilon1. To see it, we examine how the\nUV/IR mixing in the base theory with /epsilon1= 0 affects the\nperturbative /epsilon1-expansion. In theories with /epsilon1 >0, the\nUV/IR mixing disappears since the diagrams in Fig.\n2 are no longer divergent in Λ y, as large-momentum\nscatterings are further suppressed. Instead, the double-\nlog in Eq. (3) is translated to a double pole in /epsilon1as5\nΠ(kτ= 0,ky)∼1\nN|ky|\n˜e28˜α4\nπ2×\n/parenleftbigg1\n27/epsilon12+1\n9/epsilon1ln(µ/|ky|) +1\n6(ln(µ/|ky|))2/parenrightbigg\n,(5)\nwhere ˜e2=e2µ−/epsilon1is the dimensionless coupling and\n˜α= ˜e2/(4π). Since 1//epsilon1-poles cannot be absorbed by\nterms already present in the action, the naive pertur-\nbative expansion appears ill-defined. Moreover, this\nsingular self-energy suggests that /epsilon1is renormalized to\na larger value, further indicating that the /epsilon1-expansion\nmay break down. This calls for alternative control\nschemes for nFLs. See Refs. [47–49] for the dimen-\nsional regularization scheme that has no UV/IR mix-\ning, and Refs. [50–53] for other proposals.\nSummary and discussion. We provide strong evi-\ndence that a (2 + 1)-dimensional MFL exhibits UV/IR\n5See Ref. [46] for a different but related calculation at /epsilon1= 1.5\nmixing, caused by virtual Cooper pairs that spread\nover the entire FS due to large-momentum scatterings.\nOur finding suggests the breakdown of the patch the-\nory for MFLs and a potential issue in the /epsilon1-expansion\nthat uses MFLs as the base theory for nFLs.\nWe conclude with a few final remarks. First, the\nUV/IR mixing identified in (2 + 1)-dimensional MFLs\ncan be extended to more general cases. Consider met-\nals withm-dimensional FS (e.g., a spherical or cylin-\ndrical FS has m= 2 and a Weyl nodal line has m= 1)\ncoupled to a critical boson whose kinetic energy goes as\nk1+/epsilon1\ny. The contribution of virtual Cooper pairs to loop\ncorrections scales as ∼/integraltextdmky\nk1+/epsilon1\ny, which suggests that for\nm/greaterorequalslant1 +/epsilon1, there exist UV divergences associated with\nthe extended size of FS, and UV/IR mixing can arise.\nSo we expect virtual-Cooper-pair-induced UV/IR mix-\ning in (3 + 1)-dimensional gauge theories with m= 2\nand/epsilon1= 1 (on top of the UV/IR mixing identified in\nRef. [39]). This is relevant to quantum spin liquids\n[54, 55] and mixed-valence insulators [56]6.\nSecond, our result is obtained within the standard\npatch theory. To understand the full consequences of\nthe UV/IR mixing caused by large-momentum scatter-\nings, one should consider a general theory that keeps\ntrack of how the boson-fermion coupling is renormal-\nized at large boson momenta. For this, instead of\na coupling constant, one should take into account a\nmomentum-dependent coupling function , reminiscent\nof the familiar form factors in the interaction vertices in\nvarious settings [57]. Moreover, the four-fermion cou-\nplings, which should also be described by a coupling\nfunction, are not considered here, but they should in\nprinciple be studied on equal footing as the gauge cou-\npling. Whether there is UV/IR mixing can depend on\nthe microscopic details of the physical system. What\nwe have shown is the presence of a UV/IR mixing in\nsystems where the two effects above are negligible. In\nthe future, it will be of great interest to understand\nwhether such UV/IR mixing exists in systems where\nthese effects are significant and should be incorporated\ninto the theory. In any case, our results suggest that\nthe physics of MFLs is richer than originally expected,\nand mandates a qualitative improvement of the current\ntheoretical understanding.\nAcknowledgement. We thank Tobias Holder, Subir\nSachdev and T. Senthil for helpful discussion. W.Y.\nwould like to especially thank Timothy Hsieh for intro-\nducing the topic of non-Fermi liquid, Haoran Jiang for\nexplaining double-log behavior in phenomenology and\n6Since/epsilon1= 1 corresponds to local kinetic term for gauge boson,\nwe expect that UV/IR mixing identified here does not origi-\nnate from the non-analyticity of the kinetic term of the gauge\nboson.Minyong Guo for hospitality during his stay in Beijing\nwhere the bulk calculation was carried out. Research\nat Perimeter Institute is supported in part by the Gov-\nernment of Canada through the Department of Innova-\ntion, Science and Economic Development Canada and\nby the Province of Ontario through the Ministry of\nColleges and Universities. S.L. acknowledges the sup-\nport of the Natural Sciences and Engineering Research\nCouncil of Canada.\n[1] T. Senthil, “On non-Fermi liquid quantum crit-\nical points in heavy fermion metals,” Annals\nof Physics 321, 1669–1681 (2006), arXiv:cond-\nmat/0604240 [cond-mat.str-el].\n[2] Sung-Sik Lee, “Recent Developments in Non-Fermi\nLiquid Theory,” Annual Review of Condensed Mat-\nter Physics 9, 227–244 (2018), arXiv:1703.08172 [cond-\nmat.str-el].\n[3] Erez Berg, Samuel Lederer, Yoni Schattner, and Si-\nmon Trebst, “Monte Carlo Studies of Quantum Critical\nMetals,” Annual Review of Condensed Matter Physics\n10, 63–84 (2019), arXiv:1804.01988 [cond-mat.str-el].\n[4] Joseph Polchinski, “Effective Field Theory and the\nFermi Surface,” arXiv e-prints , hep-th/9210046\n(1992), arXiv:hep-th/9210046 [hep-th].\n[5] B. L. Altshuler, L. B. Ioffe, and A. J. Millis, “Low-\nenergy properties of fermions with singular interac-\ntions,” Phys. Rev. B 50, 14048–14064 (1994).\n[6] Yong Baek Kim, Patrick A. Lee, and Xiao-Gang Wen,\n“Quantum Boltzmann equation of composite fermions\ninteracting with a gauge field,” Phys. Rev. B 52,\n17275–17292 (1995), arXiv:cond-mat/9504063 [cond-\nmat].\n[7] Ar. Abanov and A. Chubukov, “Anomalous scaling at\nthe quantum critical point in itinerant antiferromag-\nnets,” Phys. Rev. Lett. 93, 255702 (2004).\n[8] Sung-Sik Lee, “Low-energy effective theory of Fermi\nsurface coupled with U(1) gauge field in 2+1\ndimensions,” Phys. Rev. B 80, 165102 (2009),\narXiv:0905.4532 [cond-mat.str-el].\n[9] Max A. Metlitski and Subir Sachdev, “Quantum phase\ntransitions of metals in two spatial dimensions. I.\nIsing-nematic order,” Phys. Rev. B 82, 075127 (2010),\narXiv:1001.1153 [cond-mat.str-el].\n[10] Shouvik Sur and Sung-Sik Lee, “Chiral non-\nFermi liquids,” Phys. Rev. B 90, 045121 (2014),\narXiv:1310.7543 [cond-mat.str-el].\n[11] Andres Schlief, Peter Lunts, and Sung-Sik Lee, “Ex-\nact critical exponents for the antiferromagnetic quan-\ntum critical metal in two dimensions,” Phys. Rev. X\n7, 021010 (2017).\n[12] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink,\nE. Abrahams, and A. E. Ruckenstein, “Phenomenol-\nogy of the normal state of cu-o high-temperature\nsuperconductors,” Phys. Rev. Lett. 63, 1996–1999\n(1989).\n[13] B. I. Halperin, Patrick A. Lee, and Nicholas Read,\n“Theory of the half-filled landau level,” Phys. Rev. B\n47, 7312–7343 (1993).6\n[14] T. Senthil, “Theory of a continuous Mott transition\nin two dimensions,” Phys. Rev. B 78, 045109 (2008),\narXiv:0804.1555 [cond-mat.str-el].\n[15] Chetan Nayak and Frank Wilczek, “Non-Fermi liquid\nfixed point in 2 + 1 dimensions,” Nuclear Physics B\n417, 359–373 (1994), arXiv:cond-mat/9312086 [cond-\nmat].\n[16] David F. Mross, John McGreevy, Hong Liu, and\nT. Senthil, “Controlled expansion for certain non-\nFermi-liquid metals,” Phys. Rev. B 82, 045121 (2010),\narXiv:1003.0894 [cond-mat.str-el].\n[17] Sung-Sik Lee, “Stability of the U(1) spin liquid with a\nspinon Fermi surface in 2+1 dimensions,” Phys. Rev. B\n78, 085129 (2008), arXiv:0804.3800 [cond-mat.str-el].\n[18] Max A. Metlitski, David F. Mross, Subir Sachdev,\nand T. Senthil, “Cooper pairing in non-fermi liquids,”\nPhys. Rev. B 91, 115111 (2015).\n[19] Patrick A. Lee and Naoto Nagaosa, “Gauge theory of\nthe normal state of high-Tc superconductors,” Phys.\nRev. B 46, 5621–5639 (1992).\n[20] Sung-Sik Lee and Patrick A. Lee, “U(1) Gauge Theory\nof the Hubbard Model: Spin Liquid States and Possi-\nble Application to κ-(BEDT-TTF) 2Cu2(CN) 3,” Phys.\nRev. Lett. 95, 036403 (2005), arXiv:cond-mat/0502139\n[cond-mat.str-el].\n[21] Dam Thanh Son, “Is the Composite Fermion a\nDirac Particle?” Physical Review X 5, 031027 (2015),\narXiv:1502.03446 [cond-mat.mes-hall].\n[22] Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda,\nand G. Saito, “Mott Transition from a Spin Liquid to a\nFermi Liquid in the Spin-Frustrated Organic Conduc-\ntorκ-(ET) 2Cu2(CN) 3,” Phys. Rev. Lett. 95, 177001\n(2005), arXiv:cond-mat/0504273 [cond-mat.str-el].\n[23] Tetsuya Furukawa, Kazuya Miyagawa, Hiromi\nTaniguchi, Reizo Kato, and Kazushi Kanoda,\n“Quantum criticality of mott transition in organic\nmaterials,” Nature Physics 11, 221–224 (2015).\n[24] Tingxin Li, Shengwei Jiang, Lizhong Li, Yang Zhang,\nKaifei Kang, Jiacheng Zhu, Kenji Watanabe, Takashi\nTaniguchi, Debanjan Chowdhury, Liang Fu, Jie Shan,\nand Kin Fai Mak, “Continuous Mott transition in semi-\nconductor moir´ e superlattices,” Nature (London) 597,\n350–354 (2021), arXiv:2103.09779 [cond-mat.str-el].\n[25] Augusto Ghiotto, En-Min Shih, Giancarlo S. S. G.\nPereira, Daniel A. Rhodes, Bumho Kim, Jiawei\nZang, Andrew J. Millis, Kenji Watanabe, Takashi\nTaniguchi, James C. Hone, Lei Wang, Cory R.\nDean, and Abhay N. Pasupathy, “Quantum critical-\nity in twisted transition metal dichalcogenides,” Na-\nture (London) 597, 345–349 (2021), arXiv:2103.09796\n[cond-mat.mes-hall].\n[26] Hiroaki Shishido, Rikio Settai, Hisatomo Harima,\nand Yoshichika Onuki, “A drastic change of\nthe fermi surface at a critical pressure in cer-\nhin5: dhva study under pressure,” Journal of the\nPhysical Society of Japan 74, 1103–1106 (2005),\nhttps://doi.org/10.1143/JPSJ.74.1103.\n[27] Yawen Fang, Gael Grissonnanche, Anaelle Legros, Si-\nmon Verret, Francis Laliberte, Clement Collignon,\nAmirreza Ataei, Maxime Dion, Jianshi Zhou, David\nGraf, M. J. Lawler, Paul Goddard, Louis Taillefer,\nand B. J. Ramshaw, “Fermi surface transformation\nat the pseudogap critical point of a cuprate super-\nconductor,” arXiv e-prints , arXiv:2004.01725 (2020),arXiv:2004.01725 [cond-mat.str-el].\n[28] Liujun Zou and Debanjan Chowdhury, “Deconfined\nmetallic quantum criticality: A U(2) gauge-theoretic\napproach,” Physical Review Research 2, 023344\n(2020), arXiv:2002.02972 [cond-mat.str-el].\n[29] Liujun Zou and Debanjan Chowdhury, “Deconfined\nMetal-Insulator Transitions in Quantum Hall Bilay-\ners,” Physical Review Research 2, 032071 (2020),\narXiv:2004.14391 [cond-mat.str-el].\n[30] Ya-Hui Zhang and Subir Sachdev, “From the pseu-\ndogap metal to the Fermi liquid using ancilla\nqubits,” Physical Review Research 2, 023172 (2020),\narXiv:2001.09159 [cond-mat.str-el].\n[31] Ya-Hui Zhang and Subir Sachdev, “Deconfined criti-\ncality and ghost Fermi surfaces at the onset of antifer-\nromagnetism in a metal,” Phys. Rev. B 102, 155124\n(2020), arXiv:2006.01140 [cond-mat.str-el].\n[32] See the Supplementary Material for additional details\nof the analysis, and additional references, including\nRefs. [58, 59].\n[33] Tobias Holder and Walter Metzner, “Fermion loops\nand improved power-counting in two-dimensional crit-\nical metals with singular forward scattering,” Phys.\nRev. B 92, 245128 (2015), arXiv:1509.07783 [cond-\nmat.str-el].\n[34] Sedigh Ghamari, Sung-Sik Lee, and Catherine Kallin,\n“Renormalization group analysis of a neck-narrowing\nLifshitz transition in the presence of weak short-range\ninteractions in two dimensions,” Phys. Rev. B 92,\n085112 (2015), arXiv:1407.6732 [cond-mat.str-el].\n[35] Nathan Seiberg and Shu-Heng Shao, “Exotic U(1)\nsymmetries, duality, and fractons in 3+1-dimensional\nquantum field theory,” SciPost Physics 9, 046 (2020),\narXiv:2004.00015 [cond-mat.str-el].\n[36] Yizhi You, Julian Bibo, Taylor L. Hughes, and Frank\nPollmann, “Fractonic critical point proximate to a\nhigher-order topological insulator: How does UV blend\nwith IR?” arXiv e-prints , arXiv:2101.01724 (2021),\narXiv:2101.01724 [cond-mat.str-el].\n[37] Henry Shackleton, Alex Thomson, and Subir Sachdev,\n“Deconfined criticality and a gapless Z 2spin liquid in\nthe square-lattice antiferromagnet,” Phys. Rev. B 104,\n045110 (2021), arXiv:2104.09537 [cond-mat.str-el].\n[38] Zheng Zhou, Xue-Feng Zhang, Frank Pollmann, and\nYizhi You, “Fractal Quantum Phase Transitions: Crit-\nical Phenomena Beyond Renormalization,” arXiv e-\nprints , arXiv:2105.05851 (2021), arXiv:2105.05851\n[cond-mat.str-el].\n[39] Ipsita Mandal and Sung-Sik Lee, “Ultraviolet/infrared\nmixing in non-Fermi liquids,” Phys. Rev. B 92, 035141\n(2015), arXiv:1407.0033 [cond-mat.str-el].\n[40] Ipsita Mandal, “UV/IR mixing in non-Fermi liq-\nuids: higher-loop corrections in different energy\nranges,” European Physical Journal B 89, 278 (2016),\narXiv:1608.06642 [cond-mat.str-el].\n[41] Leon N. Cooper, “Bound electron pairs in a degenerate\nfermi gas,” Phys. Rev. 104, 1189–1190 (1956).\n[42] Huajia Wang, Srinivas Raghu, and Gonzalo Torroba,\n“Non-Fermi-liquid superconductivity: Eliashberg ap-\nproach versus the renormalization group,” Phys. Rev.\nB95, 165137 (2017), arXiv:1612.01971 [cond-mat.str-\nel].\n[43] Michael E. Peskin and Daniel V. Schroeder, An In-\ntroduction to quantum field theory (Addison-Wesley,7\nReading, USA, 1995).\n[44] Iain Stewart, 8.851 Effective Field The-\nory. Spring 2013. Massachusetts Institute\nof Technology: MIT OpenCourseWare,\nhttps://ocw.mit.edu/courses/physics/\n8-851-effective-field-theory-spring-2013/ .\nLicense: Creative Commons BY-NC-SA.\n[45] Dimitri Pimenov, Alex Kamenev, and Andrey V.\nChubukov, “One-dimensional scattering of two-\ndimensional fermions near quantum criticality,” Phys.\nRev. B 103, 214519 (2021), arXiv:2104.13802 [cond-\nmat.str-el].\n[46] Tobias Holder and Walter Metzner, “Anomalous dy-\nnamical scaling from nematic and U(1) gauge field fluc-\ntuations in two-dimensional metals,” Phys. Rev. B 92,\n041112 (2015), arXiv:1503.05089 [cond-mat.str-el].\n[47] Denis Dalidovich and Sung-Sik Lee, “Perturbative non-\nFermi liquids from dimensional regularization,” Phys.\nRev. B 88, 245106 (2013), arXiv:1307.3170 [cond-\nmat.str-el].\n[48] Shouvik Sur and Sung-Sik Lee, “Quasilocal\nstrange metal,” Phys. Rev. B 91, 125136 (2015),\narXiv:1405.7357 [cond-mat.str-el].\n[49] Peter Lunts, Andres Schlief, and Sung-Sik Lee, “Emer-\ngence of a control parameter for the antiferromagnetic\nquantum critical metal,” Phys. Rev. B 95, 245109\n(2017), arXiv:1701.08218 [cond-mat.str-el].\n[50] Jeremias Aguilera Damia, Shamit Kachru, Srinivas\nRaghu, and Gonzalo Torroba, “Two dimensional\nnon-Fermi liquid metals: a solvable large N limit,”\nPhys. Rev. Lett. 123, 096402 (2019), arXiv:1905.08256\n[cond-mat.str-el].\n[51] Erik E. Aldape, Taylor Cookmeyer, Aavishkar A.\nPatel, and Ehud Altman, “Solvable Theory of a\nStrange Metal at the Breakdown of a Heavy FermiLiquid,” arXiv e-prints , arXiv:2012.00763 (2020),\narXiv:2012.00763 [cond-mat.str-el].\n[52] Jaewon Kim, Ehud Altman, and Xiangyu Cao, “Dirac\nfast scramblers,” Phys. Rev. B 103, L081113 (2021),\narXiv:2010.10545 [cond-mat.str-el].\n[53] Ilya Esterlis, Haoyu Guo, Aavishkar A. Patel,\nand Subir Sachdev, “Large-N theory of critical\nFermi surfaces,” Phys. Rev. B 103, 235129 (2021),\narXiv:2103.08615 [cond-mat.str-el].\n[54] Yi Zhou, Patrick A. Lee, Tai-Kai Ng, and Fu-\nChun Zhang, “Na 4Ir3O8as a 3D Spin Liquid with\nFermionic Spinons,” Phys. Rev. Lett. 101, 197201\n(2008), arXiv:0806.3323 [cond-mat.str-el].\n[55] Michael J. Lawler, Arun Paramekanti, Yong Baek\nKim, and Leon Balents, “Gapless Spin Liquids\non the Three-Dimensional Hyperkagome Lattice of\nNa4Ir3O8,” Phys. Rev. Lett. 101, 197202 (2008),\narXiv:0806.4395 [cond-mat.str-el].\n[56] Debanjan Chowdhury, Inti Sodemann, and T. Senthil,\n“Mixed-valence insulators with neutral Fermi sur-\nfaces,” Nature Communications 9, 1766 (2018),\narXiv:1706.00418 [cond-mat.str-el].\n[57] Siddharth A. Parameswaran, Rahul Roy, and Shiv-\naji L. Sondhi, “Fractional quantum Hall physics in\ntopological flat bands,” Comptes Rendus Physique 14,\n816–839 (2013), arXiv:1302.6606 [cond-mat.str-el].\n[58] R. Shankar, “Renormalization-group approach to in-\nteracting fermions,” Reviews of Modern Physics 66,\n129–192 (1994), arXiv:cond-mat/9307009 [cond-mat].\n[59] L. G. Aslamazov and A. I. Larkin, “Effect of Fluctua-\ntions on the Properties of a Superconductor Above the\nCritical Temperature - Sov. Phys. Solid State 10, 875\n(1968),” in 30 Years of the Landau Institute , Vol. 11,\nedited by Isaak M. Khalatnikov and Vladimir P. Mi-\nneev (1996) p. 23.Supplemental Material for “UV/IR Mixing in Marginal Fermi Liquids”\nWeicheng Ye,1, 2Sung-Sik Lee,1, 3and Liujun Zou1\n1Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5\n2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1\n3Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1\nContents\nI. Framework for Renormalization Group Analysis 1\nII. Summary of the RG Results 4\nIII. Feynman Diagram Calculations 4\nA. Second Leading Order Result of Self-Energy 5\nB. Third Leading Order Result of Fermion Self-Energy 7\nC. Third leading Order Result of Boson Self-Energy 10\n1. General Strategies of Calculation and Assumptions 10\n2. All Diagrams at the Third Leading Order 12\n3. Benz Diagram 12\n4. 3-String Diagram 17\nReferences 22\nThis Supplemental Material contains the framework for the renormalization group (RG) analysis (Section I), a\nsummary of the RG results (Section II), and the details of the Feynman diagram claculations (Section III).\nI. Framework for Renormalization Group Analysis\nIn this section, we present the framework for our field-theoretic RG analysis. In terms of the renormalized\nfields and couplings, the action of the patch theory for NFermi surfaces (FS) coupled to a U(1) gauge field is\nwritten as\nS=Sψ+Sint+Sa, (1)\nwhere\nSψ=/integraldisplay\n[dk]/summationdisplay\ni,pψ†\nip(k)/bracketleftbig\n−iZ1kτ+Z2(pkx+k2\ny)/bracketrightbig\nψip(k),\nSint=/integraldisplay\n[dk1][dk2]/summationdisplay\ni,pZ2λpa(−k1)ψ†\nip(k2)ψip(k1+k2),\nSa=/integraldisplay\n[dk]Z3N\n2e2µ/epsilon1|ky|1+/epsilon1a(−k)a(k)(2)\nwith [dk]≡dkτdkxdky\n(2π)3,λ±=±1,µa renormalization scale with the same dimension as ky, andZ1,2,3some\nrenormalization factors. In the above, the rotational symmetry of the patch theory is invoked for the absence of\nthe relative renormalization between kxandk2\ny, and the Ward identity is used to guarantee that the multiplicative\nrenormalization of Sintis the same as that of ( pkx+k2\ny) part of the fermion kinetic term [1]. For latter convenience,\ndefineα≡γe2withγ≡1/(4π) as a proxy for the coupling constant e2. The renormalized fields, couplings andarXiv:2109.00004v2  [cond-mat.str-el]  15 Mar 20222\nmomentum are related to the bare ones via\nkτB=Z1\nZ2kτ≡Zτkτ, kxB=kx, kyB=ky,\nψipB(kB) =/radicalbigg\nZ2\nZτψip(k)≡/radicalbig\nZψψip(k), aB(kB) =1\nZτa(k)≡/radicalbig\nZaa(k),\nαB≡γe2\nB=1\nZ3Zτγe2µ/epsilon1≡Zααµ/epsilon1(3)\nwhere the subscript Bstands for bare. Zτis introduced to account for a nontrivial dynamical exponent, Zψand\nZaare introduced to account for anomalous dimensions of the fermion and gauge fields, respectively, and Zαis\nintroduced to account for the flow of α. At the tree level, the fields and couplings have the following scaling\ndimensions in momentum space:\n[kτ]0= 2,[kx]0= 2,[ky]0= 1,\n[ψip(k)]0=−7\n2,[a(k)]0=−3,[e2]0= 0.(4)\nTheZfactors are chosen so that the following renormalization conditions are satisfied:\n∂Σ(kτ=µ2,kx=ky= 0)\n∂kτ= 0,\nΣ(ky=µ,kτ=kx= 0) = 0,\nΠ(ky=µ,kτ=kx= 0) = 0,(5)\nwhere Σ and Π are the 1-particle irreducible self-energy of the renormalized fermionic and gauge fields, respectively,\nandµis the renormalization scale. Below we focus on the case with /epsilon1= 0. In this case, it turns out that physical\nobservables have divergence as Λ y, the cutoff of ky, goes to infinity. Notice that Λ ymeasures the abundance\nofgapless degrees of freedom (DOF) on the FS, so it is not a cutoff as in a usual field theory, which separates\nhigh-energy DOF from low-energy ones. Accordingly, it is more appropriate to view Λ yas a parameter that is on\nequal footing as the couplings of the theory.\nDenote a correlation function involving nψfermionic fields and nagauge fields by G(nψ,na), which is generically\nwritten as a function of a set of momenta, α,µand Λy. ThenG(nψ,na)\nB·δ({kB}) =Znψ\n2\nψZna\n2aG(nψ,na)({k})·δ({k}),\nwhere{k}denotes the momenta this correlation function depends on, and the δ-functions impose momentum\nconservation. Due to kτB=Zτkτ,δ({k}) =Zτδ({kB}),G(nψ,na)\nB =Znψ\n2\nψZna\n2aZτG(nψ,na)({k}). Because all bare\nquantities and fields are independent of µ,µd\ndµG(nψ,na)\nB ({kB}) = 0, which leads to a Callan-Symanzik equation\nfor the renormalized correlation function:/bracketleftbigg\nµ∂\n∂µ+bαα∂\n∂α+γzkτ∂\n∂kτ+nψγψ\n2+ (na−1)γz/bracketrightbigg\nG(nψ,na)= 0 (6)\nwith\nbα≡µ\nαdα\ndµ=−µ\nZαdZα\ndµ, γz≡−µ\nZτdZτ\ndµ, γψ≡µ\nZψdZψ\ndµ. (7)\nNote that the beta function of αisβ(α) =bαα.\nOn the other hand, suppose the engineering dimension of G(nψ,na)is ∆ 0. By dimensional analysis, it must have\nthe formG(nψ,na)=µ∆0g/parenleftBig\n{kτ}\nµ2,{kx}\nµ2,{ky}\nµ,α,Λy\nµ/parenrightBig\n, wheregis some function. Thus,\n/parenleftbigg\nµ∂\n∂µ+ 2kτ∂\n∂kτ+ 2kx∂\n∂kx+ky∂\n∂ky+ Λy∂\n∂Λy−∆0/parenrightbigg\nG(nψ,na)= 0. (8)\nCombining this equation with Eq. (6) yields another form of the Callan-Symanzik equation:\n/bracketleftbigg\n(2−γz)kτ∂\n∂kτ+ 2kx∂\n∂kx+ky∂\n∂ky+ Λy∂\n∂Λy−bαα∂\n∂α−∆0−nψγψ\n2−(na−1)γz/bracketrightbigg\nG(nψ,na)= 0. (9)\nAt the fixed point, bα= 0, and the above equation implies the following scaling structure of the correlation\nfunction\nG(nψ,na)(sz{kτ},s2{kx},s{ky},sΛy) =s∆G(nψ,na)({kτ},{kx},{ky},Λy) (10)3\nwithz≡2−γzthe dynamical exponent, and ∆ ≡∆0+nψγψ\n2+ (na−1)γzthe total scaling dimension of this\ncorrelation function. It is clear that γψand 2γzcan be interpreted as the anomalous dimensions of the fermionic\nand gauge fields, respectively. Note that the anomalous dimension of the gauge field a(k) is entirely from a\nrescaling of time (by setting kτB=Zτkτ), while its real-space counterpart a(x) has no anomalous dimension\ndue to the Ward identity. Also note that the renormalized correlation function would not depend on Λ yin the\nabsence of any UV/IR mixing, but one of the central results of this paper is that there is UV/IR mixing.\nBefore finishing this section, we describe an equivalent way to analyze the theory, which is also often used in the\nliterature and differs from the above one by some convention [1–6]. We will mainly use the previous convention,\nand readers not interested in this part can skip it. In the alternative convention, the bare action is written as\nSψ=/integraldisplay\n[dk]/summationdisplay\ni,pψ†\nip(k)/parenleftbig\n−iZfZηηkτ+Zf(pkx+k2\ny)/parenrightbig\nψip(k),\nSint=/integraldisplay\n[dk1][dk2]/summationdisplay\ni,pZfλpa(−k1)ψ†\nip(k2)ψip(k1+k2),\nSa=/integraldisplay\n[dk]1\nZeN\n2e2µ/epsilon1|ky|1+/epsilon1a(−k)a(k),(11)\nand [dk] =dkτdkxdky\n(2π)3,λ±=±1. Again, the Ward identity is used to guarantee that the renormalization factor\ninSintisZf, the same as that of the pkx+k2\nypart of the fermionic kinetic term. The difference between\nthis convention and our previous one is that in this case kτhas the same dimension as kx. Instead, another\ndimensionless parameter ηis introduced to keep track of a nontrivial dynamical exponent. Below we will see\nthat, not surprisingly, these two conventions are equivalent. The bare quantities and fields are related to the\nrenormalized ones via\n(kτB,kxB,kyB) = (kτ,kx,ky), ψipB=/radicalbig\nZfψip, aB=a, ηB=Zηη= 1, e2\nB≡Zee2. (12)\nThe engineering dimensions are also given by (4) with the extra [ η]0= 0. We also define ˜ α≡γe2\nηwithγ= 1/(4π).\nSimilar to the previous convention, the Zfactors are determined by demanding the renormalization conditions\ngiven in Eq. (5), but with µ2changed to µ2/ηin the first renormalization condition:\n∂Σ(kτ=µ2/η,kx=ky= 0)\n∂kτ= 0. (13)\nNow consider a correlation function involving nffermionic fields and ncgauge fields, denoted by G(nf,nc).\nOne can show that G(nf,nc)=ηnf\n2+nc−1µ∆0g/parenleftBig\nη{kτ}\nµ2,{kx}\nµ2,{ky}\nµ,˜α,Λy\nµ/parenrightBig\n, where ∆ 0is the engineering dimen-\nsion of the correlation function, and gis the same function as in the previous convention. This implies that\nη∂\n∂ηG(nf,nc)=/parenleftBig\nnf\n2+nc−1 +kτ∂\n∂kτ−e2∂\n∂e2/parenrightBig\nG(nf,nc). Combining this result and analogous derivations as be-\nfore, similar Callan-Symanzik equations can be obtained\n/parenleftbigg\nµ∂\n∂µ+bee2∂\n∂e2+bηη∂\n∂η+nf\n2γf/parenrightbigg\nG(nf,nc)= 0,\n/bracketleftbigg\nµ∂\n∂µ+ (be−bη)e2∂\n∂e2+bηkτ∂\n∂kτ+nf\n2(γf+bη) + (nc−1)bη/bracketrightbigg\nG(nf,nc)= 0,\n/bracketleftbigg\n(2−bη)kτ∂\n∂kτ+ 2kx∂\n∂kx+ky∂\n∂ky+ Λy∂\n∂Λy−(be−bη)e2∂\n∂e2−∆0−nf\n2(γf+bη)−(nc−1)bη/bracketrightbigg\nG(nf,nc)= 0,(14)\nwhere\nbη≡µ\nηdη\ndµ, be≡µ\ne2de2\ndµ, γf≡µ\nZfdZf\ndµ. (15)\nIt is not hard to see that bη,be−bηandγf+bηshould be identified as γz,bαandγψin the previous convention,\nrespectively, and the two conventions ultimately lead to identical results. Note that in this convention, the beta\nfunction for ˜ αisβ(˜α) = (be−bη) ˜α. In this paper, we will mainly use the previous convention.4\nII. Summary of the RG Results\nTheZfactors introduced in Eq. (2) are calculated in Sec. III. Here we summarize the results. We expand Z\nin powers of 1 /N, and at each order of 1 /N, we keep results relevant to the calculations of bα,γzandγψto the\nleading order in Λ y/µ:\nZ1= 1−1\nN·2α\nπlnΛy√αµ+1\nN2·2α3\nπ2/parenleftbig\nSa(α) +Ta(α) + 2Ga(α)−Sb(α)−Tb(α)−2Gb(α)/parenrightbig\nlnΛy\nµ,\nZ2= 1 +1\nN2·2α3\nπ2/parenleftbig\nSa(α) +Ta(α)−Sb(α)−Tb(α)/parenrightbig\nlnΛy\nµ,\nZ3≈1−1\nN·2α3\nπF(α) +1\nN2·4α4\n3π2/parenleftbigg\nlnΛy\nµ/parenrightbigg2\n,(16)\nwhere the functions F(α),Ga(α),Sa(α),Ta(α),Gb(α),Sb(α),Tb(α) can be found in Eqs. (33), (47), (48), (50),\n(55), (56), (57), respectively. Using these expressions, Eqs. (3) and (7), bα,γzandγψare obrained to be\nbα≈1\nN·2α\nπ−1\nN2·8α4\n3π2lnΛy\nµ,\nγz=−1\nN·2α\nπ+1\nN2·/parenleftbigg4α3\nπ2/parenleftbig\nGa(α)−Gb(α)/parenrightbig\n−2α2\nπ2/parenrightbigg\n,\nγψ=−1\nN·2α\nπ+1\nN2·/parenleftbigg2α3\nπ2/parenleftbig\n2Ga(α)−Sa(α)−Ta(α)−2Gb(α) +Sb(α) +Tb(α)/parenrightbig\n−2α2\nπ2/parenrightbigg\n.(17)\nHere we ignore1\nN2lnΛy\nµterm inZ3and keep only terms proportional to lnΛy\nµat order 1/N2inbα. Note that\nalmost all terms in bα,γzandγψcome from explicit µdependence in Zexcept for the1\nN2α2term, which comes\nfrom the ln√αterm inZ1.\nIII. Feynman Diagram Calculations\nIn this section, we present the details of the Feynman diagram calculations leading to the results summarized\nin Section II. Parts of the relevant calculations can be found in Refs. [1, 3–5, 7, 8]. Here we focus on the case\nwhere/epsilon1= 0.1We will also assume that external kyto be positive for self-energies unless mentioned otherwise.\nFirst, we collect results of all the Feynman diagrams contributing to the self-energy of boson and fermion to the\nthird leading order in the large- Nexpansion. We will see that all of the divergent diagrams give the traditional\nlogarithmic divergence except for the Benz diagram and the 3-String diagram, which give the unconventional\ndouble logarithmic divergence. In any case, Z1,2,3can be expanded in 1 /Nas\nZi= 1 +1\nNδ(1)\ni+1\nN2δ(2)\nii= 1,2,3 (18)\nsuch that renormalization conditions Eq. (5) are satisfied. The propagator of a, denoted asD(k), and the\npropagator of ψ, denoted asG(k), to the third leading order in 1 /Nare written as\nD−1=ND−1−Π(1)−1\nNΠ(2)+...\nG−1=G−1−1\nNΣ(1)−1\nN2Σ(2)+...,(19)\nwhereDis the leading-order propagator for aas in e.g. Ref. [1]\nD(kτ,ky) =1\nγ|kτ|\n|ky|+|ky|\ne2(20)\n1As discussed in Ref. [3], if we use the second convention, we need to set η∼O(1) such that the large- Nexpansion has no extra\nfactors ofNfromη.5\nwithγ= 1/(4π), andGis the bare fermion propagator,\nG±(kτ,kx,ky) =1\n−ikτ±kx+k2y. (21)\nWe can also write down a large- Nexpansion of bα,γzandγψto the order of 1 /N2,\nbα=1\nNb(1)\nα+1\nN2b(2)\nα+...\nγz=1\nNγ(1)\nz+1\nN2γ(2)\nz+...\nγψ=1\nNγ(1)\nψ+1\nN2γ(2)\nψ+...(22)\nwhich can be calculated through solving the Callan-Symanzik equation Eq. (6) from the propagator D,Gat the\nthird leading order of 1 /N, or through Eq. (7).\nA. Second Leading Order Result of Self-Energy\nThe second leading order result of the fermion self-energy is given by the standard one-loop diagram shown in\nFig. 1. The diagram has been computed in [4, 5]. Here we write down the result of the diagram at /epsilon1= 0 with\nFIG. 1: One-loop diagram for fermion self-energy.\ny-momentum cutoff Λ y,\nΣ1LF(k) =/integraldisplay\n[dl]Gf(k−l)D(l) =ikτe2\n2π2/parenleftBigg\n1\n2+ ln/parenleftBigg\nΛy/radicalbig\nγe2|kτ|/parenrightBigg/parenrightBigg\n. (23)\nTo impose the renormalization conditions in Eq. (5), we take\nδ(1)\n1=−2α\nπln/parenleftbiggΛy\nµ√α/parenrightbigg\n, δ(1)\n2= 0 (24)\nwithα≡e2/(4π).2\nLet us then turn to the second leading order result of the self-energy of boson, Π(1). At this order only the\nso-called Aslamazov-Larkin (AL) diagram [9] in Fig. 2 gives non-zero but finite contribution as argued in Ref.\n[4]. The AL diagram reads [1]\nFIG. 2: Aslamazov-Larkin (AL) diagram for boson self-energy.\nΠAL(k) =1\n2/integraldisplaydlτdlxdly\n(2π)3Γ3(k,l,−(k+l))Γ3(−k,−l,k+l)D(l)D(k+l) (25)\n2We note that in the simple minimal subtraction scheme, one would take δ(1)=−2α\nπlnΛy\nµ, which amounts to imposing a different\nRG condition.6\nwhere Γ3is the fermion-induced three-boson vertex and receives contributions from both fermion patches as well\nas both directions in which fermions can travel in the loop,\nΓ3= Γ3\n++ Γ3\n−, (26)\nΓ3\ns(l1,l2,−(l1+l2)) =λ3\ns(fs(l1,l2,−(l1+l2)) +fs(l2,l1,−(l1+l2))), (27)\nfs(l1,l2,−(l1+l2)) =−/integraldisplaydpτdpxdpy\n(2π)3Gs(p)Gs(p−l1)Gs(p+l2). (28)\nFirst let us compute f±(k,l,−(k+l)). We perform the integral in the sequence of px,pyandpτto obtain\nf±(k,l,−(k+l)) =1\n4π/parenleftbig\nky(ilτ∓lx−l2y)−ly(ikτ∓kx+k2y)/parenrightbig\n×/parenleftbigg\nkτθ/parenleftbiggkτ\nky/parenrightbigg\n+lτθ/parenleftbigglτ\nly/parenrightbigg\n−(k+l)τθ/parenleftbigg(k+l)τ\n(k+l)y/parenrightbigg/parenrightbigg\n.(29)\nHereθ(x) is the unit step function. An important property is that if we take the static limit kτ→0, the\nfermion-induced three-boson vertex is non-zero only when the internal boson satisfies,\n−ky<ly<0,whenky>0 or 0 <ly<−ky,whenky<0, (30)\ni.e. the integration of the internal boson momentum only has nonzero contribution from a restricted finite region\nin the static limit.3Note that the imaginary part of the pole of lxfor bothfs(k,l,−(k+l)) andfs(l,k,−(k+l))\nwith a fixed s=±are ats(lτ−kτly/ky). Considering the integration of lxin Eq. (25), we see that Π ALis\nnon-zero only when the two fermion loops come from different patches. After the integration of lx, we see that\nthekxdependence drops out and we have\nΠAL(kτ,ky) =−1\n64π4k2y/integraldisplay\ndlτdlyD(l)D(k+l)/parenleftbigg\nkτθ/parenleftbiggkτ\nky/parenrightbigg\n+lτθ/parenleftbigglτ\nly/parenrightbigg\n−(k+l)τθ/parenleftbigg(k+l)τ\n(k+l)y/parenrightbigg/parenrightbigg2\n×/parenleftbig\nl2\ny+kyly/parenrightbig2\n|lτ−kτly/ky|/parenleftBig\n(lτ−kτly/ky)2+/parenleftbig\nl2y+kyly/parenrightbig2/parenrightBig.(31)\nIt is straightforward to see that this integral is convergent from the kinematic constraint in Eq. (30). Here, we\nfocus on Π AL(kτ= 0,ky). Also we temporarily assume ky>0, which indicates that −ky<ly<0. After a change\nof variable with lτ=k2\ny∆ (−∞<∆<∞) andly=kyδ(−1<δ< 0), the final result is given as follows,\nΠAL(kτ= 0,ky) =−|ky|\ne22α3\nπF(α), (32)\nwhere\nF(α)≡/integraldisplay1\n0dδ/integraldisplay∞\n0d∆δ\nδ2+α∆1−δ\n(1−δ)2+α∆∆/parenleftbig\nδ2−δ/parenrightbig2\n∆2+ (δ2−δ)2. (33)\nTo impose the renormalization conditions in Eq. (5), we take the counterterm to be\nδ(1)\n3=−2α3\nπF(α) (34)\n3This is the reason why we do not see the double log divergence in lower order diagrams. In U(1)×U(1) orU(2) gauge theory [5],\nthis also leads to the fact that there is a fixed line instead of several isolated fixed points at the leading order of the RG equation. It\nis straightforward to see that the fixed line will in principle be lifted because of the third leading order results of (boson) self-energy.7\n(a) Two-Loop Fermion Diagram.\n(b) Nested Diagrams of One-Loop Fermion Diagram with AL diagram and its\ncorresponding counterterm.\n(c) 3-Loop Fermion Diagram.\nFIG. 3: Diagrams contributing to the third leading order of fermion self-energy.\nB. Third Leading Order Result of Fermion Self-Energy\nNext, we move to the self-energies at the third leading order of 1 /N. Because the goal is to compute the beta\nfunctions to the order of 1 /N2, it is sufficient to keep only the divergent parts in the self-energies. We start from\nthe fermion self-energy. There are four non-zero diagrams as shown in Fig. 3. The two-loop fermion diagram in\nFig. 3a is convergent, while the divergence of the two nested diagrams shown in Fig. 3b cancel with each other.\nThus, the divergent part only comes from the three-loop diagram, shown in Fig. 3c.\nIn the three-loop diagram, the fermion in the top loop can run in the clockwise or counterclockwise direction.\nMoreover, only when the fermion running in the loop lies in a different ±patch than the external fermion [1, 10]\nwill the diagram give logarithmic divergence. Let us assume that the external fermion lies in the + patch. First,\nconsider the case in which the fermion in the loop travels in counterclockwise direction. Its contribution reads\nΣa\n3LF(k) =−λ3\n+λ3\n−/integraldisplay[dl1][dl2][dp]\n(2π)9G+(k−l1)G+(k−l2)G−(p)G−(p+l1)G−(p+l2)\n×D(l1)D(l1−l2)D(l2).(35)\nIntegrating over l1x,l2xresults in\nΣa\n3LF(k) =−/integraldisplaydpτdpxdpydl1τdl1ydl2τdl2y\n(2π)7θ(l1τ−kτ)−θ(−l1τ−pτ)\ni(kτ−pτ−2l1τ) + 2(ky+py)l1y−px+p2y−kx−k2y\n×θ(l2τ−kτ)−θ(−l2τ−pτ)\ni(kτ−pτ−2l2τ) + 2(ky+py)l2y−px+p2y−kx−k2y·1\n−ipτ−px+p2yD(l1)D(l1−l2)D(l2).(36)\nIntegrating over pxandpy, we obtain\nΣa\n3LF(k) =/integraldisplay\ndpτdl1τdl1ydl2τdl2y\n(2π)5D(l1)D(l1−l2)D(l2)\n2((l2−l1)yG+(k)−1+2i(l2yl1τ−l1yl2τ))T(pτ;l1τ,l2τ;l1y,l2y;kτ), (37)\nwhereG+(k)−1is just the inverse of bare + patch fermion propagator and\nT(pτ;l1τ,l2τ;l1y,l2y;kτ) = (θ(l1τ−kτ)−θ(−l1τ−pτ)) (θ(l2τ−kτ)−θ(−l2τ−pτ))\n×/bracketleftBigg/parenleftbigg\nθ/parenleftbigg2l2τ−kτ\nl2y/parenrightbigg\n−θ/parenleftbigg2l1τ−kτ\nl1y/parenrightbigg/parenrightbigg\nθ(−pτ) +/parenleftbigg\nθ/parenleftbigg2l1τ−kτ\nl1y/parenrightbigg\n−θ/parenleftbiggl2τ−l1τ\nl2y−l1y/parenrightbigg/parenrightbigg\nθ(kτ−pτ−2l1τ)\n−/parenleftbigg\nθ/parenleftbigg2l2τ−kτ\nl2y/parenrightbigg\n−θ/parenleftbiggl2τ−l1τ\nl2y−l1y/parenrightbigg/parenrightbigg\nθ(kτ−pτ−2l2τ)/bracketrightBigg\n.(38)\nSupposekτ>0. Separating l1y,l2yplane into six regions as in Fig. 4a and doing the integration of pτ, we obtain\nT(l1τ,l2τ;l1y,l2y;kτ)≡/integraldisplay\ndpτT(pτ;l1τ,l2τ;l1y,l2y;kτ) (39)8\n(a) Our separtation of l1y-l2yplane.\n(b) Result of Tin each region. The units in the l1τand l2τaxes are both kτ.\nFIG. 4: Result of Tgiven different sections on the l1y−l2yplane. Different sections of l1y−l2yplane are\nlabeled in Fig. 4a and the result of Tin each region is given in Fig. 4b where each graph from left to right then\nfrom top down corresponds to the result in section 1-6 respectively.\nin each region shown in Fig. 4b ( kτis set to be 1 in these graphs).\nNote that\nT(l1τ,l2τ;−l1y,−l2y;kτ) =−T(l1τ,l2τ;l1y,l2y;kτ) (40)\nand therefore regions connected by l1y,l2y⇐⇒−l1y,−l2ygives the same result. We can thus restrict to regions\nwithl1y>0. First consider region 1. In order to isolate the divergent terms, let us do the following change of\nparameters:\nl2y=l1yδ, l 1τ=l2\n1y∆1, l 2τ=l2\n1y∆2, kτ=l2\n1y∆0. (41)\nMoreover, we can write down Tas\nT(l2τ,l1τ;l2y,l1y;kτ) =l2\n1yT0(∆1,∆2;δ; ∆0). (42)\nThen we have\nΣa\n3LF(k) =iα3\n2π2/integraldisplay\ndl1ydδd∆1d∆2l1y\n(∆2−δ∆1)−1\n2i(1−δ)G+(k)−1\nl2\n1yT0\n×1\n1 +α|∆1||δ|\n|δ|2+α|∆2||1−δ|\n|1−δ|2+α|∆1−∆2|.(43)\nThe divergent terms come from the integration region l1y→∞ , so let us do the integration of l1yfromky, the\ndynamically generated IR cutoff, to infinity instead. We can thus first do a large l1yexpansion in this region,\nΣa\n3LF(k) =iα3\n2π2/parenleftbigg\nσ1/integraldisplay\nl1ydl1y+iσ2G+(k)−1/integraldisplay\nl−1\n1ydl1y+σ3kτ/integraldisplay\nl−1\n1ydl1y+.../parenrightbigg\n, (44)\nwhere\nσ1=/integraldisplay\ndδd∆1d∆21\n∆2−δ∆11\n1 +α|∆1||δ|\n|δ|2+α|∆2||1−δ|\n|1−δ|2+α|∆1−∆2|T0(∆1,∆2;δ; 0),\nσ2=/integraldisplay\ndδd∆1d∆21−δ\n2(∆ 2−δ∆1)21\n1 +α|∆1||δ|\n|δ|2+α|∆2||1−δ|\n|1−δ|2+α|∆1−∆2|T0(∆1,∆2;δ; 0),\nσ3=d\nd∆0/parenleftbigg/integraldisplay\ndδd∆1d∆21\n∆2−δ∆11\n1 +α|∆1||δ|\n|δ|2+α|∆2||1−δ|\n|1−δ|2+α|∆1−∆2|T0(∆1,∆2;δ; ∆0)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n∆0=0.(45)9\nThe integration of the ... terms is convergent. The term proportional to σ1is polynomially divergent and does\nnot contribute to the logarithmic divergence, so it should give an unimportant shift of chemical potential. The\nterm proportional to σ2andσ3give the following logarithmically divergent contribution to Σa\n3LF(k):\n/parenleftbiggα3\n2π2Sa(α)G+(k)−1−α3\nπ2Ga(α)ikτ/parenrightbigg\nln/parenleftbiggΛy\nky/parenrightbigg\n, (46)\nwhereGa(α) andSa(α) are defined as follows\nGa(α)≡/integraldisplay1\n0dδ/integraldisplay∞\n0d∆δ\nδ2+α∆1−δ\n(1−δ)2+α∆, (47)\nSa(α)≡/integraldisplay1\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∞\n0d∆2(1−δ)∆2\n(∆2+δ∆1)21\n1 +α∆1δ\nδ2+α∆21−δ\n(1−δ)2+α(∆1+ ∆ 2). (48)\nThe calculation in the region 2 is completely analogous to the calculation in the region 1. If 1 and 2 are exchanged\nin Eqs. (37) and (38), one can see that the result in region 2 is identical to the result in region 1. For region 3,\nit is clear that σ3= 0 while from σ2we obtain the divergent part\nα3\nπ2G+(k)−1Ta(α) ln/parenleftbiggΛy\nky/parenrightbigg\n, (49)\nwhere\nTa(α)≡/integraldisplay∞\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∆1\n0d∆2(1 +δ)∆2\n(∆2+δ∆1)21\n1 +α∆1δ\nδ2+α∆21 +δ\n(1 +δ)2+α(∆1−∆2). (50)\nAdding the results of all regions up, we get the result for the case in which the fermion travels in counterclockwise\ndirection:\nΣa\n3LF(k)→2α3\nπ2(Sa(α) +Ta(α))G+(k)−1ln/parenleftbiggΛy\nky/parenrightbigg\n−4α3\nπ2Ga(α)ikτln/parenleftbiggΛy\nky/parenrightbigg\n. (51)\nThe calculation of the case in which the fermion travels clockwise is completely analogous with similar expres-\nsions. The diagram reads\nΣb\n3LF(k) =−λ3\n+λ3\n−/integraldisplay[dl1][dl2][dp]\n(2π)9G+(k−l1)G+(k−l2)G−(p)G−(p−l1)G−(p−l2)\n×D(l1)D(l1−l2)D(l2).(52)\nIntegrate over l1x,l2x,pxandpyin order, we have\nΣb\n3LF(k) =−/integraldisplay\ndpτdl1τdl1ydl2τdl2y\n(2π)5D(l1)D(l1−l2)D(l2)\n2[(l2−l1)yG+(k)−1+2i(l2yl1τ−l1yl2τ−i(l2\n1yl2y−l1yl2\n2y))]T(pτ;l1τ,l2τ;l1y,l2y;kτ),(53)\nwhereT(pτ;l1τ,l2τ;l1y,l2y;kτ) is precisely the same as what was defined in Eq. (38). Following closely the\nprocedure outlined, we have\nΣb\n3LF(k)→−2α3\nπ2/parenleftbig\nSb(α) +Tb(α)/parenrightbig\nG+(k)−1ln/parenleftbiggΛy\nky/parenrightbigg\n+4α3\nπ2Gb(α)ikτln/parenleftbiggΛy\nky/parenrightbigg\n, (54)\nwhereGb(α) is defined as\nGb(α)≡/integraldisplay1\n0dδ/integraldisplay∞\n0d∆δ\nδ2+α∆1−δ\n(1−δ)2+α∆∆2\n∆2+ (δ2−δ)2, (55)\nandSb(α) andTb(α) are defined as\nSb(α)≡/integraldisplay1\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∞\n0d∆2(1−δ)∆2/parenleftbig\n(∆2+δ∆1)2−(δ2−δ)2/parenrightbig\n((∆2+δ∆1)2+ (δ−δ2)2)2\n×1\n1 +α∆1δ\nδ2+α∆21−δ\n(1−δ)2+α(∆1+ ∆ 2),(56)10\nTb(α)≡/integraldisplay∞\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∆1\n0d∆2(1 +δ)∆2/parenleftbig\n(∆2+δ∆1)2−(δ2+δ)2/parenrightbig\n((∆2+δ∆1)2+ (δ2+δ)2)2\n×1\n1 +α∆1δ\nδ2+α∆21 +δ\n(1 +δ)2+α(∆1−∆2).(57)\nCollecting the results for fermion self-energy at this order, we obtain the counter terms:\nδ(2)\n1=2α3\nπ2/parenleftbig\nSa(α) +Ta(α) + 2Ga(α)−Sb(α)−Tb(α)−2Gb(α)/parenrightbig\nlnΛy\nµ\nδ(2)\n2=2α3\nπ2/parenleftbig\nSa(α) +Ta(α)−Sb(α)−Tb(α)/parenrightbig\nlnΛy\nµ(58)\nwhereGa(α),Sa(α),Ta(α),Gb(α),Sb(α),Tb(α) are defined in Eqs. (47), (48), (50), (55), (56), (57) respectively.\nBefore we continue, let us mention that from the Ward identity [1], the fermion field strength divergence is the\nsame as the divergence of fermion-gauge vertex, whose contribution to order 1 /N2comes solely from Fig. 5 [8].\nFIG. 5: Diagram contributing to the divergence of fermion-fermion-boson vertex at order 1 /N2.\nC. Third leading Order Result of Boson Self-Energy\nIn this subsection, we evaluate the boson self energy at the third leading order and calculate δ(2)\n3, the contri-\nbution toZ3at the order of 1 /N2. Again, it is sufficient to keep only the divergent parts. It turns out that δ(2)\n3\nis double-logarithmic (double-log) divergent, i.e., δ(2)\n3∝/parenleftBig\nlnΛy\nµ/parenrightBig2\n, withµthe renormalization scale. The leading\ncontribution in the limit of µ/lessmuchΛyis\nδ(2)\n3=4α4\n3π2/parenleftbigg\nlnΛy\nµ/parenrightbigg2\n(59)\nThis result can be obtained by combining Eqs. (93), (114) and the renormalization condition in Eq. (5). The\nrest of this subsection is devoted to the computational details.\n1. General Strategies of Calculation and Assumptions\nSince the higher-loop diagrams are much more difficult to compute explicitly, we begin by outlining the general\nstrategies we use and spell out assumptions we make to simplify our calculations.\nWe draw all diagrams contributing to the third leading order, i.e. order 1 /N, of boson self-energy (see Fig. 6).\nIt turns out that some diagrams have internal boson momenta restricted as in Eq. (30). Using the calculation of\nthe AL diagram and the results in quantum field theory (QFT), we make the following local divergence assumption\n:integration with restricted y-momentum is convergent when the Feynman diagram contains no divergent subdia-\ngrams . Hence, only two sets of diagrams can contribute to double-log divergence, which we call Benz diagram and\n3-String diagram, and denote the results of these two sets of diagrams by Π Benz(ky) and Π 3String (ky), respectively.11\n(a) Benz Diagram.\n(b) 3-String Diagram.\n(c) Feynman diagrams with one fermion loop attached to three boson lines.\n(d) Feynman diagrams involving counterterms.\nFIG. 6: Diagrams contributing to the third leading order of boson self-energy. We do not repeat all the possible\nreshuffling of boson lines when attached to a fermion loop. We also do not include Feynman diagrams with only\none fermion loop, which are deemed to be zero in the static limit kτ→0 [1, 3].\nFor these two sets of diagrams, we can first use the residue theorem to integrate the x-momentum of all\ninternal propagators, as well as frequency and y-momentum of the two fermion loops. Then we are left with the\nintegration of internal boson frequency and y-momentum, denoted as liτandliy, withi= 1,2 for two internal\nboson lines. It turns out that at kτ= 0 these two contributions to the self energy take the form of Π Benz(ky) =\n1\n|ky|/integraltext\ndl1τdl1ydl2τdl2yB(ky,l1τ,l1y,l2τ,l2y) and Π 3String (ky) =1\n|ky|/integraltext\ndl1τdl1ydl2τdl2yS(ky,l1τ,l1y,l2τ,l2y), where\nBandScan be written as\nB(ky,l1τ,l1y,l2τ,l2y) =B(0,l1τ,l1y,l2τ,l2y) +aB(l1τ,l1y,l2τ,l2y)·ky+fB(ky,l1τ,l1y,l2τ,l2y)·k2\ny,\nS(ky,l1τ,l1y,l2τ,l2y) =S(0,l1τ,l1y,l2τ,l2y) +aS(l1τ,l1y,l2τ,l2y)·ky+fS(ky,l1τ,l1y,l2τ,l2y)·k2\ny,(60)\nwhereaB,Sis the coefficient of the ky-term in the Taylor expansion of B,S(ky,l1τ,l1y,l2τ,l2y), andfB,S(ky,l1τ,l1y,l2τ,l2y)\nare some analytic functions of kydefined by the above equations. It turns out that the integrations of both\nB(0,l1τ,l1y,l2τ,l2y) and S(0,l1τ,l1y,l2τ,l2y) are polynomially divergent for the individual diagrams. In the end,\nthey are canceled among all diagrams related to each other by rearranging the order in which photon lines attach\nto the fermion loops, e.g. between the Benz diagram in Fig. 6a and its twin diagram in Fig. 7, and among the six\ncases of 3-String diagrams shown in Fig. 8 [7]. Furthermore, Π( k) is an even function of kydue to the inversion\nsymmetry of the system, so the second terms (i.e., the terms proportional to ky) in the above equations do not\ncontribute. Hence we just need to integrate fB,S(ky,l1τ,l1y,l2τ,l2y), with a cutoff Λ yforl1yandl2y.\nTo extract potential double-log divergence in a simpler way, we apply the following expansion trick similar\nto the calculation of fermion self-energy. Since we are interested in the divergent part of the integration as\nΛy→∞ , it is enough to consider the region with sufficiently large liyandliτ. Therefore, we can simply integrate\nf(ky,l1τ,l1y,l2τ,l2y) atky= 0 here, and impose suitable dynamically generated IR cutoff for the integrals, i.e.,12\nthe scale above which f(0,l1τ,l1y,l2τ,l2y) is comparable with f(ky,l1τ,l1y,l2τ,l2y). In this way, the coefficient\nof the double-log divergent term can be extracted. To check the validity of this strategy, we compute the Benz\ndiagram in Sec. III C 3 explicitly and confirm that it gives the same answer as the one obtained from the method\nexplained above.\n2. All Diagrams at the Third Leading Order\nAll diagrams contributing to the boson self-energy to the third leading order in 1 /Nare listed in Fig. 6. For\nbrevity, we do not repeat diagrams related to the ones listed through rearranging the order in which boson lines\nare attached to a fermion loop. These diagrams are related to each other by the fact that their summation\nvanishes when external momentum vanishes [7]. Moreover, since we are mostly concerned with divergence in\nfront of the term |ky|of boson self-energy, we do not include Feynman diagrams which have only one fermion\nloop because they vanish in the static limit kτ→0 [1, 3].\nBecause of the kinetic constraint mentioned in the calculation of AL diagram in (30) [8], at the static limit\nkτ= 0, at least one boson’s y-momentum in Fig. 6c is bounded by external boson y-momentum ky. Hence, we\nconjecture that diagrams in Fig. 6c with no divergent subdiagrams are convergent. In fact, the first two diagrams\nhave divergent subdiagrams corresponding to one-loop fermion self-energy diagram as in Fig. 1, whose divergence\nis cancelled by the third and fourth diagrams in Fig. 6d involving counterterms.4We conjecture that the rest\nsix diagrams are simply finite like the AL diagram. The first and second diagram in Fig. 6d only give divergence\nto the Landau damping term, i.e., divergence in the term |kτ|/|ky|, and cancel with each other due to the Ward\nidentity. The fifth diagram in Fig. 6d is convergent.\nFinally, the Benz diagram as in Fig. 6a is explicitly calculated in Section III C 3 while the calculation of the\n3-String diagram in Fig. 6b is outlined in Section III C 45, with their double-log divergence given in Eqs . (93)\nand (114) (note that they are identically the same).\n3. Benz Diagram\nLet us briefly sketch the calculation of the Benz diagram. Before the calculation, we mention that rearanging\nthe order in which photon propagators attach to fermion loops results in a “twin” diagram of the Benz diagram\n(see Fig. 7), which is simply the nested diagram of the 3-loop fermion self-energy diagram in Fig. 3c together\nwith a single fermion loop. It only gives divergence to the Landau damping term |kτ|/|ky|that is canceled by\nthe second diagram in Fig. 6d involving counterterms corresponding to fermion field strength renormalization.\nSimilarly, the|kτ|/|ky|divergence of the Benz diagram is canceled by the first diagram in Fig. 6d involving\ncounterterms corresponding to vertex renormalization, which can be explicity checked.\nFIG. 7: The twin diagram of the Benz diagram after rearranging the order photon propagators attach to the\nfermion loop.\n4When we apply the Callan-Symanzik equation (6) to the third leading order of 1 /N,µ∂\n∂µacted on these two diagrams involving\ncounterterms is nonzero, yet it is cancelled by γzkτ∂\n∂kτacted on AL diagram.\n5See also Ref. [7] for the calculation of 3-String diagram with quadratic boson kinetic term, i.e. at /epsilon1= 1, where some (lnΛy\nky)5\ndivergence is identified.13\nConsidering different directions fermions in the loops of Benz diagram can travel as well as different patches\nthe fermions can belong to, there are altogether 16 different contributions from the Benz diagram, which we\ntemporarily denote as Π Benz =/summationtext(Π±,C/A ;±,C/A), where Π +C,−Arepresents the Benz diagram whose outer\nfermion loop lies on the + patch and runs in the clockwise (C) direction, while the inner fermion loop lies\non the−patch and runs in the anti-clockwise (A) direction, etc. Unless the two fermion loops belong to the\nopposite patches, the diagram vanishes [3, 10]. Since the photon propagator is independent of the x-component\nmomentum kx, it is convenient to take kx= 0, and it is then straightforward to see that simultaneously flipping\nthe patch labels of the two fermion loops does not change the result of the diagram, i.e., Π +C,−A= Π−C,+A, etc.\nFurthermore, simultaneously flipping the directions in which fermions run in the two loops will complex conjugate\nthe result, i.e., Π +A,−C= Π∗\n+C,−A. Therefore, the sum of all Benz diagrams can be written as\n4Re(Π +C,−C+ Π +C,−A). (61)\nTherefore, only two cases need to be considered : we can first set the outer loop fermion to be in the + patch\ntravelling clockwise, while the inner loop fermion is set to be in the −patch travelling either clockwise (case a)\nor anti-clockwise (case b), corresponding to Π +C,−Cand Π +C,−Arespectively.\nThe Benz diagram in the two cases read\nΠa\nBenz(k) =λ5\n+λ3\n−/integraldisplay\n[dl1][dl2][dp][dq]G+(q)G+(q+k)G+(q+k−l1)G+(q−l1)G+(q−l2)\n×G−(p)G−(p+l1)G−(p+l2)D(l1)D(l2)D(l1−l2),(62)\nΠb\nBenz(k) =λ5\n+λ3\n−/integraldisplay\n[dl1][dl2][dp][dq]G+(q)G+(q+k)G+(q+k−l1)G+(q−l1)G+(q−l2)\n×G−(p)G−(p−l1)G−(p−l2)D(l1)D(l2)D(l1−l2).(63)\nUsing the residue theorem to calculate the integration of l1x,l2x,px,qx,pyandqyin order, and taking the real\npart of the expression, we obtain the following expression,\nΠ/prime\nBenz(k)≡Re(Πa\nBenz+ Πb\nBenz) =1\n16(2π)6|ky|/integraldisplay\ndl1τdl1ydl2τdl2ydpτdqτ/parenleftBigg6/summationdisplay\ni=1Bi/parenrightBigg\nD(l1)D(l2)D(l1−l2),(64)\nwhere Biare defined as follows\nB1≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−kτ−qτ)−θ(−l1τ−pτ)) (θ(kτ−2l1τ−pτ+qτ)−θ(−pτ))\n×(θ(kτ−2l1τ+qτ)−θ(kτ+qτ))/parenleftbigg\nθ/parenleftbiggl1τ\nl1y/parenrightbigg\n−θ/parenleftbigg−kτ+ 2l1τ−2l2τ\n2l1y−2l2y/parenrightbigg/parenrightbigg\n×/parenleftbigg\nθ/parenleftbiggkτl1y+ 2l1yl2τ−2l1τl2y\n2l1y/parenrightbigg\n−θ(kτ)/parenrightbigg\nB1,(65)\nB2≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−kτ−qτ)−θ(−l1τ−pτ)) (θ(−2l2τ−pτ+qτ)−θ(−pτ))\n×(θ(−2l2τ+qτ)−θ(kτ+qτ))/parenleftbigg\nθ/parenleftbigg−kτ+ 2l1τ−2l2τ\n2l1y−2l2y/parenrightbigg\n−θ/parenleftbiggkτ+ 2l2τ\n2l2y/parenrightbigg/parenrightbigg\n×/parenleftbigg\nθ/parenleftbiggkτl1y+ 2l1yl2τ−2l1τl2y\n2l1y/parenrightbigg\n−θ(kτ)/parenrightbigg\nB1,(66)\nB3≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−kτ−qτ)−θ(−l1τ−pτ)) (θ(kτ−2l1τ−pτ+qτ)−θ(−pτ))\n×(θ(kτ−2l1τ+qτ)−θ(qτ))/parenleftbigg\nθ/parenleftbigg−kτ+ 2l1τ−2l2τ\n2l1y−2l2y/parenrightbigg\n−θ/parenleftbigg−kτ+ 2l1τ\n2l1y/parenrightbigg/parenrightbigg\n×/parenleftbigg\nθ/parenleftbiggkτl2y+ 2l1yl2τ−2l1τl2y\n2l2y/parenrightbigg\n−θ(kτ)/parenrightbigg\nB2,(67)\nB4≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−kτ−qτ)−θ(−l1τ−pτ)) (θ(−2l2τ−pτ+qτ)−θ(−pτ))\n×(θ(−2l2τ+qτ)−θ(qτ))/parenleftbigg\nθ/parenleftbiggl2τ\nl2y/parenrightbigg\n−θ/parenleftbigg−kτ+ 2l1τ−2l2τ\n2l1y−2l2y/parenrightbigg/parenrightbigg\n×/parenleftbigg\nθ/parenleftbiggkτl2y+ 2l1yl2τ−2l1τl2y\n2l2y/parenrightbigg\n−θ(kτ)/parenrightbigg\nB2,(68)14\nB5≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−qτ)−θ(−l1τ−pτ)) (θ(−2l1τ−pτ+qτ)−θ(−pτ))\n×(θ(−2l1τ+qτ)−θ(kτ+qτ))/parenleftbigg\nθ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg\n−θ/parenleftbiggkτ+ 2l1τ\n2l1y/parenrightbigg/parenrightbigg\n×/parenleftbigg\nθ/parenleftbiggkτ(l1y−l2y) + 2l1yl2τ−2l1τl2y\n2(l1y−l2y)/parenrightbigg\n−θ(kτ)/parenrightbigg\nB3,(69)\nB6≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−qτ)−θ(−l1τ−pτ)) (θ(−2l2τ−pτ+qτ)−θ(−pτ))\n×(θ(−2l2τ+qτ)−θ(kτ+qτ))/parenleftbigg\nθ/parenleftbiggkτ+ 2l2τ\n2l2y/parenrightbigg\n−θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\n×/parenleftbigg\nθ/parenleftbiggkτ(l1y−l2y) + 2l1yl2τ−2l1τl2y\n2(l1y−l2y)/parenrightbigg\n−θ(kτ)/parenrightbigg\nB3,(70)\nandBi=Ba\ni−Bb\niare defined as follows\nBa\n1≡l1y/parenleftbig\n(l1yl2τ−l1τl2y)2+k2\nyl2\n1yl2y(l1y−l2y)/parenrightbig\n/parenleftbig\n(l1yl2τ−l1τl2y)2+k2yl2\n1yl2\n2y/parenrightbig/parenleftbig\n(l1yl2τ−l1τl2y)2+k2yl2\n1y(l1y−l2y)2/parenrightbig,\nBa\n2≡l2y\n(l1yl2τ−l1τl2y)2+k2yl2\n1yl2\n2y,\nBa\n3≡l1y−l2y\n(l1yl2τ−l1τl2y)2+k2yl2\n1y(l1y−l2y)2,(71)\nBb\n1≡l1y/parenleftBig\n(l1yl2τ−l1τl2y)2−/parenleftBig\nl1yl2\n2y−l2\n1yl2y+kyl1yl2y/parenrightBig/parenleftBig\nl1yl2\n2y−l2\n1yl2y−kyl1y(l1y−l2y)/parenrightBig/parenrightBig\n/parenleftBig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y+kyl1yl2y)2/parenrightBig/parenleftBig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y−kyl1y(l1y−l2y))2/parenrightBig,\nBb\n2≡l2y/parenleftBig\n(l1yl2τ−l1τl2y)2−(l1yl2\n2y−l2\n1yl2y)(l1yl2\n2y−l2\n1yl2y+kyl1yl2y)/parenrightBig\n/parenleftBig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y)2/parenrightBig/parenleftBig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y+kyl1yl2y)2/parenrightBig,\nBb\n3≡(l1y−l2y)/parenleftBig\n(l1yl2τ−l1τl2y)2−(l1yl2\n2y−l2\n1yl2y)(l1yl2\n2y−l2\n1yl2y−kyl1y(l1y−l2y))/parenrightBig\n/parenleftBig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y)2/parenrightBig/parenleftbigg\n(l1yl2τ−l1τl2y)2+/parenleftBig\nl1yl2\n2y−l2\n1yl2y−kyl1y(l1y−l2y)/parenrightBig2/parenrightbigg.(72)\nNote thatBa\n1=Ba\n2+Ba\n3andBb\n1=Bb\n2+Bb\n3.\nSettingkτ= 0, we obtain\n6/summationdisplay\ni=1Bi=/parenleftbigg\nˆT1θ/parenleftbiggl1τ\nl1y/parenrightbigg\n−ˆT2θ/parenleftbiggl2τ\nl2y/parenrightbigg\n+ (ˆT2−ˆT1)θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\n(θ(l1y)−θ(l2y)) sgn (l1yl2τ−l1τl2y)B2\n+ (l1→−l1, l 2→l2−l1),(73)\nwhere the second term denotes the expression coming from transforming the first expression accordingly and the\nsimplification comes from the left-right flipping symmetry of the Benz diagram. Here ˆT1,2are defined as follows\nˆT1≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−qτ)−θ(−l1τ−pτ))\n(θ(−2l1τ−pτ+qτ)−θ(−pτ)) (θ(−2l1τ+qτ)−θ(qτ)),(74)\nˆT2≡(θ(l2τ−qτ)−θ(−l2τ−pτ)) (θ(l1τ−qτ)−θ(−l1τ−pτ))\n(θ(−2l2τ−pτ+qτ)−θ(−pτ)) (θ(−2l2τ+qτ)−θ(qτ)).(75)\nThen the first term and the second term should give the same contribution to Πa,b, and therefore we focus on the\nfirst term. We can now explicitly see that Πa,bare even functions of ky, as expected.\nFollowing the procedure outlined in Section III C 1, we expand Ba\n2at smallkyand focus on terms proportional\ntok2\ny,\nBa\n2→−l2\n1yl3\n2y\n(l1yl2τ−l1τl2y)2/parenleftbig\n(l1yl2τ−l1τl2y)2+k2yl2\n1yl2\n2y/parenrightbigk2\nyf(ky=0)·k2\ny−−−−−−−→−l2\n1yl3\n2y\n(l1yl2τ−l1τl2y)4k2\ny. (76)15\nWe will just use the expression f(0)k2\nyfor most of our calculation, yet the full expression is important in identifying\nthe dynamically generated IR cutoff for calculating the coefficient of the double-log. For Bb\n2, since only single-log\nis present, we just write down the expression of f(0)k2\nyhere for simplicity,\nBb\n2→l2\n1yl3\n2y/parenleftBigg\n−1\n/parenleftbig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y)2/parenrightbig2+8(l1yl2\n2y−l2\n1yl2y)2\n/parenleftbig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y)2/parenrightbig3\n−8(l1yl2\n2y−l2\n1yl2y)4\n/parenleftbig\n(l1yl2τ−l1τl2y)2+ (l1yl2\n2y−l2\n1yl2y)2/parenrightbig4/parenrightBigg\nk2\ny.(77)\nNote that the relevant terms in/summationtext6\ni=1Biare invariant if we simultaneously flip the sign of l1y,l2yorl1τ,l2τ.\nTherefore, let us restrict to the integration region where l1y>0 as well as l1τ>0. In this regime, the first\nterm of/summationtext6\ni=1Biis nonzero only when l2y<0 andl2τ>0, and after integrating over pτandqτit becomes\nB2×min(l1τ,l2τ)2. Now we are left with the integral over l1τl1y, andl2τandl2y. To perform this integral, first\nconsider the region where l1τ>l2τ. In order to isolate the divergence, we change the integration variables as\nl2y=−l1yδ, l 1τ=l2\n1y∆1, l 2τ=l2\n1y∆2. (78)\nAfter the integration of δ, ∆1,2as well asl1y, we have the following logarithmic divergent piece of Σa,bcoming\nfrom the first term in/summationtext6\ni=1Biin the region l1y>0,l2y<0,l1τ>l2τ>0,\n|ky|\ne2α4\n4π2/parenleftBigg/integraldisplaydl1y\nl1y/integraldisplay∞\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∆1\n0d∆2∆2\n2δ3(ωa−ωb)1\n1 +α∆1δ\nδ2+α∆21 +δ\n(1 +δ)2+α(∆1−∆2)/parenrightBigg\n(79)\nwhereωaandωbcomes fromBa\n2andBb\n2respectively,\nωa=1\n(∆2+δ∆1)2((∆2+δ∆1)2+ (ky/l1y)2δ2)ky=0−−−→1\n(∆2+δ∆1)4, (80)\nωb=1\n((δ2+δ)2+ (∆ 2+δ∆1)2)2−8(δ2+δ)2\n((δ2+δ)2+ (∆ 2+δ∆1)2)3+8(δ2+δ)4\n((δ2+δ)2+ (∆ 2+δ∆1)2)4. (81)\nNext consider the region where l2τ>l1τ. To compare two regions let us do the following change of parameters\nin this region:\nl2y=−|l1y|, l 1y=|l1y|δ, l 1τ=l2\n1y∆2, l 2τ=l2\n1y∆1. (82)\nThe logarithmic divergent piece in this region is then written as follows,\n|ky|\ne2α4\n4π2/parenleftBigg/integraldisplaydl1y\nl1y/integraldisplay∞\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∆1\n0d∆2∆2\n2δ2(ωa−ωb)1\n1 +α∆1δ\nδ2+α∆21 +δ\n(1 +δ)2+α(∆1−∆2)/parenrightBigg\n(83)\nAdding the contribution of all regions up is now straightforward, which is simply adding Eqs. (79) and (83)\nbefore multiplied by 4. The second term in/summationtext6\ni=1Bigives an extra factor of 2 while different directions in which\nfermions travel as well as different fermion patch numbers give an extra factor of 4. Carefully considering all\nthese factors, the Benz diagram gives the following log divergent contribution to boson self-energy\n4Π/prime\nBenz(k)→|ky|\ne28α4\nπ2/integraldisplaydl1y\nl1y(Ma(α)−Mb(α)), (84)\nwhereMa,b(α) are defined as\nMa,b(α)≡/integraldisplay∞\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∆1\n0d∆2∆2\n2δ2(1 +δ)ωa,b 1\n1 +α∆1δ\nδ2+α∆21 +δ\n(1 +δ)2+α(∆1−∆2)(85)\nwithωa,bdefined in Eqs. (80) and (81). At ky= 0, even though Mb(α) is just a regular function of α,Ma(α)\nis not, because of the divergent integration coming from the region where ∆ 1and ∆ 2go to zero simultaneously,\nwhich can be translated to the dynamical region where\n|l1y|,|l2y|/greatermuch|ky|,|l1τ|,|l2τ|≈|kyl1y|. (86)16\nNote that|kyl1y|is approximately the on-shell energy of a fermion with x-momentum l2\n1yandy-momentum\nky+l1y. Therefore the integral is actually double-log divergent instead of single-log divergent. Denoting the\nintegrand of Ma(α) asma(α), we separate Ma(α) in the following way\nMa(α)≡/integraldisplay∞\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∆1\n0d∆2ma(α)\n=/parenleftBigg/integraldisplay∞\n0dδ/integraldisplay∞\n1d∆1/integraldisplay∆1\n0d∆2ma(α)/parenrightBigg\n+/parenleftBigg/integraldisplay∞\n0dδ/integraldisplay1\n0d∆1/integraldisplay∆1\n0d∆2/parenleftbigg\nma(α)−∆2\n2δ\n(∆2+δ∆1)2((∆2+δ∆1)2+ (ky/l1y)2δ2)/parenrightbigg/parenrightBigg\n+/parenleftBigg/integraldisplay∞\n0dδ/integraldisplay1\n0d∆1/integraldisplay∆1\n0d∆2∆2\n2δ\n(∆2+δ∆1)2((∆2+δ∆1)2+ (ky/l1y)2δ2)/parenrightBigg\n.(87)\nThen only the third term is IR divergent at ky= 0, which will be related to the double-log divergence we are\nafter.\nIn order to compute the coefficient of the double-log through the expansion trick sketched in Section III C 1,\nwe need to first identify the IR cutoff for lτandly. For this, we return to the full expression of the integrand,\n/integraldisplayΛy\n0dl1y/integraldisplayΛy\n0dl2y/integraldisplayl2\n1y\n0dl1τ/integraldisplayl1τ\n0dl2τl2\n2τl1yl2y\n(l1yl2τ−lτl2y)2/parenleftbig\n(l1yl2τ−lτl2y)2+k2yl2\n1yl2\n2y/parenrightbig. (88)\nFrom Eq. (76) or (88), the crossover scale appears when ( l1yl2τ−lτl2y)2is of the same order as k2\nyl2\n1yl2\n2y, and\ntherefore the dynamically generated IR cutoff for lτshould bekyly. Again, note that |kyl1y|is approximately\nthe on-shell energy of a fermion with x-momentum l2\n1yandy-momentum ky+l1y. Accordingly, we can do the\nintegration of ∆ 1fromky/l1yinstead of 0 to 1, which gives us a log term. This log term is reminiscent of the\nBCS-log associated to the real Cooper pair [11, 12], which is also tied solely to the integration of frequency, and\nhence looks like a virtual BCS-log. Then the third term after the integration of δ,∆1,∆2as well asl1yis\n/integraldisplaydl1y\nl1yMa(α)→1\n6/integraldisplayΛy\nkydl1y\nl1y/integraldisplay1\nky/l1yd∆1\n∆1=1\n12/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n. (89)\nIt is also possible to calculate the coefficient of the relevant double-log divergent terms explicitly. Here, it is\npossible to do the explicit integration. After integrating over ∆ 1, ∆2andδ, the result is a function f(ky/l1y),\nexplicitly written out as follows,\nf(x) =x/parenleftbig/parenleftbig\nx2+ 3/parenrightbig\nln/parenleftbig\nx2+ 1/parenrightbig\n−2x2ln(x)−4/parenrightbig\n+ 4 arctan(x)\n12x3. (90)\nand the third term now becomes/integraltextdl1y\nl1yf/parenleftBig\nky\nl1y/parenrightBig\n. From the limiting behavior of f(x) at small and large x,\nf(x) =/braceleftBigg\n−1\n6ln(x)x/lessmuch1\n1\n2ln(x)\nx2x/greatermuch1(91)\nwe see that the integration/integraltextdx\nxf(1\nx) is IR convergent and UV double-log divergent. Hence the integration of l1y\nis IR convergent and UV double-log divergent. This integration can be carried out explicitly as well, which at\nlarge Λy/kyis\n1\n12/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n+5\n36ln/parenleftbiggΛy\nky/parenrightbigg\n+3π2+ 38\n432. (92)\nThe double-log term in (92) agrees with Eq. (89) that is obtained from the simplified scheme. This yields the\npromised double-log divergent term\nΠBenz(k)→|ky|\ne22α4\n3π2/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n. (93)17\n4. 3-String Diagram\nFinally, let us briefly sketch the calculation of the 3-String diagram. For simplicity, we focus on the case where\nkτ= 0 whenever possible. As in previous cases, the two fermion loops must belong to different patches to give\nrise to UV divergence. Since the photon propagator is independent of the x-component momentum kx, we can\nset the left fermion loop to be in the −patch without loss of generality. We can also fix that fermion in the left\nloop runs in counterclockwise direction and fermion in the right loop runs in clockwise direction. Then there are\n6 possible ways in which the three internal photon propagators attach to the two loops, shown in Fig. 8. After\nsimultaneously flipping the directions in which fermions run in the two loops, diagram aandbdo not change,\nwhilec,dande,ftransform into each other respectively. Therefore, the contribution of the 3-String diagram as\na whole becomes\n2(Πa+ Πb) + 4Re(Πc+ Πe), (94)\nwhere\nFIG. 8: All possible 3-String diagrams depending on how three internal photon propagators attach to the\nfermion loops. From left to right then from top down are case a,b,c,d,eandfrespectively.\nΠa\n3String (k) =λ4\n+λ4\n−/integraldisplay\n[dl1][dl2][dp][dq]G−(p)G−(p+k)G−(p+l1+k/2)G−(p+l2+k/2)\n×G+(q)G+(q−k)G+(q−l1−k/2)G+(q−l2−k/2)D(k/2 +l1)D(k/2−l2)D(l1−l2),(95)\nΠb\n3String (k) =λ4\n+λ4\n−/integraldisplay\n[dl1][dl2][dp][dq]G−(p)G−(p+k)G−(p+l1+k/2)G−(p+l2+k/2)\n×G+(q)G+(q+k)G+(q+l1+k/2)G+(q+l2+k/2)D(k/2 +l1)D(k/2−l2)D(l1−l2),(96)\nΠc\n3String (k) =λ4\n+λ4\n−/integraldisplay\n[dl1][dl2][dp][dq]G−(p)G−(p+k)G−(p+l1+k/2)G−(p+l2+k/2)\n×G+(q)G+(q+l1+k/2)G+(q+l2+k/2)G+(q+l2−k/2)D(k/2 +l1)D(k/2−l2)D(l1−l2),(97)\nΠe\n3String (k) =λ4\n+λ4\n−/integraldisplay\n[dl1][dl2][dp][dq]G−(p)G−(p+k)G−(p+l1+k/2)G−(p+l2+k/2)\n×G+(q)G+(q−l1−k/2)G+(q−l2−k/2)G+(q−l2+k/2)D(k/2 +l1)D(k/2−l2)D(l1−l2).(98)18\nAgain, we use the residue theorem to calculate the integration of l1x,l2x,px,qx,pyandqyin order. The expression\nat non-zero kτis extremely lengthy, but there is some simplification when kτ= 0:\nΠa,b,c,e\n3String (ky) =sgn(ky)\n(2π)6|ky|/integraldisplay\ndl1τdl1ydl2τdl2ydpτdqτ/parenleftBig\n−ˆT1Sa,b,c,e\n1 +ˆT2Sa,b,c,e\n2/parenrightBig\nD(k/2 +l1)D(k/2−l2)D(l2−l1),(99)\nwhere ˆT1,2are defined in (74) and (75) and\nSa,b\ni≡/parenleftbigg\nθ/parenleftbigg2liτ\n2liy−ky/parenrightbigg\n−θ/parenleftbigg2liτ\n2liy+ky/parenrightbigg/parenrightbigg/parenleftbigg\nθ/parenleftbigg2liτ\n2liy−ky/parenrightbigg\nSa,b\ni,−−θ/parenleftbigg2liτ\n2liy+ky/parenrightbigg\nSa,b\ni,+/parenrightbigg\n+/parenleftBigg/parenleftbigg\nθ/parenleftbigg2liτ\n2liy−ky/parenrightbigg\n−θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\nθ/parenleftbiggky(l1τ−l2τ) + 2l1yl2τ−2l1τl2y\nky(l1y−l2y)/parenrightbigg\n−/parenleftbigg\nθ/parenleftbigg2liτ\n2liy+ky/parenrightbigg\n−θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\nθ/parenleftbiggky(l1τ−l2τ)−2l1yl2τ+ 2l1τl2y\nky(l1y−l2y)/parenrightbigg/parenrightBigg\nSa,b\n0,(100)\nSc,e\n1≡/parenleftbigg\nθ/parenleftbigg2l1τ\n2l1y−ky/parenrightbigg\n−θ/parenleftbigg2l1τ\n2l1y+ky/parenrightbigg/parenrightbigg\nθ/parenleftbigg2l1τ\n2l1y+ky/parenrightbigg\nSc,e\n1\n+/parenleftbigg\nθ/parenleftbigg2l1τ\n2l1y+ky/parenrightbigg\n−θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\nθ/parenleftbiggky(l1τ−l2τ)−2l1yl2τ+ 2l1τl2y\nky(2l1y+ky)/parenrightbigg\nSc,e\n1\n−/parenleftbigg\nθ/parenleftbigg2l1τ\n2l1y−ky/parenrightbigg\n−θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg/parenleftBigg\nθ/parenleftbiggky(l1τ−l2τ) + 2l1yl2τ−2l1τl2y\nky(−2l2y+ky)/parenrightbigg\nSc,e\n2\n+θ/parenleftbiggky(l1τ−l2τ) + 2l1yl2τ−2l1τl2y\nky(l1y−l2y)/parenrightbigg\nSc,e\n0/parenrightBigg\n,(101)\nSc,e\n2≡/parenleftbigg\nθ/parenleftbigg2l2τ\n2l2y+ky/parenrightbigg\n−θ/parenleftbigg2l2τ\n2l2y−ky/parenrightbigg/parenrightbigg/parenleftbigg\nθ/parenleftbigg2l2τ\n2l2y−ky/parenrightbigg\nSc,e\n2,−−θ/parenleftbigg2l2τ\n2l2y+ky/parenrightbigg\nSc,e\n2,+/parenrightbigg\n+/parenleftbigg\nθ/parenleftbigg2l2τ\n2l2y+ky/parenrightbigg\n−θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\nθ/parenleftbiggky(l1τ−l2τ)−2l1yl2τ+ 2l1τl2y\nky(2l1y+ky)/parenrightbigg\nSc,e\n1\n−/parenleftbigg\nθ/parenleftbigg2l2τ\n2l2y−ky/parenrightbigg\n−θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg/parenleftBigg\nθ/parenleftbiggky(l1τ−l2τ) + 2l1yl2τ−2l1τl2y\nky(−2l2y+ky)/parenrightbigg\nSc,e\n2\n+θ/parenleftbiggky(l1τ−l2τ) + 2l1yl2τ−2l1τl2y\nky(l1y−l2y)/parenrightbigg\nSc,e\n0/parenrightBigg\n,(102)\nwhere the most important Sfactors for case aare defined as follows,\nSa\ni,±≡ky±2liy\n16kyliτ(ky(l1τ−l2τ)∓2(l1yl2τ−l1τl2y)),\nSa\n0≡l1y−l2y\n4 (ky(l1τ−l2τ) + 2l1yl2τ−2l1τl2y) (ky(l1τ−l2τ)−2l1yl2τ+ 2l1τl2y),(103)\nand for case b,\nSb\ni,±≡(ky±2liy)/slashBigg/braceleftBig\n2ky/parenleftbig\n4liτ−i(k2\ny−4l2\niy)/parenrightbig\n×/parenleftbig\n2 (ky(l1τ−l2τ)∓2 (l1yl2τ−l1τl2y))±i/parenleftbig\nk2\ny(l1y−l2y) + 4(l2\n1yl2y−l2\n2yl1y)/parenrightbig\n+ 2iky(l2\n1y−l2\n2y)/parenrightbig/bracerightBig\n,\nSb\n0≡(l1y−l2y)/slashBigg/braceleftBig/parenleftbig\n2 (ky(l1τ−l2τ) + 2 (l1yl2τ−l1τl2y))−i/parenleftbig\nk2\ny(l1y−l2y) + 4(l2\n1yl2y−l2\n2yl1y)/parenrightbig\n+ 2iky(l2\n1y−l2\n2y)/parenrightbig\n×/parenleftbig\n2 (ky(l1τ−l2τ)−2 (l1yl2τ−l1τl2y)) +i/parenleftbig\nk2\ny(l1y−l2y) + 4(l2\n1yl2y−l2\n2yl1y)/parenrightbig\n+ 2iky(l2\n1y−l2\n2y)/parenrightbig/bracerightBig\n,(104)19\nand for case c,\nSc\n1≡(−ky−2l1y)/slashBigg/braceleftBig\n2/parenleftBig\nk2\nyl1y−k2\nyl2y−2ikyl1τ+ 2kyl2\n1y+ 2ikyl2τ−2kyl2\n2y−4il1τl2y+ 4l2\n1yl2y+ 4il1yl2τ−4l1yl2\n2y/parenrightBig\n×/parenleftBig\nk2\nyl1y−k2\nyl2y−2ikyl1τ+ 2kyl2\n1y−4kyl1yl2y−2ikyl2τ+ 2kyl2\n2y+ 4il1τl2y−4l2\n1yl2y−4il1yl2τ+ 4l1yl2\n2y/parenrightBig/bracerightBig\n,\nSc\n2≡(−ky+ 2l2y)/slashBigg/braceleftBig\n2/parenleftBig\n−k2\nyl1y+k2\nyl2y+ 2ikyl1τ−2kyl2\n1y+ 4kyl1yl2y−2ikyl2τ−2kyl2\n2y−4il1τl2y+ 4l2\n1yl2y+ 4il1yl2τ−4l1yl2\n2y/parenrightBig\n×/parenleftBig\n−k2\nyl1y+k2\nyl2y+ 2ikyl1τ−2kyl2\n1y+ 4kyl1yl2y+ 2ikyl2τ−2kyl2\n2y−4il1τl2y+ 4l2\n1yl2y+ 4il1yl2τ−4l1yl2\n2y/parenrightBig/bracerightBig\n,\nSc\n0=Sc\n1−Sc\n2≡(−l1y+l2y)/slashBigg/braceleftBig/parenleftBig\nk2\nyl1y−k2\nyl2y−2ikyl1τ+ 2kyl2\n1y+ 2ikyl2τ−2kyl2\n2y−4il1τl2y+ 4l2\n1yl2y+ 4il1yl2τ−4l1yl2\n2y/parenrightBig\n×/parenleftBig\nk2\nyl1y−k2\nyl2y−2ikyl1τ+ 2kyl2\n1y−4kyl1yl2y+ 2ikyl2τ+ 2kyl2\n2y+ 4il1τl2y−4l2\n1yl2y−4il1yl2τ+ 4l1yl2\n2y/parenrightBig/bracerightBig\n,\nSc\n2,−≡i(ky−2l2y)/slashBigg/braceleftBig\n8kyl2τ/parenleftBig\n−k2\nyl1y+k2\nyl2y+ 2ikyl1τ−2kyl2\n1y+ 4kyl1yl2y+ 2ikyl2τ−2kyl2\n2y−4il1τl2y+ 4l2\n1yl2y+ 4il1yl2τ−4l1yl2\n2y/parenrightBig/bracerightBig\n,\nSc\n2,+≡i(ky+ 2l2y)/slashBigg/braceleftBig\n8kyl2τ/parenleftbig\n−k2\nyl1y+k2\nyl2y+ 2ikyl1τ−2kyl2\n1y−2ikyl2τ+ 2kyl2\n2y+ 4il1τl2y−4l2\n1yl2y−4il1yl2τ+ 4l1yl2\n2y/parenrightbig/bracerightBig\n,(105)\nand, finally, for case e,\nSe\n1≡i(ky+ 2l1y)/slashBigg/braceleftBig\n4(kyl1τ−kyl2τ+ 2l1τl2y−2l1yl2τ)\n×/parenleftbig\n2k2\nyl1y−2k2\nyl2y+k3\ny+ 2ikyl1τ−4kyl1yl2y+ 2ikyl2τ−4il1τl2y+ 4il1yl2τ/parenrightbig/bracerightBig\n,\nSe\n2≡(ky−2l2y)/slashBigg/braceleftBig\n4/parenleftbig\n−k2\nyl1y+k2\nyl2y−ikyl1τ+ 2kyl1yl2y+ikyl2τ−2kyl2\n2y+ 2il1τl2y−2il1yl2τ/parenrightbig\n×/parenleftbig\n2k2\nyl1y−2k2\nyl2y+k3\ny+ 2ikyl1τ−4kyl1yl2y+ 2ikyl2τ−4il1τl2y+ 4il1yl2τ/parenrightbig/bracerightBig\n,\nSe\n0=Se\n1−Se\n2≡−i(l1y−l2y)/slashBigg/braceleftBig\n4(kyl1τ−kyl2τ+ 2l1τl2y−2l1yl2τ)\n×/parenleftbig\n−k2\nyl1y+k2\nyl2y−ikyl1τ+ 2kyl1yl2y+ikyl2τ−2kyl2\n2y+ 2il1τl2y−2il1yl2τ/parenrightbig/bracerightBig\n,\nSe\n2,−≡(−ky+ 2l2y)/slashBigg/braceleftBig\n2ky/parenleftbig\nk2\ny+ 4il2τ−4l2\n2y/parenrightbig\n×/parenleftbig\n2k2\nyl1y−2k2\nyl2y+k3\ny+ 2ikyl1τ−4kyl1yl2y+ 2ikyl2τ−4il1τl2y+ 4il1yl2τ/parenrightbig/bracerightBig\n,\nSe\n2,+≡i(ky+ 2l2y)/slashBigg/braceleftBig\n4ky/parenleftbig\nk2\ny+ 4il2τ−4l2\n2y/parenrightbig\n(kyl1τ−kyl2τ−2l1yl2τ+ 2l1τl2y)/bracerightBig\n.(106)\nFirst consider case aandb. Because of the theta pre-factors, only Sa,b\n0contribute to UV divergence according\nto the local divergence conjecture in Section III C 1. Setting ky= 0 everywhere in the theta factors except for an\noverall sgn( ky), we obtain\n−ˆT1Sa,b\n1+ˆT2Sa,b\n2≈/parenleftbigg\n−ˆT1θ/parenleftbiggl1τ\nl1y/parenrightbigg\n+ˆT2θ/parenleftbiggl2τ\nl2y/parenrightbigg\n+ (ˆT1−ˆT2)θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\nsgn/parenleftbiggl1yl2τ−l1τl2y\nky(l1y−l2y)/parenrightbigg\nSa,b\n0.(107)\nAgain the real part of this expression is invariant if we simultaneously flip the sign of ky,l1y,l2yorl1τ,l2τ.20\nTherefore let us assume l1τ>0 as well as l1y>0. Then we have six regions where the expression is non-zero,\nregion 1: l2τ>l1τ>0, l1y>l2y>0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sa,b\n1+ˆT2Sa,b\n2/parenrightBig\n=−(l2τ−l1τ)2sgn(ky)Sa,b\n0,\nregion 2: l2τ>l1τ>0, l2y<0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sa,b\n1+ˆT2Sa,b\n2/parenrightBig\n=−l2\n1τsgn(ky)Sa,b\n0,\nregion 3: l1τ>l2τ>0, l2y>l1y>0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sa,b\n1+ˆT2Sa,b\n2/parenrightBig\n= (l2τ−l1τ)2sgn(ky)Sa,b\n0,\nregion 4: l1τ>l2τ>0, l2y<0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sa,b\n1+ˆT2Sa,b\n2/parenrightBig\n=−l2\n2τsgn(ky)Sa,b\n0,\nregion 5: l2τ<0, l2y>l1y>0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sa,b\n1+ˆT2Sa,b\n2/parenrightBig\n=l2\n1τsgn(ky)Sa,b\n0,\nregion 6: l2τ<0, l1y>l2y>0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sa,b\n1+ˆT2Sa,b\n2/parenrightBig\n=−l2\n2τsgn(ky)Sa,b\n0.(108)\nAfter interchanging l1τ,l2τas well asl1y,l2ysimultaneously, we see that the expressions in region 1 and 3, 2 and\n4, as well as 5 and 6 are identical to each other respectively, and therefore we can focus on region 1, 2 and 5. The\nanalysis of case candeare straightforward as well. Because of the theta pre-factors, Sc,e\n2,±do not contribute to\nthe divergence in the UV. Setting ky= 0 everywhere in the theta factors except for an overall sgn( ky), we obtain\n−ˆT1Sc,e\n1+ˆT2Sc,e\n2≈/parenleftbigg\n−ˆT1θ/parenleftbiggl1τ\nl1y/parenrightbigg\n+ˆT2θ/parenleftbiggl2τ\nl2y/parenrightbigg\n+ (ˆT1−ˆT2)θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\n×sgn (l1yl2τ−l1τl2y) sgn(ky) (θ(−l1y)Sc,e\n1−θ(−l2y)Sc,e\n2−θ(l1y−l2y)Sc,e\n0).(109)\nRestricting to the regime where l1y>0 as well as l1τ>0, we have four different sections on the l2y−l2τplane\nwhere the expression is non-zero,\nregion 1: l2τ>l1τ>0, l1y>l2y>0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sc,e\n1+ˆT2Sc,e\n2/parenrightBig\n= (l2τ−l1τ)2sgn(ky)Sc,e\n0,\nregion 2: l2τ>l1τ>0, l2y<0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sc,e\n1+ˆT2Sc,e\n2/parenrightBig\n=l2\n1τsgn(ky)Sc,e\n1,\nregion 4: l1τ>l2τ>0, l2y<0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sc,e\n1+ˆT2Sc,e\n2/parenrightBig\n=l2\n2τsgn(ky)Sc,e\n1,\nregion 6: l2τ<0, l1y>l2y>0,/integraldisplay\ndpτdqτ/parenleftBig\n−ˆT1Sc,e\n1+ˆT2Sc,e\n2/parenrightBig\n=l2\n2τsgn(ky)Sc,e\n0.(110)\nThe expression for single log divergence is very involved, but it is managable to calculate the double log\ndivergence as sketched in Section III C 1. First, we expand the whole expression in powers of ky. As in the\ncalculation of the Benz diagram, we collect terms at the third order in the expansion, i.e. terms proportional to\nk2\ny. Note that kymainly appears in two different places, in Sfactors as well as in D(k/2 +l1)D(k/2−l2)6. To\nsimplify expressions further, we are allowed to set ky= 0 inside the boson propagators except for the case cand\nregion 2 and 4 of case e, where we have to take into account the second term in the expansion of both Sc,e\n1and\nD(k/2 +l1)D(k/2−l2) at smallky, i.e. terms proportional to kyin two terms respectively7.\nFor the sake of presentation, let us first identify the double log divergence coming from the dynamical region\n(86)8. We just need to collect terms at the third order of the expansion that are proportional to 1 /l4\nτand\ncontribute to the IR divergence in the integration of l1τ,l2τ. These terms only come from case e, hence we write\ndown the terms proportional to kyin the series expansion of Se,\nSe\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nkylinear term=/parenleftBigg\nl2\n2y(l1y−l2y)3\n16(l1yl2τ−l1τl2y)4−(l1y−l2y)(l1τ−l2τ)2\n64(l1yl2τ−l1τl2y)4/parenrightBigg\nk2\ny,\nSe\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nkylinear term=/parenleftBigg\nl2\n2yl3\n1y\n16(l1yl2τ−l1τl2y)4−l1y(l2\n1τ+l2\n2τ) + 2l2yl1τl2τ\n64(l1yl2τ−l1τl2y)4/parenrightBigg\nk2\ny.(111)\n6kyalso appears in theta factors, but there it only contributes to a shift of integration regions in the IR that we can ignore for the\ncalculation in the UV.\n7ExpandingSfactors at small ky, symbolically we have S≈x0+x1ky+x2k2\ny. Moreover, D(k/2+l1)D(k/2−l2)≈d0+d1ky+d2k2\ny.\nWe can explicitly check that the integration of x0d0, i.e. the constant term, is cancelled among the six cases. The same calculation\nalso shows that the integration of x0d2is cancelled among the six cases. What we now have are x2d0as well asx1d1. We then\nnotice that x1= 0 expect forSc\n0,1andSe\n1.\n8Here we need to distinguish between integration region 1-6, and dynamical region on the boundary of respective integration region\nthat contributes mostly to the double log divergence.21\nWe see that ky-linear terms ofSeare separated into two terms Se,(1)+Se,(2). The first terms Se,(1)of the two\nexpansions are proportional to 1 /l4\nτwhile the second terms Se,(2)are not, and we isolate the first terms,\nSe,(1)\n1=l3\n1yl2\n2y\n16(l1yl2τ−l1τl2y)4k2\ny,\nSe,(1)\n2=l3\n2y(l2\n2y−3l1yl2y+ 3l2\n1y)\n16(l1yl2τ−l1τl2y)4k2\ny,\nSe,(1)\n0=l2\n2y(l1y−l2y)3\n16(l1yl2τ−l1τl2y)4k2\ny.(112)\nJust considering these terms coming from case e, we see that, to our surprise, Eq. (109) becomes identical to Eq.\n(73) after interchanging l1τ,l1ywithl2τ,l2y. Namely, after substituting Sein (109) withSe,(1)and interchanging\nlabell1τ,l1yandl2τ,l2y, we have\n/parenleftBig\n−ˆT1Se\n1+ˆT2Se\n2/parenrightBig(1)\n=sgn(ky)k2\ny\n16/parenleftbigg\nˆT1θ/parenleftbiggl1τ\nl1y/parenrightbigg\n−ˆT2θ/parenleftbiggl2τ\nl2y/parenrightbigg\n+ (ˆT2−ˆT1)θ/parenleftbiggl1τ−l2τ\nl1y−l2y/parenrightbigg/parenrightbigg\n×/parenleftBigg\n−(θ(l1y)−θ(l2y)) sgn (l1yl2τ−l1τl2y)l2\n1yl3\n2y\n(l1yl2τ−l1τl2y)4+ (l1→−l1, l 2→l2−l1)/parenrightBigg\n,(113)\nwhich is identical to the expression we obtain after substituting (76) into (73), except for an extra factorsgn(ky)\n16\nwhich can be accounted for from different factors in front of the whole expression as shown in (64) and (99).\nMoreover, both contribute to photon self-energy with an extra factor of 4 as shown in (61) and (94). Therefore,\nwe calculate the double log divergence of the 3-String diagram originating from the same source as the Benz\ndiagram, i.e. coming from the dynamical region Eq. (86),\nΠ3String (k)→|ky|\ne22α4\n3π2/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n. (114)\nAfter subtracting these terms out, now we aim to show that the double log divergences of the rest of the 3-String\ndiagram cancel among the six cases. These double-log divergences that are canceled among themselves come from\ndynamical regions with large l1τorl2τ. The UV divergences from these high-energy modes can only renormalize\nlocal terms in the action. As a result, both single-log and double-log, which require non-local counter terms, are\nexpected to be canceled. One interesting conclusion we can draw from this surprising result is that the double log\ndivergence actually does not appear at this order in the /epsilon1= 0 version of Ising-nematic transition in a metal [1],\nwhich is very closely related to the nFL theory we discussed here except λ+=λ−= +1. Here λ+=λ−because\nthe critical Ising-nematic order parameter, which is the analog of the field adiscussed here, couples to a fermion\nbilinear operator whose symmetry form factor requires λ+=λ−.\nLet us do the calculation in region 4 of case eas an example. To unearth the double-log divergence from the\nsecond term in the expansion (111), we do the following change of parameters:\nl2y=−l1yδ, l 1τ=l2\n1y∆1, l 2τ=l2\n1y∆2, (115)\nand integrate|l1y|fromkyto Λy. The logarithmic divergent piece in this region from the second term is then\nwritten as follows,\n(Πe(k))(2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nregion 4≈|ky|\n(2π)6/integraldisplay∞\n0dl1y/integraldisplay0\n−∞dl2y/integraldisplay∞\n0dl1τ/integraldisplayl1τ\n0dl2τl2\n2τSe,(2)\n1D(k/2 +l1)D(k/2−l2)D(l2−l1)\n=−|ky|\ne2α4\n16π2/parenleftBigg/integraldisplaydl1y\nl1y/integraldisplay∞\n0dδ/integraldisplay∞\n0d∆1/integraldisplay∆1\n0d∆2∆2\n2(∆2\n1+ ∆2\n2+ 2δ∆1∆2)\n(∆2+δ∆1)4\n×1\n1 +α∆1δ\nδ2+α∆21 +δ\n(1 +δ)2+α(∆1−∆2)/parenrightBigg\n.(116)\nThe above integration is IR divergent when ∆ 2andδgo to zero simultanesouly, which can be translated to the\ndynamical region where\n|l1y|/greatermuch|ky|,|l1τ|/greatermuch|kyl1y|,|l2y|≈|ky|,|l2τ|≈|kyl1y|. (117)22\nTo simplify the expression further, we change ∆ 2to ∆/prime\n2δ, then at small δthe integrand is approximately\n∆/prime\n2∆2\n1\nα(∆1+ ∆/prime\n2)4(1 +α∆1)21\nδ, (118)\nwhich gives the following double log divergence in the bracket\n1\n6α2/integraldisplayΛy\nkydl1y\nl1y/integraldisplay1\nky/l1ydδ\nδ=1\n12α2/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n. (119)\nThe calculation in the other integration regions and other cases are as straightforward, and we see that the\ndouble-log divergent terms in region 2, 4, 5 and 6 of case aandbare all the same, i.e.\nΠa,b(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nregion 2,4,5,6→|ky|\ne2α2\n192π2/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n, (120)\nwhile the double log divergent terms in region 4 and 6 of case cand the second term of eare all the same, i.e.\nΠc(k),(Πe(k))(2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nregion 4,6→−|ky|\ne2α2\n192π2/parenleftbigg\nln/parenleftbiggΛy\nky/parenrightbigg/parenrightbigg2\n, (121)\nand in all the other regions there is no additional double log divergence. Adding them all up we see that the\nadditional double log divergence cancels among the six cases, as promised.\nIt is worth pointing out that the calculation of the 3-String diagram relies on the “local divergence conjecture”\nwe make in Section III C 1 as well as the correctness of the expansion trick with correct dynamically generated\nIR cutoff. It will be very interesting if a more rigorous calculation of these coefficients, either analytically or\nnumerically, can be done. We defer this to future work.\n[1] Max A. Metlitski and Subir Sachdev, “Quantum phase transitions of metals in two spatial dimensions. I. Ising-nematic\norder,” Phys. Rev. B 82, 075127 (2010), arXiv:1001.1153 [cond-mat.str-el].\n[2] Sung-Sik Lee, “Stability of the U(1) spin liquid with a spinon Fermi surface in 2+1 dimensions,” Phys. Rev. B 78,\n085129 (2008), arXiv:0804.3800 [cond-mat.str-el].\n[3] Sung-Sik Lee, “Low-energy effective theory of Fermi surface coupled with U(1) gauge field in 2+1 dimensions,” Phys.\nRev. B 80, 165102 (2009), arXiv:0905.4532 [cond-mat.str-el].\n[4] David F. Mross, John McGreevy, Hong Liu, and T. Senthil, “Controlled expansion for certain non-Fermi-liquid\nmetals,” Phys. Rev. B 82, 045121 (2010), arXiv:1003.0894 [cond-mat.str-el].\n[5] Liujun Zou and Debanjan Chowdhury, “Deconfined metallic quantum criticality: A U(2) gauge-theoretic approach,”\nPhysical Review Research 2, 023344 (2020), arXiv:2002.02972 [cond-mat.str-el].\n[6] Liujun Zou and Debanjan Chowdhury, “Deconfined Metal-Insulator Transitions in Quantum Hall Bilayers,” Physical\nReview Research 2, 032071 (2020), arXiv:2004.14391 [cond-mat.str-el].\n[7] Tobias Holder and Walter Metzner, “Anomalous dynamical scaling from nematic and U(1) gauge field fluctuations in\ntwo-dimensional metals,” Phys. Rev. B 92, 041112 (2015), arXiv:1503.05089 [cond-mat.str-el].\n[8] Tobias Holder and Walter Metzner, “Fermion loops and improved power-counting in two-dimensional critical metals\nwith singular forward scattering,” Phys. Rev. B 92, 245128 (2015), arXiv:1509.07783 [cond-mat.str-el].\n[9] L. G. Aslamazov and A. I. Larkin, “Effect of Fluctuations on the Properties of a Superconductor Above the Critical\nTemperature - Sov. Phys. Solid State 10, 875 (1968),” in 30 Years of the Landau Institute , Vol. 11, edited by Isaak M.\nKhalatnikov and Vladimir P. Mineev (1996) p. 23.\n[10] Shouvik Sur and Sung-Sik Lee, “Chiral non-Fermi liquids,” Phys. Rev. B 90, 045121 (2014), arXiv:1310.7543 [cond-\nmat.str-el].\n[11] Joseph Polchinski, “Effective Field Theory and the Fermi Surface,” arXiv e-prints , hep-th/9210046 (1992), arXiv:hep-\nth/9210046 [hep-th].\n[12] R. Shankar, “Renormalization-group approach to interacting fermions,” Reviews of Modern Physics 66, 129–192\n(1994), arXiv:cond-mat/9307009 [cond-mat]."    },    {        "title": "2210.02902v1.Spatial_segregation_of_substitutional_B_atoms_in_graphene_patterned_by_the_moiré_superlattice_on_Ir_111_.pdf",        "content": " 1Spatial Segregation of Substitutional B Atoms in \nGraphene Patterned by the Moiré Superlattice on Ir(111) \nMarc G. Cuxart,‡* Daniele Perilli,‡ Sena Tömekce, Joel Deyerling, Felix Haag, Matthias \nMuntwiler, Francesco Allegretti, Cristiana Di Valentin,* and Willi Auwärter* \nFabrication of ordered structures at the nanoscale limit poses a cornerstone challenge for modern \ntechnologies. In this work we show how naturally occurring moiré patterns in Ir(111)-supported \ngraphene template the formation of 2D ordered arrays of substitutional boron species. A \ncomplementary experimental and theoretical approach provides a comprehensive description of \nthe boron species distribution, bonding configurations, interfacial interaction with Ir(111) and the \nimpact on graphene’s electronic structure. Atomically-resolved scanning tunnelling microscopy \nimages and density functional theory calculations reveal that boron preferably forms small \naggregates of substitutional defects in geometrically low regions of the moiré superlattice of \ngraphene, by inducing local bending of graphene towards the underlying Ir(111). Surprisingly, \ncalculations reveal that the incorporation of electron deficient boron does not lead to an enhanced \np-type doping, as the local rippling of the graphene layer prompts electron uptake from the iridium \nsubstrate that compensates the initial electron loss due to substitutional boron. Scanning tunnelling \nspectroscopy and angle-resolved photoemission spectroscopy measurements corroborate that the \narrays of boron species do not modify the electronic structure of graphene near the Fermi level, \nhence preserving the slight p-type doping induced by Ir(111).  2KEYWORDS: Boron doping, graphene, nano-fabrication, moiré superlattice, CVD, XPS, STM, \nSTS, ARPES, DFT. \nPeer-reviewed version accepted for publication in Carbon (30 September 2022).  \n1. Introduction \nGraphene grown on metallic substrates commonly reveals moiré interference patterns due to the \nmismatch between the constituting atomic lattices. The graphene carbon atoms occupy \ninequivalent positions on the underlying metal, inducing interaction of distinct strengths or even \nnature, concomitantly promoting a periodic modulation of graphene’s corrugation and electron \ndensity [1]. As a result, moiré superlattices constitute ordered nanoscale structures, otherwise \nexceptionally challenging to achieve at such small scales [2], which possess intriguing quantum \nproperties [3,4], and catalytic or templating functionalities. In this context, extensive research \nefforts have been invested on site-selective adsorption of atomic clusters [5–7] or organic \nmolecules on graphene [8–11] whereas far less attention has been devoted to templating the \nincorporation of heteroatoms into the epitaxial monolayers. \nUsing moiré patterns to achieve control over the distribution of heteroatoms embedded in \ngraphene shall constitute a way to fabricate very robust ordered nanoscale structures [12], and to \nmodify graphene’s properties permanently [13]. Boron and nitrogen are the most commonly \nexplored substitutional species because of their proximity to carbon in the periodic table, the \nsimilar size, and interest as acceptor or donor centers, respectively [14]. Diverse experimental \nmethods have been proposed based on simultaneous graphene synthesis and integration of \nheteroatoms [15–20], or post-synthetic incorporation of such species [21–26].  While in the vast \nmajority of cases heteroatoms distribute randomly within graphene, the use of strongly corrugated  3graphene supported on Ru(0001) has shown promising results towards site-selective pinning of \nsubstitutional nitrogen species at low densities [27]. \nHere, we show how the strongly corrugated moiré patterns present in graphene on Ir(111) \ntemplate the assembly of substitutional boron species, thus creating a 2D array of small boron \naggregates within graphene. . Through a comprehensive combined experimental and theoretical \nstudy, based on a single-atom and surface-averaged characterization and density functional theory \ncalculations, a complete description and rationalization of heteroatom distribution, B-C bonding \nconfiguration, interfacial interaction, and impact on graphene’s electronic structure is provided. \nInterestingly, we find that the site-selective incorporation of boron species induces local bending \nof the graphene layer towards the Ir(111) substrate, which reverses the charge transfer at the \ngraphene-Ir(111) interface and compensates for the hole doping induced by substitutional boron. \nSuch a unique doping mechanism results in a 2D periodic modulation of enhanced geometric \nrippling and charge accumulation in graphene, highly appealing for site-selective manipulation of \nmolecular states [28], and applications such as compact photo-catalytic devices and gas detectors \n[2,29]. \n \n2.  Experimental and theoretical methods \nThe experiments were performed in two different ultra-high vacuum setups, one at the Technical \nUniversity of Munich (TUM, Garching, Germany) dedicated to the low-temperature scanning \ntunnelling microscopy (STM) measurements, and the other one at the PEARL beamline (Swiss \nLight Source, Villigen, Switzerland) [30] devoted to the X-ray photoelectron spectroscopy (XPS) \nand angle-resolved photoemission spectroscopy (ARPES) experiments. Both chambers were  4equipped with a low energy electron diffraction (LEED) apparatus, which provides a link between \nthe experiments. Samples were reproducibly grown in-situ in each chamber. \n2.1. Sample preparation  \nThe Ir(111) single crystal (6 mm diameter surface, Mateck GmbH) was prepared by repeated \ncycles of sputtering (Ar+ ions at an energy of 1 keV), followed by two-step annealing at 1173 K \nunder O 2 atmosphere (3×10-8 mbar) and at 1233 K subsequently. The boron-doped graphene \nsamples were grown by dosing 5.2 L of borane tetrahydrofuran (BTHF, BH 3OC4H8) on Ir(111), \npre-heated at temperatures between 1073 - 1233 K. The resulting graphene monolayer presented \na manifold of rotational domains described in the text, with typical lateral sizes of tens of nm, \nwhose occurrence followed the same temperature dependency as when grown from ethylene \nprecursor [31]. No temperature-dependent variation in the distribution and density of boron species \nwas observed. Un-doped graphene was grown by dosing 5.4 L of ethylene on Ir(111) pre-heated \nat 1123 K. Commercially available BTHF (1.0 M in THF solution, Sigma-Aldrich) and ethylene \n(99.95%, CAN-Gas) were used. \n2.2. STM and STS characterization  \nSTM and scanning tunneling spectroscopy (STS) data were acquired using a CreaTec STM \noperating at 6 K and P < 2 × 10−10 mbar. STM images were taken in constant current mode, at the \nbias voltage stated in the figure captions, and processed using the WSxM software [32]. d I/dV \nspectra were acquired in constant height mode with a lock-in technique at 961 Hz and 5 mV peak-\nto-peak modulation voltage. \n2.3. XPS and ARPES characterization  \nPhotoelectron intensity was monitored by a Scienta EW4000 hemispherical electron energy \nanalyzer with two-dimensional detection, with the sample held at 77 K and operating at P < 2 ×  510−10 mbar base pressure. Spectra were taken in normal emission (unless stated otherwise in the \nfigure captions) with photon energies of 680 eV (O 1s), 390 eV (C 1s and B 1s) and 60 eV (valence \nband). Binding energies were calibrated using the Fermi edge subsequently measured at the same \nconditions. Core-level and valence-band spectra were measured at 10 and 20 eV pass energy, \nrespectively.  Fits to the core-level spectra were performed using the XPST macro for IGOR (M. \nSchmid, Philipps University Marburg), using Voigt-like functions on a Shirley background. Fitting \nparameters are summarized in Table S2. \n2.4. Computational details  \nDFT calculations have been performed using the plane-wave-based Quantum ESPRESSO \npackage (QE) [33,34]. The ultrasoft pseudopotentials have been adopted to describe the electron-\nion interactions with C (2s, 2p), B (2s, 2p), and Ir (4s, 3d) treated as valence electrons [35]. Energy \ncutoffs of 43 and 361 Ry (for kinetic energy and charge density expansion, respectively) have been \nadopted for all calculations. The convergence criterion of 0.026 eV/Å for the forces has been used \nduring geometry optimization, and the convergence criterion for the total energy has been set at \n10–6 Ry. To properly take into account weak interactions, the van der Waals density functional \nvdW-DF2C09x has been used [36,37], as implemented in the QE code. \nThe Ir(111) surface was modeled by a three-layer slab model with the two bottom layers fixed \nto the bulk positions during the geometry relaxation to mimic a semi-infinite solid. \nFor the simulation of the graphene/Ir(111) interface, we used the most commonly experimentally \nobserved domain (R0), which is the (10 × 10) graphene cell on a (9 × 9)  Ir(111) surface. A \nMonkhorst-Pack [38] k-Points mesh of 2 × 2 × 1 and 6 × 6 × 1 was used for the geometry relaxation \nand density of states (DOS) evaluation, respectively. To avoid interactions between adjacent \nperiodic images, a vacuum space of 17 Å (along the z-direction) has been included in the supercell  6model. Dipole correction was employed to check field effects but found to be negligible and \ntherefore it was not included. \nSTM simulations were performed using the Tersoff-Hamann approach [39], according to which \nthe tunneling current is proportional to the energy-integrated Local Density of States (ILDOS). \nSTM images were rendered with Gwyddion software [40]. Constant-current and voltage values \nfor the STM simulations have been chosen to match the experimental values. \nThe XPS spectra of boron-doped graphene on Ir have been calculated through the ΔSCF \napproach [41], which includes a pseudopotential generated with a full core hole for each ionized \ninequivalent carbon atom and allows the calculation of relative changes of binding energies, called \ncore-level shifts (CLSs). In order to compare the CLSs of different carbon atoms bounded to boron, \nwe considered a carbon atom in the TOP region as a reference, which is far away from the dopants \nand therefore is not perturbed by the substitutional boron atoms in graphene, as discussed in the \nmain text. \n \n3. Results and discussion \nThe boron-doped graphene monolayer was grown on the (111) surface of an iridium single \ncrystal, which was kept at 1073 K during exposure to 5.2 Langmuir (L) of BTHF gas under ultra-\nhigh vacuum conditions. BTHF was chosen as molecular precursor as it contains both carbon and \nboron elements, hence enabling the introduction of boron substitutional atoms simultaneously to \nthe growth process of graphene. Ir(111) was chosen as support because of the strong periodic \ncorrugation that it naturally induces to graphene, which constitutes a promising platform where \nthe segregation of boron substitutional atoms can be templated.  7The resulting material was first characterized in situ by core level XPS in order to elucidate its \nelemental composition. The spectrum in Figure 1a shows an intense C 1s peak featuring a main \ncomponent (C 1) centered at a binding energy of 284.09 eV, which is characteristic for \ngraphene/Ir(111), as confirmed on a reference sample grown by CVD using ethylene [42] (see \nFigure S1  in the Supplementary Materials). Thus, the novel BTHF precursor also affords the \nformation of monolayer graphene on Ir(111). In comparison to the reference graphene, the C 1s \npeak presented in Figure 1a exhibits a broader signature, which can be modelled by considering \ntwo additional contributions (C 2 and C 3) emerging due to the presence of multiple rotational \ndomains concomitant with an increased density of domain boundaries [43]. Their presence is \nconfirmed by real-space visualization provided by atomically-resolved STM ( Figure S4) , and by \nthe emergence of a ring-like feature in the corresponding LEED pattern, ( Figure 2e). In contrast, \nthe reference graphene sample presents sharp spots aligned with the Ir(1 × 1) pattern (inset in \nFigure S1 ), consequence of the presence of a single rotational domain [44]. In addition to C 1, C2 \nand C3, a tiny component (C 4) is attributed to carbon bonded to boron, as it appears -1.59 eV below \nC1, in line with the lower binding energies predicted by the core level shift (CLS) calculations \npresented for boron species bonding to carbon via sp2 -hybrid orbitals ( Figure S2 ) [19,45,46]. \nThe presence of boron is confirmed by the B 1s core level observed in Figure 1b that accounts \nfor a concentration of ∼ 2.5 %. It presents a slight asymmetry that can be modelled by two \ncomponents, B 1 and B 2, positioned at 187.77 and 188.63 eV, respectively. The main B 1 component \nappears at the characteristic binding energy of substitutional boron atoms forming sp2 bonds with \ncarbon, as reported for boron-doped graphene supported on diverse substrates [19,45,47]. The \nminority B 2 component emerging at 188.63 eV can be attributed to boron bond to boron in the \npresence of an Ir(111) substrate [48]. Its small intensity compared to that of B 1 is consistent with  8the low occurrence of boron directly bonded to neighboring boron observed by atomically resolved \nSTM, as discussed below based on Figure 5. The Ir 4f 7/2 signal shown in Figure 1d is modelled \nby a main component that accounts for the Ir atoms in the bulk (Ir 1), and two additional components \nthat originate from the outermost Ir atoms weakly (Ir 2) and strongly (Ir 3) interacting with the \noverlayer [43]. The possibility that the graphene layer is oxidized due to the presence of oxygen \nin the precursor molecule is ruled out due to the absence of  any O 1s signal ( Figure 1c), and of a \ncarbon – oxygen component that would be expected at between +1.5 and +2.5 eV above C1 \n(Figure 1a). \n  \nFigure 1.  Chemical characterization of boron doped-graphene on Ir(111). High-resolution XPS of (a) carbon, \n(b) boron, (c) oxygen 1s and (d) iridium 4f 7/2 core levels measured on boron-doped graphene on Ir(111). Experimental \nraw data are represented by red lines and circles, fits of the indicated components by colored areas and the sum of the \nfits by solid black lines. Raw data and sum of the fits are vertically offset for a better visualization. Inset in (c) contains \na ball-and-stick model of the molecular precursor BTHF (carbon is depicted in dark grey, hydrogen in light grey, \noxygen in cyan and boron in red). \n \n 9Next, real-space imaging of the boron-doped graphene on Ir(111) with atomic resolution was \nperformed by low-temperature STM. In the topographic image of Figure 2a, the graphene layer is \nreadily identifiable due to its characteristic honeycomb structure (depicted in yellow) and the \nhexagonal moiré superlattice (in black) [49], which is a consequence of the lattice mismatch \n(∼9.3%) and relative azimuthal angle between the graphene lattice and the underlying Ir(111) \nsurface. On numerous metal substrates, the moiré superlattice is accompanied by a strong \ncorrugation of the graphene [50], thus defining regions in which graphene is adsorbed closer or \nfurther away from the substrate (here referred as “low” or “high” areas, respectively). In the present \ncase, several graphene rotational domains exhibiting distinct moiré superlattices were identified \nby STM (see Figure S4 ), and are reflected in the ring-like feature around the spots of the Ir(111) \nsubstrate in the LEED patterns in Figure 2d,e. Specifically, R0, R17 and R18 domains (where the \nnumber denotes the rotation angle) with large moiré periodicity (24.3 ± 1.2, 23.0 ± 1.6 and 24.3 ± \n1.2 Å respectively) and high degree of corrugation ( ∼ 20 pm), and R14 and R19 domains with \nsmaller periodicity (10.9 ± 0.5 and 7.2 ± 0.4 Å) and corrugation ( ∼ 5 pm), as calculated by Meng \net al., [44] were observed. The rotational domain shown in Figure 2a,b can be further identified \nas R17 based on the angle (16.7 ± 0.3°) and ratio (0.11 ± 0.01 versus 0.112) between qgraphene and \nqmoiré extracted from the fast Fourier transform (FFT) Figure 2c. Here, we will focus on the highly \ncorrugated domains R0, R17 and R18, and treat them indiscriminately because of their very similar \nmoiré superlattice, and equal distribution and appearance of boron species (see Figure S4 ). The \nobservation of such unique moiré superlattices, which are only visible by STM when monolayer \ngraphene is in direct contact to Ir(111), demonstrates the single-atom thickness of the studied \ngraphene.  10Voloshina et al. [51] showed that the nature of the moiré corrugation of R0 is of both geometric \nand electronic origin, with geometrically high regions appearing as protrusions when imaged at \nbias voltages near the Fermi level by STM, and as depressions at higher voltages. This bias-\ndependent contrast inversion is also observed in the other highly corrugated domains, as shown in \nFigure 2a,b, where geometrically elevated regions are highlighted by black dashed circles. In \naddition to the moiré superlattice, the graphene layer presents multiple defects that appear as tiny \n“dents” of variable geometry at low bias voltage and broader depressions at higher bias voltage \n(highlighted by red dashed circles in Figure 2a and b, respectively). They will be identified as \nsmall aggregates of boron substitutional atoms (see below). These imperfections appear to have \nthe tendency to avoid the geometrically elevated regions, that is they occupy intermediary regions \nwithin the moiré unit cell in which graphene is adsorbed closer to the supporting Ir(111). As a \nresult, the defects assemble in a quasi-hexagonal array that mimics the periodicity and orientation \nof the moiré superlattice. The very same phenomenon is also observed in the highly corrugated \ndomains R0, R17 and R18, but not in flat R14 and R19 areas, which appear completely defect-free \n(see Figure S4 ). \n  11 \nFigure 2.  Assembly of boron substitutional species in Ir(111)-supported graphene . STM images of boron-doped \ngraphene on Ir(111) imaged at (a) Vbias = 0.01 V and (b) 0.60 V (scale bars: 2nm). Brighter/darker contrast indicates \nhigher/lower topographic apparent height. Graphene honeycomb network is highlighted in yellow, moiré superlattice \nin black and boron defects in red. (c) FFT pattern calculated corresponding to the STM image in (a). qgraphene and qmoiré \nare indicated by yellow and black vectors. LEED patterns measured in the same sample at (d) 39 eV (e) 78 eV. Ir(111) \n(1×1) spots are marked by pink circles and ring-like feature from a manifold of graphene rotational domains by yellow \narrows in (d). \n \nIn order to clarify the peculiar behavior of the corrugated rotational domains, we performed a \nwide set of density functional theory (DFT) calculations to get insight into the structure and relative \n 12stability of different possible configurations of substitutional boron (B C) in the R0 domain of \ngraphene/Ir(111). B C species are the most plausible candidates for the observed defects, since this \nelement is present in the BTHF precursor and, unlike oxygen, was detected by XPS ( Figure 1b). \nTable 1 reports the relative energies obtained for single and multiple B C species occupying non-\nequivalent graphene sites (top, fcc, and hcp) within the super-cell model used, either in the \ngeometrically high ( i.e., TOP in Figure 3a) or low ( i.e., FCC and HCP in Figure 3a) regions. As \na first result, we note that geometrically high regions ( Figure 3 b) appear energetically less \nfavorable than geometrically low ones ( Figure 3c) for hosting individual B C species, forcing these \nto distribute in a specifically ordered pattern, according to the graphene moiré superlattice. \nSecondly, the aggregation of more than one B C in geometrically low regions (2 B C in FCC-hcp) is \nlargely preferred, by more than 2 eV, with respect to a more spread distribution involving both \ngeometrically low and high regions (1 B C in FCC-hcp + 1 B C in TOP-fcc) and has a negligible \nenergetic toll, which should allow the formation of small B C aggregates ( Table 1). The tendency \nfor aggregation of the B atoms in the FCC region of the moiré can be further proved by comparing \n3 BC in FCC-hcp with other configurations where one or two B atoms are moved to random \npositions far from the TOP, FCC and HPC regions (named “2 B C in FCC-hcp + 1 B C random” and \n“1 BC in FCC-hcp + 2 B C random”, respectively). These configurations are higher in energy by \n+0.46 and +0.26 eV, respectively, as reported in Table 1. \n \nTable 1. Relative energies (in eV) of different B C configurations in graphene supported on Ir(111).  Energies are \nto be compared only for systems with the same type and number of atoms. ( i.e., same value of #B C). A schematic \nrepresentation of the different B C positions reported in the table is shown in Figure S3 . \n#BC Height B C position ΔE (eV)  131 Low FCC-hcp +0.29 \n1 Low HCP-fcc 0.00 \n1 High TOP-fcc +2.21 \n2 Low FCC-hcp +0.09 \n2 Low FCC-hcp + HCP-fcc 0.00 \n2 High TOP-fcc +4.38 \n3 Low FCC-hcp 0.00 \n3 Low+Intermediate  2 FCC-hcp + 1 random +0.46 \n3 Low+Intermediate  1 FCC-hcp + 2 random +0.26 \n3 High TOP-fcc Not found \n \nIt is also worth noting how B C species in the geometrically low regions (FCC and HCP), unlike \nin the high ones (TOP), cause significant geometric distortion of the graphene sheet, reducing the \nshortest vertical separation between the carbon layer and the Ir surface from ~ 3 to ~ 2 Å. This \nobservation hints to an enhancement of the interaction between graphene and Ir(111) in the \npresence of substitutional boron atoms, as will be discussed below. \n  14 \nFigure 3.  Computed boron-doped graphene on Ir(111) structures.  Ball-and-stick models of (a) pristine and (b, c, \nd) boron-doped graphene supported on Ir(111) surface. Color coding: Carbon atoms of graphene are rendered in black \n(stick representation); substitutional boron atoms are rendered in red; iridium atoms of the first, second and third layer \nare rendered in light grey, grey and dark grey, respectively. (10×10) graphene on 9×9) Ir super-cell of the R0 domain \ndepicted by the red parallelogram. \n \nThe variable shape of the “dents” observed in Figure 1a suggests that they might host multiple \nBC species, as proposed theoretically. In order to verify this experimentally, a closer inspection of \n 15the atomistic structure of the defects was performed by STM. The atomically-resolved image in \nFigure 4a shows the characteristic topographic appearance of the smallest occurring “dent” when \nmeasured at low bias voltage, consisting of a dark pseudo-triangular depression centered in one \nparticular site of the graphene honeycomb network. This is attributed to a single B C based on the \ngood agreement with the simulated STM image presented in Figure 4d, as well as the tendency of \nsubstitutional heteroatoms in graphene to exhibit 3-fold symmetric appearance [14]. Addition of \nother B C species leads to features with more extended and irregular geometries, as can be \nappreciated by comparing the experimental ( Figure 4 b,c) and simulated ( Figure 4 e,f) STM \nimages. These findings further validate the assignment of the substitutional heteroatoms to boron \nand confirm their tendency to form small aggregates. Note that such aggregates generally do not \ninclude B-B bonds, as will be discussed below. \n \n \nFigure 4.  Topographic appearance of boron substitutional heteroatoms in graphene on Ir(111). (a-c) \nExperimental and (d-f) theoretical atomically-resolved STM images of (a, d) 1 B C, (b, e) 2 B C and (c, f) 3 B C species \noccupying graphene low regions. (a-c) Vbias = 0.01 V, and (d-f) Vbias = 0.10 V with ILDOS value of 5 × 10-5 |e-|/a03. \nSubtle 3D rendering was applied to images (a-c) for better visualization. Line profiles in (g) experimental and (h) \ntheoretical STM images passing across B C (red lines) and far away from them (black dashed lines). Inset: Comparison \nbetween calculated average (red) and minimum (blue) graphene-iridium distance. \n 16 \nThe fact that the substitutional boron species appear as depressions is surprising, as it contrasts \nthe protrusion-like appearance of such defects reported for boron-doped graphene supported on \nnumerous substrates studied to date [18,20,47,52,53]. Our DFT calculations suggest that this \nunique feature is mainly due to geometric bending of the graphene layer towards the Ir(111) \nsubstrate induced by the B C species, as detailed in Figure 3 b and d. However, line profiles \nextracted from both experimental ( Figure 4g) and simulated ( Figure 4h) STM images, show that \nthe depth of the depressions produced by 2 B C is larger than that of 1 B C, while the calculated \ngeometric bending of the graphene layer is about the same in the two cases (inset in Figure 4h). \nThis points to electronic effects also contributing to the STM appearance of the single B C and \naggregates. \nStatistical analysis of the distribution of B C species was performed by counting the number of \nneighbors of a given B C as a function of their distance measured in number of atomic sites n of the \ngraphene honeycomb lattice (see the inset in Figure 5a). For the analysis, multiple atomically-\nresolved STM images, like those shown in Figure S4 , were used. The histogram in Figure 5a \nreveals that the B C species tend to occupy atomic sites within the same graphene sub-lattice, in \nagreement with the computed lower relative energies of B C pairs occupying the same sub-lattice \nover those in different sub-lattices, as shown Figure 5c. Interestingly, this behavior contrasts with \nthe tendency to indistinctly occupy both sub-lattices predicted when boron atoms are incorporated \nin free-standing graphene, as reported in a previous computational study [54], and confirmed in \nthis work. Very few B C species occupying adjacent sites were experimentally observed ( i.e., first \nneighbors), in agreement with a computed very high relative energy (+ 1.70 eV) with respect to \ntwo BC in the same sub-lattice, thus evidencing once more that formation of B-B bonds is highly \nunlikely. Equally important, the occurrence of neighbors (in both sub-lattices) decays with n (i.e.,  17the further apart the B C species are), hence explaining their tendency to distribute heterogeneously \nforming small aggregates as discussed above and shown in the STM images in Figure 2 and \nFigure 4 . The radius of the features arising from the B C aggregates is mostly limited to 8 \nhoneycomb atomic sites (the occurrence of neighbors beyond this threshold becomes \ninsignificant), which represents a maximum extension of the aggregates of around 7×7 graphene \nunit cells, slightly smaller than that of the moiré super-cell (between 9×9 and 10×10 depending on \nthe rotational domain), hence indicating that the aggregates tend to be spatially constrained within \na limited region of the 10×10 periodic unit. In addition, Figure 5 b presents the occupancy \ndistribution of B C species per aggregate, which peaks at 2 and decays until it vanishes around 6, \nhaving an expected value of 2.5 ± 1.9 B C. Since the energy toll required to go from two isolated \nBC species (FCC-hcp + HCP-fcc) to aggregates of two (both in FCC-hcp) is small (~ 0.09 eV, see \nTable 1), it can be understood that the occupancy distribution is dictated by the number of \navailable boron atoms during the growth. \n  18 \nFigure 5.  Distribution of substitutional boron heteroatoms in graphene on Ir(111). (a) Experimental radial \ndistribution of B C species from a given B C , measured in n atomic sites of the graphene honeycomb lattice. Red and \ngreen markers depict B C occupying the same or the opposite graphene sub-lattice with respect to the given B C, as \nschematically shown in the inset. (b) B C occupancy distribution within aggregates. Statistics: 570 B C species having \n1292 B C neighbors distributed within 225 aggregates. (c) Top and side view of 2 B C species occupying opposite \n(red/green balls) and the same (red/red balls) graphene sub-lattice on Ir(111). \n \n 19Next, we will discuss how the electronic structure of graphene is affected by the incorporation \nof BC species, and by the local rippling that they produce. It is well known that substitutional boron \nin free-standing graphene induces p-type doping, due to its lower atomic number compared to \ncarbon [55]. This results in an upshift of the Dirac cone with respect to the Fermi level, whose \nextent depends on the concentration of B C species: i.e., the larger the amount of substitutional B C \nspecies in the layer, the more pronounced the p-type character of graphene. The expectation is that \nthe presence of electronic holes in the π system of graphene will enhance its reactivity, altering its \ninteraction with a metal substrate such as Ir(111). \nIn order to evaluate this effect more quantitatively, we calculated and compared the differential \nelectron charge density maps following the deposition of pristine graphene and boron-doped \ngraphene on Ir(111) (see 2D plots in Figure 6a and 3D ones in Figure S4). Starting with the un-\ndoped system, we observe that graphene donates electrons to the metal forming a strongly \ninteracting interface (see graphene/Ir(111) in Figure 6 a), although the extent of such charge \ntransfer varies within the moiré super-structure: i.e., geometrically low regions (FCC and HCP) \nare transferring more electrons than geometrically high (TOP) regions, which are almost decoupled \nfrom the metal substrate. This scenario explains why graphene is slightly p-type doped when \nsupported on Ir(111), as confirmed by calculated Bader charges (where graphene donates ~ 1.6 × \n10-3 e-/carbon atom) and projected (PDOS) density of states ( Figure 6 b). It now becomes \ninteresting to investigate what happens when B C species are present. The bottom panel of Figure \n6a shows how the substitutional boron strongly perturbs the graphene/Ir(111) interaction by \ninducing a larger charge transfer from the underlying Ir atoms to the carbon atoms directly \ncoordinated to the boron atoms (see charge accumulation in Figure 6a) and facilitating the local \ndown-bending of graphene. It is worth noting that the perturbation induced by B C is quite local,  20since, moving further away from the defect, the charge plot returns to be similar to that of \ngraphene/Ir(111). \nIn order to verify the doping scenario proposed theoretically, local and non-local measurements \nof the electronic structure of the boron-doped graphene were conducted by STS and ARPES, \nrespectively. In Figure 6c, a series of d I/dV curves measured by STS on individual aggregates \ncontaining one or more B C species and on un-doped regions is presented. It shows that the boron-\ndoped graphene remains slightly p-doped, as the minimum of intensity, identified as the Dirac \npoint, lies 24 ± 5 mV above the Fermi level regardless of the region. Note that the local maximum \nat ~ -180 mV is attributed to the Ir(111) surface state [56], slightly up-shifted due to the presence \nof graphene [57]. The fact that neither the Dirac point position nor the shape of the d I/dV curves \nvary between the un-doped and doped regions, in agreement with the calculated PDOS ( Figure \n6b), reveals that the local electronic structure near the Fermi level remains largely unaltered upon \nintroduction of B C species. Analogous conclusions can be reached based on the analysis of the \ncomputed workfunctions (Table S1) since we observe that the value for 3B-doped FCC graphene/Ir \n(3BcFCC-hcp) is essentially the same as for graphene/Ir (4.63 vs 4.67 eV, respectively). \n  21  \nFigure 6.  Local electronic structure of boron-doped graphene on Ir(111).  (a) 2D charge density difference plots \n(corresponding 3D plots are shown in Figure S5 ) for pristine graphene and 1 B CFCC-hcp supported on Ir(111) surface. \nThe color scale indicates electron depletion and accumulation ranging from blue to yellow/red, respectively. Isosurface \nlevel is 0.001 e-/bohr3 for both systems.  Geometries have been superimposed to aid interpretation. (b) PDOS on \ngraphene for pristine and boron-doped graphene supported on Ir(111). The Fermi level is scaled to zero and indicated \nby a blacked dashed line.  (c) d I/dV spectra acquired on the different regions of a boron-doped graphene domain (R18) \non Ir(111) shown in STM image of the inset ( Vbias = 0.60 V, 11 nm lateral size, gentle 3D render has been applied for \nbetter visualization). Stabilization conditions: Vbias = 0.50 V and It = 0.5 nA, lock-in modulation voltage V = 5 mV. \n 22 \nIn addition, ARPES measurements of boron-doped and un-doped layers allow to assess the \nimpact of the B C species on the electronic band structure of graphene (see Figure S1  for more \ndetails about the un-doped reference sample). In the spectra taken along the Γ→K direction of \ngraphene’s (R0) Brillouin zone [58], a prominent graphene π band appearing at ~ 8 eV that crosses \nthe Fermi level near K graphene is identified in both doped and un-doped graphene ( Figure 7a, b), \nalongside with other states of attenuated intensity attributed to the underlying Ir(111) [57,59]. Note \nthat the high-momentum branch of the Dirac cones is hardly visible due to interference between \nphotoelectrons emitted from the two sub-lattices of graphene [60], and that the increased \nbackground intensity of the boron-doped spectra is attributed to the presence of a manifold of \nrotational domains (see Figure 2 d, e and Figure S4 ). A comparison between the linearly \ndispersing π band of un-doped and boron-doped graphene near the Fermi level ( i.e., the Dirac \ncone) is presented in Figure 7c, in which the right panel has been mirrored along kx to ease \ncomparability. No appreciable change in the dispersion of the Dirac cone nor in the energy position \nof the Dirac point (from 0.24 ± 0.15 to 0.27 ± 0.15 eV, see Figure S6 ) is observed, in stark contrast \nto the binding energy shift (up to ~ 1 eV) expected for boron-doped free-standing graphene at low \nboron concentrations (< 5 %) [46]. Hence, the fact that the Dirac cone remains unperturbed upon \nthe introduction of substitutional boron atoms confirms their negligible doping effect on Ir-\nsupported graphene suggested by theoretical calculations. \n  23 \nFigure 7.  Occupied electronic structure of boron-doped graphene on Ir(111).  ARPES spectra of (a) un-doped \ngraphene/Ir(111) and (b) boron-doped graphene/Ir(111) measured along the Γ→K direction. (c) Second derivative \nplot of the magnified regions marked by the colored rectangles in (a) and (b). Grey and red lines are a guide to the \neyes that denote the position of the Dirac cone near the Fermi level for un-doped and boron-doped graphene, \nrespectively. Spectra were measured at a 22.5° tilt angle from normal emission in order to visualize the first and part \nof the second Brillouin zone of graphene. \n \nFinally, we venture to rationalize why the presented method yields formation of ordered arrays \nof BC species, in striking contrast to others reported earlier. First, the use of a highly corrugated \nmoiré superlattice, like that induced by Ir(111) or Ru(0001) [27,47], is instrumental to enable site-\nselective pinning of the B C, as deduced from our DFT calculations. Second, incorporation of the \nheteroatoms during graphene growth, promoted here by the carbon and boron-containing \nmolecular precursor, could play an important role because of an enhancement of B C segregation \nat elevated temperatures, facilitating the access to the favorable moiré regions. A similar scenario \nwas demonstrated by Zabet-Khosousi et al., [61] by comparing the incorporation of nitrogen \nduring and after graphene growth, and cleverly used by Camilli et al. [62] to engineer arrays of \n0D graphene dots embedded within a 2D BNC alloy. \n \n 244. Conclusions \nIn conclusion, we have shown how the incorporation of boron in graphene during its growth \nprocess on Ir(111) results in 2D arrays of boron species structurally embedded in graphene. \nCombining atomic-scale and surface-averaged characterization complemented by computational \nmodelling reveals that this structure is intimately related to the strong corrugation of the moiré \nsuperlattice of graphene on Ir(111), as the formation of boron substitutional species is energetically \nfavored only within its geometrically low regions. The impact of the boron species on graphene’s \ndoping level is found to be marginal, as the holes that they introduce in graphene are compensated \nby the uptake of electron charges from the substrate through a stronger interfacial interaction. This \nprocess is facilitated by a local bending of the graphene layer towards the substrate that induces \ncharge accumulation around the heteroatoms. As a result, the moiré-modulated electron density \nand rippling of graphene become greatly enhanced, making this material highly appealing for \napplications in the fields of metal-free catalysis, gas sensing, or nano-electronics. \n \nASSOCIATED CONTENT  \nSupporting material \nFigures S1 – S6: Un-doped graphene on Ir(111) reference sample (XPS and LEED); schematic of \nthe calculated configurations of B C in graphene/Ir(111) and corresponding CLS calculations; \nschematic of the calculated B C positions leading to the relative energies reported in Table 1; STM \ncharacterization of diverse rotational domains of boron-doped graphene on Ir(111); 3D charge \ndensity difference plots; extended view of the ARPES data presented in Figure 7c. Table S1: \nCalculated workfunctions for free-standing and Ir-supported pristine and B-doped graphene. Table \nS2: XPS fitting parameters  used to model the XPS core levels shown in Figure 1 and Figure S1.  25Corresponding Authors \nMarc G. Cuxart – Physics Department E20, Technical University of Munich, James-Franck-Str. 1, \n85748 Garching, Germany; orcid.org/0000-0002-9085-1225; Email: marc.gonzalez-\ncuxart@tum.de \nCristiana Di Valentin – Dipartimento di Scienza dei Materiali, Università di Milano Bicocca, via \nR. Cozzi 55, 20125 Milano, Italy; orcid.org/0000-0003-4163-8062; Email: \ncristiana.divalentin@unimib.it \nWilli Auwärter – Physics Department E20, Technical University of Munich, James-Franck-Str. 1, \n85748 Garching, Germany; orcid.org/0000-0001-9452-4662; Email: wau@tum.de \nAuthors \nDaniele Perilli – Dipartimento di Scienza dei Materiali, Università di Milano Bicocca, via R. Cozzi \n55, 20125 Milano, Italy; orcid.org/0000-0002-3082-3986 \nSena Tömeke – Physics Department E20, Technical University of Munich, James-Franck-Str. 1, \n85748 Garching, Germany. \nJoel Deyerling – Physics Department E20, Technical University of Munich, James-Franck-Str. 1, \n85748 Garching, Germany. \nFelix Haag – Physics Department E20, Technical University of Munich, James-Franck-Str. 1, \n85748 Garching, Germany; \nMatthias Muntwiler – Paul Scherrer Institut, Villigen 5232, Switzerland; orcid.org/0000-0002-\n6628-3977 \nFrancesco Allegretti – Physics Department E20, Technical University of Munich, James-Franck-\nStr. 1, 85748 Garching, Germany. orcid.org/0000-0001-6141-7166.  26Author Contributions \n‡M.G.C and D.P. contributed equally to this work. \nNotes \nThe authors declare no competing financial interest. \nAcknowledgments \nThe authors thank H. Sachdev for suggesting the use of the BTHF precursor. M.G.C. acknowledges funding from the European \nUnion’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement no. 892725 \n(WHITEMAG project). S.T. and W.A. acknowledge the MSCA-ITN-ETN STiBNite (no. 956923). The Paul Scherrer Institut, \nVilligen, Switzerland is acknowledged for providing synchrotron radiation beamtime at the PEARL beamline of the SLS. C.D.V. \nacknowledges funding from MIUR through the project “MADAM: Metal Activated 2D cArbon-based platforMs” PRIN2017 grant \nNo. 2017NYPHN8 and from European Research Council (ERC) under the European Union’s HORIZON2020 research and \ninnovation programme (ERC Grant Agreement No [647020]). \nReferences \n[1] A.B. Preobrajenski, M.L. Ng, A.S. Vinogradov, N. Mårtensson, Controlling graphene \ncorrugation on lattice-mismatched substrates, Phys. Rev. B. 78 (2008) 073401. \nhttps://doi.org/10.1103/PhysRevB.78.073401. \n[2] C. Gong, K. Hu, X. Wang, P. Wangyang, C. Yan, J. Chu, M. Liao, L. Dai, T. Zhai, C. Wang, \nL. Li, J. Xiong, 2D Nanomaterial Arrays for Electronics and Optoelectronics, Adv. Funct. \nMater. 28 (2018) 1706559. https://doi.org/10.1002/adfm.201706559. \n[3] B. Borca, S. Barja, M. Garnica, D. Sánchez-Portal, V.M. Silkin, E. V. Chulkov, C.F. \nHermanns, J.J. Hinarejos, A.L. Vázquez de Parga, A. Arnau, P.M. Echenique, R. Miranda, \nPotential Energy Landscape for Hot Electrons in Periodically Nanostructured Graphene, \nPhys. Rev. Lett. 105 (2010) 036804. https://doi.org/10.1103/PhysRevLett.105.036804. \n[4] H.G. Zhang, H. Hu, Y. Pan, J.H. Mao, M. Gao, H.M. Guo, S.X. Du, T. Greber, H.-J. Gao, \nGraphene based quantum dots, J. Phys. Condens. Matter. 22 (2010) 302001. \nhttps://doi.org/10.1088/0953-8984/22/30/302001. \n[5] A.T. N’Diaye, T. Gerber, C. Busse, J. Mysliveček, J. Coraux, T. Michely, A versatile \nfabrication method for cluster superlattices, New J. Phys. 11 (2009) 103045. \nhttps://doi.org/10.1088/1367-2630/11/10/103045. \n[6] Y. Pan, M. Gao, L. Huang, F. Liu, H.-J. Gao, Directed self-assembly of monodispersed \nplatinum nanoclusters on graphene Moiré template, Appl. Phys. Lett. 95 (2009) 093106. \nhttps://doi.org/10.1063/1.3223781. \n[7] M.D. Jiménez-Sánchez, C. Romero-Muñiz, P. Pou, R. Pérez, J.M. Gómez-Rodríguez, \nGraphene on Rh(111): A template for growing ordered arrays of metal nanoparticles with  27different periodicities, Carbon. 173 (2021) 1073–1081. \nhttps://doi.org/10.1016/j.carbon.2020.11.086. \n[8] H. Dil, J. Lobo-Checa, R. Laskowski, P. Blaha, S. Berner, J. Osterwalder, T. Greber, Surface \nTrapping of Atoms and Molecules with Dipole Rings, Science. 319 (2008) 1824–1826. \nhttps://doi.org/10.1126/science.1154179. \n[9] J. Mao, H. Zhang, Y. Jiang, Y. Pan, M. Gao, W. Xiao, H.-J. Gao, Tunability of \nSupramolecular Kagome Lattices of Magnetic Phthalocyanines Using Graphene-Based \nMoiré Patterns as Templates, J. Am. Chem. Soc. 131 (2009) 14136–14137. \nhttps://doi.org/10.1021/ja904907z. \n[10] J. Lu, P.S.E. Yeo, Y. Zheng, Z. Yang, Q. Bao, C.K. Gan, K.P. Loh, Using the Graphene \nMoiré Pattern for the Trapping of C 60 and Homoepitaxy of Graphene, ACS Nano. 6 (2012) \n944–950. https://doi.org/10.1021/nn204536e. \n[11] K. Banerjee, A. Kumar, F.F. Canova, S. Kezilebieke, A.S. Foster, P. Liljeroth, Flexible Self-\nAssembled Molecular Templates on Graphene, J. Phys. Chem. C. 120 (2016) 8772–8780. \nhttps://doi.org/10.1021/acs.jpcc.6b01638. \n[12] L. Camilli, J.H. Jørgensen, J. Tersoff, A.C. Stoot, R. Balog, A. Cassidy, J.T. Sadowski, P. \nBøggild, L. Hornekær, Self-assembly of ordered graphene nanodot arrays, Nat. Commun. 8 \n(2017) 1–8. https://doi.org/10.1038/s41467-017-00042-4. \n[13] S. Ullah, Q. Shi, J. Zhou, X. Yang, H.Q. Ta, M. Hasan, N.M. Ahmad, L. Fu, A. Bachmatiuk, \nM.H. Rümmeli, Advances and Trends in Chemically Doped Graphene, Adv. Mater. \nInterfaces. 7 (2020) 2000999. https://doi.org/10.1002/admi.202000999. \n[14] F. Joucken, L. Henrard, J. Lagoute, Electronic properties of chemically doped graphene, Phys \nRev Mater. 3 (2019) 1–20. https://doi.org/10.1103/PhysRevMaterials.3.110301. \n[15] L. Zhao, R. He, K.T. Rim, T. Schiros, K.S. Kim, H. Zhou, C. Gutiérrez, S.P. Chockalingam, \nC.J. Arguello, L. Pálová, D. Nordlund, M.S. Hybertsen, D.R. Reichman, T.F. Heinz, P. Kim, \nA. Pinczuk, G.W. Flynn, A.N. Pasupathy, Visualizing Individual Nitrogen Dopants in \nMonolayer Graphene, Science. 333 (2011) 999–1003. \nhttps://doi.org/10.1126/science.1208759. \n[16] N.J. Van Der Heijden, D. Smith, G. Calogero, R.S. Koster, D. Vanmaekelbergh, M.A. Van \nHuis, I. Swart, Recognizing nitrogen dopant atoms in graphene using atomic force \nmicroscopy, Phys. Rev. B. 93 (2016) 1–9. https://doi.org/10.1103/PhysRevB.93.245430. \n[17] Y.B. Tang, L.C. Yin, Y. Yang, X.H. Bo, Y.L. Cao, H.E. Wang, W.J. Zhang, I. Bello, S.T. \nLee, H.M. Cheng, C.S. Lee, Tunable band gaps and p-type transport properties of boron-\ndoped graphenes by controllable ion doping using reactive microwave plasma, ACS Nano. 6 \n(2012) 1970–1978. https://doi.org/10.1021/nn3005262. \n[18] M. Telychko, P. Mutombo, P. Merino, P. Hapala, M. Ondráček, F.C. Bocquet, J. Sforzini, O. \nStetsovych, M. Vondráček, P. Jelínek, M. Švec, Electronic and Chemical Properties of \nDonor, Acceptor Centers in Graphene, ACS Nano. 9 (2015) 9180–9187. \nhttps://doi.org/10.1021/acsnano.5b03690. \n[19] D.Y. Usachov, A. V. Fedorov, A.E. Petukhov, O.Y. Vilkov, A.G. Rybkin, M.M. Otrokov, A. \nArnau, E. V. Chulkov, L. V. Yashina, M. Farjam, V.K. Adamchuk, B. V. Senkovskiy, C. \nLaubschat, D. V. Vyalikh, Epitaxial B-Graphene: Large-Scale Growth and Atomic Structure, \nACS Nano. 9 (2015) 7314–7322. https://doi.org/10.1021/acsnano.5b02322. \n[20] L. Zhao, M. Levendorf, S. Goncher, T. Schiros, L. Pálová, A. Zabet-Khosousi, K.T. Rim, C. \nGutiérrez, D. Nordlund, C. Jaye, M. Hybertsen, D. Reichman, G.W. Flynn, J. Park, A.N.  28Pasupathy, Local Atomic and Electronic Structure of Boron Chemical Doping in Monolayer \nGraphene, Nano Lett. 13 (2013) 4659–4665. https://doi.org/10.1021/nl401781d. \n[21] F. Joucken, Y. Tison, J. Lagoute, J. Dumont, D. Cabosart, B. Zheng, V. Repain, C. Chacon, \nY. Girard, A.R. Botello-Méndez, S. Rousset, R. Sporken, J.C. Charlier, L. Henrard, Localized \nstate and charge transfer in nitrogen-doped graphene, Phys. Rev. B - Condens. Matter Mater. \nPhys. 85 (2012) 1–5. https://doi.org/10.1103/PhysRevB.85.161408. \n[22] M. Telychko, P. Mutombo, M. Ondráček, P. Hapala, F.C. Bocquet, J. Kolorenč, M. \nVondráček, P. Jelínek, M. Švec, Achieving high-quality single-atom nitrogen doping of \ngraphene/SiC(0001) by ion implantation and subsequent thermal stabilization, ACS Nano. 8 \n(2014) 7318–7324. https://doi.org/10.1021/nn502438k. \n[23] A. Zabet-Khosousi, L. Zhao, L. Pálová, M.S. Hybertsen, D.R. Reichman, A.N. Pasupathy, \nG.W. Flynn, Segregation of sublattice domains in nitrogen-doped graphene, J. Am. Chem. \nSoc. 136 (2014) 1391–1397. https://doi.org/10.1021/ja408463g. \n[24] Y. Tison, J. Lagoute, V. Repain, C. Chacon, Y. Girard, S. Rousset, F. Joucken, D. Sharma, \nL. Henrard, H. Amara, A. Ghedjatti, F. Ducastelle, Electronic interaction between nitrogen \natoms in doped graphene, ACS Nano. 9 (2015) 670–678. https://doi.org/10.1021/nn506074u. \n[25] J. Lagoute, F. Joucken, V. Repain, Y. Tison, C. Chacon, A. Bellec, Y. Girard, R. Sporken, \nE.H. Conrad, F. Ducastelle, M. Palsgaard, N.P. Andersen, M. Brandbyge, S. Rousset, Giant \ntunnel-electron injection in nitrogen-doped graphene, Phys. Rev. B - Condens. Matter Mater. \nPhys. 91 (2015) 1–5. https://doi.org/10.1103/PhysRevB.91.125442. \n[26] A. Martín-Recio, C. Romero-Muñiz, P. Pou, R. Pérez, J.M. Gómez-Rodríguez, Combining \nnitrogen substitutional defects and oxygen intercalation to control the graphene corrugation \nand doping level, Carbon. 130 (2018) 362–368. \nhttps://doi.org/10.1016/j.carbon.2017.12.117. \n[27] X. Fei, J. Neilson, Y. Li, V. Lopez, S.J. Garrett, L. Gan, H.-J. Gao, L. Gao, Controlled \nSynthesis of Nitrogen-Doped Graphene on Ruthenium from Azafullerene, Nano Lett. 17 \n(2017) 2887–2894. https://doi.org/10.1021/acs.nanolett.7b00038. \n[28] B. de la Torre, M. Švec, P. Hapala, J. Redondo, O. Krejčí, R. Lo, D. Manna, A. Sarmah, D. \nNachtigallová, J. Tuček, P. Błoński, M. Otyepka, R. Zbořil, P. Hobza, P. Jelínek, Non-\ncovalent control of spin-state in metal-organic complex by positioning on N-doped graphene, \nNat. Commun. 9 (2018) 2831. https://doi.org/10.1038/s41467-018-05163-y. \n[29] S. Agnoli, M. Favaro, Doping graphene with boron: A review of synthesis methods, \nphysicochemical characterization, and emerging applications, J. Mater. Chem. A. 4 (2016) \n5002–5025. https://doi.org/10.1039/c5ta10599d. \n[30] M. Muntwiler, J. Zhang, R. Stania, F. Matsui, P. Oberta, U. Flechsig, L. Patthey, C. \nQuitmann, T. Glatzel, R. Widmer, E. Meyer, T.A. Jung, P. Aebi, R. Fasel, T. Greber, Surface \nscience at the PEARL beamline of the Swiss Light Source, J. Synchrotron Radiat. 24 (2017) \n354–366. https://doi.org/10.1107/S1600577516018646. \n[31] H. Hattab, A.T. N’Diaye, D. Wall, G. Jnawali, J. Coraux, C. Busse, R. van Gastel, B. \nPoelsema, T. Michely, F.-J. Meyer zu Heringdorf, M. Horn-von Hoegen, Growth temperature \ndependent graphene alignment on Ir(111), Appl. Phys. Lett. 98 (2011) 141903. \nhttps://doi.org/10.1063/1.3548546. \n[32] I. Horcas, R. Fernández, J.M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, A.M. Baro, \nWSXM: A software for scanning probe microscopy and a tool for nanotechnology, Rev. Sci. \nInstrum. 78 (2007). https://doi.org/10.1063/1.2432410.  29[33] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G.L. \nChiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. \nGebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, \nF. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. \nSclauzero, A.P. Seitsonen, A. Smogunov, P. Umari, R.M. Wentzcovitch, QUANTUM \nESPRESSO: a modular and open-source software project for quantum simulations of \nmaterials, J. Phys. Condens. Matter. 21 (2009) 395502. https://doi.org/10.1088/0953-\n8984/21/39/395502. \n[34] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, \nR. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, \nS. de Gironcoli, P. Delugas, R.A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. \nGebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. \nKüçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N.L. Nguyen, H.-V. Nguyen, A. \nOtero-de-la-Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A.P. \nSeitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, S. Baroni, \nAdvanced capabilities for materials modelling with Quantum ESPRESSO, J. Phys. Condens. \nMatter. 29 (2017) 465901. https://doi.org/10.1088/1361-648X/aa8f79. \n[35] A. Dal Corso, Pseudopotentials periodic table: From H to Pu, Comput. Mater. Sci. 95 (2014) \n337–350. https://doi.org/10.1016/j.commatsci.2014.07.043. \n[36] K. Lee, É.D. Murray, L. Kong, B.I. Lundqvist, D.C. Langreth, Higher-accuracy van der \nWaals density functional, Phys. Rev. B. 82 (2010) 081101. \nhttps://doi.org/10.1103/PhysRevB.82.081101. \n[37] I. Hamada, M. Otani, Comparative van der Waals density-functional study of graphene on \nmetal surfaces, Phys. Rev. B. 82 (2010) 153412. \nhttps://doi.org/10.1103/PhysRevB.82.153412. \n[38] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B. 13 \n(1976) 5188–5192. https://doi.org/10.1103/PhysRevB.13.5188. \n[39] J. Tersoff, D.R. Hamann, Theory of the scanning tunneling microscope, Phys. Rev. B. 31 \n(1985) 805–813. https://doi.org/10.1103/PhysRevB.31.805. \n[40] D. Nečas, P. Klapetek, Gwyddion: an open-source software for SPM data analysis, Open \nPhys. 10 (2012). https://doi.org/10.2478/s11534-011-0096-2. \n[41] S. García-Gil, A. García, P. Ordejón, Calculation of core level shifts within DFT using \npseudopotentials and localized basis sets, Eur. Phys. J. B. 85 (2012) 239. \nhttps://doi.org/10.1140/epjb/e2012-30334-5. \n[42] A.T. N’Diaye, J. Coraux, T.N. Plasa, C. Busse, T. Michely, Structure of epitaxial graphene \non Ir(111), New J. Phys. 10 (2008). https://doi.org/10.1088/1367-2630/10/4/043033. \n[43] P. Lacovig, M. Pozzo, D. Alfè, P. Vilmercati, A. Baraldi, S. Lizzit, Growth of Dome-Shaped \nCarbon Nanoislands on Ir(111): The Intermediate between Carbidic Clusters and Quasi-Free-\nStanding Graphene, Phys. Rev. Lett. 103 (2009) 166101. \nhttps://doi.org/10.1103/PhysRevLett.103.166101. \n[44] L. Meng, R. Wu, L. Zhang, L. Li, S. Du, Y. Wang, H.-J. Gao, Multi-oriented moiré \nsuperstructures of graphene on Ir(111): experimental observations and theoretical models, J. \nPhys. Condens. Matter. 24 (2012) 314214. https://doi.org/10.1088/0953-8984/24/31/314214. \n[45] J. Gebhardt, R.J. Koch, W. Zhao, O. Höfert, K. Gotterbarm, S. Mammadov, C. Papp, A. \nGörling, H.-P. Steinrück, T. Seyller, Growth and electronic structure of boron-doped \ngraphene, Phys. Rev. B. 87 (2013) 155437. https://doi.org/10.1103/PhysRevB.87.155437.  30[46] D.Y. Usachov, A. V. Fedorov, O.Y. Vilkov, A.E. Petukhov, A.G. Rybkin, A. Ernst, M.M. \nOtrokov, E. V. Chulkov, I.I. Ogorodnikov, M. V. Kuznetsov, L. V. Yashina, E.Y. Kataev, A. \nV. Erofeevskaya, V.Y. Voroshnin, V.K. Adamchuk, C. Laubschat, D. V. Vyalikh, Large-\nScale Sublattice Asymmetry in Pure and Boron-Doped Graphene, Nano Lett. 16 (2016) \n4535–4543. https://doi.org/10.1021/acs.nanolett.6b01795. \n[47] L. Pan, Y. Que, H. Chen, D. Wang, J. Li, C. Shen, W. Xiao, S. Du, H. Gao, S.T. Pantelides, \nRoom-Temperature, Low-Barrier Boron Doping of Graphene, Nano Lett. 15 (2015) 6464–\n6468. https://doi.org/10.1021/acs.nanolett.5b01839. \n[48] M.G. Cuxart, K. Seufert, V. Chesnyak, W.A. Waqas, A. Robert, M. Bocquet, G.S. Duesberg, \nH. Sachdev, W. Auwärter, Borophenes made easy, Sci. Adv. 7 (2021) 1–8. \nhttps://doi.org/10.1126/sciadv.abk1490. \n[49] C. Busse, P. Lazić, R. Djemour, J. Coraux, T. Gerber, N. Atodiresei, V. Caciuc, R. Brako, \nA.T. N’Diaye, S. Blügel, J. Zegenhagen, T. Michely, Graphene on Ir(111): Physisorption \nwith Chemical Modulation, Phys. Rev. Lett. 107 (2011) 036101. \nhttps://doi.org/10.1103/PhysRevLett.107.036101. \n[50] M. Batzill, The surface science of graphene: Metal interfaces, CVD synthesis, nanoribbons, \nchemical modifications, and defects, Surf. Sci. Rep. 67 (2012) 83–115. \nhttps://doi.org/10.1016/j.surfrep.2011.12.001. \n[51] E.N. Voloshina, E. Fertitta, A. Garhofer, F. Mittendorfer, M. Fonin, A. Thissen, Y.S. Dedkov, \nElectronic structure and imaging contrast of graphene moiré on metals, Sci. Rep. 3 (2013) \n1072. https://doi.org/10.1038/srep01072. \n[52] R. Lv, G. Chen, Q. Li, A. McCreary, A. Botello-Méndez, S. V. Morozov, L. Liang, X. \nDeclerck, N. Perea-López, D.A. Cullen, S. Feng, A.L. Elías, R. Cruz-Silva, K. Fujisawa, M. \nEndo, F. Kang, J.-C. Charlier, V. Meunier, M. Pan, A.R. Harutyunyan, K.S. Novoselov, M. \nTerrones, Ultrasensitive gas detection of large-area boron-doped graphene, Proc. Natl. Acad. \nSci. 112 (2015) 14527–14532. https://doi.org/10.1073/pnas.1505993112. \n[53] Q. Li, H. Lin, R. Lv, M. Terrones, L. Chi, W.A. Hofer, M. Pan, Locally Induced Spin States \non Graphene by Chemical Attachment of Boron Atoms, Nano Lett. 18 (2018) 5482–5487. \nhttps://doi.org/10.1021/acs.nanolett.8b01798. \n[54] T. M. Dieb, Z. Hou, K. Tsuda, Structure prediction of boron-doped graphene by machine \nlearning, J. Chem. Phys. 148 (2018) 241716. https://doi.org/10.1063/1.5018065. \n[55] L. Ferrighi, M.I. Trioni, C. Di Valentin, Boron-Doped, Nitrogen-Doped, and Codoped \nGraphene on Cu(111): A DFT + vdW Study, J. Phys. Chem. C. 119 (2015) 6056–6064. \nhttps://doi.org/10.1021/jp512522m. \n[56] L. Meng, R. Wu, H. Zhou, G. Li, Y. Zhang, L. Li, Y. Wang, H.J. Gao, Silicon intercalation \nat the interface of graphene and Ir(111), Appl. Phys. Lett. 100 (2012). \nhttps://doi.org/10.1063/1.3687688. \n[57] A. Varykhalov, D. Marchenko, M.R. Scholz, E.D.L. Rienks, T.K. Kim, G. Bihlmayer, J. \nSánchez-Barriga, O. Rader, Ir(111) Surface State with Giant Rashba Splitting Persists under \nGraphene in Air, Phys. Rev. Lett. 108 (2012) 066804. \nhttps://doi.org/10.1103/PhysRevLett.108.066804. \n[58] E. Starodub, A. Bostwick, L. Moreschini, S. Nie, F. El Gabaly, K.F. McCarty, E. Rotenberg, \nIn-plane orientation effects on the electronic structure, stability, and Raman scattering of \nmonolayer graphene on Ir(111), Phys. Rev. B. 83 (2011) 125428. \nhttps://doi.org/10.1103/PhysRevB.83.125428.  31[59] I. Pletikosić, M. Kralj, P. Pervan, R. Brako, J. Coraux, A.T. N’Diaye, C. Busse, T. Michely, \nDirac Cones and Minigaps for Graphene on Ir(111), Phys. Rev. Lett. 102 (2009) 056808. \nhttps://doi.org/10.1103/PhysRevLett.102.056808. \n[60] M. Mucha-Kruczyński, O. Tsyplyatyev, A. Grishin, E. McCann, V.I. Fal���ko, A. Bostwick, \nE. Rotenberg, Characterization of graphene through anisotropy of constant-energy maps in \nangle-resolved photoemission, Phys. Rev. B. 77 (2008) 195403. \nhttps://doi.org/10.1103/PhysRevB.77.195403. \n[61] A. Zabet-Khosousi, L. Zhao, L. Pálová, M.S. Hybertsen, D.R. Reichman, A.N. Pasupathy, \nG.W. Flynn, Segregation of sublattice domains in nitrogen-doped graphene, J. Am. Chem. \nSoc. 136 (2014) 1391–1397. https://doi.org/10.1021/ja408463g. \n[62] L. Camilli, J.H. Jørgensen, J. Tersoff, A.C. Stoot, R. Balog, A. Cassidy, J.T. Sadowski, P. \nBøggild, L. Hornekær, Self-assembly of ordered graphene nanodot arrays, Nat. Commun. 8 \n(2017) 1–8. https://doi.org/10.1038/s41467-017-00042-4. \n "    },    {        "title": "2001.02042v2.Diverse_fundamental_properties_in_stage_n_graphite_alkali_intercalation_compounds__anode_materials_of_Li__based_batteries.pdf",        "content": "Diverse fundamental properties in stage- n graphite alkali -intercalation \ncompounds: anode materials of Li+-based batteries  \n \nWei-bang Li1, Ngoc Thanh Thuy Tran2, Shih- yang Lin3, Ming- Fa Lin1,2 \n1Department of Physics, National National Cheng Kung University, Tainan, Taiwan   \n2Hierarchical Green -Energy Materials (Hi -GEM) Research Center, National Cheng \nKung University, Tainan 70101, Taiwan  \n 3Department of Physics, National Chung Cheng University, Chiayi, Taiwan  \n \nAbstract  \n \n    The diversified essential properties of the stage -n graphite alkali -intercalation \ncompounds are thoroughly explored by the first -principles calculations. According \nto the ir main features, the lithium and non- lithium materials might be quite \ndifferent from each other in stacking configurations, the intercalated alkali -atom \nconcentrations, the free conduction electron densi ties, and the atom -dominated & \n(carbon, alkali) -co-dominated energy bands. The close relations between the \nalkali -doped metallic behaviors and the geometric symmetries will be clarified \nthrough the interlayer atomic interactions, in which the significant alkali -carbon \nchemical bondings are fully examined fr om the atom-  and orbital -decomposed van \nHove singularities. The blue shift of the Fermi level, the n- type doping, is clearly \nidentified from the low -energy features of the density of states. This study is able \nto provide the partial information about anode  of Li+-based battery. There are \ncertain important differences between AC 6/AC 8 and Li 8Si4O12. \n \n1. Introduction   \n \nA bulk graphite1, which could be regarded as the layered graphene systems 2, has \nattracted a lot of theoretical and experimental researches in the basic science 3-5, \nengineering6 and application7. Each graphene layer is a pure carbon honeycomb \nlattice8; furthermore, the graphitic  are attracted together through the weak, but \nsigni ficant van der Waals interactions9. There exist the very strong intralayer \n𝜎𝜎 bondings of C -[2s, 2px, 2py] orbitals and the important interlayer C -2pz orbital \nhybridizations10, in which the former determine the most outstanding mechanical material , and the latter dominate the low -lying energy bands and thus the \nessential physical properties . Very interesting, the 3D graphite systems might \npresent the AA11, AB12, ABC13 and tur bostratic stackings14. All of them belong to \nsemimetals under the interlayer hopping integrals of C -2pz orbitals. In general, \nsuch condensed- matter systems become the n - or p-type metals, depending on the \nkinds of intercalated atoms or molecules. For example, the intercalation of alkali \natoms15 and FeCl 3 into graphite, respectively, create many free conduction \nelectrons and valence holes, as observed in a pure metal.  \n \nIt is well known that graphite could serve as the best anode materials16 in the \ncommercialized Li+ based batteries17, mainly owing to the lowest cost, the most \nstable structure for intercalation, and the outstanding ion transport under the \ncharging and discharging processes. When the Li+ ions are released from the \ncathode, the y will transport through the electrolyte, and then intercalate into the \ngraphitic system. Most important, the flexible interlayer spacings between \ngraphene layers are capable of providing the sufficient positions for the various \nintercalant concentrations18. Any intermediate states & meta -stable \nconfigurations, which are created during the ion/atom intercalation, could survive \nthrough the very strong 𝜎𝜎 bondings in graphitic sheets. As a result, graphite is \nrather suitable for studying the structural transf ormation in the chemical reactions. \nFor example, the close relations between graphene and intercalant layers in lattice \nsymmetries are expected to present the dramatic transformation before and after \nthe intercalation/de -intercalation processes.  \n \nUp to now , there are a lot of theoretical and experimental studies on the \nfundamental properties in graphite -related systems. The former covers the \nphenomenological models and number simulation methods. For example, the \ntight-binding model, the random -phase approxi mation, and the Kubo formula are, \nrespectively, utilized to investigate their electronic properties & magnetic \nquantization behaviors19-22, Coulomb excitations23-27 & impurity screenings, and \noptical absorption spectra28. Furthermore, the first -principles calculations29 are \navailable in understanding the optimal stacking configurations, intercalant lattices, \nπ, 𝜎𝜎  & intercalation -induced energy bands, free conduction electron/valence \nhole densities, and density of states. However, certain critical physical quantities \nand pictures are absent in the previous studies, such as, the atom -dominated band structures, the spatial charge densities between intercalant and graphene layers, \nthe interlayer orbital hybridization of intercalant and carbon atom, the atom - & \norbital -projected van Hove singularities, and the spin- dependent state \ndegeneracy/magnetic moment/density distribution.  \n \nThe main focuses of this book chapter are the main features of geometric and \nelectronic properties in stage- n graphite alkali -intercala tion compounds using the \nfirst-principles method29. Very interesting, the dependences on the kinds of \nintercalants and their concentrations are thoroughly explored through the delicate \ncalculations and analyses. The stacking configurations of graphene sheets, the \nlattice symmetries of intercalant layers, the dominances of valence and \nconduction states by the carbon and/or alkali atoms, the interlayer charge density \nvariations after the alkali intercalations, and atom-  & orbital -decomposed density \nof states. Whether the spin -induced Coulomb on- site interact ions or the spin -\norbital couplings play critical roles will be examined in detail. Most important, \nthe orbital -hybridizations in chemical bonds and the atom -induced spin \nconfigurations are determined from the above -mentioned results. How to simulate \nVASP b and structures by the tight -binding model is worthy of detailed \ndiscussions, since the interlayer hopping integrals are not very complicated under \nthe small primitive unit cells of layered graphite compounds. The predicted \ninterlayer distance/intercalant lattice, band overlap, and unusual van Hove \nsingularities, could be detected using the high- resolution X -ray diffraction/low -\nenergy electron diffraction30, angle -resolved photoemission spectroscopy31, and \nscanning tunneling spectroscopy32, respectively.  \n \n2. Theoretical calculations  \n \nBoth numerical simulations and phenomenological models are available in fully \nexploring the diversified physical/chemical phenomena. For example, the former \nand the latter, respectively, cover the first -principles calculations29/molecular \ndynamics33/quantum Monte Carlo34 and the generalized tight -binding \nmodel35/Kubo formula36/random -phase approximation37. Apparently, only the \nfirst method is able to investigate the lattice symmetries of the graphite -related \nsystems with/without the  chemical modifications. As for pure graphites and \ngraphite alkali -intercalation compounds, the previous V ASP results show that the stacking configurations and the intercalant lattice symmetries &  concentrations \nplay important roles in determining the semi- metallic or metallic behaviors. \nHowever, the concise pictures, which are very useful in fully comprehending the \nfundamental properties, are required in the further studies, as done for 2D layered \ngraphenes38,39 and 1D graphene nanoribbons40-42. \n \nIt is well known that the Vienna ab initio simulation package is developed within \nthe density functional theory43. The up -to-date VASP calculations can provide the \nsufficient information on the lattice symmetries, valence & conduction bands, \ncharge density distributions/variations after chemical bondings, atom - & orbital-\ndecomposed density of states44, being accompanied with/independent of the spin-\nrelated properties. Very interesting, a lot of electron quasiparticles experience two \nvery complex interactions, when t heir wave packets propagate in any periodical \nmaterial. Beyond the classical electrodynamics, the many -body electron- electron \nCoulomb interactions could be classified into the exchange and correlation \nenergies according to the first -order and the other cor rections, respectively. The \nPerdew -Burke -Ernzerh formula, which depends on the local electron density, is \nutilized to deal with many -particle Coulomb effects. As for the frequent electron-\ncrystal scatterings, they are characterized by the projector -augment ed wave \npseudopotentials. The electron Bloch wave functions are solved from the linear \nsuperposition of plane waves, with the maximum kinetic energy of 500 eV .  The \ncurrent study on stage -n graphite compounds shows that the first Brillouin zone is \nsampled by 9 × 9 × 9 and 100 × 100 × 100 k- point meshes within the Monkhorst -\nPack scheme, respectively, for the optimal geometry and band structure. Moreover, \nthe convergence condition of the ground state energy is set to be ~10-5 eV between \ntwo consecutive evaluation steps, where the maximum Hellmann -Feynman force \nfor each ion is below 0.01 eV/Å during the atom relaxations. \n \nThe self -consistent calculations, which are closely related to the local carrier \ndensity45, account for the V ASP simulations through the Kohn- Sham equations. \nAccording to the initial charge distribution, one is able to evaluate the classical & \nmany -particle exchange- correlation Coulomb interactions, and the electron- ion \nscattering potentials. After solving the second- order differential equation s, the \nintermediate electron density and  ground state energy are obtained and then are \nutilized to examine the convergence condition on the Feynman force. The significant physical quantities, the lattice symmetries/the various bond lengths in \na unit cell, the atom- dominated valence &  conduction bands, the spatial charge \ndensities/their variations after the creation of chemical bonds, the atom-  & orbital-\nprojected van Hove singularities, the atom -induced spin distributions, the spin-\nsplit or degenerate states across the chemical potential, and the vanishing or non-\nvanishing magnetic moments, are available in making a final decision about the \ncritical multi- /single -orbital hybridizations of chemical bonds and the atom -\ninduced spin configurations in the stage -n graphite alkali -intercalation compounds . \nThe above -mentioned theoretical framework, which is built from the first -\nprinciples results, presents the successful exploration on the diversified essential \nproperties of 2D layered graphenes38,39, 1D graphene nanoribbons40-42, and 2D \nsilicene -based sys tems. Now, the developed viewpoints are suitable for the anode, \ncathode and electrolyte materials of Li+-based batteries46. For example, a detailed \ncomparison between graphite and Li 4Ti5O12 is interesting and meaningful in term \nof well -known anode compounds.  \n \nThe direct linking between numerical simulations and phenomenological models \nis a very interesting topic. According to the up- to-date results of the low -energy \nband structures, the combination of the V ASP calculations and the tight -binding \nmodel is rather successful in exploring layered graphene/graphitic systems. For \nexample, both methods can account for the linear and gapless Dirac -cone band \nstructure, two pairs of linear/parabolic valence and conduction bands, three \nvertical Dirac cones/two pairs of parabolic bands plus one separated Dirac \ncone/one pair of partially flat, sombrero- shape and linear energy \ndispersions/oscillatory, sombrero- shape and parabolic bands, respectively, for the \nmonolayer, bilayer AA -/AB-stacked, and trilayer AAA -/ABA- /ABC -/AAB -\nstacked graphene systems. Their results are consistent with each other, since the \nimportant interlayer atomic interactions are purely due to the carbon- 2pz orbitals \nwithin the specific unit cells. That is to say, the single -orbital Hamiltonians, with \nthe conc ise interlayer & intralayer hopping integrals, could be expressed in the \nanalytic forms. Furthermore, all the intrinsic interactions, being modified by the \nexternal fields, are further included in the generalized tight -binding model. This \ntheoretical frame work has been successfully in creating and explaining the rich \nand unique magnetic quantization phenomena of few -layer group- IV systems, \ne.g., the unusual quantized Landau levels in graphene, silicene47, germanene48 and tinene49. \n \n3. Unique stacking configura tions and intercalant distributions      \n \nThe normal stacking configurations, which are formed by the layered graphene \nsystems, cover AAA, ABA, ABC and AAB. For example, according the V ASP \ncalculations on the trilayer graphenes, the ground state energy/the  interlayer \ndistance is lowest and highest/shortest/longest for the ABA and AAA stackings. \nThe similar results are revealed in the AA -, AB- and ABC -stacked graphites. Each \nhoneycomb structure has two equivalent sublattices with a planar geometry, \nindicatin g the orthogonal relation between one 2pz and three [2s, 2px, 2py] \norbitals. The sp2-bonding- based graphene systems maintain the flat sheets even \nunder any interlayer couplings. The significant van der Waals interactions, the \ninterlayer single -orbital hybr idizations of 2pz -2pz, account for the stacking \nconfigurations. Also noticed that the 2D silicene50/germanene51/tinene52 systems \npossess the buckled structures, since the π  and 𝜎𝜎 bondings are non- orthogonal \nto each other. Furthermore, the sp3 bonding might play an important role in the \ninterlayer atomic interactions. Apparently, graphenes are quite different from \nsilicenes/germanenes/tinenes in various essential properties.  \n \nThe i nterlayer spacings of graphitic layers are able to provide the very suitable \nchemical environment for the alkali intercalations and de- intercalations, i.e., they \nare very useful during the charging and discharging processes. The chemical \nmodifications hardly affect the planar honeycomb lattices and thus do not change \nthe orthogonal π  and 𝜎𝜎  bondings. Very interesting, the optimal geometric \nproperties are rather sensitive to the (x, y) -plane distribution and concentration of \nalkali atoms, e.g., the interlaye r distances of neighboring planes and an enlarged \nprimitive unit cell.  According to the highest concentration, the graphite alkali -\nintercalation materials could be classified into two categories: lithium and non -\nlithium ones. There exist LiC 6n or AC 8n for the stage- n materials [Figs. 7.1(a) -\n7.1(d)], as examined from the experimental measurements and the theoretical \npredictions. The specific distance between carbon and intercalant layer is \nshortest/longest for the Li/Cs [Table 7.1], only directly reflecting the effective \natomic radius. Although all the alkali atoms present the hollow -site optimal \npositions, their distribution symmetries are dominated by Li or non- Li. The layered LiC 6n and AC 8n systems, respectively, possess the three - and four -times \nenhancem ent in the unit cell; therefore, their reduced first Brillouin zones exhibit \nthe different high- symmetry points [Fig. 7.3(a)]. This is expected to have a strong \neffect on the initial π -electronic states of the Dirac -cone structure. More chemical \nbonds, whi ch arise from the alkali atoms, are produced in graphite intercalation \ncompounds, compared with pristine systems. Such bondings need to be taken into \naccount for the diversified essential properties. In addition to the intercalant \ndistribution, the stacking configuration of two neighboring graphitic sheets also \naffects the ground state energy. The current study clearly shows that only the stage -\n1 LiC 6 presents the AA stacking, and other compounds possess the AB stacking. \nThat is to say, all the graphite alk ali-intercalation materials exhibit the AB -stacked \nconfigurations except for the former system.  \n \nThe high- resolution experimental measurements of X -ray scatterings and low -\nenergy electron diffractions30 are able to examine the intercalant distribution, \ninterlayer distance, and stacking configuration of graphite alkali -intercalation \ncompounds; furthermore, those of scanning tunneling microscopy and tunneling \nelectron microscopy are very suitable for the identifications of optimal geometric \nstructures in few -layer graphene system37. The previous experiments have shown \nthe AA/AB stacking of the stage -1 LiC 6/AC 8 material, while the predicted AB \nstacking in stage -2-4 compounds [LiC 12, LiC 18 and LiC 24] require the further \nexperimental verifications.  \n \n4. Metallic an d semi -metallic behaviors  \n \nWithout alkali atom intercalations, a pristine graphite exhibits the unique semi-\nmetallic behavior through the weak, but significant van der Waals interactions. It \nis well known that a monolayer graphitic sheet has the orthogonal  π and 𝜎𝜎 \nbondings, in which the initial electronic states, respectively, come to exist at the \nFermi level [E F=0] and E~ -3.10 eV . Obviously, the low -energy physical \nproperties are mainly determined by the intralayer atomic interactions. Such \nhopping integr als, being accompanied with the uniform three nearest neighbors \nare responsible for a gapless and isotropic Dirac cone initiated from the K/K’ \nvalleys. The density of states is vanishing at E F because of the specific Dirac \npoints. A semiconducting zero- gap graphene is dramatically changed into a semi -metallic system for any few -layer stackings, or infinite -layer graphites. For \nexample, there exist the unusual overlaps of valence and conduction bands along \nthe KH path [or the Γ path under the reduced Brillouin zone] in AA -, AB - and \nABC -stacked graphites under the single -orbital interlayer atomic interactions. \nAccording to the V ASP and tight -binding model calculations, the simple \nhexagonal/rhombohedral graphite [Fig. 7.3(a)], with the highest/lowest stacking \nsymmetry, presents the largest/lowest free carrier density of conduction electrons \nand valence holes. In general, the π  valence bands, which are purely related the \nC-2pz, show the initial state s at the stable K valley, the saddle- point structure near \nthe M point [Ev~-3eV], and its termination in the Γ valley [the band width more \nthan 7 eV .  \n \nVery interesting, the band structures of lithium - and non- lithium graphite \nintercalation compounds sharply contrast with each other, such as, those of stage1 \nLiC 6 and XC 8 compounds in Figs. 7.3(d) and 7.3(e) -7.3(h), respectively. For the \nformer, the Fermi level is transferred from the middle of Dirac -cone structures \ninto the conduction ones ne ar the Γ valley after the intercalation of lithium atoms. \nThat is to say, E F presents a blue shift. Apparently, the free carriers are due to the \noutmost 2s orbitals of lithium atoms. The occupied and unoccupied states are \nhighly asymmetric to each other about the Fermi level. The initial π /π* \nvalence/conduction states are mainly determined by the stacking configurations \nin the chemical intercalations [the distribution symmetry of Li- intercalants in Fig. \n7.2(a)], or the corresponding relation between the ori ginal and reduced first \nBrillouin zone. There exists an observable energy spacing of Ec~0.31- 0.63 eV \nalong the Γ-A path in the modified valence and conduction Dirac -cone structures. \nThe creation of discontinuous states in honeycomb lattices might arise fro m the \ndifferent ionization energies of Li -2s and C -2pz orbitals [the distinct on- site \nCoulomb potential energies. The whole π  valence- band width of LiC 6, being \ncreated by the C -2pz orbitals, could be identified to about 7.32 eV/7.40 eV from \nthe electronic energy spectrum along the Γ-M-K-Γ/A-H-L-A [or Γ-K-M-Γ/A-L-\nH-A] paths. This result clearly illustrates the well- behaved π bondings in graphite \nalkali -intercalation compounds. Another 𝜎𝜎  bondings behave so, as indicated \nfrom the initial states at the Γ point of Ev~-4.20 eV . Their orthogonal relation \nremains the same after the chemical modification, being very useful in \nestablishing the tight -binding model for graphite alkali -intercalation compounds.   \nThe main features of band structure, as clearly indic ated in Figs. 7.3(e) -7.3(h), are \ndramatically changed under the other alkali -atom intercalations. Both stage -1 \nAC 8 and LiC 6 have the totally different distribution configurations [Figs. 7.2(a) \nand 7.2(b)], and so do the reduced first Brillouin zones [Figs. 7.3(a)]. For the \nformer, the Dirac- cone structure of the π  and π* bands are initiated from the stable \nK/K’ valleys, but not the Γ ones. The energy spacing of separated Dirac points \nis small or almost vanishing; furthermore, the Fermi level is situated abo ve the \nconduction point about ~1.35- 1.50 eV . In addition to E F, whether the second \nconduction energy subband is partially occupied depends on the kinds of alkali \natoms, such as, the alkali -induced free electrons in the first and second subbands \nfor KC 8/RbC 8/CsC 8 [Figs. 7.3(f)/7.3(g)/7.3(h)]. The low -energy bands belong to \nthe single states, without the split double degeneracy [Fig. 7.3(d) for LiC$_6$]. \nThe whole π -band energy spectra are identified from the K -Γ-K-M-Γ and H -A-\nH-L-A paths, leading to the widt h of ~ 7.31- 7.41 eV . Very interesting, one pair of \nπ-valence subbands come to exist near Ev~-4.0 eV , being accompanied with the \ninitial pair of the 𝜎𝜎  valence bands. This further illustrates the zone -folding \neffects on band structure and the well separatio n of π  and 𝜎𝜎 bondings. \n \nElectronic energy spectra strongly depend on the n stage of graphite intercalation \ncompound, as clearly illustrated in Figs. 7.4(a) -7.4(h). The stage -2 LiC 12 and \nAC 16 have the different first Brillouin zones on the (k x, ky) plane, and their k z-\nranges are associated with the distances between two intercalant planes [Fig. 7.1]. \nCompared with stage -1 band structures [Figs. 7.3(d) -7.3(h)], the number of Dirac \ncones becomes double in the stage -2 cases, in which the further modifications \ncover the reduced blue shift of the Fermi level, the enhanced anisotropy, the \ninduced energy spacings between valence and conduction Dirac points, and the \ndiversified energy relations near the band- edge states.  For example, four/two \nvalence and conduction pairs come to exist from the Γ /K valley [or the A/H \nvalley] for LiC 12/AC 16 [Fig. 7.4(d) and Figs. 7.4(e) -7.4(h)]. Furthermore, the \nwhole π-band widths could be roughly estimated from the ΓKM Γ /KMK Γ  path \n[or the ALHA/HLHA path]. The low -energy essential properties are dominated \nby the π  bondings of C -2pz orbitals. The above -mentioned obvious changes \ndirectly reflect the great enhancement of the interlayer atomic interactions due to \nthe C -C bonds in t wo neighboring graphitic sheets/graphene -intercalant layers. Very interesting, all the stage -2 compounds exhibit a pair of 𝜎𝜎 bands at the Γ \nand A valleys, and the energy dispersions along Γ-A are negligible. These results \nindicate the mutual orthogonality  of the planar 𝜎𝜎 and perpendicular π  bondings. \nSuch phenomenon is expected to survive in any stage -n graphite alkali -\nintercalation compounds.  \n \nThe blue shifts about the Fermi level and the free conduction electrons declines \nquickly in the increase of the  n number. The concentration of alkali atoms is \ngreatly reduced, and do the charge transfer from their outmost s -orbitals to carbon \n2pz-ones. Such results are clearly revealed in the stage- 3 systems. Figs. 7.5(a) -\n7.5(f) show the slight modifications of the  Dirac- cone structures and the \nintersecting of the Fermi level with most of conduction bands. LiC 18 [Fig. 7.5(b)] \nand AC 24 [Figs. 7.5(c) -7.5(f)], respectively, possess six and three pairs of linear \nvalence and conduction bands near the Γ/A and K/H valleys. Apparently, there \nexist the significant changes in the observable energy spacing between valence \nand conduction Dirac points, the anisotropic Fermi velocities, and the distinct \nFermi momenta slopes. The blue shifts of E F are estimated to be 0.45 eV , 0.58, \n0.55, 0.53, and 0.5 eV’s. Very interesting, the electronic energy spectra become \nmore complicated under the stronger zone -folding effects. The main features of \nelectronic properties are closely related to the weak, but important Li -C bonds, \ne.g., the minor contributions of alkali atoms on each electronic states. That is, the \ndiversified n- type dopings are created by the critical Li -C and C -C bnondings. \n \nThe conduction electron density, which is created by the alkali -atom doping, is \nworthy of a closer exami nation. It is mainly determined by the covered volume \n[area/length] in the wave -vector space between the lowest conduction state and \nthe Fermi level for 3D graphite alkali- intercalation compounds53 [2D alkali -\nadsorbed graphhenes/1D graphene nanoribbons]. I n general, the V ASP \ncalculations are too heavy to obtain the delicate values because of the strong \nanisotropic energy dispersions [the non- linear Dirac- cone structure at higher \nenergies in Figs. 7.4(d) -7.3(h)]. That a lot of 3D k -points are required for the \naccurate calculation of Free electron density is almost impossible. The similar \nphenomenon is revealed in the alkali -adatom adsorptions on graphene. However, \n1D graphene nanoribbons are very suitable for the full exploration of charge \ntransfer and conduc tion electron density, since the Fermi momenta are proportional to the Latter. After examining the critical mechanisms, namely the \nzigzag/armchair edges, different widths, five kinds of alkali atoms, and their \nvarious distributions, the free electron densi ty is almost identical to the adatom \nconcentration54. It can be deduced that any one alkali adatom contributes the \noutmost half -occupied s -orbital as a conduction carrier, or it behaves the full \ncharge transfer of - e in the alkali- carbon bonds. Such conclusions might be \nsuitable for 2D and 3D alkali -doped graphite compounds, being supported by the \nfollowing result. According to the spatial charge distributions and their variations \nafter the chemical modifications, the  chemical bondings hardly depend on the \ndistributions and concentrations of alkali adatoms.  \n \nGenerally speaking, only the high- resolution ARPES is capable of detecting the \nwave- vector -dependent occupied energy spectra below the Fermi level. The up -to-\ndate experimental measurements have successfully verified the diversified \nphenomena for the graphene -related sp2-bonding systems, as clearly observed \nunder the various dimensions, layer numbers, stacking configurations, substrates, \nand adatom/molecule chemisor ptions. For example, there exist 1D parabolic \ndispersions with sufficiently wide energy spacings/band gaps in graphene \nnanoribbons, the linear Dirac cone for monolayer graphene, two parabolic valence \nbands in bilayer AB stacking, the linear and parabolic dispersions in twisted \nbilayer systems, one linear and two parabolic bands for tri -layer ABA stacking, the \nlinear, partially flat and Sombrero -shaped dispersions in tri -layer ABC stacking, \nthe substrate -created observable energy spacing between the separated valence and \nconduction bands in bilayer AB stacking/oscillatory bands in few -layer ABC \nstacking2, the semimetal- semiconductor/semimetal- metal transitions after the \nadaom/molecule adsorptions graphene surface, and the bilayer - and monolayer -\nlike energy di spersions, respectively, at the K and H corner points [Fig 7.3(a)] for \nthe Bernal graphite. The ARPES measurements are available in examining the \nmain features of occupied band structure in graphite alkali -intercalation \ncompounds, namely, the blue shift of  the Fermi level, the initial K or Γ valley for \nthe π & π* bands, the separated or almost close Dirac- cone structures, their double \nor single degeneracy, the whole π  band along the Γ-K-M-Γ A-L-H-A paths [or \nthe K -Γ-K-M- Γ /H-A-H-L-A paths], the formation o f 𝜎𝜎 subbands at E~ -4.0 eV \nfrom the Γ valley, and the sensitive dependences on the alkali -atom distributions \nand concentrations. The experimental verifications could provide the useful information about the zone -folding effects due to the alkali interactio n, the single -\norbital hybridizations in alkali -carbon bonds, and the almost full charge transfer \n[discussed in the previous paragraph].  \n \nDensity of states at E [DOS(E)], in which is defined the number of states within \na very small energy range of dE, is ca pable of fully understanding the valence and \nconduction energy spectra simultaneously. Generally, its special structures are \ncreated by the band -edge states with the vanishing group velocities. The critical \npoints in the energy- wave- vector space cover the local extreme points [minima &  \nmaxima], the saddle points, and all the partially flat energy dispersions. \nApparently, the van Hove singularities are greatly diversified under the different \ndimensions. Of course, the discrete energy levels in 0D quantum dot s only present \nthe delta -function- like peaks, as observed in the magnetically quantized Landau \nlevels in emergent layered materials2 [e.g., few -layer graphene systems], and any -\ndimensional dispersionless energy bands. The 1D parabolic and linear bands, \nrespectively, the asymmetric divergent peaks in the square -root form and the \nplateau structures. As for the 2D condensed -matter systems, the concave-\n/convex- form dispersions], the Dirac -cone structure [the linear valence and   \nconduction bands without energy spacing] , the saddle points, and the constant -\nenergy loops, respectively, create the broadening shoulders, V -shapes, the \nsymmetric peaks relate related to the logarithmical divergence, and the square-\nroot divergent peaks. Moreover, the higher dimension in 3D materials make them \nonly show symmetric/asymmetric peaks and the shoulder structures, mainly \nowing to the absence of quantum confinement. The latter come from the 3D \nparabolic energy dispersions.  \n \nThe delicate calculations and analyses, which are cond ucted on the atom - and \norbital -dependent density of states [Figs. 7.6(a) -7.6(l)], are capable of fully \nunderstanding the metallic behavior and the close relations among the chemical \nbondings. Although band structures, with many band- edge states, become ver y \ncomplicated under the alkali -atom intercalations, the main features of van Hove \nsingularities are sufficiently clear for the identifications of diversified phenomena \nthrough a suitable broadening factor [e.g., energy width of ~0.10 eV. This further \nillustrates a critical role of zone folding in the fundamental properties. Very \ninteresting, the low -energy DOSs in stage -1 and stage -2 graphite alkali -intercalation compounds present a prominent peak just at the Fermi level [E F =0], \nregardless of the kind of alkali atoms. Furthermore, there exists a valley structure, \nwith a minimum value, at its left- hand neighbor. When such characteristic is \ncombined with the similar ones at E F] in the pristine AA - and AB -stacked graphite \nsyste ms [Figs. 7.6(a) and 7.6(b)], the Fermi level is deduced to exhibit a blue shift. \nThat is to say, E F is situated at the conduction energy subbands [E F roughly lies in \nthe center of valence and conduction bands] roughly after [before] the alkali -atom \ninterc alations. It should be noticed that the contributions due to the alkali atoms \nare weak, but rather important. The blue shifts of the stage -1 LiC 6, NaC 8, KC8, \nRbC 8, & CsC 8, and stage -2 LiC 12, NaC 16, KC 16, RbC 16 & CsC 16 are, respectively, \nestimated to be 1.90, 1.80,1.70, 1.60, 1.50, 1.30, 1.20, 1.10, 1.00, 0.90 eV’s. \n \nThe van Hove singularities in graphite intercalation compounds [Figs. 7.6(c) -\n7.6(l)], which survive in the specific energy ranges, are mainly determined by the \ncarbon or alkali atoms and their orbitals. Most important, DOS in the critical \nenergy range of - 5.0 eV < E < 3.0 eV, being relatively easily examined from the \nexperimental STS measurements, are dominated by the carbon -2pz orbitals [the \npink curves]. Furthermore, the minor contributions rel ated to the outmost s -orbital \nof alkali atoms [the green curves], especially at the conduction energy spectrum, \nplay a critical role in determining the blue shift of the Fermi level. As for the C -\n[2px, 2py] orbitals [the red and light blue curves], their c ontributions are initiated \nfrom ~ E < -4.0 eV, while they are absent in the opposite energy range. The red \nshift of the initial 𝜎𝜎 valence bands is about 1.0 eV, compared with those of the \npristine simple hexagonal and Bernal graphites [Figs. 7.6(a) and 7/ 6(b)]. Only the \nLiC 6 and LiC 12 cases [Figs, 7.6(c) and 7.6(h)] exhibit the split contributions of \n2px and 2py orbitals. This result directly reflects the anisotropic distribution \nconfiguration. More, the C -2s orbitals com to exist at the deeper energies of  ~E< -\n5.0 eV. Apparently, the above -mentioned features indicate the well separation of \nC-2pz and C -[2s, 2px, 2py] orbital contributions, and thus the normal \nperpendicular orbital hybridizations of π and 𝜎𝜎 chemical bonds. That such \nbonding behavior is stron gly linked with the significant interlayer 2pz -orbitals due \nto the carbon- alkali bonds can account for the featured electronic properties, e.g., \nthe main features of band structures and DOSs in Figs. 7.3- 7.7].  \n \nThe clear identifications of stage -1 and stage -2 of graphite alkali -intercalation compounds could be achieved from the low -energy features of van Hove \nsingularities, as indicated in Figs. 7.6(c) -7.6(g) and 7.6(h) -7.6(l), respectively. \nCompared with those of the Former, the blue shifts of the Fermi levels are \nrelatively small, in which they are, respectively, ~1.20 eV, 1.15 eV, 1.10 eV, 1.06 \neV and 1.03 eV for LiC 12, NaC 16, KC 16, RbC 16, and CsC 16. Most important, their \ndensities of states at $E_F$ do not bel ong to the local maxima. Such result directly \nreflects whether the band -edge states of conduction bands are somewhat away \nfrom the Fermi level [Figs. 7.4(b) -7.4(f)]. This significant difference between stge -\n2 and stage -1 systems further illustrates the intersecting of E F and conduction \nbands, thus leading to the n- type doping cases after the alkali -atom intercalations. \nThe experimental measurements could be utilized to verify the free conduction \nelectrons due to the alkali -atom intercalations, e.g., the atom -dependent optical \nthreshold absorption frequencies, and the doping- enhanced electrical \nconductivities. \n \nThe high- resolution STS measurements are the only method in examining the van \nHove singularities due to the valence and conduction energy spectrum, es pecially \nfor the semi -metallic, metallic or semiconducting behaviors near the Fermi level. \nSuch examinations cover the form, energy, number, and intensity of special \nstructures in density of states. Apparently, they cannot identify the wave -vector -\ndependent energy dispersions. The up- to-date measured results have successfully \nverified the diverse electronic properties in graphene -related systems with the \ndominating sp2 bondings , e.g., 1D graphene nanoribbons, carbon nanotubes, 2D \nfew-layer graphenes, adatom -adsorbed graphenes, and 3D graphite. The main \nfeatures of graphene nanoribbons, the width-  and edge -dependent energy gaps and \nthe square -root-divergent asymmetric peaks of 1D  parabolic dispersions, are \nconfirmed from the precisely defined boundary structures. The similar strong \npeaks are displayed in seamless carbon nanotubes, in which these structures \nexhibit the chirality - and radius -dependent band gaps and energy spacings between \ntwo neighboring subbands. Even with the obvious curvature effects on a \ncylindrical surface [the misorientation of 2pz orbitals and the significant \nhybridization of carbon four orbitals], armchair nanotubes belong to 1D metals \nwith a finite density o f states at the Fermi level [a sufficiently high free carrier \ndensity]. A lot of STS measurements, which are made for few -layer and adatom -\ndoped graphenes, clearly illustrate the low -lying characteristics of van Hove singularities: a V -shape energy depende nce vanishing at the Dirac point in \nmonolayer system [a zero -gap semiconductor], the asymmetry- induced peak \nstructures in the logarithmic form for twisted bilayer graphenes, a gate -voltage -\ncreated band gap in bilayer AB stacking and tri -layer ABC stacking , a delta-\nfunction- like peak centered about the Fermi level related to surface states of the \npartially flat bands in tri- layer and penta -layer ABC stackings, a sharp dip structure \nat E F accompanied with a pair of asymmetric peaks in tri- layer AAB stacking [a \nnarrow -gap semiconductor with the low -lying constant -energy loops], and a red \nshift of Dirac point arising from the n- type electron doping of Bi adatoms. The \nmeasured density of states in Bernal graphite is shown to be finite near E F \ncharacteristic of t he 3D semi -metallic property and presents the splitting of π and \nπ* strong peaks at deeper/higher energies. It should be noticed that the well -\ndefined π and 𝜎𝜎 band widths in graphene -related systems are worthy of the \nfurther STS examinations. In addition, there are no STS results for AA - and ABC -\nstacked graphites up to now, mainly owing to few content of natural graphite and \nthe difficulties in sample growth. The experimental examinations on the current \npredictions, especially for the van Hove singularitie s near E F and the π  & 𝜎𝜎 band \nwidths, are very useful in providing the detailed information about the n- type \ndopings and the orbital hybridizations of chemical bonds. \n   \nThe close relations between the V ASP simulations and the tight -binding model/the \neffective -mass approximation in graphite -/graphene -related systems are discussed \nin detail. The previous theoretical studies show that the electronic properties of \npure carbon honeycomb lattices are well characterized by the phenomenological \nmodel. According to the calculated results of two different methods, the rich and \nunique band structures are created by the stacking configuration, layer number, \nand dimensionality, such as, those due to monolayer graphene, bilayer AA/AB \nstackings55, trilayer AAA/ABA/ABC/AAB56-58, 3D AA- /AB-/ABC -stacked \ngraphites, 1D achiral/chiral carbon nanotubes59, and graphene nanoribbons. Their \noutstanding consistence is deduced to be closely related to the good separations of \nπ and 𝜎𝜎 chemical bondings and the interlayer single -2pz-orbital hybridizations. \nVery interesting, the generalized tight -binding model, which is combined with the \nstatic & dynamic Kubo formulas60 and the modified random -phase approximation, \nis capable of fully exploring the diversified magnetic quantization. As for the \nemergent 2D group- IV and group- V materials, these theoretical frameworks are successful in studying the unusual magneto- electronic properties, the various \nmagneto -optical selection rules, the unique quantum Hall conductivities, and \ncomplex magneto- plasmon modes &  inter -Landau- level excitations61.  \n \nThe low -energy physical properties of graphites and graphite alkali -intercalation \ncompounds are dominated by the single -orbital interactions of C -2pz and A -s \norbitals [the outmost one of alkali atom]. The AA -, AB - and ABC -stacked  \ngraphites11-13 have two, f our and six carbon atoms in a unit cell, so their \nHamiltonians are built from the corresponding tight -binding functions. After the \ndiagonalization of Hermitian matrices, the π - and π*-electronic energy spectra \nstrongly depend on the stacking configurations , especially for the overlap of \nvalence and conduction bands. The highest conduction electron/valence hole \ndensity is revealed in the first/third system. Apparently, the stacking- enriched \ninterlayer hopping integrals of C -2pz orbitals are responsible for t he diversified \nsemi -metallic properties. Most important, the above -mentioned phenomenological \nmethod could cover the alkali intercalation effects, the enlarged unit cells and the \nstacking configurations. That is to say, the modified Hamiltonians of graphit e \nalkali -intercalation compounds include the ionization energy difference between \nC-2pz and Li -2s orbitals, the intralayer nearest -neighboring interactions of 2pz -\n2pz/2s -2s orbital in C -C/Li-Li bonds, and the interlayer 2pz -2s/2pz -2pz orbital \nhybridizations. Specifically, the third ones might be complicated and very sensitive \nto the change in the intercalant concentration; therefore, they play a critical role in \ncreating the diversified metallic behaviors. It should be noticed that the theoretical \nmodels, w ith the zone -folding effects, are absent up to now.  \n \nIn addition to geometric and electronic properties, free conduction electrons in \ngraphite intercalation compounds are able to create the rich and unique essential \nproperties, such as, optical absorption spectra, single-  & many -particle Coulomb \nexcitations, and electrical conductivities. Apparently, the Fermi level at the \nconduction Dirac -cone is responsible for the optical threshold frequency of 2E F \nunder the vertical transitions of valence -and conduction bands. Specifically, the \nmany -body excitonic effects might come to exist under the suitable conditions, as \nrevealed in certain optical measurements. They would reduce the optical gap and \nenhance the initial absorption structure. It is well known that conduction charge \ncarriers have the outstanding Coulomb response under the dynamic and static perturbations of electron -electron interactions. As a result of the conduction and \nvalence electrons [the π*  and π carriers], there exist the diverse [momentum. \nFrequency] -phase diagrams. That is, certain intraband &  interband electron -hole \nregions and different plasmon modes might appear simultaneously, being sensitive \nto the Fermi level/carrier density and the band structure. Moreover, the Fermi \nsurface, which is formed by the various Fermi -momentum conduction states, is \npredicted to generate the Friedel charge density oscillation at the long distance. \nApparently, it has a strong effect on transport property, e.g., the residual resistivity \nrelated to the electron -impurity elastic scatterings.  \n \nThe graphite -related materials quite differ from the 3D ternary lithium oxide \nsystems in terms of the Li+-based battery anode. For there are certain important \ndifferences between the layered LiC 6/Li+C6 and Li 8Si8O12 bulk compounds. The \nformer are metals/semimetals, while the latter belongs to an indirect- gap insulation \nwith E gi more than 5 eV. The interlayer spacings of graphitic sheets, which are \ndetermined by the van der Waals interactions of C -2p$_z$ orbitals, are  available \nfor the Li+-ion/Li -atom intercalations & deintercalations during the charging & \ndischarging processes. May be, the heterojunction of graphite and electrolyte \nbecomes an issue in greatly enhancing the performance of ion transport. This \nproblem is  under the current investigations. But for the latter, there are a lot of \nchemical bonds within a unit cell, in which the Li - and Si -O bond lengths exhibit \nthe large modulation over 20%. It is deduced to be very easy to change the internal \ngeometric struct ures [or create the intermediate configurations]. The dramatic \ntransformation between the quasi -stable and intermediate geometries is worthy of \nthe further systematic studies. Of course, the boundary of two distinct lithium \noxides needs to be examined thor oughly. Additionally, the semi -metallic, metallic \nand semiconducting properties, being associated with electrons, might be not play \ncritical roles in ion -transport batteries.  \n \nIn addition to the lithium- ion-based batte ries, there exist the aluminum-62 and iron-\n63ion-induced ones, especially for the former. The AB -stacked graphite in the \nsecond  system could serve as the efficient cathode material under the anion \ntransport. Very interesting, the co- existence of AlCl 4- and Al 2Cl7- anions64 will \ndetermine the optimal geometric symmetries between two neighboring graphitic \nlayers and thus the low -lying energy valence and conduction bands. How to mix together for two kinds of large molecular ions could be thoroughly explored by the \nfirst-princi ples calculations, e.g., the diversified distribution configurations \nassociated with the relative ratio.  As to the cathode and anode systems of \ngraphite intercalation compounds, their fundamental properties are expected to be \ntotally different from each o ther in terms of the geometric, electronic, magnetic, \noptical and transport properties. The intercalations and de -intercalations, which \nare driven by the aluminum - and chloride -related anions, should be one of the non-\nnegligible transport mechanism in the cathode material of graphite. For example, \nsuch behaviors would be responsible for initiating the ionic currents and dominate \nthe stationary ion flows.  \n \n5. Conclusions  \n \nThe fundamental properties of (Li,Na,K,Rb,Cs )-intercalated graphite under \nvarious stages1- 4 have been investigated by mean of first -principle calculations. \nThe weak, but significant Van der Walls interactions would contribute to low -\nlying π-electronic structure and thus dominate the essential physic al properties. \nThe dramatic changes cover the blue shift of the Fermi level/the red shift of the 𝜎𝜎 \nbands [the n- type doing behaviors], the greatly enhanced asymmetric electron and \nhole spectra, the obviously reduced conduction electron density for the dilute \nintercalant systems, the energy spacing between valence and conduction Dirac \ncones, the initial  K/ Γ valleys for the π -electronic state with the single/double \ndegeneracy,  the whole π  band along the K -Γ–K-M- Γ/Γ–K-M-Γ paths [or the H -\nA-H-L-A/A- L-H-A paths], and a pair of 𝜎𝜎 subbands due to carbon -[2px, 2py] \norbitals at the Γ valley of E~ -4.0 eV .  \n \nThe stable layered structures and the metallic/semi -metallic properties are \nresponsible for the important differences with the Li+-based battery anode of the \ninsulating bulk Li 4Ti5O12. The alkali charge transfers to the carbon sp2 honeycomb \nlattices is deduce d to be large, being comparable with that in 1D graphene \nnanoribbons65. That is to say, the free electron density is roughly identical to the  \nalkali -atom concentration. Conduction electrons have been predicted to have \nstrongly affect the other essential properties, e.g., the great enhancement of the \noptical threshold frequency, and the various [momentum, frequency] -phase \ndiagrams of Coulomb exc itations with the rich single - and many -particle modes. Most important, the tight -binding model, which could be built from the low -lying \nV ASP band structures under the simultaneous considerations of intralayer C -C & \nA-A bondings and interlayer C -C and A -C interactions, would be very useful in \nthe diversified phenomena, e.g., the rich magnetic quantization in layered systems \nonly available by this method.  \n \nAcknowledgements:  This work is supported by the Hi -GEM Research Center and \nthe Taiwan Ministry of Scie nce and Technology under grant number MOST 108-\n2212- M-006-022-MY3 and MOST 108- 3017- F-006-003. \n \n References:  \n[1] Lee, S. -M ; Kang, D. -S ; Roh, J.- S. Bulk graphite: Materials and manufacturing \nprocess. Carbon Letters 2015, 16. 135- 146. \n[2] Tran, N. T. T.; Lin, S. Y .; Lin, C. Y; Lin, M. F. Geometric and Electronic \nProperties of Graphene -related systems: Chemical bonding schemes. CRC Press , \nbook , ISBN 9781351368483(2017).  \n[3] Wang, Y ., Panzik, J. E., Kiefer, B. & Lee, K. K.  Crystal structure of graphite \nunder room -temperature compression and decompression.  Scientific reports , 2, \n520(2012).  \n[4] Ho, J. H., Chang, C. P. & Lin, M. F.  Electronic excitations of the multilayered \ngraphite. Physics Letters A , 352, 446- 450(2006).  \n[5] Lin, C. Y ., Chen, R. B., Ho, Y . H. & Lin, M. F.  Electronic and optical properties \nof graphite -related systems. CRC Press , book , ISBN 9781138571068(2017). \n[6] Zhang, Z., Huang, H., Yang, X. & Zang, L.  Tailoring electronic properties of \ngraphene by π -π stacking with aromatic molecules. Th e Journal of Physical \nChemistry Letters ,  2, 2897- 2905(2011).  [7] Cheng, Q. et al.  Graphene -Like-Graphite as Fast -Chargeable and High-\nCapacity Anode Materials for Lithium Ion Batteries . Scientific Reports , 7, \n14782(2017).  \n[8] Mecklenburg, Matthew; Regan, B... Spi n and the Honeycomb Lattice: Lessons \nfrom Graphene. Physical review letters . 2011, 106. 116803. 10.1103.  \n[9] Gómez- Santos, G.. Thermal van der Waals Interaction between Graphene Layers. \nPhysical Review B . 2009, 80. 10.1103. \n[10] D. D. L. Chung. Journal of Materials  Science 2002, V olume 37, 8, 1475–\n1489.  \n[11] Lin, Chiun- Yan; Wu, Jhao- Ying; Chiu, Chih- Wei; Lin, Ming -Fa. AA -\nStacked Graphenes. CRC Press, book, ISBN:  9780429277368. \n[12] Lin, Chiun- Yan; Wu, Jhao- Ying; Chiu, Chih- Wei; Lin, Ming -Fa. AB -\nStacked Graphenes. CRC Press, book, ISBN:  9780429277368.. \n[13] Lin, Chiun- Yan; Wu, Jhao- Ying; Chiu, Chih- Wei; Lin, Ming -Fa. ABC -\nStacked Graphenes. CRC Press, book, ISBN:.9780429277368. \n[14] Negishi, Ryota; Wei, Chaopeng; Yao, Yao; Ogawa, Yui; Akabori, Masashi; Kanai, \nYasushi; Matsumoto, Kazuhiko; Taniyasu, Yoshitaka; Kobayashi, Yoshihiro. \nTurbostratic Stacking Effect in Multilayer Graphene on the Electrical Transport \nProperties. physica status solidi. book, ISBN:  0370- 1972. \n[15] Natori, Akiko; Ohno, Takahisa; Oshiyama, Atsushi. Work Function of Alkali -\nAtom Adsorbed Graphite. J PHYS SOC JPN . 1985, 54. 3042- 3050. 10.1143. \n[16] Mao, C. et al.  Selecting the Best Graphite for Long- Life, High- Energy Li -Ion \nBatteries. Journal of The Elec trochemical Society 2018, 165, A1837- A1845. [17] Cheng, Q. Graphene -Like-Graphite as Fast -Chargeable and High -Capacity \nAnode Materials for Lithium Ion Batteries . Scientific Reports  2017, 7, 14782. \n[18] hih-Yang Lin; Ngoc Thanh Thuy Tran; Sheng- Lin Chang; Wu- Pei Su; M. F. Lin. \nStructure - and adatom -enriched essential properties of Graphene Nanoribbons. \nCRC Press, book, ISBN ：9780367002299. \n[19] Ho, J. H., Lai, Y . H., Tsai, S. J., Hwang, J., Chang, C. & Lin, M.F  \nMagnetoelectronic Properties of a Single -Layer Graphite (Conde nsed matter: \nelectronic structure and electrical, magnetic, and optical properties). Journal of \nthe Physical Society of Japan, 75, (2006).  \n[20] Huang, Y . K., Chen, S. C., Ho, Y . H., Lin, C. Y. & Lin, M. F.  Feature -rich \nmagnetic quantization in sliding bilayer graphenes. Scientific reports , 4, \n7509(2014).  \n[21] Koshino, M. & McCann, E.  Landau level spectra and the quantum Hall effect \nof multilayer graphene. Physical Review B , 83, 165443(2011).  \n[22] Wang, Z. F., Liu, F. & Chou, M. Y .  Fractal Landau- level spectra in twisted \nbilayer graphene. Nano letters ,  12, 3833- 3838(2012). \n[23] Ho, J. H., Chang, C. P. & Lin, M. F.  Electronic excitations of the multilayered \ngraphite. Physics Letters A , 352, 446- 450(2006).  \n[24] Lin, M. F., Chuang, Y . C. & Wu, J. Y .  Electrically tunable plasma excitations \nin AA- stacked multilayer graphene. Physical Review B ,  86, 125434(2012). \n[25] Wu, J. Y ., Chen, S. C., Roslyak, O., Gumbs, G. & Lin, M. F.  Plasma excitations \nin graphene: Their s pectral intensity and temperature dependence in magnetic \nfield. ACS nano, 5, 1026- 1032(2011).  [26] Lozovik, Y . E. & Sokolik, A. A.  Influence of Landau level mixing on the \nproperties of elementary excitations in graphene in strong magnetic field. \nNanoscale rese arch letters , 7, 134(2012). \n[27] Ohta, T. et al.  Interlayer interaction and electronic screening in multilayer \ngraphene investigated with angle -resolved photoemission spectroscopy. \nPhysical Review Letters , 98, 206802(2007)  \n[28] Ain, Q. T., Al -Modlej, A., Alshammari , A. & Anjum, M. N.  Effect of solvents \non optical band gap of silicon- doped graphene oxide.  Materials Research \nExpress , 5, 035017(2018). \n[29] Leenaerts, O., Partoens, B. & Peeters, F. M.  Adsorption of H 2O, NH 3, CO, \nNO 2, and NO on graphene: A first -principles  study. Physical Review B , 77, \n125416(2008).  \n[30] De Padova, P.; Quaresima, C.; Ottaviani, C.; Sheverdyaeva, P. M.; Moras, P.; \nCarbone, C.; Topwal, D.; Olivieri, B.; Kara, A.; Oughaddou, H. Evidence of \ngraphene -like electronic signature in silicene nanoribbons. Applied Physics \nLetters 2010, 96, 261905. \n[31] Rufieux, P.; Cai, J.; Plumb, N. C.; Patthey, L.; Prezzi, D.; Ferretti, A.; Molinari, \nE.; Feng, X.; M ullen, K.; Pignedoli, C. A. Electronic structure of atomically \nprecise graphene nanoribbons. Acs Nano 2012, 6, 6930{6935. \n[32] Kano, S.; Tadaa, T.; Majima Y . Nanoparticle characterization based on STM and \nSTS. Chemical Society Reviews 2015, 44, 970.  [33] Berdiyorov, G.; Neek- Amal, M.; Peeters, F.; van Duin, A. C. Stabilized silicene \nwithin bilayer graphene: A proposal based on molecular dynamics and density-\nfunctional tight -binding calculations. Physical Review B 2014, 89, 024107. \n[34] Richard Needs. The quantum Monte Carlo method—electron correlation from \nrandom numbers. Journal of Physics: Condensed Matter  2008, 20, 6. \n[35] Zhe Liu;  Guodong Yu;  Haibo Yao;  Lei Liu;  Liwei Jiang; Yisong Zheng. A simple \ntight-binding model for typical graphyne structures. New Journal of Physics , \n2012, 14. \n[36] Jinshan Wu;  Mona Berciu. Kubo formula for open finite -size systems. \nEurophysics Letters , 201 0, 92, 3. \n[37] Joachim Paier;  Xinguo Ren;  Patrick Rinke;  Gustavo E Scuseria; ndreas \nGrüneis;  Georg Kresse;  Matthias Scheffler. Assessment of correlation energies \nbased on the random -phase approximation. New Journal of Physics , 2012, 14. \n[38] Novoselov, K. S.  et al.  Electric field effect in atomically thin carbon films. \nScience, 306, 666 -669(2004).   \n[39] Zhang, Y ., Tan, Y . W., Stormer, H. L. & Kim, P.  Experimental observation of \nthe quantum Hall effect and Berry's phase in graphene. Nature, 438, 201(2005). \n[40] Jiao, L., Wa ng, X., Diankov, G., Wang, H.  & Dai, H.  Facile synthesis of high-\nquality graphene nanoribbons. Nature nanotechnology , 5, 321(2010).  \n[41] Kosynkin, D. V . et al.  Longitudinal unzipping of carbon nanotubes to form \ngraphene nanoribbons. Nature , 458, 872(2009).  \n[42] Cai, J. et al.  Atomically precise bottom- up fabrication of graphene nanoribbons. \nNature , 466, 470(2010).  [43] Medeiros, P. V .; Mascarenhas, A. J.; de Brito Mota, F.; de Castilho, C. M. C. A \nDFT study of halogen atoms adsorbed on graphene layers. Nanotech. 2010, 21, \n485701. \n[44] Kresse, G.; Furthm �uller, J. E_cient iterative schemes for ab initio total- energy \ncalculation using a plane -wave basis set. Physical review B  1996, 54, 11169. \n[45] Kohn, W.; Sham, L. J. Self -consistent equations including exchange and \ncorrelation e_ects. Physical review  1965, 140, A1133. \n[46] Lin, Shih- Yang, Li, Wei -Bang, Tran, Ngoc Thanh Thuy, Hsu, Wen- Dung, Liu, \nHsin -Yi, Fa -Lin, Ming. Essential properties of Li/Li$^+$ graphite intercalation \ncompounds. arXiv: 1810.11166 (2018).  \n[47] Lin, S. Y ., Chang, S. L., Tran, N. T. T., Yang, P. H. & Lin, M. F.  H –Si bonding-\ninduced unusual electronic properties of silicene: a method to identify hydrogen \nconcentration. Physical Chemistry Chemical Physics , 17, 26443- 26450(2015).   \n[48] Li, S. S.  et al.  Tunable electroni c and magnetic properties in germanene by \nalkali, alkaline -earth, group III and 3d transition metal atom adsorption. Physical \nChemistry Chemical Physics , 16, 15968- 15978(2014). \n[49] Chen, R. B., Chen, S. C., Chiu, C. W. & Lin, M. F.  Optical properties of \nmonol ayer tinene in electric fields. Scientific Reports , 7,  1849(2017). \n[50] Sahin, H. & Peeters, F. M.  Adsorption of alkali, alkaline -earth, and 3d \ntransition metal atoms on silicene.  Physical Review B , 87, 085423(2013).  \n[51] Dávila, M. E., Xian, L., Cahangirov, S., Rubio, A. & Le Lay, G. Germanene: a \nnovel two- dimensional germanium allotrope akin to graphene and silicene. New \nJournal of Physics , 16, 095002(2014).  [52] Cai, B.  et al.  Tinene: a two -dimensional Dirac material with a 72 meV band \ngap.  Physical Chemistry Chemical Physics , 17, 12634- 12638(2015).  \n[53] Y . T. Lin; S. Y . Lin; Y . H. Chiu; M. F. Lin. Alkali -created riched properties in \ngraphene nanoribbons: Chemical bondings. Science Report  2017, 7: 1722. \n[54]  Tran, N. T. T., Lin, S. Y ., Lin, C. Y ., Lin M. F., Geometric and  electronic \nproperties of graphene -related systems Chemical bonding Schemes, CRC Press  \nbook  2017, ISBN 9781138556522.  \n[55] Tran, N. T. T., Lin, S. Y ., Glukhova,O. E., Lin M. F. Configuraion- induced Rich \nElectronic Properties of Bilayer Graphene. J ournal of Physical Chemistry C,  \n2015, 119 , 10623- 10630.  \n[56] Ilham Redouani ; Ahmed Jellal ; Abdelhadi Bahaoui; Hocine Bahlouli . \nMultibands tunneling in AAA -stacked trilayer graphene . Superlattice \nand Microstructures  2018 , 116, 44-53. \n[57] S. Hattendorf ; A. Georgi; M. Liebmann; M. Morgenstern. Networks \nof ABA and ABC stacked graphene on mica observed by scanning \ntunneling microscopy. Surface Science  2013, 610 , 53-58. \n[58] Thi-Nga Do;  Cheng -Peng Chang; Po-Hsin Shih;  Jhao- Ying \nWu; Ming -Fa Lin. Stacking -enriched  magneto -transport properties \nof few -layer graphenes. Phys. Chem. Chem. Phys. , 2017,  19, 29525 -\n29533 \n[59] F. L. Shyu; C. C. Tsai; C. H. Lee; M. F. Lin . Magnetoelectronic \nproperties of chiral carbon nanotubes and tori. J. Phys.: Condens. \nMatter  2003 , 18 8313– 8324.  [60] Thi-Nga Do;  Cheng -Peng Chang; Po-Hsin Shih;; Jhao- Ying \nWu; Ming -Fa Lin. Stacking -enriched magneto- transport properties \nof few -layer graphenes. Phys. Chem. Chem. Phys.  2017,  19, 29525 -\n29533 . \n[61] Jhao- Ying  Wu; Szu -Chao  Chen; Thi- Nga Do; Wu -Pei Su; Godfrey \nGumbs; Ming-Fa Lin. The diverse magneto -optical selection rules in \nbilayer black phosphorus. Scientific Reports  2018, 8 , 13303.  \n[62] Kai Yang; Zeyun Zhao; Xin Xin; Zhongliang Tian; Ke Peng; \nYangqing Lai. Graphitic carbon materials extracted from spent \ncarbon cathode of aluminium reduction cell as anodes for lithium ion \nbatteries: Converting the hazardous wastes into value -added \nmaterials.  Journal of Taiwan Institute of Chemical Engineers 2019, \n104, 201-209.  \n[63] D. Narsimulu; Goli Nagaraju; S. Chandra Sekhar; Bhimanaboina \nRamulu; Jae Su Yu. Designed lamination of binder -freeflexible iron \noxide/carbon cloth as highcapacity and stable anode material for \nlithium -ion batteries. Applied Surface Science 2019,497 , 143795.  \n[64] Qianpeng Wang; Daye Zheng; Lixin He; Xinguo Ren. Cooperative \nEffect in a Graphite Inter calation Compound: Enhanced Mobility \nof AlCl 4 in the Graphite Cathode of Aluminum -Ion Batteries. Phys. \nRev. Applied  2019, 12, 044060 .  [65] Shen -Lin Chang ; Shih-Yang Lin ; Shih-Kang Lin;  Chi-Hsuan \nLee; Ming -Fa Lin . Geometric and Electronic Properties of Edge -\ndecorated Graphene Nanoribbons. Scientific Reports  2014 , 4, 6038 . \n \n  Table  \n \nTable 7.1: The stage -dependent ground state energies and the optimal geometric \nparameters of the stage- n graphite alkali -intercalation compounds. \n \n Stage- 1 , Alkali -doped Griphite  \n LiC6  NaC8  KC8  RbC8  CsC8  \ndistance between \nalkali metal and \ncarbon layer ( Å) 1.9075 2.3005 2.662 2.9127 3.016 \nGround State \nEnergy(eV)  -57.29 -74.96 -75.13 -75.04 -75.12 \n      \n Stage- 2 , Alkali -doped Griphite  \n LiC12  NaC16  KC16  RbC16  CsC16  \ndistance between \nalkali metal and \ncarbon layer ( Å) 1.9685 2.4833 2.8233 2.9685 3.0351 \nGround State \nEnergy(eV)  -112.52 -148.78 -148.93 -148.84 -148.94 \n      \n Stage- 3 , Alkali -doped Griphite  \n LiC18  NaC24  KC24  RbC24  CsC24  \ndistance between \nalkali metal and \ncarbon layer ( Å) 2.783/1.289 2.4295/1.9934  2.6766/1.7498  2.8382/1.8016  2.5999/2.0571  \nGround State \nEnergy(eV)  -166.99 -222.64 -222.81 -222.69 -222.79 \n \n \n \nFigure Captions   \nFigure 7.1: The stage -n graphite alkali -intercalation compounds: (a) n=1, (b) n=2, \n(c) n=3, and (d) n=4.  \n \nFigure 7.2: The planar structures of (a) LiC 6 and (b) XC 8 (X=Na, K, Rb and Cs). \n \nFigure 7.3: The rich band structures of the pristine  graphite and stage -1 graphite \nalkali -intercalation compounds (a) within the corresponding first Brillouin zones: \n(b) AA & (c) AB stackings without intercalations, (d) LiC 6, (e) NaC 8, (f) KC 8, (g) \nRbC 8 and (h) CsC 8. \n \nFigure 7.4: The unusual electronic energy spectra of the AB -stacked stage- 2 \ngraphite alkali -intercalation compounds: (a) the reduced first Brillouin zones of AB \nstacking, (b) LiC 12, (c) NaC 16, (d) KC 16, (e) RbC 16, and (f) CsC 16. \n \nFigure 7.5: The rich band structures for the AB -stacked stage- 3 graphite alkali -\nintercalation compounds: (a) the reduced first Brillouin zone, (b) LiC 18, (c) NaC 24, \n(d) KC 24, (e) RbC 24, and (f) CsC 24. \n \nFigure 7.6: The atom - and orbital -decomposed density of states for the (a) AA - & \n(b) AB -stacked graphite systems, (c) LiC 6, (d) NaC 8, (e) KC 8, (f) RbC 8, (g) CsC 8, \n(h) LiC 12, (i) NaC 16, (j) KC 16, (k) RbC 16, and (l) CsC 16. The black, red, light blue, \npink, gtreen, deep blue, and purple curves, respectively, C -2s, C -2px, C -2py, C -2pz, \nalkali, carbon, and compound.  \n \n \n \n \n \n \n \n \n \n                        \n \n \nFigure 7.1: The stage -n graphite alkali -intercalation compounds: (a) n=1, (b) n=2, \n(c) n=3, and (d) n=4.  \n \n \n \n \n \n \nFigure 7.2: The planar structures of (a) LiC 6 and (b) XC 8 (X=Na, K, Rb and Cs).  \n \nFigure 7.3: The rich band structures of the pristine graphite and stage -1 graphite \nalkali -intercalation compounds (a) within the corresponding first Brillouin zones: \n(b) AA & (c) AB stackings without intercalations, (d) LiC 6, (e) NaC 8, (f) KC 8, (g) \nRbC 8 and (h) CsC 8. \n \n  Figure 7.4: The unusual electronic energy spectra of the AB -stacked stage- 2 \ngraphite alkali -intercalation compounds: (a) the reduced first Brillouin zones of AB \nstacking, (b) LiC 12, (c) NaC 16, (d) KC 16, (e) RbC 16, and (f) CsC 16. \n \n  Figure 7.5: The rich band structures for the AB -stacked stage- 3 graphite alkali -\nintercalation compounds: (a) the reduced first Brillouin zone, (b) LiC 18, (c) NaC 24, \n(d) KC 24, (e) RbC 24, and (f) CsC 24. \n \n  Figure 7.6: The atom - and orbital -decomposed density of states for the (a ) AA - & \n(b) AB -stacked graphite systems, (c) LiC 6, (d) NaC 8, (e) KC 8, (f) RbC 8, (g) CsC 8, \n(h) LiC 12, (i) NaC 16, (j) KC 16, (k) RbC 16, and (l) CsC 16. The black, red, light blue, \npink, gtreen, deep blue, and purple curves, respectively, C -2s, C -2px, C -2py, C -2pz, \nalkali, carbon, and compound.  \n \n \n  "    },    {        "title": "1502.05659v2.Occurrence_of_flat_bands_in_strongly_correlated_Fermi_systems_and_high__T_c__superconductivity_of_electron_doped_compounds.pdf",        "content": "arXiv:1502.05659v2  [cond-mat.supr-con]  7 Jun 2015Occurrence of flat bands in strongly correlated Fermi system s and high- Tc\nsuperconductivity of electron-doped compounds\nV. A. Khodel,1,2J. W. Clark,2,3K. G. Popov,4,5and V. R. Shaginyan6,7\n1NRC Kurchatov Institute, Moscow, 123182, Russia\n2McDonnell Center for the Space Sciences & Department of Phys ics,\nWashington University, St. Louis, MO 63130, USA\n3Centro de Ciˆ encias Matem´ aticas, Universidade de Madeira , 9000-390 Funchal, Madeira, Portugal\n4Komi Science Center, Ural Division, RAS, Syktyvkar, 167982 , Russia\n5Department of Physics, St.Petersburg State University, Ru ssia\n6Petersburg Nuclear Physics Institute, NRC Kurchatov Insti tute, Gatchina, 188300, Russia\n7Clark Atlanta University, Atlanta, GA 30314, USA\nWe consider a class of strongly correlated Fermi systems tha t exhibit an interaction-induced\nflat band pinned to the Fermi surface, and generalize the Land au strategy to accommodate a flat\nband and apply the more comprehensive theory to electron sys tems of solids. The non-Fermi-\nliquid behavior that emerges is compared with relevant expe rimental data on heavy-fermion metals\nand electron-doped high- Tccompounds. We elucidate how heavy-fermion metals have extr emely\nlow superconducting transition temperature Tc, its maximum reached in the heavy-fermion metal\nCeCoIn 5does not exceed 2.3 K, and explain the enhancement of Tcobserved in high- Tcsupercon-\nductors. We show that the coefficient A1of theT-linear resistivity scales with Tc, in agreement\nwith theexperimental behavior uncoveredin theelectron-d oped materials. We havealso constructed\nschematic temperature-doping phase diagram of the copper o xide superconductor La 2−xCexCuO4\nand explained the doping dependence of its resistivity.\nPACS numbers: 71.10.Hf, 71.27.+a, 74.20.-z\nAn especially challenging task confronting present-day\ncondensed matter theory is explication of the elusive ori-\ngin of the NFL behavior observed in strongly correlated\nFermi systems beyond a critical point where the low-\ntemperaturedensityofstates N(T→0)divergeswithout\nbreaking any symmetry inherent of the ground state. In\nhomogeneous matter, the case to be addressed in this ar-\nticle, such a divergence is associated with the onset of a\ntopological transition (TT) signaled by the emergence of\nan additional root p=pbof the condition [1, 2]\nǫ(p,λt;T= 0) = 0 (1)\nfor vanishing of the single-particle energy measured from\nthe chemical potential µ, otherwise satisfied only by\np=pF, the Fermi momentum. Thus, if λtis the critical\ncoupling constant for onset of the posited TT, the curve\nǫ(p,λt) must touch the axis pat the bifurcation momen-\ntump=pb. Accordingly, at p→pbthe group velocity\nv(p) =∂ǫ(p,λt)/∂pvanishes as/radicalbig\nǫ(p) to yield\nN(T→0)∝T−1/2. (2)\nIn an original scenario advanced by I. M. Lifshitz more\nthan fifty years ago [1] and recently applied to describe\nelectronsystemsofheavy-fermionmetals[3,4],someTTs\nare assumed to occur in systems of noninteracting elec-\ntrons moving in the external field of the crystal lattice.\nIn such a case, the quasiparticle occupation numbers ad-\nhere to the discrete Fermi-liquid (FL) values n(p) = 0,1\nbeyond the Lifshitz topological transition (LTT) point,\nwhile the Fermi surface becomes multi-connected. Topo-\nlogical transitions may occur in systems of interactingfermions as well [5]. Within the FL approach, the feasi-\nbility of the Lifshitz (“bubble” or “pocket”) scenario for\nthe rearrangement of the Landau state was considered\n[6, 7] within the framework of the Landau equation [8]\n∂ǫ(p,T)\n∂p=p\nM+/integraldisplay\nf(p,p′)∂n(p′,T)\n∂p′dυ′(3)\nfor the single-particle spectrum ǫ(p). Here Mis the\nmass of a free fermion, dυ= 2d3p/(2π)3is the three-\ndimensional volume element,\nn(p,T) = (1+ eǫ(p)/T)−1(4)\nis quasiparticle momentum distribution, and f(p,p′) is\na phenomenological interaction function. Eq. (3) was de-\nrived by Landau [8] from the equality\n/integraldisplay\npn(p)dυ=M/integraldisplay∂ǫ(p)\n∂pn(p)dυ (5)\nbetween the momentum of the system, moving with ve-\nlocityδv, its mass flow stemming from Galilean invari-\nance. We note that Eq. (5) is also valid when the\nmodel of heavy-fermion liquid is applicable, as it is in\nour present case [9, 10]. Equation (3) is then obtained\nupon retaining only the leading terms in δn(p) on both\nsides of Eq. (5) while invoking the FL relation\nδǫ(p,n) =/summationdisplay\np′f(p,p′)δn(p′). (6)\nAs will be seen, Eq. (5) allows one to introduce and ex-\nplore a different, interaction-induced type of rearrange-\nment of the Landau state, often called fermion conden-\nsation and described more vividly as a swelling of the2\nFermi surface [11–13]. This phenomenon has its genesis\nin a proliferation of the number of roots of Eq. (1) to\nform acontinuum . (For recent articles on this topic, see\nRefs. [9, 10, 14–20]).\nThe emergence of such a phase transition in homoge-\nneous matter can be elucidated within the original Lan-\ndau approach to FL theory, in which the ground-state\nenergyEis postulated to be a functional of the quasi-\nparticle momentum distribution n(p). AtT= 0, the\nonset of fermion condensation in homogeneous matter is\nattributedtotheoccurrenceofanontrivialsolution n∗(p)\nof the variational condition [11]\nδE(n,λ)\nδn(p)−µ= 0, p∈[pl,pu], (7)\nin a finite momentum interval surrounding pb, the chemi-\ncal potential µbeing determined from the Landau postu-\nlate that the quasiparticle and particle numbers coincide.\nItisEq.(7)thatdescribesaflatbandpinnedtotheFermi\nsurface, here also referred to as the fermion condensate\n(FC). Outside the FC domain [ pl,pu] the usual FL occu-\npancies still apply: n∗(p) = 1 at p < plandn∗(p) = 0\natp > pu. However, the occupation numbers inside the\nFC, evaluated through Eq. (7), change continuously be-\ntween 1 and 0 with increasing p. The volume occupied in\nmomentum space by quasiparticles with nonzero proba-\nbility is no longer just p3\nF/3π2but is instead given by the\nrelation [15]\npl/integraldisplay\n0dυ+pu/integraldisplay\npln∗(p)dυ=p3\nF\n3π2. (8)\nWe now turn to the demonstration that Landau equa-\ntion (3) is to be modified in dealing with Fermi systems\nharboring a FC. It is revealed by examination of ana-\nlytic properties of solutions of this equation in systems\nwhere the interaction function f(p,p′) has no singulari-\nties in momentum space. For this case, it has been estab-\nlished [7] that the solutions of Eq. (3) are in fact analytic\nfunctions of momentum pin the full momentum space.\nHowever, this property is lost if the system hosts a FC.\nIndeed, the left side of Eq. (7) is nothing but the quasi-\nparticle energy measured from the Fermi surface, which\nvanishes identically in the FC domain. Evidently, the FC\ndomain cannot occupy the full momentum space – a fact\nconfirmed in analytically soluble models of fermion con-\ndensation. Thereby we arrive at the strong conclusion\nthat the single-particle spectrum ǫ(p,n∗) of the problem\nwith FC present must be a nonanalytic function of mo-\nmentum p, for if an any analytic function vanishes iden-\ntically in some domain, it must vanish everywhere. This\nimplies that in the case where the interaction function\nfhas no singularities in momentum space, the FC solu-\ntionsǫ(p,n∗)cannot meet Eq. (3), and therefore in sys-\ntems with a FC, Landau equation (3) must be modifiedto allow for such an eventuality. This modification can\nbe made on the basis of Eq. (5) along the same lines as\nEq. (3) was obtained in Landau theory. The profound\ndifference is that the system harboring a FC is, in fact, a\ntwo-component system, as made quite evident in Eq. (8).\nCorrespondingly, variations of the quasiparticle momen-\ntum distribution n(p) lead to the closed equation\n0 =p\nM+/integraldisplay\nf(p,p′)∂n∗(p′)\n∂p′dυ′, p,p′∈[pl,pu] (9)\nfor determining n(p) =n∗(p) inside the FC domain de-\nrived with accounting for the fact that outside the FC\nregion,dn∗(p)/dp= 0. In arriving at this relation, we\nhave ensured consistency with the central feature of the\nFC-inhabitedstate that the groupvelocity vanishes iden-\nticallyin the interval [ pl,pu], whose range is found with\nthe aid of the requirement n∗(p)<1. The quasiparticle\nspectrum of the normalcomponent obeys the Landau-\ntype equation\n∂ǫ(p)\n∂p=p\nM+/integraldisplay\nf(p,p′)∂n∗(p′)\n∂p′dυ′, p /∈[pl,pu],(10)\nand in effect p′∈[pl,pu], derived in the same way as\nEq. (3) from Eq. (5). Equation (9) can also be de-\nrived by differentiation of the basic equation (7) with\nrespect to momentum p, then adopting the relation (6)\nbetween interaction-induced variations of relevant quan-\ntities, rewritten as δǫ(p) =δǫ0+ (fδn(p)), where δǫ0is\nthe variation of the spectrum of noninteracting quasipar-\nticles. In homogeneous matter, where ∂ǫ0/∂pis simply\np/M, we are led to Eq. (9). The special convenience of\nthis route lies in the opportunity to generalize Eq. (9)\nfor analysis of the FC phenomenon in electron systems\nof solids. In doing so, one needs to replace p/Mby the\ncorresponding derivative ∂ǫ0/∂p, evaluated, say, within\nthe tight-binding model, or within a more advanced mi-\ncroscopic description of the electron spectrum ǫ0(p). A\ndefinite mathematical signature identifies those interact-\ningmany-fermionsystemswhosesingle-particlespectrum\nexhibits a flat portion. This is the topological charge\n(TC) of the corresponding ground state. For a system\ncontaining a FC, the TC must take a half-odd-integral\nvalue, whereas the TC of any unorthodox state featuring\none or more Lifshitz pockets is always integral[12].\nIt is instructive to compare changes that occur in the\nfundamental equation (3) beyond the point of fermion\ncondensation with those that occur at a second-order\nphase transition. In the latter case, some symmetry in-\nherent in the Landau ground state is broken, and a corre-\nsponding order parameter comes into play (notably, the\ngap ∆(p) in superfluid FL’s), dramatically rearranging\nEq. (3) and both the key FL quantities, i.e., the mo-\nmentum distribution nFL(p) =θ(pF−p) and the single-\nparticle spectrum ǫFL(p) =pF(p−pF)/M∗, whereM∗\nis the effective mass. In the FC topological phase transi-\ntion no such order parameter enters; nevertheless Eq. (3)3\nalters, being replaced by the set of two equations (9) and\n(10), in harmony with the two-component character of\nthe phase containing a FC. Although the equations writ-\nten above to describe a system containing a FC were\nderived for zero temperature, they do not strictly apply\nthere, because states involving a FC have the distinctive\nproperty of carrying nonzero residual entropy\nS0=−/summationdisplay\n(n∗(p)lnn∗(p)+(1−n∗(p))ln(1−n∗(p))).\n(11)\nSuch behavior would contradict the Nernst theorem\nS(T= 0) = 0, if it continued to T= 0. Resolution\nof the incipient contradiction must lie in release of the\nexcess entropy S0via one or another phase transition\nassociated, especially, with pairing or magnetic ordering\nand occurring at a respective transition temperature Tc\norTNasT→0. Even so, the above equations do yield\nan approximate description, in the same sense as Lan-\ndau FL theory furnishes a viable approximate theory of\nthe low-temperature properties of liquid3Hedespitethe\nfact that this system undergoes a normal-to-superfluid\ntransition as T→0, simply because of its extremely low\ncritical temperature Tcfor termination of superfluidity.\nBy the same token, as long as the corresponding Tcis low\nenough, Eqs. (9) and (10) support an acceptable zeroth\napproximation for a system of fermions having a FC.\nTo properly address the finite-temperature case, the\nenergyEmust be replaced by the free energy F. So\nmodified, the variational condition (7) takes the Landau-\nlike form (cf. Eq. (4))\nǫ(p,n∗) =Tln1−n∗(p,T)\nn∗(p,T). (12)\nTaking the solution n∗(p) of Eq. (7) or (9) as a zeroth\napproximation for n∗(p,T) applicable at low T, we infer\nthat at finite temperatures the dispersion of the spec-\ntrum in the momentum interval [ pl,pu] becomes propor-\ntionalto T[13]. (Since the changeofthe NFL occupation\nnumbers n∗(p) incurred by finite Tin this region remains\nquite small, we shall continue to use the term FC for the\ntotality of the corresponding single-particle states.) The\ntemperature-dependent generalization of the above FC\nequations is obtained by inserting the derivative\nv(p)≡∂ǫ(p,T)\n∂p=−T∂n∗(p,T)/∂p\nn∗(p,T)(1−n∗(p,T))(13)\ninto the left side of Eq. (9) to yield, in the FC domain,\n−T∂n∗(p,T)/∂p\nn∗(p,T)(1−n∗(p,T))=p\nM+/integraldisplay\nf(p,p′)∂n∗(p′,T)\n∂p′dυ′.\n(14)\nThis equation holds in the FC interval p∈[pl,pu], whose\nlength shrinkswith T, vanishingat anupper criticaltem-\nperature Tf. The presence of the FC phase on the dis-\nordered side of the transition, where standard FL theoryshould ordinarily exert dominion, results in the break-\ndown of virtually all thermodynamic and kinetic predic-\ntions of Landau FL theory. Further discussion will be fo-\ncused on the magnitude and temperature dependence of\nthe low-Tresistivity in the disordered, FC phase, which\nmay represent the most conspicuous departures from FL\ntheory. In elasticscattering processes involving normal\nquasiparticles and the FC, the condensate behaves as a\nsystem of impurities , giving rise to a residual resistivity\nρ0incleanmetals, whose value dependson the pressure\nP. Experimental documentation of the consequent im-\npact onρ0(an effect inconceivable within the textbook\nunderstanding of kinetic phenomena in Fermi liquids) re-\nquires high-quality samples without doping xthat intro-\nduces substantial disorder. In spite of this stricture, a\nprofound effect has been observed in measurements of\nthe resistivity of the heavy-fermion compounds CeCoIn 5\nand CeAgSb 2[21, 22]. The latter compound undergoes\nan antiferromagnetic transition, whose critical temper-\nature declines under pressure. On the ordered side of\nthis transition, ρ(T) demonstrates the usual FL behavior\nρ(T) =ρ0+A2T2, with residual resistivity ρ0below 0.5\nµΩcm. However, with increasing Pthis familiar behav-\nior ofρ(T) is disrupted. As Papproachesthe value Pcat\nwhich magnetic ordering is destroyed, the residual resis-\ntivityρ0abruptly soars upward, reaching values of about\n50µΩcmonthedisorderedsideofthetransition[22]. We\nsubmit that such a jump of ρ0amounts to irrefutable ev-\nidence for the presence of a FC on the disordered side of\nthe phase transition.\nThe heavy-fermion metal CeCoIn 5proves to be a su-\nperconductor on the ordered side of the corresponding\nphase transition. We contend that in such systems, NFL\nbehavior of ρ(T) on the disordered side of this transition\nis primarily associated with inelastic scattering processes\nin which FC quasiparticles are generated or eliminated\nsinglyso as to augment the FL formula for ρ(T) with a\nterm linear in T, whose magnitude A1is proportional to\nthe FC range Lf[23–25]:\nA1(x)∝Lf(x). (15)\nThe low- Tresistivity ρ(T,P= 0) found experimentally\nfor the normal state of CeCoIn 5is in fact consistent with\nthis behavior [26]. When the pressure Pis raised to a\ncritical value P∗≃1.6 GPa, it is found that in a nar-\nrow pressure diapason around P∗, CeCoIn 5exhibits a\ncrossover to Landau-like behavior ρ(T) =ρ0+A2T2.\nConcomitantly, the residual resistivity ρ0is reduced ten-\nfold in a topological transition from a state with a FC\nto one without, dropping to a very small value around\n0.2µΩcm that is due solely to impurity scattering. The\ncrucial difference between the effects of the FC and the\nimpurities that arepresent lies in the fact that the former\n– belonging as it does to the system of electrons – con-\ntributes to the resistivity entirely by virtue of Umklapp\nprocesses occurring in the crystal lattice, which spoils4\nelectron momentum conservation. However, these pro-\ncessesplaynosignificantroleinthe thermalconductivity.\nOne is led to the conclusion that the Lorenz number\nL0= lim\nT→0κ(T)/(Tσ(T)), (16)\nevaluated at low pressures P < P∗where the FC is\npresent, must be suppressed compared with the text-\nbook value L0=π2/3e2. It is just such a violation\nof the Wiedemann-Franz law that notoriously occurs in\nCeCoIn 5[26].\nThese new insights into the observed NFL behavior of\nρ(T) may be elaborated as follows. Replacement of a\nnormal particle in a scattering diagram by a FC quasi-\nparticle entails replacement, in the formula for the re-\nsistivity ρ(T), of the T-independent density of states\nNFLof Fermi-liquid theory by the FC density of states\nN(T)∝T−1, thereby promoting linear dependence of\nρ(T) onT. On the other hand, at T > T fwhere the\neffect of the LTT point still persists, the quantity NFLis\nreplaced by N(T)∝T−1/2in accordance with Eq. (2),\nyielding ρ(T)∝T3/2. Such behavior closely resembles\nthe NFL behavior ρ(T)∝T1.5±0.1revealed in measure-\nments of the resistivity of the normal state of CeCoIn 5as\nwellasin electron-doped high-Tcsuperconductors[27,28].\nMoreover, the experimental T−Pphase diagram of\nCeCoIn 5shown in Fig. 4 of Ref. [21] and the experimen-\ntalT−xphase diagram of La 2−xCexCuO4belonging to\nthe LCCO family presented in Fig. 1 of Ref. [28] are very\nmuchalike. Thisempiricalfactcastsseriousdoubtonthe\npremisethattheKondoeffect bearssoleresponsibilityfor\nthe NFL behaviorofthe resistivity ρ(T)ofheavy-fermion\nmetals, since the Kondoeffect is absentin electron-doped\ncompounds. In both of these compounds, release of the\nresidual entropy S0occurs throughthe antiferromagnetic\nphase transition that can be replaced by the supercon-\nducting phase transition at lowering temperatures, as it\nis seen from Fig. 1.\nLet us now discuss the enhancement of the supercon-\nducting critical temperature Tc, attributed to critical\nspin-fluctuations in Refs. [28, 29]. Within the flat-band\nscenario, this enhancement is estimated on the base of\nthe standard BCS equation\n1 =V/integraldisplay\ntanh(ǫ(p,Tc)/2Tc)/2ǫ(p,Tc)dυ. (17)\n(For simplicity, we ignore the momentum dependence of\nthe pairing interaction, replacing it by the constant V).\nUpon manipulationsbased on Eq. (12) the BCS Eq. (17)\ntakes the form\n1 =−0.5V[αηn/T c+Nn(0)ln(Ω D/Tc)] (18)\nwith total electron density n, the dimensionless FC pa-\nrameter η= (pu−pl)/pFand numerical factor α=\nO(1). In obtaining this result we retain merely lead-\ning terms that diverge at Tc→0. The first term insquare brackets comes from momentum integration over\nthe FC region. The second, usual BCS one is associated\nwith contributions from bands where normal quasipar-\nticles reside. It contains the familiar density of states\nNn(0)∝pFM∗\nn/π2. In the first approximation, the BCS\nterm can be neglected, and then from Eq. (18) one finds\nTc(x)∝η(x), (19)\ni.e. criticaltemperature Tcturnsouttobelinearfunction\nof the FC parameter η. Comparing Eqs. (19) and (15),\nwe infer that both the A1term in the resistivity ρ(T) and\ncritical temperature Tcchange with input parameters P\nandηproportionally to the FC parameter η. Thus, the\ntheoretical ratio Tc/A1is approximately independent of\nthe input, in agreement with the experimental behav-\nior uncovered in the electron-doped materials LCCO and\nPCCO [28, 29].\nWe note in passing that the departure from the bare\nmassMof the effective mass M∗\nn, extracted from ex-\nperimental data on the specific heat, might be signifi-\ncant, as e.g. in the heavy-fermion metals. This deviation\nplays important role in the magnitude of Tc. To demon-\nstrate let us slightly facilitate Eq. (18) by excluding\nthe effective coupling constant Vwith the replacement\nof the unity on the l.h.s. of Eq. (18) from BCS relation\n1 =−0.5VNn(0)ln/parenleftbig\nΩD/TBCS\nc/parenrightbig\nthat yields\nTc\nǫ0\nFln/parenleftbig\nTc/TBCS\nc/parenrightbig\n=αηM\nM∗n(20)\nwithǫ0\nF=p2\nF/2M. The BCS case Tc=TBCS\ncis realized\nprovided the FC density η→0. Curiously, the BCS sit-\nuation also takes place in the heavy fermion metals, even\nwith the flat bands, because of the smallness of the ratio\nM/M∗\nn≃10−2−10−3. This explains how heavy-fermion\nmetals have extremely low Tc, its maximum reached in\nthe heavy-fermion metal CeCoIn 5does not exceed 2.3\nK. In dealing with other electron systems where the ra-\ntioM/M∗\nnis not as small as in the heavy-fermion metals\nthe situation changes. Assuming the needed attraction\nto comefrom the electron-phononexchangeand choosing\nM/M∗\nn= 1,α= 0.1,η= 0.1, one finds\nTc≃0.3∗10−2ǫ0\nF. (21)\nthat explains the enhancement of Tcobserved in high- Tc\ncompounds and, particularly, in electron-doped high- Tc\nmaterials. FromEq. (21) weinferthatthe superconduct-\ning phase transition in 2D electron gas of MOSFETs and\nsome heterostructureswhere the needed attraction in the\nCooper channel is furnished by the electron-phonon ex-\nchange may have sufficiently high Tcprovided the FC\nonset occurs near a critical density, at which the electron\neffective mass diverges [30]. Noteworthy, in the heavy-\nfermion metal CeCoIn 5, the BCS and FC contributions\ntoTcare rendered comparable by virtue of the character-\nistic effective-mass enhancement. This entails, in turn, a5\nmarked extension of the boundary between SC and Lan-\ndau Fermi liquid (LFL) regimes in the T−Pphase dia-\ngram.\n/s110/s61/s49/s110/s61/s51/s47/s50\n/s68/s111/s112/s105/s110/s103/s32 /s120/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s84/s49/s50\n/s51\n/s52/s76/s84/s84/s32/s112/s111/s105/s110/s116\n/s65/s70/s32/s112/s111/s105/s110/s116\n/s110/s61/s50/s65/s70/s32/s114/s101/s103/s105/s109/s101\n/s61\n/s48/s43/s65\n/s110/s84/s110\n/s76/s70/s76\nFIG. 1: Schematic temperature-doping T−xphase dia-\ngram of La 2−xCexCuO4. The exponents nof the resistiv-\nityρ=ρ0+AnTnare shown for three different regimes,\nthat appear around the LTT point, labeled 1,2,3. The Lan-\ndau Fermi liquid regime with ρ(T)∝A2T2dependence is la-\nbeledbyLFL.Thetransition areas betweenthecorrespondin g\nregimes are indicated by solid curves. Both the LTT point at\nx=xLTTand the antiferromagnetic point under the super-\nconductivity dome (labeled 4) are indicated by arrows. The\nonsetofantiferromagnetism istracedbythedash-dot-dotl ine.\nIt is a remarkable feature of models of fermion conden-\nsation [11, 13, 31] that nontrivial solutions of Eq. (7) car-\nrying a residual entropy S0often emerge just beyond the\nLTT point where the resistivity ρ(T) varies as T3/2. As a\nresult, near the LTT point the two NFL regimes, propor-\ntionalrespectivelyto T3/2andT,aresituatedadjacentto\neach other, as it is shown in the schematic phase diagram\n1. It is seen from Fig. 1, that four different regimes for\nthe resistivity behavior come into play in the immediate\nvicinity of FC onset: (1) the LFL regime ρ(T)∝T2, (2)\nthe LTT-point regime ρ(T)∝T3/2, (3) the FC regime\nρ(T)∝T, and (4) the high- Tcsuperconducting regime\nwithρ= 0. To confirm our analysis ofthe phase diagram\nshown in Fig. 1, we consider the behavior of the resistiv-\nity in the regions ”1” and ”3”. In the region ”3”, as it is\nseen from Fig. 2 and follows from the above discussion,\nthe coefficient A1→0 as the doping xapproaches the\nLTT point from above, while in accordance with Eq. 20\nTc→0. On the other hand, as the doping xtends to the\nLTT point from below, the coefficient A2diverges, for in\nthe LFL regime due to the conservationofthe Kadowaki-\nWoods ratio one has A2∝(M∗)2[32, 33]. As a result,one has [9, 10]\n(M∗(x))2∝A2≃/parenleftbigg\na1+a2\nx−xLTT/parenrightbigg2\n,(22)\nwherea1anda2are constants and xLTTis the doping x\nat which the LTT point takes place. It seen from Fig. 2,\nthe right panel, that the behavior of ( M∗)2given by Eq.\n(22) is in agreement with the experimental facts.\n/s48/s46/s49/s50 /s48/s46/s49/s54 /s48/s46/s50/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s76/s84/s84/s32/s112/s111/s105/s110/s116/s65\n/s50/s32/s40/s109 /s32/s99/s109/s32/s75/s45/s50\n/s41/s48/s46/s48/s48/s54\n/s48/s46/s48/s48/s51/s32/s84\n/s99/s47/s32/s50/s53/s32/s40/s75/s41/s65\n/s49/s32/s40/s109 /s32/s99/s109/s32/s75/s45/s49\n/s41/s44\n/s68/s111/s112/s105/s110/s103/s32 /s120/s48/s46/s48/s48/s48/s76/s97\n/s50/s45/s120/s67/s101\n/s120/s67/s117/s79\n/s52\nFIG. 2: Doping dependence of Tc(x),A1(x) andA2(x) in\nzero field [28]. The left vertical axis shows the coefficient A1\nandTcdivided by 25, while the right vertical axis shows the\ncoefficient A2. In the LFL regime, as the system approaches\ntheLTTpoint, shown bythearrow, andthedoping x→xLTT\nfrom higher doping, the coefficient A2increases in accordance\nwith Eq. (22), as it is depicted by the solid curve.\nIn conclusion, prominent on the scene of this seminal\nsubfield of condensed matter physics are numerous ver-\nsions of Hertz-Millis-Moriya (HMM) theory that ascribe\nNFL behavior of strongly correlated Fermi systems to\nquantum critical fluctuations. In stark contrast to these\ntheories, experimentalstudiesofthe classofsuchsystems\naddressed herein furnish clear evidence of the persistence\nof NFL behavior in regions far enough from the lines Tc\nand/orTN(x) that the influence of critical fluctuations\nshould be minimal. The crux of the matter is that the\nsingle-particledegrees offreedom arethe real playmakers\nforNFLbehavior,andthetraditionalmissionofintegrat-\ning them out is futile, as in throwing the baby out with\nthe bath water.\nThe authors are indebted to J. Paglione, E. Saper-\nstein, and M. V. Zverev for valuable discussions. This\nwork is supported by NS-932.2014.2 and RFBR Grants:\n13-02-00085,14-02-00044. VRS is supported by the Rus-\nsian Science Foundation, Grant No. 14-22-00281. VAK\nthanks the McDonnell Center for the Space Sciences for\nsupport. JWC expresses his gratitude to Professor Jos´ e\nLu´ ıs da Silva and his colleagues at Centro de Ciˆ encias6\nMatem´ aticas for generous hospitality and fruitful dis-\ncussions during summer residence at the University of\nMadeira.\n[1] I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).\n[2] G. E. Volovik, Quantum Phase Transitions from Topol-\nogy in Momentum Space , Lect. Notes in Physics 718, 31\n(2007).\n[3] D. LeBoeuf el al., Phys. Rev. B 83, 054506 (2011).\n[4] J. Chang et al., Phys. Rev. B 84, 014507 (2011).\n[5] M. Llano and J. P. Vary, Phys. Rev. C 14, 1083 (1979).\n[6] M. V. Zverev and M. Baldo, JETP 87, 1129 (1998); J.\nPhys.: Condens. Matter 11, 2059 (1999).\n[7] S. A. Artamonov, V. R. Shaginyan, and Yu. G.\nPogorelov, JETP Lett. 68, 942 (1998).\n[8] L. D. Landau, Sov. Phys. JETP bf 3, 920 (1957)].\n[9] V.R. Shaginyan, M.Ya. Amusia, A.Z. Msezane, and K.G.\nPopov, Phys. Rep. 492, 31 (2010).\n[10] M. Ya. Amusia, K. G. Popov, V. R. Shaginyan, W.\nA. Stephanowich, Theory of Heavy-Fermion Compounds ,\nSpringer Series in Solid-State Sciences 182, (2015).\n[11] V. A. Khodel and V. R. Shaginyan, JETP Lett. 51, 553\n(1990).\n[12] G. E. Volovik, JETP Lett. 53, 222 (1991).\n[13] P. Nozi` eres, J. Phys. I France 2, 443 (1992).\n[14] S.-S. Lee, Phys. Rev. D 79, 086006 (2009).\n[15] V. A. Khodel, J. W. Clark, and M. V. Zverev, Physics of\nAtomic Nuclei 74, 1237 (2011).\n[16] G. E. Volovik, JETP Lett. 93, 66 (2011).\n[17] J. W. Clark, M. V. Zverev, and V. A. Khodel, Annals ofPhysics, 327, 3063 (2012).\n[18] D.Yudin, D.Hirschmeier, H.Hafermann, O. Eriksson, A.\nI. Lichtenstein, and M. I. Katsnelson, Phys. Rev. Lett.\n112, 070403 (2014).\n[19] A.A. Shashkin, V.T. Dolgopolov, J.W. Clark, V.R.\nShaginyan, M.V. Zverev, and V.A. Khodel, Phys. Rev.\nLett.112, 186402 (2014).\n[20] G. E. Volovik, arXiv 1409.3944.\n[21] V. A. Sidorov, M. Nicklas, P. G. Pagliuso, J. L. Sarrao,\nY. Bang, A. V. Balatsky, and J. D. Thompson, Phys.\nRev. Lett. 89, 157004 (2002).\n[22] M. Nakashima et al., J. Phys.:Condensed Matter 15,\nL111 (2003).\n[23] V. A. Khodel and M. V. Zverev, JETP Lett. 85, 404\n(2007).\n[24] V. R. Shaginyan, A. Z. Msezane, K. G. Popov, J. W.\nClark, M. V. Zverev, and V. A. Khodel, Phys. Rev. B\n86, 085417 (2012).\n[25] V. A. Khodel, V. R. Shaginyan, and P. Schuck, JETP\nLett.63, 752 (1996).\n[26] J. Paglione et al., Phys. Rev. Lett. 97, 106606 (2006).\n[27] N. P. Armitage, P. Furnier and R. L. Greene, Rev. Mod.\nPhys.82, 2421 (2010).\n[28] K. Jin, N. P. Butch, K. Kirshenbaum, J. Paglione, and\nR. L. Greene, Nature, 476, 73 (2011).\n[29] L. Taillefer, Annu. Rev. Cond. Mat. Phys. 1, 51 (2010).\n[30] A. A. Shashkin, Phys. Usp. 48, 129 (2005).\n[31] V. A. Khodel, V. R. Shaginyan, and V. V. Khodel, Phys.\nRep.249, 1 (1994).\n[32] K. Kadowaki and S. B. Woods, Solid State Commun. 58,\n507 (1986).\n[33] V.A.KhodelandP.Schuck,Z.Phys.B:Condens.Matter\n104, 505 (1997)."    },    {        "title": "1703.04104v2.Electronic_origin_of_melting_T_P_curves_of_alkali_metals_with_negative_slope_and_minimum.pdf",        "content": " \n1Electronic origin of melting T –P curves of alkali meta ls with negative slope \nand minimum  \nDegtyareva V.F.  \nInstitute of Solid State Physics, Russi an Academy of Sciences, Chernogolovka, Russia \ndegtyar@issp.ac.ru \nGroup I elements – alkali metals Li, Na, K, Rb  and Cs – are examples of simple metals \nwith one s electron in the valence band. Under pressure these elements display \nunusually complex structural behaviour transforming from close-packed to low \nsymmetry open-packed struct ures. Unexpectedly complex form was found for melting \ncurves of alkalis under compression with an initial increase in the melting temperature \nin accordance with Lindemann criterion and a further decrease to a very low melting \npoint. To understand complex and low symmetry crystal structures in compressed alkalis, a transformation of the electron ener gy levels was suggested which involves an \noverlap between the valence band and outer core electrons. Within the model of the \nFermi sphere – Brillouin zone interac tion one can understand the complex melting \ncurve of alkalis. \nKeywords: alkali metals, melting T-P curves, core electron ionization   \n1. Introduction \nGroup I elements – alkali metals Li, Na, K, Rb and Cs – are examples of simple  metals with \none s electron in the valence band. Experiment ally it was found that under pressure these \nelements display unusually complex structural  behaviour transforming from close-packed \nstructures such as body-centered cubic (bcc) and face-centered cubic (fcc) to low symmetry \nopen-packed structures (see review [1] and re ferences therein). An unexpectedly complex \nbehaviour was found for T – P melting curves of  alkalis under compre ssion with an initial \nincrease in the melting temperature in accord ance with Lindemann criterion and a further \ndecrease to very low melting point. The lowest  point of melting was f ound to be 300K at 120 \nGPa for sodium and 200K at 50 GPa for lit hium [2,3]. Heavier alkalis have also non- \n2monotonic behaviour of the melting curve on T – P diagrams.  \nComplexity of T – P phase diagrams of alkali  elements attracts attention of theoretical \nconsiderations. To mention some aspects of the theore tical approaches that  are related to the \nfocus of the present paper, the Fermi surface – Brillouin zone interaction was found to be a \n“dominant contribution” to crystal structure of  s-band elements including alkali metals [4]. \nEssential deformation of the Fermi surface from the initially spherical form was calculated for \nlithium [5]. The negative melting slope of sodium under pressure was reproduced with \ncalculations as driven by electronic transitions in the liquid state of Na [6]. The melting curve \nof dense sodium was simulated by assuming effects of inner-core 2 p states [7]. \nTo understand complex and low symmetry structures in compressed alkalis a \ntransformation of the electr on energy levels was suggested which involves an overlap \nbetween the valence band a nd outer core electrons [8,9]. The structures Na- oP8 above \n117 GPa and K- oP8 above 54 GPa are similar to the intermetallic compound AuGa at \nambient pressure with the 2 valence electrons pe r atom. From this similarity it was suggested \nthat Na and K became divalent metals under strong compression.  Within the model of the \nFermi sphere – Brillouin zone interaction on e can understand the complex melting curve of \nalkalis.  \n2. Theoretical background and method of analysis \nThe crystal structure of metallic phases is  defined by two main energy contributions: \nelectrostatic (Ewald) energy a nd the electron band structure te rm. The latter usually favours \nthe formation of superlattices and distorted structures. The en ergy of valence electrons is \ndecreased due to a formation of Brillouin planes with a wave vector q near the Fermi level k F \nand opening of the energy pse udogap on these planes if q hkl ≈ 2k F. Within a nearly free-\nelectron model the Fermi sphere radius is defined as k F = (32z/V)1/3, where z is the number of  \n3valence electrons per atom and V is the atomic volume. This effect, known as the Hume-\nRothery mechanism (or electron concentration rule ), was applied to account for the formation \nand stability of the intermetallic phases in binary simple metal systems like Cu-Zn, and then extended and widely used to explain the stability  of complex phases in various systems, from \nelemental metals to intermetallics [10]. \n The stability of high-pressure phases in alkalis is analyzed using a computer program \nBRIZ [11] that has been develo ped to construct Brillouin zone s or extended Brillouin-Jones \nzones (BZ) and to inscribe a Fermi sphere  (FS) with the free-electron radius k\nF. The resulting \nBZ polyhedron consists of nume rous planes with relatively  strong diffraction factor and \naccommodates well the FS. The volume of BZ’s and Fermi spheres can be calculated within this program.\n \nThe liquid state of metals can be consider ed within the FS – BZ model assuming that \nthe first strong structure factors correspond to a nearly-spherical BZ [12]. The interaction of \nthis BZ with the Fermi level at k F leads to the formation of the pseudo-gap and a decrease of \nthe electronic energy. The well known example of  significant increase of stability for liquid \nstate is displayed by several bina ry alloys with the eutectic t ype of phase diagrams such as \nAu – Ge and Au – Si. These alloys ha ve the melting temperature at around 360oC whereas the \npure Au melts at 1064oC. The composition of eutectic alloys is 28 at% Ge and ~20 at% Si \ncorresponding to z = 1.84 or 1.6 electron/atom, respectively.  \nMercury – the only elemental metal that exists  in the liquid state at ambient conditions \n– has two electrons in the valence band. The melting point of Hg is –38.7oC. This and the \nabove examples show the increased stability of  the liquid state in the metals with 1.6 – 2 \nvalence electrons. The structure factor for li quid metals is deformed by appearance of a \nshoulder on the high-Q side as was observed for Hg [13]. Stabil ity of the liquid state in Hg \nwas analysed by appearance of a deep minimum of density of states near the Fermi level [14].   \n4 \n3. Evaluation of experimental data \n3.1. Flat and negative slope of melting curves of alkalis \nUnusual behaviour of the T-P curves of alka lis may result from a strong increase of band \nstructure energy where the nearly spherical Fermi surface of bcc-alkalis develop necks \nattracted to BZ planes, as was shown for bcc-Li at 8 GPa [5]. However, this attraction is \nreciprocal and leads to contrac tion of BZ planes and to expans ion of the unit cel l in the real \nspace. FS-BZ interactions exist also for liquids  that have spherical BZ and non-directional \nattraction to FS. Therefore FS-BZ interactions produce more remarkable effects for solid bcc-\nstate and the liquid state becomes denser than crystal at some pressure. Consequently, the \nmelting curve display a negative slope as is seen on Figure 1 for sodium around 30 – 40 GPa. \nAt these pressures the atomic local order for bcc-Na and liquid-Na is similar and there is no electronic transfer b ecause of a lack of d-states in the valence shell in contrast to Cs, as \ndiscussed in [6]. At higher pre ssure above 65 GPa the electronic transition in liquid-Na occurs \nat lower pressures than for solid-Na [6]. \n3.2. Deep minimum of meltin g curves of alkalis \nWe suggest that the decrease of the melting temp eratures is related to an overlap of the \nvalence electron band and core electrons as  was considered for the structures Na- oP8 above \n117 GPa and K- oP8 above 54 GPa where it was assumed that Na and K become divalent \nmetals at such compression [8,9]. It is known th at the liquid reacts to the external influence \nmore easily than the solid. Similar electron tran sfer may occur in liquid state and even at \nlower pressure than for solid state.  \nExperimental diffraction data on liquid Na  near the minimum of T-P melting curve \n(from [15]) display structure f actor peaks with appearance of a shoulder similar to that of  \n5liquid Hg as shown in Figure 2.  Therefore it is reasonable to assume the valence electron \nnumber for Na at this pressures equals  two (like Hg), whereas for solid Na ( fcc and cI16 \nphases) at this pressure the vale nce electron number equals to one.  \nDiffraction data for Rb in the liquid state (from [16]) display significant changes under \ncompression as shown in Figure 3. The first structure factor under goes broadening with \nappearance of a shoulder at the high-Q side b ecoming similar to that of liquid Hg. These \nchanges can be attributed to the suggestion of th e increase of the number  of electrons in the \nvalence band at the expense of the outer core electrons. \nSimilar effects have been observed experime ntally for liquid Cs as shown on Figure 4 \n(after data from [17]). In the heaviest alkali metal Cs the electron transfer starts at significantly lower pressure just above 4 GP a, as can be suggested by analyzing the \nbroadening of the first diffracti on peak. This event of diffraction changes in the liquid state \nunder compression was observed also for potassium  [18].  Thus, the structural and electronic \ntransfer is common for all dense liquid alkalis. \n4. Conclusion \nLong sequences of crystal structures fo r alkali metals under compression with the \ntransformations from the close-packed structures to the complex open-packed structures and non-monotonic T - P curves raise the following  question: what are the physical reasons \nbehind these behaviours. One of reason that we suggest in this paper is the Fermi sphere – \nBrillouin zone interactions that increases under pressure. \nThe first stage of T – P melting curves w ith an appearance of a maximum that is \nfollowed by a negative slope can be  explained due to difference in  the attraction of FS to BZ \nthat occurred in crystal and liquid states and causes the liquid to become more compressible \nthan solid.  \n6The next stage of T - P melting curves w ith deep minimum and unusual increase of \nstability of liquid phase can be  understood by assuming the increa se of the valence electron \nnumbers due to the overlap w ith the core electrons under st rong compression.  Non-simple \nbehaviour in melting of alkali me tals under high pressure is conn ected to the essential changes \nof the electron state in the valence band. \nAcknowledgements \nThe author gratefully acknowledges Olga De gtyareva for her valuable discussion and \ncomments. This work was supported by the Program of the Russian Academy of Sciences \n“The Matter under High Pressure”.  \n \nReferences :  \n[1] McMahon MI, Nelmes RJ. High-pressure st ructures and phase transformations in \nelemental metals. Chem Soc Rev. 2005;35:943-963. \n[2] Gregoryanz E, Degtyareva  O, Somayazulu M, et al.  Melting of dense sodium.  Phys Rev \nLett. 2005;94:185502. \n[3] Guillaume CL, Gregoryanz E, Degtyareva O, et al. Cold melting and solid structures of \ndense lithium. Nature Phys. 2011;7:211-214. \n[4] Ackland GJ, Macleod IR. Origin of the co mplex crystal structures of elements at \nintermediate pressure. New Journal of Physics 2004; 6:138.  \n[5] Rodriguez-Prieto A, Bergara A, Silkin VM, and Echenique PM. Complexity and Fermi \nsurface deformation in compresse d lithium. Phys Rev B 2006, 74, 172104. \n[6] Raty J-Y, Schwegler E, and Bonev SA. Electronic and stru ctural transitions in dense liquid \nsodium. Nature  2007, 449, 448-451. \n[7] Yamane A, Shimojo F, and Hoshino K. E ffects of Inner-Core 2p States on Melting Curve \nand Structure of Dense Sodium at High Pressu res. Journal of the Physical Society of \nJapan. 2008, 77, 064603. \n[8] Degtyareva VF, Degtyareva O. Structure stability in the si mple element sodium under \npressure. New J Phys. 2009; 11:063037.    \n7[9] Degtyareva VF. Potassium unde r pressure: electronic origin of complex structures. Solid \nState Sci. 2014;36:62-72.  \n[10] Degtyareva VF. Simple metals at high pressures: the Fermi sphere – Brillouin zone \ninteraction model. Phys.-Usp. 2006;49:369-388. \n[11] Degtyareva VF, Smirnova IS. BRIZ: a visu alization program for Brillouin zone – Fermi \nsphere configuration. Z.  Kristallogr. 2007;222:718-721. \n[12] Nagel SR and Tauc J. Nearly-Free-Electr on Approach to the Theo ry of Metallic Glass \nAlloys. Phys Rev Lett. 1975;35:380-383. \n[13] Bosio L, Cortes R, and  Segaud C. X-ray diffrac tion study of liquid mercury over \ntemperature range 173 to 473 K. J Chem Phys. 1979;71:3595-3600 \n[14] Mott NF. The Electrica l Properties of Liquid Merc ury. Phil Mag. 1966;13:989-1014. \n[15] Hanfland M. Na at high temperature and pressure. High  pressure melting. ESRF report \n24093_A. (2003). \n[16] Gorelli FA, Ulivi L, Santoro M, et al. St ructure and Dynamics of liquid Rb at high \npressures and temperatures. ESRF report 53845_A. (2014). \n[17] Falconi S, Lundegaard LF, Hejny C, andMcMahon IM. X-ray Diffraction Study of \nLiquid Cs up to 9.8 GPa. Phys Rev Lett. 2005;94:125507. \n[18] Narygina O,  McBride EE, Stinton GW, and McMahon MI. Melting curve of potassium \nto 22 GPa. Phys Rev B. 2011;84:054111.  \n8      \n  \nFigure 1. The T-P melting curve for Na (after data from [2]) with the negative slope \nabove 30 GPa. In the inset, Fermi surface – Brillouin zone configurations are given \nfor Na- bcc at ambient pressure (AP) and at ~ 40 GPa in comparison with the nearly \nspherical pseudo-BZ for the liquid Na.   \n       (a)  \n  (b)  \n  \nFigure 2. (a) Diffraction data for Na in solid  and liquid state (aft er data from ESRF \nreport [15]). Evaluation of 2kF positions is given by accepting the valence electron \nnumber z = 1 for Na-fcc and z = 2 for Na- liquid at 120 GPa. (b) Diffraction data for \nliquid Hg at ambient conditi on (after data from [13] ) with indication of 2kF \npositions for z = 2.   \n9                     \n  \nFigure 3. Diffraction data for Rb in the li quid state (after data  from ESRF report \n[16]). The trace of measured points is indicated on the phase T-P diagram (upper plot). Evaluation of 2k\nF positions is given by accepting the valence electron number \nz = 1 for 4 GPa and z = 2 for 21 GPa. \n  \n  \nFigure 4. The T-P phase diagram for Cs on the left panel with the marked \nexperimental points (red) for diffraction data for Cs in th e liquid state on the right \npanel (after data from [17]). Evaluation of 2k F positions is given by accepting the \nvalence electron number z = 1 for 2.7 GPa and z = 2 for 6.8 GPa. "    },    {        "title": "1804.00244v2.Searching_for_topological_Fermi_arcs_via_quasiparticle_interference_on_a_type_II_Weyl_semimetal_MoTe__2_.pdf",        "content": "1\nSearchingfortopologicalFermi arcsviaquasiparticleinterfer ence\nonatype‐IIWeylsemimetalMoTe 2\n\nDavideIaia1,GuoqingChang2,Tay‐RongChang3,JinHu4,ZhiqiangMao4,HsinLin2,\nShichaoYan5*&VidyaMadhavan1*\n\n1DepartmentofPhysicsandFrederi ckSeitzMaterialsResearchLa boratory,UniversityofIllinoisUrbana‐\nChampaign,Urbana,Illinois61801,USA \n2InstituteofPhysics,Academ iaSinica,Taipei11529,Taiwan \n3DepartmentofPhysics,NationalChengKungUniversity,Tainan7 01,Taiwan\n4DepartmentofPhysicsandEngineeringPhysics,TulaneUniversit y,NewOrleans,Lo uisiana70018,USA\n5SchoolofPhysicalScienceandTechnology,ShanghaiTechUnivers ity,Shanghai,201210,China\n*Emails:yanshch@shanghaitech.e du.cn;vm1@illinois.edu\n\n\nWeylsemimetalsdisplayanoveltopologicalphaseofmatterwhe retheWeylnodesemergeinpairsof\noppositechiralityandcanbeseenaseitherasourceorasink ofBerrycurvature.Th eexoticeffectsin\nWeylsemimetals,suchassurface Fermiarcsandthechiralanom aly,makethemanewplaygroundfor\nexploring novel functionalities. F u r t h e r e x p l o i t i n g t h e i r p o t e n tial applications requires clear\nunderstandingoftheirtopologicalelectronicproperties,such asWeylpointsandFermiarcs.Herewe\nreportaFouriertransformscanningtunnelingspectroscopy(FT‐ STS)studyonatype‐IIWeylsemimetal\ncandidate MoTe 2whoseWeylpointsarepredicatedtobelocatedaboveFermilev el.Althoughits\nelectronicstructurebelowtheFermilevelhavebeenidentified byangleresolvedphotoemission\nspectroscopy(ARPES),bycomparingourexperimentaldatawithf irst‐principlescalculations,weare\nabletoidentifytheoriginsofthemultiplescatteringchannel satenergiesbothbelowandaboveFermi\nlevel.Ourcalculationsalsoshowtheexistenceofbothtrivial andtopologicalarclikestatesabovethe\nFermienergy.IntheFT‐STSexpe riments,wehaveobservedstron gsignalsfromintra ‐arcscatteringsas\nwellasfromthescatteringbetwe enthearc‐likesurfacestates andtheprojectedbulkstates.Adetailed\ncomparisonbetweenourexperimentalobservationsandcalculated resultsrevealsthetrivialandnon‐\ntrivialscatteringchannelsaredifficulttodistinguishinthi scompound.Interestingly,wefindthatthe\nbrokeninversionsymmetrychange stheterminatingstatesonthe twoinequivalentsurfaces,whichin\nturnchangestherelativestreng thofthescatteringchannelso bservedintheFT‐STSimagesonthetwo\nsurfaces.\n2Transitionmetaldichalcogenides (TMDs)provideafertilegroun dforexploringexoticcollectivequantum\nphenomena,suchassuperconductivityandchargedensitywaves1‐3.Recently,theMo xW1‐xTe2classof\nTMDshasattractedgreatinterestduetothetopologicalnature oftheelectronicstatesandasa\ntopologicalWeylsemimetal(TWS)4‐7.Theirlow‐energyexcitationsbehaveasWeylfermionsthatalw ays\nappearinpairswithoppositech irality.TWSsareclassifiedin totype‐Iandtype‐II:t ype‐ITWSshavepoint\nlikeFermisurfacesandrespect Lorentzinvariance,whiletype‐ IIwithtiltedWeylconesbreakLorentz\ninvariance5,8,9.Type‐ITWSswerefirstpredictedandexperimentallydiscovere dinthe(Ta,Nb)(As,P)\nfamilycompounds8‐11.Type‐IITWSshavebeenproposedtoexistintheMo xW1‐xTe2familyTMDs4‐7and\nLaAlGe12.Inthissystem,thetiltedWey lconesthatarisefromtopolog icallyprotectedcrossingsofvalence\nandconductionbandscausetouchingpointsbetweenelectronand holepocketsneartheFermilevel.\nExoticFermiarcsstatesthatjointhesetouchingpoints,aree xpectedonthesurfacesofthesematerials.\nInthelastfewyears,therehave beenaseriesofexperimental studiesaimedatverifyingtheexistenceof\ntheFermiarcs13‐21.\nFortheMo xW1‐xTe2family,theWeylnodesareusuallylocatedatenergiesabovet heFermilevelwhich\nmakesitdifficulttobeaccessedbyconventionalangleresolve dphotoemissionspectroscopy(ARPES)13‐\n15,17,20.Fouriertransformscanningtunnelingspectroscopy(FT‐STS)co ntainsinformationonthescattering\nvectorsbetweenelectronicstatesinmomentumspaceandcanpro videtheinformationonboththe\noccupiedandunoccupiedstates22whichmakesitaparticularlypowerfulprobeforthetopologic alstates\noftheMo xW1‐xTe2.Therehavebeenseveral FT‐STSstudiesontheMo xW1‐xTe215,23‐28.However,duetothe\ncomplexbandstructureandthelowsignalstrengthintheFT‐ST Simages,theexistenceofFermiarcsin\nMoxW1‐xTe2systemstillremainstobeclarified.\nHereweuselow‐temperaturescanningtunnelingmicroscopy(STM) andfirst‐principlescalculationsto\ndirectlyvisualizeandidentifytheelectronicstatesinMoTe 2,anendmemberoftheMo xW1‐xTe2system.\nTheotherendmember,WTe 2wasthefirstpredicatedtype‐IIWeylsemimetal,butthesmall momentum\nseparationbetweentheoppositeW eylpointsthatarelocatedab out50mVabovethe Fermilevel,made\nexperimentalconfirmationbyconventionalARPESdifficult5,13.MoTe 2waslaterpredicatedasanother\ncandidateoftype‐IIWeylsemimetalwheretheWeylpointshave sixtimeslargermomentumseparation\nwhichmakesexperimentalprobeofWeylnodesmucheasier6,7.ForMoTe 2,previousARPESworkshave\nfoundevidenceforthetrivialandtopologicallyprotectedFerm iarcsatenergiesbelowFermilevel18‐20.\nHowever,forvariousreasonssuchasthesizeoftheSTSmapso rthepropertiesofthescatteringimpurities,\npreviousFT‐STSsonMoTe 2didnotshowsufficientsignalstrengthtoperformaclearana lysisoftheFT‐\nSTSdata,andhavenotarrivedatanunequivocalconclusionabo utthenatureoftheWeylstates.Inthis\nwork,wepresentFT‐STSdatawithmuchhighersignalstrengtht hatallowsustocomparethevarious\nscatteringchannelsobservedint heexperimentaldatawiththeo ry,andreacharobustconclusiononthe\noriginsofthescatteringchannels.Interestingly,basedonthi shigherresolutionFT‐STSdata,wecometo\ndifferentconclusionscomparedtothepreviouswork15,23,24,26:thetrivialandnon‐trivialscatteringchannels\nareindistinguishableintheFT‐STSimages.Moreover,thebroke ninversionsymmetryinthiscompoundis\nreflectedintheFT‐STSimagest akenontwoinequivalentsurfac es.\n3Results\nTheTdphasecrystalhasanorthorhombicstructurewithvandeWaals stackingofTe‐Mo‐Tesandwich\nlayersalongthe c‐axisdirectionasshowninFig.1a.DuetotheweakvandeWaa lsinteractionbetween\ntheTelayers,theMoTe 2samplecleavesattheTelayersandthecleavedsurfaceisTe‐ terminated.The\nbulk,andsurfaceBrillouinzonesforthistermination,aresho wninFig.1b.Inhigh‐resolutionSTM\ntopographies,thetwoinequivalentTeatomicrowsinthetopTe layercanbeclearlyresolved(Fig.1d).\nTherearemainlytwokindsofatomicdefectsandtheylookdiff erentintheSTMtopographiestakenwith\npositivebiasvoltages(asshown intheredandblackcirclesi nFig.1e).ThedefectslieontheTesitesand\narelikelytobeTevacancies/im puritiesinthetwoTeatomicr owsatthetopTelayer.Inaddition,inthe\nnegativebiastopography,thereareotherfaintdefectsthatma ybeattributedtovacancies/impuritiesin\nthelayersunderneath(moredeta ilscanbefoundinSupplementa ryFig.S1andFig.S2).\nFig.1fshowsthepositiondependentd I/dVspectratakenona50nmby50nmareaonMoTe 2inthe\nenergyrangeof 500mV.Ingeneral,thed I/dVspectraareparabola‐shapedwithminimumaround10mV\nabovetheFermilevel.Despitethepresenceofimpurities,the lowenergydensityofstatesaswellasthe\noverallparabola‐shapeofthespectraremainshomogeneousandt herearenoclearimpurity‐induced\nresonancestatesinthisenergyrange26.However,thescatteringpatternofquasiparticlesbytheatom ic\nimpuritiescanbeclearlyseeninthed I/dVmaps,Fig.1c.ThenextstepistostudytheFouriertransform s\nofthedI/dVmapstolookforsignaturesoftheelectronicstatesinmoment umspace.\nFigs.2a‐ishowFT‐STSimagesatafewenergiesbelowandabov etheFemilevel(alsoseeFigs.S6andS7).\nThereisadistinctchangeintheFT‐STSaswegofromnegative topositiveenergies.Withincreasingthe\nbiasabovetheFermilevel,severalstrongscatteringvectorsg raduallyemerge.Toidentifytheoriginof\nthescatteringvectorswecarried outfirst‐principlesdensity functionaltheory(DFT)calculationsofband\nstructuresofMoTe 2whichwerethenusedtocalculateFT‐STSimages.Usingtheite rativeGreen’sfunction\nmethod,weobtainedthesurfacestatesofMoTe 2underdifferentterminations(detailsinSupplementary\nSection 8 which were then used to simulate the FT‐STS patterns using spin‐dependent scattering\nprobabilitymethod).\nAscanbeseeninFig.3,thesimulatedFT‐STSmatcheswellwit hthemeasuredFT‐STSoverawiderange\nofenergies.InthemeasuredFT‐STSatenergiesbelow 100mV(rightpanelinFig.3a),thereisalip‐like\nfeaturewithabrightcenter.BycomparingwiththesimulatedF T‐STSat140mV,weconcludethatthis\nlip‐likefeatureisduetothescatteringofthebulkstatesof MoTe 2(SupplementaryFig.S5).At 40mV,\nthemeasuredFT‐STSshowstwoshortarc‐likefeaturesatthele ftorrightsideofthebrightcenterspot\n(rightpanelofFig.3b).Duetothemultiplescatteringvector sseeninthesimulatedFT‐STSatnegative\nbiasvoltages,itisdifficulttounambiguouslyidentifytheor iginofthesetwoshortarcs.However,by\ncomparingthemomentum‐spacepositionoftheseshortarcs,wec oncludethattheymostlikelyarise\nfromthescatteringprocessesfromthetrivialsurfacestatest othehole‐pockets.Atpositivebiasvoltages,\nthescatteringpatternsbecomemuchclearer(Fig.2i).Bycomp aringthecalculatedFT‐STSwiththe\nmeasuredFT‐STSat+40mVand +100mV(Figs.3cand3dandsupp lementalFig.S9),wecanforthefirst\ntime,identifyallthescatteringpatternsas: Q1isinducedbytheintra‐scatteringofthetopologicallytrivia l4surfacestates; Q2iscausedbythescatteringbetweenthesurfacestatesandthe electronpockets. Q3and\nQ4areduetothescatteringbetweenthesurfacestatesandthep rojectedbulkstatesatthe 𝑌തpoint.\nAlthoughthehigh‐resolutionFT‐STSdataclearlyshowmultiple scatteringvectorsfromthesurfacestates\nandourexperimentalmeasurementsmatchthenumericalcalculati ons,thesedataalonedonotprovide\nunambiguousevidencefortheexistenceofnon‐trivialFermiarc s.Accordingtoourcalculations,theWeyl\npointsforMoTe 2arelocatedat4mVand57mV.Astheenergyincreasedfromth eFermilevelto+100mV,\nthescatteringvectors( Q1toQ4)graduallyemergeinthemeasuredFT‐STSs(Figs.2f‐i),andth ereisno\nsuddenchangeintheFT‐STSsneartheenergypositionsoftheW eylnodes.Thissuggeststhatthereare\nnoclearfeaturesinducedbythenontrivialFermiarcsintheF T‐STSsonMoTe 2.Wenotethatour\nconclusionsareincontr astwithpreviousstudies15,23,24,26,27.MostpreviousSTMst udieshaveclaimedthat\nMoxW1‐xTe2familymaterialisaWeylsemimetalbasedontheexistenceof thelongFermiarcsintheFT‐\nSTSdata.However,basedonour theoreticalandexperimentalst udieswefindthatthelongarcsarise\nprimarilyfromtopologicallytrivialscatteringchannels.Inad dition,forMo 0.66W0.34Te2samples,the\nabsenceofthescatteringvector( Q3)betweenthearc‐likefeatureandprojectedbulkstateswasta kenas\npoofoftheexistenceoft hetopologicalFermiarc27.Incontrast,inourhigh‐resolutiondataonMoTe 2,we\nclearlydetectthisscatteringvector( Q3)therebydeterminingthatthescatteringprocessfromtrivial\nsurfacestatesformsanimportantcomponentofthescatterings eeninourFT‐STSdata.\nDiscussionThisleadstothequestion:whyisitdifficulttodetectthen ontrivialFermiarcsinMoTe\n2evenwithhigh‐\nresolutionFT‐STSmeasurements?Weylfermionsarelocalsingula ritiesinmomentumspace(Fig.S10).\nTherefore,atenergiesormomentumspaceawayfromtheWeylnod es,thenon‐trivialpropertiesinduced\nbyWeylfermionsarenegligible.Infigure4weshowthecalcul atedsurfacestatesofMoTe 2atthree\ndifferentenergies,below,at,an dabovetheFermilevel.Figs. 4a‐cshowconstant energycontoursat\n40mV,0mVand+40mVrespectively.Figs.4d‐farethezoom‐in ofFigs.4a‐cinthemomentumregion\naroundthepositionsoftheWeylnodes(theorangerectanglear eashowninFigs.4a‐c).Fig.4dclearly\nshowsthatthelongtopologicallytrivialarcinFig.4aliesf arfromtheWeylnodesinmomentumspace,\nanditisthereforenotrelatedtotheWeylnodes.Withincreas ingenergy,thetopologicallytrivialarc\nmovestowardstheWeylnodesandstartstotouchtheWeylnodes attheFermilevel(Fig.4e).Inthe\nconstantenergycontourof E=+40meV,thelongtrivialarcisdisconnectedwiththeWeyln odesanda\nsmalltopologicalFermiarc(arc2)canbeseeninbetweenthe twoWeylnodes,asshowninFig.4f(also\nseeFig.S10).However,theseparationinmomentumspacebetwee nthenontrivialarc(arc2)andthe\ntrivialarcs(arc1and3)isonlyabout0.008Å1,whichmakesitextremelydifficulttodistinguishthe\nscatteringfeaturesinFT‐STSinducedbythenontrivialarcand thoseinducedbythelongtrivialarc.Thus,\nbycomparingtheSTMdatawithcalculations,weconcludethatt hetrivialandnon‐trivialscattering\nchannelsareindistinguishableinthiscompound.\nFinally,ourSTMdatashowsignaturesofthebrokeninversions ymmetrythat’scriticaltothephysicsof\nMoTe2.Duetothebulkbrokeninversionsymmetry,thetwosurfaceso btainedaftercleavingare5inequivalent(SupplementaryFig.S11).AsshowninFig.S11,wh ilethesurfacestructureitselfisthesame,\ntheatomicarrangementinthesecondlayerisdifferentbetween thetwosurfaces.AnearlierARPESstudy\nhadsuggestedthatthismightaffecttheelectronicstructure20.Thetwoinequivalentsurfaceswere\nrevealed in our data when different cleaves showed slightly dif ferent FT‐STS patterns (Fig.5,\nSupplementaryFig.S12andFig.S13).OurFT‐STScalculationsf orthesecondsurface,alsoshowthatthe\nterminatingelectronicstatesaremeasurablydifferentforthe twosurfaces.Thedifferencesarisefromthe\ndifferentweightsofbulkandsurfacestatesonthetwotermina tionswhichmakesthescatteringchannels\ninthemeasuredFT‐STSimagestakenonthesetwosurfaceshave differentrelativestrengths.Whilethe\nmatchbetweenthecalculationsanddataisnotasgoodforthis secondsurface,bothSTMaswellastheory\nindicatethatthebrokeninversi onsymmetrymakesthetwosurfa ceselectronicallyinequivalent(Fig.5).\nInconclusion,wehaveperformedadetailedlow‐temperatureSTM /STSstudyontheMoTe 2sample.In\nthespatiallyresolvedd I/dVmaps,thequasiparticleinterferencepatternscanbeclearlyr esolved.We\nthoroughlycharacterizedtheelectronicstructureofMoTe 2withFT‐STSstakenatenergiesbothbelowand\nabovetheFermilevel.Inthehigh‐resolutionFT‐STSsaboveFer milevel,thescatteringvectorsbetween\nthetrivialarcsandtheprojectedbulkstatescanbeclearlyr esolved.Duetothesmallmomentum‐space\nseparationbetweenthenontrivia larcandthetrivialarc,iti sdifficulttoclearlyidentifythenontrivialarc\nfeaturesintheFT‐STSs.Ourdatashowthatthebulkbrokeninv ersionsymmetryhasameasurableeffect\nonthesurfaceelectronicproperties.\nMethodsSamplesynthesis. TheMoTe\n2singlecrystalsweregrownbyusingachemicalvaportransport method.A\nmixtureofstoichiometricMoandTepowderweresealedintoan evacuatedquartztubewithiodineused\nasatransportagent.Thequartztubewasthenplacedinadoub lezonefurnacewithtemperaturegradient\nfrom900oCand800oC.Largesinglecrystalsofcentimetersizewereobtainedafter twoweeks.The\ncompositionandstructureofMoTe 2singlecrystalswerecheckedusingx‐raydiffractionandanen ergy‐\ndispersivex‐rayspectrometer.\nScanningtunnelingspectroscopymeasurements. HighqualityMoTe 2sampleswerecleavedat77Kand\nimmediatelyinsertedintotheSTMscannerat6K.Differential conductance(d I/dV)spectrawereacquired\nat6Kusingastandardlock‐intechniquewith~4mV rmsmodulationatafrequencyof987.5Hz. \n\nReferences:1. CastroNeto,A.H.Chargeden sitywave,superconductivity,a ndanomalousmetallicbehaviorin2D\ntransitionmetaldichalcogenides. Phys.Rev.Lett. 86,4382(2001).\n2. Rossnagel,K.Ontheoriginofcharge‐densitywavesinselec tlayeredtransition‐metaldichalcogenides.\nJ.Phys.Condens.Matter 23,213001(2011).63. Sooryakumar,R.&Klein,M.V.Ramanscatteringbysupercond ucting‐gapexcitationsandtheir\ncouplingtocharge‐densitywaves. Phys.Rev.Lett. 45,660(1980).\n4. Chang,T.‐R.,Xu,S.‐Y.,Chang,G.,Lee,C.‐C.,Huang,S.‐M. ,Wang,B.,Bian,G.,Zheng,H.,Sanchez,D.S.,\nBelopolski,I.,Alidoust,N.,Neupane,M.,Bansil,A.,Jeng,H. ‐T.,Lin,H.&ZahidHasan,M.Predictionof\nanarc‐tunableweylfermi onmetallicstateinMo xW1‐xTe2.Nat.Commun 7,10639(2016).\n5. Soluyanov,A.A.,Gresch,D.,Wang,Z.,Wu,Q.,Troyer,M., Dai,X.&Bernevig,B.A.Type‐IIWeyl\nsemimetals. Nature527,495(2015).\n6. Sun,Y.,Wu,S.‐C.,Ali,M.N.,Felser,C.&Yan,B.Predict ionofWeylsemimetalinorthorhombicMoTe 2.\nPhys.Rev.B 92,161107(2015).\n7. Wang,Z.,Gresch,D.,Soluyan ov,A.A.,Xie,W.,Kushwaha,S .,Dai,X.,Troyer,M.,Cava,R.J.&Bernevig,\nB.A.MoTe 2:Atype‐IIWeylt opologicalmetal. Phys.Rev.Lett. 117,056805(2016).\n8. Huang,S.‐M.,Xu,S.‐Y.,Belopolski,I.,Lee,C.‐C.,Chang, G.,Wang,B.,Alidoust,N.,Bian,G.,Neupane,\nM.,Zhang,C.,Jia,S.,Bansil,A.,Lin,H.&Hasan,M.Z.AWe ylfermionsemimetalwithsurfacefermi\narcsinthetransitionmetalmonopnictideTaAsclass Nat.Commun. 6,7373(2015).\n9. Weng,H.,Fang,C.,Fang,Z.,Bernevig,B.A.&Dai,X.Weyl semimetalphaseinnoncentrosymmetric\ntransition‐metalmonophosphides. Phys.Rev.X 5,011029(2015).\n10.Lv,B.Q.,Weng,H.M.,Fu,B. B.,Wang,X.P.,Miao,H.,M a,J.,Richard,P.,Huang,X.C.,Zhao,L.X.,\nChen,G.F.,Fang,Z.,Dai,X.,Qian,T.&Ding,H.Experimenta ldiscoveryofWeylsemimetalTaAs Phys.\nRev.X5,031013(2015).\n11.Xu,S.‐Y.,Belopolski,I.,Al idoust,N.,Neupane,M.,Bian, G.,Zhang,C.,Sankar,R.,Chang,G.,Yuan,Z.,\nLee,C.‐C.,Huang,S.‐M.,Zheng, H.,Ma,J.,Sanchez,D.S.,Wa ng,B.,Bansil,A.,Chou,F.,Shibayev,P.P.,\nLin,H.,Jia,S.&Hasan,M.Z.DiscoveryofaWeylfermionsem imetalandtopologicalFermiarcs.\nScience349,613(2015).\n12.Xu,S.‐Y.,Alidoust,N.,Chang,G.,Lu,H.,Singh,B.,Belo polski,I.,Sanchez,D.S.,Zhang,X.,Bian,G.,\nZheng,H.,Husanu,M.‐A.,Bian,Y.,Huang,S.‐M.,Hsu,C.‐H.,C hang,T.‐R.,Jeng,H.‐T.,Bansil,A.,Neupert,\nT.,Strocov,V.N.,Lin,H.,Jia,S.&Hasan,M.Z.Discoveryo fLorentz‐violatingtypeIIWeylfermionsin\nLaAlGe.SciAdv3(6),(2017).\n13.Bruno,F.Y.,Tamai,A.,Wu, Q.S.,Cucchi,I.,Barreteau, C.,delaTorre, A.,McKeownWalker,S.,Riccò,\nS.,Wang,Z.,Kim,T.K.,Hoesch,M.,Shi,M.,Plumb,N.C.,Gi annini,E.,Soluyanov,A.A.&Baumberger,\nF.FermiarcelectronicstructureandChernnumbersinthetype ‐IIWeylsemimetalcandidateMo xW1‐\nxTe2.Phys.Rev.B 94,121112Sep(2016).\n14.Belopolski,I.,Xu,S.‐Y.,Ishida,Y.,Pan,X.,Yu,P.,San chez,D.S.,Zheng,H.,Neupane,M.,Alidoust,N.,\nChang,G.,Chang,T.‐R.,Wu,Y.,Bian,G.,Huang,S.‐M.,Lee,C .‐C.,Mou,D.,Huang,L.,Song,Y.,Wang,\nB.,Wang,G.,Yeh,Y.‐W.,Yao,N. ,Rault,J.E.,LeFèvre,P., Bertran,F.,Jeng,H.‐T.,Kondo,T.,Kaminski,7A.,Lin,H.,Liu,Z.,Song,F.,Shin,S.&Hasan,M.Z.Observa tionoflargetopologica llytrivialFermiarcs\ninthecandidatetype‐IIWeylsemimetalWTe 2Phys.Rev.B 94,085127Aug(2016).\n15.Deng,K.,Wan,G.,Deng,P.,Zhang,K.,Ding,S.,Wang,E., Yan,M.,Huang,H.,Zhang,H.,Xu,Z.,\nDenlinger,J.,Fedorov,A.,Yang,H.,Duan,W.,Yao,H.,Wu,Y. ,Fan,S.,Zhang,H.,Chen,X.&Zhou,S.\nExperimentalobservationoftopologicalFermiarcsintype‐IIW eylsemimetalMoTe 2.Nat.Phys. 12,\n1105(2016).\n16.Feng,B.,Chan,Y.‐H.,Feng,Y.,Liu,R.‐Y.,Chou,M.‐Y.,K uroda,K.,Yaji,K.,Harasawa,A.,Moras,P.,\nBarinov,A.,Malaeb,W.,Bareille ,C.,Kondo,T., Shin,S.,Kom ori,F.,Chiang,T.‐C.,Shi,Y.&Matsuda,I.\nSpintextureintype‐IIWeylsemimetalWTe 2Phys.Rev.B 94,195134(2016).\n17.Huang,L.,McCormick,T.M.,Ochi,M.,Zhao,Z.,Suzuki,M. ‐T.,Arita,R.,Wu,Y.,Mou,D.,Cao,H.,Yan,\nJ.,Trivedi,N.&Kaminski,A.S pectroscopicevidenceforatyp eIIWeylsemimetallicstateinMoTe 2Nat.\nMater.15,1155(2016).\n18.Jiang,J.,Liu,Z.,Sun,Y.,Yang,H.,Rajamathi,C.,Qi,Y .,Yang,L.,Chen,C.,Peng,H.,Hwang,C.‐C.,Sun,\nS.,Mo,S.‐K.,Vobornik,I.,Fujii,J.,Parkin,S.,Felser,C., Yan,B.&Chen,Y.Signatureoftype‐IIWeyl\nsemimetalphaseinMoTe 2.Nat.Commun. 8,13973(2017).\n19.Liang,A.,Huang,J.,Nie,S. ,Ding,Y.,Gao,Q.,Hu,C.,H e,S.,Zhang,Y.,Wang,C.,Shen,B.,Liu,J.,Ai,P.,\nYu,L.,Sun,X.,Zhao,W.,Lv,S.,Liu,D.,Li,C.,Zhang,Y., Hu,Y.,Xu,Y.,Zhao,L.,Liu,G.,Mao,Z.,Jia,X.,\nZhang,F.,Zhang,S.,Yang,F.,Wang,Z.,Peng,Q.,Weng,H.,D ai,X.,Fang,Z.,Xu,Z.,Chen,C.&Zhou,\nX.J. Electronic evidence for type II Weyl semimetal state in M oTe2. Preprint at\nhttps://arxiv.org/abs/1604.01706(2016).\n20.Tamai,A.,Wu,Q.S.,Cucchi, I.,Bruno,F.Y.,Riccò,S., Kim,T.K.,Hoesch,M.,Barreteau,C.,Giannini,\nE.,Besnard,C.,Soluyanov,A.A .&Baumberger,F.Fermiarcsa ndtheirtopologicalcharacterinthe\ncandidatetype‐IIWeylsemimetalMoTe 2.Phys.Rev.X 6,031021(2016).\n21.Wang,C.,Zhang,Y.,Huang,J .,Nie,S.,Liu,G.,Liang,A. ,Zhang,Y.,Shen,B.,Liu,J.,Hu,C.,Ding,Y.,Liu,\nD.,Hu,Y.,He,S.,Zhao,L.,Yu ,L.,Hu,J.,Wei,J.,Mao,Z., Shi,Y.,Jia,X.,Zhang,F.,Zhang,S.,Yang,F.,\nWang,Z.,Peng,Q.,Weng,H.,Dai,X.,Fang,Z.,Xu,Z.,Chen, C.&Zhou,X.J.ObservationofFermiarc\nanditsconnectionwithbulkstatesinthecandidatetype‐IIWe ylsemimetalWTe 2.Phys.Rev.B 94,\n241119(2016).\n22.Hoffman,J.E.,McElroy,K., Lee,D.‐H.,Lang,K.M.,Eisak i,H.,Uchida,S.&Davis,J.C.Imaging\nquasiparticleinterferenceinBi 2Sr2CaCu2O8+δ.Science297,1148(2002).\n23.Berger,A.N.,Andrade,E.,Kerelsky,A.,Edelberg,D.,Li, J.,Wang,Z.,Zhang,L.,Kim,J.,Zaki,N.,Avila,\nJ.,Chen,C.,Asensio,M.C.,Ch eong,S.‐W.,Bernevig,B.A.& Pasupathy,A.N.Temperature‐driven\ntopologicaltransitionin1T’‐MoTe 2.npjQuantumMater. 3,2(2018).\n24.Deng,P.,Xu,Z.,Deng,K., Zhang,K.,Wu,Y.,Zhang,H.,Z hou,S.&Chen,X.Revealingfermiarcsand\nweylnodesinMoTe 2byquasiparticleinterferencemapping. Phys.Rev.B 95,245110(2017).825.Lin,C.‐L.,Arafune,R.,Liu,R.‐Y.,Yoshimura,M.,Feng,B .,Kawahara,K.,Ni,Z.,Minamitani,E.,\nWatanabe,S.,Shi,Y.,Kawai,M. ,Chiang,T.‐C.,Matsuda,I.& Takagi,N.Visualizing type‐IIWeylpoints\nintungstenditelluridebyquasiparticleinterference. ACSNano 11,11459(2017).\n26.Rüβmann,P.,Weber,A.P.,G lott,F.,Xu,N.,Fanciulli,M. ,Muff,S.,Magrez,A.,Bugnon,P.,Berger,H.,\nBode,M.,Dil,J.H.,Blügel,S. ,Mavropoulos,P.&Sessi,P.U niversalresponseofthetype‐IIWeyl\nsemimetalsphasediagram.Preprintathttps://arxiv.org/abs/170 6.00456(2017).\n27.Zheng,H.,Bian,G.,Chang,G.,Lu,H.,Xu,S.‐Y.,Wang,G. ,Chang,T.‐R.,Zhang,S.,Belopolski,I.,Alidoust,\nN.,Sanchez,D.S.,Song,F.,Jeng,H.‐T.,Yao,N.,Bansil,A., Jia,S.,Lin,H.&Hasan,M.Z.Atomic‐scale\nvisualizationofquasiparticleinterferenceonatype‐IIWeyls emimetalsurface Phys.Rev.Lett. 117,\n266804(2016).\n28.Zhang,W.,Wu,Q.,Zhang,L.,Cheong,S.‐W.,Soluyanov,A. A.&Wu,W.Quasiparticleinterferenceof\nsurfacestatesinthetype‐IIWeylsemimetalWTe 2Phys.Rev.B 96,165125(2017).\nAcknowledgements \nSTMworkwassupportedbyUSDepartmentofEnergy,ScannedProb eDivisionunderAwardNumberDE‐\nSC0014335. S.Y. acknowledges the fi nancial support from Science  and Technology Commission of\nShanghaiMunicipality(STCSM)( GrantNo.18QA1403100)andthes tart‐upfundingfromShanghaiTech\nUniversity.TheworkatTulaneUniversitywassupportedbythe USDepartmentofEnergyunderGrantNo.\nDE‐SC0014208(supportforsi nglecrystalgrowth).\n\nAuthorcontributionsS.Y.andV.M.conceivedtheexperiment.SamplesweregrownbyJ .H.andobtainedfromZ.M..S.Y.and\nD.I.carriedouttheSTMstudies.G.C.,T.‐R.C.andH.L.dida llthecalculations.S.Y.,V.M.,andD.I.wrote\nthepaper.\n9Figure1:\n\nFigure1.STMtopographiesandspatiallyresolvedd I/dVspectraonMoTe 2.(a)CrystalstructureofTd‐\nphaseMoTe 2.(b)SchematicsofthesurfaceBrillouinzones(SBZ)andbulkBrillo uinzones(BZ)ofMoTe 2.\n(c)dI/dVmapat30mV.(d)High‐resolutionSTM topographytakenwith+100mVand150pA. (e)Large‐\nscaleSTMtopographiestakenwith+100mVand300pA.Theblack andredcirclesindicatetwodifferent\nkindsofatomicimpuritiesonthetopTe‐layer.( f)Spatially‐resolvedd I/dVspectra(blue)takenovera\n50nmby50nmareawith 100mVand750pAsetpoint.Theredcurveistheaveragedd I/dVspectrum\noverthisarea.\n\n10Figure2:\n\nFigure2.QuasiparticleinterferenceonMoTe 2.(a‐i)Mirror‐symmetrizedFT‐STSsatenergiesof 200mV,\n40mV,00mV,0mV,40mV,+20mV,+40mV,+60mVa nd+100mVrespectively.Forthe FT‐\nSTSswithpositivesamplebiasvoltages,beforeperformingthe Fouriertransform,thed I/dVmapsarefirst\nnormalizedwiththetunneli ngcurrentmapat+200mV.\n\n11Figure3:\n\n  \nFigure3.ComparisonbetweenthecalculatedFT‐STSsandthemeasuredFT‐S TSs.(a)Constantenergy\ncontoursofMoTe 2attheenergy E=40mV(leftpanel);theoreticallysimulatedFT‐STSat 40mV\n(middlepanel);experimentallymeasuredFT‐STSat 40mV(rightpanel).( b‐c)Thesameas a,butfor\nenergyE=40mVand+40mV,respectively.Thearrowsintherightpanelo fbindicatethetwoshortarc‐\nlikefeatures.( d)Thesameas atoc,butforenergy E=100mV.Q1,Q2,Q3andQ4arethescattering\nvectors(seethemaintextfordetails). \n12Figure4:\n\nFigure4.TrivialFermiarcsandWeylnodesinMoTe 2.(a‐c)CalculatedconstantenergycontoursofMoTe 2\nattheenergy E=40mV,0mVand+40mV,respectively.( d‐f)Thezoom‐inoftheorangeboxinpanel a,\nbandc.ThelongarcawayfromtheWeylnodes(blackandwhitedotsl abelledasW 1andW 2)is\ntopologicallytrivial.Thetrivialarcindicatedbytheorange arrowsin dandeiscutintothreepieces,\nlabeledas“1’’,“2”and“3”in f,where“2”isthetopologicalFermiarc.Thesizeof d‐fisabout0.08Å‐1by\n0.1Å‐1.\n\n13Figure5:\n\nFigure5.CalculatedFT‐STSsandthemeasuredFT‐STSstakenonthesecond surface.(a)Constantenergy\ncontoursofMoTe 2attheenergy E=00mV(leftpanel);theoreticallysimulatedFT‐STSat 00mV\n(middlepanel);experimentallymeasuredFT‐STSat 00mV(rightpanel).( b‐c)Thesameas a,butfor\nenergyE=40mV,+40mVand+100mV,respectively.\n"    },    {        "title": "1212.4183v3.Anomalously_small_resistivity_and_thermopower_of_strongly_compensated_semiconductors_and_topological_insulators.pdf",        "content": "Anomalously small resistivity and thermopower of strongly compensated\nsemiconductors and topological insulators\nTianran Chen and B. I. Shklovskii\nFine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA\n(Dated: June 16, 2021)\nIn the recent paper, we explained why the maximum bulk resistivity of topological insulators\n(TIs) such as Bi 2Se3is so small1. Using the model of completely compensated semiconductor we\nshowed that when the Fermi level is pinned in the middle of the gap the activation energy of\nresistivity \u0001 = 0 :3(Eg=2), whereEgis the semiconductor gap. In this paper, we consider a strongly\ncompensated n-type semiconductor. We \fnd the position of the Fermi level \u0016calculated from\nthe bottom of the conduction band Ecand the activation energy of resistivity \u0001 as a function of\ncompensation K, and show that \u0001 = 0 :3(Ec\u0000\u0016) holds at any 0 <1\u0000K\u001c1. In the same range of\nrelatively high temperatures, the Peltier energy (heat) \u0005 is even smaller: \u0005 '\u0001=2 = 0:15(Ec\u0000\u0016).\nWe also show that at low temperatures, the activated conductivity crosses over to variable range\nhopping (VRH) and \fnd the characteristic temperature of VRH, TES, as a function of 1 \u0000K.\nI. INTRODUCTION\nThe three-dimensional (3D) topological insulator\n(TI)2{6has gapless surface states, which host a spectrum\nof quantum transport phenomena7,8. While a number of\ncrystals have been identi\fed to be 3D TIs, most of them\nare poor insulators and the bulk of TI crystals of sub-\nstantial size ( >10\u0016m) shunts the surface conductivity.\nThe current literature9{17broadly discusses how one can\nachieve a bulk-insulating state.\nTypically as-grown TI crystals such as Bi 2Se3are heav-\nily dopedn-type semiconductors. (It is believed that\nBi2Se3is doped by Se vacancies.) To make them insulat-\ning, these TIs are compensated by acceptors. With in-\ncreasing compensation K=NA=ND, whereNDandNA\nare the concentrations of monovalent donors and accep-\ntors, the Fermi level shifts from the conduction band to\ninside the gap and then into the valence band at K > 1.\nWhen compensation of donors is complete, K= 1, the\nFermi level is in the middle of the gap and the most in-\nsulating state of TI is achieved. For a TI with a gap\nEg\u00180:3 eV the resistivity is expected to obey the acti-\nvation law\n\u001a=\u001a0exp(\u0001=kBT) (1)\nwith activation energy \u0001 = Eg=2\u00180:15 eV, so that the\nTI is a good insulator at room temperatures and below.\nHowever, the current experimental situation near K=\n1 is frustrating16. In the temperature range from 100\nand 300 K, although resistivity is activated, the activa-\ntion energy \u0001\u001850 meV, which is three times smaller\nthan expected. At T\u0018100 K the activated transport\ncrosses over to variable range hopping (VRH). When\ntemperature is further decreased, resistivity grows even\nmore slowly and below 50 K, resistivity saturates around\n\u001a(T)<10 \ncm. This means that in spite of complete\ncompensation, even at helium temperatures conductance\nof TI samples thicker than 10 \u0016m is dominated by the\nbulk.\nIn the recent paper1, we suggest an explanation of\nanomalously large bulk conductivity of TI at K= 1. Weassume that both donors and acceptors are shallow and\nrandomly positioned in space and we use the theory of\ncompletely compensated semiconductor (CCS)18,19. The\nidea that at K= 1, when almost all donors and accep-\ntors are charged, random spatial \ructuations in the local\nconcentration of impurities lead to large \ructuations of\ncharge. Because the average concentration of screening\nelectronsn=ND\u0000NA\u001cND, the random potential is\npoorly screened and has huge \ructuations. These \ruc-\ntuations bend conduction and valence band edges and in\nsome places bring them to the Fermi level, creating elec-\ntron and hole puddles, which in turn non-linearly screen\nthe random potential. Therefore, the amplitude of po-\ntential \ructuations is limited by Eg=2. The ground state\nof the completely compensated semiconductor shown in\nFig. 1a therefore reminds a network of p-njunctions18,19.\nThe characteristic size Rof thesep-njunctions in Bi 2Se3\nwithEg'0:3eV,ND= 1019cm\u00003, and dielectric con-\nstant\u0014= 30 isR'150 nm\u001dN\u00001=3\nD = 4:6 nm1, i.e.,\nwe deal with a very long range potential. As a result,\nthe resistivity can be dramatically di\u000berent from the one\nfor the \rat bands picture of TI1. First, at relatively high\ntemperatures, the activated conduction is due to the elec-\ntrons and holes being activated from the Fermi level to\ntheir corresponding classical percolation levels (classical\nmobility edges), EeandEh, in the conduction and the\nvalence bands. According to numerical modeling1at\nK= 1 the activation energy \u0001 '0:15Eg, becauseEe\nandEhare substantially closer to the Fermi level \u0016than\nthe unperturbed by a random potential bottom of the\nconduction band Ecand ceiling of the valence band E\u0017\n[see Fig. 1a]. Second, at low enough temperatures, elec-\ntrons and holes can hop (tunnel) between puddles, so that\nvariable range hopping replaces activated transport. We\nshowed that the activated resistivity crosses over directly\nto Efros-Shklovskii (ES) law of VRH20,\n\u001a=\u001a0exp(TES=T)1=2; (2)\nwhereTES=Ce2=kB\u0014\u0018,eis the electron charge, \u0018is\nthe localization length of the states with the Fermi levelarXiv:1212.4183v3  [cond-mat.str-el]  18 Apr 20132\nenergy and C= 4:4 is a numerical coe\u000ecient. Together,\nour results for the activated and VRH resistivity estab-\nlished the universal upper limit of resistivity \u001a(T) one can\nachieve for a 3D TI compensated by shallow inpurities.\nFIG. 1. Energy diagram of (a) completely compensated semi-\nconductor ( K= 1) and (b) strongly compensated semicon-\nductor (1\u0000K\u001c1) with gap Eg. The upper and the lower\nstraight lines indicate the unperturbed positions of bottom\nof the conduction band, Ec, and ceiling of the valence band\nE\u0017; the middle straight line corresponds to the Fermi level \u0016.\nMeandering lines represent the band edges, which are modu-\nlated by the \ructuating potential of charged impurities. Ris\nthe characteristic size of potential \ructuations. Percolation\nlevelsEefor electrons and Ehfor holes are shown by dashed\nlines. Puddles occupied by carriers are shaded. Shallow im-\npurities levels are not shown because they practically merge\nwith band edges.\nIn this paper, we change our focus from a possible max-\nimum bulk resistivity of a completely compensated semi-\nconductor at K= 1 to the more practical question of the\ndependence of bulk resistivity of a strongly compensated\nsemiconductor (SCS) on Kat 0<1\u0000K\u001c1. Indeed,\nwith existing methods of growth of TI samples one can\nnot getK= 1 exactly. It is important to know how sta-\nble the resistivity results at K= 1 are for the case of\n1\u0000K\u001c1. For example, one can ask at which 1 \u0000Kthe\nactivation energy \u0001 is twice smaller than at K= 1. For\nde\fniteness, we consider n-type SCS, where the concen-\ntration of electrons n=ND\u0000NA\u001cNDand 1\u0000K\u001c1.\nWe model numerically the ground state of such SCS andits resistivity using algorithms similar to Ref.1. We \fnd\nthat in agreement with the analytic theory18, when 1\u0000K\ngrows, the screening of the random potential improves\nand its correlation length Rdecreases. The amplitude\nof the random potential decreases as well. As a result,\nhole puddles shrink and eventually vanish and the chem-\nical potential \u0016moves up, so that Ec\u0000\u0016decreases. One\ncan say that with increasing 1 \u0000K, the screening due\nto bending of the conduction band occurs only while all\nacceptors remain occupied by electrons and negatively\ncharged. All these changes are illustrated by transition\nfrom (a) to (b) in Fig. 1.\nAs a result of these changes, the activation energy \u0001\ndecreases with growing 1 \u0000K. We \fnd that the relation\n\u0001 = 0:3(Ec\u0000\u0016) obtained in Ref.1forK= 1 remains\nvalid for 1\u0000K\u001c1 (see Fig. 7 below) as well. [In p-type\nsemiconductor where K=ND=NA, a similar relation-\nship \u0001 = 0 :3(\u0016\u0000Ev) takes place.] By K= 0:97 the\nactivation energy \u0001 is about two times smaller than at\nK= 1. This result shows that achieving maximum resis-\ntivity with \u0001 = 0 :15Egis problematic. It also explains\nthe origin of large scatter of magnitude of \u0001 among TI\nsamples16.\nIn principle, our prediction that \u0001 = 0 :3(Ec\u0000\u0016) can be\ndirectly compared with experiments in TIs. Indeed, for\neachK, the position of the Fermi level can be found via\nmeasurements of the surface concentration of electrons\nin the gapless surface state using Shubnikov-de-Haas os-\ncillations. On the other hand, at low temperatures, we\n\fnd numerically a direct cross-over from activation to\nES VRH. We also \fnd how TESbeing correlated with \u0001\ndecreases with 1\u0000K.\nOur assumption of random distribution of impurities\nis crucial for this theory. Usually, for samples made by\ncooling from melt, the distribution of impurities in space\nis a snapshot of the distribution the impurities have at\nhigher temperature when the di\u000busion of impurities prac-\ntically freezes. In semiconductors with a narrow enough\ngap at this temperature, there is a concentration of in-\ntrinsic carriers larger than the concentration of impuri-\nties. Intrinsic carriers screen the Coulomb interaction\nbetween impurities, so that impurities remain randomly\ndistributed in space. At lower temperatures, when in-\ntrinsic carriers recombine, impurities are left in random\npositions18,21. If di\u000busion freezes at T\u00181000K, it is\nreasonable to assume that impurities are randomly posi-\ntioned in a semiconductor with Eg\u00140:3eV. This justi-\n\fes the use of this theory for typical TIs. Our results are\napplicable to other narrow gap semiconductors, for exam-\nple, InSb. (Historically, large e\u000bort was made to make\nInSb insulating via strong compensation. The goal was\nto improve characteristics of InSb based photo-detectors.\nResults were again frustrating: the dark resistivity was\ntoo small. Our results are in reasonable agreement with\ntransport experiment data for InSb22,23.)\nThe plan of the paper is as follows. In Sec. II, we\nformulate the model, explain the algorithm of numerical\nsimulation of the pseudoground state and present results3\nfor the density of states (DOS). In Sec. III, we present\nour algorithm for computation of hopping conductivity,\nanalyze our results and arrive at a small activation en-\nergy for conduction band resistivity \u0001 = 0 :3(Ec\u0000\u0016).\nWe also evaluate the localization length of states with\nenergy close to Fermi energy and estimate the charac-\nteristic temperature of ES law TES. In Sec. IV, we es-\ntimate the thermopower of strongly compensated semi-\nconductor and show that the Peltier energy (heat) is\n\u0005'\u0001=2 = 0:15(Ec\u0000\u0016), in qualitative agreement with\na recent experimental paper24. We conclude in Sec. V,\nwhere we comment on predictions of this model for the\nHall e\u000bect measurements and compare these predictions\nwith experimental data16.\nII. THE MODEL, PSEUDOGROUND STATES,\nAND THE DENSITY OF STATES\nTo model a heavily doped SCS, we create a cube \flled\nwith 20000 donors and 20000 Kacceptors that are ran-\ndomly positioned in space. We numerate all donors and\nacceptors by index iand useni= 0 or 1 for the number\nof electrons residing on a donor or an acceptor. In addi-\ntion, we use a variable fito discriminate between donors\n(fi= 1) and acceptors ( fi=\u00001). The Hamiltonian of\nour system is\nH=X\niEg\n2fini+X\nhijiV(rij)qiqj; (3)\nwhereqi= (fi+ 1)=2\u0000niis the net charge of site iand\nall energies are de\fned relative to the Fermi level. The\n\frst term contains the energies of shallow donors and ac-\nceptors, which is very close to the semiconductor gap Eg.\nThe second term of His the sum of interaction energies\nof charged impurities. If two impurities are at distance\nr>>aB, whereaBis the Bohr radius of impurity states,\none can use the Coulomb interaction V(r) =e2=\u0014r. For\na pair of empty donors, one donor shifts down the energy\nof the electron on the other by an energy V(r) =\u0000e2=\u0014r.\nThis classical form for V(r) is good for a lightly doped\nSCS. But in a heavily doped SCS, where aB> N\u00001=3\nD,\nmost impurities have at least one neighbor at distance\nr < aBand quantum-mechanical averaging over elec-\ntron wave function becomes important. (This is why an\nuncompensated heavily doped semiconductor is a good\nmetal). For example, such a pair of donors cannot create\na state deeper than that of the helium-like ion with a\nbinding energy 4 EB, whereEB=e2=2\u0014aBis the bind-\ning energy of the shallow donor state. Here, we deal\nwith heavily doped SCS, where ( Ec\u0000\u0016)>4EBand\nquantum e\u000bects limit the role of short-range potential.\nTo model such a case, we continue to use the classical\nHamiltonian Eq. (3), but truncate the Coulomb potential\ntoV(r) =e2=\u0014(r2+a2\nB)1=2. Note that Eq. (3) does not\ninclude the kinetic energy of electrons and holes in con-\nduction and valence bands and, therefore, aims only atdescription of the low temperature ( kBT\u001cEg) physics\nof SCS.\nBelow, we use dimensionless units for r,aB,H,Eg,\nandkBT, measuring all distances in units of N\u00001=3\nD and\nall energies in units of e2N1=3\nD=\u0014. Thus Eq. (3) now\ncan be understood as dimensionless, where Eg\u001d1 and\nV(r) = (r2+a2\nB)\u00001=2. For TI with Eg= 0:3 eV,\n\u0014= 30, and ND= 1019cm\u00003, we haveN\u00001=3\nD = 4:6\nnm ande2N1=3\nD=\u0014'10 meV, so that the dimensionless\ngapEg= 30. We could not model Eg= 30, because in\nthis case, the very large correlation length of long-range\npotential,R, leads to large size e\u000bect. Instead, we run\nmore modest Eg= 15, for which the size e\u000bect requires\nextrapolation only at K= 11. Our goal is to \fnd the\nactivation energy \u0001 and estmate TESas a function of K\nor\u0016.\nWe search for the set fni;figthat minimizes Hand\nuse such a set to calculate the DOS and the conductivity.\nWe start from the neutral system of all populated by elec-\ntrons (negatively charged) acceptors ( ni= 1;qi=\u00001),\nof equal number of randomly chosen 20000 Kempty\n(positively charged) donors ( ni= 0;qi= 1), and of\n20000(1\u0000K) \flled (neutral) donors ( ni= 1;qi= 0).\nCharged donors and acceptors create a random poten-\ntial whose magnitude exceeds Eg. In order to screen the\nCoulomb potential \ructuations, some electrons leave ac-\nceptors for donors. At any stage of this process, there are\ntwo types of occupied states { neutral donors and nega-\ntively charged acceptors, and two types of empty states {\npositively charged donors and neutral acceptors, respec-\ntively. Electrons may hop from an occupied impurity to\nan empty one. If the proposed move lowers the total\nsystem energy H, then it is accepted, otherwise it is re-\njected. To check whether Hgoes down, for a given set of\nelectron occupation numbers fni;fig, it is convenient to\nintroduce the single-electron energy state, \"i, at a given\nimpurityi:\n\"i=Eg\n2fi\u0000X\nj6=iV(rij)qj: (4)\nFor alli,jwithni= 1 andnj= 0, we check that ES\npseudoground state stability criterion is satis\fed:\n\"j\u0000\"i\u0000V(rij)>0: (5)\nIf this criterion is not satis\fed, we move the electron from\nimpurityitojand recalculate all \"i. This process is done\nby looping all possible pairs of impurities i;jwithni= 1\nandnj= 0 and is continued until no single-electron\ntransfers can be made to lower H. The \fnal arrange-\nment of electrons can be called a pseudoground state,\nbecause the higher stability criteria of ground state are\nnot checked. Pseudoground states are known to describe\nthe properties of the real ground state pretty well18,25.\nThe results below are obtained at Eg= 15,aB= 1 for\nK= 1;0:99;0:98;0:97;0:96, and 0:95 (averaged over 100\nrealizations of impurities coordinates).4\n0 0.01 0.02 0.03 0.04 0.05468\n1−KEc − µ\n(Ec − µ) ~ 1.4(1− K )−1/3\nFIG. 2. (Color online) Fermi level \u0016as a function of 1 \u0000K\nforaB= 1 andEg= 15. The size of dots characterizes the\nuncertainty.\nFor a pseudoground state, we \fnd the Fermi energy\n\u0016as a half distance between the minimum empty and\nmaximum occupied energy \". Fig. 2 shows how the Fermi\nlevel\u0016(K) shifts from the middle of the gap towards the\nconduction band bottom with growing 1 \u0000K. At 1\u0000K >\n0:01, this dependence is in reasonable agreement with the\nprediction of single-band theory (the theory that ignores\nvalence band and acceptors)18thatEc\u0000\u0016=A(1\u0000\nK)\u00001=3. However, note that for heavily doped SCS, the\ncoe\u000ecientAh'1:4 is twice smaller than the coe\u000ecient\nAl'2:8 obtained in Ref.18for a lightly doped SCS,\nwhereaB\u001c1. In this case, the short-range Coulomb\ninteraction at distance r\u001cN\u00001=3\nD leads to an additional\ncontribution to \u0016of the same order of magnitude.\nTo con\frm our understanding of results for 1 \u0000K >\n0:01, we obtained the same results for the position of\nFermi level \u0016(and DOS of donors and conductivity, see\nbelow) using a simpli\fed one-band model where all ac-\nceptors are assumed to be negative. Such program is\nsimilar to the classical impurity band program used in\nChapter 14 of Ref.18, but uses the rede\fned V(r).\nThe resulting DOS of impurities is shown in Fig. 3 for\nK= 1 andK= 0:95. AtK= 1, the almost constant\nsymmetric DOS between \u0000Eg=\u000015 andEg= 15 re-\n\rects a practically uniform distribution of random poten-\ntial from\u0000Eg=2 toEg=2, and a corresponding uniform\ndistribution of band edges Ecbetween 0 and EgandE\u0017\nbetween 0 and\u0000Eg[see Fig. 1(a)]. In the middle (at the\nFermi level) one sees the ES Coulomb gap20.\nAtK < 1, the DOS of impurities loses the donor-\nacceptor symmetry it has at K= 1. As mentioned in\nIntroduction (see Fig. 1), with growing 1 \u0000K, acceptors\nbecome all \flled and disengaged from screening. Accep-\ntor DOS (leftmost peak) splits from the donor one, which\nin turn has two peaks separated by the Fermi level. The\nlarge right peak belongs to empty donors, while the small\nand narrow left peak belongs to occupied donors. The\ndonor peaks are separated by the ES Coulomb gap.\n−10 0 1000.040.080.12DOS, g*(ε)\nelectron energy, εK = 1K = 0.95FIG. 3. (Color online) Dimensionless single-electron DOS\ng\u0003(\") in units of [(1 + K)ND=(e2N1=3\nD=\u0014)] as a function of\n\"calculated from the Fermi level for aB= 1 andEg= 15 at\nK= 0.95 (blue) and 1 (red). Impurity states with \"<0 are\noccupied and with \">0 are empty. At K= 1, the total DOS\nof impurities has donor-acceptor symmetry, which is lost with\ngrowing 1\u0000K.\n−1 0 100.010.02\nelectron energy, εDOS, g*(ε)\nK = 1K = 1\nFIG. 4. (Color online) Dimensionless DOS g\u0003(\") for neu-\ntral (occupied by electrons) donors with \" < 0 and neutral\n(empty) acceptors with \" >0 foraB= 1 andEg= 15 atK\n= 0.98 (blue) and 1 (red).\nGrowing with 1\u0000Kthe disengagement of acceptors\nfrom screening is also illustrated in Fig. 4, where we\nshow the DOS g\u0003(\") for neutral donors and acceptors.\nIf atK= 1, the total number of electrons and holes in\npuddles are equal, with growing 1 \u0000K, the total num-\nber of electrons in electron paddles grows, while the to-\ntal number of holes in hole puddles decreases. Thus, at\n1\u0000K\u00150:02, valence band practically plays no role in\nscreening.5\nIII. NUMERICAL MODELING OF HOPPING\nCONDUCTIVITY\nOnce the energies f\"igare known, we evaluate the re-\nsistivity using the approach of the Miller-Abrahams re-\nsistor network18. Each pair of impurities i;jis said to\nbe connected by the resistance Rij=R0exp[2rij=\u0018+\n\"ij=kBT], where the activation energy \"ijis de\fned18as\nfollows:\n\"ij=(\nj\"j\u0000\"ij\u0000V(rij); \"j\"i<0;\nmax [j\"ij;j\"jj]; \"j\"i>0:(6)\nThe resistivity of the system as a whole is found using a\npercolation approach. Speci\fcally, we \fnd the minimum\nvalueRcsuch that if all resistances RijwithRij< Rc\nare left intact, while others are eliminated (replaced with\nR=1), then there exists a percolation pathway con-\nnecting opposite faces of the simulation cube. The sys-\ntem resistivity \u001a(T) is de\fned as RcN\u00001=3\nD. Here, we\nconcentrate on the exponential term of resistivity \u001aig-\nnoring details of the prefactor18.\nForK= 0.95, 0.97, 0.98, and 1 at aB= 1 andEg= 15,\nthe computed dependence of (ln \u001a)\u0003= (\u0018=2) ln(Rc=R0) is\nshown as a function of ( T\u0003)\u00001=2in the huge range of tem-\nperatures 0:03<T\u0003<200 in Fig. 5. Here, T\u0003= 2kBT=\u0018\nis yet another dimensionless temperature. These nota-\ntions are introduced to exclude any explicit dependence\non\u0018. One can see at low temperatures 0 :03<T\u0003<0:3\nthe resistivity is well described by ES law Eq. (2) (with\nC'4:4 atK= 1). The higher temperature range\n1<T\u0003<200 is plotted separately as a function of 1 =T\u0003\nin Fig. 6. We \fnd two activated regimes of hopping con-\nductivity. At high temperatures 50 < T\u0003<200, we see\nthe large activation energy Ea\u0018Ec\u0000\u0016, while in the\nrange of intermediate temperatures 1 <T\u0003<Eg, we see\nmuch smaller activation energy \u0001 = 0 :3(Ec\u0000\u0016).\nThe \frst activation energy Eadoes not have any phys-\nical meaning for a real SCS, because at kBT > Egcon-\nductance of SCS is actually not due to hopping but free\ncarriers with high energy, which are not taken into ac-\ncount by energy Eq. (3) (see Ref.1). In contrary to Ea,\nthe second activation energy \u0001 = 0 :3(Ec\u0000\u0016) makes full\nphysical sense and should be seen in real experiment. The\norigin of this activation energy for the hopping transport\nis also explained in Chapter 8 of Ref.18. AtT\u001cEg, elec-\ntrons optimize their conductivity by using for hopping\nimpurities energetically close to the Fermi level. Even-\ntually at very low temperatures, such opitmization leads\nto ES conductivity. However, when donor energies are\nslowly modulated by the long-range potential, there are\nlarge areas that do not have donors with energies close to\nthe Fermi level and the tunneling through them is slow.\nTherefore, there is a range of temperatures where elec-\ntrons use only nearest-neighbor donors for hopping, while\nactivating to donors is located at the percolation level of\nnearest-neighbor percolation. We then \fnd the activation\nenergy from the Fermi level to the nearest-neighbor per-\n0 1 2 3 4 5 604812\n(T*)−1/2resistivity, (ln ρ)*\n  \nK = 0.95\nK = 0.97\nK = 0.98\nK = 1FIG. 5. (Color online) The temperature dependence of the\nresistivity in the whole temperature range 0 :03< T\u0003<\n200. The dimensionless resistance (ln \u001a)\u0003is plotted against\n(T\u0003)\u00001=2to illustrate that the resistivity follows the ES law\nat low temperatures. The dashed lines are the best linear \fts.\n0 0.2 0.4 0.6 0.8 1123\n(T*)−1resistivity, (ln ρ)*\n  \nK = 0.95\nK = 0.97\nK = 0.98\nK = 0.99\nK = 1\nFIG. 6. (Color online) The temperature dependence of the re-\nsistivity in the high temperature range 1 <T\u0003<200. (ln\u001a)\u0003\nis plotted against ( T\u0003)\u00001to illustrate that the resistivity is\nactivated at high temperatures. The dashed lines are the best\nlinear \fts.\ncolation level by studying the hopping activation energy\n\u0001. In a heavily doped semiconductor, this energy is indis-\ntinguishable from the activation energy of electrons from\nthe Fermi level to the conduction band percolation level\nEe. [Of course, holes are activated from the Fermi level\nto their percolation Ehas well so that \u0001 = 0 :3(\u0016\u0000Eh)].\nWe veri\fed that hopping conduction modeling cor-\nrectly predicts the activation energy of the band trans-\nport by direct calculation of the percolation level Ee.\nFor this purpose, we created a cubic lattice with a small\nlattice constant N\u00001=3\nD=3. At every site of this lattice,\nwe calculated the potential of all charged impurities and\nthen found lowest energy Eeat which percolation over\nthis lattice takes place. The activation energy of the band\ntransport was again close to \u0001 = 0 :3(Ec\u0000\u0016). This re-6\n3 4 5 6 711.522.5\nEc − µactivation energy, ∆\n∆ ~ 0.3(Ec − µ)\nFIG. 7. (Color online) The activation energy \u0001 at K=\n1;0:99;0:98;0:97;0:96, and 0:95 (from right to left). The\ndashed line is the best linear \ft \u0001 '0:3(Ec\u0000\u0016).\n.\nsult is also close to what was obtained in Ref.26based on\nan estimate of percolation level for a generic long-range\nrandom potential18.\nIn Fig.7, we plot \u0001 as a function of Ec\u0000\u0016for all the\nvalues\u0016(K) obtained at K= 1;0:99;0:98;0:97;0:96, and\n0:95. We see that the relation \u0001 '0:3(Ec\u0000\u0016) holds well\nfor allKin this interval.\nSo far, we emphasized the results that do not explic-\nitly depend on \u0018. Actually, a magnitude of \u0018is necessary\nto calculate TES. We argue now that in a TI \u0018is quite\nlarge leading to the prominent role of VRH. If an elec-\ntron with an energy close to the Fermi level were tunnel-\ning from an electron puddle to a distant one along the\nstraight line, it would tunnel through high barriers and\nits wave function would decay with \u0018\u001caB. Actually, a\ntunneling electron can use the same geometrical path as\na classical percolating electron with energy \u0001 above the\nFermi level that avoids large barriers. We assume that\nalong such a path tunneling barriers Vare uniformly dis-\ntributed in the range 0 \u0014V\u0014\u0001 and neglect contribu-\ntion of curvature of this path into action. Integration\noverVthen gives (here we return to normal units) \u0018=\n~=(8m\u0001=9)1=2andkBTES= 4:2(e2=\u0014~)(m\u0001)1=2. For a\nTI withaB=N\u00001=3\nD, we getTES= 4:2[(e2N1=3\nD=\u0014)\u0001]1=2.\nFor \u0001 varying between 1 and 2.5 e2N1=3\nD=\u0014as shown in\nFig. 7,TESchanges from 4.2 to 6.6 e2N1=3\nD=\u0014. For\u0014= 30,\nND= 1019cm\u00003, ande2N1=3\nD=\u0014kB'100 K,TESvaries\nfrom 420 to 660 K. In order to study VRH in TI samples\nexperimentally, one has to deal with large enough sam-\nples, where surface conductance is smaller than the bulk\none.27IV. THERMOPOWER\nIn the recent paper24, the authors studied activation\nenergy of the bulk resistivity of series of samples of\nBi2Te3\u0000xSexwith di\u000berent xand thereby di\u000berent po-\nsitions of the Fermi level in the TI gap. They found that\nwhen the Fermi level sinks into the gap, the activation en-\nergy of resistivity \u0001 grows and reaches a maximum at 40\nmeV and then decreases. The increase of the activation\nenergy \u0001 on both sides of the maximum is accompanied\nby the increase of the absolute value of the thermopower\nS. However, near the maximum of \u0001, the thermopower\nabruptly changes its sign. These \fndings are in agree-\nment with what one can expect when a semiconductor\ngoes through the point of complete compensation. Here,\nwe would like to concentrate on the maximum absolute\nvalue of the thermopower, for example, at n-type side of\nthe maximum.\nIt is known that for \rat bands n-type semiconductor\nwith the Fermi level \u0016inside its gap the thermopower\nS= \u0001=eT, where the activation energy \u0001 = Ec\u0000\u0016. For\nbended bands of a strongly compensated n-type semicon-\nductor, one could think that S= \u0001=eT, where the acti-\nvation energy \u0001 = Ee\u0000\u0016is determined by the activation\nto percolation level Ee. Actually, it was argued28{30that\nthe Peltier energy (heat) \u0005 = eTS is determined by the\naverage potential energy of electrons E(conduction band\nbottom) along most conducting one-dimensional percola-\ntion paths, \u0005 = <E\u0000\u0016>. (We call a percolation path\nany line where the potential energy of electron is smaller\nthanEeand we call a set of the least resistive of these\npaths, which carry most of the current, the most conduct-\ning percolation paths.) The thermopower of an open cir-\ncuit following an individual percolation path can be ob-\ntained by integrating E\u0000\u0016along this path. Among two\nparallel paths connecting points A and B, the more resis-\ntive one has a somewhat larger open circuit thermopower\nand, therefore, drives circular current back through the\nleast resistive one. This current reduces thermopower of\nthe resistive path so that the voltage between A and B\nis determined by the more conducting path.\nIf the probability distribution of potential energy Eon\nmost conducting paths is the same as for the uncondi-\ntional probability distribution of E, which we call DOS\ng\u0003(E) above, we can use g\u0003(E) to calculate \u0005 and S. For\nexample, in the case of a constant g\u0003(E) for\u0016<E <E e,\nwe getES=<E\u0000\u0016>= \u0001=2 = (Ee\u0000\u0016)=2. This con-\nclusion was con\frmed by the numerical experiment28for\nthe case of a constant g\u0003(E).\nIn a strongly compensated semiconductor, one can use\nthe realg\u0003(E) found above. For example, at K= 0:95\none can use Fig. 7 to \fnd that \u0001 = Ee\u0000\u0016'1. Then\nusing DOS shown in Fig. 3 one can check that the average\nenergy in the range of 0 <E < 1 is<E\u0000\u0016>'\u0001=2 =\n0:5. Thus our simple approximate prediction is that the\nlargest achievable \u0005 '\u0001=2. This conclusion is valid for\nallK\u00140:98 we studied.\nFor the data of the paper24, our prediction means that7\natT= 100 K the largest thermopower S= \u0005=eT ob-\nserved should be of the order 25 mV/100 K = 0.25 mV/K\nin resonable agreement with the observed value S= 0:4\nmV/K.\nIn this paper, we are not considering the additional\ncontribution to thermopower of activated electrons from\nphonon drag31,32. This e\u000bect becomes signi\fcant only\nat temperature T\u0014TD=3, whereTDis the Debye tem-\nperature, because at larger temperatures, the low-energy\nphonons interacting with electrons are strongly scattered\nby thermal phonons, which in turn are strongly inter-\nacting with imperfections of the crystal. In Bi 2Se3,\nTD\u0018150 K, so that phonon drag should get important\nonly below 50 K (where electron transport is already via\nhopping), while the activated transport we are interested\nin happens at T\u0015100 K.\nIn order to go beyond the above approximation that\nthe distribution of energies on paths contributing to \u0005 is\ngiven by the density of states g(E), we calculate currents\nIijin every Miller-Abrahams resistor Rijand the total\ncurrentI(U) for a small applied voltage Uby solving\nKirchho\u000b equations for the ground state of impurities\nobtained by our algorithm. Following Ref.33, we then\ncalculate the energy \rux through a cross-section of the\nsampleQ(U) as a sum of energy \ruxes carried by re-\nsistorsqij= (Ei+Ej)Iij=2eand found \u0005 = Qe=I . We\nsimplify the implementation of this procedure by modify-\ning our algorithm in the following way: instead of dealing\nwith completely randomly positioned donors and accep-\ntors, we randomly position them on all sites that are\nappropriate to their number cubic lattice. To \fnd the\nenergiesEi, we use a simple Coulomb potential. (There\nis no need in truncation at small distances via \fnite aB.)\nWe concentrate on the range of relatively high temper-\natures, where the conductivity is characterized by acti-\nvated behavior. We checked that the conductance I=U\nhas the same activation energy \u0001 as obtained by the\npercolation algorithm. We found that in the range of\n0:95\u0014K\u00140:98, where the asymmetry of the density of\nstates is large and donors dominate the transport, Peltier\nenergy \u0005=\u0001'0:40\u00060:05, not too far from the simpli-\n\fed theories and the experimental data24. ForK > 0:98,\ngrowing donor-acceptor symmetry reduces \u0005 and brings\nit to zero at K= 1, in agreement with the data of thepaper24.\nV. CONCLUSION\nIn this paper, we applied the model of strongly com-\npensated semiconductor to a bulk TI with narrow gap.\nWe calculated the activation energy of the bulk resistiv-\nity \u0001 and showed that it grows as \u0001 = 0 :3(Ec\u0000\u0016), when\nthe compensation degree K!1 and the Fermi level sinks\ninto the gap. If one of the two carriers still dominates and\nthe thermopwer is still monopolar the Peltier energy is\n\u0005'\u0001=2. Both predictions seem to agree with most of\nthe TI data.\nWe would like to mention that the same model is able\nto interpret measurements of the Hall E\u000bect obtained for\nthe same samples. The Hall constant RHis expected to\ngrow exponentially with decreasing temperature with the\nsame activation energy \u0001 as the resistivity28,34,35. The\nreason for such growth is that RHis dominated by nodes\nof percolation path network that occur at energy close\nto the percolation level. Such nodes are relatively rare\nat low temperatures. Therefore RH(T) =\u001a(T)u(T)=c\ngrows with decreasing T, where mobility u(T)/Tmand\nm\u00152. The observed behavior of RH(T) does not con-\ntradict this prediction16. Indeed, the largest activation\nenergy ofRHwas found to be on average \u001815 meV\nlarger than the largest \u0001 \u001850 meV. This di\u000berence is\nof the order of 1 :5kBTat the characteristic measurement\ntemperature of activation law T= 100Kand, therefore,\nthe experimental data is compatible with a power law\nu(T). In future work, we plan to narrow the range of\ntheoretical predictions by a numerical evaluation of RH\nfor the simulated above potential of our model.\nACKNOWLEDGMENTS\nWe are grateful to Brian Skinner with whom this work\nwas started, for reading the manuscript and helpful ad-\nvices. We acknowledge useful discussions with A. Akrap,\nY. Ando, A. L. Efros, Y. M. Gal'perin, J. Kakalios and\nI. P. Zvyagin. T. Chen was partially supported by the\nFTPI.\n1B. Skinner, T. Chen, and B. I. Shklovskii, Phys. Rev. Lett.\n109, 176801 (2012).\n2L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,\n106803 (2007).\n3J. E. Moore and L. Balents, Phys. Rev. B 75, 121306\n(2007).\n4L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).\n5X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B\n78, 195424 (2008).\n6R. Roy, Phys. Rev. B 79, 195322 (2009).\n7M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045(2010).\n8X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n9D.-X. Qu, Y. S. Hor, J. Xiong, R. J. Cava, and N. P. Ong,\nScience 329, 821 (2010).\n10J. G. Analytis, R. D. McDonald, S. C. Riggs, J.-H. Chu,\nG. S. Boebinger, and I. R. Fisher, Nat. Phys. 6, 960 (2010).\n11J. G. Checkelsky, Y. S. Hor, M.-H. Liu, D.-X. Qu, R. J.\nCava, and N. P. Ong, Phys. Rev. Lett. 103, 246601 (2009).\n12N. P. Butch, K. Kirshenbaum, P. Syers, A. B. Sushkov,\nG. S. Jenkins, H. D. Drew, and J. Paglione, Phys. Rev. B8\n81, 241301 (2010).\n13J. G. Analytis, J.-H. Chu, Y. Chen, F. Corredor, R. D.\nMcDonald, Z. X. Shen, and I. R. Fisher, Phys. Rev. B 81,\n205407 (2010).\n14K. Eto, Z. Ren, A. A. Taskin, K. Segawa, and Y. Ando,\nPhys. Rev. B 81, 195309 (2010).\n15Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando,\nPhys. Rev. B 84, 075316 (2011).\n16Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando,\nPhys. Rev. B 84, 165311 (2011).\n17Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando,\nPhys. Rev. B 85, 155301 (2012).\n18A. L. Efros and B. I. Shklovskii, Electronic Properties of\nDoped Semiconductors (Springer-Verlag, New York, 1984)\nAvailable from http://www.tpi.umn.edu/shklovskii.\n19B. I. Shklovskii and A. L. Efros, Sov. Phys.-JETP 35, 610\n(1972).\n20A. L. Efros and B. I. Shklovskii, J. Phys. C: Solid State\nPhys. 8, L49 (1975).\n21Y. S. Gal'perin and A. L. Efros, Sov. Phys. - Semicond. 6,\n941 (1972).\n22E. M. Gershenzon, I. N. K. V A Il'in, and L. B. litvak\nGorskaya, Sov. Phys. - Semicond. 9, 874 (1975).\n23N. G. Yaremenko, Sov. Phys. - Semicond 9, 554 (1975).24A. Akrap, A. Ubaldini, E. Giannini, and L. Forro, arXiv\narXiv:1210.3901 , arXiv:1210.3901 (2012).\n25A. M obius, M. Richter, and B. Drittler, Phys. Rev. B 45,\n11568 (1992).\n26V. F. Mitin, Journal of Applied Physics 107, 033720\n(2010).\n27Historically VRH between puddles was studied in Ref.?.\nThis paper was written before Ref.20and claimed Mott\nVRH. Now it is clear that resistivity obeys Eq. (2). The\ntheory?of the transition from activated transport to ES\nlaw is to be modi\fed as well, but we are not dwelling on\nthis transition range, because it is di\u000ecult to study details\nof such a transition in experiment.\n28H. Overhof and W. Beyer, Philos. Mag. B 43, 433 (1981).\n29D. Quicker and J. Kakalios, Phys. Rev. B 60, 2449 (1999).\n30H. Overhof and M. Schmidtke, Phys. Rev. B 61, 12977\n(2000).\n31L. Gurevich, J. Phys. (U.S.S.R) 9, 10 , 477, 67 (1945, 1946).\n32C. Herring, Phys. Rev. 96, 1163 (1954).\n33I.P.Zvyagin, Phys. Stat. Sol. (b) 58, 443 (1973).\n34V. G. Karpov, A. Y. Shik, and B. I. Shklovskii, Sov. Phys.\n- Semicond 16, 901 (1982).\n35A. Y. Shik, Electronic Properties of Inhomogeneous Semi-\nconductors (Gordon and Breach, 1995)."    },    {        "title": "1804.04478v1.Bulk_and_surface_characterization_of_In__2_O__3__001__single_crystals.pdf",        "content": " 1 Bulk and surface characterization  of In2O3(001) \nsingle crystals  \n \nDaniel R. Hagleitner,1 Peter Jacobson,1 Sara Blomberg, 2 Karina Schulte, 2 \nEdvin Lundgren,2 Markus Kubicek,3 Jürgen Fleig,3 Frank Kubel,3 Christoph \nPuls,3 Andreas Limbeck ,3 Herbert Hutter,3 Lynn A. Boatner,4 Michael \nSchmid,1 Ulrike Diebold1 \n1 Institute of Applied Physics, TU Wien, Wiedner Hauptstr. 8 -10/134, 1040 Wien, Austria  \n2 Division of Synchrotron Radiation Resea rch, Lund University, Box 118, SE -22100, Sweden  \n3 Institute of Chemical Technologies and Analytics, TU Wien, Getreidemarkt 9/164, 1060 \nWien, Austria  \n4 Materials Science and Technology Division , Oak Ridge National Laboratory, Oak Ridge, \nTennessee 37831  \n \nA comprehensive bulk and surface investigation of high -quality In 2O3(001) \nsingle crystals is reported. The transparent -yellow, cube -shaped single \ncrystals were grown using  the flux method. ICP-MS measurements reveal \nsmall residues of Pb, Mg and Pt in the crystals. Four-point -probe \nmeasurements show a resistivity of 2  ± 0.5  105 Ω cm, which translates into \na carrier concentration of  ≲ 1012 cm-3. The results from X -ray diffraction (XRD) \nmeasurements revise the lattice constant to 10.1150(5) Å from the previously  \naccepted value of 10.117 Å. Scanning Tunneling Microscopy (STM) images of \na reduced (sputtered/annealed) surface show a step height of 5 Å, which \nindicates a preference for one type of surface termination. A combination of \nlow-energy ion scatter ing (LEIS)  and atomically resolved STM indicate s an \nindium -terminated surface with small islands of 2.5 Å height, which \ncorresponds to a strongly distorted indium lattice.  Scanning Tunneling \nSpectroscopy (STS) reveals a pronounced surface state at the Ferm i Level \n(EF). Photoelectron Spectroscopy (PES) show s additional, deep -lying band \ngap states, which can be removed by exposure of th e surface to activated \noxygen. Oxidation also results in a shoulder at the O  1s core level at a higher \nbinding energy, possib ly indicative of a surface peroxide species. A downward \nband b ending of 0. 4 eV and an upward band bending of ~0.1 eV is observed \nfor the reduced and oxidized surface s, respectively.  \n \n1 Introduction  \nIndium oxide, In 2O3, has drawn increased  attention from rese archers over \nrecent years. When doped with SnO 2, the material is commonly referred to as \nIndium Tin Oxide (ITO), which is the prototypical Transparent Conducting \nOxide (TCO). ITO combines high optical transparency in the visible range with \nconductivity app roaching that of a metal.1 As a consequence of its interesting  2 physical and chemical characteristics, it is widely used in several technical \napplications including transparent films for Organic Light Emitting Diodes \n(OLEDS) and organic photovoltaic cells (OPVC), gas sensing, and \ntransparent infrared reflectors. In 2O3 is also used in heterogeneous catalysis2 \nand in chemical gas sensing .3 Despite its technological importance, \nsurprisingly little is known about the fundamental surface properties of ITO or \nthose of pure In 2O3. Even basic material characteristics, such as the \nfundamental band gap , are the subject of some continuing  controver sy.4,5 The \nuse of a value of 3.75 eV6 for In2O3 (which is the optically -determined band  \ngap) instead of the actual value of around 2.9 eV4 has led to many incorrect \nconclusions - especially regarding band bending effects and defect chemistry.  \nAnother problem associated with  previous investigations is that the samples \nwere produced mainly by magnetron sputt ering, evaporation, or by pressing \npellets of powder, which usually  leads to polycrystalline, highly defective \nsamples7 - although new information could be gained from epitaxially -grown \nIn2O3 and ITO thin films.4,8-10 In the present  work , high quality, single crystals \nof In 2O3 are examined.  \nIn2O3 is an n -type semiconductor. The n -type conduction arises from intrinsic \ndonor defect s, which also explains the pronounced non -stoichiometry \nobserved under highly reducing conditions.11 So far , In2O3 was regarded as a \nvery good conductor even without extrinsic dopants due to its high carrier \nconcentration.12  \nIn2O3 can exist in three different modifications.  In thi s work , only single -crystal \nsamples crystallized in the bixbyite structure (space group  \nIa3 ) are utilized . \nThe bixbyite structure can be derived from a 2 22 fluorite lattice.13 To obtain \nthe bixbyite structure, 12 oxygen atoms per unit cell are removed from the \nanion sublattice in a systematic way. Each {001} anion layer contains four of \nthese vacant oxygen positions. The cubic unit cell of In 2O3 has – according to \nthe literature14 – a lattice parameter of 10.117 Å (for a revised value see Sec. \n2.1.2 ) and consists of 80 atoms; 32 indium atoms and 48 oxy gen atoms \n(Figure 1). Two  distinct indium positions can be found in the crystal, that have \ndifferent symmetr ies. These are commonly referred to as In -b (8 atoms) and \nIn-d (24 atoms). The In -b sites are more  regularly coordinated than the In -d \nsites. \nAlong the (001) direction , the crystal can be thought of as a stacking of \nthree different layers. The se can be identified as a mixed layer, which consists \nof In-b and In -d atoms (M -layer), a layer which contains only In -d atoms (D -\nlayer), and an oxygen layer. An important characteristic of In 2O3(001) is the \npolarity of its surface (Tasker type -III)15,16: The alternating indium and oxygen \nlayers lead to a net dipole moment perpendicular to the surface. However, \npolar surfaces cannot have bulk -like surface terminations: in this case the \nelectrostatic potential  would diverg e for a macroscopic material. A \ndepolarization field is , therefore , required to stabilize the surface. This can be  3 achieved by differen t mechanisms such as reducing  the top and bottom layer \nsurface charges, changes in surface stoichiometry, surface reconst ructions, \nabsorption or faceting.16 Recent STM measurements of epitaxial ITO films \nshowed that the non -polar (111) surface has a (1x1) structure with a simple \nbulk-like termination8, while the polar (001) surface is considerably more \ncomplex.9 These previous measurements were , however , complicated by the \nfact that (001) -oriented In 2O3 films tend to facet, which leads to rough \nsurfaces.10 The availability of high -quality, (001) -oriented single crystals, in \ncombination with a recent DFT work17 that describes possible terminations of \nlow-index In 2O3 surfaces, has motivated us to undertake the present study.  \n \nFigure 1 (Color online) Unit cell of the In 2O3 bixbyite structure. Large ( red) \nballs oxygen, s mall (dark ) balls In -b and In -d (bright).  \n \n2 Experiments  \n2.1 Bulk Characteristics  \n2.1.1  Crystal Growth  \nThe crystals were grown using a  flux method. The flux (solvent) used was \nB2O3 and PbO. In addition, a small amount of MgO was added. This mixture \nof In 2O3, B2O3, PbO and MgO powders was melted in a platinum crucible and \nmaintained  at a temperature of 1200 °C for 4 to 10  hours. The temperature \nwas then programmed  to decrease by 3  °C per hour. As soon as a \ntemperature of 500°C was reached, the furnace was turned  off. Th e crystals \nwere extracted from the solidified flux using a 1:4 solution of nitride acid and \nwater.  \nThe resulting cubic crystals are yellow -transparent and have si de lengths  \nbetween 1 and 2 mm as show in Figure 2. Polarized light microscopy  showed \nthat the crystals  can be characterized as heterogeneous single crystals with \nmultiple growth domain s and  stress birefringence (see Figure 2).   4 \n  \nFigure 2 (Color  online) The investigated In 2O3 single crystals. A polarized light \nmicroscope image of one of the crystal is shown.  \n2.1.2  ICP-MS/SIMS  \nICP measurements  were performed  to determine whe ther the crystals have \nresidues  from the flux  or show other unkn own contaminations. The \nmeasurements were performed using an NWR -213 laser ablation system from \nNew Wave Research (ESI, Fremont, CA) coupled to a X -Series  2 ICP -MS \nSystem from Thermo Fisher Scientific (Bremen, Germany). A total of 18 \ndifferent spots were a blated to minimize the influence of local irregularities in \nthe sample. Trace  impurities w ere quantified  using a mass balance model \nbased on the assumption that the target is 100 % In 2O3 with a natural isotope \ndistrib ution. Element concentrations were calculated from the acquired \nintensity counts with reference to the 113In signal and the respective natural \nisotope abundance. Results were averaged from all 18 measurements.  The \nhighest  concentrations were found for  Pb (4307 ± 74  ppm), Mg ( 1388 ± 42  \nppm) and Pt ( 155 ± 28  ppm) . Trace  amounts  (< 50 ppm) of Zr, Sn, Sb, Nd and \nBi were  identified . Boron has  a low signal  intensity  in ICP -MS an d only a smal l \namount  was detected . \nThe TOF-SIMS measurements were carried out on a TOFSIMS.5 (ION -TOF \nGmbH, Münster, Germany) instrument. Depth profiles  (~2000 nm)  for the \nmost abundant  impurities in the crystals (Pb, Mg, Pt)  showed a homogenous \ndistribution and no significant reduction of intensity with sputter time.  \n2.1.3  XRD  \nA small  fragment broken from the corner region of the platelet was used for \nsingle crystal analysis. This crystal was controlled optically and showed no \nbirefringence or domains. Structure analysis was made using a Bruker  5 SMART diffractometer with a graphite mono chromator and a SMART APEX \ndetector using molybdenum K α radiation ( λ = 0.71073 Å) at room temperature \n(293 K). Frames were integrated using SAINT PLUS (Bruker, 2008) and \nabsorption correction was performed using a multi -scan approach with \nSADABS (Bruker, 2008). The lattice parameters and systematic extinction s \nclearly indicate  an orthorhombic space group \nIa3 . The unit cell size was \ndetermined to 10.1150(5) Å . The structure analysis was made using the XTAL \nrefinement package and converged to R= 0.017.18 The atomic coordi nates are \ngiven in Table 1. Details of the refinement are given in the Table A.1, atomic \ndisplacement parameters can be found i n Table A.2  and inter -atomic \ndistances in Table A.3.  \n \nSite  Wyckoff Symbol  x Y Z \nIn1 8b 0.25 0.25 0.25 \nIn2 24d 0.466393(2)  0 0.25 \nO 48e 0.39037(2)  0.15470(2)  0.381852(19)  \nTable 1: In 2O3 Atomic co ordinates determined by XRD.  The occupation of all \nsites is 1.  \n2.1.4  Resistivity Measurements  \nThe resistivity of the In 2O3 crystals was measured in an  ambient atmosphere  \nusing the four -point -probe method (frequency range: 1 - 10 Hz) without any \npretreatment of the sample . The resistivity at room  temperature is (2 ± 0.5)  \n105 Ω cm.  \n2.2 Surface Characteristics  \n2.2.1  Surface Quality  \nThe sample surface preparation consist ed of several sputter and annealing \ncycles in an UHV -chamber (base pressure < 5  10-10 mbar). A sputtering \ncycle included a ten-minute surface bombardment with  2 keV argon ions \n(current density ≈ 10-5 A/cm2). During the annealing excursions the sample \nwas kept at a temperature of 500°C for 10 minutes. Annealing to higher \ntemperatures (600°C) caused an irreversible color change of the crystal to \norange and was th erefore avoided ; this change is likely  caused by the \ndesorption of oxygen and the formation of additional oxygen vacancies.19 \nAfter several sputter and annealing cycles the surface showed a sharp LEED \npattern as shown  in Figure 3 (a). These results compare well with the LEED \nresults reported earlier by Morales and Diebold ,9 which were obtained on pure \nand tin-doped In2O3 (001) thin films grown  with oxygen -plasma assisted MBE \non Yttria -Stabilized Zirconia (YSZ). Similar to their results, a p4g glide plane \nsymmetry is observed, which causes the extinction of the (0, n + 1/2) and (n + \n1/2, 0) spots indicated b y white boxes in Figure 3 (a).  6 Auger electron spectroscopy  (AES), X -ray photoelectron spectroscopy (XPS) \nand LEIS  with 1 keV helium ions and a scattering angle of  105° confir med that \nthe surface is free of contamination. LEIS, which is sensitive to strictly the top -\nmost layer , showed an indium peak that is far larger than the oxygen peak \n(Figure 3 (b)). \n       \n  \nFigure 3 (a) LEED results of an In 2O3(001) single crystal ( EElectron = 100 eV) \nprepared by sputtering and annealing. Marked with squares are the (0, n + \n1/2) and (n + 1/2, 0) spots that are missing due to the p4g glide plane \nsymmetry. (b) He+ LEIS spectrum  that shows a far more pronounced indium \npeak ( E = 916.1  eV) compared to the oxygen peak ( E = 525 eV)  \n2.2.2  STM  \nScanning Tunneling Microscopy was performed in UHV (base pressure < 5  \n10-10 mbar) at room temperature with two different instruments (SP ECS \nAarhus design and a customized Omicron micro -STM ) using \nelectrochemically etched tungsten tips. Imaging was possible with both \npositive and negative bias with no apparent difference in the appearance of \natomically -resolved images. A 100 100 nm² image o f the sputtered/annealed \nsurface is shown in Figure 4. It exhibits wide terraces with relatively regular \nstep edges . The line profile shows that the terraces have a step height of 5 Å. \nAfter exposing the su rface to activated oxygen with a plasma source, the \nsame step height distribution was found, although the resulting  surface \nappeared more complex in higher resolution images. The detailed results of \nthese oxygen -exposure  experiments will be reported in a f uture publication.  \nFigure 5 (a) shows a 20 15.75 nm² image with higher resolution. It is \ndominated by bright, atomic -sized protrusions, which at first seem to be \narranged in a disordered way. Analysis with  Fourier transform s, however, \nclearly reveals a square 3.6 -Å lattice; this length equals a quarter of the \ndiagonal of the  In2O3 bulk unit cell. We have extracted this lattice by creating a \nbinary image with only one nonzero pixel at the center of each atom and \nextracting the 3.6 -Å periodicity in the Fourier domain. The resulting lattice \npositions are shown as dots in Figure 5 (b). It is obvious that most atoms are \nclose to a lattice site of this regular 3.6 -Å grid, but not directly on it. Only a few  7 atoms are far from lattice sites  - sometimes close to the center of a grid cell.  \nThe coverage of the protrusions in Figure 5 amounts to about two per unit \ncell, whi ch is ¼ of the atomic density of the 3.6 -Å lattice. The dark areas \nwithout protrusions are situated  2.5 Å  below the protrusion level . \n \n \n \nFigure 4: (Color online) STM image of a sputtered/annealed In 2O3(001) \nsurface taken with a sample bias  of VSample Bias  = -3 V and tunneling current \nITunnel = 0.225 nA.  The arrow indicates the location of the line profile.   8 \n \nFigure 5 (Color online) : (a) STM  image with atomic resolution  (VSample bias  = -\n2.6 V, ITunnel = 0.17 nA). (b) This shows a magnification of the framed part \nabove. The 3.6 Å lattice acquired from the  Fourier transform is overlaid.   \n2.2.3  STS \nSTS was applied to measure the local density of states (LDOS) of the sample \naround EF. A total of 400 spectroscopy point s from a 20  20 grid in a 30  30 \nnm2 area were acquired. A lock -in amplifier was used to directly obtain the \ndifferential conductance dI/dV , which is roughly proportional to the LDOS. The \nSTS spectrum in Figure 6 shows the average over the 400 dI/dV  curves. A \nstrong state at EF is clearly visible.  \n \nFigure 6 Scanning Tunneling Spectroscopy of a sputtered/annealed \nIn2O3(001) surface.  \n2.2.4  PES \nHigh-resolution photoelectron spectroscopy was c arried out at the MAX II \nstorage ring at MAX -lab in Lund, Sweden. The SX -700 monochromator at  9 beamline I3 1120 supplies photons in the energy range from 43 to 1500 eV. \nThe spectroscopy end station is used for high -resolution XPS and NEXAFS \nmeasurements under UHV conditions (base pressure: < 3  10-10 mbar). The \npreparation chamber includes , among others , an argon ion sputter gun, LEED \noptics, mass spectrometer, and an atomic oxygen source ( MBE -\nKomponenten OBS -40). A hemispherical energy analyzer (SCIENTA \nSES200) was used for the photoemission exp eriments. Measurements were \nperformed at room temperature with the photon beam incident at 55° to the \nsample surface and normal emission.  \nFor the se measurements , the In 2O3 crystal was mounted on a rhodium \ncrystal, which  allowed us to calibrate the Fermi e nergy by measuring the \nrhodium surface next to the In2O3 sample . It is defined as  0 eV binding energy \nfor all of the spectra shown here. The valence band (VB) region was acquired \nwith photon energies of 60, 85, 105 and 185 eV as shown  in Figur e 7 (b). Due \nto the small size of the In2O3 crystals , it turned out to be difficult to completely \navoid rhodium peaks (the rhodium counts could be  minimized to about 1% of \nthe In 3d5/2 counts) . The photon energy o f 105 eV was chosen because the \nRh 4d-levels  have a cross -section  minimum at this energy.21  \nSample preparation was identical to th at used  for STM  analysis . To oxidize \nthe surface , the cr ystal was treated with atomic oxygen for 30 minutes at an \noxygen pressure of 3  10-7 mbar and a flux of ~ 5  1014 oxygen atoms cm-2 \ns-1. While dosing, the sample was kept at a temperature of 200  °C or 400  °C.  \nFigur e 7 (a) shows a comparison of the VB of the reduced and oxidized \nsurface s. The valence band maximum (VBM) is determined by extrapolating a \nlinear fit of the leading edge of the valence band photoemission to the \nbackground level.22 The VBM was determined using the data displayed in \nFigur e 7 (a) and lies at 2.9 eV and 2.4 eV below EF for the reduced and the  \noxidized surface s, respectively. A shift of the same magnitude can be seen for \nthe In 3d5/2 core levels (0.5 2 ± 0.11 eV, Figure 8 (a)) and the O 1s core level \n(0.54 ± 0.1 1 eV, Figure 8 (b)). The In 4d semicore levels are shifted by a \nsmaller amount (≈ 0.3 eV); probably they are influenced by the VB.  \nA closer look at the O 1s peak of the oxidized surface reveals a shoulder at \nthe higher binding energy, shifted by 1.84 eV with respect to the main peak as \nshown in Figure 8 (c). Further  noteworthy features  in Figur e 7 (a) are the \npronounced gap states of the sputtered/annealed surface, which completely \ndisappear for the oxidized surface.  \nFigur e 7 (b) shows  the VB of the reduced surface at different photon energies. \nIt is apparent that the intensity of the gap states decreases with increasing \nphoton energy, i.e., with decreasing surface sensitivity.   10 \n \nFigur e 7 (Color online) (a) Normalized valence band spectra of the oxidized \nand sputtered/annealed In 2O3(001) sample at 105 eV photon energy . Linear \nextrapolation s for determining  the leading edge of the VB are indicated.  (b) \nVB of  the reduced surface at different photon energies.  \n \n \n  \nFigure 8 (Color online) PES of the (a) indium  3d5/2 and (b)  O 1s core levels, \ntaken with a photon energy of 610 eV . (c) Line fits of the O 1s peak of the \noxidized surface at 610 eV (c).   \n3 Discussion  \n3.1 Surface Structure  \n As shown in Sec. 2.2.2  the step edges of the In 2O3(001) surface have a \nheight of exclusively 5 Å. This has an important consequence if one considers  \nthe crystal stru cture of In 2O3 (Figure 1): the distance between equivalent \nlayers (e.g. from M - to M-layer) is roughly 5 Å. However, adjacent layers (i.e., \nfrom M - to D-layer) have a distance of about 2.5 Å. The observatio n of 5 Å -\nhigh steps implies that the large terraces have one particular termination, \neither M or D .   11 \n \nFigure 9 (Color online) Models of the In 2O3(001) surfaces with (a) D-layer, (b) \nM-layer (b), and (c) O-layer termination. Shown are bulk -terminated layers \nwith all the indium and oxygen  atoms present. Note that these terminations \nrepresent polar surfaces; for stoichiometric surfaces only half of the atoms \nshown here are expected to be present.  \n \nIn a recently published paper by Agoston and Alb e,17 possible terminations for \nlow-index In 2O3 surfaces w ere investigated in detail by DF T calculations. \nBoth, stoichiometric  (i.e., half of the indium/oxygen atoms missing from the \nbulk-terminated surfaces that are shown in Figure 9) and  non-stoichiometric \nsurfaces were considered as a functi on of the oxygen chemical potential. For \nthe stoichiometric surfaces , these calculations show that the half -oxygen \nterminations are energetically preferable compared to the half -indium \nterminations, in agreement with previous DFT results from Walsh and \nCatlow.1 The differences are very small, however. Due to the small energy \ndifferences between the different ter minations , Agoston and Albe conclude \nthat considerable disorder has to be expected and that even a coexistence of \ndifferent terminations could be possible.  \nThe surface stoichiometry depends on the oxygen chemical potential. \nAgoston and Albe found that dif ferent  surface terminations become \nenergetically preferable , when going from highly reducing to highly oxidizing \nconditions : (i) additional indium nucleates on top of the surface ; (ii) 2 oxygen \natoms per unit cell (M-layer), (iii) 4 oxygen atoms per unit c ell (D-layer); ( iv) \nstoichiometric termination with six surface oxygen atoms; (v) partial dimerized  \ntermination with  two dimers  (i.e. peroxo species)  and four oxygen atoms ; (vi) \n6 dimers  (M-layer) . A dimerized surface would be in agreement with other \nDFT-calculations, which predict that surface oxygen atoms undergo a \ndimerization.3,23 \nFor the sputtered/annealed surface, we can safel y rule out  a (partial ) peroxide \ntermination  for the following reasons : (1) the preparation included annealing to \n800 K in an oxygen background pressure of less than 10-12 mbar . This  \ncorrespond to an oxygen chemical potential of approximately ~-2 eV \nassumin g equilibrium with the residual gas (which is unlikely, however). This  12 would lead to a metallic termination according to Ref. 17. An oxygen -rich \ntermination can be ruled out because oxygen has been depleted by \nsputtering; (2) the top -layer consists predominantly of indium according to \nLEIS ( Figure 3 (b)); (3) according to simulated STM images , oxygen dimers \nshould appear  dark;17 (4) as suggested by Gassenbauer  et al,24 a surface \nperoxo species should show an O 1s core level with an additional structure at \nthe higher BE side. While we, indeed, see such a peak for oxidize d surface in \nFigure 8 (c), it is absent for the sputtered/annealed surface.  \nAt first sight, t he atom -sized features in Figure 5 (a) seem to be arranged in \nan unsystematic way. As shown in Figure 5 (b), most protrusions (atoms) are \nclose to lattice sites of a 3.6 Å lattice, rotated by 45° with respect to the axes \nof the unit cell . The ball model in Figure 9 (c) shows that all oxygen atoms are \nsituated close to lattice sites of a square lattice with a/4 ≈ 2.5 Å, aligned with \nthe axes of the unit cells. We have examined whether the protrusions visible \nin the STM image ( Figure 5) could  fit to such a lattice, but we found  no \nsolution  that is  compatible wit h the fourfold symmetry of the surface. The \narrangement of the indium atoms is a distorted 3.6  Å lattice rotated  by 45°, for \nboth, D - and M -terminations (see Figure 9 (a), (b)). This suggests  that the \nprotrusions visible in the STM image of the sputtered/annealed surface are \nindium atoms, but th at this  indium lattice is only partially occupied. The \nnumber of protrusions in the STM image  is determined as 2 per unit cell. This \ncorresponds to half the indium a toms expected for a stoichiometric In -\nterminated surface (or 1/4 of the density of the full indium  D- or M-layer  shown \nin Figure 9 (a) and (b)).  \nComparing the atoms in Figure 5 with the simulated STM images for the \nreduced surface , the elongated  shape of the atoms coul d be an indication th at \nwe are seeing the D -layer. The unresolved, dark areas in Figure 5, ~2.5 Å \nbelow the protrusions,  would th en be the M -layer. The surface would thus \nconsist of an M -layer at the basis, with small, irregular patches of a D -layer on \ntop. The deviations of the atomic positions from a square 3.6 -Å lattice, as \nseen by STM, are larger than  those in either the D - or the M -layer, however, \nthey might be related to either a density of states that is not centered at the \natoms or due to mutual repulsion of the positively charged indium atoms \ndistorting the lattice.  \nThe occurrence of small islands  on a polar metal -terminated oxide surface is \nreminiscent of the ZnO(0001) case, where it was found that O -terminated step \nedges provide additional negative charge, compensating for the excess \npositive charge of the zinc .25 We consider it possible that the same \nmechanism explain s the occurrence of the small indium  islands on the \nIn2O3(001) surface.  \nInterestingly, the STM results presented here show quite a different \nappearance from the results  obtained earlier by Morales and Diebold.9 They  13 investigated thin films of ITO (001) grown on YSZ  and prepared with an \noxygen plasma source , i.e., under highly oxidizing conditions. Protrusions \narranged in a  zig-zag pattern were observed and interpreted as oxygen \ndimers of the peroxide termination. Cross-shaped dark features were \nattributed  to missing dimers. The samples investigated in Ref.  9 differ in \nvarious ways from the single crystals studied here.  They have been grown in \na highly  oxidiz ing environment;  they consisted of relatively thin epitaxial films \nthat may have been inf luenced by strain induced by the YSZ substrate; and \nthey were doped with tin. All three parameters are expected to affect the \nsurface termination - although it is left to future investigations to determine to \nwhat extent.  \nThe photoelectron spectrum of the O 1s CL  of the oxidized surface  (Figure 8 \n(c)) shows a shoulder attributed to the presence of peroxide species . Agoston \nand Albe17 considered a  ‘full’ peroxide terminati on (6 dimers /unit cell ) and the \n‘partial ’ peroxide  surface (2 dimers, 4 oxygen atoms).  Assuming we probed \nthree monolayers with PES (due to the photoelectron escape length at a \nkinetic energy of ~100 eV) , the expected  ratio of oxygen in peroxid e versus  \noxygen in bulk -like positions  should be  ~1/3 and ~1/10 for the full and partial \nperoxide  termination , respectively . The intensity ratio of the  shoulder  to the \nmain O 1s peak in  Figure 8 (c) amounts to  5.3%. This indicates that the \noxidized crystal has  a considerably  lower peroxide coverage than the full \nperoxide termination.  \n3.2 Electronic Structure  \n3.2.1  Bulk Properties  \nWe will initially discuss the results from the four -point -probe measurements. \nThe accuracy of the resistivity values is limited by the fact that no distinction \nbetween bulk and surface conductivity is possible. Also surfac e states m ay \nplay a role in the surface conductivity. This makes it difficult to determine the \nresistivity accurately, but they do give an order of magnitude that will be \nsufficient for the following considerations.  \nThe carrier concentration n can be  calcu lated from the conductivity  and the \ncarrier mobility µ using the relation:26 \n\nen\n \nThe conductivity  = (5 ± 1.3)  10-6 Ω-1 cm-1 was calculated from the \nresistivity value f rom the four-point -probe  measurement. The carrier mobility \nvalue  = 32 cm² V-1 s-1 was obtained from Ref.  4. The carrier concentration \nthen results in (1 ± 0.2 )  1012 cm-3, which is about seven orders of magnitude \nsmaller than the lowest value reported previously in the literature  (see Ref.  4). \nThis means our crystals have fewer impurities than the ones investi gated in  14 previous publications. This is emphasized by the fact that our crystals are \nyellow compared to the often reported green or black coloration  of In 2O3.27 A \npossible reason for this extremely low carrier concentration might be  the Pb \nand Mg residues in the crystals ; they could compensate the natural n -type \ndoping by a p -type doping.  \nNevertheless , the carrier concentration value is probably still assessed to be \ntoo high  to represent the intrinsic value, and it gives an upper limit  for two \nreasons:  First, water or surface states could have led to an increase of the \nconductivity at th e surface. Second, the mobility of our crystals could be \nhigher compared to the value from the literature because of the lower impurity \nscattering in our crystals.  \nThe Debye screening length \nD  of the crystal was estimated using the \nrelation28 \n2/1\n20\n\nneTkbr\nD\n,  \nwhere \n0  is the vacuum permittivity, \nr  the relative permittivity , kb the \nBoltzmann constant , T the temperature, n the carrier concentration , and e the \nelementary charge. For \nr  the value 4 was used.29 The resulting Debye length \nhas a rather large value of 2.4 m due to the low carrier concentration.  \nThe bulk F ermi energy can be estimated  from the bulk carrier concentration :26  \nTkTkmnE Ebb\nC F\n\n\n\n23\n22*\n2ln\n, \nwhere EF is the Fermi energy, EC the conduction band  energy  and m* the \neffective mass  of electrons in the CB . The effective mass is 0.35 m e.30 The \nexpected position of the Fermi level in the bulk is , therefore , 0.4 eV below the \nCBM at room temperature  \n3.2.2  Surface States  \nIn general , the VB of In 2O3 is dominated by O 2p-derived states with a lower \nVB peak from In 5s.1,17,23,31 The CB is dominated by the In 5s states. There \nare two interesting effects regarding the electronic structure, which were \nobserved  by PES and STS - namely gap states  and a rigid  shift in the core \nlevels . Both  depend on the sample preparation conditions .  \nGap states of pure In 2O3 have been reported previ ously . Klein investigated \nthe surface of reactively evaporated In 2O3 films in situ by using synchrotron -\nexcited photoemission.32 He observed gap states depending on the oxygen \npressures while preparing the films. He describes an increasing intensity of \nthe gap states with decreasing pres sure, which goes along with an increase of \nthe optical absorption. He concluded that these gap states have their origin in \ndifferent stoichiometries.   15 Agoston and Albe calculated the electronic structure of the reduced, \nstoichiometric, and the full peroxide  termination.17 The reduced surface shows \nIn 5s -derived, half -filled metallic gap states w hich should have an acceptor \ncharacter and , therefore , cause an upward band bending. For the \nstoichiometric surface , they predict – similar to Walsh and Catlow1 - a split -off \nfeature from the O 2p states close to the VBM, which is caused by \nundercoordinated oxygen atoms. For the surface under oxidizing conditions , \nperoxide surface states appear in the band gap c oming from the anti -bonding \np−π orbitals of the dimers. Other distinctive features for this termination are \nthe bonding p−σ orbitals at the bottom of the VB and a ~5 eV splitting of the O \n2s states of the dimers.  \nAs shown in Figur e 7 (a), the reduced surface shows low-lying gap states at \n~2.4 eV and ~1.8 eV. These m ay be the In  5s states of an indium terminated \nsurface calculated by Agoston and Albe, but in contrast to their suggestion , \nthe states are completely below EF (filled).  \nFigur e 7 (b) shows  the VB of the reduced surface at different photon energies. \nThe gap states apparently decrease with increasing energy. This indicates  \nthat the gap states are located at the surface . Another interesting feature of \nFigur e 7 (b) is the decrease of the VB peak at ~8.2 eV with increasing photon \nenergy . According to the electronic density of states (EDOS) calculations  this \npeak should mainly arise from  the In 5s states . Therefore,  it represents  \nanother argument for an  indium  termination.  \nAs EF at the surface is very close to the CBM (according to PES) the state \nmeasured only with STS at EF (Figure 6) could be ei ther a surface state or the \nedge of the CB. We believe that it cannot be the CB  edge  alone because in \nthis case we would expect  a higher slope . Unfortunately , this surface state  \ncould not be confirmed by PES.  There are two plausible explanations for th is: \nFirst, STS is only sensitive for surface states at \n)0 (||k . The PES analyzer \nhad a rather wide acceptance angle of  \n12 . The measurable parallel \ncomponent with this set up can be determined by : \n2sin2\n2*\n||\nKinEmk\n. \nThis leads to the result that areas of  \n2 2\n|| 95.0 Å k   (photon energy: 105 eV) \nor \n1.67Å2  (photon energy 185 eV) are covered. Therefore, the analyzer \ncollects electrons from an area lar ger than the first Brillouin zone (BZ): \n2\na\n\n2\n0.39Å2\n. A state at  \n)0 (||k  might not be visible if it comprises only a \nsmall fraction of the first BZ. Assuming that the state observed by STS is \nlocated clos e to EF, has a parabolic dispersion , and energy of \nE  from the \nbottom of the band to EF, we can estimate the fraction of the BZ occupied by \nthis state:   16 \n2*\n||2\nE EmkKin. \nWith an effective mass equal to that of the In2O3 CB (\nm*0.35me )30 and \nassuming  \nE2kbT  at 300 K, the area includes only ~1 % of the area \nmeasured by the PES analyzer.   \nThe second argument is related to the cross sections: The results from \nAgoston  and Albe suggest that the surface state probably arises from In 5s \nstates.17 According to the  literature , the In 5s photoionization cross sections \nshould  be rather small  compared to the ones for O 2p  (~6.7% at 105 eV \nphoton energy).21 Even at 185 eV – where the In 5s cross section is about \n25% of the O 2p cross section – the surface state could not be observed.  \nThe oxidized surface does not exhibit gap states  in PES  contrary to what we \nwould have expected from the DFT calculations :17 neither the predicted O 2s \nsplitting and the orbital at the bottom of the VB (full peroxide  termination ) nor \nthe gap st ates for the undercoordinate d oxygen (partly peroxide surface ) \ncould be  observed  \n3.2.3  Band Bending  \nA more -or-less rigid  shift of the VB (Figur e 7 (a)) and all core level s (Figure 8 \n(a) and (b) ) is usually expla ined by band bending.  Surface states in \nsemiconductors result from breaking of the translational symmetry of the bulk. \nThey can be either donor -like ( usually closer to the VBM) or acceptor -like \n(usually closer to the CBM). The energy at which the gap state  change s its \ncharacter from  predominantly donor -like to acceptor -like is the charge \nneutrality level (CNL) of the semiconductor.33 Depending on their position \nrelative to the Fermi level , they can be  either neutral (occupied donor -like or \nunoccupied acceptor -like states)  - or charged positively (unoccupied donor -\nlike states)  or negatively (occupied a cceptor -like states) .33 Charged surface \nstates cause a rearrangement of carriers close to the su rface in order to \nscreen the surface charge. This causes the upward or downward bending of \nthe CB and VB relative to the Fermi level .4  \nThe large Debye length of 2.4 μm implies that the band bending reaches far \ninto the bulk. This is in good agreement with the experiment, where no \nreduction of the band bending is observable when increasing the photon \nenergy, i.e., lowering surface sensitiv ity of the PES measurement. However, \nthe carrier concentration at the surface should be higher due to the band \nbending ( EF closer to the CBM; see discussion below) , which would decrease \nthe screening length.  \nTo determine the direction of the band bending , the bulk Fermi energy (0.4 eV \nbelow the CBM) has to be regarded. The VB and the core level s of the \noxidized surface are shifted by approximately the same value of 0.5 eV to \nlower binding energies. Assuming a fundamental band gap of 2.93 eV7, the  17 Fermi level is located 2.5 eV  above the VBM  in the bulk . At the surface,  the \nFermi level  is pinned at 2.9 eV (2.4 eV) for the reduced (oxidized) surface \nabove the VBM, according to the results from  PES. This means that the VB \nand the CB are ben t downward by 0. 4 eV for the reduced surface ( Figure 10 \n(a)), and bent upward by 0.1  eV or less for the oxidized surface ( Figure 10 \n(b)).  \n \nFigure 10: (Color online) Downward band bending of the reduced surface ( a) \nand upwards band bending of the oxidized surface ( b). \nBand bending in In 2O3 has been reported previously.  King et al. described a \ndownward band bending of 0.3 eV deter mined by XPS measurements and \ncarrier statistics calculations for thin films grown by plasma -assisted molecular \nbeam epitaxy (PAMBE) on YSZ.4 This value was revised later to 0.45 ± 0.16 \neV due to better knowledge of the band gap and an alternative method of \nanalysis.7 Their value is , nevertheless , in good agreement with our finding , \nwhere the VBM was determined by the common method of extrapolation of \nthe leading edge of the VB (see Figur e 7 (a)).  \nKlein observed that higher oxygen pressure during deposition leads to a lower \nFermi level position in the band gap,32 which is the same tr end as for our \nresults. Since  he assume d a band gap of 3.6 eV , he explain ed the PES \nresults with a surface depletion layer with an upward band bending of 1 eV.  \nFor the reduced surface , the gap states visible close to the VBM are filled \n(occupied donor -like, neutral) and cannot cause band bending. The downward \nband bending can be explained by positively charged, unoccupied , donor -like \nsurface states from the excess indium atoms. Th ese lead to an accumulation \nof electrons at the surface  that screen the positi vely charged surface state \nand cause the observed downward band bending. The excess indium is \nremoved by oxidation , and the respective surface states and downward band \nbending disappear. The reason for the slight upward band bending , which \nindicates an ele ctron depletion layer at the surface , is unclear. As it is very \nsmall , it might be  only an artifact caused by  uncertainties in determin ing the  \nbulk Fermi level. Assuming a lower carrier concentration by a factor  of 10 (as  18 the (1 ± 0.2) × 1012 cm-3 are an u pper limit) , the upward bend bending would \nalmost disappear.  \n4 Conclusions  \nThe bulk and surface properties of In 2O3(001) single crystals have been \ninvestigated. The flux -grown samples permitted an accurate determination of \nthe bulk lattice constant with XRD.  Residues of Pb, Mg and Pt were found in \nthe In 2O3 crystals, which are characterized by a low carrier concentration of ≲ \n1012 cm-3.  \nSputtering and annealing in UHV up to a maximum temperature of 500 °C \n(higher temperatures cause an irreversible orange discoloration) likely \nproduces indium -terminated surfaces. These are characterized by low -lying \ngap states at ~2.4 eV and ~1.8 eV located at the surface. The position of the \nVBM at a binding energy of 2.9 eV implies a strong downward band -bending \nby 0.4 eV. A pronounced surface state was observed in STS. STM shows the \npresence of wide terraces that are separated by 5  Å high s tep edges. The \nterraces are covered with 2.5  Å high regions with bright, atomic -sized features \nin a square pattern aligned with the [110] direction. A possible interpretation is \nthat the terraces consist of an M -layer with small “patches” of indium atoms o f \nthe D -layer.  \nWhen the surfaces are exposed to activated oxygen, the electronic structure \nchanges. The low -lying band gap states disappear and the downwards band -\nbending is removed or even slightly reversed. A shoulder at the O 1s core \nlevel indicates th e presence of peroxo species  but probably not a completely \ndimerized termination .  \n \n \n \n5 Appendix  \nDetails of the XRD refinement are given in the Table A.1. Atomic \ndisplacement parameters can be found i n Table A.2 ; inter-atomic distances in \nTable A.3.  \n \nRefinement on F²  18 parameters  \nR[F² > 2σ(F²)] = 0.016  Weighting scheme based on measured s.u.'s 1/ σ²  \nwR(F²) = 0.017  (Δ/σ)max = 0.20  \nS = 3.61  Δρmax = 2.66 e Å−3  \n910 reflections  Δρmin = −2.77 e Å−3  \nTable A.1: Refinement parameters.  \nSite Wyckoff \nSymbol  U11 U22 U33 U12 U13 U23 \nIn1 8b 0.004290(9)  0.004290(9)  0.004290(9)  0.000580(5)  0.000580(5)  0.000580(5)  \nIn2 24d 0.003971(11)  0.004030(10)  0.004162(10)  0 0 0.000542(5)   19 O 48e 0.00615(5)  0.00592(5)  0.00511(5)  -0.00046(4)  -0.00096(4)  -0.00027(4)  \nTable  A.2: Atomic displacement components.  \nIn1—O1i 2.1734 ( 2) In2—O1 3.8167 (3)  \nIn1—O1ii 2.1734 (2)  In2—O1iii 3.8167 (3)  \nIn1—O1 2.1734 (2)  In2—In2xii 3.8392 (1)  \nIn1—O1iii 2.1734 (2)  In2—In2 3.8392 (1)  \nIn1—O1iv 2.1734 (2)  In2—In2 3.8392 (1)  \nIn1—O1v 2.1734 (2)  In2—In2 3.8392 (1)  \nIn1—In2vi 3.3445 (1)  In2—O1 4.0096 (3)  \nIn1—In2vii 3.3445 (1)  In2—O1 4.0096 (3)  \nIn1—In2viii 3.3445 (1)  In2—O1i 4.1535 (3)  \nIn1—In2i 3.3445 (1)  In2—O1xiii 4.1535 (3)  \nIn1—In2ii 3.3445 (1)  In2—O1 4.2583 (3)  \nIn1—In2 3.3445 (1)  In2—O1 4.2583 (3)  \nIn1—In2 3.8241 (1)  In2—O1 4.2766 (3)  \nIn1—In2 3.8241 (1)  In2—O1 4.2766 (3)  \nIn1—In2 3.8241 (1)  In2—O1 4.2912 (3)  \nIn1—In2 3.8241 (1)  In2—O1 4.2912 (3)  \nIn1—In2 3.8241 (1)  In2—O1vi 4.3265 (3)  \nIn1—In2 3.8241 (1)  In2—O1 4.3265 (3)  \nIn1—O1 3.9926 (3)  In2—O1vii 4.5346 (3)  \nIn1—O1 3.9926 (3)  In2—O1xiv 4.5346 (3)  \nIn1—O1 3.9926 (3)  In2—O1xv 4.7559 (3)  \nIn1—O1 3.9926 (3)  In2—O1xvi 4.7559 (3)  \nIn1—O1 3.9926 (3)  In2—O1 4.9599 (3)  \nIn1—O1 3.9926 (3)  In2—O1 4.9599 (3)  \nIn1—O1 4.1002 (3)  O1—O1 2.7986 (3)  \nIn1—O1 4.1002 (3)  O1—O1 2.7986 (3)  \nIn1—O1 4.1002 (3)  O1—O1xvii 2.8154 (3)  \nIn1—O1 4.1002 (3)  O1—O1vii 2.8154 (3)  \nIn1—O1 4.1002 (3)  O1—O1 2.9387 (3)  \nIn1—O1 4.1002 (3)  O1—O1i 3.2856 (3)  \nIn1—O1vi 4.5334 (3)  O1—O1 3.2856 (3)  \nIn1—O1vii 4.5334 (3)  O1—O1 3.3119 (3)  \nIn1—O1viii 4.5334 (3)  O1—O1 3.3119 (3)  \nIn1—O1ix 4.5334 (3)  O1—O1 3.5488 (3)  \nIn1—O1x 4.5334 (3)  O1—O1 3.5488 (3)  \nIn1—O1xi 4.5334 (3)  O1—O1 3.7116 (3)  \nIn2—O1 2.1249 (2)  O1—O1 4.1121 (3)  \nIn2—O1 2.1249 (2)  O1—O1vi 4.3137 (3)  \nIn2—O1 2.1952 (2)  O1—O1 4.3137 (3)  \nIn2—O1xi 2.1952 (2)  O1—O1xv 4.3468 (3)  \nIn2—O1 2.2236 (2)  O1—O1 4.4732 (3)  \nIn2—O1iv 2.2236 (2)  O1—O1iv 4.4732 (3)  \nIn2—In2 3.3617 (1)  O1—O1 4.5196 (3)  \nIn2—In2vi 3.3617 (1)  O1—O1 4.6173 (4)  \nIn2—In2 3.3617 (1)  O1—O1ii 4.6173 (4)  \nIn2—In2vii 3.3617 (1)      \nTable A.3: Interatomic distances.  \nSymmetry codes: (i) z, x, y; (ii) y, z, x; (iii) − z+1/2, − x+1/2, − y+1/2; (iv) − y+1/2, − z+1/2, − x+1/2; (v) − x+1/2, − y+1/2, \n−z+1/2; (vi) z, −x+1/2, y+1/2; (vii) y+1/2, z, −x+1/2; (viii) − x+1/2, y+1/2, z; (ix) − z+1/2, x, −y; (x) − y, ��z+1/2, x; (xi) x, \n−y, −z+1/2; (xii) z+1/2, − x+1/2, − y; (xiii) z, −x, −y+1/2; (xiv) y+1/2, − z, x; (xv) x+1/2, y, −z+1/2; (xvi) x+1/2, − y, z; (xvii) \nz+1/2, x, −y+1/2.  \n \n \n \n6 Acknowledgments  \nThis work was supported in part by the Austrian Science Fund (FWF; project \nF45) and the U.S. Department of Energy. Research at the Oak Ridge National \nLaboratory for one author (LAB) is sponsored by the U.S. Department of  20 Energy, Basic Energy Sciences, Materials Sciences and Engineering Division.   \nUseful discu ssions with Péter Ágoston und Karsten Albe ar e gratefully \nacknowledged .  \n \n \n7 Bibliography  \n1 Aron Walsh and C. Catlow, Journal of Materials Chemistry 20, 10438 \n(2010).  \n2 T. Bielz, H. Lorenz, W. Jochum, R. Kaindl, F. Kl auser, B. Kloetzer, and S. \nPenner, The Journal of Physical Chemistry C 114, 9022 –9029 (2010).  \n3 G. Korotcenkov, V. Brinzari, A. Cornet, and J. Morante, Sensors and \nActuators B: Chemical 106, 563 -571 (2005).  \n4 P.D.C. King, T. Veal, D. Payne, A. Bourlange, R . Egdell, and C. McConville, \nPhys. Rev. Lett 101, 116808 (2008).  \n5 Christoph Janowitz, Valentina Scherer, Mansour Mohamed, Alica Krapf, \nHelmut Dwelk, Recardo Manzke, Zbigniew Galazka, Reinhard Uecker, Klaus \nIrmscher, Roberto Fornari, Marcel Michling, Diete r Schmeißer, Justin R \nWeber, Joel B Varley, and Chris G Van De Walle, New Journal of Physics 13, \n085014 (2011).  \n6 R. L. Weiher, Journal of Applied Physics 37, 299 (1966).  \n7 P.D.C. King, T. Veal, F. Fuchs, and C.Y. Wang, Physical Review B 79, \n205211 (2009).  \n8 Erie H. Morales, Yunbin He, Mykola Vinnichenko, Bernard Delley, and Ulrike \nDiebold, New Journal of Physics 10, 125030 (2008).  \n9 Erie H. Morales and Ulrike Diebold, Applied Physics Letters 95, 253105 \n(2009).  \n10 A. Bourlange, D.J. Payne, R.G. Palgrave, J. S. Foord, R.G. Egdell, R.M.J. \nJacobs, A. Schertel, J.L. Hutchison, and P.J. Dobson, Thin Solid Films 517, \n4286 -4294 (2009).  \n11 J. De Wit, Journal of Solid State Chemistry 13, 192 -200 (1975).   21 12 Peter Ágoston, Paul Erhart, Andreas Klein, and Karsten Albe, J ournal of \nPhysics: Condensed Matter 21, 455801 (2009).  \n13 Peter Agoston and Karsten Albe, Physical Review B 81, 195205 (2010).  \n14 M. Marezio, Acta Crystallographica 20, 723 -728 (1966).  \n15 P.W. Tasker, Journal of Physics C: Solid State Physics 12, 4977 (197 9). \n16 Jacek Goniakowski, Fabio Finocchi, and Claudine Noguera, Reports on \nProgress in Physics 71, 016501 (2008).  \n17 Peter Agoston and Karsten Albe, Physical Review B 84, 045311 (2011).  \n18 S. R. Hall, H. D. Flack, and Stewart J. M., Xtal 3.2 Reference Manu al and \nUser’s Guide  (Universities of Western Australia, Geneva and Maryland, \n1992).  \n19 L. E. Halliburton, N. C. Giles, N. Y. Garces, Ming Luo, Chunchuan Xu, Lihai \nBai, and L. A. Boatner, Applied Physics Letters 87, 172108 (2005).  \n20 R. Nyholm, J. N. Anders en, U. Johansson, B. N. Jensen, and I. Lindau, \nNuclear Instruments and Methods in Physics Research 468, 520 -524 (2001).  \n21 J. J. Yeh and I. Lindau, Atomic Data and Nuclear Data Tables 32, 1-155 \n(1985).  \n22 S. A. Chambers, T. Droubay, T. C. Kaspar, and M. Gu towski, Journal of \nVacuum Science & Technology B: Microelectronics and Nanometer \nStructures 22, 2205 (2004).  \n23 Chenggang Zhou, Jiaye Li, Su Chen, Jinping Wu, Kevin R. Heier, and \nHansong Cheng, Journal of Physical Chemistry C 112, 14015 -14020 (2008).  \n24 Y. Gassenbauer, R Schafranek, Andreas Klein, S. Zafeiratos, M. Hävecker, \nA. Knop -Gericke, and R. Schlögl, Physical Review B 73, 245312 (2006).  \n25 Olga Dulub, Ulrike Diebold, and G. Kresse, Physical Review Letters 90, 1-4 \n(2003).  \n26 Charles Kittel, Einführung  in Die Festkörperphysik , 8th ed. (Oldenbourg, \n1989).  \n27 J. De Wit, Journal of Solid State Chemistry 20, 143 -148 (1977).   22 28 Peter Y. Yu and M. Cardona, Fundamentals of Semiconductors , 4th ed. \n(Springer, 2010).  \n29 J. R. Bellingham, W. A. Phillips, and C. J.  Adkins, Journal of Physics: \nCondensed Matter 2, 6207 (1990).  \n30 I. Hamberg and C.G. Granqvist, Journal of Applied Physics 60, R123 –R160 \n(1986).  \n31 F. Fuchs and F. Bechstedt, Physical Review B 77, 155107 (2008).  \n32 Andreas Klein, Applied Physics Letters 77, 2009 -2011 (2000).  \n33 Winfried Mönch, Semiconductor Surfaces and Interfaces , 1st ed. (Springer, \n1993).  \n \n "    },    {        "title": "2008.02237v1.Effects_of_Sr_doping_on_the_electronic_and_spin_state_properties_of_infinite_layer_nickelates.pdf",        "content": "E\u000bects of Sr-doping on the electronic and spin-state properties of in\fnite-layer\nnickelates\nJyoti Krishna,1Harrison LaBollita,1Adolfo O. Fumega,2, 3Victor Pardo,2, 3and Antia S. Botana1,\u0003\n1Department of Physics, Arizona State University, Tempe, AZ 85287, USA\n2Departamento de Fisica Aplicada, Universidade de Santiago de Compostela, Santiago de Compostela E-15782, Spain\n3Instituto de Investigacions Tecnoloxicas,Universidade de Santiago de Compostela, Santiago de Compostela E-15782, Spain\n(Dated: August 6, 2020)\nThe recent discovery of high-T csuperconductivity (HTS) in Sr-doped NdNiO 2has sparked a\nrenewed interest in investigating nickelates as cuprate counterparts. Parent cuprates [Cu2+: d9] are\nantiferromagnetic charge transfer insulators with the involvement of a single dx2\u0000y2band around the\nFermi level and strong p\u0000dhybridization. In contrast, isoelectronic NdNiO 2[Ni+: d9] is metallic\nwith a dx2\u0000y2band self-doped by Nd-d states. Using \frst principles calculations, we study the\ne\u000bect of Sr-doping in the electronic and magnetic properties of in\fnite-layer nickelates as well as\nthe nature of the holes. We \fnd that hole doping tends to make the material more cuprate-like as it\nminimizes the self-doping e\u000bect, it enhances the p\u0000dhybridization, and it produces low-spin (S=0,\nnon-magnetic) Ni2+dopants in analogy with the S=0 Zhang-Rice singlets that appear in cuprates.\nI. BACKGROUND\nA possible route to address the origin of high-\ntemperature superconductivity (HTS) is to \fnd cuprate\nanalog families, which might help unveil what is relevant\nfor HTS1. One plausible strategy to \fnd cuprate analogs\nis to replace2Cu2+with isoelectronic Ni1+: d9. This\noxidation state formally takes place in in\fnite layered\nnickelates RNiO 2(R= La, Nd)3{7. After 30 years of try-\ning, Sr-doped NdNiO 2has recently been reported to be\na superconductor with T c\u001815 K8and a dome-shaped\ndoping dependence9.\nAs discussed in the literature, the parent phase of 112\nnickelates (at d9\flling) is quite di\u000berent from that of\ncuprates. Experimentally, transport data indicate that\nRNiO 2materials are not insulating and there is no ex-\nperimental evidence for antiferromagnetic order in any\nRNiO 2material3{6. In addition, electronic-structure cal-\nculations of RNiO 2indicate signi\fcant di\u000berences from\ncuprates due to the presence of low-lying R-5d states\ncrossing the Fermi level that e\u000bectively self-dope the\ndx2\u0000y2band so that the formal electron count for the\nNi becomes d8+\u000e. Also, because Ni is to the left of Cu\nin the periodic table, it has an increased charge-transfer\nenergy10{14(twice as large than a prototypical cuprate\nvalue). However, both in cuprates and in\fnite-layer nick-\nelates superconductivity appears upon hole doping, so a\nrelevant question to address is how do the above param-\neters change upon doping as well as the nature of holes.\nIn cuprates, one starts with a parent phase in which\nthe Cu2+ions have one active d x2\u0000y2orbital hybridized\nwith theporbitals of the neighboring in-plane oxygens15.\nWhen such a system is doped by holes, one would for-\nmally create states Cu3+(d8) from the initial Cu2+(d9).\nHowever, Cu3+is a state with negative charge-transfer\ngap so the situation is e\u000bectively di\u000berent and holes pre-\ndominantly go to the O-p orbitals, that is, the state\nCu2+(d9)L(where L stands for a ligand hole, e.g., hole\non oxygen). This allows the formation of Zhang-Rice\nFIG. 1. Schematic representation of the energy level dia-\ngrams for high-spin and low-spin states of Ni2+(d8)(left) and\nNi2+\u000e(d8+\u000e)(right) ion in a square-planar environment. t 2gla-\nbel refers to non-degenerate xy,xz,yzorbitals for simplicity.\nGreen arrows refer to a partial occupation of that particular\norbital.\nsinglets16. As mentioned above, in nickelates the charge-\ntransfer energy is much larger11so one may expect Ni2+\nto be dominant with holes residing on the Ni, rather\nthan in the O-p band. Ni2+can be in two di\u000berent spin\nstates, as depicted in Fig. 1: i) A low-spin state (LS)\nif the electron from the higher-lying d x2\u0000y2orbital is re-\nmoved, in which case the resulting con\fguration is (t 2g)6\n(dz2)2with S=0; ii) a high-spin (HS) state if the elec-\ntron from the d z2orbital is removed, in which case the\nresulting con\fguration is (t 2g)6(dz2)1(dx2\u0000y2)1with S\n= 117,18. It has recently been argued that hole-doping\nNd112 should produce Ni2+with spin S = 119{21. Here,\nwe \fnd from \frst-principles calculations that include ex-\nplicit Sr-doping in RNiO 2(R= La, Nd) supercells, that a\nlow-spin state (S=0) is preferred instead due to the large\ncrystal \feld splitting of the e gstates in a square pla-\nnar environment. This is consistent with what is found\nin other square planar layered nickelates17,22{25. We also\n\fnd that upon hole doping, the electronic structure of in-\n\fnite layer nickelates becomes more cuprate-like with re-arXiv:2008.02237v1  [cond-mat.supr-con]  5 Aug 20202\nFIG. 2. Crystal structure of undoped RNiO 2(R=La, Nd)\n(a) and Sr-doped RNiO 2supercells of di\u000berent size 2 \u00022 (b),\n3\u00023 (c), and 4 \u00024, corresponding, respectively, to average Ni-\nd \fllings: 8.75, 8.89 and 8.75. R atoms in green, Sr-atoms in\nblue, oxygen atoms in red, Ni atoms in gray. Note that (d)\ncorresponds to a particular type of 25% Sr-doping, where all\nthe Sr dopants are clustered around a particular Ni cation,\nthat one can study as the closest possible approximation to a\nnominally Ni2+impurity.\nduced charge transfer energy and suppressed self-doping.\nII. COMPUTATIONAL METHODS\nWe have performed calculations in both LaNiO 2(with\na= 3.96 \u0017A,c= 3.37 \u0017A), and NdNiO 2(a= 3.92 \u0017A,c= 3.28\n\u0017A), two experimentally available members in\fnite-layer\nnickelates, in order to check for chemical pressure e\u000bects.\nSupercells of size 2 \u00022, 3\u00023 and 4 \u00024 relative to the prim-\nitive P4/mmm cell were employed to study the e\u000bect of\n11% and 25% Sr doping in both LaNiO 2and NdNiO 2.\nThey give rise to an average Ni- d\flling of 8.75, 8.89 and\n8.75, , respectively (with 4 Sr dopants in the unit cell clus-\ntered around a particular Ni atom in the latter, which be-\ncomes nominally a Ni2+). The corresponding structures\nare shown in Fig. 2. The structures of the Sr-doped\nsupercells were fully relaxed using the pseudo-potential\ncode Vienna ab-initio simulation package (VASP)26,27\nwithin Perdew-Burke-Ernzerhof version of the general-\nized gradient approximation (GGA)28. Electronic struc-\nture calculations were performed using the all-electron,\nfull potential code WIEN2k29based on the augmented\nplane wave plus local orbitals (APW + lo) basis set.The missing correlations beyond GGA at Ni sites were\ntaken into account through LDA+U calculations30. Two\nLDA+Uschemes were used: the `fully localized limit'\n(FLL) and the `around mean \feld' (AMF)31,32. For both\nschemes, we have studied the evolution of the electronic\nstructure with increasing U(UNi= 1.4 to 6 eV, J= 0.8\neV).\nIII. STRUCTURAL PROPERTIES\nThe structure of RNiO 2materials consists of NiO 2\nplanes (with 180\u000eO-Ni-O bond angles) that are sepa-\nrated by a single layer of R ions. The Sr-doped structures\nexperience a small increase in volume with respect to the\nparent materials \u00181-2 % for the levels of doping we are\nconsidering, up to 25%. This trend is the expected one\nas Sr2+is larger than La3+and Nd3+. Also, in the struc-\ntures containing Sr-doping, there is a slight deviation in\nbond-lengths and bond-angles after structural relaxation.\nIn particular, there is a distortion of the Ni-O distances\nin the NiO 2planes consisting of a modulation of the Ni-O\nbond length: shorter around the Ni ions closer to a 2+\nstate, those proximate to the Sr atoms (1.95- 1.96 \u0017A) and\nlonger around Ni ions closer to a 1+ state (1.98-1.99 \u0017A),\nthose away from the dopants. Finally, there is signi\fcant\nbuckling close to the Sr-substituted atom with Ni-O-Ni\nbond angles \u0018175\u000e.\nIV. ELECTRONIC STRUCTURE AND\nMAGNETISM\nUsing spin-polarized calculations, we have performed\na full study of the stability of di\u000berent spin states for\ndi\u000berent magnetic con\fgurations in the Sr-doped RNiO 2\nsupercells shown in Fig. 2 of size 2 \u00022, 3\u00023, and 4 \u00024\nwith e\u000bective d\fllings of d8:75d8:89, and clustered-d8:75,\nrespectively. In the smaller cells (2 \u00022, 3\u00023), two possi-\nble spin states can occur in Sr-doped RNiO 2. The (on\naverage) d8+\u000ecations sit in a square planar environment\nthat leads to a large splitting between the d x2\u0000y2and\ndz2bands. The crystal \feld splitting (\u0001 cf) within the e g\nstates can be then comparable to the Hund's rule cou-\npling (JH); if the former is larger, a low spin (LS) state\n(with S=\u000e/2 and a moment \u000eper nickel) develops and if\nthe latter is larger, a high spin (HS) state would be more\nstable (with S= (2- \u000e)/2 and a moment 2- \u000eper nickel).\nIn contrast, in a large enough cell, a d8con\fguration can\nbe stabilized for the Ni closer to the Sr dopants. In this\nsituation\u000ewould e\u000bectively be zero in the above descrip-\ntion so that a LS state with S=0 or a HS state with S=1\ncould be obtained for that speci\fc Ni atom surrounded\nby \frst neighbor Sr atoms. This distinction is depicted\nin Fig. 1 with the HS and LS states leading to drasti-\ncally di\u000berent properties: the former is non-cuprate like\n(in that the d z2states are the dominant ones around the\nFermi level) and the latter is more cuprate-like (in that3\nFIG. 3. Evolution of the lowest-energy spin state (either HS\nor LS states) with U for the two types of LDA+U schemes:\nFLL (fully localized limit) and AMF (around mean \feld).\nResults indicate that only at large U and only for the FLL\nfunctional, the HS state could be stabilized, otherwise the LS\nstate is more stable for both dopings studied for the La and\nNd cases.\nthedx2\u0000y2states dominate around the Fermi level). We\nnote that for either LS and HS d8+\u000ecases, an AFM cou-\npling between neighboring Ni atoms is always preferred,\nmediated by the close to half-\flling d x2\u0000y2band.\nThe energetics for these two di\u000berent spin states in\nthe di\u000berent supercells have been analyzed within two\ndi\u000berent LDA+U schemes: around mean \feld (AMF)\nand fully localized limit (FLL). There is a well-known\ntendency of these di\u000berent LDA+U \ravors to make LS\n(AMF) or HS (FLL) more stable, respectively33. This\nis a direct consequence of the double counting term in\nthe AMF scheme giving magnetic states a larger energy\npenalty than FLL does. In addition, we have taken into\nconsideration the possible e\u000bect of chemical pressure in\nthe stability of HS vs LS state by performing calculations\nfor both La112 and Nd112. We have also studied the\nevolution with U of the energy di\u000berences as it is also\nexpected that a large U would tend to favor the HS state.\nFor the 2 \u00022 and 3 \u00023 cells, the magnetic moments areconsistent with the above description in terms of Ni- d8+\u000e\nions with\u0016=(2-\u000e)\u0016Bfor the HS state and \u0016=\u000e\u0016Bfor\nthe LS state with \u000e\u00180.8-0.9 (see Table A.1 for more\ndetails). We note that within FLL, at low U values a\nHS state cannot be obtained, whereas at high U values\n(above 4.2 eV), a LS state cannot be converged. Within\nAMF, at low U values a HS state cannot be obtained\neither. These trends agree with the tendencies described\nabove for the di\u000berent LDA+U \ravors. Even though in\nthe smaller cells all Ni ions have the same moment (since\nthe number of inequivalent Ni ions is not enough for one\nof them to nominally achieve a d8con\fguration), they\ndo allow us to establish trends for the two di\u000berent spin\nstates as a function of U.\nFigure 3 summarizes the phase diagram of the two pos-\nsible (HS and LS) states as a function of U for AMF and\nFLL schemes, for both LaNiO 2and NdNiO 2at two dif-\nferent Sr doping levels- 25% (2 \u00022 supercell), and 11%\n( 3\u00023 supercell). Within the AMF LDA+U \ravor, the\nLS state is favored, independent of the chosen U value.\nThe FLL \ravor also favors the LS state at low values of\nU, whereas at high values of U the high-spin solution is\nfavored. These results can again be understood from the\nfact that the AMF scheme is known to favor the stabi-\nlization of low-spin con\fgurations, whereas FLL tends to\nfavor high-spin con\fgurations, as explained above. These\ntrends are consistent with earlier results on the trilayer\nmembers of the layered nickelate family17,22. Given that\nin square planar trilayer nickelates (with the Pr variant\nalso being metallic) the AMF scheme was deemed to pro-\nvide a better description of the electronic structure in\ncomparison to experiments, the same can likely be antic-\nipated for metallic in\fnite-layer systems so these results\nsuggest that indeed a LS state would be the most sta-\nble situation for these smaller cells. The only remarkable\ndi\u000berence observed upon a change in R in Fig. 3 is that\nfor NdNiO 2, the transition to a HS state within the FLL\nscheme occurs at a slightly larger U when compared to\nthe La material. This result is expected: the Nd-based\nunit cell is smaller, and the LS state tends to be typically\nfavored at smaller volumes22.\nUsing these AMF-LS results for 2 \u00022 and 3 \u00023 Sr-doped\ncells, at an e\u000bective U= 5 eV we analyze the evolution of\nthe La-d, O-p and Ni-d orbital-resolved densities of states\n(DOS) upon increasing Sr-doping (see Fig. 4). We show\nthe DOS of LaNiO 2but an identical picture is obtained\nfor its Nd-counterpart. In a simpli\fed picture, hole-\ndoping would simply shift down the Fermi level, but this\nis a self-doped system, and hence moving the La-d bands\nto higher energies would reduce the self-doping e\u000bect as a\nfunction of Sr doping. We describe how this mechanism\nworks analyzing the actual DOS plots in which the La-d\ncharacter at E Fcan be seen to decrease with doping, as\nthe La-d bands clearly shift to higher energies. This tends\nto reduce the self-doping e\u000bect mentioned above giving\na more pure single-band cuprate-like picture. Reinforc-\ning this picture, O-p states are pushed to higher energies\nwith doping. As the centroid of the Ni-d bands does not4\n-2-1012\n-4-2024DOS (states/eV)\n-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6\nE (eV)-4-2024undoped\n11% Sr doping\n25% Sr dopingO p (a)\n-1-0.500.5\n-2-1012DOS (states/eV)\n-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6\nE (eV)-2-1012undoped\n11% Sr doping\n25% Sr dopingLa d (b)\n-3-1.501.53\n-3-1.501.53DOS (states/eV)\n-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6\nE (eV)-3-1.501.53undoped\n11% Sr doping\n25% Sr dopingNi d (c)\nFIG. 4. Evolution of the orbital resolved DOS for O-p, La-d,\nand Ni-d states upon increasing Sr-doping in LaNiO 2. We\nsee that, as Sr doping is introduced, the centroid of the Ni-\nd bands does not signi\fcantly change while the O-p bands\nmove closer to the Fermi level, e\u000bectively reducing the charge-\ntransfer energy. The La d bands move away from the Fermi\nlevel, reducing the self-doping e\u000bect.\nget shifted noticeably, this reduces the p\u0000denergy split-\nting (charge-transfer energy) by more than 1 eV when\ngoing from the undoped compound to 25% Sr-doping.\nWe note that the e\u000bect on the Ni-d states is more com-\nplicated, since on the one hand, Sr introduces holes but\non the other, it reduces the self-doping e\u000bect from the La-\nd electron pocket that now moves away from the Fermi\nlevel. The overall result is a lower weight of the Ni d z2\nband around the Fermi level as Sr-doping is introduced,\nFIG. 5. DOS of the nominally Ni2+dopant in the 4 \u00024 su-\npercell with 25% Sr-doping in the two di\u000berent spin states\nstudied. HS (LS) con\fguration is shown in the top (bottom)\npanel. It can be noticed that the lower-energy LS con\fgura-\ntion leads to a depletion of Ni dz2states around the Fermi\nlevel.\nmaking the system more cuprate-like in that respect (see\nFig. A.1 for further details). All of the above described\ntrends imply that, as Sr dopants are introduced in RNiO 2\nmaterials, some of their electronic-structure features be-\ncome closer to those of the cuprates: low-spin dopant\nstates, reduced charge-transfer energy, and a single Ni\ndx2\u0000y2band around the Fermi level.\nWe now move to the 4 \u00024 supercells with an average\nd8:75\flling given by a 25% Sr substitution that allow,\nvia clustering of all the Sr dopants, for one Ni in the cell\nto be nominally d8(the Ni atom completely surrounded\nby \frst-neighbor Sr cations in Fig.2). In this scenario,\nwe \fnd that a LS (S=0) state is preferred in both RNiO 2\n(R= La, Nd), for the Ni ion surrounded by Sr atoms even\nwithin FLL- within AMF the low-spin solution is not only\nthe lowest in energy, but a high-spin solution does not5\neven exist, as attempts to start the self-consistency pro-\ncedure with a S=1 state of Ni2+ion lead to a vanishing\nmagnetic moment. The rest of the Ni atoms preserve the\nexpected magnetic moments (see Table A.1 for more de-\ntails). Figure 5 contrasts the orbital resolved density of\nstates for the nominally d8Ni cation (that is surrounded\nby Sr atoms) in both a LS state and a HS state in this\n4\u00024 supercell within FLL. We choose once again to\nshow calculations for Sr-doped LaNiO 2but the situation\nis identical in the Nd-material. In the low-spin state the\nt6\n2gd2\nz2con\fguration is clear with the two d x2\u0000y2or-\nbitals remaining unoccupied for both spin channels. In\nthe high-spin case, the t 2gorbitals are also completely oc-\ncupied, but now one electron occupies the majority spin\ndz2and dx2\u0000y2orbitals. The LS state for the Ni2+is\nstrongly connected to a reduction in the d z2character\naround the Fermi level. This can be seen in Fig. 5 where\nthe Ni dz2band crossing the Fermi level for the HS state\nbecomes fully occupied, well below the Fermi level for the\nLS state. This e\u000bect is concomitant with the reduction\nof the La-d self-doping e\u000bect upon increasing Sr-doping\ndescribed above. Overall, the stable LS state solution we\n\fnd gives then rise to an explicit cuprate-like scenario\nwith planes of S=1/2 ions that are lightly doped with\nmobile low-spin S=0 ions, a con\fguration that is directly\nanalogous to the low-spin S=0 Cu3+ion situation, medi-\nated by O-p holes as explained above. We note that in\nother intensively studied nickelates (such as in La 2NiO 4)\nthe high-spin (S=1) con\fguration of Ni2+is favored34,35\nand not the low-spin (S=0) con\fguration as we \fnd here.\nLa2NiO 4is structurally di\u000berent to RNiO 2as it preserves\napical oxygen atoms and has an octahedral environmentfor its Ni2+cations. We show here that if the Ni2+ions\nare forced into a square planar local environment as hap-\npens in the 112 materials, they prefer a low-spin d8(S=0)\ncuprate-like state instead.\nV. FINAL REMARKS\nUsing ab-initio calculations for Sr-doped RNiO 2(R=\nLa, Nd) systems we have shown that explicit Sr-doping\ngives rise to important changes in electronic structure\nand magnetic properties with respect to their undoped\ncounterparts: 1) It reduces the self-doping e\u000bect by shift-\ning the R-d bands up in energy away from the Fermi\nlevel leading to a more single-band-like picture, with the\nNi dx2\u0000y2band dominating. 2) It gives rise to low-spin\nNi2+dopants irrespective of the method used, doping\nlevel (11% or 25%) or rare-earth cation (both La and Nd\nyield comparable results). These low-spin dopants make\nthe situation similar to that in cuprates. 3) It drasti-\ncally reduces the charge-transfer energy (by up to 1 eV\nfor %25 doping) bringing it close to values observed in\ncuprates. Hence, our calculations show that the appear-\nance of superconductivity in in\fnite-layer nickelates upon\nSr-doping might be accompanied by a more cuprate-like\nelectronic structure and spin-states.\nACKNOWLEDGEMENTS\nThis work is supported by the MINECO of Spain\nthrough the project PGC2018-101334-B-C21. A.O.F.\nthanks MECD for the \fnancial support received through\nthe FPU grant FPU16/02572. AB and JK acknowledge\nASU for startup funds.\n\u0003antia.botana@asu.edu\n1M. R. Norman, Reports on Progress in Physics 79, 074502\n(2016).\n2V. I. Anisimov, D. Bukhvalov, and T. M. Rice, Phys. Rev.\nB59, 7901 (1999).\n3A. Ikeda, T. Manabe, and M. Naito, Physica C: Super-\nconductivity 495, 134 (2013).\n4A. Ikeda, Y. Krockenberger, H. Irie, M. Naito, and H. Ya-\nmamoto, Applied Physics Express 9, 061101 (2016).\n5M. Crespin, P. Levitz, and L. Gatineau, J. Chem. Soc.,\nFaraday Trans. 2 79, 1181 (1983).\n6M. A. Hayward, M. A. Green, M. J. Rosseinsky, and\nJ. Sloan, Journal of the American Chemical Society 121,\n8843 (1999).\n7M. Hayward and M. Rosseinsky, Solid State Sciences 5,\n839 (2003), international Conference on Inorganic Mate-\nrials 2002.\n8D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R.\nLee, Y. Cui, Y. Hikita, and H. Y. Hwang, Nature 572,\n624 (2019).\n9D. Li, B. Y. Wang, K. Lee, S. P. Harvey, M. Osada, B. H.\nGoodge, L. F. Kourkoutis, and H. Y. Hwang, Phys. Rev.\nLett. 125, 027001 (2020).10X. Wu, D. Di Sante, T. Schwemmer, W. Hanke, H. Y.\nHwang, S. Raghu, and R. Thomale, Phys. Rev. B 101,\n060504 (2020).\n11A. S. Botana and M. R. Norman, Phys. Rev. X 10, 011024\n(2020).\n12K.-W. Lee and W. E. Pickett, Phys. Rev. B 70, 165109\n(2004).\n13M.-Y. Choi, K.-W. Lee, and W. E. Pickett, Phys. Rev. B\n101, 020503 (2020).\n14M. Hepting, D. Li, C. J. Jia, H. Lu, E. Paris, Y. Tseng,\nX. Feng, M. Osada, E. Been, Y. Hikita, Y.-D. Chuang,\nZ. Hussain, K. J. Zhou, A. Nag, M. Garcia-Fernandez,\nM. Rossi, H. Y. Huang, D. J. Huang, Z. X. Shen,\nT. Schmitt, H. Y. Hwang, B. Moritz, J. Zaanen, T. P.\nDevereaux, and W. S. Lee, Nat. Mater. 19, 381 (2020).\n15D. I. Khomskii, Transition Metal Compounds (Cambridge\nUniversity Press, 2014).\n16F. Zhang and T. Rice, Phys. Rev. B 37, 3759 (1988).\n17J. Zhang, A. S. Botana, J. W. Freeland, D. Phelan,\nH. Zheng, V. Pardo, M. R. Norman, and J. F. Mitchell,\nNature Physics 13, 864 (2017).\n18H. Zhang, L. Jin, S. Wang, B. Xi, X. Shi, F. Ye, and J.-W.\nMei, Phys. Rev. Research 2, 013214 (2020).6\n19F. Lechermann, Phys. Rev. B 101, 081110 (2020).\n20F. Lechermann, arXiv , 2005.01166 (2020).\n21F. Petocchi, V. Christiansson, F. Nilsson, F. Aryasetiawan,\nand P. Werner, ArXiv , 2006.00394 (2020).\n22V. Pardo and W. E. Pickett, Phys. Rev. B 85, 045111\n(2012).\n23J. Zhang, Y.-S. Chen, D. Phelan, H. Zheng, M. Norman,\nand J. Mitchell, Proceedings of the National Academy of\nSciences 113, 8945 (2016).\n24V. Pardo and W. E. Pickett, Phys. Rev. Lett. 105, 266402\n(2010).\n25A. S. Botana, V. Pardo, W. E. Pickett, and M. R. Norman,\nPhys. Rev. B 94, 081105 (2016).\n26G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169\n(1996).\n27G. Kresse and J. Furthm uller, Computational materials\nscience 6, 15 (1996).\n28J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n29P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and\nJ. Luitz, Vienna University of Technology, Austria (2001).\n30V. I. Anisimov, F. Aryasetiawan, and A. Lichtenstein,\nJournal of Physics: Condensed Matter 9, 767 (1997).\n31V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T.\nCzy_ zyk, and G. A. Sawatzky, Phys. Rev. B 48, 16929\n(1993).\n32M. T. Czy_ zyk and G. A. Sawatzky, Phys. Rev. B 49, 14211\n(1994).\n33E. R. Ylvisaker, W. E. Pickett, and K. Koepernik, Phys.\nRev. B 79, 035103 (2009).\n34J. Rodriguez-Carvajal, M. Fernandez-Diaz, and J. Mar-\ntinez, Journal of Physics: Condensed Matter 3, 3215\n(1991).\n35D. Buttrey, J. Honig, and C. Rao, Journal of Solid State\nChemistry 64, 287 (1986).7\nAppendix A: Magnetic moments for all Sr-doped\nsupercells at di\u000berent U values and orbital resolved\nNi-d DOS for di\u000berent Sr-dopings\nFigure A.1 shows the orbital-resolved DOS for Ni\ndx2\u0000y2and dz2. As Sr-doping is introduced, the Ni d z2\nband moves away the Fermi level, leaving a dominant\ndx2\u0000y2contribution that makes Sr-doped 112 nickelates\na more cuprate-like, single-band system than their un-\ndoped counterparts.\n-1-0.500.51\n-4 -3 -2 -1 0 1 2 3 4\nE (eV)-1-0.500.51DOS (states/eV)-1-0.500.51undoped\n11% Sr doping\n25% Sr dopingNi-dz2\n-1-0.500.51\n-8 -6 -4 -2 0 2 4 6\nE (eV)-1-0.500.51DOS (states/eV)-1-0.500.51 undoped\n11% Sr doping\n25% Sr dopingNi-dx2-y2\nFIG. A.1. Evolution of the orbital resolved DOS for Ni-dz2\nand Ni-dx2\u0000y2states upon increasing Sr-doping in LaNiO 2,\nobtained using the AMF scheme with U= 5 eV.\nTable A.1 shows a summary of the magnetic moments\nobtained for the di\u000berent calculations at di\u000berent Sr-\ndoping concentrations and dopant con\fgurations, for all\nthe U values considered in the two LDA+U schemes uti-\nlized, as explained in the main text.TABLE A.1. Ni atomic magnetic moments (MM) (in \u0016B) for\nthe di\u000berent DFT+U methods used in this work, for di\u000berent\nU values for both HS and LS states. We denote by (-) a\nsolution that cannot be converged.\n2\u00022 La\nU-J (eV) DFT+U \ravor \u0016(HS) \u0016(LS)\n0 FLL (AMF) 0.40 (0.40) 0.50 (0.50)\n1.3 FLL (AMF) 1.02 (0.89) 0.76 (0.72)\n2.3 FLL (AMF) 1.16 (1.04) 0.79 (0.72)\n3.0 FLL (AMF) 1.21 (1.11) 0.83 (0.72)\n4.2 FLL (AMF) 1.25 (1.18) 1.23 (0.71)\n5.2 FLL (AMF) 1.28 (1.25) 1.38\n2\u00022 Nd\n0 FLL (AMF) - (-) 0.29 (0.29)\n1.4 FLL (AMF) - (-) 0.64 (0.60)\n2.7 FLL (AMF) - (-) 0.70 (0.62)\n4.1 FLL (AMF) - (-) 0.74 (0.61)\n5.4 FLL (AMF) - (-) 0.81 (0.58)\n6.8 FLL (AMF) 1.42 (-) - (0.29)\n8.2 FLL (AMF) 1.53 (-) - (0.20)\n3\u00023 La\n0 FLL (AMF) 0.63 (0.63) 0.36 (0.36)\n1.3 FLL (AMF) 0.83 (0.76) 0.84 (0.80)\n2.3 FLL (AMF) 0.93 (0.8) 0.87 (0.8)\n3.3 FLL (AMF) 1.15 (0.82) 0.91 (0.79)\n4.2 FLL (AMF) 1.24 (1.02) 0.96 (0.77)\n5.2 FLL (AMF) 1.30 (1.14) 1.00 (0.73)\n3\u00023 Nd\n0 FLL (AMF) - (-) 0.52 (0.53)\n1.4 FLL (AMF) - (-) 0.42 (0.51)\n2.7 FLL (AMF) - (-) 0.31 (0.50)\n4.1 FLL (AMF) 1.41 (-) - (0.58)\n5.4 FLL (AMF) 1.57 (-) - (0.80)\n6.8 FLL (AMF) 1.54 (1.19) - (0.85)\n8.2 FLL (AMF) 1.49 (1.51) - (1.10)\n4\u00024 La\n2.3 FLL (AMF) 1.10 (-) 0.74 (0.70)\n3.3 FLL (AMF) 1.20 (-) 0.85 (0.80)\n4.3 FLL (AMF) 1.25 (-) 0.95 (0.82)\n5.2 FLL (AMF) 1.37 (-) 1.10 (0.26)\n4\u00024 Nd\n0 FLL (AMF) - (-) 0.46 (0.46)\n1.3 FLL (AMF) 1.01 (-) 0.75 (0.71)\n2.3 FLL (AMF) 1.14 (1.02) 0.78 (0.72)\n3.0 FLL (AMF) 1.19 (1.09) 0.81 (0.72)\n4.2 FLL (AMF) 1.28 (1.16) 1.20 (0.70)\n5.4 FLL (AMF) 1.57 (1.19) - (0.79)\n6.8 FLL (AMF) 1.53 (1.19) - (0.85)"    },    {        "title": "2005.08327v2.Remarkable_low_energy_properties_of_the_pseudogapped_semimetal_Be__5_Pt.pdf",        "content": "Remarkable low-energy properties of the pseudogapped semimetal Be 5Pt\nLaura Fanfarillo,1, 2J. J. Hamlin,1R. G. Hennig,3, 4Ajinkya C. Hire,3, 4P. J. Hirschfeld,1\nJungsoo Kim,1Jinhyuk Lim,1Yundi Quan,1, 3, 4G. R. Stewart,1and Stephen R. Xie3, 4\n1Department of Physics, University of Florida, Gainesville, Florida 32611, USA\n2Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy\n3Department of Materials Science and Engineering,\nUniversity of Florida, Gainesville, Florida 32611, USA\n4Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA\n(Dated: October 27, 2020)\nWe report measurements and calculations on the properties of the intermetallic compound Be 5Pt.\nHigh-quality polycrystalline samples show a nearly constant temperature dependence of the electrical\nresistivity over a wide temperature range. On the other hand, relativistic electronic structure\ncalculations indicate the existence of a narrow pseudogap in the density of states arising from\naccidental approximate Dirac cones extremely close to the Fermi level. A small true gap of order \u0018\n3 meV is present at the Fermi level, yet the measured resistivity is nearly constant from low to room\ntemperature. We argue that this unexpected behavior can be understood by a cancellation of the\nenergy dependence of density of states and relaxation time due to disorder, and discuss a model for\nelectronic transport. With applied pressure, the resistivity becomes semiconducting, consistent with\ntheoretical calculations that show that the band gap increases with applied pressure. We further\ndiscuss the role of Be inclusions in the samples.\nI. INTRODUCTION\nIntermetallic compounds involving Be have not been\nthe subject of intensive study, in part because of the el-\nement's toxicity. However, several recent measurements\nin the Be-Pt series have displayed remarkable properties.\nBe21Pt5is a complex metallic alloy that crystallizes in\nthe cubic space group F\u001643m, where polyhedral 26-atom\nPt-Be clusters decorate four Wyco\u000b sites, resulting in a\n416-atom conventional unit cell. Unlike most complex\nmetallic alloys, this material displays superconductivity\nbelow a critical temperature of Tc= 2:06 K [1].\nBe5Pt, on the other hand, crystallizes in the same\nspace group but into a much simpler structure with only\n24 atoms per conventional unit cell. The electronic struc-\nture is usually reported as semiconducting, with a gap\nof 190 meV deduced from the high-temperature resis-\ntivity. This is consistent with the energy scale found\nwithin density functional theory (DFT) [2] correspond-\ning to the excitation of electrons from the top of the\nvalence band to a sharp peak in the conduction band.\nHowever, there is a general depletion of the density of\nstates over a wider range, decreasing down to the Fermi\nlevel, and it is di\u000ecult to judge from the calculations pre-\nsented in Ref. 2 whether a true gap exists at some lower\nenergy scale. Experimentally, the resistivity is metal-\nlic but with a nearly temperature-independent behavior\nover the entire measured temperature range from 3 K -\n300 K. Though this suggests that the system is a metal\nat the lowest energies, quantitative estimates are di\u000e-\ncult, as the samples contains a \flamentary secondary\nphase of Be metal. The low-temperature electrical re-\nsistivity of Be (6 :4\u000210\u00009µ\n cm) can be substantially\nsmaller than that of Cu (3 :4\u000210\u00007µ\n cm) [3]. The ex-\nistence or nonexistence of a true gap, and the origin of the\nextremely \rat temperature dependence of the resistivitybelow 100 K, remain puzzles.\nIn the pioneering work of Ref. 2, the intrinsic resis-\ntivity of Be 5Pt crystals, T-independent over a 200 K\nrange, was reported on samples tens of microns in size,\nand electronic structure calculations on a large energy\nscale were presented. Here we follow up these intriguing\nresults with a closer examination of the low-energy band\nstructure, an analysis of the unusual transport behav-\nior, and studies of the pressure dependence of electronic\nproperties up to 30 GPa.\nFirst, using an arc melting technique, we have grown\npolycrystalline samples which we characterize by x-ray\ndi\u000braction, resistivity, Hall e\u000bect, and speci\fc heat mea-\nsurements. We present basic sample characterization of\nour polycrystals in Sec. II.\nWe also have performed DFT calculations for this sys-\ntem, focusing in Sec. III on the low-energy band struc-\nture near the Fermi level. The unusual structure in the\nDOS turns out to be due to light-mass valence and con-\nduction bands, consisting of Pt dstates and Be pstates,\nthat nearly touch the Fermi level and exhibit a tiny indi-\nrect band gap of order only 3 meV. This makes Be 5Pt the\nonly binary, nontopological intermetallic system with cal-\nculated band gap signi\fcantly smaller than room temper-\nature [4]. Even more intriguingly the two bands near the\nFermi level provide a quasilinear DOS until nearby heav-\nier mass bands are reached; these determine the band gap\npreviously stated in the literature. Finally, at slightly\nhigher energies we note the existence of doubly degener-\nate Weyl loops within 100 meV of the Fermi level. This\nsuggests that electron doping with, e.g.relatively small\namounts of Au, could create a system with Weyl states\nat the Fermi level.\nNext, we provide in Sec. IV a simple model to explain\nthe low temperature transport, including the remarkable\nfact that the resistivity is nearly \rat in temperature,arXiv:2005.08327v2  [cond-mat.mtrl-sci]  26 Oct 20202\nwhereas the Hall coe\u000ecient decreases rapidly with de-\ncreasing temperature. The model relies on the anomalous\nscattering behavior of conduction electrons from static\ndisorder in a pseudogap state, analogous to Dirac met-\nals like graphene, although the quasi-Dirac point in this\nsystem is accidental in nature.\nFinally, in Sec. VI we present calculations and mea-\nsurements of Be 5Pt under pressure. DFT calculations\nin theF\u001643mstructure predict that the tiny gap near the\nFermi level further opens with pressure, with concomi-\ntant enhancement of resistivity and eventually true semi-\nconducting behavior. This is precisely what is observed\nin the measurements reported here. No structural transi-\ntions are anticipated theoretically, nor does the electrical\nresistivity data suggest such transitions up to 30 GPa.\nII. MATERIALS CHARACTERIZATION\nWe prepared polycrystalline samples of Be 5Pt follow-\ning a strategy guided by the phase diagram of Pt-Be,\nwhich shows the phases (starting from the Be-rich side of\nthe phase diagram) Be, Be 5Pt, and then possibly Be 21Pt5\n(or Be 4:2Pt) [1]. The Be 21Pt5phase is described [1] as\nsuperconducting (resistive onset) at Tc= 2:06 K. When\nwe prepared Be 5Pt via arc-melting, we therefore added\nextra Be (which has a high vapor pressure at the elevated\nmelting temperature) to compensate for the Be mass loss\nduring the melting process. As long as the sample re-\nmains Be-rich after arc-melting, the x-ray characteriza-\ntion shows single phase cubic Be 5Pt structure (Fig. 1),\nwithout detectable impurity phases\nFIG. 1. Measured and calculated x-ray pattern for Be 5Pt.\nThe smaller peak on the high angle side of the measured\nhigher angle lines is due to the incident Cu K-alpha xray\nradiation having two wavelengths (K-alpha1 and K-alpha2)\nwhile the calculated pattern in red is for a single incident\nwavelength. A small o\u000bset has been introduced to displace\nthe patterns.\nIf su\u000ecient Be is evaporated in the production processto produce a sample more Be poor than the 5:1 ratio\nin Be 5Pt, the x-ray pattern becomes much more com-\nplicated, and superconductivity (absent in the stoichio-\nmetric or super-stoichiometric Be 5Pt samples) with a Tc\nresistive onset of 2 :47 K is detected. We presume this is\nindicative of the presence of a minor amount of the Amon,\net al. Be21Pt5phase [1], although our Tconset is slightly\nhigher. In our typical preparations of Be-rich Be 5Pt, the\nexcess Be tends to form connected regions (see micro-\ngraph in Fig. 2), leading to a low temperature electrical\nresistivity signi\fcantly lower than `ideal' stoichiometric\nBe5Pt, where according to Amon et al. [2],\u001a0asT!0\nis\u0018360 m\n-cm. In Fig. 3, we show the resisitivity of\na sample of Be 5Pt, together with the low-temperature\nHall coe\u000ecient as a function of temperatures. Our sam-\nples of Be-rich Be 5Pt have\u001a0asT!0 between 0.45 and\n2.0 m\n-cm.\nThe presence of 5 percent second phase of Be threaded\nthrough the sample as illustrated in Fig. 2 is not an\nimpediment to determining the temperature dependence\nof the majority phase - see Fig. 3. Similarly, the tem-\nperature dependence of the Hall e\u000bect (inset to Fig. 3)\nand the pressure response of the resistivity (discussed be-\nlow in section VI) can also be determined. The essential\npoint is that the resistivity of Be[5, 6] is so much smaller\nthan the intrinsic resistance of Be 5Pt[2] that the mea-\nsured resistance of our composite sample, treated in a\nresistor network model, would be much smaller than our\nmeasured value of 2 m\n-cm if the Be percolated across\nthe sample.\n40 μm\nFIG. 2. 600x magni\fcation image of a slightly Be-rich ( \u00185%)\nBe5Pt sample showing Be second phase regions (blue/purple).\nIII. ELECTRONIC STRUCTURE\nA. Structure and Method\nBe5Pt crystallizes in the non-centrosymmetric space\ngroupF\u001643m(#216). There are four formula units per\nconventional cell. Pt, Be1 and Be2 sit at Wycko\u000b posi-\ntions 4a, 4c and 16e respectively. See Fig. 4. The lin-\nearized augmented plane wave method, as implemented3\nFIG. 3. Resistivity and low temperature Hall coe\u000ecient (in-\nsert).\nFIG. 4. The crystal structure of Be 5Pt is face-centered cu-\nbic. Be atoms sit at two non-equivalent Wycko\u000b positions\n16e (Be1) and 4c (Be2). Be1 form networks of corner sharing\ntetrahedra.[2] The Be1-Be1 bond length is 2.10 /RingA, while Be1-\nBe2 bond length is 2.48 /RingA. The distance between two nearest\nBe2 atoms is 4.22 /RingA.\nin WIEN2k[7], is used to carry out \frst-principles calcu-\nlations. The lattice constant of Be 5Pt is 5:975/RingA. The\ngeneralized gradient approximation (GGA) [8] is chosen\nas the exchange correlation functional. Relativistic ef-\nfects are included and spin-orbit coupling is treated with\nthe second variational method. Mu\u000en-tin radii of Pt\nand Be are 2.47 and 1.98 a.u., respectively. Rmin\nMTKmax,\nwhich determines the plane wave cuto\u000b in the interstitial\nregion, is kept at 7. To properly describe the electronic\nstructure of Be 5Pt near the Fermi level, we adopt a \u0000-\ncentered k-mesh with 11921 k-points in the irreducible\nBrillouin zone wedge. For the structure prediction calcu-\nlations, we used the Vienna Ab initio Simulation Package\n(VASP). The electronic structures and the band gaps of\nBe5Pt obtained using VASP and WIEN2k are similar.Site x y z\nPt 4a 0. 0. 0.\nBe1 16e 0.124 0.124 0.624\nBe2 4c 0.25 0.25 0.25\nTABLE I. Coordinates of Pt and Be under ambient pressure\nobtained from DFT structure relaxation.\nB. Results and discussions\nFIG. 5. (a) electronic structure (left) and density of states\n(right) of Be 5Pt from DFT. (b) electronic structure (left) and\ndensity of states of Be 5Pt within 0.4 eV of the Fermi level.\nDOSs are in units of states per eV per primitive cell. The two\nlowest conduction bands are linear along X!Wand they\ntouch at the X point (Weyl point).\nThe calculated density of states (DOS) of Be 5Pt has a\n\\pseudogap\" Epgof roughly 400meV, where the DOS is\nstrongly suppressed, and in addition, a true gap \u0001 of \u00183\nmeV (see Fig.5). Low temperature transport properties\nof Be 5Pt are therefore dominated by the states in close\nvicinity to the Fermi level, i.e.the states near the valence\nband maximum and the conduction band minimum. The\nvalence band maximum at (0, 0.28, 0.28) and its equiv-\nalent k-points are trivial M3type van Hove singularities\nwith light e\u000bective masses mx= 0:27me,my= 0:15me\nandmz= 0:17me. At theWpoint, the lowest two unoc-\ncupied states are split by 200 meV due to spin-orbit cou-\npling. In addition, dispersion of the lowest unoccupied\nconduction band near Wis strongly anisotropic, with rel-4\natively \rat dispersion along W!Xand a steep slope\nalongW!K.\nBe5Pt crystallizes in space group (#216) with broken\ninversion symmetry. The inversion symmetry breaking\nof the bulk Be 5Pt results in Dresselhaus spin-orbit cou-\npling (SOC) which lifts spin degeneracy at general k-\npoints in the Brillouin zone. Symmetry analysis of space\ngroup (#216) [9] reveals that SOC vanishes along \u0000 !X\npath, which our DFT calculations con\frm. Bands along\n\u0000!Xare therefore doubly degenerate, see Fig. 6 (b). In\nFig. 6 (a), the band dispersions of the two lowest conduc-\ntion bands near Xare shown, demonstrating the linear\ndispersion near the Weyl point at X.\nBecause the DOS has a sharp edge above the Fermi\nlevel, a small amount of electron doping can shift states\nnearWandXbelow the Fermi level. In Fig. 6, we\nplot the band dispersions of the two lowest conduction\nbands near the Xpoint on the kx\u0000kyplane. We have\nchosen (001) as the Xpoint. Interestingly, local band\ndispersions on kx\u0000kyplane are linear near X, which is\nin agreement with Ref. [10]. The e\u000bective Hamiltonian\nnearXcan therefore be written as\nHD=\f(kx\u001bx\u0000ky\u001by) (1)\n\u001bxand\u001byare Pauli matrices and kx=yare crystal mo-\nmenta along conventional reciprocal lattices. As a result\nof the linear Dresselhaus SOC, the expectation values of\nsxandsyare dependent on kxandky. We compute\nthe spin textures of the two lowest conduction bands us-\ning the WIEN2k code[11] and \fnd that the DFT spin\ntextures are exactly what would be obtained from the\nlinearizedHDin Eq. (1).\nFIG. 6. (a) DFT low energy dispersion of the two lowest un-\noccupied bands on the kx\u0000kyplane, i.e. the plane that's\nperpendicular to the \u0000 \u0000Xdirection. (b) band dispersion of\nthe two lowest unoccupied bands on the kx-kzplane. Dres-\nselhaus spin-orbit coupling vanishes along \u0000 \u0000Xwhich leaves\nthe two fold spin degeneracy intact. Spin textures of the low-\nest and the next lowest unoccupied bands are shown in (c)\nand (d).To understand the e\u000bect of pressure on the electronic\nstructure of Be 5Pt, we carry out DFT calculations of\nBe5Pt under 5;10;20;30;40;50;70;90;110;130 and\n150 GPa. Under ambient pressure, the indirect band gap\nof Be 5Pt is only 3 meV. As pressure increases from 0 to\n70 GPa, the indirect band gap goes up quickly to 78 meV.\nAbove 70 GPa, it starts to slowly decrease with higher\npressure. The pseudogap, on the other hand, increases\nwith pressure monotonically from 0.4 eV under ambient\npressure to 1.2 eV under 150 GPa.\nWe note that pressure has a dramatically di\u000berent im-\npact on occupied vs unoccupied states near the Fermi\nlevel. The occupied states within 0.25 eV of the Fermi\nlevel are insensitive to changes in pressure. For the unoc-\ncupied states within 1 eV above the Fermi level, pressure\nhas the e\u000bect of transferring a signi\fcant amount of spec-\ntral weight away from the Fermi level, thereby widening\nthe pseudogap.\nIt is interesting to consider what may happen to the\nproperties of this material with a small amount of elec-\ntron doping, e.g.with Au. First, we expect the electronic\nproperties to change rather rapidly, due to the steep rise\nin the calculated DFT density of states on the electron\ndoped side at ambient pressure. These calculations sug-\ngest that as little as 2% Au could change the density\nof states at the Fermi level by orders of magnitude. In\naddition, slightly more Au doping should raise the chem-\nical potential to the \u00180:1 eV level where the Weyl point\nwould be at the Fermi surface, assuming a rigid band\nshift. We plan to study this doping sequence in a subse-\nquent study.\nIV. INTERPRETATION OF TRANSPORT\nThe rather high value of the measured resistivity of our\nsamples is consistent with a semiconducting behavior, as\nclaimed in Ref. [2], but the temperature dependence of\nthe resistivity, constant over more than 100 K at low T,\nis not. Electronic structure calculations presented in Sec.\nIII (see Fig. 5) suggest, in fact, an asymmetric pseudogap\nin the density of states from about \u00000:35 eV to 0:05 eV,\nwith a peak in the unoccupied density of states at the\nupper edge created by a \rat band. Closer to the Fermi\nenergy, a tiny true indirect gap of about 3 meV opens.\nAt \frst glance, such a pseudogap/full-gap combination\nmight be expected to lead to a semiconducting tempera-\nture dependence of the DC transport, but we argue here\nthat this may be misleading.\nLet us assume that we can model the low energy\nphysics by an asymmetric, very slightly gapped V-shaped\ndensity of states with a tiny true gap, \u0001 = \u0001 e+\n\u0001h\u00183 meV, as illustrated in the schematic Figure\n8. We model the electronic structure relevant for low-\ntemperature transport with two Dirac-like bands: an\nelectron-like band, \u000fe\nk=\rekand a hole-like one, \u000fh\nk=\n\u0000\rhk, where kis measured with respect to the near-\ntouching point of the band and we assume \rh> \re.5\nFIG. 7. DFT calculations of (a) the indirect band gap (red)\nand the pseudogap (blue) of Be 5Pt versus pressure. (b) DOS\nof Be 5Pt under pressure.\nSuch a model yields to a density of state linear in energy\nN(\u000fk)\u0018\u000fksimilar to the band structure calculations.\nComparing with Fig. 8, the model discussed in this sec-\ntion corresponds to zero gap \u0001 \u000b= 0,\u000b=e;h, and zero\nchemical potential \u0016, but it can be easily generalized to\ninclude hole doping or gapping the band, both of which\nare discussed in the Appendix.\nFIG. 8. Cartoon of Be 5Pt band structure near Fermi level at\nambient pressure.\nElectronic transport in a typical metallic system with\nlarge nonzero density of states at the Fermi level may\nnormally be well-described by assuming a constant car-rier scattering time \u001c; this leads to a temperature-\nindependent resistivity and Hall coe\u000ecient. Both the\nband structure calculations and the experimental Hall\ndata for this system presented here suggest that this pic-\nture is incorrect. In fact, since the density of states is\nstrongly energy-dependent near the Fermi level, a con-\nstant relaxation time approximation is no longer valid.\nLet us assume that the transport at low temperatures is\ndominated by a small number of weakly scattering point-\nlike defects with concentration niand potential u. As\nshown in the Appendix the scattering rate from point-\nlike weak impurities for such a model is given by [12]\n1\n\u001c\u000b(!)= 2\u000600(!) = \u00000\n\u000bj!j; (2)\nwhere we introduced the dimensionless parameter \u00000\n\u000b=\nniu2=\r2\n\u000b, characterizing the scattering strength. Note\nthat we have set both the lattice parameter a= 1 and\nthe reduced Planck constant \u0016 h= 1 in this and subsequent\nexpressions.\nThe longitudinal conductivity is given by the sum over\nthe bands of the hole and electron contributions that are\ngiven by\n\u001b\u000b\nxx\u0018e2\n2\u0019\u00000\u000b; (3)\nFrom (3), we see that the T-dependence of the H= 0\nlongitudinal conductivity vanishes to leading order, as\nexpected. Note there is no universal term in the con-\nductivity, due to the neglect of self-consistency in the\nBorn approximation, which is relevant only over an ex-\nponentially small temperature range. The T-dependence\nof the conductivity from inelastic scattering will of course\ncontribute at su\u000eciently high temperatures, but this de-\npends on details of the system, such as Debye frequency,\nphonon distribution, etc. that we do not model here.\nThe analysis of the Hall coe\u000ecient requires the com-\nputation of the transverse component of the conductiv-\nity tensor,\u001bxy. Also in this case, the total conductivity\nis given by the sum of the electron and hole contribu-\ntions. As detailed in the Appendix and discussed in [13],\nthe cyclotronic frequency !cfor a Dirac-like electronic\nstructure is singular at zero energy, thus the denomina-\ntor (1 +!2\nc\u001c2) cannot be neglected even if the magnetic\n\feld is small. Details of the derivation can be found in\nthe Appendix, the \fnal result for the transverse conduc-\ntivity of the \u000b-band reads\n\u001b\u000b\nxy=\u0006e2\n2\u0019\u00000\u000b\u00142\n\u000bZ0\n\u00001d\u000f\u000f2\n\u000f4+\u00144\u000b\u0012\n\u0000@f\n@\u000f\u0013\n:(4)\nwhere the parameter \u0014\u000bhas the dimensions of energy and\nit is de\fned as \u00142\n\u000b=eHv2\n\u000b=\u00000\n\u000bwithv\u000b=\r\u000bthe band\nvelocity. The denominator determines the \u001bxy\u0018T2de-\npendence as T!0 and 1/T2asT!1 . The crossover\nbetween low and high Tbehaviors depends on the value\nof\u0014\u000b. Now we discuss combining the contributions of\nhole and electron bands. Notice that in a completely6\nsymmetric model for the Dirac bands, the system would\nbe compensated i.e. \u001bh\nxy=\u0000\u001be\nxyand the Hall coe\u000ecient\nwould be zero. However, in our model in order to repro-\nduce the particle-hole asymmetry of the density of states\nwe assume \rh>\re, as a consequence here \u0014h>\u0014e. The\nasymmetry of the Dirac bands of our model implies that\nthe total conductivity is dominated by the hole contri-\nbution since \rh> \re. This e\u000bect is enhanced by the\nT-dependence of the chemical potential, which moves to\nnegative values as Tincreases (see Appendix).\nDespite the fact that \u001bxyincreases with increasing T, a\nconsistent explanation of the Hall coe\u000ecient temperature\ndependence reported in Fig. 3 is di\u000ecult using this model\nalone. This is because the Hall coe\u000ecient is given by\nRH=\u001bxy\nH(\u001b2xx+\u001b2xy); (5)\nand theTdependence of RHdepends not only on \u001bxy,\nbut in principle also on \u001bxx(H) (see Eq.A6). Applying\nEq. (5) directly with the current model dominated by the\nhole band, gives an RHwith positive sign that increases\nasT!0 in contrast to the experiment. The strong\nTdependence of \u001bh\nxx(H)\u0018T4at lowTis at the root\nof this discrepancy. As discussed in the Appendix, the\ninclusion of a small gap in the model does not changes\nthe Hall coe\u000ecient T-behavior. However, if one assumes\nthe Fermi level lies a few meV inside the electron band,\nwe recover a non-negligible contribution of the electron\nband at low temperature. This e\u000bectively cuts o\u000b the\nsingular behavior of RHat low temperature and, within a\ncertain range of parameters, and can reproduce a RH(T)\nsimilar to the one experimentally observed as shown the\nAppendix.\nHowever this result requires \fne tuning, which is prob-\nably unreasonable given the other uncertainties in the\nanalysis. The main source is the existence of Be inclu-\nsions in the samples, as shown in Fig. 2. Although the\nprecise values of the resistivity of the Be inclusions in\nour samples are unknown, they clearly correspond to a\nconductivity much larger [5, 6] than the intrinsic Be 5Pt\nmaterial, in agreement with the conclusions of Ref. 2. If\nwe crudely model our samples as consisting of two paral-\nlel conducting channels, we expect that \u001b=\u001bBe+\u001bBe5Pt,\nwith\u001bBe>>\u001bBe5Pt. Note\u001bBeis the e\u000bective conductiv-\nity of the Be conducting network. Because \u001bBe5Pt\nxx is also\nT-independent and smaller than \u001bBe\nxx, we may neglect it\nentirely in the estimation of RH, and assume \u001bxx\u0019\u001bBe\nxx,\nquiteT-independent over 100K, and furthermore only\nweakly dependent on Hdue to its strong metallic char-\nacter. On the other hand, if \u001bBe\nxyis not too much larger\nthan\u001bBe5Pt\nxy , its temperature dependence will dominate\nthe extremely weak temperature dependence of \u001bBe\nxy. Fur-\nthermore, while the Hall coe\u000ecient of pure hcp Be metal\nhas a positive or negative sign according to the direc-\ntion of the magnetic \feld perpendicular to or parallel to\nthe basal plane[14], the former value is signi\fantly larger,\nand is expected to dominate in the Be inclusions present\nhere; we therefore assume that \u001bBe\nxyis constant in Tandhas a positive sign, like \u001bBe5Pt\nxy , which is dominated by\nthe light hole states. The temperature dependence of\nthe latter dominates, however, so we conclude that for\nour samples,\nRH\u0019\u001bBe\nxy+\u001bBe5Pt\nxy\nH(\u001bBexx)2\u00181 +cT2; (6)\nwherecis a positive constant. This result agrees qualita-\ntively with the measured Hall coe\u000ecient of the composite\nsample. If samples can eventually be prepared without\nthe Be inclusions, the Appendix contains more detailed\npredictions for the expected intrinsic behavior.\nV. CALCULATIONS UNDER PRESSURE\nWe used the Genetic Algorithm for Structure and\nPhase Prediction (GASP)[15, 16] to search for possible\nphase transitions in Be 5Pt under pressure at \fxed compo-\nsition. Two GASP searches were performed at 50 GPa\nand 150 GPa. The ambient pressure atomic structure,\nalong with 20 randomly generated structures, were used\nto initialize the GASP searches. GASP uses a genetic al-\ngorithm to perform global optimization, minimizing the\nenthalpy of formation in candidate structures. The algo-\nrithm iteratively creates `o\u000bspring' structures from `par-\nent' structures by using genetic operators like mutation\nor mating, corresponding to operations like adding and\nremoving atoms or splicing two structures, respectively.\nStructural features or `genes' which tend to lower the en-\nthalpy of a structure are promoted in later structures as\nthe algorithm progresses.\nVASP[17][18] was used to relax the structures and cal-\nculate the enthalpies. The cuto\u000b energy for the plane-\nwave basis set was set to 520 eV. We used the pro-\njector augmented wave (PAW) pseudopotentials[19] and\nthe Perdew-Burke-Ernzerhof (PBE) generalized gradi-\nent approximation[8] (GGA) for the exchange-correlation\nfunctional. A k-point density of 20 k-points per /RingA-1\nwas used for all the DFT relaxations. A stopping cri-\nterion of 600 DFT relaxations was used for both GASP\nsearches. Setting the maximum number of atoms in gen-\nerated structures limits the search space to structures\ncontaining 6 or 12 atoms. In a previous work by the\nauthors of GASP, it was noted that increasing the maxi-\nmum system size also exponentially increases the number\nof local minima in the energy landscape [20]. E\u000eciency\nis further reduced because individual energy calculations\nare much more expensive for larger structures. In an-\nother work by Lyakhov et al., it was noted that randomly-\ngenerated large cells often have very poor formation ener-\ngies; many are glass-like structures [21]. Based on these\npoints, we have enforced a maximum of 12 atoms per\ngenerated con\fguration, equivalent to two formula units,\nbased on available computational resources.\nFig. 9 shows the enthalpy of structures generated by\nGASP at 50 GPa and 150 GPa. During the GASP7\n50 150\nPressure (GPa)0.00.10.20.30.4H-Hr (eV/atom)\nFIG. 9. Enthalpy di\u000berence of structures generated in GASP\nat 50 GPa and 150 GPa with respect to the lowest enthalpy\nstructure, H r, shown in red. All of the low enthalpy structures\nhaveF\u001643mspacegroup. The inset shows the primitive lattice\nwith the lowest enthalpy.\nsearches, structures that were predicted to be more sta-\nble relaxed to structures with F\u001643mspacegroup, which\nis the same as that of the ambient condition ground-state\nstructure. This indicates that no structural phase tran-\nsition is seen in PtBe5 under pressure at 50 GPa and 150\nGPa. Fig 10 shows the enthalpy and volume as a function\nof the pressure across the lowest enthalpy structures from\nthe GASP run at 50GPa and 150GPa. Four of the \fve\nlowest enthalpy structures from both the GASP runs had\nthe same symmetry. The P63mcstructure shows almost\nno variation in enthalpy and volume with respect to the\nF\u001643mstructure because they are related through stack-\ning faults. The Pt atoms in F\u001643mform an ABC type\nclosed pack stacking whereas they form an ABA type\nclosed pack stacking in P63mc. The structure with F\u001643m\nspace group has the lowest enthalpy and thus the ambient\ncondition ground-state structure remains the most stable\nstructure up to 150 GPa.\nThe absence of pressure induced structural transitions\nis not surprising given that the ambient pressure struc-\ntures of both Pt and Be individually exhibit a remarkable\nstability under pressure. Platinum remains in the ambi-\nent pressure fcc structure to at least 304 GPa [22]. The\nstability of fcc Pt under pressure can be attributed to the\nfact that its neighbor to the right in the periodic table,\nAu, also adopts the fcc structure. Pressure is well known\nto induce a transfer of s-electrons into d-states in several\nelements. This s!dtransfer often leads to structural\ntransitions. As s!dtransfer occurs in Pt, it becomes\nmore Au-like. However, Au is also in the fcc structure, so\nno structural transition occurs [23]. Beryllium remains in\nthe ambient pressure HCP phase to at least 174 GPa [24].\nIn the heavier alkaline earth elements (Ca, Sr, and Ba),\npressure-induced s!dtransfer leads to the appearance\nof complex crystal structures at high pressure [25]. How-\never, Be has no nearby dstates. Lithium, adjacent to\nBe, also has no nearby dstates, yet exhibits a number ofcomplex crystal structures under relatively low pressures,\ndue to the overlap of the 1 s2core electrons. However, the\n1s2core of Be (1 :38 pm) is substantially smaller than that\nof Li (1:86 pm) [26], and hence core overlap will not begin\nto occur until much higher pressures than for Li.\nThe arguments above involve solids formed from the\nisolated elements Be and Pt. Of course, in a compound,\nthere could always be another structure that exhibits a\nsmaller volume of lower energy, hence decreasing the en-\nthalpy. However, the PtBe 5compound is a comparably\nclose packed. Therefore, a priori, we did not anticipate\na phase transformation and con\frmed this expectation\nby a genetic algorithm search. A concrete example of\nsuch a mechanism is the structural transition in Hume-\nRothery phases [27], driven by Fermi-sphere Brillouin-\nzone interactions, as well as pressure induced structural\ntransitions [28]. The basic idea is that when the Fermi\nsphere is close to the Brillouin zone boundary, a gap can\nopen, thus lowering the energy. A structural transition\ncan then occur if the transition brings additional portions\nof the Brillouin zone into close proximity with the Fermi\nsphere (thus further lowering the energy). Due to the\nlow carrier density of Be 5Pt, this mechanism is unlikely\nto be relevant, however.\nVI. PRESSURE DEPENDENCE OF\nRESISTIVITY\nFor the high-pressure resistivity measurements, a\nmicro-sized Be 5Pt sample (\u001840\u000240\u00025µm3) was\ncut from a larger piece of polycrystalline sample and\nplaced in a gas-membrane-driven diamond anvil cell\n(OmniDAC from Almax-EasyLab) along with a ruby ( \u0018\n10µm in diameter) for pressure calibration [29]. Two op-\nposing diamond anvils (0.15 and 0.5 mm central \rats)\nwere used, one of which was a designer-diamond anvil\n(0.15 mm central \rat) with six symmetrically deposited\ntungsten microprobes in the encapsulated high-quality-\nhomoepitaxial diamond [30]. A 316 stainless steel metal\ngasket was pre-indented from \u0018150 to 25 µm in thick-\nness with a hole ( \u001880µm in diameter), which was \flled\nwith soapstone (steatite) for insulating the sample and\nwhich also served as the pressure-transmitting medium.\nFor the resistivity calculation, we used the van der Pauw\nmethod, (assuming an isotropic sample in the measure-\nment plane), \u001a=\u0019tR/ln 2, where t is the sample thick-\nness (\u00185µm) with a current of 1 mA. The high-pressure\nresistivity cell was placed inside a customized continuous-\n\row cryostat (Oxford Instruments). A home-built opti-\ncal system attached to the bottom of the cryostat was\nused for the visual observation of the sample and for the\nmeasurement of the ruby manometer. The pressure was\napplied from\u00181 (initial loading) to 32 GPa slowly for \u0018\n5 hours for the pressure-dependent resistivity measure-\nment at room temperature and then released to each\npressure (Fig. 11) for the temperature-dependent resis-\ntivity measurements in the range from \u0018295 down to8\n0 25 50 75 100 125 150\nPressure (GPa)0.00.10.2H-H\nr (eV/atom)\n25 50 75 100 125 150\nPressure (GPa)0.00.10.2V-V\nr  (Å3\n/atom)  \nP6m2P6\n3mc P62m\nR32\nF43mP6\n3mc\nP62m\nR3mF43m R3m\n P6m2 R32\n(a) (b)\nFIG. 10. (a) Enthalpy and (b) volume relative to the F\u001643mambient condition ground state structure as a function of pressure\nof the lowest enthalpy structures from the GASP structure search at 50 GPa and 150 GPa.\n51 01 52 02 53 010203040500\n102030050100150Resistivity (mΩ cm)P\nressure (GPa)Be5PtU\nnloading2\n91 K\nρ (5 K)ρ\n (291 K)ρ\n (mΩ cm)P\n (GPa)\nFIG. 11. Resistivity of Be 5Pt versus pressure from \u001832 down\nto 2 GPa (unloading) at room temperature. Inset represents\nthe resistivity values at \fve di\u000berent pressures for 5 and 291\nK. Notice that pressure change due to change in temperature\n(291 down to 5 K) is relatively small, for example \u00181 GPa\nat 2.8 GPa and\u00182.6 GPa at 31 GPa, indicating the stability\nof the pressure cell. Bottom right inset shows the photograph\nof Be 5Pt sample (\u001840\u000240\u00025µm3) along with a ruby\nfor pressure calibration, a soapstone insulation (a bright area\nsurrounding the sample), a 316 stainless steel metal gasket\nwith a hole (\u001880µm in diameter), and six leads (tungsten)\ncon\fguration. Leads 3 and 4 were used to measure voltage\ndrop, whereas leads 2 and 5 were used to provide current \row.\n1.8 K (Fig. 12).\nFigure 11 shows room-temperature resistivity data un-\nder high pressure from \u001832 down to 2 GPa during de-\ncompression. The resistivity data during compression is\nnot shown because lead 1 (see Fig. 11 inset), used for\none of the current leads, had a short with the metal gas-\nket, which might include additional resistance in the data\nother than the sample. The leads for unloading pressure\nare without any short. It is known that resistivity ob-\n05 01 001502002503000204060801001201401601800\n.00360.00400.00442.53.03.54.0Resistivity (mΩ cm)T\nemperature (K)Be5PtC\nooling 31 GPa \n22 GPa \n11 GPa \n5.7 GPa \n2.8 GPa \n0 GPal\nnρ1\n/T (K-1)2782 502 27T (K)FIG. 12. Resistivity versus temperature curves (cooling) for\nBe5Pt at 31, 22, 11, 5.7, 2.8, and 0 GPa (measured at room\ntemperature). Data at 0 GPa is adopted from Fig. 3. Inset\nshows ln\u001aversus 1/Tplot to calculate the band gap energy\n(Eg) using the Arrhenius relation, ln \u001a(T) = ln\u001a0+Eg=2kBT.\nCyan lines are the linear \fts to the data from 0.0034 K-1\n(294 K) to 0.0044 K-1(227 K), where each slope gives the\ncorresponding Eg=2kBvalue.\ntained from decompression provides more accuracy than\nfrom compression because sample thickness is relatively\nwell-kept during decompression [31]. The resistivity of\nBe5Pt at room temperature increases under high pres-\nsures, which indicates the band gap energy ( Eg) also in-\ncreases according to the Arrhenius relation, \u001a(T) =\u001a0\nexp (Eg=2kBT) at a \fxed T. This is consistent with\nthe band structure calculations for greater than ambi-\nent pressure in Fig. 7, showing the band gap opens fur-\nther with increasing pressures to \u001827 GPa. The inset\nin Fig. 11 compares the resistivity at 5 and 291 K for\nselected pressures. The anomaly \u001823 GPa in Fig. 11,\nwhich shows pressure reversing, is from the adjustment of\npressure determination of the ruby manometer, whereas9\nthe discontinuities in the resistivity curve are due to\nthe changes in pressures before and after temperature-\ndependent resistivity measurements.\nThe temperature-dependent resistivity curves of Be 5Pt\nare shown in Fig. 12 for \fve unloading pressures at 31,\n22, 11, 5.7, and 2.8 GPa, including a separate ambient\npressure measurement (see Fig. 3). Remarkably, the re-\nsistivity curve at ambient pressure is nearly \rat over a\nwide temperature range compared with the high pressure\nresistivity curves, as discussed above. The resistivity at\n2.8 GPa begins to increase monotonically with decreas-\ning temperature, which corresponds to a typical semicon-\nducting behavior. With further increasing pressure, the\nincrease in temperature-dependent resistivity gets larger\nand larger (see Fig. 11 inset). This enhancement in the\nsemiconducting behavior under high pressures to 31 GPa\nis again in a good agreement with the DFT calculations\nof pressure e\u000bects on Be 5Pt in Sec. III. The \fts of the\nArrhenius equation to the high pressure resistivity curves\n(ln\u001aversus 1/Tplot) are shown in the Fig. 12 inset in the\ntemperature range from \u0018227 to 294 K to calculate the\nband gap energy ( Eg). The estimated Egvalues are\u0018\n35, 38, 39, 43, and 32 meV at 2.8, 5.7, 11, 22, and 31 GPa,\nrespectively, which shows a good qualitative agreement\nwith the calculated true gaps shown in Fig. 7.\nVII. CONCLUSIONS\nThrough a series of measurements and theoretical cal-\nculations, we have argued that the little-studied inter-\nmetallic semiconductor Be 5Pt is in fact an extremely\nunusual member of its materials class. While not super-\nconducting like its cousin Be 21Pt5, our electronic struc-\nture calculations suggest that at ambient pressure it is\nsemiconducting with one of the smallest gaps of any in-\ntermetallic, an indirect gap of order 3 meV according\nto DFT calculations we presented here. Insulating tem-\nperature dependence is notobserved in low-temperature\ntransport, however; rather the resistivity is \rat at ambi-\nent pressure over a range of roughly 100K, together with\na stronglyT-dependent Hall e\u000bect.\nThe calculated band structure near the Fermi level has\nhole and electron bands resembling Dirac cones whose\nextrema are accidentally nearly degenerate. In addition,\nthere is a Weyl loop structure quite close to the Fermi\nlevel, such that a small amount of doping might allow\nobservation of topological quantization. Here we studied\nthe transport in the quasi-Dirac cone bands and devel-\noped a model to explain the T-independent resistivity\ntogether with the sign and unusual Tdependence of the\nobserved Hall coe\u000ecient. Agreement with experiment\nrequires assumptions about the in\ruence of the Be inclu-\nsions found in current samples, however.\nCalculations were performed under pressure using a\ngenetic algorithm structural relaxation method, which\nfailed to \fnd any phase transitions from the ambient pres-\nsureF\u001643mstructure up to 150 GPa, consistent with re-sistivity measurements performed in a diamond anvil cell\nup to this pressure. However, the temperature depen-\ndence of the resistivity evolved signi\fcantly, consistent\nwith the opening of the gap found in DFT calculations.\nIn summary, we have performed a close investigation\nof the properties of the intermetallic semiconductor\nBe5Pt, and shown that it is a remarkable member\nof this class, primarily because of an extraordinarily\nsmall accidental gap, surrounded by a somewhat larger\npseudogap feature in the density of states. Additional\nunusual features of the band structure include Weyl\nloops close to the Fermi level; we have suggested that\ndoping the compound with a small amount of gold may\nallow one to bring this feature to the Fermi level.\nAcknowledgements. The authors are grateful for\nuseful discussions with Y. Grin and Young-Joon Song.\nWork on this project was supported by the US Depart-\nment of Energy Basic Energy Sciences under Contract\nNo. DE-SC-0020385. L. F. acknowledges support by the\nEuropean Union's Horizon 2020 research and innovation\nprogramme through the Marie Sk lodowska-Curie grant\nSuperCoop (Grant No 838526).\nAppendix A: Transport\nWe consider a two-band toy model with two Dirac-like\nbands. For the moment, we neglect the tiny band gap\npresent in the Be 5Pt system, i.e. we \fx \u0001 e= \u0001h= 0,\nand further assume that \u0016= 0 at zero temperature. The\nband dispersion is then \u000f\u000b=\u0006\r\u000bk, where\u000b=e;hand\nkis measured with respect to the touching point of the\nband. This will yield a linear density of states in en-\nergyN\u000b(\u000f) =j\u000fj=(\r2\n\u000b\u0019). To account for the particle-\nhole asymmetry of the density of states found in the\nband structure calculations for Be 5Pt we need \rh>\re.\nNote that because the Dirac cone is now asymmetric, the\nchemical potential \u0016depends signi\fcantly on Tand must\nbe calculated self-consistently.\nAssuming that the transport at low temperatures is\ndominated by a small number of weakly scattering point-\nlike defects with concentration niand potential uthe\nscattering rate reads [12]\n1\n\u001c\u000b(!)= 2\u000600(!) = 2niu2X\nkImG(k;!)\n=\u0019niu2Z\nd\u000f N\u000b(\u000f)\u000e(!+\u000f) = \u00000\n\u000bj!j;(A1)\nwhere we introduced the dimensionless parameter \u00000\n\u000b=\nniu2=\r2\n\u000b, characterizing the scattering strength. In this\nand subsequent expressions we have set both the lattice\nparametera= 1 and the reduced Planck constant \u0016 h= 1.\nThe longitudinal conductivity is given by the sum of\nthe electron and hole contribution \u001bxx=P\n\u000b\u001b\u000b\nxxwith10\n\u001b\u000b\nxx=e2v2\n\u000b\n2Zd2k\n(2\u0019)2\u001c\u000b(\u000fk)\u0012\n\u0000@f\n@\u000fk\u0013\n=e2v2\n\u000b\n2Z\nd\u000f N\u000b(\u000f)\u001c\u000b(\u000f)\u0012\n\u0000@f\n@\u000f\u0013\n=e2\u0015\u000b\n2\u0019\u00000\u000b; (A2)\nwherev\u000b=\u0006\r\u000bis the velocity for the \u000b=e;hband\nand\u0015\u000ba numerical factor coming from the evaluation of\nthe integral and that in our case is \u0015h>\u0015e. We recover\nthe result quoted in the main text that shows that the\nT-dependence of the longitudinal conductivity vanishes\nto leading order. The total conductivity is dominated\nby the contribution of the holes that is in our model\n\u00189 times larger than that of the electrons. This e\u000bect\nresults from the strong hole-electron anisotropy of the\nbands that forced us to assume \rh> \reand pushes the\nchemical potential to negative value as the temperature\nincreases.\nσxx  [105 (Ωm)-1 ] \nk = 3 K \nk = 10 K \nT(K)10 20 30 40 0σxy  [105 (Ωm)-1 ] \nT(K)10 20 30 40 0hole\nelectron\ntotk = 3 K \nk = 10 K 00.050.10\n00.050.100.020.04\n0\n0.020.04\n0\nFIG. 13. Temperature dependence of the hole and electron\nband contributions to \u001bxx(H) and\u001bxyfor\u0014e= 3;10 K.\nTo estimate the temperature independent term e2=(2\u0019\u00000\n\u000b),\nwe \fx \u00000\n\u000bsuch that the total longitudinal conductivity at\nat zero \feld, Eq.(A2), \fts the experimental value \u001bxx=\n0:5 105(\nm)\u00001. The anisotropy of band structure, encoded\nin the parameters \r2\nh\u00184\r2\ne, and in the temperature shift of\nthe chemical potential to negative values, makes the transport\ndominated by the hole band over the whole range of temper-\nature.\u00142\n\u000bacts as a cut-o\u000b for the divergence of the \u001b\u000b\nxy\nThe Hall coe\u000ecient in a two band system is given by\nRH=\u001bxy\nH(\u001b2xx+\u001b2xy); (A3)\nwhereHis the magnetic \feld, \u001bxx=P\n\u000b\u001b\u000b\nxxis the total\nlongitudinal conductivity and \u001bxy=P\n\u000b\u001b\u000b\nxythe trans-\nverse one. Following the same procedure as in Eq.(A2),the transverse conductivity for each band is given by[13]\n\u001b\u000b\nxy=\u0000e2v2\n\u000b\n2Z\nd\u000f N\u000b(\u000f)!\u000b\nc(\u000f)\u001c2\n\u000b(\u000f)\n1 + (!c\u000b(\u000f)\u001c\u000b(\u000f))2\u0012\n\u0000@f\n@\u000f\u0013\n=e2\n2\u0019\u00000\u000b\u00142\n\u000bZ0\n\u00001d\u000f\u000f2\n\u000f4+\u00144\u000b\u0012\n\u0000@f\n@\u000f\u0013\n: (A4)\nHere!\u000b\nc=eHv2\n\u000b=\u000fis the energy dependent cyclotron\nfrequency. It is negative for the hole band, \u000fh<0, and\npositive for the electron one, \u000fe>0. We further intro-\nduced the parameter \u0014\u000bsuch that!\u000b\nc(\u000f)\u001c\u000b(\u000f) =\u00142\n\u000b=\u000f2,\ni.e.\n\u00142\n\u000b=eHv2\n\u000b\n\u00000\u000b: (A5)\nSince 1=\u001c\u000b(\u000f) andN\u000b(\u000f) vary as\u000f, the integral without\nthe (!\u000b\nc\u001c\u000b)2term in the denominator is actually diver-\ngent as 1=\u000f2. Thus this term is required, unlike the or-\ndinary Hall e\u000bect for metallic systems, even in the low\n\feld limit, and determines the \u001b\u000b\nxy\u0018T2dependence as\nT!0. To determine the crossover between low and\nhigh-temperature behavior, we must estimate the value of\n\u0014\u000bfrom Eq.(A5). Notice that the T-behavior of the two\nbands change over di\u000berent energy scales as one can see\nin Fig.(13) where we show the temperature-dependence\nof the transverse conductivity together with the electron\nand hole contributions for two representative values of\n\u0014e. Here we used \rh\u00184\re, as extracted from the den-\nsity of states in Fig.(5) approximated as linear as shown\nin the schematic of Fig.(8). As a consequence we have\nthat\u0014h\u00184\u0014e, thus the low-high temperature crossover\nappears at lower temperature for the electron term with\nrespect to the hole term. The sign of the transvers con-\nductivity is positive due to the dominant contribution of\nthe hole band.\nSince in the evaluation of \u001bxywe account for the full\nmagnetic \feld dependence, i.e. we retain the full denom-\ninator in Eq.(A4), when computing the Hall coe\u000ecient\nvia Eq.(A3), we need to use the same approximation for\nthe \feld-dependent longitudinal conductivity \u001bxx, i.e.\n\u001b\u000b\nxx(H) =\u0000e2v2\n\u000b\n2Z\nd\u000f N\u000b(\u000f)\u001c(\u000f)\n1 +!c2(\u000f)\u001c2(\u000f)\u0012\n\u0000@f\n@\u000f\u0013\n=e2\n2\u0019\u00000\u000bZ0\n\u00001d\u000f\u000f4\n\u000f4+\u00144\u000b\u0012\n\u0000@f\n@\u000f\u0013\n(A6)\nwhose temperature-dependence is shown in Fig.(13 for\nthe same values of \u0014\u000bused in the \u001bxycalculation. It\nis evident that the total contribution of both the lon-\ngitudinal and transverse conductivities come essentially\nform the hole band regardless of the value of \u0014\u000b. The\nstrong anisotropy of the Dirac cone considered here re-\nduces the two-band model to a single hole-like band\nmodel. As a consequence, the Hall coe\u000ecient reduces\ntoRH\u0018\u001bh\nxy=(\u001bh\nxy)2+ (\u001bh\nxx)2, that regardless of the\nexact value of the \u0014\u000bparameter used, present a diver-\ngent behavior as T!0 is approached, see dotted lines\nin Fig.(14). The anomalous temperature dependence of11\n\u001bxx(H)\u0018T4at lowTis at the root of this behavior in\nthe current model.\nT(K)10 20 30 40 0k = 3.0 K k =1.5 K RH [a.u]\n0.10.20.3\n00.40.5\nFIG. 14. Temperature dependence of RHfor the two-band\nDirac model. Dotted lines: result obtained using \u0001 e= \u0001h=\n0,\u0016(T= 0) = 0 for \u0014e= 1:5;3:0 K. We \fnd that the transport\nis dominated by the hole band and we get a divergent RH(T)\nforT!0. Continuous lines: result obtained using \u0001 e=\n\u0001h= 0:5 meV and \u0016(T= 0) = 3:4 meV. If the chemical\npotential at T= 0 lies inside the electron band the divergence\nat low-T is suppressed by the contribution of the electron\nband and we get an RHthat increases as T is increased over\nthe 3 - 20 K range.\nThe inclusion of a small gap in the model, does not\nchange qualitatively the low-temperature behavior of the\nconductivities and of the Hall coe\u000ecient. This new en-\nergy scale could change the relative weight of the bands\nespecially if \u0014\u000bis of the same order of the gap. How-\never, this e\u000bect is small with respect the suppression of\nthe electron band contribution due to the strong tem-\nperature dependent chemical potential shift to negative\nvalues that makes the physics still dominated by the hole-\nlike band over the whole temperature range.\nAn e\u000bective parameter that crucially a\u000bects the tem-\nperature dependence of the Hall coe\u000ecient is, instead,\nthe doping. In fact, if we assume \u0016(T= 0)>0, such that\nthe Fermi level at zero temperature lays at the edge of\nthe electron band, or further inside the band, we recover\na not negligible contribution of the electron band at very\nlow T that e\u000bectively cuts o\u000b the divergence of the the\nHall coe\u000ecient at low temperature, see Fig.(14) where we\nshow the results for a speci\fc case with \u0001 e= \u0001h= 0:5\nmeV and\u0016(T= 0) = 3:4 meV for two di\u000berent values of\n\u0014e. As a consequence the Hall coe\u000ecient is now increas-\ning as T is increased up to \u001820 K as seen in experiment.\nNotice that the crossover between the divergent RH(T) of\nthe un-doped model to the RH(T) increasing with tem-\nperature of the doped case is controlled by the position\nof the Fermi level at zero temperature with respect the\nelectron band, thus it is also a\u000bected by the gap values.\nMoreover the values of \u0014\u000bused in the calculation in\ru-\nences the range of temperature over which RHincreases,\ni.e. the position of the peak as shown in Fig.(14). This\nresult thus requires \fne tuning, which is probably unrea-\nsonable given the other uncertainties in the analysis. In\nthe main text we explicitly discuss the possible role ofthe existence of Be inclusions in the samples.\nAppendix B: Pressure dependence of the lowest\nunoccupied DOS peak\nWe have shown in Fig. 7 (b) of the main text that\nthe \frst peak of the unoccupied DOS is strongly pres-\nsure dependent. To understand the origin of the pres-\nsure dependence, we plot the band dispersion of the two\nlowest conduction bands along X!Win Fig. 15. The\n\frst DOS peak above the Fermi level is the result of the\ntwo conduction bands (Pt 5d bands) touching at X. As\npressure increases, the two bands touch at progressively\nhigher energy. As a result, the DOS peak is pushed up\nand a signi\fcant amount of spectral weight is transferred\naway from the Fermi level.\nFIG. 15. In the \fgure above, the band dispersions of the two\nlowest unoccupied bands under eleven di\u000berent pressures are\nshown. The two fold degeneracy at Xponit gives rise to the\n\frst DOS peak above the Fermi level.\nAppendix C: Dependence of the bandgap on the\nexchange-correlation functional\nBecause the true spectral (indirect) gap of Be 5Pt\nof 3 meV found in the calculations presented is un-\nusually small, we have checked this result by calculat-\ning the same quantity using several exchange-correlation\nfunctionals, including LDA, PBE, the strongly con-\nstrained and appropriately-normed exchange-correlation\nfunctional (SCAN) [32, 33], as well as the hybrid func-\ntional HSE06 [34]. The predicted bandgaps shown in\nTab. II are consistently very small, ranging from 3 to\n22 meV, except for the HSE06 result of 297 meV. We\nnote further that in Ref. [2], a larger bandgap of 85 meV\nwas estimated from the DOS calculated using the FPLO12\nTABLE II. Calculated bandgaps in meV of Be 5Pt using\nseveral exchange-correlation functionals and the VASP and\nWien2k codes. For the hybrid functional HSE06, we vary the\namount of HF mixing from the default value of \u000b= 1=4 to a\nsmaller value of 1 =8. All results include spin-orbit coupling\ninteractions.\nLDA PBE SCAN HSE06 HSE06\n\u000b= 1=4\u000b= 1=8\nVASP 22 8 12 297 160\nWIEN2k 13 3 4 { {\ncode. It is well known that while HSE06, with the con-ventional choice of Hartree-Fock (HF) mixing coe\u000ecient\nof\u000b= 1=4, works well for molecular systems and sim-\nple semiconductors, it can signi\fcantly overestimate the\nbandgap in systems with large dielectric constants and\nsmall bandgaps [35]. In fact, from our Arrhenius anal-\nysis of the resistivity presented in Fig. 11, the larger\nHSE06 bandgap is inconsistent with the data, which sug-\ngest a gap of at most \u001830 meV. Reducing the amount\nof HF mixing brings the bandgap closer to the experi-\nmental value, indicating that Be 5Pt has a sizeable elec-\ntric susceptibility. We note further that the shape of the\nbandstructure and the density of states does not change\nsigni\fcantly among all these di\u000berent functionals, such\nthat the analysis presented here is expected to be robust\nagainst small errors in the gap value.\n[1] A. Amon, A. Ormeci, M. Bobnar, L. G. Akselrud,\nM. Avdeev, R. Gumeniuk, U. Burkhardt, Y. Prots,\nC. Hennig, A. Leithe-Jasper, and Y. Grin, Accounts of\nChemical Research 51, 214 (2018).\n[2] A. Amon, E. Svanidze, A. Ormeci, M. K onig, D. Kasi-\nnathan, D. Takegami, Y. Prots, Y.-F. Liao, K.-D. Tsuei,\nL. H. Tjeng, A. Leithe-Jasper, and Y. Grin, Angewandte\nChemie International Edition 58, 15928 (2019).\n[3] F. R. Fickett, Electrical properties of materials and their\nmeasurement at low temperatures , NBS Technical Note\n1053 (U.S. National Bureau of Standards, 1982).\n[4] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards,\nS. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder,\nand K. a. Persson, APL Materials 1, 011002 (2013).\n[5] M. A. Mitchell, Journal of Applied Physics 46, 4742\n(1975), https://doi.org/10.1063/1.321550.\n[6] F. M. Mueller, K. A. Johnson, W. J. Medina, H. D.\nLewis, D. S. Phillips, M. F. Hundley, J. D. Thomp-\nson, Z. Fisk, L. A. Jacobson, C. J. Maggiore, J. F.\nSmith, E. M. Honig, W. L. Gordon, J. E. Schirber, and\nR. Kossowsky, Applied Physics Letters 57, 240 (1990),\nhttps://doi.org/10.1063/1.103702.\n[7] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,\nJ. Luitz, R. Laskowski, F. Tran, and L. D. Marks,\nWIEN2k: An Augmented Plane Wave plus Local Orbitals\nProgram for Calculating Crystal Properties (Vienna Uni-\nversity of Technology, Austria, 2018).\n[8] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n[9] G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[10] S. Mishra, S. Thulasi, and S. Satpathy, Phys. Rev. B 72,\n195347 (2005).\n[11] Y.-J. Song, K.-H. Ahn, W. E. Pickett, and K.-W. Lee,\nPhys. Rev. B 94, 125134 (2016).\n[12] N. H. Shon and T. Ando, Journal of the\nPhysical Society of Japan 67, 2421 (1998),\nhttps://doi.org/10.1143/JPSJ.67.2421.\n[13] T. Ando, Journal of the Physical Society of Japan 75,\n074716 (2006), https://doi.org/10.1143/JPSJ.75.074716.\n[14] I. Shiozaki, Journal of Physics F: Metal Physics 5, 451\n(1975).\n[15] W. W. Tipton and R. G. Hennig, Journal of Physics:\nCondensed Matter 25, 495401 (2013).\n[16] B. C. Revard, W. W. Tipton, A. Yesypenko, and R. G.\nHennig, Physical Review B 93, 054117 (2016).[17] G. Kresse and J. Furthm uller, Computational Materials\nScience 6, 15 (1996).\n[18] G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169\n(1996).\n[19] P. E. Bl ochl, Phys. Rev. B 50, 17953 (1994).\n[20] B. C. Revard, W. W. Tipton, and R. G. Hennig, Predic-\ntion and Calculation of Crystal Structures: Methods and\nApplications , , 181 (2014).\n[21] A. O. Lyakhov, A. R. Oganov, and M. Valle, Computer\nPhysics Communications 181, 1623 (2010).\n[22] H. K. Mao, Y. Wu, L. C. Chen, J. F. Shu, and R. J.\nHemley, High Pressure Research 5, 773 (1990).\n[23] R. Ahuja, P. S oderlind, J. M. Wills, B. Johansson, and\nO. Eriksson, High Pressure Research 12, 161 (1994).\n[24] M. I. McMahon and R. J. Nelmes, Chemical Society Re-\nviews 35, 943 (2006).\n[25] J. J. Hamlin, Physica C: Superconductivity and its Ap-\nplications 514, 59 (2015).\n[26] J. T. Waber and D. T. Cromer, The Journal of Chemical\nPhysics 42, 4116 (1965).\n[27] N. F. Mott and H. Jones, The theory of the properties of\nmetals and alloys (Dover Publ., New York, 1958).\n[28] V. F. Degtyareva, Physics-Uspekhi 49, 369 (2006).\n[29] A. D. Chijioke, W. J. Nellis, A. Soldatov, and I. F.\nSilvera, Journal of Applied Physics 98, 114905 (2005).\n[30] S. T. Weir, J. Akella, C. Aracne-Ruddle, Y. K. Vohra,\nand S. A. Catledge, Applied Physics Letters 77, 3400\n(2000).\n[31] C. T. Seagle, E. Cottrell, Y. Fei, D. R. Hummer, and\nV. B. Prakapenka, Geophysical Research Letters 40, 5377\n(2013).\n[32] J. Sun, J. P. Perdew, and A. Ruzsinszky, Proceedings\nof the National Academy of Sciences 112, 685 (2015),\nhttps://www.pnas.org/content/112/3/685.full.pdf.\n[33] J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev.\nLett. 115, 036402 (2015).\n[34] J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal\nof Chemical Physics 124, 219906 (2006).\n[35] M. Gerosa, C. E. Bottani, C. D. Valentin, G. Onida, and\nG. Pacchioni, Journal of Physics: Condensed Matter 30,\n044003 (2017)."    },    {        "title": "1005.0843v2.Anisotropies_in_the_Diffuse_Gamma_Ray_Background_from_Dark_Matter_with_Fermi_LAT__a_closer_look.pdf",        "content": "arXiv:1005.0843v2  [astro-ph.HE]  6 Sep 2011Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 8 November 2018 (MN L ATEX style file v2.2)\nAnisotropies in the Diffuse Gamma-Ray Background from\nDark Matter with Fermi LAT: a closer look\nA. Cuoco1,2, A. Sellerholm1, J. Conrad1and S. Hannestad2\n1The Oskar Klein Centre for Cosmo Particle Physics, AlbaNova , SE-106 91 Stockholm, Sweden\n2Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, Bygn. 1520 8000 Aarhus Denmark\n8 November 2018\nABSTRACT\nWe perform a detailed study of the sensitivity to the anisotropies re lated to Dark\nMatter (DM) annihilation in the Isotropic Gamma-Ray Background (I GRB) as mea-\nsured by the Fermi Large Area Telescope (Fermi-LAT). For the fir st time, we take into\naccount the effects of the Galactic foregrounds and use a realistic representation of\nthe Fermi-LAT. We implement an analysis pipeline which simulates Fermi-L AT data\nsets starting from model maps of the Galactic foregrounds, the F ermi resolved point\nsources, the extra-galactic diffuse emission and the signal from DM annihilation. The\neffects of the detector are taken into account by convolving the m odel maps with the\nFermi-LAT instrumental response. We then use the angular power spectrum to char-\nacterize the anisotropy properties of the simulated data and to st udy the sensitivity\nto DM. We consider DM anisotropies of extra-galactic origin and of Ga lactic origin\n(which can be generated through annihilation in the Milky Way sub-str uctures) as\nopposed to a background of anisotropies generated by sources o f astrophysical origin,\nblazars for example. We find that with statistics from 5 years of obs ervation Fermi is\nsensitive to a DM contribution at the level of 1%-10% of the measure d IGRB depend-\ning on the DM mass mχand annihilation mode. In terms of the thermally averaged\ncross section /angbracketleftσAv/angbracketright, this corresponds to ∼10−25cm3s−1, i.e. slightly above the typi-\ncal expectations for a thermal relic, for low values of the DM mass mχ<∼100 GeV.\nThe anisotropy method for DM searches has a sensitivity comparab le to the usual\nmethods based only on the energy spectrum and thus constitutes an independent and\ncomplementary piece of information in the DM puzzle.\n1 INTRODUCTION\nCombined studies of the cosmic microwave background ra-\ndiation, supernova cosmology and large Galaxy redshift sur -\nveys provide nowadays compelling evidence of the existence\nof non-baryonic Dark Matter (DM) (Komatsu at al. 2008;\nTegmark et al. 2006). Consistency with the observed struc-\nture of the Universe, especially at large (above galactic)\nscales favorsColdDarkMatter(CDM),i.e.theparticles con -\nstituting the cosmologically required dark matter have to b e\nmoving non-relativistically. This property is always fulfi lled\nby particles of mass in the GeV to TeV region that interact\nwith the weak interaction strength, i.e. weakly interactin g\nmassive particles (WIMPs). WIMPs can annihilate or decay\ntodetectablestandardmodelparticles, inparticulargamm a-\nrays, giving rise to the possibility of “indirect detection ”\n(Bertone et al. 2004; Jungman et al. 1995; Bergstrom 2000;\nBaltz et al. 2008). A standard approach is to look for spec-\ntral signatures in the energy spectrum of gamma-rays from\ncelestial objects, either targeting the continuum spectru m\nproduced by decay of secondary pions or the more feeble\nsignature of a line due to DM annihilating directly into pho-\ntons.\nAnother possible experimental signature is given bydifferences in the spatial distribution of gamma-ray signal s\ninduced by DM as compared to conventional astrophysical\nsources. In particular, the small scale fluctuations in an\notherwise isotropic gamma-ray background (IGRB) might\nbe different in the case of annihilating DM, since its flux\nscales as the squared density. After the pioneering study\nof Ando and Komatsu (Ando & Komatsu 2006) , followed\nby two other more detailed works (Ando et al. 2006;\nAndo et al. 2007), the issue of DM annihilation and\nanisotropies in the isotropic gamma background has raised\nincreasing interest. In connection to the extra-galactic\ngamma-ray background, anisotropy has been further stud-\nied in (Cuoco et al. 2006; Cuoco et al. 2007; Cuoco 2008)\nwhile predictions from the Millenium II simulation have\nrecently been described in Zavala et al. (2009). It has also\nbeen realized that Galactic substructures can produce an\nalmost isotropic gamma background with similar fluctuation\nproperties as the extragalactic background and with promis -\ning chances of detection (Siegal-Gaskins 2008; Ando 2009;\nSiegal-Gaskins and Pavlidou 2009; Hensley et al. 2009);\nsee also (Pieri et al. 2007; Pieri et al. 2009)). Other\nworks have investigated the anisotropy pattern result-\ning from both the Galactic and extragalactic contri-\nc∝circlecopyrt0000 RAS2A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\nbution (Fornasa et al. 2009; Hooper & Serpico 2007;\nIbarra et al. 2010) or from the the DM annihilation\naround intermediate mass black holes (Taoso et al. 2009).\nAnisotropies from DM annihilation appearing in the radio\nsky have been investigated in (Zhang and Sigl 2008).\nFinally,– apart from the power spectrum – an interesting\napproach is to investigate the probability distribution of\nthe fluctuations in order to detect possible departures\nfrom Gaussianity (or, rather, from Poisson statistics, in\nthe common situation of low number of counts), thereby\nproviding another handle on the separation of conven-\ntional astrophysical and DM contribution (Lee et al. 2008;\nDodelson et al. 2009). Anisotropy properties of astro-\nphysical processes other than DM contributing to the\nisotropic gamma background include resolved point\nsources (Ando et al. 2006), galaxy clusters and shocks\n(Miniati et al. 2007; Keshet et al. 2003) and emission from\nnormal galaxies (Ando & Pavlidou 2009).\nThe Fermi Large Area Telescope (Fermi-LAT), which\nwas launched June 11, 2008, is the most suitable tool\nfor these kind of studies due to its large field of view\n(∼2.4 sr), energy coverage from 100 MeV to 300 GeV\n(Atwood et al. 2009) and very good angular resolution. We\nwill thus focus in the following on the prospect to detect\nDM anisotropies with Fermi-LAT.\nAll of the above studies assume either that the impact\nof Galactic foregrounds on the DM induced anisotropies is\nnegligible or that the Galactic foregrounds can be cleaned a t\na suitable level to allow DM anisotropies studies. Further,\nthe effects of the Fermi-LAT detector are generally taken\ninto account only in an approximate way, though a detailed\nrepresentation of the Fermi instrument is publicly availab le.\nThe aim of this work is to address the above\nissues in detail. In particular our approach will\nbe the following: we include the Galactic fore-\ngrounds explicitly using a model based on the\nGALPROP code (Strong and Moskalenko 1998;\nMoskalenko and Strong 1997; Strong et al. 2004b) and\nwe then use this model to build masks to select the most\npromising regions in the sky suitable for the analysis of the\nanisotropies. In these regions we study the anisotropies in\nthe scenario in which a DM contribution is present over the\nGalactic foregrounds and the extra-galactic emission from\nblazars compared to the case without DM and we derive\nthe sensitivity to the DM signal employing anisotropy\nstudies with Fermi-LAT data. Both the cases of Galactic\nDM in substructures and extra-galactic DM are considered\nseparately. The effect of the Fermi-LAT response, including\nthe effects of the point spread function, the effective area,\nas well as non-uniformities in exposure, are included in\nall calculations at a realistic level by using the instrumen t\nresponse functions provided by the Fermi Science Tools.\nThis approach is suitable for a sensitivity study like the\none presented here, and does not aim to fully reproduce a\nreal data analysis for which foreground cleaning (not per-\nformed in this study) most likely will be required. One of\nthe findings of our analysis is indeed that even with a very\ngenerous masking, foregrounds still have a sizeable contri bu-\ntion to the overall anisotropies and represent an important\nlimitation to DM sensitivity. This implies that foreground\ncleaning will be a crucial step in the data analysis pipeline .\nOn the other hand, if cleaning can be achieved in an effec-tive way, this also means that the sensitivity to DM can be\nfurther improved with respect to the results derived in this\npaper. Ourresults can thusbe regarded as conservative from\nthis point of view. Foreground cleaning will be the subject\nof forthcoming investigations.\nThe paper is structured as follows. In section II we in-\ntroduce and discuss the various components contributing to\nthe Galactic and extragalactic gamma-ray emission. In sec-\ntions III and IV we review their anisotropy properties and\nwe describe the tools and methods for their analysis. In sec-\ntion V we briefly describe the Fermi-LAT instrument and\nthe characteristics relevant for our analysis. In section V I we\ndiscuss the results of our data simulation pipeline. Finall y,\nin section VII we evaluate Fermi’s sensitivity to distingui sh\na scenario in which the diffuse gamma-ray emission has a\nDM contribution (either extragalactic or galactic in origi n)\nwith respect to the case in which DM is absent. Conclusions\nare presented in section VIII.\n2 GAMMA-RAY SKY COMPONENTS\n2.1 Isotropic Gamma-Ray Background\nIn the following, for clarity, we will refer to the isotropic ex-\nperimentally detected emission as the Isotropic Gamma-Ray\nBackground (IGRB) given that in principle its origin may\nbe not fully extra-galactic. Indeed, we will also consider t he\npossibility of a nearly isotropic emission from DM annihila -\ntion in the Milky Way substructures. We will use the term\nExtra-Galactic Gamma Background (EGB) when referring\nto predictions from modelsof extra-galactic emission either\nfrom DM annihilation or from astrophysical processes.\nA first detection of the IGRB in the GeV range was re-\nported by the EGRET experiment (Sreekumar et al.1998)\n(see also the analysis by Strong, Moskalenko and Reimer\n(Strong et al. 2004a) ). The recent measurement from The\nFermi collaboration (Abdo et al. 2009b) finds that the\nIGRB energy spectrum has an almost perfect power law\nwithslope s≈ −2.4whichisapproximatelywhatisexpected\nfrom the contribution of unresolved blazars thus suggestin g\nan extragalactic origin of the IGRB.\nA (possibly incomplete) list of contributions to\nthe IGRB is given by the emission in astrophysi-\ncal processes from Blazars, AGNs, Starburst Galax-\nies, Star Forming Galaxies, Galaxy clusters, Clusters\nShocks, Gamma-Ray Bursts (Stecker and Salamon 1996;\nPavlidou and Fields 2002; Gabici and Blasi 2002;\nTotani 1998). A contribution is also expected from ultra\nhigh energy photons and electrons cascading to GeV-TeV\nenergies through interactions with the CMB or the Infrared-\nOptical Background (Kalashev et al. 2007). Each of these\nsource classes is expected to produce anisotropies althoug h\nall them should more or less trace the filamentary pattern\nof the cosmological Large Scale Structures. Anisotropies\nwill be discussed in more detail in the next two sections.\n2.2 DM Annihilation\nApart from the astrophysical contributions, an exotic com-\nponent arising from WIMP annihilation could also be ex-\npected. In the case of uniformly distributed DM, the an-\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 3\n102103104105\nE[MeV]10-610-510-410-310-2E2·dφ/dE0[MeVcm−2s−1sr−1]|b|>30◦Total GDE\nIGRB\npi0\nSources\nIC\nBremss\n193 GeV WIMP, b¯b\n102103104105\nE[MeV]10-610-510-410-310-2E2·dφ/dE0[MeVcm−2s−1sr−1]|b|>60◦Total GDE\nIGRB\npi0\nSources\nIC\nBremss\n193 GeV WIMP, b¯b\nFigure 1. Energy spectrum of the various components contributing to t he gamma sky above 30◦(left) and above 60◦(right) galactic\nlatitude. The isotropic component is the determination fro m Abdo et al. (2009b). Also shown is an example of extragalact ic WIMP\nemission normalized with the model of Ullio et al. (2002) (wi th NFW halo profile and no halo substructures). The spectrum o f the\nresolved point sources is calculated from the Fermi 11-mont h source catalogue (Ballet 2009).\nnihilation flux for a thermal relic cross section ( ∝angbracketleftσAv∝angbracketright ∼\n3×10−26cm3s−1) is many orders magnitude below the\nIGRB. However, given that the DM distribution on cos-\nmological scales could be very clumpy, the enhancement\nfactor (denoted as ∆2(z)) due to the quadratic depen-\ndence of the annihilation signal on the DM density has\nto be taken into account. This enhancement is typically\nseveral orders of magnitude, thus boosting the DM signal\nto a level comparable to the IGRB (Bergstrom et al. 2001;\nUllio et al. 2002; Taylor and Silk 2002). The exact boost de-\npends on the details of the modeling of the DM clustering,\nthe DM halo profile and the structure formation history.\nWe fix the absolute normalization of the DM spectra fol-\nlowing the DM haloes clustering model described in Ullio et\nal.(2002). Two versions of the model are employed: a con-\nservative version assuming a Navarro-Frenk-White (NFW)\nprofile (Navarro et al. 1995) for the haloes and no substruc-\ntures, which gives a low DM normalization, and a more op-\ntimistic version where the haloes still follow a NFW profile\nbut the DM signal is enhanced by the presence of substruc-\ntures in the DM haloes. It should be noted that the clus-\ntering properties of the DM haloes are still very uncertain,\nespecially at small scales (approximately below the typica l\ngalactic halo scale), and even larger normalizations are po s-\nsible. For example recent results from the Millenium-II sim -\nulation (Zavala et al. 2009) indicate that, depending on the\nextrapolation employed for the low mass haloes (non re-\nsolved in the simulation), an order of magnitude enhance-\nment with respect to the above “optimistic” case can be\nachieved.Overall, Zavala et al. (2009) place thepossible v al-\nues of the boost ∆2(z=1) in the range 104to 107(see also\n(Abdo et al. 2010a), in particular Fig.1).\n2.3 Galactic Foregrounds\nThe main contribution to the Galactic foregrounds is given\nby the decay of pions produced in the interaction of CosmicRay (CR) nucleons, by the inverse Compton (IC) scattering\nofCR electrons on the interstellar radiation field (ISRF)an d\nby bremsstrahlung of CR electrons on the interstellar gas. A\nfurther contribution from pulsars has also been considered\nrecently (Faucher-Giguere & Loeb 2009). According to this\nanalysis, the contribution from Galactic millisecond puls ars\ncan be quite isotropic (i.e. can contribute to high Galac-\ntic latitudes) and constitute a large fraction of the IGRB.\nHowever, lacking more precise studies of the subject, we wil l\nneglect this contribution for the present analysis.\nThe normalization of the Galactic foregrounds is\nclearly important to identify the regions where the\nIGRB is dominant, i.e. the regions most promis-\ning for the anisotropy analysis. We employ the so\ncalled “conventional model” (Strong and Moskalenko 1998;\nMoskalenko and Strong 1997; Strong et al. 2004b) which is\nderived under the assumption that the CR electron and nu-\ncleon spectra in the galaxy can be normalized by the lo-\ncal (solar system) measurements. This model represents a\nnice fit of the Fermi data, at least at Galactic latitudes\n|b|>10◦, where the emission is mostly local in origin (com-\ning within a few kpc’s of the solar position) (Porter 2009a;\nAbdo et al. 2009c; Porter 2009b). We will, indeed, focus on\nthisregionoftheskyinthefollowing. TheGalactic centerr e-\ngion and the Galactic plane are anyway strongly dominated\nby the Galactic emission and they are thus not suitable for\nthe anisotropy analysis. A description of the diffuse emissi on\nin terms of the conventional model is thus accurate enough\nfor the present analysis.\n2.4 Resolved Point Sources\nThe number of resolved point sources has grown consider-\nably in the Fermi era relative to the roughly 270 in the 3rd\nEGRET catalogue (Hartman et al.(1999)). The first year\nFermi catalogue includes 1451 sources with a significance\nlarger than about 4 sigma (Abdo et al. 2009d). We include\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–0004A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\nexplicitly these sources in our analysis. As will be clear in\nthe following, it is important to include and, in addition,\nto model this component correctly since it is a considerable\nsource of anisotropy which needs to be reduced to gain sen-\nsitivity to the anisotropies of the diffuse component.\nWhileweusethefirstyearcatalogue, inthefollowing we\nproduce a forecast over 5 years of data taking, thus should\nin principle be using an associated 5 years catalogue. The\neffect of using a five year catalogue, however, is expected to\nbe small: the masked area will increase, but on the other\nhand the Poisson noise will be reduced. Both effects are\nsmall since increasing the data taking time for the catalogu e\nwill add sources on the faint end, requiring small masking\narea and producing a small decrease in Poisson noise (since\nthe brightest sources are already removed). On the other\nhand, modeling the 5 year catalogue requires an assumption\non the nature of the unresolved sources, introducing a model\nuncertainty.Wetherefore decidedtousethe1yearcatalogu e\nonly.\n2.5 Unresolved Point Sources\nPart of the IGRB is made of unresolved point sources . It is\nstill a matter of debate if unresolved point sources can make\nup all the IGRB or only part of it. This issue is likely to be-\ncome clearer when population studies of the Fermi resolved\npoint sources become available. Present studies indicate that\nunresolved blazars can only account for at most 30 % of the\nIGRB measured by the Fermi-LAT (Abdo et al. 2010b).\nUnresolved point sources can also be a further source of\nanisotropy intheIGRBitself, producingaPoisson noise–li ke\ncontribution to the anisotropy spectrum. It is worthwhile t o\npointoutthatthisPoisson noise is differentfrom thePoisso n\nshot noise in the angular power spectrum (see below and\nappendix A), which is related to the finite number of counts.\nThe contribution of unresolved point sources depends both\non the ability of the Fermi-LAT to detect point sources and\nremove them from the diffuse emission and on the intrinsic\nnumber density of the sources themselves. Both these terms,\nin turn, are quite model dependent but they are expected to\nbe reasonably constrained from population studies of sourc e\nclasses with resolved members.\nFor the moment, for the sake of studying the impor-\ntance of the effect, we consider, beside the model described\nin the next section, a model fully made of unresolved point\nsources. For the Poisson noise we assume a value of CP=\n5.0×10−5which is in the middle between the estimate of\n10−4(Ando et al. 2007) (see their Fig.4) and the prelim-\ninary measurement of 10−5from the Fermi collaboration\n(Siegal-Gaskins 2010).\nIt is also worth noting that the Poisson noise of unre-\nsolved sources is a time varying term slowly decreasing in\ntime. The brighter unresolved sources, in fact, get finally\nresolved, as Fermi continues to collect more statistics, an d\nonce they are removed, and/or masked, the “new” remain-\ning unresolved signal has a slightly lower level of Poisson\nnoise. A detailed study of this effect requires extrapolatio n\nfrom population studies which we leave for future work.\nIn Figure 1 we show the energy spectrum of the various\nGalactic and extra-galactic components for the model we\nemploy. In particular, we show the average spectrum in the\nregions|b|>30◦and|b|>60◦, where the IGRB has thehighest relative contribution. We show the power law fit to\nof the IGRB from (Abdo et al. 2009b). Also shown is an\nexample of extragalactic WIMP emission normalized to the\nmodel of Ullio et al. (2002) (with NFW halo profile and no\nhalo substructures). The contribution from resolved point\nsources detected by Fermi (Abdo et al. 2009a),(Ballet 2009)\nis also present. It can be seen that, combined, these sources\nconstitute a significant fraction of the diffuse background.\nFor simplicity, the spectrum of each single source is modele d\nas a simple power law and thus the behavior at high energies\n(>∼10 GeV) may have some biases with respect to the real\nspectrum.\n3 ANISOTROPIES: GENERAL\nCONSIDERATIONS\nWe will follow two approaches to quantify the anisotropies\nof all the gamma-ray sky components. For the components\nfor which maps as a function of energy are available (i.e. for\nthe Galactic foregrounds and for the resolved point sources )\nwe calculate the angular power spectra directly from the\nmaps using HEALpix (Gorski et al. 2005)(see appendix A\nfor more details on HEALpix and the definition of power\nspectra).Fortheremainingcomponents(DMandastrophys-\nical EGB) we model the power spectra from theory. From\nthe theoretical power spectra we then also produce simu-\nlated sky-maps (under the assumption that the fluctuations\nare gaussian). In this way we have effective sky maps for all\nthe components and we can combine them, convolve them\nwith the Fermi IRFs, simulate finite statistics of events, ap -\nply masks, etc. The final maps will thus represent a sim-\nulated gamma-ray sky as seen through the Fermi detector\nand where all the components are entangled. Power spec-\ntra are then recovered at the end from the final maps using\nHEALpix. In this way the sensitivity to DM can be studied\nas described in section 7 by comparing the power spectra of\nthe maps with and without the DM component.\nFig.2 shows separately the all-sky angular spectra of\nthe various components: Galactic foregrounds, astrophysi cal\nEGB, WIMPs EGB and point sources. The spectra are cal-\nculated from simulated maps at E >1 GeV after convolving\nwith the Fermi IRFs (see the next sections). For illustrativ e\npurposes, the flux of the DM component is arbitrarily nor-\nmalized to the level of the IGRB detected by Fermi and an\nobservation time of 5 years is assumed. It can be seen that\nthe Galactic foreground anisotropies have a decreasing spe c-\ntrum in land dominate over the extragalactic anisotropies\n(either astrophysical or WIMP) up to l∼200−300. Luck-\nily,however, thiscan beremediedusinganappropriate mask\nexcluding the Galactic plane region where most of the fore-\ngrounds are concentrated. In this way the foregrounds can\nbe reduced such as to become subdominant at a multipole\nofl∼30 as shown in the lowest panel of fig.2. The spec-\ntrumofpointsources highlydominates boththeforegrounds\nand the extragalactic signals. This indicates that an efficie nt\npoint source masking is crucial to extract the interesting e x-\ntragalactic signal. These steps will be explored in the next\nsections. The figure also illustrates the effect of the Poisso n\nnoise (first panel) coming from finite statistics present in t he\ncalculated raw power spectra. Customary power spectra are\nplotted with the noise removed (second panel).\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 5\nFigure 2. All sky angular spectra for E >1 GeV of the employed\nmodels forpointsources (red), Galactic foregrounds (gree n), EGB\n(magenta) and EGB DM (blue) after convolution with the Fermi -\nLAT Point Spread Function. For illustration, the flux of the D M\ncomponent is arbitrarily normalized to the level of the IGRB de-\ntected byFermi-LAT,and5 yearsofFermi-LATobservations h ave\nbeen assumed. The power spectrum of the exposure map is also\nshown (black line). In the top panel the spectra before shot n oise\nremoval are shown (shot noise is represented by the dashed li nes)\n. All the spectra are calculated with HEALpix. The last panel\nshows the angular power spectra after the application of a su it-\nable mask to cover the low latitude Galactic foregrounds and the\npoint sources. Since the mask is effective in suppressing the point\nsources signal the related power spectrum has been removed.The following section describes in detail how the power\nspectra of DM and astrophysical EGB are modeled and how\nthe related maps are simulated. The section can be skipped\nby the reader not interested in these details.\n4 DARK MATTER AND ASTROPHYSICAL\nANISOTROPIES\n4.1 Modeling the EGB\nApart the Poisson term coming from the unresolved point\nsources, the remaining source of anisotropies of the IGRB is\ngiven by the anisotropic spatial distribution of the source s\nthemselves. To derive the anisotropy we will assume, as a\nreasonable first approximation, that the gamma ray sources\nare distributed as the matter density of the universe ρ(/vector x),\ni.e. following the cosmological Large Scale Structures (LS S).\nIn principle ρs, the density distribution of astrophysical\nsources, should be used instead of ρ:ρsin general exhibits\na scale and time dependent bias with respect to the matter\ndensity. However, specific classes of astrophysical gamma-\nray sources have different biases. For example, blazars are\nwell known to concentrate at the center of clusters of galax-\nies, thus presenting an over-bias with respect to galaxies a t\nhigh densities. On the other hand, galaxies and clusters of\ngalaxies reasonably trace the matter density, at least in th e\nrecent cosmic epoch. The assumption ρs=ρis thus general\nenough to approximately describe emission from astrophys-\nical sources.\nGiven these assumptions the extragalactic cosmic\ngamma-ray signal can be written as (Ullio et al. 2002;\nBergstrom et al. 2001; Cuoco et al. 2006)\nIγ(Eγ,ˆn)∝/integraldisplay∞\n0z.ρ(z,ˆn,r(z))g[Eγ(1+z)]e−τ(Eγ,z)\nH(z)(1+z)3,(1)\nwhereg(E) =dNγ/dEis the photon spectrum of the\nsources, Eγis the energy we observe today, ρ(z,ˆn,r) is the\nmatter density in the direction ˆ nat a comoving distance\nr, and the redshift zis used as time variable. The Hub-\nble expansion rate is related to its present z=0 value H0\nthrough the matter and cosmological constant energy den-\nsities as H(z) =H0/radicalbig\nΩM(1+z)3+ΩΛ, and the reduced\nHubble expansion rate h(z) is given by H(z) = 100 h(z)\nkm/s/Mpc. We will in the following use Ω M= 0.3, ΩΛ= 0.7\nandH0= 70 km/s/Mpc. The quantity τ(Eγ,z) is the op-\ntical depth of photons to absorptions via pair production\n(PP)ontheExtra-galactic BackgroundLight(EBL).Weuse\nthe parametrization of τ(Eγ,z) from (Stecker et al. 2006)\nfor 0< z <5, where the evolution of the EBL is included\nin the calculation. The EBL is expected to be negligible at\nredshifts larger than z≈5 corresponding to the peak of\nstar formation. Thus, gamma photons produced at earlier\ntimes experience an undisturbed propagation until z≈5,\nwhile only in the recent epoch they start to lose energy\ndue to scattering on the EBL. Correspondingly, we assume\nτ(Eγ,z) =τ(Eγ,5) forz >5 (see also formula (A.6) in\n(Cuoco et al. 2006)).\nIn the case of cosmological DM annihilation, the re-\nsulting spatial distribution of the gamma signal follows th e\nsquare of the matter distribution ρ2(/vector x) through\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–0006A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\nIχ(Eγ,ˆn) =<σχv>\n8πm2χ/integraldisplay∞\n0z.ρ2\nχ(z,ˆn,r(z))g[Eγ(1+z)]e−τ(Eγ,z)\nH(z)(1+z)3,(2)\nwhich, taking the spatial average, reduces to the expressio n\nIχ(Eγ) =<σχv>¯ρ02\n8πm2χ/integraldisplay∞\n0z.∆2(z)(1+z)3g[Eγ(1+z)]e−τ(Eγ,z)\nH(z),(3)\nwhere ∆2(z) is the clumpiness enhancement factor and ¯ ρ0\nis the average DM density at z= 0.\nWe can write the astrophysical EGB and DM in a com-\npact form:\nIγ(E1,E2,ˆn)∝/integraldisplay∞\n0z.Wγ(E1,E2,z)ρ(z,ˆn), (4)\nIχ(E1,E2,ˆn)∝/integraldisplay∞\n0z.Wχ(E1,E2,z)ρ2(z,ˆn), (5)\nwhereWγ(E1,E2,z) andWχ(E1,E2,z) are the astrophysi-\ncal and DM window functions, which contain the informa-\ntion about gamma-ray propagation, injection spectra and\ncosmological effects\nW(E1,E2,z)≡/integraldisplayE2\nE1E.g[E(1+z)](1+z)3α−3\nH(z)e−τ(E,z),(6)\nwhereα= 1,2 applies in the astrophysical and DM cases,\nrespectively and E1andE2are the boundaries of the energy\nbands considered. We are using the notation ρ(z,ˆn,r(z)) =\n(1+z)3×ρ(z,ˆn)tounderlinethatthewindowfunction is only\ndependent on the two variables, direction and redshift, and\nto make the (1+ z)3behavior of the matter density explicit.\nIn particular, choosing E1=EcutandE2= +∞, when\nproperly normalized W(Ecut,z) represents the probability\nof receiving a photon of Eγ> Ecutemitted at a redshift z.\nThis can be used to define an effective horizon, zH, beyond\nwhich the probability of receiving a photon is negligible. I n\npractice, above about 10 GeV, pair production attenuation\nsets the scale of the horizon which is about zH∼0.5 and\nzH∼2 for the α= 1,2 cases respectively. In this case,\nDMannihilation producesamuchlarger anisotropythanthe\nastrophysical sources. In contrast, below 10 GeV the horizo n\nis set by cosmological effects and zH∼1 andzH∼10 for\nthe two cases. In the low energy regime we thus see that the\nDM anisotropies are averaged over a much larger horizon\nand the final signal can be smoother than the astrophysical\ncase. Further details are discussed in (Cuoco et al. 2007).\nThe injection energy spectrum for DM annihila-\ntiong(Eγ) =dNγ/dE, given the annihilation chan-\nnel and the WIMP mass mχ, is calculated with Dark-\nSUSY (Gondolo et al. 2004), which, in turn, uses a tabu-\nlation derived from simulation of the particle processes pe r-\nformed with Pythia(Sjostrand et al. 2007). Wewill consider\nin the following the b¯b,τ+τ−andµ+µ−annihilation chan-\nnels.\n4.2 Angular Power Spectra\nGiven the above expressions we can write the angular\npower spectra of the various dimensionless fluctuation field s\nδI/<I>\nCl\nγ=/integraldisplay\ndr\nr2W2\nγ(r)Pρ/parenleftBig\nk=l\nr,z(r)/parenrightBig\n, (7)Cl\nχ=/integraldisplay\ndr\nr2W2\nχ(r)Pρ2/parenleftBig\nk=l\nr,z(r)/parenrightBig\n, (8)\nwhere we have used the Limber approximation\n(Limber 1953), which is accurate for all but the very\nlowest multipoles, while Pρ(k,z(r)) andPρ2(k,z(r)) are\n3D power spectra of the matter field and of its square. We\nare also interested in the spectra of cross correlation amon g\ntwo different energy bands, which can be written as\nCl\n12=/integraldisplay\ndr\nr2W1(r)W2(r)Pi/parenleftBig\nk=l\nr,z(r)/parenrightBig\n, (9)\nwherei=ρ,ρ2for the astrophysical and DM case.\nWe take the 3D power spectra PρandPρ2from an\nN-body simulation of Large Scale Structures formation\n(Cuoco et al. 2007). We note that Pρ2has a dependence\non the clustering properties of DM below galactic scales\n(see (Ando & Komatsu 2006) for an analytical derivation\nofPρ2) which are not resolved in the N-body simulation.\nThe anisotropy properties in the case of cosmological DM\nannihilation with substructures are therefore not fully co n-\nsistent with its energy spectrum normalization, which does\ntake into account the effects of substructures. This has to be\ntaken into account when interpreting the sensitivities in t he\ncosmological substructures model shown in the following.\nAs described in the previous section we will consider\nin the following two different anisotropy models for the as-\ntrophysical EGB in order to bracket our uncertainties and\nestimate the effect on the DM sensitivity. The first one is a\nmodel dominated by the correlation term of the power spec-\ntrum (Eq.7), while the second one is a model dominated by\nthe Poisson term CPfrom unresolved point sources. In this\ncase the angular power spectrum is simply:\nCl\nγ=CP (10)\nwithCP= 5×10−5.\nFinally we will also consider the possibility that\npart of the IGRB is produced by DM annihilation in the\nsub-structures of our galaxy as found by recent N-body\nsimulations (Diemand et al. 2007; Springel et al. 2008a;\nSpringel et al. 2008b). In this case, compared to the cosmo-\nlogical annihilation scenario, the situation is simpler si nce,\nin the absence of redshift effects, the pattern of anisotropi es\nis independent of energy (and thus of the particle physics\nmodel). This can be seen more clearly in Eq.6 where W\nbecomes a constant independent of energy and redshift. The\nanisotropies thus only depend on the clustering properties\nof DM in the Milky Way (MW). We will take as a reference\nmodel the Cl\nχgalfrom (Siegal-Gaskins 2008) where the\nanisotropies have been calculated from a simulated MW\nsubstructure map. In particular, we will use the spectrum\nshown in their Fig.5 for a minimum clump mass of 10 M⊙.\nCompared with the cosmological DM anisotropies this\nmodel gives a normalization of the angular power spectrum\na factor ∼100 larger (or more) depending on the energy.\nMore precisely, the shapes of the power spectra are sim-\nilar for the two cases, while the absolute normalization\nfor the Galactic case is l(l+ 1)Cl/2π≈10 atl= 100\nindependent of energy, whereas for the extragalactic case\nl(l+ 1)Cl/2π≈0.02 at 10 GeV. Fig.3 shows the energy\ndependence of l(l+1)Cl/2πatl= 100 for the various cases\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 7\nFigure 3. Energy dependence of the anisotropies for our models\nof cosmological EGB DM, astrophysical EGB with and without\na dominant Poisson term and DM in Galactic substructures. Th e\nanisotropy curve for the DM refers to a mχ=1 TeV WIMP anni-\nhilating into b¯b.\nwhile Fig.4 shows l(l+1)Cl/2πfor the DM EGB case for 4\ndifferent energies.\n4.3 Model Sky Maps\nTo produce a realistic simulation of data from Fermi\nwe have to simulate the expected sky pattern of every\ncomponent. In the case of Galactic foregrounds we em-\nploy the GALPROP package (Strong and Moskalenko 1998;\nMoskalenko and Strong 1997) which produces as output the\nsky maps of the foregrounds as a function of energy. As\nalready described above the conventional diffuse model is\nused. It is worth noticing that energy dependent CR diffu-\nsion effects imply an energy dependent sky pattern as well.\nIn general, at low energies the diffusion length of CRs is\nlarge and the resulting maps are thus smoother than the\nhigher energy counterparts where the diffusion length de-\ncreases. Thus the relative importance of the Galactic fore-\ngrounds anisotropies at small scales ( l>∼100) increases with\nenergy. Of course, projection effects along the line of sight\ncan sometimes complicate the picture and give small scale\nanisotropies also at low energies although this in practice\nhappens only along the Galactic plane where there is a con-\ntribution from structures many kpc away from the solar po-\nsition.\nFor the remaining components, i.e. the astrophysical\nEGB and the DM EGB, we create template maps as gaus-\nsian realizations of the theoretical angular power spectra Cl\nderived above. The situation is however complicated by the\nfact that the angular spectra are energy dependent and, fur-\nther, that there are non-zero cross-correlation power spec -\ntra between different energy bands. Energy dependent maps\nwith the correct cross-correlation properties among differ -\nent energies need thus to be realized. In practice, we choose\nNE= 60 logarithmically spaced energy bands between 10\nMeV and 10 TeV (i.e 10 bands per energy decade). We\nthen calculate the angular auto-correlation power spectra\nfor each bandandthecross-correlation power spectraamong\nall the possible pairs of energy bands according to equation s\n7-10. The result is the model power spectrum matrix Cij\nli,j= 1,NEfor the DM or astrophysical EGB. We then\ngenerate the maps in harmonic space ( ai\nlm, (see definitions\nin the appendix)) up to l≈500 sampling from the 60x60\nmatrix of the multi-gaussian distribution given by the Cij\nl\nfor eachl. In practice the energy behavior is generally highly\ncorrelated so thatonly the first 4-5principal components ar e\nrequiredforthesimulation ofthe60maps.The synfast tool\nfrom HEALpix is then employed to combine the harmonic\ncomponents into the real space maps fEi(ˆΩ). These energy\ndependent anisotropy maps are then weighted and normal-\nized at each energy according to the energy spectrum of the\ncomponent itself (astrophysical or DM EGB), giving the fi-\nnal template maps. These can now be added to the model\nof Galactic foregrounds and to the model of point sources\nto give a simulation of the gamma-ray sky where all the\ncomponents are entangled.\nFor the case of Galactic DM the procedure is simpler\ngiven that the anisotropy pattern is independent of the en-\nergy and only one map needs to be simulated. The same\nholds for the astrophysical EGB model with pure Poisson\nnoise.\nConvolution with the instrumental effects is described\nin the next section.\n5 THE FERMI-LAT AND SIMULATED\nDATASETS\nIn the following we describe the various instrumental effect s\nintroduced in the Gamma-ray sky when observed through\nthe eyes of the Fermi-LAT telescope.\n5.1 Charged particles contamination\nCharged particles, mainly protons, electrons and positron s,\nas well as a smaller number of neutrons and Earth albedo\nphotons, presentamajor instrumental backgroundtopoten-\ntial DM signals, in particular when considering the isotrop ic\ngamma-ray signal. These background particles greatly dom-\ninate the flux of cosmic photons incident on the LAT, and a\nmultivariate technique, employing information from all LA T\ndetector systems, is used to reduce the remaining charged\nparticle background to only a fraction of the expected\nflux of extragalactic diffuse emission (Atwood et al. 2009).\nThe analysis of the extragalactic background from Fermi\n(Abdo et al. 2009b) however indicates that, with the stan-\ndard data selection cuts, the residual contamination is com -\nparable to the photon background itself above a few GeV\nand dominates at higher energies. Generally, this back-\ngroundisexpectedtobebasically isotropicsoitcanbeeasi ly\ntaken into account in the anisotropy analysis. Alternative ly,\nstringent cuts can be applied on the photon events decreas-\ning the contamination but also somewhat the exposure and\nthus the signal (Abdo et al. 2009b). For our simulation we\nwill use the effective area for the “diffuse” event class as de-\nfinedin(Atwood et al. 2009).Usingmore stringentselection\ncuts to reduce the effects of CR contamination reduces the\neffective area by a factor of ∼25% (Abdo et al. 2009b). The\nsensitivity curves should change accordingly, approximat ely\nby the small factor√\n1.25∼1.1. This estimate is reasonably\naccurate for a sensitivity study as the present one. We do\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–0008A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\nnot attempt a detailed simulation of the CR background it-\nself, postponing a more careful investigation of the issue t o\nthe data analysis work.\n5.2 Point Spread Function and Exposure\nThe effects of the Fermi-LAT detector can be summarized\nin terms of the IRFs which include the point spread func-\ntion (PSF) and the energy dependent exposure maps. For\nthis analysis we use the PSF and exposure as available from\nthe Fermi Science tools p6v31. We neglect the effect of en-\nergy dispersion (which is about 10 %) and assume that the\nphoton energy is perfectly reconstructed. In this respect, for\nthis analysis we will divide the simulated datasets in wide\nenergy bands so that this instrumental effect is negligible\nfor our purposes. To obtain simulated maps we use the en-\nergy dependent model input maps and then convolve them\nwith the Fermi PSF andsimulated exposure of 5 years. More\nspecifically, theexposurewas generatedfrom 1yrofpointin g\nhistory simulated with gtobssim and then rescaled to 5 yrs.\nThe convolution with the PSF and the exposure was then\nperformed usingthe GaDGET (Ackermann et al. 2009)tool\n(see also Abdo et al. (2009b)). The resulting maps have\na normalization in units of counts per steradian which is\nfurther re-normalized to counts per pixel ( ∼10−5sr for a\nNside=256 HEALpix pixelization). Finally Poisson random\nnoise is added to simulate the effect of finite statistics.\nThe Fermi PSF is energy dependent and improves at\nhigh energies. It corresponds to an angular resolution of\nabout 3◦at 100 MeV, 0 .6◦at 1 GeV and 0 .1◦above 10 GeV\n(see (Atwood et al. 2009) for more details). The PSF thus\nheavily affects the anisotropies especially below 1 GeV, sup -\npressing the medium-high multipoles in the angular power\nspectrum. The sensitivity to these multipoles is not com-\npletely lost however. In principle, from the knowledge of th e\nPSF it is possible to recover the true power spectrum even\nat higher multipoles although the error grows with larger\nattenuation (it grows exponentially with multipole l: the\nsuppression factor is given by Wl=exp(−l2σ2\nb/2), for a\ngaussian PSF with variance σb).\nThe effect of the PSF can be seen in Fig.4 where our\nmodel power spectra for the DM EGB component at vari-\nous energies are compared with the corresponding PSF con-\nvolved spectra. The latter are obtained by convolving the\nmodel DM EGB skymaps (simulated from the angular spec-\ntra with the procedure described in the previous section)\nwith the PSF and extracting the power spectra from the\nresulting maps. As expected, the PSF heavily suppresses\nanisotropies at l >100 for E <∼1GeV. Notice also that there\nis a certain suppression at l>∼200 even for E= 300 GeV,\nwhich is slightly more than the factor Wl=exp(−l2σ2\nb/2)\nexpected at this energy for a gaussian PSF with width 0 .1◦.\nThis extra effect is likely related to the non-gaussian form\nof the PSF which even at very high energies ( >100 GeV)\nexhibits a tail towards large angular spreads. The figure als o\nshows thedifference inPSF for eventsconvertinginthefront\nor the back of the detector where the reconstruction perfor-\nmances are different. It can be seen that the PSF is better\nfor events converting in the front, giving a sensitivity to\n1See http://fermi.gsfc.nasa.gov/ssc/Figure 4. Power spectra at various energies of the model of DM\nEGB emission (for a mχ=1 TeV WIMP annihilating into b¯b) and\ntheir corresponding PSF convolved counterparts (fromtop t o bot-\ntom atE= 300 GeV, E= 30 GeV, E= 3 GeV, E= 300 MeV).\nThe PSF suppresses anisotropies at l >100 for E <∼1GeV. The\nriseof the power spectrum at high multipoles in the E= 300 MeV\ncase is a numerical HEALpix artifact. The lower panel shows t he\npower spectra resulting from the different PSFs for events co n-\nverting in the front or the back of the detector and for the ave rage\nPSF. The dipole has not been included in the simulated maps,\nhence the sharp rise at l= 2.\nslightly higher multipoles especially at low energies. Abo ve\na few GeV, instead, the difference is less important. We will\nuse the average PSF for our analysis.\nThe exposure maps are very uniform over an averaging\nperiod of 1 year (see again (Atwood et al. 2009)) so that the\nconvolution with the exposure only adds power to the very\nlowest multipoles l<∼10. This can seen in fig.2 where we plot\nthe angular power spectrum of the exposure which, indeed,\nfalls rapidly at l= 10. The exposure pattern is also energy\ndependent, although only weakly. The integrated exposure,\nhowever,rises steeplystartingfrom about100MeVandthen\nflattens above about 1 GeV.\n6 RESULTS OF THE SIMULATION\n6.1 Simulated Sky-Maps\nThe maps produced as described in the previous sections are\nsummed up to build a simulation of the observed gamma-\nraysky.Areference no-DMmodel is derivedaddingGalactic\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 9\nFigure 5. (Left) 5 year averaged counts map (in HEALpix nside=256 form at) of our reference no-DM model, which contains astrophysi cal\nEGB, Galactic foreground and resolved point sources. An ene rgy dependent mask is applied to suppress Galactic foregrou nds and point\nsources (see the text). The simulation has been divided into five energy bands (in the energy ranges indicated in the title s) and only the\nregion outside the masks is shown. The EGB energy spectrum is harder than the Galactic diffuse one, so at high energies ther e is more\nsky area available for the analysis. (Right) A random realiz ation of the expected counts.c∝circlecopyrt0000 RAS, MNRAS 000, 000–00010A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\nFigure 6. Power spectra matrix for the 5 average counts maps (times the masks) of the previous figure. Before computing the spectra,\nthe maps have been (re)normalized so that to represent fluctu ation maps. Auto-correlations are in the diagonal and cross -correlations\nare off-diagonals. The dashed lines in the auto-correlation plots indicate the level of shot noise ∝1/Ncounts. The noise in different energy\nbands, in contrast, is uncorrelated so that no shot noise is p resent in the cross-correlations. The blue lines represent the power spectrum\nmatrix of the masks alone. It can be seen, that the masks give t he “bulk” level of anisotropies. The anisotropies of the mas ked no-DM\nreference model then add over the mask level (black curves). At higher energies the angular resolution of Fermi-LAT incr eases and the\nanisotropy excess with respect to the masks is better visibl e (in the lower right part of the plot) at medium-high multipo les. To be at a\nsignificant level, the difference has to exceed the errors bar s. Statistical errors are represented by the gray shaded are a for a confidence\nlevel of two sigma. Finally, the red curves represent the ang ular power spectra when a DM EGB component is added on top of th e\nprevious contributions. For illustration mχ≈200 GeV is chosen and the DM flux is normalized to the same level as the astrophysical\nEGB at 10 GeV. The bump visible above the statistical error le vel in the 10 GeV-100 GeV autocorrelation spectrum (and the a bsence\nof it in the other energy bins) would make this model detectab le.\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 11\nforegrounds, astrophysical EGB and point sources. The as-\ntrophysical EGB is normalized to the level of the observed\nIGRB. The normalization of the Galactic foregrounds is the\none given as output by GALPROP. Point sources are nor-\nmalized to their observed flux. The overall statistics is nor -\nmalized to 5 years of data taking by Fermi. The maps we\nobtain are average counts maps , but a realization of effec-\ntive counts maps can be easily obtained drawing poisson\ncountsin each pixelfrom theaverage counts maps. Although\nwe use 60 energy bands to produce the simulated maps, it\nis convenient to rebin them into fewer maps, in order to\nhave a reasonable amount of statistics in each band. We\nthus divide the dataset in 5 energy bands from 400 MeV\nto 800 GeV. Finally, a crucial step is the construction of\nthe masks which will suppress the Galactic foregrounds and\nthe resolved point sources. For each energy band the mask\nare defined as the region where the foregrounds do not ex-\nceed twice the EGB (200%), and the point sources emission\ndoes not exceed more than 20% of the EGB itself. Simu-\nlated sky maps (outside the masked regions) for both the\naverage counts and one particular realization of the counts\nare shown in Fig.5. Indeed, outside the masks the remaining\nsignal is quite isotropic, indicating that the IGRB makes\nmost of the signal, but still a certain residual foreground\ncontamination is apparent, especially near the edges of the\nmask. Clearly this is expected given the loose cut employed\nto build the masks (only 200% of the EGB). A tighter cut\non the foregrounds can be applied but then the available\narea for the analysis would shrink to close to zero since the\nGalactic foregrounds never go below 20-30% of the EGB\neven at the Galactic poles. For the present forecast anal-\nysis a substantial foreground contamination is still accep t-\nable since the foreground model is known in advance and\nwhat is required for a significant detection is that the EGB\nanisotropy exceeds the anisotropy of the residual Galactic\nforegrounds. In general, for a real data analysis some kind\nof foreground cleaning must be performed to obtain useful\nresults. We postpone a detailed treatment of this issue to\nthe analysis of the real data.\nOn the other hand, the cut of 20% of the EGB for point\nsources is rather effective and removes most of the gamma-\nray events from resolved point sources, except perhaps few\nevents around the most powerful sources where, statistical ly,\nsome events very far from the source position are expected\ndue to the tails of the PSF. From Fig.5 it can also be seen\nthatthefraction ofskywheretheisotropic componentdomi-\nnates increases with energy. This is expected considering t he\nfact that the IGRB has a spectrum harder than the Galac-\ntic emission (see also Fig.1). Also it is clear that the area\nmasked for the point sources substantially decreases at hig h\nenergy due to the improvement of the PSF. This is true for\nall but the highest energy band, where the fraction of sky\ncovered by the point sources increases again. This can be\nunderstood looking at the energy spectrum of the resolved\nsources in Fig.1. This becomes very hard for E>∼10 GeV\nso that, even if the PSF at this energies is very narrow, its\ntail still contains a flux which is a significant fraction of th e\nIGRB.6.2 Angular Power Spectra\nIn Fig.6 we plot the various auto-correlation and cross-\ncorrelation spectra in a “triangle plot”, where the diagona l\nrepresents the auto-correlations and the the off-diagonals\nthecross-correlations. The bluelines representthespect raof\nthe masks only, which thusprovide the zero-level anisotrop y.\nThe black line represents the average spectrum of the ref-\nerence no-DM model convolved with masks, i.e. the maps\nin the left column in Fig.5. As explained above, outside the\nmask thereference model is toa large extentgivenbythe as-\ntrophysicalEGB,althoughasubstantialresidualcontamin a-\ntion from the Galactic foregrounds is still present. The res id-\nual point source contamination, on the other hand, is neg-\nligible. The grey shaded area is the two sigma error region\nin the power spectrum accounting for the effect of the finite\nnumber of events. A logarithmic binning of the multipoles\nin 18 bands is used. The level of shot noise is shown with a\ndashed line. It can be seen that the shot noise increases in\nthe high energy bands where the statistics decrease, as ex-\npected. Finally, the red curves represent the angular power\nspectra if a EGB DM component is added on top of the pre-\nvious contributions. For illustration aWIMP with mχ≈200\nGeV is chosen and the DM flux is normalized to the same\nlevel of the EGB at 10 GeV. It can be seen, indeed, that in\nthe 4th energy band (approximately 10-100 GeV), where the\nDM emission peaks, the anisotropy has a noticeable bump\ncompared to the EGB anisotropy and provides a clear sig-\nnature for this model. The DM anisotropy signal disappears\nin the 5th band (100-800 GeV) where the DM contribution\ndrops.This “energy modulation intheanisotropy spectrum”\nis also discussed in (Hensley et al. 2009)).\nIt has to be noted that the spectra look quite different\nfrom the all-sky spectra of the astrophysical and DM EGB\nshown in Fig.2, both in the shape and normalization. This\nis of course due to the effects of the residual foregrounds\nand especially of the mask which distorts the original spec-\ntrum. The mask effects can in principle be removed with\nspecifically designedalgorithms like theMASTERalgorithm\n(Hivon et al. 2002). This comes, however, at the price of fur-\nther errors in the final spectrum which need to be addressed\nwith Montecarlo simulations (a package implementing the\nfull procedure is described for example in (Lewis 2008) ).\nFor the present purpose, it suffices to use the masked sky\npower spectrum without attempting to deconvolve it from\nthe mask itself. This is fine as long as all the power spectra\nare calculated on the same sky region in order to allow a\nconsistent treatment.\n7 DM SENSITIVITY\nIt is in principle straightforward to calculate the sensiti vity\nto DM requiring that the DM anisotropy does not exceed\nthe anisotropies of the reference no-DM model. Let us de-\nfineCij\nEGBl′as the binned multipoles power auto- and cross-\ncorrelation spectra of the reference model2. The indices i,j\nrun from 1 to 5 and indicate the energy bands and the index\n2We label it for simplicity EGB although, besides the true as-\ntrophysical EGB, there are some residual contaminations fr om\nGalactic foregrounds and point sources (see previous secti on).\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–00012A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\nFigure 7. Exclusion plots in the mχ-∝angbracketleftσAv∝angbracketrightplane for various\nannihilation channels: b¯b,τ+τ−,µ+µ−. Each panel shows the\nsensitivity curves at the 2 σ, 3σand 5σlevels. Two different nor-\nmalizations for the flux of the cosmological DM signal have be en\nemployed using an updated version of the (Ullio et al. 2002)\nmodel: NFW Halo profile with substructures (lower curves) an d\nno-substructures (higher curves). The curves are the sensi tivities\nconsidering the auto-correlation spectra information alo ne.\nl′indicates the multipoles bins. Let us also define ∆ Cij,mn\nEGBl′\nas the covariance matrix of the Cij\nEGBl���, i.e. ∆Cij,mn\nEGBl′is the\ncovariance between Cij\nEGBl′andCmn\nEGBl′. We divide the mul-\ntipole range 2 −512 into 18 logarithmically spaced multipole\nbands. Adding to the previous signal, the contribution for\na given DM model, specified through the annihilation chan-\nnel, the velocity averaged annihilation cross section and t he\nDM mass, we further obtain Cij\nDMl(∝angbracketleftσAv∝angbracketright,mχ) which canbe compared with the no-DM hypothesis as:\nχ2(∝angbracketleftσAv∝angbracketright,mχ) =\nΣl′ij,mn/parenleftbig\nCij\nEGBl′−Cij\nDMl′/parenrightbig/parenleftbig\n∆Cij,mn\nEGBl′/parenrightbig−1(Cmn\nEGBl′−Cmn\nDMl′) (11)\nwhere/parenleftbig\n∆Cij,mn\nEGBl′/parenrightbig−1is the inverse of the covariance matrix.\nAs a first step, however, we might want to consider a simple\nexpression, for example considering as observables only th e\nauto-correlation spectra Cii\nEGBl′. Unfortunately, the naive\nexpression\nχ2(∝angbracketleftσAv∝angbracketright,mχ) = Σ l′i/parenleftbig\nCii\nEGBl′−Cii\nDMl′/parenrightbig2//parenleftbig\n∆Cii,ii\nEGBl′/parenrightbig\n(12)\nturns out to be incorrect because the two spectra Cii\nEGBl′\nandCmm\nEGBl′are correlated, i.e. ∆ Cii,mm\nEGBl′∝negationslash= 0. The correla-\ntion, indeed, is given by ∆ Cii,mm\nEGBl′= 2/(2l+1)(Cim\nEGBl′)2, i.e.\nby the cross correlation between the bands iandm, as intu-\nitively expected. The variance of Cii\nEGBl′can be also easily\ncalculated and gives ∆ Cii,ii\nEGBl′= 2/(2l+1)(Cii\nEGBl′+CN)2,\nwhereCN= 4π/Neventsis the poisson shot noise. The cor-\nrect expression is then\nχ2(∝angbracketleftσAv∝angbracketright,mχ) =\nΣl′ii,mm/parenleftbig\nCii\nEGBl′−Cii\nDMl′/parenrightbig/parenleftbig\n∆Cii,mm\nEGBl′/parenrightbig−1(Cmm\nEGBl′−Cmm\nDMl′) (13)\nwhere/parenleftbig\n∆Cii,mm\nEGBl′/parenrightbig−1is the inverse of ∆ Cii,mm\nEGBl′and\n∆Cii,ii\nEGBl′= (δCii\nEGBl′)2=2\n2l+1(Cii\nEGBl′+CN)2(14)\n∆Cii,mm\nEGBl′= (δCii\nEGBl′δCmm\nEGBl′) =2\n2l+1(Cim\nEGBl′)2(15)\nTo take into account multipole binning and partial sky cov-\nerage, in the above expressions we have to substitute\n2\n2l+1→2\nfsky(2l+1)∆l\nCN→/parenleftBigδNcounts\nδΩ/parenrightBig−1\nfsky (16)\nwhere ∆ lis the number of binned multipole in the band l’,\nandδNcounts/δΩ is the density of the counts per steradian.\nNotice that in the above formulae no corrections for the\nangular resolution (the exp( l2σb) term) are required since\nwe are referring to raw, PSF uncorrected spectra. Finally,\nwe disregard the information at l <10 to minimize the\neffects of the exposure. Let us also notice that, after all, th e\nvalue of χ2calculated with Eq.12 turns out to be generally\nsimilar to the one of Eq.13. This behavior may be due to\nthe different levels of noise in the various energy bands and\nin the autocorrelation spectra which makes the off-diagonal\ncomponents in the covariance matrix (which do not have\nshot noise) generally subdominant.\nIn case we want to exploit all the available informa-\ntion, i.e. including both the auto-correlations and the cro ss-\ncorrelations as observables, the full expression Eq.11 for the\nχ2has tobeemployed.Inthiscase thefull covariance matrix\nbecomes more complicated and we give the corresponding\nexpressions in appendix B.\nSome further comments are in order regarding the cal-\nculation of χ2. We use always the same set of DM anisotropy\nmaps to calculate the Cls for the cosmological DM case, de-\nspite the fact that they in principle depend on mχ(and the\nannihilation mode) through Eqs.6 and 7. Consequently, a\ndifferent set of maps should be generated for each value of\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 13\nmχand for each channel. Fortunately, this dependence is\nweak above about 10 GeV where the DM horizon does not\ndepend much on the DM annihilation energy spectrum so\nthe effects on the sensitivity plots above 10 GeV are small,\naccordingly. Tobetter justify this approximation we can al so\nmake a comparison with the case of DM annihilation from\nsubstructures in the Galactic halo where we use a very dif-\nferent set of maps. The sensitivities generally change by a\nfactor of a few in this case (see below). The slight depen-\ndence of the maps on DM model for the cosmological case\nis thus likely to be smaller than that.\nWe can see the results in Fig. 7, where the sensitivities\nare reported for various annihilation channels. The abso-\nlute normalization of the DM spectra are obtained following\nthe DM haloes clustering model of (Ullio et al. 2002) as de-\nscribed in the introduction. Two versions of the model are\nshown: a conservative version with a NFW profile for the\nhaloes and no substructures which gives a low DM normal-\nization, and thus weaker constraints, and a more optimistic\nversion where the DM signal is enhanced by the presence\nof substructures in the DM haloes. In principle, using the\nrecent results of(Zavala et al. 2009) from the Millenium-II\nsimulation, an orderofmagnitudeenhancementwithrespect\nto the above “optimistic” case can be achieved with a corre-\nspondingly better sensitivity (see also (Abdo et al. 2010a) ,\nin particular Fig.1).\nThe various channels have generally similar behavior.\nTheb¯bchannel produces quite smooth limits. The µ+µ−\nandτ+τ−sensitivity curves in contrast have a steeper slope\nand exhibit more structure. This is due to the fact that for\nthesechannels thegammaemission is concentrated athigher\nenergies in a narrow peak near the energy corresponding\nto the DM mass. The sensitivity is thus more sensitive to\nthe coarse binning in energy chosen for this analysis. The\nsensitivity in the τ+τ−channel and especially the µ+µ−\nchannel at higher DMmasses somewhat decreases duetothe\nlower number of photons per annihilation (and thus lower\nstatistics) available.\nIncludingthe cross-correlation spectra information does\nnot seem to improve the sensitivity. This probably has to\nbe ascribed to the fact that the χ2for the auto-correlation\ncase already includes some cross-correlation information in\nthe off-diagonal terms of the covariance matrix. The level\nof anisotropies is also very low at low energy for this case\n(see Fig.3) and this somewhat reduces the importance of the\ninformation contained in the cross-correlation. The situa tion\nis different for the Galactic DM scenario as discussed below.\nWe also checked the case in which only the highest en-\nergy bands are employed for the analysis. We would expect\nan improvement in sensitivity given that very little DM sig-\nnal is expected in the 0.4-0.8 GeV band and thus scan-\nning in this energy range introduces a statistical penalty\nfactor. We find that the sensitivity indeed improves, but\nonly marginally. On the other hand, the lowest energy band\nis useful because being mostly clean of DM it represents a\ngood calibration for the background anisotropies.\nIn Fig. 8 we show the case in which the DM anisotropy\npattern does not vary with energy. This case is supposed\nto mimic the DM emission from unresolved substructures in\nthe Milky Way. We thus simulate a single anisotropic map\nwhich we use for all energies. The angular spectrum for DM\nemployed to simulate the map is normalized as in the case ofFigure 8. Same as previous figure but for the Galactic substruc-\ntures case. For a direct comparison with the cosmological ca se we\nuse for the DM signal the same normalization(s) employed for the\nextra-galactic case. The solid curves are the sensitivitie s consid-\nering the auto-correlation spectra information alone. The dotted\nones are derived considering also the information containe d in the\ncross-correlation spectra.\nsubstructures with a minimum mass of 10 M⊙as derived in\n(Siegal-Gaskins 2008) and it roughly corresponds to the ex-\ntragalactic anisotropies for an energy of about 300 GeV (see\nsection 2). For a direct comparison with the cosmological\ncase we use for the Galactic signal the same normalization\nemployed for the extragalactic case, which is justified by th e\nfactthatmorephysicallymotivatedmodels fortheDMemis-\nsion in the MW have a large degree of freedom (Ando 2009)\nand can easily be tuned to match the extragalactic case.\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–00014A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\nThe inclusion of the information from the cross-\ncorrelation spectra in this case, contrary to the cosmologi cal\none, gives a significant improvement, especially for higher\nDM masses. The reason for this difference is likely to be\nfound in the very different set of anisotropy maps employed\nfor the Galactic substructures case. In particular, the ver y\nhigh degree of anisotropy present also at low energy for this\ncase enhances the importance of a direct measurement of\nthe cross-correlation between low and high energy bands.\nIn contrast, the very low DM anisotropy at low energy for\nthe cosmological case (lower than the astrophysical EGB\nanisotropy) makes this information almost irrelevant. We\nalso find in this case that excluding the lowest energy band\ndoes not improve the sensitivity significantly.\nThe sensitivity increases substantially with respect\nto the cosmological case, especially for high DM masses.\nThe slope of the sensitivity curves is indeed slightly less\nsteep. Assuming higher levels of anisotropy for the Galacti c\ncase, which are possible, (Ando 2009; Siegal-Gaskins 2008;\nSiegal-Gaskins and Pavlidou 2009)acorrespondinglyhighe r\nsensitivity could be achieved. However, the scaling is not l in-\near. It is worth noting for example that although at about\n10 GeV the Galactic anisotropies are about 2 orders of mag-\nnitudes larger than the extragalactic ones, the sensitivit y is\nonly a factor 2-3 better. The scaling however is better for\nmχ>∼100 GeV where the difference in sensitivity is roughly\na factor of 10. This non linear scaling is due to the lim-\nited limited amount of detected photons (as it is the case at\nthe edges of the sensitivity curve). Here, it is very difficult\nto distinguish between the two (galactic and extra-galacti c)\nlevels of anisotropies even if the latter is a factor 100 larg er,\nsince the statistical error bars are also very large. The muc h\nlarger galactic anisotropy thus only modestly decrease the\nminimum amount of events required to detect the signal.\nFinally in Fig. 9 we show the impact on the DM sensi-\ntivity of changing the underlying astrophysical EGB model\nto the Poisson dominated one. For illustration, only the ex-\ntragalactic τ+τ−case with NFW profile and substructures\nis shown. The interesting results (perhapssomewhat counte r\nintuitive)is thatevenwith thisincreased levelofbackgro und\nin the anisotropy spectrum, still the sensitivity to the DM\ncomponent remains roughly unchanged. The reason is that a\nPoisson noise at the level of 5.0e-5 is anywayclearly detect ed\nwith a statistics of 5 years, so the DM component can still\nbe easily detected as a modulation (as usual, in multipole\nand in energy) of this dominant Poisson noise anisotropy\nspectrum. The underlying assumption here, however, is that\nthis new background is fully under control and well charac-\nterized. The extent to which this will be true is however\nnot clear and a better understanding may require dedicated\nsimulations and analyses which are beyond the scope of this\npaper.\n8 SUMMARY AND CONCLUSIONS\nIn the present analysis we have calculated the sensitivity\nto Dark Matter annihilation exploiting the characteristic\nanisotropy properties that DM would imprint in the Extra-\nGalactic Gamma Background. We performed a realistic sim-\nulation of the Fermi-LAT including the effect of the Point[t]\nFigure 9. DM sensitivity for annihilation into into b¯b, extragalac-\ntic anisotropies and NFW Halo profile with substructures, fo r the\ntwo cases of EGB background anisotropies from the correlati on\nterm only (same case of Fig.8, solid contours) and from a pure\nPoisson term only (dashed contours).\nSpread Function, effective area and of the anisotropic sky\nexposure.\nMore importantly, we investigated the impact of Galac-\ntic foregrounds on the DM anisotropy sensitivity using a\nGALPROP model to produce the sky maps of the Galactic\ngamma emission. The contribution of Galactic foregrounds\nis generally non negligible even above Galactic latitudes o f\nabout 60◦. Foreground cleaning will thus be an important\n(and likely challenging) issue in the analysis of the real da ta.\nFor the present work we do not attempt foreground cleaning\n(except for masking the Galactic plane and point sources)\nand we only require that the anisotropy from DM has to ex-\nceed the level of foregrounds anisotropy to be detectable.\nFurther, we only analyze a high Galactic latitude region\nwhere the model dependence of the foreground is expected\nto be moderate. In this respect our analysis can be regarded\nas a conservative one: In reality, if foreground cleaning ca n\nbe effectively implemented, a sensitivity to a lower level of\nanisotropy can be achievable. Also, a larger fraction of the\nsky can be employed than the small conservative one used in\nthe present analysis. We leave a more careful investigation\nof the issue for future work.\nDespite the difficulties involved in the foreground anal-\nysis, we find that the prospects for detecting DM anisotropy\nsignatures are promising. For some WIMP masses, sen-\nsitivity to ∝angbracketleftσAv∝angbracketright ∼3×10−26cm3s−1(i.e. the thermal\nrelic value) can be achieved using the anisotropy infor-\nmation alone. Interestingly, these constraints are compar a-\nble to what is achievable using the energy spectrum alone\n(Abdo et al. 2010a) . The anisotropy information is thus\ncomplementarytotheenergyspectrumandprovides across-\ncheck and an increase in the overall sensitivity to the DM\nsignal.\nWe also investigated the recently proposed scenario of\na population of DM substructures in the Milky Way halo as\npredicted by N-body simulations. These subhaloes give rise\nto an annihilation signal which is almost uniform in the sky\nand which can be comparable in intensity to the observed\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 15\nisotropic gamma background. For this case, the anisotropy\nsignal is predicted to be even higher than for the extragalac -\ntic case due to the huge clumpiness of the DM distribution\nthat correspondingly gives an enhanced sensitivity. We find ,\nindeed, that when increasing the anisotropy the sensitivit y\nto the DM signal increases as well with an improvement of\na factor of ∼10 formχ>∼100 GeV.\nIn summary, simulations indicate that ,with the sensi-\ntivity of the Fermi-LAT, detailed anisotropies studies of t he\ngamma-ray sky can be performed. The next obvious step\nwill be the analysis of the actual data. Anisotropy analyses\nwill soon provide a complementary source of information\nwhich will give further insights into the DM problem and\nthe properties of high-energy astrophysical sources.\nACKNOWLEDGMENTS\nThe authors would like to thank Toby Burnett, Jennifer\nSiegal-Gaskins and Eiichiro Komatsu for their numerous\ncomments and suggestions for improvement, and Chris Sav-\nage for a carefully reading of the manuscript. We also wish\nto thank Gianfranco Bertone for organizing the workshop\n“Gamma in Zurich” which stimulated many interesting dis-\ncussions on gamma-ray anisotropies.\nTheFermiLAT Collaboration acknowledges generous\nongoing support from a number of agencies and institutes\nthat have supported both the development and the oper-\nation of the LAT as well as scientific data analysis. These\ninclude the National Aeronautics and Space Administration\nand the Department of Energy in the United States, the\nCommissariat ` al’Energie AtomiqueandtheCentreNational\nde la Recherche Scientifique / Institut National de Physique\nNucl´ eaire et de Physique des Particules in France, the Agen -\nzia Spaziale Italiana and the IstitutoNazionale di Fisica N u-\ncleare in Italy, the Ministry of Education, Culture, Sports ,\nScience and Technology (MEXT), High Energy Accelerator\nResearch Organization (KEK) and Japan Aerospace Explo-\nration Agency (JAXA) in Japan, and the K. A. Wallenberg\nFoundation, the Swedish Research Council and the Swedish\nNational Space Board in Sweden.\nAdditional support for science analysis during the op-\nerations phase is gratefully acknowledged from the Istitut o\nNazionale di Astrofisica in Italy and the Centre National\nd’´Etudes Spatiales in France.\nREFERENCES\nA.A.Abdo et al.[FermiLATCollaboration],Astrophys.J.Suppl.\n183(2009a) 46 [arXiv:0902.1340 [astro-ph.HE]].\nA. A. Abdo et al.[Fermi LAT Collaboration], arXiv:1002.3603\n(2009b).\nA. A. Abdo et al.[Fermi LAT Collaboration], Phys. Rev. Lett.\n103(2009c) 251101 [arXiv:0912.0973 [astro-ph.HE]].\nA. A. Abdo et al.[Fermi LAT Collaboration], arXiv:1002.2280\n(2009d).\nA. A. Abdo et al.[Fermi-LAT Collaboration], arXiv:1002.4415\n(2010a).\nA. A. Abdo et al.[Fermi-LAT Collaboration], arXiv:1003.0895\n(2010b).\nM. Ackermann, G. Johannesson, S. Digel, I. V. Moskalenko,\nT. Porter, O. Reimer and A. Strong [Fermi LAT Collabo-\nration], AIP Conf. Proc. 1085(2009) 763.S. Ando, arXiv:0903.4685 [astro-ph.CO].\nS. Ando and E. Komatsu, Phys. Rev. D 73(2006) 023521\n[arXiv:astro-ph/0512217].\nS.Ando and V.Pavlidou, Mon. Not. Roy. Astron. Soc. 400(2009)\n2122 [arXiv:0908.3890 [astro-ph.HE]].\nS. Ando, E. Komatsu, T. Narumoto and T. Totani, Mon. Not.\nRoy. Astron. Soc. 376(2007) 1635 [arXiv:astro-ph/0610155].\nS. Ando, E. Komatsu, T. Narumoto and T. Totani, Phys. Rev. D\n75(2007) 063519 [arXiv:astro-ph/0612467].\nW. B. Atwood et al.[LAT Collaboration], Astrophys. J. 697\n(2009) 1071 [arXiv:0902.1089 [astro-ph.IM]].\nJ. Ballet [Fermi Collaboration], Talk Given at the Washingt on\nFermi Symposium 2009\nE.A. Baltz et al., JCAP0807(2008) 013 [arXiv:0806.2911 [astro-\nph]].\nL. Bergstrom, Rept. Prog. Phys. 63(2000) 793\n[arXiv:hep-ph/0002126].\nL. Bergstrom, J. Edsjo and P. Ullio, Phys. Rev. Lett. 87, 251301\n(2001) [astro-ph/0105048].\nG. Bertone, D. Hooper and J. Silk, Phys. Rept. 405(2005) 279\n[arXiv:hep-ph/0404175].\nA. Cuoco, PoS IDM2008 (2008) 114.\nA. Cuoco, S. Hannestad, T. Haugbolle, G. Miele, P. D. Serpico\nand H. Tu, JCAP 0704(2007) 013 [arXiv:astro-ph/0612559].\nA. Cuoco, J. Brandbyge, S. Hannestad, T. Haugboelle and\nG. Miele, Phys. Rev. D 77(2008) 123518 [arXiv:0710.4136\n[astro-ph]].\nJ. Diemand, M. Kuhlen and P. Madau, halo,” Astrophys. J. 657\n(2007) 262 [arXiv:astro-ph/0611370].\nS. Dodelson, A. V. Belikov, D. Hooper and P. Serpico,\narXiv:0903.2829 [astro-ph.CO].\nC. A. Faucher-Giguere and A. Loeb, arXiv:0904.3102 [astro-\nph.HE].\nM. Fornasa, L. Pieri, G. Bertone and E. Branchini,\narXiv:0901.2921 [astro-ph].\nS. Gabici and P. Blasi, Astropart. Phys. 19, 679 (2003)\n[astro-ph/0211573].\nGondolo P.,Edsjo J.,UllioP.,Bergstrom L.,Schelke M.and B altz\nE.A., JCAP 0407(2004) 008 [astro-ph/0406204].\nK. M. Gorski, E. Hivon, A. J. Banday, B. D. Wandelt,\nF. K. Hansen, M. Reinecke and M. Bartelman, Astrophys.\nJ.622(2005) 759 [arXiv:astro-ph/0409513].\nHartman, R. C., et al. 1999, VizieR Online Data Catalog, 212,\n30079\nB. S. Hensley, J. M. Siegal-Gaskins and V. Pavlidou,\narXiv:0912.1854 [astro-ph.CO].\nE. Hivon, K. M. Gorski, C. B. Netterfield, B. P. Crill,\nS. Prunet and F. Hansen, Astrophys. J. 567(2002) 2,\narXiv:astro-ph/0105302.\nD. Hooper and P. D. Serpico, JCAP 0706(2007) 013\n[arXiv:astro-ph/0702328].\nHunter S.D. et al., Astrophys. J. 481(1997) 205.\nA. Ibarra, D. Tran and C. Weniger, Phys. Rev. D 81(2010)\n023529 [arXiv:0909.3514 [hep-ph]].\nG. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267\n(1996) 195 [arXiv:hep-ph/9506380].\nO. E. Kalashev, D. V. Semikoz and G. Sigl, Phys. Rev. D 79\n(2009) 063005 [arXiv:0704.2463 [astro-ph]].\nU. Keshet, E. Waxman, A. Loeb, V. Springel and L. Hernquist,\nAstrophys. J. 585(2003) 128 [arXiv:astro-ph/0202318].\nE. Komatsu et al.[WMAP Collaboration], arXiv:0803.0547\n[astro-ph].\nS.K. Lee, S. Ando and M.Kamionkowski, arXiv:0810.1284 [ast ro-\nph].\nA. Lewis, Phys. Rev. D 78(2008) 023002 [arXiv:0804.3865 [astro-\nph]].\nD. N. Limber, Astrophys. J. 117(1953)\nJ. F. Navarro, C. S. Frenk and S. D. White, Astrophys. J. 462\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–00016A. Cuoco, A. Sellerholm, J. Conrad & S. Hannestad\n(1996) 563\nF. Miniati, S. M. Koushiappas and T. Di Matteo, Astrophys. J.\n667(2007) L1 [arXiv:astro-ph/0702083].\nMoskalenko I.W. and Strong A.W., Astrophys. J. 493(1998) 694\nV. Pavlidou and B. D. Fields, Astrophys. J. 575, L5 (2002)\n[astro-ph/0207253].\nL. Pieri, G. Bertone and E. Branchini, Mon. Not. Roy. Astron.\nSoc.384(2008) 1627 [arXiv:0706.2101 [astro-ph]].\nL. Pieri, J. Lavalle, G. Bertone and E. Branchini, arXiv:090 8.0195\n[astro-ph.HE].\nT. A. Porter [Fermi Collaboration], arXiv:0907.0294 [astr o-\nph.HE].\nT. A. Porter [Fermi Collaboration], Talk Given at TeV Partic le\nAstrophysics 2009\nJ. M. Siegal-Gaskins, JCAP 0810(2008) 040 [arXiv:0807.1328\n[astro-ph]].\nJ. M. Siegal-Gaskins and V. Pavlidou, arXiv:0901.3776 [ast ro-\nph.HE].\nJ.M.Siegal-Gaskins,forthe Fermi-LATCollaborationand E .Ko-\nmatsu, [arXiv:1012.1206 [astro-ph.HE]].\nSjostrand T., Mrenna S. and Skands P., Comput. Phys. Commun.\n178(2008) 852 [arXiv:0710.3820].\nV. Springel et al., arXiv:0809.0894 [astro-ph].\nV. Springel et al., arXiv:0809.0898 [astro-ph].\nSreekumar P. et al.[EGRET Collaboration], Astrophys. J. 494\n(1998) 523 [astro-ph/9709257].\nF. W. Stecker and M. H. Salamon, Astrophys. J. 464, 600 (1996)\n[astro-ph/9601120].\nF. W. Stecker, M. A. Malkan and S. T. Scully, Astrophys. J.\n648, 774 (2006) [astro-ph/0510449]. See also F. W. Stecker,\nM. A. Malkan and S. T. Scully, Astrophys. J. 658(2007) 1392\n[arXiv:astro-ph/0612048].\nF. W. Stecker, S. D. Hunter and D. A. Kniffen, Astropart. Phys.\n29(2008) 25 [arXiv:0705.4311 [astro-ph]].\nStrong A.W. and Moskalenko I.W., Astrophys. J. 509(1998) 212\nStrong A.W., Moskalenko I.W. and Reimer O., Astrophys. J. 613\n(2004a) 956 [astro-ph/0405441].\nStrong A.W., Moskalenko I.W. and Reimer O., Astrophys. J. 613\n(2004b) 962 [astro-ph/0406254].\nM. Taoso, S. Ando, G. Bertone and S. Profumo, Phys. Rev. D 79\n(2009) 043521 [arXiv:0811.4493 [astro-ph]].\nJ. E. Taylor and J. Silk, “The clumpiness of cold dark matter: Im-\nplications for the annihilation signal,” Mon. Not. Roy. Ast ron.\nSoc.339(2003) 505 [arXiv:astro-ph/0207299].\nM. Tegmark et al., Phys. Rev. D 74, 123507 (2006)\n[arXiv:astro-ph/0608632].\nT. Totani, Astropart. Phys. 11, 451 (1999) [astro-ph/9810207].\nA. A. Abdo et al.[Fermi LAT Collaboration], arXiv:1003.0895v1\n[astro-ph.CO]\nP. Ullio, L. Bergstrom, J. Edsjo and C. G. Lacey, Phys. Rev. D\n66, 123502 (2002) [arXiv:astro-ph/0207125].\nJ. Zavala, V. Springel and M. Boylan-Kolchin, arXiv:0908.2 428\n[astro-ph.CO].\nL. Zhang and G. Sigl, JCAP 0809(2008) 027 [arXiv:0807.3429\n[astro-ph]].\nAPPENDIX A: ANGULAR POWER\nSPECTRUM\nWe summarize in this appendix the conventions and nota-\ntionusedfortheanisotropyanalysis. Theanisotropyofasi g-\nnal on the sky can be quantified in terms of the power spec-\ntrum i.e. the spherical harmonic transform on the sphere.\nMore specifically, a map on the sphere f(ˆΩ) can be decom-\nposed in spherical harmonics as f(ˆΩ) = Σ lmalmYl\nm(ˆΩ) and\nthe angular power spectrum can then be defined asCl≡<|alm|2> (A1)\nwhere< ... >indicates the statistical ensemble average. The\nquantity\nˆCl=/summationdisplay\nm|alm|2\n2l+1(A2)\nis an unbiased estimator of the true power spectrum Cl, i.e.\n<ˆCl>=Cl.\nIn the case of two maps, besides the two autocorrelation\npowerspectra C1\nlandC2\nl,furtherinformation iscontainedin\nthecross-correlation power spectrum whichcanbeestimated\nas\nˆCl12=/summationdisplay\nma1\nlma2∗\nlm\n2l+1=/summationdisplay\nma1∗\nlma2\nlm\n2l+1(A3)\nwhere the equivalence of the two expressions comes from\nthe equality: ai\nlm= (−1)mai∗\nl−m,i= 1,2. The three spectra\nare of course not independent but are part of a covariance\nmatrix:/parenleftbigg\nC1\nlC12\nl\nC12\nlC2\nl/parenrightbigg\n(A4)\nwhich has to be taken into account when evaluating the\nerrors for the forecast. This is described in more details in\nthe next section.\nIn practice, to calculate the power spectra, the\nharmonic transform almis performed with HEALpix\n(Gorski et al. 2005)after binningtheeventsintomapsofthe\nHEALpix format itself. Due to the finite number Nof col-\nlected events beside the intrinsic anisotropies the data al so\ncontain noise (shot noise or white noise) which appears in\nthe power spectrum as a constant whose value is 4 π/N. The\nnoisesets thelimitabovewhichtheintrinsic anisotropies can\nbe detected. It is standard to remove the shot noise from the\nspectrum showing only the intrinsic power spectrum. It is\nalso customary to show not the power spectrum Clbut the\nquantity l(l+1)Cl/2π.\nAPPENDIX B: χ2COVARIANCE MATRIX\nIn the general case in which we include as observables the\ncross-correlation power spectra the covariance matrix can\nbe calculated by propagation of the errors from the expres-\nsions A2 and A3 and then taking the ensemble average. The\nvariance ∆ Cij,ij\nEGBl′of the cross-correlation spectrum Cij\nEGBl′\n(i∝negationslash=j) can be written as\n∆Cij,ij\nEGBl′=<(δCij\nEGBl′)2> =\n1\n2l+1/parenleftbig\n(Cij\nEGBl′)2+(Cii\nEGBl′+CNi)(Cjj\nEGBl′+CNj)/parenrightbig\n(B1)\nWe then have the case of covariance ∆ Cii,mn\nEGBl′between\nan auto-correlation spectrum Cii\nEGBl′and cross-correlation\nspectrum Cmn\nEGBl′. For this case we have\n∆Cii,mn\nEGBl′=< δCii\nEGBl′δCmn\nEGBl′> =\n2\n2l+1/parenleftbig\n(Cii\nEGBl′+CNi)(Cmn\nEGBl′)/parenrightbig\n(B2)\nFor the case of covariance between two cross-correlation\nspectra we have to distinguish two cases. In the case of co-\nvariance ∆ Cij,jn\nEGBl′between Cij\nEGBl′andCjn\nEGBl′where there\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000Anisotropiesinthe DiffuseGamma-RayBackgroundfromDarkM atter with FermiLAT: a closerlook 17\nis an energy band (an index) in common we have:\n∆Cij,jn\nEGBl′=< δCij\nEGBl′δCjn\nEGBl′> =\n1\n2l+1/parenleftbig\n(Cjj\nEGBl′+CNj)(Cin\nEGBl′)+(Cij\nEGBl′)(Cjn\nEGBl′)/parenrightbig\n(B3)\nIf instead there is no energy band (no index) in common\n∆Cij,mn\nEGBl′finally we have\n∆Cij,mn\nEGBl′=< δCij\nEGBl′δCmn\nEGBl′> =\n1\n2l+1/parenleftbig\n(Cim\nEGBl′)(Cjn\nEGBl′)+(Cin\nEGBl′)(Cjm\nEGBl′)/parenrightbig\n(B4)\nExpression (14) and (B1) can also be rewritten in the\nmore familiar form\nδCl\nCl=/radicalbigg\n2(1+CN/Cl)2\n(2l+1)∆lfsky, (B5)\nfor the auto-correlation spectra and by\nδC12\nl\nC12\nl=/radicalbigg\n1\n(2l+1)∆lfsky×\n/radicalBigg/parenleftbigg\n1+C1\nlC2\nl\nC12\nlC12\nl(1+CN1/C1\nl)(1+CN2/C2\nl)/parenrightbigg\n,(B6)\nfor the cross-correlation spectra, where again\nCNi=/parenleftbigg\nδNi\ncounts\nδΩ/parenrightbigg−1\nfsky (B7)\nand ∆lis the number of binned multipole in the band l′,\nandδNi\ncounts/δΩ is the density of the counts per steradian\nin the map i.\nc∝circlecopyrt0000 RAS, MNRAS 000, 000–000"    },    {        "title": "0806.3826v1.Reconstruction_of_the_Fermi_surface_in_the_pseudogap_state_of_cuprates.pdf",        "content": "arXiv:0806.3826v1  [cond-mat.supr-con]  24 Jun 2008Pis’ma v ZhETF\nReconstruction of the Fermi surface in the pseudogap state o f cuprates\nE.Z.Kuchinskii, M.V.Sadovskii1)\nInstitute for Electrophysics, RAS Ural Branch, 620016 Ekat erinburg, Russia\nSubmitted *\nReconstruction of the Fermi surface of high-temperature su perconducting cuprates in the pseudogap state\nis analyzed within nearly exactly solvable model of the pseu dogap state, induced by short-range order fluctu-\nations of antiferromagnetic (AFM, spin density wave (SDW), or similar charge density wave (CDW)) order\nparameter, competing with superconductivity. We explicit ly demonstrate the evolution from “Fermi arcs”\n(on the “large” Fermi surface) observed in ARPES experiment s at relatively high temperatures (when both\nthe amplitude and phase of density waves fluctuate randomly) towards formation of typical “small” electron\nand hole “pockets”, which are apparently observed in de Haas - van Alfen and Hall resistance oscillation\nexperiments at low temperatures (when only the phase of dens ity waves fluctuate, and correlation length of\nthe short-range order is large enough). A qualitative crite rion for quantum oscillations in high magnetic fields\nto be observable in the pseudogap state is formulated in term s of cyclotron frequency, correlation length of\nfluctuations and Fermi velocity.\nPACS: 71.10.Hf, 74.72.-h\nPseudogap state of underdoped copper oxides [1, 2,\n3,4]isprobablythemainanomalyofthenormalstateof\nhigh temperature superconductors. Especially striking\nis the observation of “Fermi arcs” in ARPES experi-\nments, i.e. parts on the “large” Fermi surface around\nthe diagonalofthe Brillouinzone(BZ) with moreorless\nwell defined quasiparticles, while the parts of the Fermi\nsurface close to BZ boundaries are almost completely\n“destroyed” [5, 6, 7].\nHowever, the recent observation of quantum oscilla-\ntion effects in Hall resistance [8], Shubnikov - de Haas\n[9] and de Haas - van Alfen (dHvA) oscillations [9, 10]\nin the underdoped YBCO cuprates, producing evidence\nfor rather “small” hole or electron [11] pockets of the\nFermi surface, seemed to contradict the well established\nARPES data on the Fermi surface of cuprates.\nQualitatvive explanation of this apparent contra-\ndiction was given in Ref. [12] within very simplified\nmodel of hole-like Fermi surface evolution under the ef-\nfect of short-range AFM fluctuations. Here we present\nan exactly solvable model of such an evolution, which\nis able to describe continuous transformation of “large”\nARPESFermisurfacewithtypical“Fermiarcs”athigh-\nenough temperatures into a collection of “small” hole-\nlike and electron-like “pockets”, which form due to elec-\ntroninteractionwithfluctuationsofSDW(CDW)short-\nrange order at low temperatures (in the absence of any\nkind ofAFM (or charge)long-rangeorder). We also for-\nmulate a qualitative criterion for observability of quan-\ntum oscillation effects in high-magnetic field in this,\nrather unusual, situation.\n1)e-mail: sadovski@iep.uran.ruWebelievethat thepreferable“scenario”forpseudo-\ngap formationcan be most likely based on the picture of\nstrong scattering of the charge carriers by short–ranged\nantiferromagnetic (AFM, SDW) spin fluctuations [2, 3],\ni.e. fluctuations of the order parameter competing with\nsuperconductivity. In momentum representation this\nscattering transfers momenta of the orderof Q= (π\na,π\na)\n(a– lattice constant of two dimensional lattice). This\nleads to the formation of structures in the one-particle\nspectrum, which are precursors of the changes in the\nspectraduetolong–rangeAFMorder(perioddoubling).\nAs a result we obtain non–Fermi liquid like behavior\nof the spectral density in the vicinity of the so called\n“hot spots” on the Fermi surface, appearing at intersec-\ntions of the Fermi surface with antiferromagnetic Bril-\nlouinzoneboundary[2,3], whichinthelowtemperarure\n(large correlation length of the short-range order) can\nlead to a significant Fermi surface reconstruction, sim-\nilar to that appearing in the case of AFM long-range\norder.\nWithin this approach we have already demonstrated\n[13, 14] the formation of “Fermi arcs” at high-enough\ntemperatures, when AFM fluctuationscanbe effectively\nconsidered as static and Gaussian [15, 16]. Here we\npresent an exactly solvable model, quite similar to that\nanalyzed qualitatively in Ref. [12], which is capable to\ndescribe a crossover from “Fermi arc” picture at high\ntemperatures (typical for most of ARPES experiments)\nto that of small “pockets” at low temperatures (typical\nfor quantum oscillation experiments).\n12 E.Z.Kuchinskii, M.V.Sadovskii\nWe shall consider a two - dimensional generalization\nof an exactly solvable model proposed in one - dimen-\nsion in Ref. [17] (and also analyzed in a simplified two-\ndimensional approach in Ref. [18]), which is physically\nequivalent to the model of Ref. [12], but can produce\na complete picture of Fermi surface reconstruction and\nformation of both hole and electron “pockets”.\nWe consider electrons in two-dimensionalsquare lat-\ntice with nearest ( t) and next nearest ( t′) neighbour\nhopping integrals, which leads to the usual “bare” dis-\npersion:\nε(k) =−2t(coskxa+coskya)−4t′coskxacoskya−µ ,\n(1)\nwhereais the lattice constant, µ— chemical potential,\nand assume that these electronsare scatteredby the fol-\nlowing (static) random field, imitating AFM(SDW) (or\nsimilar CDW) short-range order:\nV(l) =Dexp(iQl−iql)+D∗exp(−iQl+iql) (2)\nwherel= (nxa,nya) numerates lattice sites and D=\n|D|eiφdenotes the complex amplitude of fluctuating\nSDW (or CDW) order parameter, while q= (qx,qy) is\na small deviation from the dominating scattering vector\nQ= (Qx,Qy) = (π\na,π\na).\nGeneralizingthe approachofRefs. [17, 18] (compare\nwith Ref. [12]) we consider a specific model of disorder,\nwhere both qxandqyare random and distributed ac-\ncording to:\nP(qx,qy) =1\nπ2κ\nq2x+κ2κ\nq2y+κ2(3)\nwhereκ=ξ−1is determined by the inverse correlation\nlength of short-range order. Phase φis also considred\nto be random and distributed uniformly on the interval\n[0,2π].\nFactorized form of (3) is not very important physi-\ncally, but allows for an analytic solution for the Green’\nfunction which takes the form [18]:\nGD(ε,k) =ε−ε(k+Q)+ivκ\n(ε−ε(k))(ε−ε(k+Q)+ivκ)−|D|2\n(4)\nwherev=|vx(k+Q)|+|vy(k+Q)|,withvx,y(k) =∂ε(k)\n∂kx,y.\nSpectral density A(ε,k) =−1\nπImGD(ε,k) at the\nFermi level ( ε= 0), is shown in Fig. 1, and demonstrate\nthe formation of small “pockets” instead of large “bare”\nFermi surface. Here and in the following we have as-\nsumedrathertypical(for cuprates)valuesof t′/t=−0.4\nand doping n= 0.9 (10% hole doping), corresponding\ntoµ=−1.08t.\nFig.1. Reconstruction of the Fermi surface in the low\ntemperature (large correlation length) regime of pseu-\ndogap fluctuations ( n= 0.9,t′/t=−0.4. Shown are\nintensity plots of spectral density for ε= 0: (a) –\nD= 0.2t,κa= 0.01; (b) – D= 0.7t,κa= 0.01; (c)\n–D= 1.5t,κa= 0.01; (d) – D= 0.7t,κa= 0.1;\nDashed line denotes “bare” Fermi surface, dotted line\n— shadow Fermi surface.\nThe poles of the Green’s function (4), determining\nthe quasiparticle dispersion and damping the limit of\nlarge enough correlation length ( vκ≪t, low tempera-\nture), are given by:\n˜E(±)=E(±)\nk−ivκ\n2/parenleftBigg\n1∓ε(−)\nk\nEk/parenrightBigg\n(5)\nwithε(±)\nk=1\n2[ε(k)±ε(k+Q)],Ek=/radicalBig\nε(−)2\nk+|D|2,\nand\nE(±)\nk=ε(+)\nk±/radicalBig\nε(−)2\nk+|D|2 (6)\nwhich is just the same as dispersion in the case of the\npresence of long-range AFM order. Equation E(−)\nk=\n0 determines the hole “pocket” of the Fermi surface,\naround the point (π\n2a,π\n2a) in the Brillouin zone, while\nE(+)\nk= 0 defines the electronic “pockets”, centered\naround (π\na,0) and (0 ,π\na), as shown in Fig. 1 (a).\nQuasiparticle damping as given by the imaginary\npart of (5) is, in fact, changing rather drastically as par-\nticle moves around the “pocket” of the Fermi surface.\nBeing practically zero in the nearest to point Γ = (0 ,0)\nnodal (i.e. on the diagonal of the Brillouin zone) point\nofthistrajectoryonthe hole“pocket”,it becomesofthe\norder of ≈vn\nFκin the far (from Γ) nodal point. Here\nwe have introduced vn\nF=|vx(k|)+|vy(k)||ε(k)=0,kx=ky\n— particle velocity at the nodal point of the “bare”Reconstruction of the Fermi surface 3\nFermi surface. On the trajectory around the electronic\n“pocket” quasiparticle damping changes from nearly\nzeronearthe crossingpoints ofthe “bare”Fermi surface\nwith Brillouin zone boundary up to ≈va\nFκat points\nclose to the similar crossing points of the “shadow”\nFermi surface. Here va\nF=|vx(k|)+|vy(k)||ε(k)=0,kx=π\nais the velocity in the antinodal point of “bare” Fermi\nsurface.\nOf course, the complete theory of quantum (Shub-\nnikov - de Haas or de Haas - van Alfen) oscillations for\nsuchpeculiarsituationcanbe rathercomplicated. How-\never, a rough qualitative criterion for the observability\nof quantum oscillations in our model can be easily for-\nmulated as follows. Effective width of spectral densities\nin our model, which determines smearing of the Fermi\nsurfaces, can be roughly compared to impurity scatter-\ning contribution to Dingle temperature and estimated\nasτ−1∼<vF>\nξ, where < vF>is the velocity aver-\naged over the Fermi surface. In fact it gives a kind of\nthe upper boundary to pseudogap scattering rate in our\nmodel. Then our criterion takes the obvious form:\nωHξ\n< vF>∼ωH\ntξ\na≫1 (7)\nwhereωHis the usual cyclotron frequency.\nAs the most unfavourable estimate (overestimating\nthe effective damping) we take:\n< vF>=/braceleftBigg\nvn\nFfor hole “pocket”\nva\nFfor electronic “pocket”(8)\nExperimentally oscillations become observable in mag-\nnetic fields larger than 50 T [8, 9, 10, 11]. Taking\nthe large correlation length ξ= 100aand magnetic\nfieldH=50T we get ωHτ≈0.8 for hole “pocket”\nandωHτ≈1.3 for electronic “pockets” in our model.\nThus we need rather large values of correlation length\nξ∼50−100afor oscillations to be observable. How-\never, this value may be smaller in the case of cyclotron\nmass larger than the mass of the free electron used in\nthe above estimates.\nFrom Luttinger theorem it follows that the number\nof electrons per cell is given by n= 2a2Sfs\nπ2, whereSfs\nis the area of the “bare” Fermi surface ( ε(k) = 0) in the\nquarter of the Brillouin zone. Similarly, we can deter-\nmine this concentration as n= 2a2Ssh\nπ2calculating the\nareaSshof the “shadow” Fermi surface ( ε(k+Q) = 0)\naroundthepoint M(π\na,π\na). Obviously Ssh=Sfs. Then,\nin the limit of |D| →0, for hole doping we get [19, 20]:\np= 1−n=a2Sh−S′\ne\nπ2=a2Sh−Se/2\nπ2(9)\nwhereShis the area of hole “pocket” and S′\neis the area\nof the parts of electronic pocket inside the quarter of/s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48 /s49/s44/s50/s48/s44/s48/s48/s48/s44/s48/s50/s48/s44/s48/s52/s48/s44/s48/s54/s48/s44/s48/s56/s48/s44/s49/s48/s48/s44/s49/s50/s48/s44/s49/s52/s48/s44/s49/s54/s48/s44/s49/s56\n/s32/s97/s50\n/s83\n/s104 /s47/s50\n/s32/s97/s50\n/s83\n/s101/s39/s47/s50\n/s61/s97/s50\n/s83\n/s101/s47/s50/s50\n/s32/s112/s61/s97/s50\n/s40/s83\n/s104 /s45/s83\n/s101/s39/s41/s47/s50/s97/s50\n/s83/s47/s50\n/s44/s32/s112\n/s68/s47/s116\nFig.2. The area of hole ( a2Sh\nπ2) and electronic ( a2S′\ne\nπ2)\n“pockets” in the quarter of Brillouin zone and “doping”\np=a2(Sh−S′\ne)\nπ2as fuctions of the pseudogap amplitude\nD/t(n= 0.9 (µ=−1.08t),t′/t=−0.4).\nthe Brillouin zone (which is a half of the total area of\nelectronic “pocket” Se).\nHowever, these expressions are valid only for |D| →\n0. With the growth of the pseudogap amplitude |D|\nthe area of both hole and electronic “pockets” diminish\n(as can be seen from Fig.1 and Fig.2). In the presence\nof electronic “pocket” this suppression of the area of\nboth “pockets”compensate each other, leaving the dop-\ning given by Eq. (9) almost unchanged (Fig.2). After\nthe disappearance of electronic “pocket”, taking place\nat|D|=µ−4t′= 0.52t(i.e. when E(+)\nk=(π/a,0)= 0),\nthere is no way to compensate the suppression of the\narea the hole “pockets” with the growth of |D|and the\nnumber of carriers, determined by (9), will also be sup-\npressed, going to zero with the disappearance of the\nhole “pocket”, taking place at |D|=−µ= 1.08t(which\nis defined by E(−)\nk=(π/2a,π/2a)= 0) and dielectric (AFM)\ngap “closes” the whole Fermi surface (Fig.1(c)). Thus,\nthe doping calculated according to Eq. (9) in the case\nof large enough pseudogap amplitude (in the absence\nof electronic “pocket”) will be significantly underesti-\nmated.\nExperimentally, only one frequency of quantum os-\ncillations F≈540Twas observed in YBCO [9]. As-\nsuming it corresponds to the presence of only the\nhole “pocket”, we obtain for the area of this “pocket”\na2Sh/π2= 0.078, which, accordingto Fig.2 corresponds\nto|D| ≈0.7t.\nGreen’s function (4) describes the “low tempera-\nture” regime of pseudogap fluctuations, when the am-\nplitude fluctuations of the random field (2) are “frozen\nout”. In the “high temperature” regime both the phase\nand the amplitude |D|of (2) are fluctuating. Assum-\ning these fluctuations Gaussian we take the probabilityFig.3. Formation of the Fermi “arcs” in the high-\ntemperature regime of pseudogap fluctuations ( n= 0.9,\nt′/t=−0.4,κa= 0.01). Shown are intensity plots\nof spectral density for ε= 0. (a) – ∆ = 0 .2t; (b) –\n∆ = 0.4t; (c) – ∆ = 0 .7t; (d) – ∆ = 1 .5t; Dashed line\ndenotes “bare” Fermi surface.\ndistribution of amplitude fluctuations given by Rayleigh\ndistribution [18]:\nPD(|D|) =2|D|\n∆2exp/parenleftbigg\n−|D|2\n∆2/parenrightbigg\n(10)\nThen the averaged Green’s function takes the form:\nG∆(ε,k) =/integraldisplay∞\n0d|D|PD(|D|)GD(ε,k) (11)\nProfilesofthespectraldensityattheFermilevel( ε= 0),\ncorresponding to (11) and different values of the pseu-\ndogap width ∆ are shown in Fig. 3. The growth of\nthe pseudogap width leads to the “destruction” of the\nFermisurfaceclosetoBrillouinzoneboundariesandfor-\nmation oftypical Fermi “arcs”, qualitatively(and quan-\ntitatively) similar to that obtained in our previous work\n[13, 14] and in accordance with the results of ARPES\nexperiments, which are typically done at much higher\ntemperatures, than experiments on quantum fluctua-\ntions.\nThis work is supported by RFBR grant 08-02-\n00021andRASprograms“Quantummacrophysics”and\n“Strongly correlated electrons in semiconductors, met-\nals, superconductors and magnetic materials”. MVS is\ngratefully acknowledges a discussion with L. Taillefer at\nGRC’07 Conference on Superconductivity, which stim-\nulated his interest in this problem.1. T. Timusk, B. Statt, Rep. Progr. Phys, 62, 61 (1999).\n2. M. V. Sadovskii, Usp. Fiz. Nauk 171, 539 (2001)\n[Physics – Uspekhi 44, 515 (2001)].\n3. M. V. Sadovskii. Models of the pseudogap state in\nhigh-temperature superconductors. In “Strings, branes,\nlattices, networks, pseudogaps and dust”, Scientific\nWorld, Moscow 2007, p.p. 357-441 (in Russian); arXiv:\ncond-mat/0408489.\n4. M. R. Norman, D. Pines, C. Kallin. Adv. Phys. 54, 715\n(2007)\n5. M. R. Norman, H. Ding, J. C. Campuzano, T. Yokoya,\nT. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki,\nP. Guptasarma, D. G. Hinks. Nature 392, 157 (1998)\n6. A. Damascelli, Z. Hussain, Z.-X. Shen. Rev. Mod. Phys.\n75, 473 (2003)\n7. J. C. Campuzano, M. R. Norman, M. Randeria. In\n“PhysicsofSuperconductors”, Vol.II,Ed.byK.H.Ben-\nnemann and J. B. Ketterson, Springer, Berlin 2004, p.p.\n167-273\n8. N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois,\nJ.-B. Bonnemaison, R.Liang, D. A.Bonn, W. N. Hardy,\nL. Taillefer. Nature 447, 565 (2007)\n9. A. F. Bangura, J. D. Fletcher, A. Carrington, J. Leval-\nlois, M. Nardone, B. Vignolle, P. J. Heard, N. Doiron-\nLeyraud, D. LeBoeuf, L. Taillefer, S. Adachi, C. Proust,\nN. E. Hussey. Phys. Rev. Lett. 100, 047004 (2008)\n10. C. Jaudet, D. Vignolles, A. Audouard, J. Levallois,\nD. LeBoeuf, N. Doiron-Leyraud, B. Vignolle, M. Nar-\ndone, A. Zitouni, R. Liang, D. A. Bonn, W. N. Hardy,\nL. Taillefer, C. Proust. Phys. Rev. Lett. 100, 187005\n(2008)\n11. D. LeBoeuf, N. Doiron-Leyraud, J. Levallois, R. Daou,\nJ.-B. Bonnemaison, N. E. Hussey, L. Balicas, B. J.\nRamshaw, R. Liang, D. A. Bonn, W. N. Hardy, S.\nAdachi, C. Proust, L. Taillefer. Nature 450, 533 (2007)\n12. N. Harrison, R. D. McDonald, J. Singleton. Phys. Rev.\nLett.99, 206406 (2007)\n13. E. Z.Kuchinskii, I.A.Nekrasov, M. V.Sadovskii. Pisma\nZh. Eksp. Teor. Fiz. 82, 217 (2005) [JETP Lett. 82, 198\n(2005)]\n14. E. Z. Kuchinskii, M. V. Sadovskii. Zh. Eksp. Teor. Fiz.\n130, 477 (2006) [JETP 103, 415 (2006)]\n15. E. Z. Kuchinskii, M. V. Sadovskii, Zh. Eksp. Teor. Fiz.\n115, 1765 (1999) [JETP 88, 347 (1999)].\n16. J. Schmalian, D. Pines, B. Stojkovic. Phys. Rev. B 60,\n667 (1999)\n17. L. Bartosch, P. Kopietz. Eur. J. Phys. B 17, 555 (2000)\n18. E. Z. Kuchinskii, M. V. Sadovskii. Zh. Eksp. Teor. Fiz.\n121, 758 (2002) [JETP 94, 654 (2002)]\n19. S. Chakravarty, H.-Y. Kee. arXiv: 0710.0608\n20. T. Morinari. arXiv: 0805.1977"    },    {        "title": "1406.6431v1.Superconductivity_in_the_vicinity_of_antiferromagnetic_order_in_CrAs.pdf",        "content": "Superconductivity in the vicinity of  antiferromagnetic  order in CrAs \nWei Wu1†, Jinguang Cheng1,2†*, Kazuyuki Matsubayashi2, Panpan Kong1, Fukun Lin1, \nChangqing Jin1, Nanlin Wang1, Yoshiya Uwatoko2*, and Jianlin Luo1* \n1 Beijing National Laboratory for Condensed Matter Physics an d Institute of Physics, Chinese Academy of Sciences, \nBeijing 100190, China \n2 Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan \n \n��These authors contributed equally to this work. \n*E-mail:  jgcheng@iphy.ac.cn ; uwatoko@issp.u-tokyo.ac.jp ; jlluo@iphy.ac.cn  \n \nOne of the common features of unconventional,  magnetically mediat ed superconductivity \nas found in the heavy-fermions, hi gh-transition-temperature (high-T c) cuprates, and iron \npnictides superconductors is that  the superconductivity emerges  in the vicinity of long-\nrange antiferromagnetically ordered state.1 In addition to doping charge carriers, the \napplication of external physical pressure has b een taken as an effective and clean approach \nto induce the unconventional superconductivity  near a magnetic qu antum critical point \n(QCP).2,3 Superconductivity has been observed in  a majority of 3d transition-metal \ncompounds,4-9 except for the Cr- and Mn-based compo unds in the sense that the low-lying \nstates near Fermi level are dominated by th eir 3d electrons. Herein, we report on the \ndiscovery of superconductivity on the verge of  antiferromagnetic order in CrAs via the \napplication of external high pre ssure. Bulk superconductivity with T c  2 K emerges at the \ncritical pressure P c  8 kbar, where the first-order anti ferromagnetic transition at T N  265 \nK under ambient pressure is comp letely suppressed. Abnormal  normal-state properties \nassociated with a magnetic QCP have been observed nearby P c. The close proximity of \nsuperconductivity to an antiferromagnetic or der suggests an unconventional pairing \nmechanism for the superconducting state of Cr As. The present finding opens a new avenue \nfor searching novel superconductors in the Cr and other transitional -metal based systems. \n CrAs adopts the orthorhombic MnP-type stru cture (inset of Fig. 1) with unit-cell \nparameters a = 5.649Å, b = 3.463 Å, and c = 6.2084 Å at room temperat ure. Its crystallographic \nand magnetic properties have been extensively investigated during 1970s due to the observation \nat T N = 270  10 K of a first-order antiferromagnetic transition,10-12 which is accompanied with \ndiscontinuous changes of lattice parameters. The mo st dramatic change is a sudden expansion of \nlattice constant b by ~ 4% below T N.13 Neutron diffraction measur ements established a double-\nhelical magnetic structure propagating along the orthorhombic c axis with the Cr moment of 1.7 \nμB lying essentially within the ab plane.11,12 The acquisition of  la rge Cr moment below T N has \nbeen rationalized as an itineran t- to localized-electron transiti on as a consequence of the sudden \nelongation of Cr-Cr seperation along the b direction.10 The coupled structural, magnetic, and \nelectronic degrees of freedom at T N make high pressure a very effective knob in fine tuning its \nground-state properties. Indeed, earlier high- pressure studies ha ve shown that the \nantiferromagnetic transition at T N can be readily suppressed by pre ssure and vanishes completely \nat P ~ 5 kbar.14 Since exotic phenomena such as the unconventional s uperconductivity are \nfrequently observed in the vicinity of a magne tic quantum critical point, we are motivated to \nexplore in great detail the physical pr operties of antiferromagnetic CrAs as T N is suppressed \nprogressively under hydrostatic pressure.  \nWe measured the resistivity (T) of a number of CrAs single crystals under various \nhydrostatic pressures up to 21.4 kbar in a wide  temperature range from 300 K down to 70 mK. \nThe (T) data of one represented sa mple shown in Fig. 1(a) illustrated the variation of T N with \npressure (See the Method  section about how the pressure values at T N are estimated). As seen in \nFig. 1(a), the first-order antiferro magnetic transition is manifested as a sudden drop of resistivity \nat T N = 264 K upon cooling at ambient pressure. As indicated by the arrows, T N decreases \nprogressively with increasing pressu re; the resistivity anomaly at T N changes from a sudden drop \nfor P < 3 kbar to a smoothly-varied kink whose ma gnitude is also dimini shed gradually within \nthe pressure range 3 < P < 7 kbar. Such an evolution of the resistivity anomaly at T N suggests a \ngradual crossover from strong to we ak first-order transition, especia lly near the crit ical pressure \nwhere T N vanishes. It should be noted that in  the previous high-pressure study on a \npolycrystalline sample T N terminated at about 150 K under 5 kbar,14 whereas our measurements on a high-quality single-crystalline  sample enabled us to track T N down to 70 K at P = 6.97 kbar. \nAbove this pressure, no anomaly can be discerned in resistivity above 3 K. \nOur key finding of this study is the observation of superc onductivity in CrAs with the \napplication of high pressure. The (T) data shown in Fig. 1(b, c) depict explicitly how the \nsuperconductivity emerges and then evolves with  pressure at low te mperatures. Below T N, (T) \nmeasured outside the pressure ce ll does not show any anomaly down to 350 mK, thus confirming \nthe absence of superconductivity in CrAs at am bient pressure. Interestingly, when the sample \nwas subjected to a minute pressure below 3 kbar, (T) starts to decrease below ~2.5 K and \ndisplays multi-step drops before reaching a consta nt, non-zero value near 1 K. The amplitude of \nresistivity drop grows progressively with pressure , and eventually reaches zero resistivity near 1 \nK at P  3 kbar, signaling the occurr ence of superconductivity. Upon further increasing pressure \nin the range 3 < P < 7 kbar where T N remains finite, the superconducting transition is featured by \nsimilar multi-step drops; the onset temperature firs t increases and then decrease with pressure, \nwhile the zero-resistivity temperature T c0 1 K keeps nearly constant. The multi-step feature \nfades away with further increasing pressure, and is  finally changed to a si ngle transition for P > 8 \nkbar where T N vanishes completely. As seen in Fig. 1(c), T c0 first increases and reaches a \nmaximum of 1.47 K at P = 10.88 kbar, above which it decreases gradually and reaches 1.17 K at \n21.37 kbar, the maximum pressure of the present study. It should be noted that the pressure-\ninduced superconductivity in CrAs as well as the p eculiar features of resistivity shown in Fig. 1 \nare well reproducible as confirmed on three indepe ndent CrAs single crystals of similar quality, \ni.e. the residual resistivity ratio RRR > 200.  \nThe above resistivity result s revealed that superconducti ng state coexists with the \nantiferromagnetically ordered state within a broa d pressure range below 7 kbar; the progressive \ndevelopment of the resistivity drop and the mu lti-step feature of superconductivity transition \nsignal the growth of supe rconducting state at the expense of th e antiferromagnetic state. In order \nto gain insights into the competing nature of thes e two states and to further verify the bulk nature \nof the observed superconductivity, we resorted to ac magnetic susc eptibility measurements that \ncan probe the diamagnetic sign al due to superconducting tran sition. As shown in Fig. 2, the \ndiamagnetic signal starts to appear at P > 3 kbar, and the superconducting shielding fraction \ngrows steadily with pressure, reaching over 90%  of the sample volume above 8 kbar. These results thus confirmed directly the bulk nature of pressure-indu ced superconductivity in CrAs. In \naddition, the onset temperatures of the diamagnetic  signal are also in ex cellent agreement with \nthe (T) data shown in Fig. 1(c).  \nBased on the above results, we can construct a temperature-pressure (T-P) phase diagram \nfor CrAs single crystal. As s een in Fig. 3, the first-order antif erromagnetic transition temperature \nTN = 264 K at ambient pressure can be suppressed qui ckly by the application of external pressure; \nan extrapolation of T N(P) curve gives a critical pressure P c  8 kbar where T N approaches zero \ntemperature. At P  Pc, bulk superconductivity with  shielding fraction more than 90% is realized \naccompanied with an order of magnitude decr ease of the 10-90% superconducting transition \ntemperature width Tc, as seen in Fig. 3(b). In this pressure region, the superconducting \ntransition temperature T c exhibits a broad maximum around 11 kbar and then decreases gradually \nwith further increasing pr essure. For 3 kbar < P < P c, the superconducting state coexists with the \nantiferromagnetic state, and the vol ume fraction grows gradually at th e expense of the other. It is \ninteresting to note that the supe rconducting state in th is pressure region is characterized by a \nhigher onset temperature for superconductivity, Fig. 1(b).  \nThe phase diagram of CrAs shown in Fig. 3 resembles that of many unconventional \nsuperconducting systems,2,15,16 including the heavy-fermion, high-T c cuprate, and iron-pnictide \nsuperconductors. As a universal trend in these sy stems, superconductivity em erges in the vicinity \nof a quantum critical point (QCP) where a high-t emperature ordered state involving spin, charge, \nor lattice degrees of freedom is suppresse d via applying extern al tuning parameter δ, such as \ndoping charge carrier, chemical or physical pres sure. In addition, the superconducting transition \ntemperature T c always passes through a maxi mum value at some critical δc, leading to a dome-\nshaped T c-δ phase diagram. The proximity of supe rconductivity to a QCP lends strong support \nfor the unconventional mechanism for Cooper pairing via the criti cal fluctuations. Despite the \nfirst-order nature of the antiferromagnetic transition in CrAs, the following observations \nsuggested the presence of strong antiferromagnetic fluctuations n ear the critical pressure P c and \nthe possible unconventional pairing mechanism. \nFirst, we found that within a large pr essure regime 3 < P < 21 kbar where \nsuperconductivity takes pl ace the normal-state (T) below 10 K follows the power-law relationship: (T) = 0 + ATn with the exponent n falling in the range 1.5  0.1, which is very \nclose to the value predicated fo r incoherent scattering of quasipa rticle via magnetic interactions \nin three-dimensional antiferromagnets .17,18 One of the representative (T) data at 9.5 kbar is \nplotted in the form of  vs T1.5 in Fig. 3(c). In our previous study,19 we have shown that (T) of \nCrAs single crystal at ambient pressure follows the Fermi-liquid behavior below 10 K, with the \nKadowaki-Woods ratio A/ 2 falling onto the universal line of many correlated metals, where  is \nthe electronic specific-heat coefficient. Our fitting to the ambient (T) confirms the previous \nresults with n = 2.15(4). The observa tion of n < 2 has been taken as  a characteristic signature for \nnon-Fermi-liquid metals near a magnetic QCP.20 \nSecond, the antiferromagnetic fluctuati ons seem to be abundant above T N in that the \nmagnetic susceptibility of CrAs  keeps increasing with temperat ure up to at least 700 K under \nambient pressure.19 As a matter of fact, such an increasi ng behavior of magne tic susceptibility \nhas been found to be universal in the recentl y discovered iron-based superconductors, and \nregarded as an indication for antiferromagnetic fluctuations.16 In this sense, these spin \nfluctuations would be optimized near the critical pressure where the antiferromagnetic order of \nCrAs is suppressed completely. Therefore, the cr itical spin fluctuations associated with the \nquantum criticality could act as an im portant glue medium for Cooper pairing. \nLast but not least, th e pressure-induced superconductivity in  CrAs is found to be sensitive \nto the residual resistivity 0. Our initial high-pressure studies on the CrAs crystals with 0  10 \nμ cm (RRR  40-50) displayed indications of s uperconductivity below 1 K, but no zero \nresistivity can be ach ieved. We finally confirmed the bu lk superconductivity with higher T c on \nhigh-quality CrAs crystals having much reduced 0  1-2 μ cm. Such a sensitivity of \nsuperconductivity and the associated T c to disorders (impurities and/or defects) would imply an \nunconventional (non s-wave) paring mechanism as found in heavy-fermion superconductors2 and \nthe p-wave superconductor Sr 2RuO 4.21 In these cases, superconductivity is expected to be \ndestroyed when the electron mean free path lmfp is smaller than the superconducting coherent \nlength . Our rough estimations of lmfp  and  seem to verify such a scenario. We first estimated  \nnear the optimal pressure fr om the upper critical field μ0Hc2, which is determined from the (T) \ncurves under different magnetic  fields. Fig. 4 displayed the μ0Hc2 as a function of T c at P = 9.5 \nkbar. Here, we determined T c as the middle point between 10% and 90% drop of resistivity with the error bar being the 10-90% width.  As shown in Fig. 4, μ0Hc2 versus T c can be fitted well with \nthe empirical formula: μ0Hc2(T) = μ0Hc2(0) {1-[T c/Tc(0)]α}, with  μ0Hc2(0) = 0.96(2) T, T c(0) = \n1.506(6) K, and  = 1.41(6), respectively. The obtained μ0Hc2(0) allows us to estimate the \nGinzburg-Landau coherence length  = 185 Å according to the relationship: μ0Hc2(0) = 0/22, \nwhere 0 = 2.067  10-15 Web is the magnetic flux quantum. Based on the measured Hall \ncoefficient |R H| = 2.6 10-10 m3/C at 5 K, on the other hand, lmfp values of about 766 Å and 153 Å \nare obtained for the clean ( 0 = 2 μ cm) and dirty ( 0 = 10 μ cm) samples, respectively, by \nassuming a spherical Fermi surface with one kind of  charge carrier. In spite of large uncertainty \nin the above estimations, we confirmed that bulk  superconductivity can be seen only in samples \nwith lmfp  > , a signature for unconventional pairing mechanism.  \nAlthough superconductivity has been reported for some chromium alloys,22 e.g. Cr-Rh \nand Cr-Re, to the best of our knowledge, CrAs is the first superconductor among the Cr-based  \ncompounds . Unlike the alloys, most of the density of states near Fermi level for CrAs is \nattributed to the Cr-3d states,23 which makes the magnetic degree of freedom relevant for the \nobserved superconductivity. As a matter of fact, the itinerant- electron antiferromagnetism of \nchromium metal itself has been the subj ect of much interest over fifty years.24 To date, no \nunambiguous evidence of supercon ductivity has been observed near the quantum critical point \nwhen the spin-density-wave antiferromagnetism of Cr is suppressed by  either high pressure25,26 \nor chemical doping27. In this regard, further theoretical an d experimental studies are needed to \nelucidate the pairing symmetry and the dominant mechanism, especially the role of magnetism, for the observed superconductivity in CrAs.  \nCrAs belongs to the large family of transitio n-metal pnictides with a general formula MX \n(M = transition metal, X = P, As, Sb), which form in either hexagonal NiAs-type (B8\n1) or \northorhombic MnP-type (B31) stru cture (for a review, see Ref.  28 and references therein). \nPrevious studies have demonstrated a close re lationship between the magnetism of 3d-element \nand the M-M interatomic distance due to the direct orbital overlap across the shared MX 6 edges. \nWith decreasing the size of X atom in the CrX se ries, the Cr moment decreases from the nearly \nfull moment of 3 μB/Cr for X = Sb to 1.7 μB/Cr for X = As, and finally to nonmagnetic for X = \nP.28 It seems that the magnetism of Cr in CrAs is  located very close to the boundary of localized- \nto itinerant-electron transition because a 5%-P doping has been shown to suppress completely the long-range antiferromagnetic transition. This fact can expl ain why the antiferromagnetic \ntransition is so sensitive to the external pressure as we observe d in this study. Beside, chemical \nsubstitutions for Cr in Cr 1-xMxAs can also suppress effectively the antiferromagnetic transition. \nTherefore, our discovery of s uperconductivity in CrAs single cr ystal would stimulate revival \ninterest in this family of compounds in the context of possi ble unconventional superconductivity.  \n \nMethods \nCrAs single crystals were grown out of th e Sn flux. Details about  the crysta l growth \nprocedures and sample characterizations at ambient pressure have been given elsewhere.19 High \npressures were generated with a self-clamped piston-cylinder cell made of non-magnetic BeCu \nand NiCrAl alloys.29 Glycerol with a room-temperature so lidification pressure far above 2 GPa \nwas chosen as the pressure transmitting medium in order to minimize the shear stress when changing pressures at room temperature. The pr essure (P) inside the piston-cylinder cell was \nmonitored by detecting the superc onducting transition temperature (T\nc) of a piece of Pb \naccording to the equation: P (kbar) = (T 0 - T c)/0.0365, where T 0 = 7.20 K is the T c o f  P b  a t  \nambient pressure. It should be noted that the pre ssure inside the clamp-t ype pressure cell is not \nconstant within the whole temperature range. It has been shown that the pressure drops about 3 \nkbar upon cooling down from ro om temperature to 4.2 K.30 Since the antiferromagnetic \ntransition of CrAs spans over a large temperature range under pressures, we have to estimate the \npressure at different temperatures  properly. According to Ref. 30,  th e pressure is nearly constant \nbelow 90 K and it increases line arly to 300 K with a slope of 0.012 kbar/K for pressure below 8 \nkbar. The pressure values at T N shown in Fig. 1(a) are estimated by the extrapolation: P(T N) = \nP(Pb) + 0.012 (T N-90).  \nThe low temperatures down to 70 mK were re ached by attaching the pressure cell to a \nHeliox-AC insert (Oxford Instru ment) or dilution refrigerator. The resistivity was measured by \nthe conventional four-probe method with the current applied along the orthorhombic b axis.  Ac \nmagnetic susceptibility was meas ured with a mutual induction method at a fixed frequency of \n317 Hz with a modulation field of about 2 Oe. The diamagnetic signal due to the superconductivity transition was estimated by compar ing to the diamagnetic signal of Pb, which \nserved as a pressure manometer, with n early the same size as the CrAs sample. \n \nAcknowledgements \nWe thank Y. P. Wang, Z. Fang, J. Q. Yan, J. S. Zhou, J. B. Goodenough, N. Mori, T. Kato, T. \nKanomata, M. Matsuda, F. Steglich, Q. Si, P. J. Sun, G. M. Zhang and L. Yu for fruitful \ndiscussions. Work at IOP/CA S was supported by the National Science Foundation of China \n(Grant Nos. 11025422, 11304371), the National Basic Re search Program of China (Grant Nos. \n2014CB921500,  2011CB921700), and the Strategic Priority Research Program of the Chinese \nAcademy of Sciences (Grant Nos. XDB01020300, XDB07000000). Work at ISSP/UT was \npartially supported by Grant-in-Aid for Scie ntific Research, KAKENHI (Grant Nos. 23340101, \n252460135), the Grant-in-Aid for Young Scientis t B (No. 24740220) from the Ministry of \nEducation, Culture, Sports, Science and T echnology, and the JSPS fellowship for foreign \nresearchers (Grant No. 12F02023).  \n \nAuthor contributions \nW.W., J.L. and N.W. synthesized the high-quality CrAs single crystals and initiated the present \nstudy; J.C., and K.M. measured the resistivity and ac magnetic susceptibili ty under high pressure \nwith piston-cylinder cell and confirmed the bulk superconductivity; F.L. pe rformed resistivity, dc \nmagnetic susceptibility, and Hall coefficient meas urements at ambient pressure; P.K. and C.J. \ncarried out the initial high-pressure resistivity measurements with diamond anvil cell; J.C. and \nJ.L. wrote the paper with contribu tions from all authors. J.L. a nd Y.U. supervised the project. \nCompeting financial interests \nThe authors declare no compe ting financial interests.  \nFigure 1  (color online) Resistivity (T) of the CrAs single crys tal under vari ous hydrostatic \npressures. (a) (T) data up to P = 6.97 kbar in the whole temperatur e range highlighting the \nvariation with pressure of the antif erromagnetic transition temperature T N indicated by the arrows, \n(b) and (c) (T) below T = 4 K in the whole investig ated pressure range highlighting the \nevolution with pressure of the superconducting transition. The pressu re values listed in (a) are \nthe estimated pressure at T N, see Method  for details. The pressure values given in (b) and (c) are \ndetermined directly from the superconducting tran sition of Pb. A schematic crystal structure of \nMnP-type CrAs is displayed in the inset of Fig. 1(a).  2.0\n1.5\n1.0\n0.5\n0.0 ( cm)\n3.0 2.0 1.0\nT (K)1 bar\n2.013.28 \n4.476.97 \n(b)400\n300\n200\n100\n0 ( cm)\n300 200 100 0\nT (K)1 bar\n1.73 kbar\n3.15\n4.06\n4.98\n5.75\n6.97\n(a)1.5\n1.0\n0.5\n0.0 ( cm)\n2.0 1.5 1.0\nT (K)21.37\n8.309.50\n10.8815.0218.55\n(c)\n \nFigure 2 (color online)  Temperature dependence of th e ac magnetic susceptibility 4 χ of a CrAs \nsingle crystal under various hydrostatic pressures.  \n -1.0-0.8-0.6-0.4-0.20.04\n2.0 1.5 1.0 0.5\nTemperature (K)2.86 kbar4.09\n4.99\n6.40\n7.859.69\n20.86\n16.19 \nFigure 3 (a) Temperature-pressure phase diagram of CrAs; (b) the su perconducting shielding \nfraction and transition temperature width as a func tion of pressure; the sym bols of circle (red), \nsquare (blue), and triangle (gr een) in (a, b) repres ent three independent samples with RRR = 240, \n327, and 250, respectively. (c) A plot of  vs T1.5 for the normal-state (T) curve at P = 9.5 kbar.  \n \n \nFigure 4  Temperature dependence of  the upper critical field μ0Hc2 for CrAs single crystal at P = \n9.5 kbar. Solid line is the fitting cu rve, see the text for details.  \n \nReferences \n1 Norman, M. R. The Challenge of Unconventional Superconductivity. Science  332, 196-\n200 (2011). \n2 Mathur, N. D.  et al.  Magnetically mediated superconductivity in heavy fermion \ncompounds. Nature  394, 39-43 (1998). \n3 Sefat, A. S. Pressure Effects on Two Superconducting Iron-Based Families. Rep. Prog. \nPhys. 74, 124502 (2011). \n4 Hardy, G. F. & Hulm, J. K. The superconduc tivity of some transition metal compounds. \nPhys. Rev.  93, 1004-1016 (1954). \n5 Schooley, J. F., Hosler, W.  R. & Cohen, M. L. Superconductivity in semiconducting \nSrTiO 3. Phys. Rev. Lett. 12, 474-475 (1964). \n6 Bednorz, J. G. & Muller, K. A. Possible high Tc superconductivity in the Ba-La-Cu-O \nsystem. Z. Phys. B  64, 189-193 (1986). \n7 He, T.  et al.  Superconductivity in the non-oxide perovskite MgCNi 3. Nature  411, 54-56 \n(2001). \n8 Takada, K.  et al.  Superconductivity in two-dimensional CoO 2 layers. Nature  422, 53-55 \n(2003). 1.0\n0.8\n0.6\n0.4\n0.2\n0.00Hc2 (T)\n1.6 1.2 0.8 0.4 0.0\nTc (K)P  = 9. 5 kbar\n \n0Hc2(0) = 0.96(2) T\nTc(0) = 1.506(5) K\n = 1.41(6)9 Kamihara, Y., Watanabe, T., Hirano , M. & Hosono, H. Iron-based layered \nsuperconductor La[O 1-xFx]FeAs (x = 0.05-0.12) with T c = 26 K. J. Am. Chem. Soc.  130, \n3296-3297 (2008). \n10 Boller, H. & Kallel, A. First order crystall ographic and magnetic phase transition in CrAs. \nSolid State Comm.  9, 1699-1706 (1971). \n11 Selte, K., Kjekshus, A., Jamison, W. E., Andr esen, A. F. & Engebretsen, J. E. Magnetic \nstructure and properties of CrAs. Acta Chem. Scand.  25, 1703-1714 (1971). \n12 Watanabe, H., Kazama, N., Yamaguchi, Y. & Ohashi, M. Magnetic structure of CrAs and \nMn-substituted CrAs. J. Appl. Phys.  40, 1128-1129 (1969). \n13 Kazama, N. & Watanabe, H.  Magnetic properties of Cr 1-xMn xAs system. J. Phys. Soc. \nJpn. 30, 1319-1329 (1971). \n14 Zavadskii, E. A. & Sibarova, I. A. Some  features of phase tr ansition in chromium \narsenide at high pressures. Sov. Phys. JETP  51, 542-547 (1980). \n15 Jin, K., Butch, N. P., Kirshenbaum, K., Pagli one, J. & Greene, R. L. Link between spin \nfluctuations and electron pairi ng in copper oxide superconductors. Nature  476, 73-75 \n(2011). \n16 Paglione, J. & Greedan, J. E. High-temperat ure superconductivity in iron-based materials. \nNature Physics  6, 645-658 (2010). \n17 Moriya, T. Spin Fluctuations in I tinerant Electron Magnetism .  (Springer-Verlag, Berlin, \n1985). \n18 Moriya, T. & Takimoto, T. Anomalous pr operties around magnetic instability in heavy-\nelectron systems. J. Phys. Soc. Jpn.  64, 960-969 (1995). \n19 Wei, W.  et al.  Low temperature properties of  pnictide CrAs single crystal. Science China  \n53, 1207-1211 (2010). \n20 Lohneysen, H. v., Rosch, A., Vojta, M. & Wolfle, P. Fermi-liquid instabilities at \nmagentic quantum phase transition. Rev. Mod. Phys.  79, 1015 (2007). \n21 Mackenzie, A. P.  et al.  Extremely strong dependence of superconductivity on disorder in \nSr2RuO 4. Phys. Rev. Lett. 80, 161-164 (1998). \n22 Matthias, B. T., Genalle, T. H., Compton, V. B., Corenzwit, E. & Hull, J. G. W. \nSuperconductivity of chromium alloys. Phys. Rev.  128, 588-590 (1962). \n23 Ito, T., Ido, H. & Motizuki, K. Electronic st ructure and magnetic properties in CrX (X=P, \nAs, and Sb). J. Mag. Mag. Mater.  310, e558-e559 (2007). \n24 Fawcett, E. Spin-density-wav e antiferromagnetism in chromium. Rev. Mod. Phys.  60, \n209-283 (1988). \n25 Jaramillo, R.  et al.  Breakdown of the Bardeen-Cooper-Schrieffer ground state at a \nquantum phase transition. Nature  459, 405-409 (2009). \n26 Jaramillo, R., Feng, Y., Wang, J. & Rosenbaum, T. F. Signatures of quantum criticality in \npure Cr at high pressure. PNAS  107, 13631-13635 (2010). \n27 Yeh, A.  et al.  Quantum phase transition in a common metal. Nature  419, 459-462 (2002). \n28 Motizuki, K., Ido, H., Itoh, T. & Morifuji, M. Electronic structure and magnetism of 3d-\ntransition metal pnictides.  131 (2009). \n29 Uwatoko, Y. Design of a piston cylinder type high-pressure cell. Rev. High Press. Sci. \nTech.  12, 306-311 (2002). \n30 Thompson, J. D. Low-temperature pressure va riation in a self-cla mping pressure cell. Rev. \nSci. Instrum.  55, 231-234 (1984). \n"    },    {        "title": "1003.4580v2.High_frequency_radio_properties_of_sources_in_the_Fermi_LAT_1_year_Point_Source_Catalogue.pdf",        "content": "The Astrophysical Journal , 718:587–595, 2010 August 1 doi: 10.1088/0004-637X/718/2/587\nC/circlecopyrt2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.\nHIGH-FREQUENCY RADIO PROPERTIES OF SOURCES IN THE FERMI -LAT\n1 YEAR POINT SOURCE CATALOG\nElizabeth K. Mahony1,2, Elaine M. Sadler1, Tara Murphy1,3, Ronald D. Ekers2, Philip G. Edwards2,\nand Marcella Massardi4\n1Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia\n2Australia Telescope National Facility, CSIRO, P.O. Box 76, Epping, NSW 1710, Australia\n3School of Information Technologies, The University of Sydney, NSW 2006, Australia.\n4INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy\nReceived 2010 March 26; accepted 2010 May 5; published 2010 July 2\nABSTRACT\nThe high-frequency radio sky, like the gamma-ray sky surveyed by the Fermi satellite, is dominated by flat spectrum\nradio quasars and BL Lac objects at bright flux levels. To investigate the relationship between radio and gamma-ray\nemission in extragalactic sources, we have cross-matched the Australia Telescope 20 GHz survey catalog (AT20G)with the Fermi -LAT 1 year Point Source Catalog (1FGL). The 6.0 sr of sky covered by both catalogs ( δ< 0\n◦,\n|b|>1.◦5) contains 5890 AT20G radio sources and 604 1FGL gamma-ray sources. The AT20G source positions\nare accurate to within ∼1 arcsec and, after excluding known Galactic sources, 43% of Fermi 1FGL sources have\nan AT20G source within the 95% Fermi confidence ellipse. Monte Carlo tests imply that at least 95% of these\nmatches are genuine associations. Only five gamma-ray sources (1% of the Fermi catalog) have more than one\nAT20G counterpart in the Fermi error box. The AT20G matches also generally support the active galactic nucleus\n(AGN) associations in the First LAT AGN Catalog. We find a trend of increasing gamma-ray flux density with\n20 GHz radio flux density. The Fermi detection rate of AT20G sources is close to 100% for the brightest 20 GHz\nsources, decreasing to 20% at 1 Jy, and to roughly 1% at 100 mJy. Eight of the matched AT20G sources have noassociation listed in 1FGL and are presented here as potential gamma-ray AGNs for the first time. We also identify\nan alternative AGN counterpart to one 1FGL source. The percentage of Fermi sources with AT20G detections\ndecreases toward the Galactic plane, suggesting that the 1FGL catalog contains at least 50 Galactic gamma-ray\nsources in the southern hemisphere that are yet to be identified.\nKey words: galaxies: active – gamma rays: galaxies – radio continuum: galaxies\nOnline-only material: color figures, machine-readable table\n1. INTRODUCTION\nThe recent release of the First Fermi -LAT (1FGL; Abdo et al.\n2010a ) catalog provides an unprecedented view of the gamma-\nray sky. Many of these high energy sources (particularly thoseat high Galactic latitudes) are attributed to active galactic nuclei\n(AGNs), where the gamma-ray emission is thought to originate\nfrom collimated jets close to the core (Blandford & Rees 1978 ).\nAs a result, flat spectrum radio-loud QSOs (FSRQs) and BL Lac\nobjects, where the jet is more closely aligned with our line of\nsight, dominate the population of extragalactic sources selected\nin gamma-ray surveys. Further properties of the AGN detected\nbyFermi are presented in more detail in the First LAT AGN\ncatalog (1LAC; Abdo et al. 2010b ).\nThe 1FGL catalog lists AGN associations for 41% of Fermi\nsources with |b|>1.\n◦5, but there are many that remain\nunidentified. The relatively large uncertainty in the Fermi\npositions (typically 540 arcsec) makes it necessary to search at\nother wavelengths in order to identify the correct astronomicalsource, and the Fermi team have used a Bayesian analysis\nmethod to cross-match the 1FGL with a range of catalogs (Abdo\net al. 2010a ).\nSearching high-frequency radio catalogs provides an alterna-\ntive and potentially more direct way to identify the extragalactic\nFermi sources. The dominant emission at high radio frequencies\noriginates from the radio jets close to the central core, and thus\nprovides a snapshot of the most recent AGN activity (Mattox\net al. 2001 ). This is very different from lower radio frequen-cies near 1.4 GHz, where extended radio lobes, built up over\nlarge periods of time, dominate the emission. The source den-\nsity of high-frequency radio sources on the sky is also lower,\n5\ngreatly reducing the number of spurious radio sources within\naFermi error box. Since most Fermi sources are expected to\nhave a flat radio spectrum, this makes radio identification with ahigh-frequency catalog much more straightforward and reliable.\nThis was recognized by Healey et al. ( 2008 ), who assembled\nan all-sky catalog of 8 GHz measurements of candidate blazars(selected on the basis of their radio spectral index at 1–5 GHz) to\nhelp identify Fermi sources. The 20 GHz AT20G catalog, which\nhas no spectral pre-selection, provides an independent check ofradio–gamma-ray associations for southern sources.\nThe Australia Telescope 20 GHz (AT20G) survey (Murphy\net al. 2010 ) is a blind radio survey of the southern sky (excluding\nthe Galactic plane at |b|<1.\n◦5) down to a limiting flux density\nof 40 mJy at 20 GHz. Observations for this survey were carriedout using the Australia Telescope Compact Array from 2004\nto 2008. This 20 GHz selection produced a catalog dominated\nby flat spectrum radio sources (Sadler et al. 2006 ; M. Massardi\net al. 2010, in preparation), which is ideal in searching for Fermi\ncounterparts. The AT20G radio positions are accurate to ∼1\narcsec, which also allows us to make optical identifications in areliable way.\n5The surface density of sources stronger than 100 mJy at 20 GHz is\n0.15 deg−2(Murphy et al. 2010 ), a factor of 17 lower than the value\n2.65 deg−2for sources stronger than 100 mJy at 843 MHz (Bock et al. 1999 ).\n587588 MAHONY ET AL. V ol. 718\nTable 1\nSource Statistics for the Fermi –AT20G Catalog Comparison\nRegion of Sky Fermi\nSources\nFullFermi -LAT catalog (1FGL) 1451\nSouthern hemisphere only: δ<0a n d|b|>1.◦5 604\nExcluding 16 hr <R.A.<18 hr and −15◦<δ< 0 577\nExcluding known Galactic objects 540(1FGL-South data set)\nNote. Since the AT20G catalog only covers the southern hemisphere,\nour analysis used a subset of the 1FGL catalog covering the sameregion of sky.\nIn this paper, we present an analysis of the AT20G sources de-\ntected in the 1FGL catalog. Section 2details the cross-matching\nprocedures between 1FGL and AT20G, while Section 3presents\nthese results. This is followed by a discussion of the multiwave-\nlength properties, including a comparison between the 20 GHz\nand gamma-ray fluxes in Section 4. Concluding remarks are\npresented in Section 5.\n2. CROSS-MATCHING 1FGL WITH THE AT20G\nCATALOG\nSince the AT20G catalog only covers the southern sky, we\nused a subset of 1FGL sources south of the equator (and\nexcluding the Galactic plane |b|<1.◦5; see Table 1)f o rt h e\ncross-matching. An incomplete region in the AT20G survey atR.A. 16–18 hr and −15\n◦<δ< 0◦, due to bad weather during\nthe observations, was also excluded. Finally, since the AT20G\ncatalog is best suited for identifying extragalactic AGN, sourcesclassified as Galactic\n6in the 1FGL catalog (including pulsars,\npulsar wind nebula, globular clusters, supernova remnants, and\nX-ray binaries) were removed from the sample. Table 1gives\na breakdown of the numbers left in the 1FGL catalog at each\nstage in this process. We have used the second version of the\n1FGL catalog (updated on 2010 February 4), which was themost recent version at the time of writing.\nUsing this set of Fermi sources (referred to here as the\n1FGL-South data set), we then searched the AT20G catalogfor radio sources within the 95% confidence ellipse given in\nthe 1FGL catalog. This resulted in 234 (43%) Fermi sources\nwith AT20G counterparts. In a few cases, there were two ormore AT20G sources within the Fermi ellipse, as discussed in\nSection 3.1, giving a total of 240 AT20G matches to the 540\nFermi sources.\nTo investigate the likely number of spurious matches due to\nchance, we performed a simple Monte Carlo test. This involved\ngenerating a fake 1FGL-South data set, with the same number ofsources and 95% confidence parameters, and then searching for\nAT20G sources. The fake data sets were produced by displacing\ntheFermi positions by 10\n◦in right ascension. This was repeated\nfive times, resulting in an average of 10.6 matches to the 540\nFermi sources. Thus, we would expect around 4.4% of the\nAT20G– Fermi matches to be chance associations. By the nature\nof our cross-matching, we are also missing the 5% of real\nassociations that lie outside the 95% Fermi confidence region.\nWe have not searched for these, since our priority is to make\na reliable set of matches that provides a “clean” sample. We\ntherefore estimate that our final list of Fermi –AT20G matches\nis 95% complete and at least 95% reliable.\n6We also excluded six sources in the LMC and SMC fields.Figure 1. Fermi detection rate of AT20G radio sources as a function of 20 GHz\nflux density. The vertical error bars show the counting errors for each bin while\nthe horizontal error bars indicate the bin size.\nFigure 2. Fermi detection rate of AT20G radio sources as a function of 20 GHz\nflux density split into flat and steep spectrum sources. The bins are the same\na ss h o w ni nF i g u r e 1with the exception of the brightest bin, which has not\nbeen plotted as there was only one source above 5 Jy (Centaurus A which hasa flat spectrum core). Where there are no detections, an upper limit is plotted\nindicating the detection rate of one source in that bin.\n(A color version of this figure is available in the online journal.)\nTheFermi detection rate for AT20G sources increases with\nincreasing 20 GHz flux density, as shown in Figure 1. Here,\nwe binned the AT20G catalog in flux density and plotted the\npercentage of AT20G sources detected in the 1FGL-South data\nset for each bin plotted. Further correlations with 20 GHz fluxdensity are discussed in Section 4.\nAlthough the AT20G source population is dominated by flat\nspectrum radio sources, the number of steep spectrum sourcesincreases at lower 20 GHz flux densities (M. Massardi et al.\n2010, in preparation). Figure 2shows the Fermi detection rates\nfor the flat and steep spectrum sources, respectively. As ex-pected, the detection rate is much higher for the flat spectrum\nsources, and there are very few steep spectrum sources detected\nin gamma rays. Here, we have used the traditional dividing linebetween the flat and steep spectrum sources of α=−0.5( g i v e n\nbyS∝ν\nα) as originally determined by Wall et al. ( 1977 ).\nThe physical basis for this definition is discussed further inNo. 2, 2010 HIGH-FREQUENCY RADIO PROPERTIES OF FERMI -LAT SOURCES 589\nTable 2\nList of Southern Fermi Sources That have More than One AT20G Source Inside the Fermi 95% Error Ellipse\nFermi -LAT Name AT20G Offset S20 α20\n5Listed 1FGL\nAssociation (deg) (mJy) Association\n1FGL J0141.7 −0929 J014125 −092843 0.068 764 ... PKS 0139 −09\nJ014137 −093001* 0.022 99 ... ...\n1FGL J0625.9 −5430 J062552 −543850* 0.133 304 −0.30 CGRaBS J0625 −5438\nJ062648 −543214 0.132 106 −1.39 ...\n1FGL J0636.1 −7521 J063515 −751418 0.128 47 ... ...\nJ063546 −751616* 0.085 3142 −0.30 PKS 0637 −75\n1FGL J1159.4 −2149 J115910 −215005 0.074 46 ... ...\nJ115921 −214246 0.113 224 −0.20 CGRaBS J1159 −2142\nJ115951 −214853* 0.087 308 −0.71 PKS 1157 −215\n1FGL J1441.7 −1538 J143956 −153150* 0.458 569 ... PKS 1437 −153\nJ144145 −152336 0.244 120 ... ...\n1FGL J2207.1 −0021 J220643 −003103* 0.194 477 ... CGRaBS J2206 −0031\nJ220755 −000215 0.373 59 ... ...\nNotes. The sources that have been used in the following analysis are marked by an asterisk. This is the brightest AT20G counterpart,\nwhich also corresponds to the 1FGL AGN association, except in the case of 1FGL J0141.7 −0929 where the brightest source falls outside\nthe 95% error ellipse.\nFigure 3. AT20G detection rate of the 1FGL-South data set as a function of\ngamma-ray flux density. The vertical error bars show the counting errors foreach bin while the horizontal error bars indicate the bin size.\nM. Massardi et al. (2010, in preparation). The spectral index be-\ntween 8.4 and 20 GHz was used to separate flat and steep spec-trum sources. As a result, AT20G sources with no 4.8 or 8.4 GHz\ninformation have been excluded in Figure 2. It should also be\nnoted that there are only 12 steep spectrum Fermi –AT20G ob-\njects that are comparable to the false identification rate of 10.6\nsources.\nWe can also turn this around and look at the AT20G detection\nrate of the 1FGL-South data set as a function of gamma-ray flux\ndensity (Figure 3). The AT20G detection rate also increases\nwith increasing gamma-ray flux density. Here and throughoutthe paper, we use the gamma-ray flux density given in the 1FGL\ncatalog. This is the differential flux at the pivot energy, which is\ndefined as the energy at which the error on the differential fluxis minimal.\nA total of 110 of the sources in the 1FGL-South data set\nare classified as variable in gamma rays, with a less than 1%\nprobability of being a steady source (Abdo et al. 2010a ). Of\nthese, 96 (87%) are detected at 20 GHz. On the other hand, only138 (32%) of the non-variable gamma-ray sources are detected\nin AT20G. This is discussed further in Section 4.1.3. AT20G SOURCES WITH FERMI COUNTERPARTS\n3.1. Fermi Sources with Multiple AT20G Matches\nTable 2lists the five cases where more than one AT20G\nsource falls within the Fermi 95% confidence ellipse. In each\ncase, the brightest 20 GHz source also has a listed association\nin the 1FGL catalog. These associations have been used in the\nfollowing analysis. In these cases, it is possible that more thanone AT20G source contributes to the gamma-ray flux observed\nbyFermi .\nTable 2also lists the Fermi source 1FGL J0141.7 −0929,\nwhich has two possible AT20G matches but one is outside\ntheFermi 95% confidence ellipse. The Fermi team associate\n1FGL J0141.7 −0929 with the radio source PKS 0139 −09\n(=AT20G J014125 −092843), which has a 20 GHz flux of\n764 mJy and a spectral index of α\n20\n1=0.06, but the AT20G\nposition of this source places it just outside the Fermi 95%\nconfidence ellipse. The AT20G catalog also lists a second\nAT20G match (AT20G J014137 −093001) with a smaller offset\nof 0.022 deg. This source is fainter at 20 GHz (99 mJy), but its\n1.4 GHz flux density of 150 mJy in the NRAO VLA Sky Survey\n(NVSS; Condon et al. 1998 ) implies a flat spectral index of\nα20\n1=−0.15. We identify AT20G J014137 −093001 as another\npotential counterpart for the Fermi source 1FGL J0141.7 −0929,\nand we use this source rather than PKS 0139 −09 in the following\nanalysis.\n3.2. New Fermi–Radio Associations\nCross-matching the 1FGL catalog with the AT20G catalog\nrevealed a number of new AGN associations, listed in Table 3.\nThese sources do not have any listed counterpart in the 1FGL\ncatalog, though some of them are listed as “affiliations” in the\n1LAC (Abdo et al. 2010b ). These are classified by the Fermi\nteam as candidate matches because the statistical Bayesian\nanalysis, which provides a quantitative association probability,\ncould not be carried out. The fact that these sources appear\nin the AT20G survey provides further evidence that they are\nthe correct AGN counterparts. The eight sources that do notappear in any of the 1LAC tables are marked by an asterisk in\nTable 3.590 MAHONY ET AL. V ol. 718\nTable 3\nAT20G Counterparts of Fermi Sources with No Listed AGN Association in the 1FGL Catalog\nFermi -LAT Name AT20G R.A. Decl. Offset lb S 20 α20\n5Optical\nAssociation (J2000) (deg) (mJy) ID\n1FGL J0124.6 −0616 J012450 −062501 01 24 50.40 −06 25 01.0 0.145 145 .4 −67.85 1 ... Y\n1FGL J0315.6 −5109 J031425 −510431 03 14 25.76 −51 04 31.7 0.218 264 .3 −54.0 112 −0.06 Y\n1FGL J0445.2 −6008 J044501 −601500 04 45 01.60 −60 15 00.2 0.105 270 .0 −38.95 50 .05 Y\n1FGL J0608.1 −0630c J060746 −062307a06 07 46.16 −06 23 07.7 0.155 213 .7 −12.6 4157 ... N\n1FGL J0648.6 −6052 J064740 −605805 06 47 40.87 −60 58 05.2 0.146 270 .9 −23.97 90 .01 N\n1FGL J0746.5 −0711 J074627 −070951* 07 46 27.33 −07 09 51.1 0.045 225 .7+ 8 .89 3 ... N\n1FGL J0814.5 −1011 J081411 −101210 08 14 11.71 −10 12 10.8 0.096 231 .9+ 1 3 .29 8 ... Y\n1FGL J0904.7 −3514 J090442 −351423* 09 04 42.34 −35 14 23.8 0.018 259 .4+ 7 .8 300 0 .19 N\n1FGL J1029.2 −6422 J103042 −641526* 10 30 42.66 −64 15 26.2 0.200 288 .4 −5.49 80 .15 N\n1FGL J1153.4 −8108 J115546 −810117 11 55 46.09 −81 01 17.7 0.154 300 .7 −18.47 5 −0.42 Y\n1FGL J1234.0 −5736 J123407 −573552* 12 34 07.04 −57 35 52.9 0.020 300 .6+ 5 .24 9 −0.22 N\n1FGL J1256.1 −5922 J125604 −591943* 12 56 04.93 −59 19 43.4 0.051 303 .5+ 3 .54 1 −0.81 N\n1FGL J1314.9 −5338 J131504 −533436* 13 15 04.24 −53 34 36.0 0.063 306 .5+ 9 .1 865 ... N\n1FGL J1553.5 −3116 J155333 −311832 15 53 33.49 −31 18 32.3 0.038 342 .6+ 1 7 .24 7 −0.62 Y\n1FGL J1656.2 −3257 J165616 −330207* 16 56 16.83 −33 02 07.7 0.085 350 .6+ 6 .4 287 0 .19 N\n1FGL J1823.5 −3454 J182338 −345412* 18 23 38.58 −34 54 12.0 0.016 358 .7 −9.98 4 −0.24 N\n1FGL J2014.5 −0047 J201428 −004723 20 14 28.62 −00 47 23.0 0.021 42 .0 −18.84 9 ... Y\n1FGL J2022.5 −4532 J202226 −451329 20 22 26.40 −45 13 29.2 0.311 354 .9 −34.5 173 ... N\nNotes. The sources marked with an asterisk are not identified in either of the 1LAC secondary tables.\naThis source (1FGL J0608.1 −0630c) may be a spurious Fermi detection and is discussed in Section 3.3\nMany of the new AT20G identifications in Table 3are at\nlow Galactic latitude ( |b|<10◦). This is not surprising as the\nAGN catalogs used by the Fermi team to identify the source\nof the gamma-ray emission are known to be incomplete at low\nGalactic latitudes (Abdo et al. 2010b ).\n3.3. 1FGL J0608.1 −0630c\nThe AT20G source J060746 −062307 (PKS 0605 −06) was\nfound within the 95% confidence region of the Fermi source\n1FGL J0608.1 −0630c. This source is in a region of high-\nmass star formation (known as Mon R2) and was identified\nby Jackson et al. ( 2002 ) as a Galactic H iiregion. There is also\nan ultra-compact H iiregion with associated methanol masers\n(Walsh et al. 1998 ), and it is listed as a Galactic source in\nthe AT20G catalog. This source has been flagged as confused\nin the Fermi catalog (denoted by the “c” in the name), meaning\nthat the detection is highly sensitive to the model used to\nsubtract diffuse Galactic gamma-ray emission. As noted in\nSection 4.7 of the 1FGL catalog paper (Abdo et al. 2010a ),\nthis particular source is located close to the Orion nebula, and\nthere is a high chance that it is a spurious Fermi detection.\nIf real, 1FGL J0608.10630c is almost certainly Galactic ratherthan an extragalactic source and has been removed from the\nfollowing analysis, but the possibility of gamma-ray emission\nrelated to extreme mass outflows in this region should beinvestigated.\n3.4. The Fermi–AT20G Catalog\nThe complete list of all 233 Fermi –AT20G matches is\nincluded as additional material in the electronic version of thispaper (see Table 4). The content of each column is as follows.\n(1)Fermi name (1FGL JHHMM.m-DDMM).\n(2) and (3) Fermi right ascension and declination.\n(4) and (5) Semimajor and semiminor axes of 95% confidence\nellipse.\n(6) Position angle of 95% confidence ellipse.(7) Pivot energy in units of MeV . This is the energy at which\nthe error on the differential flux is minimal.\n(8) and (9) Gamma-ray flux density and uncertainty in\ncm\n−2MeV−1s−1. This is the differential flux at the pivot energy.\n(10) Gamma-ray variability index. Values greater than 23.21\nhave<1% probability of being a steady source (Abdo et al.\n2010a ).\n(11) Fermi Bright Source Catalogue identification where\navailable (0FGL JHHMM.m-DDMM).\n(12) Association as listed in 1FGL.(13) Class as listed in 1FGL.\n(14) AT20G name (AT20GJHHMMSS-DDMMSS).\n(15) and (16) AT20G right ascension and declination.(17) and (18) AT20G Galactic longitude and latitude.\n(19) Offset between the Fermi and AT20G positions in\ndegrees.\n(20)–(22) 20 GHz, 8.4 GHz, and 4.8 GHz flux densities\nin mJy.\n(23) Radio spectral index between 4.8 and 20 GHz.(24) Quality flag of the AT20G flux. “p” denotes a poor flux\nmeasurement for a variety of reasons explained in Murphy et al.\n(2010 ). “g” refers to a good flux measurement.\n(25) 1 GHz flux density in mJy from either the NVSS or\nSydney University Molonglo Sky Survey (SUMSS) catalogs.\n(26) and (27) Position of optical identification where avail-\nable.\n(28) b\nJmagnitude from the SuperCOSMOS database.\n(29) and (30) Redshift and the corresponding reference.\n3.5. Galactic Latitude Distribution of Fermi–AT20G Matches\nFigure 4shows the overall Fermi detection rate for extragalac-\ntic sources in the AT20G catalog, binned by Galactic latitude.\nRadio sources identified as Galactic H iiregions and planetary\nnebula in the AT20G catalog were not included in this analysis.\nDue to the much smaller area of sky at b> 0 in the southern\nhemisphere, the data were binned in |b|rather than bto improve\nthe statistics.No. 2, 2010 HIGH-FREQUENCY RADIO PROPERTIES OF FERMI -LAT SOURCES 591\nTable 4\nTheFermi -AT20G Catalog\nFermi -LAT Name AT20G Name RA Dec S 20 α zR e f .\n(J2000) (J2000)\n1FGL J0000.9-0745 AT20GJ000118-074626 00:01:18.04 −07:46:26.8 177 ··· ··· ···\n1FGL J0004.7-4737 AT20GJ000435-473619 00:04:35.65 −47:36:19.0 868 −0.03 0.875 E. K. Mahony 2010, in prep.\n1FGL J0013.1-3952 AT20GJ001259-395426 00:12:59.89 −39:54:26.4 1609 −0.16 ··· ···\n1FGL J0017.4-0510 AT20GJ001735-051241 00:17:35.84 −05:12:41.5 186 ··· 0.227 1LAC\n1FGL J0029.9-4221 AT20GJ003017-422446 00:30:17.51 −42:24:46.3 478 −0.05 0.495 Jackson et al. 2002\n1FGL J0038.4-2504 AT20GJ003814-245901 00:38:14.72 −24:59:01.9 1125 0.46 0.498 Jones et al. 2009\n1FGL J0047.3-2512 AT20GJ004733-251717 00:47:33.08 −25:17:17.7 608 −0.90 0.001 1LAC\n1FGL J0048.0-8412 AT20GJ004426-842241 00:44:26.26 −84:22:41.8 484 −0.16 1.032 1LAC\n1FGL J0049.8-5738 AT20GJ004959-573827 00:49:59.48 −57:38:27.6 1872 −0.05 1.797 Peterson et al. 1976\n1FGL J0050.0-0446 AT20GJ005021-045221 00:50:21.59 −04:52:21.0 279 ··· 0.920 1LAC\n(This table is available in its entirety in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and co ntent.)\nFigure 4. Percentage of AT20G sources that have Fermi counterparts binned in\nequal area bins of Galactic latitude. The region close to the plane ( |b|<1.◦5)\nis excluded here as that area was not covered in the AT20G survey. Once againthe vertical error bars are the counting errors, while the horizontal error barsshow the size of each bin. The lower plot shows the distribution of sources\nflagged as Galactic in the AT20G catalog. These sources have been removed\nwhen calculating the detection rate. The dashed line at |b|=1.\n◦5 indicates the\nlimit of the AT20G survey, and lower latitudes have been excluded from thisanalysis.\nThe Fermi detection rate of ∼4% is roughly constant with\nGalactic latitude, as would be expected if the AT20G sourcesbeing matched are predominantly AGNs. A linear regression\nanalysis reveals a gradient of 0.009, and we cannot rule out\nthe null hypothesis that the slope is flat. The flat distributionprovides further evidence that the AT20G– Fermi associations\nare genuine matches. If there were many spurious identifica-\ntions, we would expect to see a peak close to the Galactic plane,where the surface density of Fermi sources is higher than at the\nGalactic poles.\nWe can now turn this around to look at the AT20G detection\nrate of sources in the 1FGL-South data set (Figure 5), after first\nremoving the 37 1FGL sources that are associated with Galactic\nobjects such as pulsars and pulsar wind nebulae (none of theseFigure 5. AT20G detection rate binned in Galactic latitude. The black circles\nshow the detection rate of AT20G sources in the 1FGL-South data set (where\nknown Galactic sources have been removed), while the red squares show thedetection rate when these sources are included. The lower plot shows thehistogram of sources defined as Galactic in the 1FGL catalog that have been\nadded when plotting the red squares. The dotted line shows the linear fit to the\nblack points.\n(A color version of this figure is available in the online journal.)\n37 Galactic Fermi sources are detected in the AT20G survey).\nThe AT20G detection rate of 1FGL sources is notconstant\nwith Galactic latitude, but decreases at lower Galactic latitudes.Here, a linear regression fit has a slope of 0.23, and we can\nrule out that the slope is flat at the 99% confidence level. This\nstrongly suggests that the 1FGL catalog contains a mixture ofextragalactic and (unidentified) Galactic sources, whose relative\nnumbers change with |b|. The red squares in Figure 5show the\ndetection rate when the known Galactic sources are added backto the 1FGL-South sample.\nTo calculate the minimum number of Galactic sources still\nunidentified, we assumed that there were no unidentified Galac-tic sources toward the Galactic poles. Taking the detection rate\nof 52% given in the last bin in Figure 5to be the expected AT20G\ndetection rate across the whole sky means that there are at least592 MAHONY ET AL. V ol. 718\nFigure 6. Minimum number of Galactic Fermi sources needed to make the\nAT20G detection rate constant with the Galactic latitude.\n50 high-latitude ( |b|>1.◦5) Galactic gamma-ray sources in the\nsouthern hemisphere that still remain to be identified. The esti-\nmated distribution of these sources is shown in Figure 6. Here,\nthe total number of sources in each bin is the amount needed tomake the AT20G detection rate constant with Galactic latitude\n(Figure 5). The Galactic sources already identified are shown\nby the solid line.\nWe performed a two-dimensional (2D) Kolmogorov–Smirnov\ntest comparing the 1FGL-South Galactic latitude distribution\nwith the distribution of Fermi –AT20G matches (the solid and\ndashed lines shown in Figure 7). This showed that they were not\ndrawn from the same population at the 99% significance level.\nAs the AT20G matches are all AGNs, this provides further evi-\ndence that not all the Galactic sources in the 1FGL catalog have\nbeen identified.\n4. RESULTS\n4.1. Flux Comparison\nTo investigate any correlations between the radio and gamma-\nray properties, we plotted the 20 GHz flux against the gamma-\nray flux density as shown in Figure 8, revealing a trend of\nincreasing gamma-ray flux density with increasing 20 GHz\nflux. This was also hinted at by the Fermi -LAT Bright Source\nCatalogue (Abdo et al. 2009b ), but the fainter gamma-ray\nsources were needed to confirm that the trend did exist. Some\nadditional scatter may be introduced due to the fact that theAT20G observations were taken up to 2–5 years previous to\ntheFermi observations. However, it is still notable that there are\nno bright 20 GHz sources with weak gamma-ray fluxes. Theupper limits shown on the left side of the plot represent the\n1FGL-South sources that do not have an AT20G counterpart.\nAs discussed in Section 3.5, some of these may be unidentified\nGalactic sources for which we would not expect to see any\ncorrelation with 20 GHz flux.\nSpearman-rank correlation tests reveal correlation coeffi-\ncients of ρ=0.44 and ρ=0.46 for the variable and non-\nvariable sources, respectively. The probabilities of finding these\ncoefficients by chance are P=9.0×10\n−6for the variable\nsources and P=1.5×10−8for the non-variable sources.\nFor all the Fermi –AT20G sources, the correlation coefficient is\nρ=0.49 with a chance probability of P=7.5×10−16. TheseFigure 7. Galactic latitude distribution of all 1FGL-South sources (solid line),\nsources with AT20G matches (dashed line), and sources that have been newly\nidentified as counterparts (dotted line).\nFigure 8. 20 GHz flux against Fermi flux density for sources classified as\nvariable in 1FGL (red “+”) and those not classified as variable (blue “ ×”).\nThere is a clear trend for the non-variable sources. The Fermi sources not\ndetected in AT20G are shown as triangles at the 40 mJy flux limit of the survey(also shown by the dashed line). The median gamma-ray fluxes for the upperlimits are denoted by the stars, but it should be noted that these upper limits\ninclude a mixture of populations and are not limited to AGN.\n(A color version of this figure is available in the online journal.)\nnumbers are similar to the correlation coefficient of ρ=0.42\nwith chance probability P=4.5×10−6between the 8.4 GHz\nand peak gamma-ray flux given in Giroletti et al. ( 2010 ) and\nAbdo et al. ( 2009a ). The above references also note that a num-\nber of biases and selection effects need to be taken into accountto test the significance of these correlations. One such exam-\nple is the bias introduced due to the selection of flat spectrum\nsources from lower frequency catalogs taken at different epochs,as is done for the CRATES catalog. Using the AT20G survey to\nfindFermi counterparts provides a clean and unbiased sample\nas it is a blind survey at 20 GHz. Due to the large number ofupper limits, a survival analysis approach is needed to investi-\ngate the significance of these correlations. However, since many\nof the upper limits are likely to be Galactic sources that wouldcontaminate any correlation, we have not attempted that in this\npaper.No. 2, 2010 HIGH-FREQUENCY RADIO PROPERTIES OF FERMI -LAT SOURCES 593\nFigure 9. 20 GHz flux against, the Fermi flux density for the different classes\nof sources given in the 1FGL catalog. The stars on the left once again show the\nmedian gamma-ray fluxes for the Fermi sources not detected in AT20G.\n(A color version of this figure is available in the online journal.)\nFigure 8also shows that the variable Fermi sources have\nhigher gamma-ray fluxes than the non-variable sources. This\ncan account for the higher AT20G detection rate for the variable\nsources mentioned in Section 2, as the correlation between\ngamma-ray and 20 GHz flux implies that a larger fraction of\nthe brighter gamma-ray sources will be above the 40 mJy fluxlimit of the AT20G survey.\nFigure 9shows that this trend splits into different population\nclasses. These are classified in the 1FGL catalog (Abdo et al.2010a ) as blazars (either BL Lac objects (bzb) or flat spectrum\nradio quasars (bzq)), non-blazar active galaxies (agn), active\ngalaxies of unknown type (agu), or starburst galaxies (sbg). Thenew AT20G associations (those listed in Table 3)a r es h o w n\nby the purple diamonds. Optical spectroscopy is needed to\nproperly classify these objects. Spearman-rank correlation testsgive a correlation coefficient of ρ=0.31 and a corresponding\nchance probability of P=0.0006 for the FSRQ and ρ=0.43,\nP=0.002 for the BL Lac objects. In comparison, Abdo\net al. ( 2009a ) reported correlation coefficients of ρ=0.19,\nP=0.080 and ρ=0.49,P=0.00055 for the FSRQ and\nBL Lac objects, respectively. This work used the 8.4 GHz fluxand the peak gamma-ray flux above 100 MeV for sources in the\nFermi -LAT Bright AGN Sample (LBAS).\nThese results indicate that the basic emission mechanism\nfor the radio and gamma-rays is coupled, with both related\nto the relativistic jets of the AGN. The radio emission results\nfrom synchrotron emission by relativistic electrons while thegamma-ray emission is thought to be created by inverse Comp-\nton scattering of photons by similar, or possibly the same, high\nenergy particles (Maraschi et al. 1992 ; Ghisellini & Madau\n1996 ; Bloom & Marscher 1996 ). A similar correlation between\nthe gamma-ray and high-radio frequency fluxes was found in\nKovalev et al. ( 2009 ), who compared the average 100 MeV–1\nGeV photon flux with quasi-simultaneous Very Long Baseline\nInterferometry (VLBI) 15 GHz flux measurements for objects\nin the Fermi -LAT Bright Source Catalogue. There is also the\npossibility that the Doppler boosting effects are similar in both\nthe radio and gamma-ray regimes. This could account for thestrength of the correlation seen in Figures 8and9given that the\nobservations were not simultaneous.Figure 10. Comparison of spectral indices between 4.8 to 8.4 GHz and 8.4 to\n20 GHz. The majority of Fermi sources are clustered toward the center of the plot\nimplying that they have flat spectral indices from 4.8 to 20 GHz. The gray pointsshow the distribution of all AT20G sources with simultaneous measurementsat the three frequencies. The tail of AT20G sources in the lower left corner is\nthe steep spectrum radio sources. The dashed line marks the division of flat and\nsteep spectrum sources.\n(A color version of this figure is available in the online journal.)\n4.2. Radio Spectral Indices\nThe highly variable nature of flat spectrum radio sources\nmakes it difficult to study the spectral index properties withoutsimultaneous data. The AT20G survey is therefore very suitable\nfor investigating these properties since many objects have near\nsimultaneous measurements at 20, 8.4, and 4.8 GHz (separatedby approximately one month). Figure 10shows the comparison\nof radio spectral indices from 4.8 to 8.4 GHz and from 8.4\nto 20 GHz for those objects with simultaneous AT20G data.Comparing the Fermi detections with the entire AT20G catalog\n(shown in gray in Figure 10) highlights the fact that the gamma-\nray AT20G sources are generally confined to the flat spectrumpopulation, typical of FSRQ and BL Lac objects.\nWe have also cross-matched with lower radio frequency\ncatalogs using either the 1.4 GHz NVSS (Condon et al. 1998 )\nor the 843 MHz SUMSS (Mauch et al. 2003 ) catalogs.\n7These\ntwo surveys are ideal, since when they are combined, they cover\nthe entire southern sky at a similar resolution (45 arcsec) to asimilar flux limit (approximately 2 and 6 mJy, respectively).\nFigure 11shows the distribution in spectral index for a range of\ndifferent frequencies. As expected the vast majority of sources\nhave flat spectra for all frequency ranges. The median spectral\nindices for each distribution are α\n20\n8=−0.16,α8\n5=−0.01,and\nα5\n1=0.00,where S∝να.\n4.3. Optical Properties\nThe accuracy of the AT20G positions allows us to search for\noptical counterparts in a straightforward manner. The Super-COSMOS Science Archive (Hambly et al. 2001 ) was searched\nfor the closest source within a 2.5 arcsec radius down to\nthe completeness limit of b\nJ=22 mag. The Galactic plane\n(|b|<10◦) was excluded in the analysis due to the high density\n7The SUMSS catalog excludes |b|<10◦so the second-epoch Molonglo\nGalactic Plane Survey (MGPS-2; Murphy et al. 2007 ) was also searched in\norder to have complete sky coverage. For simplicity, we refer to both of thesecatalogs combined as the SUMSS catalog.594 MAHONY ET AL. V ol. 718\nFigure 11. Radio spectral index distributions between different frequencies\nwhere S∝να.\nFigure 12. Redshift distribution for variable Fermi –AT20G sources (top) and\nnon-variable sources (bottom). The solid line shows the redshifts that were foundby searching existing databases, and the dashed line shows the distribution of\nthe 1LAC redshifts.\nof foreground stars and Galactic dust extinction. There were 35\nFermi –AT20G matches excluded as a result. Using this simple\napproach introduces selection effects, meaning that some iden-\ntifications may be missed. For example, very nearby galaxies\n(i.e., Centaurus A, NGC 253, and NGC 4945) are not foundbecause the search radius is too small compared to the angular\nsize of the galaxy. High redshift quasars may also be missed\ndue to the Ly αforest moving into the b\nJband and therefore\nfalling below the magnitude limit. A more rigorous method of\nfinding optical counterparts for AT20G sources is ongoing and\nis discussed in E. K. Mahony et al. (2010, in preparation).\nOur analysis resulted in 172 (87%) optical counterparts with\nbJ/lessorequalslant22 mag. Of these, 95 also have redshift information\nfrom either the NASA Extragalactic Database,8the 6dF Galaxy\nSurvey (Jones et al. 2009 ), or were obtained from follow-up\nobservations (E. K. Mahony et al. 2010, in preparation). Therelease of the 1LAC catalog (Abdo et al. 2010b ) provided\nadditional redshifts that were not available in existing databases.\n8http://nedwww.ipac.caltech.edu/Figure 13. Redshift distribution for Fermi –AT20G sources. Flat spectrum radio\nQSOs are shown in the top panel and BL Lac objects in the bottom panel.\nThese are included in the data table with the reference “1LAC”\nand are shown by the dashed line in Figures 12and13.\nFigure 12shows the distribution of redshifts for both the\nvariable Fermi sources and the non-varying sources. Median\nredshifts are z=1.09 and z=0.79,respectively. Figure 13\nshows the redshift distribution for the flat spectrum radio quasarsand BL Lac objects. Due to the many unknown biases and\nselection effects, it is difficult to interpret the significance of\nthese results. For example, many variable sources are classified\nas flat spectrum radio quasars for which obtaining redshifts at\nhigher redshifts is much easier due to the presence of strongemission lines. It can be more difficult for other extragalactic\npopulations, such as BL Lac objects where the continuum often\ndominates any features in the optical spectrum. This is shown inFigure 13where the quasars span a much larger range in redshift\nwith a median of z=1.19 whereas the BL Lac objects have\nlower redshifts with a median of z=0.32.\nThese selection biases, along with the incompleteness of the\nredshift sample, mean that these results are not necessarily\nrepresentative of the entire Fermi –AT20G population.\n5. CONCLUSIONS\nThis paper presents results comparing the First Fermi -LAT\ncatalog (1FGL) with the AT20G survey. A blind survey at high\nradio frequencies is an ideal catalog to search for extragalactic\ngamma-ray counterparts as, like the gamma-ray sky, it isdominated by flat spectrum radio sources (either quasars or\nBL Lac objects).\nCross-matching the 1FGL catalog with the AT20G catalog\nresulted in 233 (43%) associations within the 95% Fermi\nconfidence ellipse. Monte Carlo tests show that we expect 4.4%\nto be chance associations, meaning that the final catalog of\nFermi –AT20G matches is 95% complete and 95% reliable. The\ndetection rate of Fermi sources in the AT20G catalog increases\nstrongly with 20 GHz flux density. It ranges from approximately\n1% at 100 mJy to almost 100% at 20 GHz fluxes above 10 Jy.\nA number of sources were identified that did not have\nany association information in the published 1FGL catalog.\nWhile some of these have been cataloged as “affiliations”\nin the 1LAC catalog, eight of them are identified here aspotential Fermi counterparts for the first time. The AT20G\nsource J014137 −093001 is also identified as an alternativeNo. 2, 2010 HIGH-FREQUENCY RADIO PROPERTIES OF FERMI -LAT SOURCES 595\ncounterpart to the Fermi source 1FGL J0141.7 −0929. The\nremaining AT20G matches are in agreement with the Fermi\nAGN source association.\nThis confirms that the majority of gamma-ray sources have\nflat radio spectra as the AT20G survey provides an unbiased\nsample with no spectral pre-selection. It also verifies that thereis no missing gamma-ray population of steep spectrum radio\nsources.\nThe distribution of AT20G detections as a function of Galactic\nlatitude reveals that the percentage of AT20G matches decreases\ntoward the Galactic plane. We calculate that there are at least\n50 unidentified Galactic gamma-ray sources in the southernhemisphere at latitudes above |b|=1.\n◦5, as the AGN detection\nrate should not vary with Galactic latitude.\nWe find a trend of increasing gamma-ray flux density with\n20 GHz flux density, which is slightly stronger for the non-\nvariable Fermi sources. The majority of the Fermi –AT20G\nmatches have flat radio spectral indices across a range offrequencies from 1 GHz to 20 GHz. Of the Fermi sources\nwith AT20G detections, 87% also have optical counterparts\nin SuperCOSMOS. The median redshifts are z=1.09 and\nz=0.79 for the variable and non-variable Fermi sources,\nrespectively. The flat spectrum radio quasars have a median\nredshift of z=1.19 and the BL Lac objects have a median\nredshift of z=0.32.\nThe AT20G survey provides a clean catalog to search for\nextragalactic gamma-ray sources observed in the southernhemisphere. As such it provides a useful database for identifying\ngamma-ray counterparts as the Fermi mission continues to\ndiscover many more of these objects.\nThis research made use of data provided by the NASA /\nIPAC Extragalactic Database (NED) which is operated by\nthe Jet Propulsion Laboratory, California Institute of Technol-\nogy, under contract with the National Aeronautics and SpaceAdministration. It has also made use of data obtained from the\nSuperCOSMOS Science Archive, prepared and hosted by the\nWide Field Astronomy Unit, Institute for Astronomy, University\nof Edinburgh, which is funded by the UK Science and Technol-\nogy Facilities Council. The authors thank Andrew Walsh forhis helpful advice. We also thank the referee for very useful\ncomments. E.M.S. acknowledges support from the Australian\nResearch Council through the award of an ARC Australian Pro-fessorial Fellowship. Partial financial support for this research\nhas been provided to M.M. by the Italian ASI /INAF Agreement\nI/072/09/0 for the Planck LFI Activity of Phase E2 and I /016/\n07/0 “COFIS.”\nNote Added in Proof . A recent preprint by Ghirlanda et al.\n(2010 ) also cross matches the 1FGL and AT20G catalogues.\nThese authors find a similar (though not identical) list of Fermi -\nAT20G counterparts, but their paper focuses on different aspects\nof the sample properties and includes SEDs of a few interesting\nAT20G sources.\nREFERENCES\nAbdo, A. A., et al. 2009a, ApJ,700, 597\nAbdo, A. A., et al. 2009b, ApJS ,183, 46\nAbdo, A. A., et al. 2010a, arXiv: 1002.2280\nAbdo, A. A., et al. 2010b, ApJ,715, 429\nBlandford, R. D., & Rees, M. J. 1978, in Pittsburgh Conf. on BL Lac Objects,\ned. A. M. Wolfe (Pittsburgh: Univ. Pittsburgh Press), 328\nBloom, S. D., & Marscher, A. P. 1996, ApJ,461, 657\nBock, D., Large, M. I., & Sadler, E. M. 1999, AJ,117, 1578\nCondon, J. J., Cotton, W. D., Greisen, E. W., Yin, Q. F., Perley, R. A., Taylor,\nG. B., & Broderick, J. J. 1998, AJ,115, 1693\nGhirlanda, G., Ghisellini, G., Tavecchio, F., & Foschini, L. 2010, MNRAS,\nsubmitted (arXiv: 1003.5163 )\nGhisellini, G., & Madau, P. 1996, MNRAS, 280, 67\nGiroletti, M., Reimer, A., Fuhrmann, L., Pavlidou, V ., & Richards, J. L. 2010,\narXiv: 1001.5123\nHambly, N. C., et al. 2001, MNRAS ,326, 1279\nHealey, S. E., et al. 2008, ApJS ,175, 97\nJackson, C. A., Wall, J. V ., Shaver, P. A., Kellermann, K. I., Hook, I. M., &\nHawkins, M. R. S. 2002, A&A ,386, 97\nJones, D. H., et al. 2009, MNRAS ,399, 683\nKovalev, Y . Y ., et al. 2009, ApJ,696, L17\nMaraschi, L., Ghisellini, G., & Celotti, A. 1992, ApJ,397, L5\nMattox, J. R., Hartman, R. C., & Reimer, O. 2001, ApJS ,135, 155\nMauch, T., Murphy, T., Buttery, H. J., Curran, J., Hunstead, R. W.,\nPiestrzynski, B., Robertson, J. G., & Sadler, E. M. 2003, MNRAS ,342,\n1117\nMurphy, T., Mauch, T., Green, A., Hunstead, R. W., Piestrzynska, B., Kels, A.\nP., & Sztajer, P. 2007, MNRAS ,382, 382\nMurphy, T., et al. 2010, MNRAS ,402, 2403\nPeterson, B. A., Jauncey, D. L., Condon, J. J., & Wright, A. E. 1976, ApJ,L5\nSadler, E. M., et al. 2006, MNRAS ,371, 898\nWall, J. V ., Pearson, T. J., & Longair, M. S. 1977, in IAU Symp. 74, Radio\nAstronomy and Cosmology, ed. D. L. Jauncey (Cambridge: Cambridge Univ.\nPress), 269\nWalsh, A. J., Burton, M. G., Hyland, A. R., & Robinson, G. 1998, MNRAS ,\n301, 640"    },    {        "title": "1307.1291v2.State_Counting_for_Excited_Bands_of_the_Fractional_Quantum_Hall_Effect__Exclusion_Rules_for_Bound_Excitons.pdf",        "content": "State Counting for Excited Bands of the Fractional Quantum Hall E\u000bect: Exclusion\nRules for Bound Excitons\nAjit C. Balram,1Arkadiusz W\u0013 ojs,2and Jainendra K. Jain,1\n1Department of Physics, 104 Davey Lab, Pennsylvania State University, University Park PA, 16802 and\n2Institute of Physics, Wroclaw University of Technology, 50-370 Wroclaw, Poland\n(Dated: June 15, 2021)\nExact diagonalization studies have revealed that the energy spectrum of interacting electrons in\nthe lowest Landau level splits, non-perturbatively, into bands, which is responsible for the fascinating\nphenomenology of this system. The theory of nearly free composite fermions has been shown to be\nvalid for the lowest band, and thus to capture the low temperature physics, but it over-predicts the\nnumber of states for the excited bands. We explain the state counting of higher bands in terms of\ncomposite fermions with an in\fnitely strong short range interaction between an excited composite-\nfermion particle and the hole it leaves behind. This interaction, the form of which we derive from\nthe microscopic composite fermion theory, eliminates con\fgurations containing certain tightly bound\ncomposite-fermion excitons. With this modi\fcation, the composite-fermion theory reproduces,for\nall well-de\fned excited bands seen in exact diagonalization studies, an exact counting for \u0017 >1=3,\nand an almost exact counting for \u0017\u00141=3. The resulting insight clari\fes that the corrections to\nthe nearly free composite fermion theory are not thermodynamically signi\fcant at su\u000eciently low\ntemperatures, thus providing a microscopic explanation for why it has proved successful for the\nanalysis of the various properties of the composite-fermion Fermi sea.\nPACS numbers:\nI. INTRODUCTION\nThe eigenspectrum of interacting particles contains\nfundamental information about its physics. A non-\nperturbative restructuring of the low energy spectrum by\nthe interaction sometimes signals the formation of com-\nplex quasiparticles that are in general distinct from the\noriginal particles and may even carry unusual quantum\nnumbers1. An important example is that of interacting\nelectrons in the lowest Landau level (LL). The most strik-\ning aspect of its exact Coulomb spectrum is that it splits,\nnonperturbatively, into mini-bands2, referred below to\nas \\\u0003 bands.\" The \u0003 bands owe their existence entirely\nto interactions, and their formation lies at the heart of\nmany phenomena, most notably the fractional quantum\nHall e\u000bect (FQHE)3. Therefore, an understanding of the\nphysical origin of these bands and their structure as a\nfunction of the \flling factor \u0017is a central problem of the\nFQHE.\nThe non-perturbative physics of bands within the low-\nest LL is understood as a consequence of the formation\nof composite fermions. The composite fermion (CF)\ntheory4{6postulates a mapping of the problem of in-\nteracting electrons at \flling factor \u0017into that of com-\nposite fermions at \flling factor \u0017\u0003, the two related by\n\u0017=\u0017\u0003=(2p\u0017\u0003\u00061) wherepis a positive integer. The un-\nderlying physics is that electrons capture 2 pvortices each\nas a result of the repulsive interaction; this, in turn, pro-\nduces a dynamics, due to the Berry phases produced by\nthe bound vortices, as though composite fermions experi-\nenced a reduced magnetic \feld given by B\u0003=B\u00002p\u001a\u001e0,\nwhere\u001e0=hc=e is the \rux quantum. If one neglects the\ninteractions between composite fermions, then the CF\ntheory predicts that the \u0003 bands at \u0017have a one to onemapping to the familiar \\Landau bands\" at \u0017\u0003\nfree CF model : \u0003 bands at \u0017,Landau bands at \u0017\u0003\nHere, the term \\Landau bands\" refers to the bands in the\nmany particle spectrum , to be distinguished from Landau\nlevels , which are the energy levels of a single particle;\nof course, all eigenstates of the Landau bands can be\nenumerated straightforwardly from the knowledge of the\nsingle particle Landau levels. Should this mapping be\ncon\frmed, one can interpret the \u0003 bands as arising from\nsingle-particle \\\u0003 levels\" (\u0003Ls) of composite fermions,\nwhich are analogous to the electron LLs at \u0017\u0003, thus en-\nabling a \\single-particle-like\" explanation of the FQHE.\nIt should be stressed that, unlike LLs, the \u0003Ls are a non-\ntrivial emergent concept that cannot be derived at a sin-\ngle particle level. The only way to con\frm their reality is\nto compare the above correspondence between the exact\nmany particle spectra of interacting electrons at \u0017and\nthe Landau bands at \u0017\u0003. In other words, while we con-\nstruct many body states of electrons starting from single\nparticle LLs, for composite fermions we must carry out\nthe inverse program: we only have the many body states\nto begin with, from which we must deduce the existence\nof \u0003 levels. A single particle derivation of \u0003 levels is\nnot possible, because composite fermions themselves are\ncollective entities de\fned only inside the background of\nother composite fermions.\nThe important question, of course, is: How well does\nthe mapping work? Previous studies have shown that\nthelowest \u0003 band at\u0017does indeed resemble the lowest\nLandau band at \u0017\u0003. The correct account of the states\nbelow the gap is fortunately su\u000ecient for the explanation\nof many phenomena occurring at low temperatures. In\nparticular, the FQHE at the Jain fractions \u0017=n=(2pn\u0006arXiv:1307.1291v2  [cond-mat.str-el]  28 Nov 20132\n1) is explained as the IQHE of composite fermions, with\nthe ground state represented by n\flled \u0003Ls of composite\nfermions.\nThe mapping, however, breaks down qualitatively for\nexcited bands. As shown by Wu and Jain7, the analogy to\nfree fermions at \u0017\u0003predicts spurious \\unphysical\" states\nin higher bands, i.e. states without counterparts in the\n\u0003 bands seen in exact diagonalization. The number of\nsuch unphysical states grows with increasing energy.\nWhy should we be concerned about excited bands? We\nwould of course like to gain as complete an understanding\nof this quintessential many-body system as possible, but\nthere are also at least two experimental motivations be-\nhind our current study. One, while many phenomena are\ngoverned by the low lying states, high energy excitations\ncan also be probed, e.g., by light scattering. In particular,\nthe existence of higher \u0003 bands has been invoked for an\nexplanation8of the high energy collective modes observed\nin Raman experiments of Hirjibehedin et al.9and Rhone\net al.10Two, the distinction between the lowest and the\nhigher \u0003 bands vanishes in the limit of \u0017= 1=2; here the\ngaps separating the \u0003 bands collapse and all\u0003Ls par-\nticipate in the low energy manifold. One may therefore\nquestion the applicability of the model of nearly free com-\nposite fermions for the thermodynamics of the CF Fermi\nsea state at \u0017= 1=211. Notwithstanding, the model of\nnearly free composite fermions has proved successful in\nunderstanding experiments. (The Chern Simons theory\nof the 1/2 state developed by Halperin, Lee and Read11\nalso does not contain the physics of the missing states.)\nA number of experimental results, such as the Shub-\nnikov de Haas oscillations12,13, spin polarization14,15,\nvalley polarization16, commensurability oscillations17{22,\nthermopower23and cyclotron resonances24,25of compos-\nite fermions have been successfully analyzed in terms of\na Fermi sea of nearly free composite fermions. This sug-\ngests that the model of nearly free composite fermions\nought to be valid to at least some extent, but that is not\nobvious, a priori , in light of the above discussion of the\nmissing states.\nThis raises several conceptual questions. How do we\nunderstand the excited \u0003 bands of the exact spectra?\nDoempty \u0003Ls really exist above the CF Fermi level?\n(The existence of \flled \u0003 levels has been conclusively\ndemonstrated by a variety of means, but relatively few\nstudies have probed the empty \u0003 levels10.) How do we\nreconcile the experimental successes of the nearly free-\nCF model with the theoretical discrepancies mentioned\nabove? We answer these questions in this paper.\nOur principal result is to show that the the concept of\nempty \u0003 levels above the CF Fermi level is perfectly sen-\nsible, and that the excited \u0003 bands can be constructed\nfaithfully within the CF theory, provided we impose\na strong short-range repulsion between a CF particle\n(CFP) and a CF hole (CFH) to eliminate certain tightly\nbound CF excitons. Here \\CFP\" refers to a composite\nfermion in an otherwise empty \u0003L, and \\CFH\" to a miss-\ning composite fermion in an otherwise full \u0003L. Elimina-tion of these CF excitons produces an excellent account\nof the \u0003 bands as seen in exact diagonalization studies.\nThis understanding clari\fes that the number of elim-\ninated states is thermodynamically insigni\fcant at low\ntemperatures, when the density of CFPs and CFHs is\ndilute. Our calculations also show, surprisingly, that\nthe residual interactions between composite fermions be-\ncomes weaker as we approach the CF Fermi sea at the\nhalf \flled LL. These \fndings provide a microscopic jus-\nti\fcation for the adequacy of the nearly-free CF model\nfor the low temperature thermodynamics of the FQHE\nstates as well as of the CF Fermi sea.\nThe picture that has emerged reveals two kinds of in-\nteractions between composite fermions. One is a weak\ninteraction that causes \u0003 band broadening but does not\nchange the number of states in each \u0003 band. Accurate\nmethods have been developed to treat that interaction.\nThe other interaction, the one identi\fed in this paper, is\nin\fnitely strong and eliminates some states by sending\nthem to in\fnity { this interaction acts as a constraint\nthat projects out certain con\fgurations, thereby altering\nthe structure of the \u0003 bands. The latter interaction will\nbe seen to have a strange form, in that it acts only be-\ntween a CFP and a CFH occupying di\u000berent \u0003Ls. With\nthese two interactions, the CF theory provides an ex-\ncellent quantitative description of the \u0003 bands wherever\nthey can be clearly identi\fed in exact Coulomb spectra\n(we have tested three lowest bands at 1/3, 2/5 and 3/7,\nand two lowest bands at \flling factors in between).\nThe plan of the paper is as follows. In the next sec-\ntion, we show results that illustrate the discrepancy be-\ntween the \\naive\" prediction from the non-interacting CF\ntheory and exact diagonalization. Section III contains\na postulate for the strong interaction between compos-\nite fermions. This postulate is shown to be qualitatively\nvalid in Section IV and quantitatively accurate in Section\nV. The paper is concluded with some remarks on the rele-\nvance of our results to experiments. We assume fully spin\npolarized electrons throughout this study, as our goal is\nto establish the principles governing the counting of the\nlevels in the excited bands.\nII. THE PUZZLE OF \\MISSING STATES\" IN\nTHE EXCITATION SPECTRUM\nThe essential puzzle can be seen in Figs. 1, 2 and 3.\nThese \fgures show Coulomb spectra of interacting elec-\ntrons in the lowest LL in the spherical geometry26as a\nfunction of N, the number of electrons, and 2 Q, the \rux\nthrough the surface of the sphere (in units of \u001e0=hc=e).\nLis the total orbital angular momentum. The eigenener-\ngies are obtained exactly by a brute force diagonalization\nof the Coulomb Hamiltonian. The energies are shown in\nunits ofe2=\u000f`, where`=p\n~c=eB is the magnetic length\nand\u000fis the dielectric constant of the background mate-\nrial.\nThe top row of Fig. 1 depicts several exact Coulomb3\nFIG. 1: (color online) Emergent \\\u0003 bands\" in the FQHE spectra. The exact Coulomb energies are shown by dashes, obtained\nin the spherical geometry for Nelectrons at total \rux 2 Q(hc=e). The left, middle and right panels correspond to \flling factors\n\u0017= 1/3, 2/5, and 3/7, respectively. The top panel shows the spectra for the incompressible state, while the middle and bottom\npanels show spectra of systems that contain a single CF particle or CF hole in the ground state. The energy is quoted in units\nofe2=\u000f`, where`=p\n~c=eB is the magnetic length and \u000fis the dielectric constant of the host medium. Alternate bands are\ncolored in red and blue. The integers in brackets give the number of Lmultiplets in the Landau bands of free fermions at\n(N;2Q\u0003) withQ\u0003=Q\u0000N+ 1. The integers outside the bracket show the number of multiplets that remain after eliminating\ntheL= 1 CF exciton; these agree with the actual number of multiplets except in two cases in panels (a) and (g), marked in\ngreen. The dots are the CF spectra obtained by CF diagonalization.\nspectra for the incompressible states at 1/3, 2/5 and 3/7;\nthe middle and bottom rows show spectra for systems\neach of whose ground states contains a single CFP or\na single CFH. (Note that in all cases, the excited states\ncontain one or more CFPs and CFHs.) In each case, we\nhave taken the largest Nfor which the relevant bands are\nwell de\fned; for larger Nthe bands do exist, but begin\nto overlap, making it di\u000ecult to identify them clearly as\nwould be needed for an unambiguous counting of states\nwithin each band. Fig. 2 shows spectra for systems for\nwhich the ground band contains several CFPs or CFHs,\nas indicated in the \fgure caption. In some cases, the sec-\nond excited bands are also identi\fable; two such exam-\nples are shown in Fig. 3 where the ground band contains\na single CFP or CFH in the 3/7 state.\nIn all cases, the spectrum contains \u0003 bands that are\nidenti\fable by inspection. For clarity, we have shown the\nstates of successive bands with di\u000berent colors. In theabsence of interelectron interaction, allstates would de-\ngenerate (because all electrons are occupying the lowest\nLL), producing a degenerate Landau band. It is expected\nthat the Coulomb interaction will broaden the Landau\nband, but there is no a priori reason to suspect that it\nwould split into \u0003 bands. The emergence of \u0003 bands\nis a nonperturbative e\u000bect, persisting as we reduce the\nstrength of the Coulomb interaction, say by taking the\ndielectric function \u000fto be very large.\nThe CF theory explains the formation of \u0003 bands\nwithin the lowest Landau band as follows. It postu-\nlates the formation of composite fermions, bound states\nof electrons and 2 pvortices, which experience an e\u000bec-\ntive magnetic \feld B\u0003=B\u00002p\u001a\u001e0. It further postulates\nthat composite fermions can be treated, to a \frst approx-\nimation, as noninteracting. The FQHE problem is thus\nmapped into the problem of noninteracting fermions at\nB\u0003. Translating into the spherical geometry in the stan-4\ndard manner, the problem of interacting electrons at 2 Q\nis mapped into that of noninteracting fermions at an ef-\nfective \rux 2 Q\u0003= 2Q\u00002p(N\u00001). We allow fermions at\n2Q\u0003to occupy higher LLs. The \u0003 bands at 2 Qare thus\npostulated to be related to the Landau bands at 2 Q\u0003.\nAssuming free composite fermions, the number of\nstates in \u0003 bands can thus be enumerated straightfor-\nwardly. Given the symmetry of the problem, these or-\nganize themselves into Lmultiplets, with each multiplet\nconsisting of 2 L+ 1 states. The values of Lin a given\nband can be determined from the angular momentum al-\ngebra. For this purpose, we \frst list all con\fgurations\nof fermions that correspond to a given band. A compos-\nite fermion in the lowest \u0003 level has angular momentum\nl=jQ\u0003j; a composite fermion in the \frst excited LL has\nl=jQ\u0003j+1; a composite fermion in the second excited LL\nhasl=jQ\u0003j+2; and so on. Given the occupations of dif-\nferent \u0003 levels, we can determine all possible total angu-\nlar momenta Lby a simple calculation. Filled bands can\nbe left out in this calculation, as they do not contribute\nto the total angular momentum, and for bands that are\nalmost full, it is easier to consider the CFHs rather than\nCFPs. We must be careful here to take account of the\nfermionic nature of the CFPs or CFHs. Fig. 4 shows the\nvarious con\fgurations for the \frst four bands at \u0017\u0003= 3\n(i.e.,\u0017= 3=7).\nFor speci\fcity, let us take the 3/7 state ( N;2Q) =\n(18;37) which maps into ( N;2Q\u0003) = (18;3) of composite\nfermions. We ask what is the spectrum of free composite\nfermions here. The ground state is unique, in which the\nlowest three \u0003 levels, corresponding to angular momen-\ntum shells with l= 1:5, 2.5 and 3.5, are full, producing\nL= 0. For the \frst excited band, we create a CFP\nwithlqp= 4:5 and a CFH with lqh= 3:5, which pro-\nduces a single multiplet at each of L= 1;2;\u0001\u0001\u0001;8. The\nsecond excited \u0003 band consists of (i) a CFP-CFH pair\nwithlqp= 5:5 andlqh= 3:5; (ii) a CFP-CFH pair with\nlqp= 4:5 andlqh= 2:5; or (iii) two CFPs with lqp= 4:5\nand two CFHs with lqh= 3:5. The predicted number of\nLmultiplets is given in Fig. 1c inside brackets.\nThe analysis for other fractions is analogous. The num-\nber of multiplets for each Lin each band is shown in Figs.\n1, 2 and 3 inside brackets. We make the following obser-\nvations:\n\u000fFor the lowest \u0003 band, the structure predicted by\nthe free CF theory is in perfect agreement with that\nseen in exact diagonalization. That has been con-\n\frmed for a large many other con\fgurations, and,\nto date, there is no known exception to this cor-\nrespondence in the range of \flling factors in the\nlowest LL where FQHE is seen. (We note that the\ncorrespondence does not always hold in the second\nLL, or at\u0017 <1=5 in the lowest LL where the crystal\nphase becomes relevant.)\n\u000fFor higher \u0003 bands, the predicted Lvalues in gen-\neral do not agree with the observed ones. The dis-\nagreement is minimal for the \frst excited band atthe Jain fractions n=(2pn\u00061), i.e.the single exciton\nbranch, where the only deviation is that the free-CF\ncounting predicts one multiplet at L= 1 whereas\nnoL= 1 multiplet is observed in the exact spectra.\nFor some cases, the deviation can be much larger.\nFor example, for ( N;2Q) = (16;32), there are 16\nL= 5 multiplets, whereas the correspondence with\nthe noninteracting system ( N;2Q\u0003) = (16;1) pre-\ndicts 27L= 5 multiplets.\n\u000fThe number of multiplets predicted by the free CF\ntheory is never less than that observed in numerical\nexperiments. The free CF theory thus over-predicts\nthe number of excited states.\nThis illustrates the puzzle of the missing states, an\nexplanation of which is the goal of the present article.\nWe note that it is clear that the mapping between free\ncomposite fermions and the interacting electrons in the\nlowest LL must break down at some energy, because, for\na system with \fnite N, the FQHE system in the lowest\nLL has a \fnite number of linearly independent states,\nwhereas the free fermion system with an in\fnite number\nof Landau levels has an in\fnite number of linearly inde-\npendent states. However, this observation by itself does\nnot explain why the mapping breaks down already at the\n\frst excited band; there are certainly su\u000eciently many\nstates available in the lowest Landau band to allow the\nmapping to hold for several lowest bands. A possibility,\nwhich would a priori seem more likely, would be that the\nmapping would continue to hold, as we go to higher and\nhigher bands, until all lowest LL states are exhausted. It\nis also unclear why the number of missing states depends\non the \flling factor and the angular momentum Lin a\nseemingly haphazard fashion as seen above.\nWith more work, the CF theory actually explains the\nexcited bands as well. While we will be concentrating\non the counting rules, the CF theory also tells us how to\nconstruct explicit wave functions for the \u0003 band states at\n2Qstarting from the Landau band states at 2 Q\u0003. In this\nmethod, we \frst construct all wave functions in the qth\nLandau band of Nelectrons at 2 Q\u0003, and denote this basis\nbyf\bN;2Q\u0003;q;L;\u000bg, where we have chosen the basis func-\ntions to have de\fnite total orbital angular momentum L,\nand\u000blabels the di\u000berent orthogonal wave functions with\nthe sameLquantum numbers. Now we construct Jain's\nCF wave function corresponding to each basis function\nas (in Haldane's spherical geometry26)\n\tCF\nN;2Q;q;L;\u000b =PLLLY\nj<k(uivj\u0000viuj)2p\bN;2Q\u0003;q;L;\u000b (1)\nwhich produces a correlated basis f\tCF\nN;2Q;q;L;\u000bgat 2Q=\n2Q\u0003+ 2p(N\u00001). Hereu= cos(\u0012=2)e\u001e=2andv=\nsin(\u0012=2)e\u0000i\u001e=2denote the position of a particle on the\nsphere (\u0012and\u001eare the angular coordinates); qlabels\nthe \u0003 bands; and it can be proven that the angular\nmomentum quantum number Lremains invariant un-\nder this mapping5. The symbolPLLLdenotes lowest5\nFIG. 2: Exact Coulomb spectra are shown for several systems, whose ground band corresponds to several QPs or QHs: (a)\ntwo QPs of 2/5; (b) two QPs of 3/7; (c) two QHs of 3/7; (d) two QHs of 2/5; (e) three QPs of 3/7; (f) three QHs of 3/7.\nThe energies levels are shown by dashes and alternate \u0003 bands are colored in red and blue for ease of identi\fcation of di\u000berent\nbands. The black dots show the CF energies obtained from CF Diagonalization in the basis derived from all states up to the\n\frst excited Landau band. The integers in the brackets give the number of Lmultiplets in the Landau bands of free fermions\nat (N;2Q\u0003), where 2Q\u0003= 2Q\u00002(N\u00001). The integers outside the brackets show the number of multiplets that remain after\nexcluding the single excitons listed in Eq. 2, and agree with the actual numbers of multiplets in all cases.\nFIG. 3: Exact Coulomb spectra for ( N;2Q) = (17;35) corresponding to a single QH at 3/7 (left panel) and (16 ;32) corresponding\nto a QP at 3/7 (right panel); energies levels are shown by dashes and alternate \u0003 bands are colored in red and blue. The\nblack dots show the CF energies obtained from CF Diagonalization in the basis derived from all states up to the second excited\nLandau band. The integers in the brackets give the number of Lmultiplets in the Landau bands of free fermions at ( N;2Q\u0003),\nwhere 2Q\u0003= 2Q\u00002(N\u00001). The integers outside the brackets show the number of multiplets that remain after excluding the\nsingle excitons listed in Eq. 2, and agree with the actual numbers of multiplets in all cases.6\nFIG. 4: (color online) This \fgure shows all con\fgurations up\nto energy less than or equal to 3 units of CF cyclotron energy\n(~!\u0003\nc) at\u0017\u0003= 3 (which corresponds to the electron \flling\n\u0017= 3=7). The angular momentum of each \u0003 level is shown;\nit isjQ\u0003jfor the lowest \u0003 level and increases by one unit for\neach successive \u0003 level.\nLL projection, which is accomplished either by the Dev-\nJain method2(which can be implemented for small sys-\ntems with, typically, N\u001410) or by the Jain-Kamilla\nmethod27(which enables a treatment of large systems as\nwell). We then orthogonalize this basis, and diagonalize\nthefullCoulomb Hamiltonian in this basis. This pro-\ngram is called \\CF diagonalization\" (CFD). For small\nsystems, orthogonalization can be performed by express-\ning the wave functions in the Slater determinant basis;\nfor larger systems, a Monte Carlo method has been devel-\noped by Mandal and Jain28to perform CFD (the eval-\nuation of various multi-dimensional matrix elements is\nperformed by the Monte Carlo method, in which a de-\nsired accuracy can be obtained by running the computerfor a su\u000eciently long time). The various steps are com-\nplicated, but have been given in the literature27,28and\nwill not be repeated here, except to note that no approx-\nimations are made.\nA remarkable observation made by Wu and Jain7(also\nsee Ref. 2) was:\n1. When we composite-fermionize all (orthogonal)\nstates of a Landau band at 2 Q\u0003, the resulting states\nat 2Qare often not all linearly independent. The\nresulting linearly independent basis at 2 Qis thus,\nin general, smaller than that at 2 Q\u0003. Such lin-\near dependences are surprising because the actual\nwave functions are very complex, and indicate some\nhidden mathematical structures in Jain's CF wave\nfunctions that are yet to be discovered.\n2. The number of linearly independent states at 2 Q\nagrees with that seen in exact spectra for all cases\nstudied so far.\nWe have found this to be valid for all additional cases\nstudied during the course of the present work. This tells\nus that the CF theory allows, in principle, a calculation\nof the number of states in each band. Nevertheless, the\nway things stand at this stage, to get the correct count-\ning it is necessary to construct the orthogonal CF basis,\nwhich requires a complex computer calculation. That is\nunsatisfying, because: it can be accomplished only for\nsmall systems; it does not give us an insight into the\norigin of the missing states; and it fails to clarify if the\nmissing states are thermodynamically signi\fcant. Our\naim in this article is to bring further clarity to this issue.\nIII. EXCLUSION RULES FOR SINGLE\nEXCITONS\nHaving shown that the model of noninteracting com-\nposite fermions produces spurious unphysical states in\nthe excitation spectrum, it is natural to ask if the ex-\ncited \u0003 bands can be explained in terms of interacting\ncomposite fermions. This interaction must be in\fnitely\nstrong, so that it will eliminate some states by sending\nthem to in\fnity. Further, it must do so in precisely such\na manner as to bring the \\modi\fed Landau bands\" into\none-to-one correspondence with the \u0003 bands. It will also\nbe most satisfying, and also meaningful, if the interaction\nis a relatively simple 2-body interaction.\nHow do we identify the form of the interaction that will\naccomplish the task? Let us \frst note that no spurious\nstates appear in the lowest Landau band, which implies\nthat no interaction is needed for fermions in the \\ground\"\nmanifold. (Please recall that we are concerned here with\na strong interaction that would eliminate states; there is\nof course a weak residual interaction between composite\nfermions that would broaden the ground manifold, but\nwithout altering the number of states.) For the incom-\npressible states at 1/3, 2/5 and 3/7 (upper panels) the7\nL= 1 exciton is missing in the \frst excited \u0003 band.\nThis suggests that we should impose an interaction that\neliminates the L= 1 CF exciton. As noted above, the\nmicroscopic CF theory actually predicts this elimination,\nbecause a \\composite-fermionization\" of the L= 1 exci-\nton wave function at \u0017\u0003= 1, 2 or 3 annihilates it2(also\nsee Ref. 29).\nWe make the simplest assumption that the interaction\nacts only on particle-hole pairs of composite fermions. A\njusti\fcation of this assumption can come only a poste-\nriori. If it does not work, or works only partially, we\nwill need to return to this assumption and modify it.\nGiven that the microscopic CF theory contains informa-\ntion about the missing states, we are motivated to use the\nCF theory itself to derive the constraints on single exci-\ntons. We use the notation in which jn!n+k;Lilabels a\nsingle exciton at L\u0015kin which an electron has been ex-\ncited from the nthLL (n= 1;2;\u0001\u0001\u0001) byk= 1;2;\u0001\u0001\u0001units\nof CF cyclotron energy ~!\u0003\nc. (No exciton with L < k is\npossible for an exciton in which a composite fermion has\nbeen excited across k\u0003 levels because the angular mo-\nmenta of the CFP and CFH di\u000ber by k.) We construct\nwave functions of single CF excitons with progressively\nhigher cyclotron energies and ask how many are elimi-\nnated upon composite-fermionization. We have carried\nout this program for k\u00143 and our \fndings are summa-\nrized in the following paragraph and in Fig. 5.\nWe \fnd that for the fractions \u0017=n=(2n+ 1) with\nn= 2;3, the CF exciton jn!n+L;Liis to be excluded\nbecause its wave function is linearly dependent on the CF\nexciton wave functions at the same Lin lower bands. (We\nnote that there is \rexibility as to which of the linearly\ndependent CF excitons we exclude; we have made choices\nwhich we \fnd to be most convenient.) For \u0017= 1=3 all\nexcitonsj1!1 +k;LiwithL\u00142k\u00001 are eliminated.\nThus the following CF excitons are excluded for k\u00143\n(shown pictorially in Fig. 5):\n\u0017=1\n3:j1!2; 1i;j1!3; 2i;j1!3; 3i;\nj1!4; 3i;j1!4; 4i;j1!4; 5i\n\u0017=2\n5:j2!3; 1i;j2!4; 2ij2!5; 3i (2)\n\u0017=3\n7:j3!4; 1i;j3!5; 2i;j3!6; 3i\nThese exclusion rules have been demonstrated exactly\n(numerically) by performing CFD on all Nvalues up to\n14, 16 and 18 for \u0017= 1=3, 2/5 and 3/7, respectively, and\nare expected to be valid for arbitrary N. We do not have\na simple understanding of these exclusion rules or for the\ndi\u000berence between the rules at 1/3 and those at other\nfractions of the form n=(2n+ 1). Nonetheless, it should\nbe stressed that these exclusion rules are not ad hoc but\nare a consequence of exact linear dependences of the CF\nexciton wave functions. This extends previous works in\nRefs. 2,7,8 on the elimination of single excitons.\nIn the following, we explore the consequences of these\nexclusion rules. These will surely modify the spectrum,\nbut the question is to what extent the modi\fed spectrum\nagrees with that seen in exact diagonalization studies.IV. MODIFIED SPECTRUM\nWe now proceed to construct the modi\fed \u0003 bands\nin which the above excitons are excluded. For a gen-\neral case, we \fnd it convenient to diagonalize the basis\nf\bN;2Q\u0003;q;L;\u000bgfor an interaction which imposes a very\nheavy penalty for the excitons enumerated above, and\nretain only the zero energy states. (To obtain Leigen-\nstates, we also add other pseudopotentials with a very\nsmall coe\u000ecient of order 10\u00006. This interaction is small\nenough that it allows us clearly to identify the zero en-\nergy states.) Take for example the second excited band\nof the states with n\u00152 \flled \u0003 levels. This contains two\nkinds of states. In one, a single CFP is excited across two\n\u0003 levels; we already know, from our rules above, which of\nthese are to be eliminated. The nontrivial part consists\nof the two CF excitons, each produced by exciting a sin-\ngle particle across one \u0003 level. To project out all states\ncontaining thejn!n+ 1; 1iexciton, we diagonalize our\ninteraction in the basis of two particles and two holes\nwith appropriate angular momenta and identify the zero\nenergy states. Combining the results produces the num-\nber of multiplets shown outside brackets in the top panels\nof Fig. 1. All other systems can be treated analogously,\nand Figs. 1, 2 and 3 show the number of remaining L\nmultiplets in each modi\fed Landau band.\nFIG. 5: This \fgure depicts the missing CF excitons (their an-\ngular momenta Lare shown) up to 3 units of \\CF cyclotron\nenergy.\" The top, middle and bottom panels are for excita-\ntions out of one, two and three \flled \u0003 levels, respectively.8\nFor some simple cases, the counting can be done\nstraightforwardly, as we illustrate by taking a couple of\nexamples. It is convenient directly to count the number\nof eliminated states. First consider the second excited\nband of the \u0017\u0003= 3 system with ( N;2Q\u0003) = (18;3) [i.e.\n(N;2Q) = (18;37)]. Here, there are two kinds of states,\nthose containing two excitons and those containing a sin-\ngle exciton. In the \frst kind, we wish to eliminate all con-\n\fgurations in which one of the two excitons has L= 1.\nWe therefore ask how many multiplets can be formed by\ncombining two excitons, with one exciton at L= 1 and\nthe other at L= 1;2;\u0001\u0001\u0001;8. We use the relations:\n1\n1 = 0\b2\n1\n2 = 1\b2\b3\n1\n3 = 2\b3\b4\n\u0001\u0001\u0001\n(In the \frst equation above the two excitons are identi-\ncal bosons, which is the reason why L= 1 is missing on\nthe right hand side; for all other cases, the two excitons\nhave di\u000berent angular momenta and hence are distin-\nguishable particles.) This predicts 1 ;1;3;3;3;3;3;3;2;1\nmissing multiplets at L= 0;1;\u0001\u0001\u0001;9. In the second kind\nof states with a single exciton (in which a single com-\nposite fermion has been excited across two \u0003 levels), the\nL= 2 multiplet is missing as per our exclusion rules. To-\ngether, this produces 1 ;1;4;3;3;3;3;3;2;1 missing mul-\ntiplets atL= 0;1;\u0001\u0001\u0001;9, which are precisely the number\nof eliminated multiplets observed in exact diagonaliza-\ntion (i.e., the di\u000berence between the numbers inside and\noutside the brackets in Fig. 1c for the second excited\nband). For the system in Fig. 1f that contains a CFP\nin the ground state, we add an L= 1 exciton to a CFP\nwithL= 4 to predict a single missing multiplet at each of\nL= 3;4;5. For the the system in Fig. 1f which contains\na CFH in the ground state, we add an L= 1 exciton to a\nCFH withL= 3:5 to obtain a single missing multiplet at\neach ofL= 2:5;3:5;4:5. These also agree perfectly with\nthe correct counting. The analysis for the other panels\ncan be performed analogously.\nOur calculations show that for \u0017\u0003>1 (\u0017 >1=3), the\nelimination of the above mentioned excitons brings the\ncounting of states into perfect agreement with the exact\nresults. This is true even of the second excited band,\nwherever it is identi\fable in the exact spectra. We have\nconsidered all Nvalues accessible to our exact diago-\nnalization study (up to 16 and 18 particles for 2/5 and\n3/7, respectively), and con\frmed the agreement when-\never well de\fned \u0003 bands can be identi\fed.\nFor\u0017\u0003\u00141 (\u0017\u00141=3), the modi\fed counting is in\nmuch better, but not perfect, agreement with the exact\nresults. There is slight mismatch for the second excited\nband at\u0017= 1=3 (Fig.1a) and for the \frst excited band\nof the CFH of \u0017= 1=3 (Fig. 1g); the predicted numbers\nthat do not match the exact ones are shown by green\ncolor. (The exact number can be obtained by counting\nthe number of multiplets in the spectrum.) The predictednumber is still slightly greater than that observed, which\npossibly indicates the need for an additional three-body\nconstraints, e.g. on the 2CFH+1CFP trion state at \u0017\u0003=\n1. We have not pursued this further.\nV. QUANTITATIVE CONFIRMATION\nAs mentioned above, we can go beyond the count-\ning of multiplets of various \u0003 bands to construct ex-\nplicit basis functions for them by CFD. As also men-\ntioned above, composite-fermionization of the \u0003 bands of\nfreecomposite fermions produces the correct counting; in\nthis case, the number of linearly independent states af-\ntercomposite-fermionization is less than that before, and\nprecisely equal to the number of physical states in the \u0003\nbands. We have con\frmed that composite-fermionization\nof the modi\fed spectrum does not eliminate any states.\nThis shows how we can identify the surviving states with-\noutactually doing the full CFD.\nIn either case, CFD produces a detailed prediction for\nthe actual spectra. The spectra obtained from CFD are\nshown by dots in Fig. 1, quantitatively establishing the\nvalidity of the CF description for the excited \u0003 bands.\nWhile the agreement for the excited \u0003 bands is less ac-\ncurate than that for the lower \u0003 bands, it is still excellent\nand captures the energy splittings very nicely. The slight\ndiscrepancy arises because in the actual system, compos-\nite fermions mix with higher \u0003 level states to lower their\nenergy. Such mixing is small, as indicated by the small-\nness of the deviation between the exact and the CFD\nenergies. However, if need be, more accurate energies\nmay be obtained from the CF theory by allowing \u0003L\nmixing by incorporating yet higher \u0003 bands in the CFD\ncalculation. An explicit example is shown in Figure 6 for\n\u0017= 3=7, which gives a much more accurate account of\nthe second and third excited \u0003 bands that than that in\nFig. 1c .\nAs we mentioned above, in addition to the in\fnite\nhard-core interaction that eliminates certain excitons,\nthere exists a weak residual interaction between compos-\nite fermions that is responsible for a splitting of the states\nin a given \u0003 band. This is best captured by the process of\nCFD, which produces the splittings very accurately. One\nmay ask if there is a simple e\u000bective interaction between\nCFP-CFP, CFP-CFH and CFH-CFH that will describe\nthat splitting30,31. That question is outside the scope of\nthis paper.\nVI. CONCLUDING REMARKS\nIn summary, we have shown that the entire many parti-\ncle excitation spectrum of the FQHE can be understood\nin terms of a model of interacting composite fermions.\nThere are two kinds of interaction between composite\nfermions: (i) A strong interaction that eliminates some\nstates by pushing them to in\fnity; this acts as a con-9\nFIG. 6: CF spectra for ( N;2Q) = (18;37) obtained by per-\nforming CF Diagonalization in the basis derived from all\nstates up to the third excited Landau band. This corresponds\nto\u0017\u0003= 3 (\u0017= 3=7) and should be compared with Fig. 1c\nwhich includes all states up to the second excited band.\nstraint on the allowed space of states. (ii) A weak interac-\ntion that removes the degeneracy of the remaining states\nand results in a broadening of the \u0003 bands. The sec-\nond interaction can be treated accurately by the method\nof composite-fermion diagonalization. Our main accom-\nplishment in this article is to identify the strong inter-\naction and show how it produces a correct counting of\nstates in various \u0003 levels.\nIt is remarkable that such an excellent and detailed\nunderstanding is possible for a strongly correlated system\nwith inherently non-perturbative physics. We close with\na few observations.\nThe strong interaction has a strange form, in that it\ncontains no CFP-CFP or CFH-CFH term; it only acts\nbetween an excited composite fermion (CFP) and the\nhole left behind (CFH). Also, while the interaction has\na 2-body form for composite fermions, it represents very\ncomplex many-particle structures of electrons. To begin\nwith, composite fermions and their \u0003 levels are them-\nselves complicated, emergent entities. The CF hole rep-\nresents a collective state of composite fermions in which a\ncomposite fermion is missing in an otherwise full \u0003 level.\nOur work has implications for experiments. It con\frms\nin detail the reality of higher unoccupied \u0003Ls, thus pro-\nviding support for the identi\fcation of the high energy\nmodes observed in by Rhone et al.10as single CF exci-\ntons across more than one \u0003L8,10. It is noted that this\nexperiment measures modes at relatively large wave vec-\ntors (i.e., large L), whereas the eliminated CF excitons\nhave small wave vectors (i.e. small L).\nThe following two observations shed light on why\nFQHE states along n=(2n+ 1) with high nare so sta-\nble, and why the CF theory has proved successful for\nthe CF Fermi sea. (i) A remarkable feature emerging\nfrom the exact spectra is that the \u0003 bands become bet-\nter de\fned for n=(2n+ 1) states with higher n. For ex-\nample, at\u0017\u0003= 1 the second excited band merges withthe third excited band for N > 6. In contrast, the sec-\nond excited band at \u0017\u0003= 3 is sharply de\fned even for\nN= 18. This demonstrates that the e\u000bective inter-\naction between composite fermions becomes weaker as\ntheir \flling factor is increased. This is surprising, be-\ncause one expects that as the gaps get smaller, there\nwould be a stronger mixing between di\u000berent \u0003 levels.\nOur results show that the residual interaction between\ncomposite fermions (which is what causes broadening\nof a \u0003 band) decreases faster than the CF cyclotron\ngaps separating the \u0003 levels as we increase n. Some\ninsight into this behavior can be gained from the ob-\nservation that the residual interaction between compos-\nite fermions is proportional to e\u00032= (2n+ 1)\u00002, where\ne\u0003is magnitude of the local charge of the CFP or the\nCFH, whereas the CF cyclotron energy is proportional11\nto~!\u0003\nc=eB\u0003=m\u0003c=eB=(2n+ 1)m\u0003c, wherem\u0003is the\nCF mass and B\u0003is the e\u000bective magnetic \feld sensed by\ncomposite fermions. (ii) We can further deduce from our\nresults that the number of eliminated states is thermo-\ndynamically insigni\fcant, at least at low temperatures\nwhen the density of excited CF particles and CF holes\nis small. The reason is that only those states are elim-\ninated in which a CFP and a CFH are close to one an-\nother, which is not a probable con\fguration at su\u000eciently\nsmall densities. (We note that this conclusion requires\nthe understanding of the missing states developed above.\nOtherwise, in small system studies, it appears as though\na large fraction of states is missing.) It can also be seen\nthat the exclusion rules do not a\u000bect the counting of\nstates at large momenta, which is where most states ap-\npear (including the 2 L+ 1 multiplicity), and which are\nalso the states probed by certain experiments. These ob-\nservations altogether o\u000ber insight into why the analysis\nin terms of nearly free composite fermions has proved so\nsuccessful for the physics of the CF Fermi sea.\nWe expect similar physics at the Jain fractions\nn=(2pn\u00001), but have not pursued that in this work be-\ncaue these fractions involve reverse \rux attachment32,\nwhich is technically more di\u000ecult to deal with. It would\nbe natural to seek a generalization of the above consid-\nerations to include spin; the situation is likely be more\ncomplicated there because the Hund's rule for compos-\nite fermions already eliminates some of the states pre-\ndicted by the model of free composite fermions33. It\nwould also be interesting to explore the origin of miss-\ning states within the Chern-Simons11,34, Hamiltonian35,\nand the conformal \feld theory36treatments of composite\nfermions.\nVII. ACKNOWLEDGMENTS\nWe are grateful to S.S. Mandal for many useful discus-\nsions and for help with computation. We acknowledge \f-\nnancial support from the NSF grants DMR-1005536 and\nDMR-0820404 (Penn State MRSEC), the Polish NCN\ngrant 2011/01/B/ST3/04504 and the EU Marie Curie10\nGrant PCIG09-GA-2011-294186 (AW). We thank Re-\nsearch Computing and Cyberinfrastructure, a unit of\nInformation Technology Services at Pennsylvania StateUniversity, as well as Wroclaw Centre for Networking\nand Supercomputing and Academic Computer Centre\nCYFRONET, both parts of PL-Grid Infrastructure.\n1P. W. Anderson, Physics Today 50, 42 (1997).\n2G. Dev and J. Jain, Phys. Rev. Lett. 69, 2843 (1992).\n3D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev.\nLett.48, 1559 (1982)\n4J. K. Jain, Phys. Rev. Lett. 63, 199 (1989).\n5J. K. Jain, Composite Fermions (Cambridge University\nPress, Cambridge, UK, 2007).\n6J. J. Quinn, A W\u0013 ojs, K.-S. Yi, and G. Simion, Physics\nReports 481, 29 (2009).\n7X. G. Wu and J. Jain, Phys. Rev. B 51, 1752 (1995).\n8D. Majumder, S. S. Mandal, and J. K. Jain, Nature Physics\n5, 403 - 406 (2009).\n9C. F. Hirjibehedin, I. Dujovne, A. Pinczuk, B. S. Dennis,\nL. N. Pfei\u000ber, and K. W. West, Phys. Rev. Lett. 95, 066803\n(2005).\n10T. D. Rhone, D. Majumder, B. S. Dennis, C. Hirjibehedin,\nI. Dujovne, J. G. Groshaus, Y. Gallais, J. K. Jain, S. S.\nMandal, A. Pinczuk, L. Pfei\u000ber, and K. West, Phys. Rev.\nLett.106, 096803 (2011).\n11B. I. Halperin, P. A. Lee, N. Read, Phys. Rev. B 477312\n(1993).\n12R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfei\u000ber, K. W.\nWest, Solid State Comm. 90, 71 (1994).\n13D. R. Leadley, R. J. Nicholas, C. T. Foxon, and J. J. Harris,\nPhys. Rev. Lett. 72, 1906 (1994).\n14S. Melinte, N. Freytag, M. Horvati\u0013 c, C. Berthier, L. P.\nL\u0013 evy, V. Bayot, and M. Shayegan, Phys. Rev. Lett. 84,\n354 (2000).\n15N. Freytag, M. Horvati\u0013 c, C. Berthier, M. Shayegan, and L.\nP. L\u0013 evy, Phys. Rev. Lett. 89, 246804 (2002).\n16N. C. Bishop, M. Padmanabhan, K. Vakili, Y. P.\nShkovnikov, R. P. De Poortere, and M. Shayegan, Phys.\nRev. Lett. 98, 266404 (2007); M. Padmanabhan, T. Goke-\nmen, and M. Shayegan, Phys. Rev. B 81, 113301 (2010).\n17R.L. Willett, R.R. Ruel, K. W. West, and L. N. Pfei\u000ber,\nPhys. Rev. Lett. 71, 3846 (1993).\n18W. Kang,H. L. Stormer, L. N. Pfei\u000ber, K. W. Baldwin,\nand K. W. West, Phys. Rev. Lett. 71, 3850 (1993).\n19V.J. Goldman, B. Su, and J. K. Jain, Phys. Rev. Lett. 72,2065 (1994).\n20J.H. Smet, D. Weiss, R. H. Blick, G. Lutjering, K. von\nKlitzing, R. Fleischmann, R. Ketzmerick, T. Geisel, and\nG. Weimann, Phys. Rev. Lett. 77, 2272 (1996).\n21D. Kamburov, M. Shayegan, L.N. Pfei\u000ber, K.W. West, and\nK.W. Baldwin, Phys. Rev. Lett. 109, 236401 (2012); D.\nKamburov, Y. Liu, M. Shayegan, L.N. Pfei\u000ber, K.W. West,\nand K.W. Baldwin, Phys. Rev. Lett. 110, 206801 (2013).\n22D. Kamburov, M. A. Mueed, M. Shayegan, L. N. Pfei\u000ber,\nK. W. West, K. W. Baldwin, J. J. D. Lee, and R. Winkler,\nPhys. Rev. B 88, 125435 (2013).\n23X. Ying, V. Bayot, M. B. Santos, and M. Shayegan, Phys.\nRev. B 50, 4969 (1994).\n24I. V. Kukushkin, J. H. Smet, K. von Klitzing and W.\nWegscheider, Nature 415, 409 (2002).\n25I. V. Kukushkin, J. H. Smet, D. Schuh, W. Wegscheider,\nand K. von Klitzing, Phys. Rev. Lett. 98, 066403 (2007).\n26F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983).\n27J. K. Jain and R. K. Kamilla, Int. J. Mod. Phys. B 11,\n2621 (1997); Phys. Rev. B 55, R4895 (1997).\n28S. S. Mandal and J. K. Jain, Phys. Rev. B 66, 155302\n(2002).\n29Song He, S. H. Simon, and B. I. Halperin, Phys. Rev. B\n50, 1823 (1994).\n30P. Sitko, S. N. Yi, K. S. Yi, J. J. Quinn, Phys. Rev. Lett.\n76, 3396 (1996).\n31S. -Y. Lee, V. W. Scarola, J. K. Jain, Phys. Rev. Lett. 87,\n256803 (2001).\n32X. G. Wu, G. Dev and J. K. Jain, Phys. Rev. Lett. 71, 153\n(1991).\n33X. G. Wu and J. K. Jain, Phys. Rev. B 49, 5085 (1994);\nPhys. Rev. B 49, 7515 (1994).\n34A. Lopez and E. Fradkin, Phys. Rev. B 44, 5246 (1991).\n35Ganpathy Murthy and R. Shankar, Rev. Mod. Phys. 75,\n1101 (2003).\n36T. H. Hansson, C.-C. Chang, J. K. Jain, and S. Viefers,\nPhys. Rev. B 76, 075347 (2007)"    },    {        "title": "0806.4750v2.Ab_initio_Derivation_of_Low_energy_Model_for_Iron_Based_Superconductors_LaFeAsO_and_LaFePO.pdf",        "content": "arXiv:0806.4750v2  [cond-mat.supr-con]  20 Aug 2008Typeset with jpsj2.cls <ver.1.2> Letter\nAb initio Derivation of Low-Energy Model for Iron-Based Superconduc tors LaFeAsO\nand LaFePO\nKazuma Nakamura1∗, Ryotaro Arita1,2, and Masatoshi Imada1,2\n1Department of Applied Physics, University of Tokyo, 7-3-1 H ongo, Bunkyo-ku, Tokyo 113-8656, Japan\n2CREST, JST, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n(Received November 8, 2021)\nEffective Hamiltonians for LaFeAsO and LaFePO are derived fr om the downfolding scheme\nbased on first-principles calculations and provide insight s for newly discovered superconduc-\ntors in the family of LnFeAsO 1−xFx, Ln = La, Ce, Pr, Nd, Sm, and Gd. Extended Hubbard\nHamiltonians for five maximally localized Wannier orbitals per Fe are constructed dominantly\nfrom five-fold degenerate iron-3 dbands. They contain parameters for effective Coulomb and\nexchange interactions screened by the polarization of othe r electrons away from the Fermi level.\nThe onsite Coulomb interaction estimated as 2.2-3.3 eV is co mpared with the transfer integrals\nbetween the nearest-neighbor Fe-3 dWannier orbitals, 0.2-0.3 eV, indicating moderately stron g\nelectron correlation. The Hund’s rule coupling is found to b e 0.3-0.6 eV. The derived model\noffers a firm basis for further studies on physics of this famil y of materials. The effective mod-\nels for As and P compounds turn out to have very similar screen ed interactions with slightly\nnarrower bandwidth for the As compound.\nKEYWORDS: first-principles calculation, effective Hamilto nian, downfolding, constrained RPA method,\nLaFeAsO, LaFePO, Oxypnictide, high-temperature supercon ductivity\nRecent discovery of a new superconductor,\nLaFeAs(O,F),1)has triggered extensive studies on\nthe family of layered iron arsenide compounds with\nZrCuSiAs-type or ThCr 2Si2-type structure, whose\nsuperconducting critical temperature Tcis now raising\nup to∼55 K.2)A mother compound LaFeAsO shows\nantiferromagnetic order3)with bad metallic transport\nproperties1)supporting a significant role of electron\ncorrelations. The antiferromagnetic ordered moment\n∼0.36µBis unexpectedly small, implying large quantum\nfluctuations arising from electron correlations with com-\npeting ground states. Superconductivity appears when\ncarriers are doped by the substitution of F for O,1)or\nthe introduction of O vacancies.4)Many of experimental\ndata support a significant role of electron correlation\nin realizing the superconductivity.5–8)Although the\nAs compounds have the maximum Tc∼56 K, the P\ncompounds show one order of magnitude lower Tc.9)\nConventional density-functional calculations with the\nlocal density approximation (LDA) or the generalized\ngradient approximation (GGA) have clarified entangled\nten-band structure near the Fermi level mainly originat-\ning from five-fold degenerate iron-3 dorbitals contained\nin each of two iron atoms in the unit cell.10–14)For the\nmothermaterial,the initial LDAcalculation10)predicted\nthe nonmagnetic ground state in close proximity to a fer-\nromagnetic metal while the recent results show antifer-\nromagnetic order.11–13)In particular, the stripe-type an-\ntiferromagnetic order is correctly reproduced.12,13)The\ncalculated ordered moment obtained so far ranges be-\ntween 1.2 and 2.6 µB,11–13,15)in contrast to the tiny\nordered moment discussed above. Broad peak structures\nof magnetic Lindhard function calculated by using the\nLDA/GGA Fermi surface suggest competitions of sev-\n∗Electronic mail: kazuma@solis.t.u-tokyo.ac.jperal different ordering tendencies.14,16–18)\nAlthough overall experimental results suggest notice-\nable correlation effects, realistic roles of electron correla-\ntions on theoretical grounds are not clear. The relevance\nofthe correlationeffect is underactivedebates.19–21)It is\nimperative to estimate at least the effective Coulomb in-\nteraction from first principles. Furthermore, since all the\n3dbands of Fe are as a first look wholly involved near\nthe Fermi level, it is important to elucidate interplay of\norbitaldegeneracyandelectroncorrelation,which canbe\nstudied only by a model for degenerate bands. A reliable\ntheoretical model derived for this family of compounds\nthus provides us with a firm basis for understanding the\nsuperconductivity and also a starting point for exploring\nfurther possibility of higher Tccompounds.\nIn this letter, we present ab initio low-energy effective\nHamiltonians of LaFeAsO and LaFePO. Implications for\nthe superconductivity in F-doped materials are also dis-\ncussed. A reliable downfolding scheme has recently been\nestablished which has enabled derivation of low-energy\neffective Hamiltonians from ab initio density-functional\ncalculationofrealmaterials.22–24)Thelow-energyHamil-\ntonian is derived after eliminating higher-energy degrees\nof freedom and estimating the renormalization effect\nfrom the high-energy part onto the bands near the Fermi\nlevel. The accuracy of the downfolding procedure has\nbeen established in various cases.24–26)Here, we employ\nthis downfolding scheme in deriving the extended Hub-\nbardmodelsofLaFeAsOandLaFePO,consistingofband\ndispersion (kinetic energy) of electrons at maximally lo-\ncalizedWannierorbitals(MLWOs),27)aswellasscreened\nCoulomb and exchange interactions. Since the ten-fold\nFe-3dbands are basically isolated from other bands, we\nderive effective models for these ten bands.\n12 J. Phys. Soc. Jpn. Letter Author Name\nThe extended ten-band Hubbard model reads\nH=/summationdisplay\nσ/summationdisplay\nRR′/summationdisplay\nnmtmRnR′aσ†\nnRaσ\nmR′\n+1\n2/summationdisplay\nσρ/summationdisplay\nRR′/summationdisplay\nnm/braceleftbigg\nUmRnR′aσ†\nnRaρ†\nmR′aρ\nmR′aσ\nnR\n+JmRnR′/parenleftbig\naσ†\nnRaρ†\nmR′aρ\nnRaσ\nmR′+aσ†\nnRaρ†\nnRaρ\nmR′aσ\nmR′/parenrightbig/bracerightbigg\n,(1)\nwhereaσ†\nnR(aσ\nnR) is a creation (annihilation) opera-\ntor of an electron with spin σin thenth MLWO cen-\ntered on Fe atoms in the unitcell at R.tmRnR′con-\ntains single-particle levels and transfer integrals, given\nbytmRnR′=/an}bracketle{tφmR|H0|φnR′/an}bracketri}htwith|φnR/an}bracketri}ht=a†\nnR|0/an}bracketri}htand\nH0being the one-body part of H.UmRnR′andJmRnR′\nare screened Coulomb and exchange interactions, respec-\ntively, expressed as\nUmRnR′=/an}bracketle{tφmRφmR|W|φnR′φnR′/an}bracketri}ht(2)\nand\nJmRnR′=/an}bracketle{tφmRφnR′|W|φnR′φmR/an}bracketri}ht(3)\nwithWbeing a screened Coulomb interaction. There are\nalreadyattempts to estimate tmRnR′using MLWO.11,14)\nWe here focus on an ab initio derivation of the many-\nbody part of H; we estimate the interaction parameters\nin eqs. (2) and (3), based on a constrained random-phase\napproximation (cRPA).22,23)The cRPA has several ad-\nvantagesoverothermethodssuchasconstrainedLDA.28)\nWe can precisely exclude screening processes among the\nFe-3dMLWOs being the bases of the effective model.\n(This screening should be considered when we solve the\neffectivemodel.) In addition,wecancalculatematrixele-\nments ofWas a function of RandR′;29)i.e., we can ob-\ntainonsiteandoffsiteinteractionsatonetime.ThecRPA\nbecomes a good approximation, when the high-energy\neliminated bands are well separated from the Fermi level\nandthescreenedCoulombinteractionsbetweenthehigh-\nenergy electrons and the low-energy electrons are weak.\nAs is noted above, quantitative accuracies of our down-\nfolding including cRPA has already been confirmed in a\nnumber of examples.22–25,29)In the present case of the\niron compounds, the condition above is equally satisfied.\nUmRnR′of eq. (2) is practically calculated in the re-\nciprocal space by using a Fourier transform of W,\nW(r,r′) =/summationdisplay\nqGG′ei(q+G)rWGG′(q)e−i(q+G′)r′.(4)\nHere,Gis a reciprocal lattice vector and qis a wave\nvector in the first Brillouin zone. We define WGG′(q) as\nWGG′(q) =4π\nΩ1\n|q+G|ǫ−1\nGG′(q)1\n|q+G′|,\nwhere Ω is the crystal volume and ǫ−1\nGG′(q) is the inverse\ndielectric matrix which is related to the irreducible po-\nlarizability χbyǫGG′(q) =δGG′−v(q+G)χGG′(q),\nwherev(q) = 4π/Ω|q|2is the bare Coulomb interaction.The polarization matrix χGG′(q) is calculated as\nχGG′(q)=/summationdisplay\nk/summationdisplay\nαβ′\n/an}bracketle{tψαk+q|ei(q+G)r|ψβk/an}bracketri}ht\n×/an}bracketle{tψβk|e−i(q+G)r|ψαk+q/an}bracketri}htfαk+q−fβk\nEαk+q−Eβk,\nwhere{ψαk}arethe Blochstatesand the prime attached\ntothebandsumindicatesthatweexcludethe3 d-3dband\ntransitions in the calculation of χ. By inserting eq. (4)\ninto eq. (2), we obtain the form of\nUm0nR=4π\nΩ/summationdisplay\nqGG′e−iqRρmq(G)ǫ−1\nGG′(q)ρ∗\nnq(G′),(5)\nwhere\nρnq(G) =1\nN|q+G|N/summationdisplay\nk/an}bracketle{t˜ψnk+q|ei(q+G)r|˜ψnk/an}bracketri}ht\nwith|˜ψnk/an}bracketri}ht=/summationtextN\nR|φnR/an}bracketri}hte−ikR.It should be noted here\nthat the quantities /an}bracketle{t˜ψmk+q|ei(q+G)r|˜ψnk/an}bracketri}htcan be easily\nevaluatedwiththefastFouriertransformationtechnique.\nMatrix elements of the bare (or unscreened) Coulomb\ninteraction as Ubare\nm0nR=/an}bracketle{tφm0φm0|v|φnRφnR/an}bracketri}htare cal-\nculated with replacing ǫ−1\nGG′(q) of eq. (5) by δGG′. The\nparalleltreatmentis applied tothe derivationofscreened\nexchange interactions in eq. (3). The result is\nJm0nR=4π\nΩ/summationdisplay\nqGG′ρmnRq(G)ǫ−1\nGG′(q)ρ∗\nmnRq(G′) (6)\nwith\nρmnRq(G) =1\nN|q+G|N/summationdisplay\nke−ikR/an}bracketle{t˜ψmk+q|ei(q+G)r|˜ψnk/an}bracketri}ht.\nWe implemented this scheme in Tokyo Ab initio\nProgram Package .30)With this program, electronic-\nstructure calculations with the GGA exchange-\ncorrelation functional31)were performed using a\nplane-wave basis set and the Troullier-Martins norm-\nconserving pseudopotentials32)in the Kleinman-\nBylander representation.33)Our iron pseudopotential\nwas constructed under the reference configuration\n(3d)6.0(4s)1.8(4p)0.0by employing the cutoff radii for\nthe 3d, 4s, and 4pstates at 2.1 Bohr. The energy cutoff\nwas set to 64 Ry and a 9 ×9×5k-point sampling was\nemployed. The experimental crystal-structure data were\ntaken from ref. 3 for LaFeAsO and ref. 9 for LaFePO.\nThe dielectric matrices were expanded in plane waves\nwith an energy cutoff of 64 Ry and the total number of\nbands included in the sum in eq. (5) was set to 70. The\nsum ofkin eq. (5) was evaluated by the tetrahedron\nmethod. The additional terms in the long-wavelength\ndielectric matrix due to nonlocal terms in the pseudopo-\ntentials was explicitly considered following ref. 34. The\npointq=G=0in eqs. (5) and (6) requires special\nhandling because of the singularity in the Coulomb\ninteraction, which was treated in the manner described\nin ref. 34.\nThe upper two panels of Fig. 1 shows ab initio band\nstructures ofLaFeAsO (left) and LaFePO(right). We see\na good agreement between the original GGA band (redJ. Phys. Soc. Jpn. Letter Author Name 3\nline)andtheWannier-interpolatedband(bluedots).The\nlower panels visualize our calculated MLWOs with the\nyz(left) andz2(right) symmetry. In principle, the band\ndispersion thus obtained should be renormalized by the\ninteractionbetweenelectronsinthe Fe-3 dandeliminated\nbands in the downfolding procedure.22)In this letter, we\nassume that this self-energy effect is small and thus we\nemploy the same dispersion for the downfolded Hamil-\ntonian. The overall band structure of LaFeAsO shows a\nfeature similar to that of LaFePO with slightly ( ∼20%)\nnarrower bandwidth for the As compound. Around the\nΓ point, in addition to two hole cylinders with the dyz\nanddzxcharacters, a 2D-like dx2−y2band crosses near\nthe Fermi level for the As compound, in contrast to a\n3D-likedz2band of the P compound.\nr6r4r2 0 2 4 6\nΓ       X      M          Γ  ZΓ       X       M          Γ  ZEnergy (eV)\nFig. 1. Upper panels: Ab initio band structures of LaFeAsO (left)\nand LaFePO (right). Red line and Blue dots are original-GGA\nand Wannier-interpolated bands, respectively. The zero of energy\nis the Fermi level. Lower panels: Isosurface contours of yz- (left)\nandz2- (right) MLWOs in LaFeAsO. The amplitudes of the con-\ntour surface are +1.5/√v(blue) and −1.5/√v(red), where vis\nthe volume of the primitive cell. Fe and As nuclei are illustr ated\nby yellow and silver spheres, respectively.\nIn Fig. 2, we plot diagonalelements ofthe macroscopic\ncRPA dielectric matrix, ǫM(q+G) = 1/ǫ−1\nGG(q), calcu-\nlated for LaFeAsO, as a function of |q+G|. We note\nthat the resulting dielectric constant ǫ0\nMatq+G→0\nexhibits a rather high value of 6.3, which is compared to\nthat of transition metal oxides such as SrVO 3(6.5). It is\nalso interesting to note that ǫM(q+G) does not severely\ndepend on the direction of q+G, which means that the\nscreening response is quite isotropic.\nFig. 3 plots matrix elements of the screened Coulomb\ninteraction, Um0nRin eq. (5), denoted by green dots, for\nLaFeAsO as a function of the distance between the cen-\nters of the MLWOs; r=|/an}bracketle{tφnR|r|φnR/an}bracketri}ht − /an}bracketle{tφm0|r|φm0/an}bracketri}ht|.\nIn this plot, we set mto adxyMLWO and display\nonly the interactions between it and other MLWOs.\nFor comparison, we also plot bare Coulomb interactions 0 1 2 3 4 5 6 7\n 0 0.5  1 1.5  2 2.5\n|q+G| (a.u.)εM(q+G)\nFig. 2. Macroscopic dielectric function of LaFeAsO as a func tion\nof|q+G|obtained by the constrained RPA method.\nUbare\nm0nRas red dots, which decay as 1 /r(solid line) be-\nyond the nearest-neighborFe-Fe distance ( ≥2.65˚A). It is\nclear that the bare Coulomb interaction is significantly\nscreened. In addition, as expected from Fig. 2, Um0nR\ndecays as an isotropic function of 1 /(ǫ0\nMr) (dotted line).\nSince the offsite interactions ( Um0nR/negationslash=0) are more than\nfive times smaller than the onsite Uvalues, we may ne-\nglect them in the first step and just start with the onsite\nHubbard model. The screened exchange interactions of\nJm0nRin eq. (6) are found to decay very quickly; the\nmagnitude is nearly zero, except for the onsite values.\n  0  4  8 12 16\n 0  4  8 12\nr (Angstrom)Interaction (eV)\nFig. 3. Calculated screened Coulomb interactions of LaFeAs O as\na function of the distance between the centers of MLWOs. Only\nthe interactions between a dxyMLWO at the homecell and other\nMLWOs are plotted. The red and green dots represent the bare\nand screened interactions, respectively. The solid and dot ted\ncurves denotes 1 /rand 1/(ǫ0\nMr), respectively.\nWe summarize in Table I the list of the important\nparameters in the low-energy effective Hamiltonian of\neq. (1). The transfer integral between nearest-neighbor\niron Wannier orbitals, t, is typically 0.2-0.3 eV, whereas\nthe onsite screened Coulomb interaction, U, exhibits 2.2-\n3.3 eV. The strong orbital dependence comes from the\nfact that each orbital has different amount of leakage\non the neighboring As atoms. The onsite exchange in-\nteraction (Hund’s rule coupling) is 0.3-0.6 eV. We note\nthat the resulting UandJvalues are smaller than the\nbare values by the factors of about 1/5 and 4/5, respec-\ntively. We note in passing that, for the two-degenerate\ndyzanddzxorbitals, our computed Jyz,zx(= 0.45 eV)\nis close to the value estimated from ( Uyz,yz−Uyz,zx)/2\n(0.50 eV). The interaction parameters for the P com-\npound are found to be very similar; for example, the\nonsiteUvalues range from 1.9 eV for dx2−y2to 3.3 eV\nfordz2anddxyas in the case of the As compounds.\nThepresentvalueof Uissubstantiallysmallerthanthe4 J. Phys. Soc. Jpn. Letter Author Name\nTable I. List of important parameters (in eV) in the present\nextended Hubbard Hamiltonian in eq. (1). From the top, the\nthree 5×5 matrices represent transfer integrals between nearest-\nneighbor sites, onsite screened Coulomb interactions, and onsite\nexchange interactions of LaFeAsO. The bottom 5 ×5 matrix is\ntransfer integrals of LaFePO. The onsite energies for the fiv e\norbitals of LaFeAsO are ( ǫxy,ǫyz,ǫz2,ǫzx,ǫx2−y2) = (−0.12,\n+0.10,−0.19, +0.10, +0.16) eV.\nt(LaFeAsO) xy yz z2zx x2−y2\nxy−0.32−0.25−0.30−0.25 0.00\nyz−0.25−0.21−0.08−0.13 0.18\nz2−0.30−0.08 0.08 −0.08 0.00\nzx−0.25−0.13−0.08−0.21−0.18\nx2−y20.00 0.18 0.00 −0.18−0.18\nU(LaFeAsO) xy yz z2zx x2−y2\nxy3.31 1.95 1.89 1.95 2.09\nyz1.95 2.77 2.20 1.78 1.67\nz21.89 2.20 3.27 2.20 1.65\nzx1.95 1.78 2.20 2.77 1.67\nx2−y22.09 1.67 1.65 1.67 2.20\nJ(LaFeAsO) xy yz z2zx x2−y2\nxy - 0.54 0.64 0.54 0.27\nyz0.54 - 0.41 0.45 0.43\nz20.64 0.41 - 0.41 0.50\nzx0.54 0.45 0.41 - 0.43\nx2−y20.27 0.43 0.50 0.43 -\nt(LaFePO) xy yz z2zx x2−y2\nxy−0.35−0.31−0.32−0.31 0.00\nyz−0.31−0.24−0.04−0.13 0.18\nz2−0.32−0.13 0.13 −0.04 0.00\nzx−0.31−0.13−0.04−0.24−0.18\nx2−y20.00 0.18 0.00 −0.18−0.27\nvalues(∼4eVorlarger)employedinmodelstudies,19,20)\nwhich, as a first look, suggests that much smaller corre-\nlation effects should be expected in reality. However, it\nshould be cautioned that the DMFT employed in the lit-\nerature19,20)ignoringspatialcorrelationsmaylargelyun-\nderestimate correlation effects. By considering the ratio\nU/t∼10 (the value itself is comparable to the cuprates)\nand the presence of the five-degenerate orbitals per Fe\nsite, electron correlations in these compounds are mod-\neratelystrong. In addition, the present Uvalues have\norbital dependence with the order of 1 eV, which will\ngenerate nonnegligible orbital dependence of renormal-\nization factor; it can be a critical parameter that affects\nthe Fermi surface in quasipariticle band structure. Our\ndownfoldedeffective Hamiltonianthus posesvariouscon-\nstraints on modeling of this family of materials and care-\nful analyses by accurate low-energy solvers should be re-\nquired for reliable discussions of magnetic and supercon-\nducting mechanisms.\nThe effective models for LaFeAsO and LaFePO are\nbasically similar in the band dispersion as well as in the\nscreened Coulomb interaction. A main differences is the\nband width (see Fig.1) and this point can also be con-\nfirmed from tvalues of LaFePO somewhat larger than\nthose of LaFeAsO (see Table I). Another difference is a\nbandclosetotheFermisurfaceatΓpoint;i.e.,whetherit\nhasa two-dimensionaldispersion with the dx2−y2charac-\nter or it has a three-dimensional one with the dz2charac-ter. These differences can be systematically understood\nin terms of the Fe-Fe and Fe-pnictogen distances.35)On\nthe other hand, in our calculations, the interaction pa-\nrameters do not exhibit noticeable differences between\nLaFeAsO and LaFePO. The mechanism of the supercon-\nductivityhastoclarifyhowthesesubtledifferencesinthe\none-body and many-body terms of the effective Hamilto-\nnian lead to alargedifference in the critical temperature,\n26-55 K for As compounds and 4 K for P compounds.\nThe present model may be used for further stud-\nies by using low-energy solvers such as the dynamical\nmean-field theory36)and path-integral renormalization\ngroup.37,38)Our ten-band model may also offer a firm\nstarting point for further downfolding to derive effective\nmodels with fewer number of bands near the Fermi level.\nThis work is supported from MEXT Japan under the\ngrant numbers 16076212,17071003,17064004,19019012,\nand 19014022. We also thank the facilities at the Super-\ncomputer Center, Institute for Solid State Physics, Uni-\nversity of Tokyo. We thank Yoshihide Yoshimoto, Taichi\nKosugi, and Takashi Miyake for useful discussions.\n1) Y. Kamihara, T. Watanabe, M. Hirano and H. Hosono: J. Am.\nChem. Soc. 130(2008) 3296.\n2) Z.A.Ren,W.Lu, J.Yang, W.Yi,X.L.Shen, Z.C.Li,G.C.Che,\nX.L. Dong, L. L.Sun, F.Zhou, Z.X.Zhao: arXiv:0804.2053v1.\n3) C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcliff II, J.\nL. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang\nand P. Dai: Nature 453(2008) 899.\n4) Z. A. Ren, G. C. Che, X. L. Dong, J. Yang, W. Lu, W.\nYi, X. L. Shen, Z. C. Li, L. L. Sun, F. Zhou, Z. X. Zhao:\narXiv:0804.2582v1.\n5) Y. Nakai, K. Ishida, Y. Kamihara, M. Hirano and H. Hosono:\nJ. Phys. Soc. Jpn. 77(2008) 073701; K. Ahilan, F. L. Ning,\nT. Imai, A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, D.\nMandrus: arXiv:0804.4026v2.\n6) J. P. Carlo, Y. J. Uemura, T. Goko, G. J. MacDougall, J. A.\nRodriguez, W.Yu, G.M.Luke, P.Dai,N.Shannon, S.Miyasaka,\nS. Suzuki, S. Tajima, G. F. Chen, W. Z. Hu, J. L. Luo, N. L.\nWang: arXiv:0805.2186v1.\n7) T. Sato, S. Souma, K. Nakayama, K. Terashima, K. Sugawara,\nT.Takahashi, Y.Kamihara, M.Hirano and H.Hosono: J.Phys.\nSoc. Jpn. 77(2008) 063708; Y. Ishida, R. Eguchi, M. Mat-\nsunami, K. Horiba, M. Taguchi, A. Chainani, Y. Senba, H.\nOhashi, H. Ohta, S. Shin: arXiv:0712.3083.\n8) H. Liu, X. Jia, W. Zhang, L. Zhao, J. Meng, G. Liu, X. Dong,\nG. Wu, R. H. Liu, X. H. Chen, Z. A. Ren, W. Yi, G. C. Che,\nG. F. Chen, N. L. Wang, G. Wang, Y. Zhou, Y. Zhu, X. Wang,\nZ. Zhao, Z. Xu, C. Chen, X. J. Zhou: arXiv:0805.3821v1.\n9) Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H.\nYanagi, T. Kamiya, and H. Hosono: J. Am. Chem. Soc. 128\n(2006) 10012.\n10) D. J. Singh and M. H. Du: arXiv:0803.0429v1.\n11) C. Cao, P. J. Hirschfeld, and H. P. Cheng: arXiv:0803.323 6v1\n12) S. Ishibashi, K.Terakura and H. Hosono: J. Phys. soc. Jpn .77\n(2008) 053709.\n13) F. Ma and Z. Y. Lu: arXiv:0803.3286v1.\n14) K.Kuroki, S.Onari, R.Arita, H.Usui, Y.Tanaka, H.Konta ni,\nH. Aoki: arXiv:0803.3325v1.\n15) I.I.Mazin, M.D.Johannes, L.Boeri,K.Koepernik, D.J.S ingh:\narXiv:0806.1869v1.\n16) I. I. Mazin, D. J. Singh, M. D. Johannes, M. H. Du:\narXiv:0803.2740v2.\n17) J.Dong, H.J.Zhang, G.Xu,Z.Li,G.Li,W.Z.Hu, D.Wu, G.F.\nChen,X.Dai,J.L.Luo,Z.Fang, N.L.Wang:arXiv:0803.3426v 1.\n18) T. Yildirim: arXiv:0804.2252v1.\n19) A.O.Shorikov,M.A.Korotin,S.V.Streltsov, S.L.Skorn yakov,J. Phys. Soc. Jpn. Letter Author Name 5\nD. M. Korotin, V. I. Anisimov: arXiv:0804.3283v1.\n20) K. Haule, J. H. Shim, and G. Kotliar: Phys. Rev. Lett. 100\n(2008) 226402.\n21) L. Craco, M. S. Laad, S. Leoni and H. Rosner:\narXiv:0805.3636v1\n22) F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Bie r-\nmann, and A. I. Lichtenstein: Phys. Rev. B 70(2004) 195104.\n23) I. V. Solovyev and M. Imada: Phys. Rev. B 71(2005) 045103.\n24) Y.Imai, I.Solovyev, and M.Imada: Phys.Rev.Lett. 95(2005)\n176405; Y. Imai and M. Imada: J. Phys. Soc. Jpn. 75(2006)\n094713.\n25) Y.Otsuka and M.Imada: J.Phys.Soc. Jpn. 75(2006) 124707.\n26) K. Nakamura, Y. Yoshimoto, R. Arita, S. Tsuneyuki, and M.\nImada: Phys. Rev. B 77(2008) 195126.\n27) N. Marzari and D. Vanderbilt: Phys. Rev. B 56(1997) 12847;\nI.Souza, N.Marzari and D.Vanderbilt: Phys.Rev.B 65(2002)\n035109.\n28) O. Gunnarsson, O. K. Andersen, O. Jepsen, and J. Zaanen:\nPhys. Rev. B 39(1989) 1708.\n29) F. Aryasetiawan, K. Karlsson, O. Jepsen, and U. Schonber ger:Phys.Rev.B 74(2006) 125106; T.Miyake and F.Aryasetiawan:\nPhys. Rev. B 77(2008) 085122.\n30) M. Tsukada et al., computer program package TAPP, Univer-\nsity of Tokyo, Tokyo, JAPAN, 1983 −2000: J. Yamauchi, M.\nTsukada, S. Watanabe, and O. Sugino: Phys. Rev. B 54(1996)\n5586.\n31) J.P.Perdew, K.Burke, and M.Ernzerhof: Phys.Rev. Lett. 77\n(1996) 3865.\n32) N. Troullier and J. L. Martins: Phys. Rev. B 43(1991) 1993.\n33) L. Kleinman and D. M. Bylander: Phys. Rev. Lett. 48(1982)\n1425.\n34) M.S.Hybertsen and S.G.Louie: Phys.Rev.B 34(1986) 5390;\nPhys. Rev. B 35(1987) 5585.\n35) V. Vildosola, L. Pourovskii, R. Arita, S. Biermann, and A .\nGeorges: arXiv:0806.3285v1\n36) A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg: Rev .\nMod. Phys. 68(1996) 13.\n37) M. Imada and T. Kashima: J. Phys. Soc. Jpn. 69(2000) 2723.\n38) T. Kashima and M. Imada: J. Phys. Soc. Jpn. 70(2001) 2287."    },    {        "title": "0905.3305v1.Topological_change_of_the_Fermi_surface_in_ternary_iron_pnictides_with_reduced_c_a_ratio__A_dHvA_study_of_CaFe2P2.pdf",        "content": "arXiv:0905.3305v1  [cond-mat.supr-con]  20 May 2009Topologicalchange ofthe Fermisurface internary iron-pni ctides\nwithreduced c/aratio: AdHvA study ofCaFe 2P2\nAmalia I. Coldea,1C.M.J. Andrew,1J.G. Analytis,2,3R.D. McDonald,4\nA.F. Bangura,1J.-H. Chu,2,3I.R. Fisher,2,3and A. Carrington1\n1H.H. Wills Physics Laboratory, University of Bristol, Tynd all Avenue, Bristol, BS8 1TL, UK\n2Stanford Institute for Materials and Energy Sciences,\nSLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA\n3Geballe Laboratory for Advanced Materials and Department o f Applied Physics, Stanford University, USA\n4Los Alamos National Laboratory, Los Alamos, NM 87545, USA\n(Dated: November 2, 2018)\nWe report a de Haas-van Alphen effect study of the Fermi surfa ce of CaFe 2P2using low temperature torque\nmagnetometry up to 45 T. This system is a close structural ana logue of the collapsed tetragonal non-magnetic\nphase of CaFe 2As2. We find the Fermi surface of CaFe 2P2to differ from other related ternary phosphides in\nthatits topology is highly dispersive inthe c-axis, being three-dimensional incharacter andwithident ical mass\nenhancement onboth electronand hole pockets ( ∼1.5). The dramatic change intopology of the Fermi surface\nsuggests that in a state with reduced ( c/a) ratio, when bonding between pnictogen layers becomes impo rtant,\nthe Fermisurface sheets are unlikely tobe nested.\nThenatureoftheFermisurfacedimensionalityanditsprox-\nimity to nesting in the iron-pnictide superconductors is at\nthe core of understanding the mechanism of superconductiv-\nity. An important finding is that under hydrostatic pressure,\nBaFe2As2andSrFe 2As2[1]superconductwhereasCaFe 2As2\ndoes not [2]. Instead it undergoes a structural transition t o a\nphase with a much reduced c-axis length which is known as\nthe‘collapsedtetragonal’(cT)state. Ifpressureisappli edus-\ninga non-hydrostaticmediumCaFe 2As2doesbecomesuper-\nconducting [1, 3]. This suggests that small uniaxial pressu re\ncomponentsstabilizethesuperconductivityinCaFe 2As2. Un-\nderstanding why this happens in these materials and whether\nthe Fermi surface nesting or the strong coupling with the lat -\ntice are the relevant parameters will be important for under -\nstandingtheoriginofsuperconductivityinironpnictides .\nCaFe2As2hasthreedistinctmagneticandstructuralphases.\nLikemanyotheriron-arsenides,atzeropressure,thehight em-\nperaturetetragonalstatebecomesorthorhombicandantife rro-\nmagnetically ordered below TN≃170K. At low tempera-\ntures under a pressure of 0.35GPa there is a transition to the\ncT state in which there is a ∼10% decrease in the c-axis lat-\ntice parameter and a ∼2% increase in the a-axis [4]. Recent\nbandstructurecalculations[5,6]suggestthatthesephase tran-\nsitionshaveadramaticeffectontheFermisurface. Inthehi gh\ntemperaturetetragonalphasetheFermisurfaceispredicte dto\nbesimilartotheother122pnictideconsistingoftwostrong ly\nwarped electron cylinders at the Brillouin zone corners and\nholecylindersat the centerofthe zone( Γ). Calculationssug-\ngest that these hole pockets are sensitive to structural det ails\noftheAspositionandthe c-latticeconstant,whereastheelec-\ntronpocketsarelessaffected[5]. However,inthecTstatet he\ncalculated Fermi surface suffers a major topological chang e;\nthe two electron cylinders at the zone corners transform int o\nasinglewarpedcylinderwhereastheholecylindersbecomea\nlargethreedimensionalsheet.\nTheFermisurfaceofmanyiron-pnictideshavebeenexten-\nsively studied by angle resolvedphotoemissionspectrosco py,howeverso-farthecTphaseofCaFe 2As2hasbeeninaccessi-\nble to this technique because of the high pressures involved .\nQuantumoscillation(QO)studieshavetheadvantageofprob -\ning precisely the bulk three dimensionalFermi surface but s o\nfarhaveonlybeenpossibleinthemagneticallyorderedphas e\nof the iron-arsenides [7] as the tetragonal superconductin g\nphasecannotbeaccessedbecauseofthehighvaluesoftheup-\npercriticalfield( ∼100T).AlthoughQOstudiesarepossible\nunderhighpressure, these measurementsare technicallyve ry\nchallenging. One route to provide an insight into the Fermi\nsurface properties of the iron-arsenides is to study the ana lo-\ngousnon-magneticphosphidematerials. Thesematerialsha ve\nalmost identical calculated Fermi surfaces to the arsenide s in\ntheir non-magnetic state but are either non-superconducti ng\nor have a low Tc(and low Hc2) and single crystals can be\ngrown with very high purity making them ideal for QO stud-\nies. Quantum oscillations have been measured in supercon-\nductingLaFePO [8] (whichis analogousto the non-magnetic\n‘1111’arsenidese.g., LaFeAsO) and SrFe 2P2[9] (which is a\nstructural analogue to the non-magnetic tetragonal ‘122’ a r-\nsenides).\nHere we report a de Haas-van Alphen effect study of the\nFermi surface of CaFe 2P2which is a very close structural\nanalogue of the cT phase of CaFe 2As2. The isoelectric sub-\nstitution of As with P does not change the number of Fe 3d\nelectrons, but enhances P-P hybridization causing the latt ice\ntocontractin the cdirection. Consequentlythe interlayerP-P\ndistance approaches the molecular bond length [10], just as\ntheAs-AsdistancedoesinthecTphase[11]. Weshowexper-\nimentally that the Fermi surface of CaFe 2P2is differentfrom\nthe multiband electron-hole structure found in SrFe 2P2[9]\nandotherpcnitides,andisformedofasinglewarpedcylinde r\nand a large three dimensional hole sheet. We find a isotropic\nmassenhancementsonthe electronandholesheets( ∼1.5).\nHighqualitysinglecrystalsofCaFe 2P2weregrownfroma\nSnfluxsimilartopreviousreports[9]. Theresidualresisti vity\nratioρ(300K)/ρ(1.8K) was measured to be greater than 120.2\n0 5 10 15 20 Δ\n\u0000 ✁ ✂ ✄ ☎ ✆ ✄ ✝\nββ ββ✞αα αα✟δδ δδ\nγγ γγαα αα✠θ✡ ☛\n☞ ✌ ✍\n☛\n☞\n✞ ✞ ☛\n☞ ✍ ✎\n☛✏αα αα✟αα αα✑αα αα✟ββ ββ\nF (kT)b) a) \nτosc  (a.u.) FFT Amplitude (a.u.) 30 35 40 45 -1x10 -2 01x10 -2 \nθ ✒ ✓ ✔ ✕ ✖\na\nbCaFe 2P2\nB (T) \nFIG. 1: (color online) a) Oscillatory part of torque at T= 0.4K\nfor a single crystal of CaFe 2P2(sample A). The inset shows sample\nA attached to a piezocantilever. b) The Fourier transform sp ectra\nshowingtheevolutionoftheextremalareasoftheFermisurf acewith\nthe fieldangle, θ. The positions of different pockets, α1,α2,β,γ,δ\nand harmonics ( 2α1,3α1) as well as a weak peak, α’ are indicated\nbyarrows.\nTorquemagnetometrywasperformedusingpiezoresistivemi -\ncrocantilevers in high fields 18 T in Bristol (sample B) and\n45T(sampleA)attheNHFML,Tallahassee. Wecompareour\ndata with the predictions of bandstructurecalculations wh ich\nwe performed using an augmented plane wave plus local or-\nbital method as implementedin the WIEN2K code [12]. The\nexperimentallydeterminedlatticeparametersandinterna lpo-\nsitionswereusedforthese calculations[13].\nFig. 1(a) shows the oscillatory part of raw torque signal\nfor a CaFe 2P2crystal up to 45T. The fast Fourier transform\n(FFT)forseveralotherorientationsasthefieldisrotatedf rom\nB||c(θ= 0) towards B||a(θ= 90◦) is shown in Fig.1b.\nThe FFT frequencies Fof oscillatory torque data (in the\n1/Bdomain) are related to extremal Fermi surface areas by\nF= (/planckover2pi1/2πe)Ak. Close to B||cthe strongest amplitude peak\n(α1) is at 4.35kT and we also observe several harmonics of\nα1(see Fig.1b); we can distinguish a frequency α2∼7.3kT\nand a tiny feature α’ at 6 kT [14]. A small pocket γis ob-\nserved at ∼420T; at higher angles ( θ= 15◦) the amplitude\nof anotherfrequency, β, becomessignificant and the position\nof this peak varies strongly with increasing angle. When we\nrotate close to B||aa strong amplitude signal, δ, correspond-\ningtoanextremalareaof3.2kTisfound.\nTheangulardependenceoftheobserveddHvAfrequenciesallows us to identify the shape of the Fermi surface sheets\nfrom which they originate (see Fig. 2). Rotating away from\nB||ctheα1andα2orbits display a much stronger angular\nvariation compared to extremal orbits on a simple cylinder\nwith a weak kzdispersion ( F(θ)∼1/cosθ), [8]. This sug-\ngeststhat theseorbitsoriginatefroma stronglywarpedcyl in-\ndrical Fermi surface [14]. The size of the βorbit changes\ndramatically with θsuggesting that this Fermi surface sheet\nhas an prolate ellipsoidal shape with a maximum at θ= 0◦.\nTheαorbits are well reproduced in both samples, but the β\norbitwasonlyobservedinsampleAinmuchhigherfields(up\nto45T).Thiscanbeexplainedastheimpuritydampingofthe\ndHvAsignalisproportionalto exp(−kF/(ℓB))sothathigher\nfield are needed to see the larger βorbit (kF∝√\nF) for the\nsamemean-free-path( ℓ).\nWe nowcomparedtheexperimentaldatainCaFe 2P2tothe\npredictionsof the bandstructurecalculations. The Fermi s ur-\nface (see Fig. 2 and Fig. 3) is quasi-three dimensional and is\ncomposed of a large hole sheet in the form of a flat pillow at\nthe top of the zone whereas the electron sheets are strongly\ndistorted quasi-two-dimensional tubes centered on the zon e\ncorners. There are also two tiny hole pockets centered at Z\n(seethebottompanelofFig.3)containingonly ∼0.008holes\ncompared with the large hole and electron sheets which each\ncontains∼0.41holes/electrons;ifthePpositionisoptimized\ninthecalculation(byminimisingthetotalenergy, zP=0.3890)\ntheydisappear.\nFig. 2showsgoodagreementbetweendata andcalculation\ninthecaseofthe βfrequencywhichcorrespondstocrosssec-\ntions on the hole sheet (band 3) and the α1andα2frequen-\ncieswhichcorrespondtotheminimumandmaximalextremal\nareas of the strongly warped electron cylinder (band 4) (the\ndegree of warping is roughly ∆F/F∼50%in CaFe 2P2as\ncomparedwith ∼23%inSrFe 2P2[9]). Thecomplexin-plane\nand interplane corrugations of the Fermi surface could give\nrise to additional branches not predicted by our calculatio ns.\nThis could explain the origin of the δbranch, which is ob-\nserved within ∼25◦ofθ= 90◦(B||a) (Fig. 2) and may\naccount for the small shift found for α2as well as the pres-\nence of the weak feature at α’. It is worth emphasizing that\nthe agreement between data and calculations ( ±10meV) for\nthemainsheetsisgoodincontrasttoourfindingsforLaFePO\nand SrFe 2P2[8, 9] where rigid band shift of up to 100meV\nwere needed to bring the bandstructure into agreement with\nexperiment.\nAny small band shifts would mainly affect the small pock-\nets of the Fermi surface centeredat the Z point, as stated ear -\nlier. For example, by shifting the hole bands down the tiny\npockets at the Z point disappear and the large hole pillow\ntransforms into a large torroid [15]; when B||awe could ex-\npected orbits from extremal areas on such torroid which may\ngive rise to the δbranch (which is about half the size of the\nβorbit close to B||a) (see bottom panel of Fig. 3). The γ\nfrequency increases to about θ= 42(3)◦and then decrease\nsuggesting that it has a short closed cylinder shape (see Fig .\n2). Byshiftingtheholebandsup(by ∼40meV)thesmall3D3\nTABLE I: Experimental and calculated Fermi surface paramet ers\nof CaFe 2P2close toθ= 0◦(B/bardblc) similar to those predicted for\nCaFe2As2inthe cT phase [6].\nExperiment Calculations\nF(kT)m∗\nmeℓ(nm) Orbit F(kT)mb\nmem∗\nmb\nα1(e)4.347 2.05(4) 190 4min4.439 1.38 1.49(3)\nα2(e)7.295 3.48(7) 71 4max7.837 2.40 1.45(3)\nβ(h)18.360 4.0(2) 86 3min18.407 2.65 1.51(8)\npocket centered at the Z point (band 2) could be assigned to\ntheγbranch (Fig.3). Alternatively, γwhich has a low effec-\ntivemass( m∗= 0.4me) couldoriginatefroman Snimpurity\nphase which has orbits with similar frequencies and masses\n[16],butx-raydiffractionmeasurementsdoesnotidentify any\nSn impurities at the level of ∼1%. In any case this pocket,\nγ, accounts for only a tiny fraction of holes ( ∼0.03) and we\nbelieveit isunlikelyto playanymajorrole.\nFig. 3 shows a comparison between the Fermi surface\nof CaFe 2As2in the tetragonal phase ( c/a=2.98), CaFe 2As2\nin the cT phase ( c/a=2.66 at P=0.63 GPa) and CaFe 2P2\n(c/a=2.59). As mentioned before, it is clear that there is a\nremarkable similarity between CaFe 2P2and the cT phase of\nCaFe2As2. Thus in CaFe 2As2applying chemical pressure\n(by the isoelectronic substitution of As by P) is equivalent to\napplied hydrostatic pressure , as found in other ternary pnic-\ntides [17]. This state of reduced c/aratio has a different\nFermi surface topology compared to CaFe 2As2or SrFe 2P2\n(c/a=3.03). Yildirim [11] has argued that the cT phase of\nCaFe2As2occurs when, by reducing the Fe moment, the Fe-\nAs bonding weakens and the (inter and intra-planar) As-As\nbondinggetsstrongercausingthesignificantstrongreduct ion\nin thecaxis [4]. Similarly, in non-magnetic phosphides, the\nreductionin the c-axis(orthe c/aratio)resultsin anincrease\nP-P hybridization between pnictogen ions along the cdirec-\ntion (close to the single bond distance) [10]. The spacer be-\ntween the iron layers (Sr or Ba) limits the degree of this hy-\nbridization between layers and such a state with strong pnic-\ntogenbonding is unlikelytooccur[5, 17].\nThe effective masses, m∗, of CaFe 2P2for each Fermi sur-\nface orbit extracted from the temperature dependence of the\ndHvA signals, using the conventional Lifshitz-Kosevich fo r-\nmula [18], are shown in Table I and compared to the corre-\nsponding bandstructure values. The masses for the electron\nand hole sheets are enhanced by the same factor of ∼1.5in\ncontrast to the sheet dependent variation of the enhancemen t\nobservedrecentlyinSrFe 2P2(whichhastwoelectronandtwo\nhole Fermi surface pockets). By comparing the quasiparticl e\nenhancement on the electron sheets (which have the largest\nmean free path and often match better to the bandstructure\ncalculations being less sensitive to structural changes co m-\npared with the hole pockets) we observe that in LaFePO the\naverageenhancementis 2.38(2.2forinner and2.54for outer\npocket) [8] whereas in SrFe 2P2is 1.85 (1.6 for inner and 2.1\nfor the outer electron sheet)[9]. The conventional electro n--90 -45 0 45 90 0.0 0.5 γ\nθ (degrees)  band 2 \n 35810 13 15 18 \nδ δα2CaFe 2P2\nβ\nα1α' band 3 \n band 4 \n dHvA Frequency F (kT) \nFIG.2: (color online) Angle dependence of allobserved freq uencies\ncompared withthe band structure predictions (solidlines) . Different\nsymbols correspond to sample A (filled circles) and sample B ( open\ncircles). Solidgreypointsindicatethepositionofverywe akfeatures.\nThe bottom panel shows an expanded low frequency scale for th e\nγpocket and the dotted line is a guide to the eye. The calculate d\nFermi surface of CaFe 2P2is also shown. The solid lines delimits\ntheBrillouinzone andtheFermisurface sheets are represen tedinan\nextended zone.\nphononcoupling, λe−ph, in the cT phase of CaFe 2As2is cal-\nculated to be 0.23 [11] but such calculationsfor CaFe 2P2are\nnot yet available. Due to their structural and electronic si m-\nilarities it is likely that in CaFe 2P2a mass enhancement of\n1.23couldbeduetoaconventionalelectron-phononcouplin g,\nbut other effects could also be important [19]. The further\nanisotropic mass enhancement observed in LaFePO and pos-\nsibly in SrFe 2P2could have another origin and may be re-\nlatedtothenestingoftheFermisurface. Themeanfreepaths ,\nℓ, of the minimum electronic orbit is a factor 2 larger than\nthat of large hole sheet and of the maximum electronic orbit.\nThis suggests both anisotropy in scattering between electr on\nand holes but also along kz, as also observed in SrFe 2P2and\nLaFePO[8, 9].\nIn the case of the superconducting LaFePO and SrFe 2P24\nFIG. 3: (color online) Comparison of the calculated Fermi su rface\ntopology of CaFe 2As2(tetragonal phase), CaFe 2As2(cT phase) and\nCaFe2P2. SlicesthroughthecenteroftheBrillouinzone(solidline s)\ninthe (110) plane are shown.\n[8, 9] the energies in the band structure were shifted in op-\nposite directionforthe electronandhole pocketsto matchu p\nthe experimentaldata. Theseasymmetricbandshifts aresug -\ngested to result from geometric nesting [22]. In the present\nmeasurements on CaFe 2P2no band shifts were required ( ∼\n±10meV) to achieve agreement with experiment and con-\nsidering this model it would imply the absence of nesting in\nCaFe2P2(or the lack of long-range order in the cT phase of\nCaFe2As2[21].)\nIn conclusion, we have experimentally determined the\nFermi surface of CaFe 2P2which is closely related to the col-\nlapsed tetragonal phase of CaFe 2As2. We find that the Fermi\nsurface is composed of a single highly dispersive electron\ncylinderatthezonecornersandalargethreedimensionalpi l-\nlowshapedholesurface. Themassenhancementduetomany-\nbody interactions is isotropic ( ∼1.5) and may be dominated\nby the electron-phonon coupling. The features of this Fermi\nsurface whichdoesnot fulfill nesting conditionis likely to be\nshared by the non-magneticcT phase of CaFe 2As2and other\nternarypcnitideswithreduced c/aratioandmayexplainwhysuperconductivityis absentin suchastate.\nWe thank M. Haddow and E. Yelland for technical sup-\nport. We acknowledgefinancialsupportfromtheRoyal Soci-\nety, EPSRC, and EU 6th Framework contract RII3-CT-2004-\n506239. Work at Stanford was supported by the U.S. DOE,\nOffice of Basic Energy Sciences under contract DE-AC02-\n76SF00515. Work performed at the NHMFL in Tallahassee,\nFlorida, was supported by NSF Cooperative Agreement No.\nDMR-0654118,bythe StateofFlorida,andbytheDOE.\n[1] P.Alireza, et al.,J. Phys.Cond. Mat. 21, 012208 (2008).\n[2] W.Yu, etal., Phys.Rev. B 79, 020511 (2009).\n[3] M. S. Torikachvili, S. L. Bud’ko, N. Ni, and P. C. Canfield,\nPhys. Rev. Lett. 101, 057006 (2008).\n[4] A. Kreyssig, etal.,Phys. Rev. B 78, 184517 (2008).\n[5] Y.-Z. Zhang, H. C. Kandpal, I. Opahle, H. O. Jeschke, and\nR. Valentip. arXiv:0812.2920 (2008).\n[6] D. A.Tompsett and G.G.Lonzarich, arXiv:0902.4859.\n[7] J. G.Analytis, etal., arXiv:0902.1172.\n[8] A. I.Coldea, et al.,Phys.Rev. Lett. 101, 216402 (2008).\n[9] J. G.Analytis, etal., arXiv:0904.2405.\n[10] E. Gustenau, P. Herzig, and A. Neckel, J. Solid State Che m.\n129, 147 (1997).\n[11] T. Yildirim,Phys.Rev. Lett. 102, 037003 (2009).\n[12] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and\nJ. Luitz, WIEN2K, An Augmented Plane Wave + Local Or-\nbitals Program for Calculating Crystal Properties (Karlheinz\nSchwarz, Techn. Universit¨ at Wien,Austria,2001).\n[13]a=3.855˚A,c=9.985˚A, andzP=0.3643 for CaFe 2P2[24],\na=3.8915˚A,c=11.690˚A,andzP=0.372forCaFe 2As2[4],and\na=3.978˚A,c=10.607 ˚A, andzP=0.3663 for CaFe 2As2in the\ncT phase ( p=0.63GPa)[4].\n[14] The X-raydiffractionshows that sample A containa sing le do-\nmain with a ( ∼1◦) mosaic spread and about one percent of\nanother phase missaligned inthe ( ac)plane.\n[15] P.M. C.Rourke, etal.,Phys. Rev. Lett. 101, 237205 (2008).\n[16] J. E.Craven,Phys. Rev. B 182, 693(1969).\n[17] A. Kimber, S. A. J.and Kreyssig, et al., Nature Materials\n(2009).\n[18] D. Schoenberg, Magnetic Oscillations in Metals (Cambridge\nUniversity Press,London, 1984).\n[19] Many-body interactions could be responsible for a futh er en-\nhancement of the electron-phonon interaction due the stron g\npolarization of pictogen ions [20].\n[20] M. Kulic andA. Haghighirad, arXiv:0904.3512.\n[21] A. I.Goldman, et al.,Phys. Rev. B 79, 024513 (2009).\n[22] L. Ortenzi, E. Cappelluti, L. Benfatto, and L. Pietrone ro,\narXiv:0903.0315.\n[23] T. Kondo, et al.,arXiv:0905.0271.\n[24] A. Mewis, Z.Naturforsch. B 35B, 391 (2009)."    },    {        "title": "1503.08951v1.Room_Temperature_Quantum_Spin_Hall_Insulators_with_a_Buckled_Square_Lattice.pdf",        "content": "Room Temperature Quantum Spin Hall Insulators  with  a Buckled Square Lattice  \nWei Luo1 and Hongjun Xiang1,2* \n1Key Laboratory of Computational Physical Sciences (Mini stry of Education), \nState Key Laboratory of Surface Physics, and Department of Physics, Fudan \nUniversity, Shanghai 200433, P. R. China  \n2Collaborative Innovation Center of Advanced Microstructures, Fudan University, \nShanghai 200433, China  \nEmail: hxiang@fudan.edu.cn  \nABSTRACT : Two-dimensional (2D) topological insulators (TIs), also  known as \nquantum spin Hall (QSH) insulators, are excellent candidate s for coherent spin \ntransport related  applications because the edge states of 2D TIs  are robust against \nnonmagnetic impurities since the only available backscattering channel is forbidden. \nCurrently, most known 2D TIs are based on a hexagonal (specifically, honeycomb) \nlattice.  Here, we propose that there exists th e quantum spin Hall effect  (QSHE)  in a \nbuckled square lattice. Through performing global structure optimization, we predict a \nnew three -layer quasi -2D (Q2D)  structure which has the lowest energy  among all \nstructures with the thickness less than 6.0  Å for t he BiF system . It is identified to be a \nQ2D TI with a large band gap (0.69 eV). The electronic states of the Q2D BiF system \nnear the Fermi level are mainly contributed by the middle Bi square lattice, which are \nsandwiched by two inert BiF 2 layers. This is beneficial since the interaction between a \nsubstrate and the Q2D material may not change the topological properties of the \nsystem, as we demonstrate in the case of the NaF substrate. Finally, we come up with \na new tight -binding model for a  two-orbital  system with the buckled square lattice to \nexplain the low -energy physics of the Q2D BiF  material . Our study not only predicts \na QSH  insulator for realistic room temperature  applications, but also provide s a new \nlattice system for engineering t opological states such as quantum anomalous Hall \neffect.  \n \nKeywords: Quantum spin Hall effect, Topological insulator, Square lattice, \nFirst-principles calculations, Structure prediction, Two -dimensional material  Introduction  \nTopological insulators  (TIs) ,1-7 as a new class of quantum materials , hold great \npotential for applications in quantum information , spintronics , field effect transistors,  \nas well as thermoelectrics.8-14 The most interesting character of TIs is the presence of \nrobust topological Dirac  surface states in three -dimensional ( 3D) TIs or helical edge  \nstates in two-dimensional ( 2D) TIs with spin locked to momentum as protected by \ntime-reversal symmetry . For coherent nondissipative spin transport related  \napplications, 2D TIs are  better than 3D TIs  as the electrons at the edges of 2D TIs can  \nonly move along two directions with opposite spins and thus  free from backscattering \ncaused by nonmagnetic defects,  whereas the surface states of 3D TI s are only free \nfrom exact  180° -backscattering and suffer from scattering of other angles.  \nCurrently, quantum spin Hall effect ( QSHE) was experimentally observed only \nat very low temperature s.6,7 There is great interest in searching for room temperature  \n(RT) 2D TIs. Among all predicted or synthesized 2D TIs, the hexagonal honeycomb  \nlattice is the most common motif. For instance, g raphene  was predicted to be the first \n2D topological insulator (TI) .15 However, QSHE in graphene may  only be observed  at \nan extremely low temperature due to its tiny band gap (about  \n48 10  meV16) opened \nby spin-orbit coupling ( SOC ) effect . Subsequently, silicene with a buckled \nhoneycomb lattice was predicted to be a 2D TI with a larger band gap (about 1.55 \nmeV) due to its stronger  intrinsic SOC effect and buckling  effect .10 Unfortunately, the \nband gap is still too small for RT application , and free -standing silicene or silice ne \ngrown  on an insulating substrate have not been obtained  experimentally.  Wang et al.  \npredicted that a 2D organometallic triphenyl -metal  compound with the hexagonal \nlattice is a new organic TI with a band gap of about 8.6 meV .14 Very recently, Song  et \nal. proposed that  2D honeycomb Bi halide systems are huge -band -gap (up to 1.08 eV) \nTIs. These giant -gaps are entirely due to the result of strong on -site SOC  being related \nto the  \nxp and \nyp  orbital s of the Bi atoms around the two valley K and K' of the \nhoneycomb lattice .17 Experiments are awaited to check the stability of 2D honeycomb \nBi halides . We note  that Wu et al.  studied systematically the physics of the honeycomb lattice  with \nxp  and \nyp  orbitals.18-20 Although the presence of the Dirac  \nband structure makes the honeycomb lattice the most common  platform for \nengineering the topological states, a large gap 2D TI suitable for realistic RT \napplications remains to be synthesized experimentally. A nature question then arises: \nDoes QSHE exist in other simple lattices ? Addressing this question successfully will \nnot only enrich our understanding of topological phenomena, but also pave new ways \nfor designing topological materials for realistic applications.  \nIn this work, we suggest that  QSHE can occur in the buckled square lattice . With  \nour newly developed global optimization approach, we predict that a new quasi -2D \n(Q2D)  material (i.e., Bi 4F4) with a distorted square lattice of Bi is much more stable \nthan the previously proposed honeycomb BiF structure .17 This Bi 4F4 system is \nidentified to be a room temperature TI with a large band gap (0.69 eV). A new \ntight-binding ( TB) model for the buckled square lattice with two orbitals per site is \nproposed to understand the origin of the topological nontrivial electronic propert ies of \nBi4F4. \n \nResults  and discussions  \nGeometrical and electronic structures of the Q2D Bi 4F4 material . As we \ndiscussed above, the hypothetical 2D honeycomb BiF compound is found to be \nlarge -gap TIs17. However, it  may not be the lowest energy structure.  With our newly \ndeveloped global optimization approach for predicting Q2D materials (see \nMethod s),21 we perform extensive search to find more stable Q2D BiF structures. Our \ncalculations show that the  Q2D BiF structure [shown in Fig. 1(a)] has the lowest \nenergy among all structures with the thickness less than 6.0 Å. Since the unit cell \ncontains  four formula units of BiF, we  refer to this structure as Bi 4F4. Instead of the \nhexagonal lattice, Bi4F4 takes a distorted  square lattice with two slightly different \nlateral lattice constants  (4.520 Å and 4.527 Å ). The total energy of this structure is \nlower than that of 2D honeycomb  BiF17 by about 0.4 2 eV/atom.  The formation energy \nof Bi 4F4 is calculated to be 1.62 eV/atom  with respect t o bulk Bi and fluoride molecule , indicating no  phase separation  behavior . Our Q2D Bi 4F4 structure can be \nseen as a stacking of three layers: A bismuth layer with the distorted buckled square \nlattice is sandwiched between two BiF 2 layers.  The computed phonon frequencies22,23 \n(see Supplementary  Fig. S 1) indicate that Bi 4F4 is dynamically stable. Furthermore, \nfirst-principles molecular dynamics simulations indicate that Bi 4F4 is thermally stable \nat least up to 500 K  (see Supplementary Fig. S 2). \nThe local density approximation ( LDA ) band structure of Bi 4F4 is shown in the \nleft panel of Fig. 1(b). We can see a Dirac  cone at the Fermi level close to the Y point . \nThe detailed analysis shows that t he bands around the Fermi level  are mainly \ncontributed by \nxp  and \nyp  orbital s of the middle Bi layer of Bi 4F4. The \nzp  orbitals \nof the middle Bi layer lie at around -1.5 eV and 2.0 eV with respect to the Fermi level  \n(see Supplementary Fig. S 3). We note that the electron -phonon coupling removes the \nDirac  cone near one X point  and the other cone lying between \n  and M point  due to \nthe presence of the in-plane di stortion of the middle Bi layer. After including  the SOC \neffect, the band gap opens at the Dirac  point [see Fi g. 1(b)]. The whole system has an \nindirect band gap of 0.37 eV with the conduction band minimum ( CBM ) near (0, 0.5). \nSince LDA usually underestimates the band gap, we adopt the more reliable hybrid \nfunctional HSE0624 to find that the indirect band gap is 0.6 9 eV (see Supplementary \nFig. S 4). It is large enough for RT application s. Because Bi 4F4 has an inversion \nsymmetry, we can calculate the topological Z 2 invariant by counting th e parity of the \noccupied states25 and find that Bi 4F4 is a 2D TI (Z 2 = 1).  \nFor realistic application s, the Q2D Bi 4F4 material should lie on a substrate, or \nBi4F4 can be directly grown on a substrate. We find that the lattice constant of Bi 4F4 is \nvery close to that of the insulating substrate NaF. The lattice mismatch between NaF  \nand Bi 4F4 is about 2.0%. We suggest that the [001] NaF surface can be a good \nsubstrate for Bi 4F4 [see Supplementary Fig. S 5(a)]. Our band structure calculation \nshows that the whole system has a band gap and states near the Fermi level are mostly \ncontribut ed by Bi \nxp  and \nyp  orbitals of Bi 4F4 [see Supplementary Fig. S 5(b)]. We \nfind that Bi 4F4 on the NaF substrate is still a TI. In order to confirm this, we gradually decrease the distance between B i4F4 and the substrate NaF and find that the band gap \ndoes not close. In other words , the interaction between the insulating NaF substrate \nand Bi 4F4 is weak so that the topological nontrivial electronic properties of Bi 4F4 \nremain intact. It is noted that the t wo inert outer layers makes the topological \nproperties of Bi 4F4 more insensitive to the substrate, different from the cases of \nsilicene and bi -layer Bi.  We also check the robustness of the TI electronic structure of \nBi4F4 with respect to an epitaxial strain. Our LDA+SOC calculations show that the \nindirect band gap of Bi 4F4 increases with strain and there is a finite gap even when a 5% \ncompressive strain is present  (see Fig. 2). Note that the quasi -particle band gap should \nbe larger since LDA underestimates the band gap.  \nElectronic structures of a simple  model system Bi2F2. The main physics of \nBi4F4 can be understood by considering  a simple r system. As already mentioned \nabove, Bi 4F4 is composed by two BiF 2 layer and a middle Bi layer. The BiF 2 layer is \ninert since their electronic states are far from the Fermi level. The chemical bonding \nin the outer BiF 2 layer can be analyzed as follows. In a BiF 2 layer, each Bi atom is \nsurrounded by four F atoms and forms  a covalent bond with one Bi atom from the \nmiddle Bi layer; each F atom  forms bonds with two Bi atoms of the BiF 2 layer. \nAmong the five valence electrons (6s26p3) of a Bi atom, two electrons transfer to the \nneighboring F atoms, one electron is used to form a \nzp -\nzp \n bond with th e Bi \natom from the middle layer, and the remaining two electrons occupy the lone -pair 6s \norbital . For a Bi atom in the middle Bi layer, one electron is used to form the \nzp -\nzp \n\n bond with Bi atom of BiF 2 layer, two electrons occupy the lone -pair 6s orbital , \nand the remaining two low -energy electrons will occupy the \nxp  and \nyp  orbital s. \nTherefore, we can adopt a simple model structure Bi 2F2 (see Fig. 3 ) to understand the \nlow-energy physics in B i4F4. In this Bi 2F2 structure, the Bi atoms form the buckled \nsquare lattices , neglecting the in -plane distortion. Each Bi atom is saturated with a \nfluorine atom to shift the \nzp  orbital  far from the Fermi level.   \nThe LDA band structure of Bi2F2 is shown in the left panel of Fig. 3. Similarly to the case of Bi 4F4, there are Dirac  cones at the Fermi level. One difference is that there \nare more Dirac  cones in Bi 2F2 due to the absence of the in -plane distortion. Our \nanalysis shows that the electronic states near the Fermi level are mostly contributed \nby Bi \nxp  and \nyp  orbital s. After i ncluding the SOC effect, the band gap is opened in \nBi2F2, similar to the case of Bi 4F4. Furthermore, the Z 2 invariant of Bi2F2 is also 1. \nTherefore, the simple Bi2F2 structure carries  the main physical properties of the Bi4F4 \nstructure , as we expected . To confirm further the TI behavior in Q2D BiF  system with \na square lattice, we carry out direct LDA+SOC calculations to obtain the edge band \nstructure of a Bi 2F2 nano ribbon  with the width of 5.4 nm [see Supplement Fig. S6 (a)]. \nWe find there is a Dirac  state with the linear dispersion  mainly contributed by the \nedge Bi atoms at Г point, evidencing the topological nontrivial behavior  [see \nSupplement Fig. S6 (b)]. \nTB model for the buckled square lattices . In order to understand the \ntopological electronic structures of Bi 2F2 and Bi 4F4, we propose a new TB model for \nthe buckled square lattice , as shown in Fig.  4(a). Because of the presence of buckling, \nthere are two lattice points in the primitive cell and the  lattice is divided into two \nsublattices [A (top) and B (bottom) ]. This  square  lattice has an in -plane lattice spacing \nd and, for simplicity, we  assume d = 1 in the following discussion.  We can see  that the \nlateral mirror symmetry is broken due to the buckling, but the system still has spatial \ninversion symmetry at the bond centers between the nearest -neighbor (NN) A and B \nsites. At each lattice point, there are two orbitals \nxp  and \nyp . The number of \nelectrons in a primitive cell is four, i.e., half -filled case. The total effective low -energy \nHamiltonian contains three parts:  \n1 t so H H H H   , \nwith \nˆ ˆ ˆ ˆ , , , , , , , ,\n,[ ( ) ( )] . .t pp i xs i x xs i ys i y ys pp i xs i y xs i ys i x ys\ni A sH t c c c c t c c c c H c   \n   \n     ,  \n, , , ,\n{ , }(22) . .so i x i y i x i y\ni A BH c c c c H cii\n   \n  \n, and \nˆ ˆ 1, ,,ˆ, , ,..eff\nis d s i d s\ni A d sH t c c H c\n\n\n , \nwhere \n,isc  destructs a particle with spin s (\n  or \n  along the z axis) on the \n  (\nxp \nor \nyp ) orbital at the site i; \nˆ(1,0)x  and \nˆ(0,1)y  represent the unit vectors in the x and y  directions, respectively.  The first term \ntH  represents the usual NN hopping \nterm between the p orbitals with the Slater -Koster interaction parameters \nppt  (>0) \nand \nppt  (<0).26 The second term is the on -site SOC (\nLSu r u r  with \n  as the \nmagnitude of SOC effect) between the \nxp  and \nyp  orbitals written in a second \nquantization form.   \nThe third term \n1H  is a new additional effective NN hopping caused by the SOC \neffect , where \ns  represents the spin state that is opposite to \ns , \n labels the orbital \ndifferent from \n , and \nˆd  is the in -plane distance vector between two NN lattice \npoints. The strength of the effective hopping is described by sixteen interaction  \nparameters  \nˆ,eff\ndst . These parameters are related to each other by symmetry: \nˆ,eff eff\nxxt it\n, \nˆ,eff eff\nyxtt , \n*\nˆ ˆ ,,()eff eff\nd s d stt , \nˆ ˆ ,,eff eff\nd s d stt  , \n*\nˆ ˆ ,,()eff eff\nd s d stt , where \nefft  \n(>0) represents the magnitude of the effective hopping . Physically, this effective \nhopping may originate from a two -step process. As an example, we show how this \nworks for the case of \nˆ,eff\nxyt  in Fig. 4(c). Although the basis of our low -energy \nHamiltonian only contains \nxp  and \nyp  orbitals, there are high -lying \nzp  orbitals in \nreality. In the first step, a spin -up electron on the \nyp  orbital of the A sublattice hops \ninto the spin -down \nzp  orbital due to the on -site SOC. Then, this electron transfers to \nthe spin -down \nyp  orbital of the NN B-site, which is possible due to the presence of \nthe buckling. Note that the second step is forbidden in a planar square lattice, which \nmakes the effective NN hopping term vanishing there.  Overall, the effective NN \nhopping term is a first -order SOC effect with \nefft . This two-step process is \ndifferent from the cases of systems with the honeycomb  lattice. In graphene and \nsilicene, the relevant orbital is \nzp  and there is effective next -NN hopping due to the \nsecond -order16 or first -order SOC10 effect, respectively. On the other hand, the on -site \nSOC was found to be responsible for the band gap opening in honeycomb Bi halides with the low -energy \nxp  and \nyp  orbitals .17 In the following, we will show that the \neffective NN hopping \n1H  plays a key role in realizing the TI state.  \nWe now discuss the electronic structure of our newly proposed model. First, we \nonly consider the usual hopping term \ntH . A typical band structure with \n1.0ppt  \nand \n0.2ppt  is shown in the left panel of Fig. 4(b). We can see that there are two \nkinds of Dirac cones at the Fermi level; one locates at X point, another lies between Г \nand M. These Dirac cones can be seen as a result of the Brillouin zone folding since \ntH\n does not break the translation symmetry along \nˆx  and \nˆy . We find that all four \ndegenerate occupied bands at Г have even parity, while there are two occupied \neven -parity bands and two occupied odd -parity bands for the other three \ntime-reversal -invariant k -points (two X and one M). The fact that the states with even \nparity have lower energy than those with odd parity at Г can be easily understood by \nthe orbital interaction picture (see Supplementary Fig. S 7). Second, we add the on -site \nSOC term  \nsoH . The band structure in the case of \n0.6  is shown in  the middle \npanel of Fig. 4(b). Obviously, the Dirac cones are kept although their positions are \nshifted. Therefore , the on -site SOC effect can not open a band gap in the buckled \nsquare lattices, in contrast to the case of honeycomb Bi halides .17 Some bands split \ndue to the  on-site SOC  interaction . In particular, the four -fold degenerate occupied \nbands at Г split into t wo two -fold degenerate states whose parities remain even.  \nFinally, we find that the effective NN hopping \n1H  can open a band gap at the \nDirac points, as shown in  the right  panel of  Fig. 4(b) for  the case of \n0.2efft . The \nmagnitude of the band gap is almost proportional to \nefft  when fixing other \nparameters [see Fig. 4(e)]. Note that there is no gap if we switch off the on -site SOC \nterm. An important finding is that all four occupied states at Г have even parity, while \nthe number of occupied states with even -parity is the same as that with odd -parity for \nX and M. According to the parity criteria ,25 the system is a 2D  TI (Z 2 = 1). Therefore, \nwe find that the effective NN hopping due to SOC is indis pensable for the occurrence of the QSHE.  \nWe now investigate the dependence of the ground state on the parameters of the \nmodel Hamiltonian in details. As we discussed above, the system is not a TI  whenever\n0efft\n. Now, we consider the case of \n0efft . We find that the system becomes a \ntrivial insulator when the on -site SOC is too strong.  This can be understood as follows. \nSince the parities of the occupied states at X and M do not depend on \n , we can \nfocus on the eigenstates at Г. By solving exactly the full model Hamiltonian at Г, we \nobtain four two -fold degenerate states: EI=\n/ 2 2( )pp pptt    , E II=\n/ 2 2( )pp pptt \n, EIII=\n/ 2 2( )pp pptt    , EIV=\n/ 2 2( )pp pptt  . States I and \nII have even parity, while states III and IV have odd parity. Interestingly, these \neigenvalues are independent on \nefft . Since \n0  and usually \n0pp pptt , state I \nwith the lowest energy is always occupied, while state IV with the highest energy is \nalways empty. Whether state II has a lower energy than state III depends on the \nrelative magnitude of sign of  \n/4  and\npp pptt . When the on -site SOC is weak, i.e., \n/4pp pptt\n, state II with even parity is occupied and the system is in a TI state. \nOn the contrary, state III with odd parity is occupied and the system is a trivial \ninsulator. The phase diagram in t he case of \n0efft  is shown in Fig. 4(d). Noted that \nour model is different from that of FeSe system  proposed by Hao et al.27 Their model \nis intended to describe the interactions between d orbitals, the band gap induced by \nthe SOC is small, and the appearance of the topological  phase need s the help of a \nsubstrate . \nWe find that the band structure of Bi 2F2 is very similar to that of our new TB \nHamiltonian  [compare Fig. 4(b) and Fig. 3] . It is anticipate d that the low -energy \nphysics of Bi 2F2 can be described by our model. In order to prove this point, we obtain \nthe low -energy effective Hamiltonian by using the maximally -localized Wannier \nfunction  (MLWF).28-30 We choose \nxp  and \nyp  orbitals  as the projection functions \nfor each Bi atom . The band structure near the Fermi level is reproduced rather well by the MLWF  Hamiltonian  (see the right panel of Fig. 3, denoted by blue  lines). With the \nMLWF basis, we can obtain the parameters of our above TB model Hamiltonian:  \n1.34 eVppt\n, \n0.26 eVppt , \n1.40 eV , \n0.13 eVefft . Other interactions such \nas further neighboring hopping are found to be much weaker. Therefore, Bi 2F2 and \nthus Bi 4F4 can be well described by our model  for the buckled square lattice . \nConclusions. Using the new particle swarm optimization ( PSO) approach \nadapted for the Q2D system s, we predict that the most stable Q2D structure with the \nthickness less than 6 Å is a three -layer structure (i.e., Bi 4F4) for BiF, in which the \nmiddle Bi layer takes a distorted square lattice. We find that Bi 4F4 is a RT TI with a \nband gap of 0.69 eV . The TI state in Bi 4F4 is robust with respect to the interaction with \nsubstrate and epitaxial strain. In order to understand its topological electronic \nstructures, we proposed a TB model for the buckled square lattice with two orbitals \n(i.e., \nxp  and \nyp ) per site. We show that the band inversion is due to the chemical \nhybridization between the  \nxp and \nyp  orbital s. Different from the cases of 2D \nsystems (graphene, silicene , honeycomb Bi halides ) with a honeycomb lattice, a \nnewly discovered effective NN hopping term due to the SOC is crucial for the gap \nopening of Dirac cone.  Bonding analysis and ML WF calculations suggest  that the \nlow-energy electronic structure of Bi 4F4 can be described rather well by the new TB \nmodel.  In this work, we not only predict a RT quantum spin Hall  insulator with a \ndistorted square lattice for realistic spintronic applications, but also  provide a new \nbasic TB model for investigating topological  states in the simple square lattice.  \n \nMethods.  \nDFT calculations.  In our work, density functional theory (DFT) method is used \nfor structural relaxation and electronic structure calculation. The ion -electron \ninteraction is treated by the projector augmented -wave31 technique as implemented in \nthe Vienna ab initio simulation package.32 The exchange -correlation potential is \ntreated by LDA.33,34 For structural relaxation, all the atoms are allowed to relax until \natomic forces are smaller than 0.01 eV/Å. The HSE06 functional is adopted to compute the accurate band gap.24 \nPSO Algorithm for Q2D Systems. To obtain the most stable Q2D structure of \nBiF, we adopt our newly developed global optimization PSO approach for predicting \nQ2D materials.21 In our implementation, we generate random Q2D structures with the \nthickness along the c -axis smaller than a given thickness. For each Q2D structure, we \nfirst ra ndomly select a layer group instead of a planar space groups.35 The lateral \nlattice parameters and atomic positions are then randomly generated but confined \nwithin the chosen layer group symmetry. Subsequently, local optimization including \nthe atomic coord inates and lateral lattice parameters is performed for each of the \ninitial structures. In the next generation, a certain number of new structures (the best \n60% of the population size) are generated by PSO.36 The other structures are \ngenerated randomly, which is critical to increase the structure diversity. When we \nobtain the new Q2D structure by the PSO operation or random generation, we make \nsure that the thickness of the Q2D structure is smaller than the given thickness.  \nIn our simulation for the BiF system, we set the population size to 30 and the \nnumber of generations to 20. The ratio of bismuth and fluoride is fixed to 1:1 and the \nchemical formula ranges from Bi 1F1 to Bi 8F8. We consider six different initial \nthickness (between 1 and 6 Å) for each system. In addition, we repeat twice of each \ncalculation in order to make results reliable.  \n \nASSOCIATED CONTENT  \nSupplementary Information   \nSupplementary Information accompanies this paper at http://pubs.acs.org . \nCorrespondence and  requests for materials should be addressed to H.X. \n \nAUTHOR INFORMATION  \nCorresponding Author  \n*E-mail: hxiang@fudan.edu.cn  (H. J. Xiang ). \n \nACKNOWLEDGMENTS  Work was supported by NSFC, the Special Funds for Major State Basic Research, \nFANEDD, NCET -10-0351, Research Program of Shanghai Municipality and MOE, \nProgram for Professor of Special Appointment (Eastern Scholar), and Fok Ying Tung \nEducation Foundation.  \n \nREFERENCES  \n1. Hasan , M. Z. ; Kane, C. L.  Rev. Mod. Phys.  2010,  82, 3045 .  \n2. Qi, X. L. ; Zhang,  S. C.  Rev. Mod. Phys. 2011,  83, 1057 .  \n3. Yan, B. H. ; Zhang,  S.-C. Rep. Prog. Phys. 2012,  75, 096501 . \n4. Qian , X.; Liu, J.; Fu, L.; Li, J. Science 2014, 346, 1344 . \n5. Bernevig, B. A. ; Hughes, T. L.; Zhang, S. C. Science  2006, 314, 1757 . \n6. Kö nig , M.; Wiedmann , S.; Brü ne , C.; Roth , A.; Buhmann , H.; Molenkamp , L. W.; \nQi, X. L.; Zhang, S. C.  Science  2007,  318, 766 . \n7. Knez, I.; Du, R. R.; Sullivan, G. Phys. Rev. Lett. 2011,  107, 136603  \n8. Weng, H. M. ; Dai, X.; Fang, Z. Phys. Rev. X 2014, 4, 011002.  \n9. Kane , C. L. ; Mele, E. J. Phys. Rev. Lett. 2005, 95, 146802 . \n10. Liu, C. C. ; Feng, W. X. ; Yao, Y. G. Phys.  Rev. Lett. 2011,  107, 076802.  \n11. Murakami, S. Phys. Rev. Lett.  2006, 97, 236805 . \n12. Chuang , F. C. ; Yao, L. Z.; Huang, Z. Q.; Liu, Y. T.; Hsu, C. H.; Das, T.; Lin, H.; \nBansil, A. Nano Lett. 2014, 14, 2505.  \n13. Xu, Y.; Yan, B. H.; Zhang, H. J.; Wang, J.; Xu, G.; Tang, P. Z.; Duan, W. H.; \nZhang, S. C. Phys. Rev. Lett. 2013, 111, 136804 . \n14. Wang, Z. F. ; Liu, Z.; Liu, F. Nat. Commun. 2012,  4, 1471 . \n15. Kane , C. L. ; Mele, E. J. Phys. Rev. Lett. 2005, 95, 226801 . \n16. Yao, Y. G. ; Ye, F.; Qi, X. L.; Zhang, S. C.; Fang, Z. Phys. Rev. B. 2007, 75, \n041401 . \n17. Song , Z. G. ; Liu, C. C.; Yang, J. B.; Han, J. Z.; Fu, B. T.; Ye, M.; Yang, Y. C.; \nNiu, Q.; Lu, J.; Yao, Y. G. NPG Asia Materials , 2014,  6, e147.  \n18. Wu, C. J.; Bergman,  D.; Balents, L.; Sarma , S. D.  Phys. Rev. Lett. 2007 , 99, \n070401.  19. Wu, C. J. Phys. Rev. Lett.  2008 , 101, 186807 . \n20. Zhang, G. F.; Li, Y.; Wu, C. J. Phys. Rev. B . 2014 , 90, 075114 . \n21. Luo, W.; Ma, Y. M.; Gong, X. G.; Xiang, H. J. J. Am. Chem. Soc. 2014, 136, \n15992 . \n22. Parlinski, K.; Li, Z. Q.; Kawazoe, Y. Phys. Rev. Lett. 1997,  78, 4063 . \n23. Togo, A.; Oba, F.; Tanaka, I. Phys. Rev. B 2008,  78, 134106.  \n24. Heyd , J. Scuseria, J. E. J. Chem. Phys. 2003, 118, 8207.  \n25. Fu, L.; Kane , C. L. Phys. Rev. B. 2007,  76, 045302 . \n26. Slater , J. C.; Koster, G. F. Phys. Rev.  1954, 94, 1498 . \n27. Hao, N. N.; Hu, J. P. Phys. Rev. X  2014,  4, 031053.  \n28. Mostofi, A. A. ; Jonathan, R. Y. ; Lee, Y. S.; Souza, I.; Vanderbilt, D.; Marzari,  N. \nComput. Phys. Commun. 2008, 178, 685.  \n29. Marzari , N.; Vanderbilt, D. Phys. Rev. B  1997,  56, 12847 . \n30. Souza, I.; Marzari, N.; Vanderbilt, D. Phys. Rev. B  2001,  65, 035109 . \n31. Blö chl, P. E. Phys. Rev. B 1994, 50, 17953 . \n32. Kresse,  G.; Hafner, J. Phys. Rev. B  1994,  49, 14251.  \n33. Perdew, J. P. ; Zunger, A.  Phys. Rev. B  1981,  23, 5048.  \n34. Ceperkey, D. M. ; Alder, B. J.  Phys. Rev. Lett. 1980, 45, 566.  \n35. Luo, X. Y.; Yang, J. H.; Liu, H. Y.;  Wu, X. J.; Wang, Y. C.; Ma, Y. M.; Wei, S. \nH.; Gong, X. G.; Xiang , H. J.  J. Am. Chem. Soc. 2011, 133, 16285 . \n36. Wang, Y. C. ; Lv, J.; Zhu, L.; Ma, Y. M. Phys. Rev. B  2010,  82, 094116 .  \nFigure 1. Geometrical  and electronic  structures of  Bi4F4. (a) Top and side view of \nBi4F4. The top  and bottom  layer s are inert BiF 2 layer s. The square lattice of the middle \nBi layer is slightly distorted. (b) LDA band structure s of Bi 4F4 without (left panel) and \nwith (right panel) SOC . The contribution from  \nxp and \nyp  orbital s of the middle  Bi \nlayer  is highlighted.  \n \n \nFigure 2. Band gap of Bi 4F4 as a function of epitaxial  strain  from the LDA+SOC \ncalculations . The indirect band gap increas es with strain Δ (\n00 ( ) /a a a  , where \na and \n0a  is the lateral latt ice constants of the strained and unstrained systems, \nrespectively).  Even a large compressive strain (about -5%) can not  destroy the \nnontrivial topological properties, indicating the robust TI behavior in Bi 4F4. Note that \nLDA underestimates the band gap.  \n \n \nFigure 3. Geometrical and electronic structures of  Bi2F2 with a buckled square lattice . \nLeft panel: LDA band structure  without SOC. There are more Dirac  cones compared \nwith Fig. 1 (b) since the square lattice in Bi2F2 is not distorted . The contribution from  \nxp\n and \nyp  orbitals of Bi is highlighted. The inset  shows  the top and side view s of \nBi2F2. Right panel: Band structure from the LDA+SOC calculation s. The band \nstructure obtained by the MLWF Hamiltonian is shown in blue.  \n \n \nFigure  4. Description and electronic properties of our new TB model for a buckled \nsquare lattice. (a) Schematic illustration of the buckled squar e lattice . The red and \nblue balls represent sublattices  A (top)  and B (bottom) , respectively. One space \ninversion center  in the middle of NN A and B sites is  denoted  by “ i”. (b) Band \nstructure s of the buckled square lattice in the cases of the usual hopping (H t, left \npanel), usual hopping and on-site SOC  (Ht+H so, middle panel ), and all three terms \n(Ht+H so+H 1, the right panel), respectively. The parities of states at \n  are marked . (c) \nIllustration  of how a first order SOC  effect induces the effective NN hopping for the \ncase of \nˆ,eff\nxyt . (d) The phase diagram of the TB model for a positive \nefft . The olive \nregion  represents TI . The red dot denote s the case of Bi 2F2. (e) Band gap as a function \nof the effective NN hopping \nefft  when fixing other parameters.  \n \n \n \n \n \n \n \nSupplementary Information for  \n “Room Temperature Quantum Spin Hall Insulators with a Buckled Square Lattice”  \n \nWei Luo1 and Hongjun Xiang1,2* \n1Key Laboratory of Computational Physical Sciences (Ministry of Education), \nState Key Laboratory of Surface Physics, and Department of Physics, Fudan \nUniversity, Shanghai 200433, P. R. China  \n2Collaborative Innovation Center of Advanced Microstructures, Fudan University, \nShanghai 200433, China  \n \n \n[*]Email: hxiang@fudan.edu.cn  \n \n \nSupplementary Figure S1. The phonon dispersions of Bi 4F4 from the LDA \ncalculations . There are no imaginary frequencies in phonon dispersion, indicating its \ndynamical stability.  \n \n \nSupplementary Figure S2. The t hermal stability of Bi 4F4. The left figure is the \nfluctuation of total energy (128 -atoms/cell) as a function of simulation time f rom \nfirst-principles canonical molecular dynamics simulations at 500 K. The right figure  \nshows the final structure after  the molecular dynamics simulation . One can see that \nthe basic framework is kept intact, indicating that Bi4F4 is thermally stable at 500K.  \n \nSupplementary Figure S3. B and structures of Bi 4F4 obtained from the LDA (left \npanel) and LDA+SOC (right panel) calculations, respectively. The contribution \nfrom \nzp  orbitals of the middle Bi layer  is highlighted.  \n \nSupplementary Figure S4. The band structure of Bi 4F4 calculated by \nHSE06+SOC. One can see that the b and structure is similar to that in the case of \nLDA+SOC. The accurate band gap of Bi4F4 is about 0.69 eV from the HSE06+SOC \ncalculation. This is large enough for RT application. Note that LDA+SOC gives a \nsmaller band gap since LDA usually underestimates band gaps.  \n \n \n \nSupplementary Figure S5. Geometrical and electronic structures  of Bi4F4 on NaF \nsubstrate.  (a) The structure of Bi 4F4 on the [001] NaF surface after structural \noptimization.  NaF takes a rock salt structure. The Bi (F) atom of Bi 4F4 is on top of the \nF (Na) atom of the NaF substrate due to the electrostatic interaction. The distance \nbetween Bi 4F4 and the surface is about 2 .62 Å. (b) Ba nd structure of the whole system \ncalculated by LDA+SOC. The states near the Fermi level is mostly contributed by \nxp  \nand \nyp  orbitals (denoted by red circles ) of the middle Bi layer of Bi 4F4. This is \nbecause NaF is a large band gap (about 5.90 eV from LDA) insulator with a very low \nvalence band and a very high conduction band. The band structure of the whole \nsystem near the Fermi level is similar to that of the free -standing Bi 4F4 except for \nsome band splitting due to the extrinsic Rashba effect. This suggests that Bi 4F4 on \nNaF substrate is still a TI.  \n \n \nSupplementary Figure S6. Geometrical and electronic structures of a Bi 2F2 \nnanoribbon. (a) Geometry of a nanoribbon constructed along the x axis. The width is \n12 times of lattice constant, about 5.4 nm. (b) LDA+SOC band structure of the ribbon. \nOne can clearly see that two gapless edge states appear in bulk band gap, which cross \neach other linearly at \n  point. These edge states are mainly contributed by the edge \nBi atoms, which further prove that Bi 2F2 is a TI.  \n \n \nSupplementary Figure S7. Schematic illustration wave functions at \n  for the \nsquare lattices. Here, only the usual hopping term is considered. The eigenvalues can \nbe expressed as EI = E II = \n2( )pp pptt  and EIII = E IV =\n2( )pp pptt . States I and II \nwith even parity have lower energy than states III and IV with odd parity since \n  \ninteraction is stronger than \n  interaction. This is the key to the TI behavior in the \nbuckled square latti ces. \n \n \n \n \n "    },    {        "title": "0906.3169v1.Correlated_electron_physics_in_multilevel_quantum_dots__phase_transitions__transport__and_experiment.pdf",        "content": "arXiv:0906.3169v1  [cond-mat.str-el]  17 Jun 2009Correlated electron physics inmultilevelquantum dots:\nphase transitions,transport, and experiment.\nDavid E. Logan, Christopher J. Wright, and Martin R. Galpin\nOxford University, Chemistry Department, Physical & Theor etical Chemistry, South Parks Road, Oxford, OX1 3QZ, UK.\n(Dated: October 26, 2018)\nWe study correlated two-level quantum dots, coupled in effe ctive 1-channel fashion to metallic leads; with\nelectron interactions including on-level and inter-level Coulomb repulsions, as well as the inter-orbital Hund’s\nrule exchange favoring the spin-1 state in the relevant sect or of the free dot. For arbitrary dot occupancy, the\nunderlyingphases,quantumphasetransitions(QPTs),ther modynamics,single-particledynamicsandelectronic\ntransportproperties are considered; anddirectcompariso n ismade toconductance experiments onlateralquan-\ntumdots. Twodistinctphases arisegenerically, onecharac terisedbyanormalFermiliquidfixedpoint(FP),the\notherbyanunderscreened (USC)spin-1FP.AssociatedQPTs, whichoccur ingeneral ina mixedvalent regime\nofnon-integral dot charge, are foundtoconsist of continuo us lines of Kosterlitz-Thouless transitions,separated\nby first order level-crossing transitions at high symmetry p oints. A ‘Friedel-Luttinger sum rule’ is derived and,\ntogetherwithadeducedgeneralizationofLuttinger’stheo remtotheUSCphase(asingularFermiliquid),isused\nto obtain a general result for the T=0 zero-bias conductance, expressed solely in terms of the do t occupancy\nandapplicable toboth phases. Relatedly, dynamical signat ures of the QPTshow twobroad classes of behavior,\ncorrespondingtothecollapseofeitheraKondoresonance,o rantiresonance,asthetransitionisapproachedfrom\nthe Fermi liquid phase; the latter behavior being apparent i n experimental differential conductance maps. The\nproblem is studiedusing the numerical renormalization gro up method, combined withanalytical arguments.\nI. INTRODUCTION\nTheKondoeffectisoneoftheenduringparadigmsofquan-\ntummany-bodytheory1. Formostofitshistoryithasbeenas-\nsociatedwith bulk condensedmatter,notablytransitionme tal\nimpuritesdissolvedincleanmetals,andcertainheavyferm ion\nrare earth compounds1. In recent years, however, the ad-\nvent of quantum dot systems – with the impressive control\nandtunabilitypossible for‘artificial atoms’– hasgenerat eda\nstrong resurgence of interest in Kondo and related physics i n\nnanoscaledevices(forreviewssee e.g.[2,3]).\nIn odd-electron quantum dots the spin-1 /2 Kondo ef-\nfect arises. Manifest experimentally4,5as a strong low-\ntemperature enhancement of the zero bias conductance, indi -\ncating the formationof the local Kondo singlet below a char-\nacteristicKondotemperature,thebasictheoreticalmodel here\nisofcoursetheAndersonimpuritymodel6: asingledotlevel,\nwith a single on-levelCoulombinteraction,tunnelcoupled to\nnon-interactingmetallicleads. MoreovertheAndersonmod el\ncapturesnotonlytheKondoregime–arisingtowardsthecen-\nter of the associated Coulomb blockade valley where the dot\nlevel is singly occupied – but also the mixed valent regimes\nof non-integraloccupancyoccurringtowardsthe edgesof th e\nvalley. As such, it encompassesessentially all the physics as-\nsociatedwitha single‘active’dotlevel.\nThe situation is naturally more complex, and richer, if two\nactivedotlevelsareintegraltoelectronictransport. For exam-\nple, higher dot spin states now become possible, in this case\na 2-electron triplet stabilised by the inter-orbital Hund’ s rule\nexchange7,8. This state has been observed experimentally in\neven-electron dots, for both lateral9,10,11,12and vertical13de-\nvices(aswellasinasingle-moleculedot14). Ittooismanifest\ninastrongenhancementofthezero-biasconductance,indic a-\ntive15,16ofproximitytoanunderscreenedspin-1fixedpoint17\ninwhichthespin-1isquenchedtoaneffectivespin-1\n2oncou-\nplingto theleads.Much important theoretical work on the problem has en-\nsued; including both the 1-channel case (see e.g.[16,18,19,\n20,21,22,23,24]) where the single screening channel yield s\nunderscreened (USC) spin-1 as the stable low-temperature\nfixed point, and the 2-channel case15,16,23,25,26,27where the\nspin-1 local moment is fully screened at the lowest temper-\natures. Further, since the USC spin-1 fixed point is clearly\ndistinct from that characteristic of a normal Fermi liquid –\nthe USC phase being a ‘singular Fermi liquid’28– quantum\nphase transitions from a normal Fermi liquid to the USC\nphase are expected, and found, to arise in the 1-channel case\n(with pristine transitions broadened into crossovers for 2 -\nchannel screening). This too has been studied quite exten-\nsively18,19,20,22,24,25,26,27. However the large majority of pre-\nvious work on these ‘singlet-triplet’ transitions has focu ssed\non a somewhat particular case – the middle of the 2-electron\nCoulombblockadevalleywhere, throughoutbothphases, the\ndotoccupancy/chargeremainscloseto2;asituationwerega rd\nas unlikely to be applicable to a transition driven by tuning a\ngatevoltageintheabsenceofamagneticfield(asintheexper -\nimentse.g.of [11]),where oneinstead expectsthe dotcharge\ntovarycontinuouslywithgatevoltage. Anotableexception is\nthe work of [22], in which low-temperature transport is con-\nsideredinaregionseparatingtwoadjacentCoulombblockad e\nvalleys with spins S=1\n2andS=1 on the dot, and where the\nresultantquantumphasetransition,drivenbygatevoltage and\narising in the limit B→0+of vanishing magnetic field, oc-\ncursina mixedvalentregimeofnon-integraldotcharge.\nIn view of the above our aim here is to consider a rather\ngeneral model of a two-level quantum dot, coupled in a 1-\nchannel fashion to metallic leads; to consider its underlyi ng\nphases, thermodynamics, single-particle dynamics and ass o-\nciated low-temperature( T) electronic transport, for arbitrary\ndot charge – spanning as such the full range of possible be-\nhavior;andultimatelytomaketangiblecomparisontoexper i-\nment11. Themodelitselfisspecifiedinsec.IIandreflectsthe2\nnaturalcomplexityofatwo-leveldot,whereinadditiontot he\none-electron dot levels electron interactions include bot h on-\nlevel and inter-level Coulomb repulsions, together with in ter-\norbital spin-exchange. We study it using Wilson’s numerica l\nrenormalizationgroup(NRG) technique29,30,31as the method\nof choice, employing the full density matrix formulation of\nthe method32,33(fora recentreview see [34]); togetherwhere\npossiblewithanalyticalarguments.\nThe intrinsic phases and associated thermodynamics are\nconsidered in sec. III. With ε1,ε2denoting the one-electron\nlevel energies, the general structure of the phase diagrams in\nthe (ε1,ε2)-plane is found to consist of a closed, continuous\nline of quantum phase transitions (QPTs) separating an USC\nspin-1 phase from a continuously connected normal Fermi\nliquid phase; although more complex topologies arise as the\nexchange coupling is driven weakly antiferromagnetic (sec .\nIIIB),leadingultimatelytodestructionoftheUSCphase. T he\ntransitions are found in general to be of Kosterlitz-Thoule ss\ntype,exceptforparticularlinesofsymmetrywherefirstord er,\nlevel-crossingtransitionsarise(sec. IIIA).\nSec. IV focusses on the T=0 zero-bias conductance Gc,\nand associated static phase shift δ. A ‘Friedel-Luttingersum\nrule’ for δis derived, applicable to both the normal Fermi\nliquid and the USC spin-1 phases, and reducing to the usual\nFriedelsumrule1,35intheFermiliquidphase. Since theUSC\nphase is a singular Fermi liquid28, and as such not perturba-\ntivelyconnectedtothenon-interactinglimitofthemodel, one\ndoes not expect Luttinger’s (integral) theorem36to apply. A\ngeneralization of it for the USC phase is however deduced,\nanditsimportantconsequencesforthezero-biasconductan ce\nconsidered;leadingtoasimpleresultwhich,for boththenor-\nmalFermiliquidandUSCphases,gives Gcintermsofthedot\noccupancy/charge(or,strictly,the‘excessimpuritychar ge’1).\nSingle-particledynamicsforbothphasesaredetailedinse c.\nV. In particular, dynamical signatures of the QPT on ap-\nproachingitfromthenormalFermiliquidarefoundtofallin to\ntwo broad classes, corresponding respectively to an ‘on the\nspot’ vanishing of either a Kondo resonance, or a Kondo an-\ntiresonance, in the single-particle spectrum; the spectra l col-\nlapse in either case being associated with a vanishing Kondo\nscaleTKasthetransitionisapproached,andintermsofwhich\nuniversalscalingofdynamicsisfoundto occur.\nFinally, in sec. VI we make explicit comparison to the ex-\nperiments of [11] on a lateral dot; in which, on continuous\ntuning of a gate voltage at zero magnetic field, both the nor-\nmal spin-1 /2 Fermi liquid and the USC spin-1 phase are ob-\nserved in adjacent Coulomb blockadevalleys. Both the zero-\nbias conductance as a function of gate voltage, and (in this\ncaseinevitablyapproximate)differentialconductancema psas\nafunctionofbothgateandbiasvoltages,arecomparedto ex-\nperiment; and the features observed related to the dynamics\nconsidered in secs. (IV,V). We believe it fair to say that the\nunderlyingtheoryaccountsratherwell forexperiment.II. MODEL\nInteracting quantum dots and other nanodevices are de-\nscribed generally by the dot ˆHD, a pair of non-interacting\nleadsˆHL, and a tunnel coupling between the subsystems:\nˆH=ˆHD+ˆHL+ˆHT. We consider in this work a two-level\ninteractingquantumdotofform:\nˆHD=∑\ni,σ/parenleftBig\nεi+1\n2Uˆni−σ/parenrightBig\nˆniσ+U′ˆn1ˆn2−JHˆs1·ˆs2(1)\nHere ˆniσ=d†\niσdiσwhered†\niσcreates a σ(=↑,↓) spin elec-\ntron in level i(=1,2), ˆni=∑σˆniσis the total number oper-\nator for level i, andˆsiis the local spin-operator with compo-\nnentsˆsα\ni=∑σ,σ′d†\niσσ(α)\nσσ′diσ′andσ(α)thePaulimatrices. The\nsingle-particle levels have energies εi, the on-level Coulomb\ninteraction (taken to be the same for both levels) is denoted\nbyU, and the inter-level interaction by U′. Finally JHis the\nexchange coupling, taken in accordance with Hund’s rule to\nbe ferromagnetic ( JH>0, although we also comment in sec.\nIIIBontheweaklyantiferromagneticcase). Thestatesaris ing\nfromˆHDitself will bediscussedinsec. IIBbelow.\nThe Hamiltonian for the two equivalent non-interacting\nleads (ν=L,R) is given by ˆHL=∑ν∑k,σεka†\nkνσakνσ. Tun-\nnel coupling to the leads is described generally by ˆHT=\n∑ν∑i,k,σViν(d†\niσakνσ+h.c.)whereViνisthetunnelcoupling\nmatrix element between dot level iand leadν. We consider\nexplicitlyinthispaperthecaseofaneffective1-channels etup,\ninwhichtheratio V2ν/V1ν≡V2/V1isindependentofthelead\nindexν;i.e.the tunnel couplings are of form ViL=αVi,\nViR=βVi(withα2+β2=1), as illustrated schematically\nin fig. 1. A simple canonical transformation to new lead\norbitals may then be performed, ckσ=αakLσ+βakRσand\n˜ckσ=−βakLσ+αakRσ, such that solely the bondingcombi-\nnationofleadstates( ckσ)couplestothedot:\nˆHT=∑\ni,k,σVi(d†\niσckσ+h.c.) (2)\nWe can thus drop the lead index νand consider one effective\nlead\nˆHL=∑\nk,σεkc†\nkσckσ, (3)\nhence the effective 1-channel description illustrated in fi g. 1.\nIn practice we consider the standard case1of a symmetric,\nflat-bandconductionbandwithhalfbandwidth D,i.e.thelead\ndensityofstates (perconductionorbital)is1 /(2D).\nIt need hardly be added that the tunnel coupling pattern\nconsidered (fig. 1) is not the most general case, which would\nby contrast involve an irreducibly 2-channeldescription15, in\ngeneral with strong channel anisotropy15,16. The richness of\nthe physics arising in the case considered, with its associ-\nated pristine QPTs, is nonethelessmore than ample to justif y\nits study (indeed for NRG calculations in practice, we focus\nlargely on the case V2=V1). It has moreover been argued3\n12\nL RαV2\nαV1βV2\nβV1≡\n12V2\nV1\nFIG.1: Schematicofthetwo-leveldot(levels1,2)coupledt otheν=\nL,Rleadswiththetunnelcouplings V2ν/V1ν≡V2/V1independentof\nν; leadingtothe equivalent one-lead description indicated .\n(seee.g.[19,20]) that the 1-channel case is generally appro-\npriateto lateral quantumdots, while a 2-channelmodelisap -\npropriateforverticaldots.\nIn considering equilibrium electronic transport per se, the\ncentralquantityisofcoursethezero-biasconductance, Gc(T),\nacross the L/Rleads (fig. 1). An expression for it is readily\nobtainedfollowingMeirandWingreen37,andis:\nGc(T) =2e2\nhG0/integraldisplay∞\n−∞dω−∂f(ω)\n∂ωπ(Γ11+Γ22)Dee(ω)(4)\nHeref(ω)=[eω/T+1]−1(kB≡1)isthe Fermifunctionand\nΓii=πρV2\ni(5)\nis the hybridization strength for level i(ρis the lead den-\nsity of states). The dimensionless conductance prefactor\nG0=(2αβ)2–orequivalently G0=4ΓLΓR/(ΓL+ΓR)2with\nΓν=πρ∑iV2\niν– reflects the relative asymmetry in tunnel\ncoupling to the L/Rleads. It is naturally maximal, G0=1,\nfor symmetric coupling where (fig. 1) α=1/√\n2=β(i.e.\nΓR=ΓL). Thekeyquantitydeterminingtheconductanceeqn.\n(4), which we analyse in detail in later sections, is the ‘eve n-\neven’ single-particle spectrum: Dee(ω) =−1\nπImGee(ω)in\nterms of the (retarded) Green function Gee(ω)(↔Gee(t) =\n−iθ(t)/an}bracketle{t{deσ(t),d†\neσ}/an}bracketri}ht). Thee-orbital creation operator is\ngivengenerallyby\nd†\neσ=1√Γ11+Γ22/parenleftBig/radicalbig\nΓ11d†\n1σ+/radicalbig\nΓ22d†\n2σ/parenrightBig\n(6)\nintermsofthe levelcreationoperators,suchthat\nGee(ω) =1\nΓ11+Γ22∑\ni,jΓijGij(ω) (7)\nin terms of the corresponding propagators for the dot levels ,\nGij(ω)(i,j∈{1,2}); andwhere\nΓ12=πρV1V2≡/radicalbig\nΓ11Γ22 (8)\nistheinter-levelhybridizationstrength.\nForthecase V2=V1all hybridizationstrengthscoincide,\nΓij≡Γ:V2=V1(9)\n(and the e-epropagator then reduces simply to Gee(ω) =\n1\n2[G11(ω)+G22(ω)+2G12(ω)]). Itisconvenientinthiscase\ntospecifythe‘bare’parametersof ˆHDintermsof Γ, defining\n˜εi=εi\nΓ,˜U=U\nΓ,˜U′=U′\nΓ,˜JH=JH\nΓ.(10)A. Symmetries\nWe will subsequently consider different phases of the dot-\nlead coupled system in the ( ε1,ε2)-plane, for given values of\ntheinteractionparameters U,U′andJHentering ˆHD(eqn.(1)).\nTo this end it is economicalto exploitsymmetry. Rather than\nthebarelevels ε1,ε2it isoftenhelpfultoemploy\nx=ε1+1\n2U+U′(11a)\ny=ε2+1\n2U+U′. (11b)\nTheir significance arises from a particle-hole transformat ion\n(p-ht)ofˆH=ˆHD+ˆHL+ˆHT(eqns.(1-3)),namely30\ndiσ→d†\niσckσ→−c†\n−kσ. (12)\nˆH≡ˆH(x,y)transformsunderthe p-htasˆH(x,y)→2(x+y)+\nˆH(−x,−y), and is hence invariantat the p-hsymmetricpoint\nx=0=y. Use of x,ythus specifies the level energies rel-\native to this point. All physical properties, thermodynami c\nand dynamic, have characteristic symmetries under the p-ht,\nwhich we exploit many times in the paper. For example the\nfree energy F(x,y) =−Tln{Tre−βˆH(x,y)}is, modulo an ir-\nrelevantconstant,equivalenttoits p-htcounterpart( F(x,y)=\n2(x+y)+F(−x,−y)), whence e.g.phase boundaries (secs.\n(IIB,III))areinvariantunderinversion (x,y)→(−x,−y);and\nthusonly y≥xneedinpracticebeconsidered.\nThe second symmetry exploited is a ‘1-2’ transformation,\nviz.thetrivialcanonicaltransformation\n(d1σ,d2σ)→(d2σ,d1σ) (13)\nunder which the dot Hamiltonian ˆHD(x,y)→ˆHD(y,x). The\nsame symmetry applies to the full ˆHfor V2=V1; whence\ne.g. F(x,y) =F(y,x)is invariant to reflection about the line\ny=x, and in consequence phase boundaries need overall be\nconsideredonlyfor y≥|x|.\nB. Groundstate phases: overview\nIt isfirst instructiveto considerbrieflythe states ofthe is o-\nlateddotinthe( x,y)-plane,asdeterminedbythegroundstates\nofˆHD(eqn. (1)). We label the dot states as ( n1,n2) (with\nni=/an}bracketle{tˆni/an}bracketri}htthe ground state charge for level i), with energies\nED(n1,n2). For allJH>0, the 2-electron dot ground state is\nthe (1,1) spin triplet with energy ET(≡ED(1,1)=ε1+ε2+\nU′−1\n4JH),centredonthe p-hsymmetricpoint (x,y)=(0,0);\nas illustrated in fig. 2(a) for the representative case U=20,\nU′=7.5andJH=5. All othergroundstates, indicatedin the\nfigure,areeitherspinsingletsordoublets.\nConsidering in particular y≥ |x|– phase boundaries be-\ning invariant to inversion and reflection as above – the (1 ,1)\ntriplet is bordered both by another 2-electron state, viz.the\nspin singlet (2 ,0), and by a 1-electronspin doublet (1 ,0) (the\nbounding lines for which are given by y=x+U−U′+1\n4JH4\n-30-20-10 0 10 20 30\n-30-20-10 0 10 20 30y\nx(1,1)(0,0)\n(2,2)(1,0)\n(0,1)\n(0,2)(2,0)\n(1,2)(2,1)a)\n-30-20-10 0 10 20 30\nx(1,1)(0,0)\n(2,2)(1,0)\n(0,1)\n(0,2)(2,0)\n(1,2)(2,1)b)\n-30-20-10 0 10 20 30\nx(1,1)(0,0)\n(2,2)(1,0)\n(0,1)\n(0,2)(2,0)\n(1,2)(2,1)c)\n-30-20-10 0 10 20 30\nxUSCFLd)\nFIG.2: (a)-(c)showgroundstatesoftheisolateddotinthe( x,y)-plane,for U=20andJH=5(inunitsof Γ≡1);with(a) U′=7.5,(b)U′=0\nand (c)U′=U. States are labelled as (n1,n2), withni=/an}bracketle{tˆni/an}bracketri}htthe charge on level i. Phase boundaries surrounding the 2-electron triplet stat e\n(n1,n2)=(1,1)) areindicated bysolidlines, andall others bydotted lines . The dashed arrowedline in(a) shows the form of anexperimen tal\n‘trajectory’ on application of a gate voltage, see text. For the same bare parameters as (a), Fig. 2(d) shows the phase dia gram obtained via\nNRG for the lead-coupled dot system (with V2=V1), as detailed in sec. III ff. It consists of a line of continuous quantum phase transitio ns\n(thick solid line) and two first order level-crossing transi tions on the line y=x(shown as dots), separating a singular Fermi liquid phase28\ncharacterisedbyanunderscreenedspin-1fixedpoint17(interior,‘USC’)fromanormalFermiliquid(‘FL’)charact erisedingeneralbyafrozen\nimpurityFP31. The hexagonal boundary of the (1,1)tripletstate for the isolateddot (fig.2(a)) isalsoshown fo r comparison.\nandy=1\n2U+1\n4JHrespectively). The dashed line in fig. 2(a)\nshows a typical ‘trajectory’, y=x+Δε(i.e.ε2=ε1+Δε),\nexpectedfromexperimenton applicationof a gate voltage Vg\nto the dot, with ε1∝Vgand fixed level spacing Δε=ε2−ε1.\nNotethatthe2-electron (1,1)tripletistherebyaccessedfrom\nthe 1-electron state (1,0)22(as relevant to comparison with\nexperiment,sec.VI).\nFig. 2(b,c) show the isolated dot ground states arising for\ninter-level Coulomb repulsion U′=0 andU′=U, respec-\ntively. For y>|x|in the former case, the (1,1)triplet state\nis bordered almost exclusively by the 1-electron state (1,0),\nwhile in the latter case it is bordered almost exclusively by\nthe 2-electron singlet (2,0). These two cases are of course\nextremes; and although aspects of the model have been con-\nsideredpreviouslyforthe case U′=U18,19,22, we knowofno\ncompellingreasonwhythe intra-andinter-levelCoulombre -\npulsions should in general be near coincident for reasonabl y\nsmall dots(indeedwe arguein sec. VI that comparisonto the\nexperimentof[11]isconsistentwiththe contrary).\nThe states of the dot per seare of course quite trivial. We\nnowconsiderthefulllead-coupledsystem,ouraimherebein g\nto give simple qualitative arguments for the general form of\nthephasediagraminthe( x,y)-plane.\nOn coupling to the leads, the effective low-energy model\ndeep in the spin-1 regime centred on (x,y) = (0,0), is nat-\nurally a 1-channel spin-1 Kondo model15,16(obtained for-\nmally by a Schrieffer-Wolff transformation1,38retaining only\nthe triplet (1 ,1)-state of the dot itself, see also Appendix A).\nIts low-energy physics is well known17: half the spin-1 is\nscreened by the conduction electrons, leading to a free spin -\n1\n2with weak residual ferromagnetic coupling to the metallic\nlead,whichresultsininnon-analytic(logarithmic)corre ctions\ntoFermiliquidbehavior;theresultantstatebeingclassifi edas\na singular Fermi liquid28. The associated low-energy fixed\npoint(FP) is ofcourse the underscreenedspin-1(USC) FP of\nNozi` eresandBlandin17.Bycontrast,deepinthe1-electron(1 ,0)-regime(fig.2),the\neffectivelow-energymodelisobviouslyspin-1\n2Kondo,anor-\nmalFermiliquidwithafullyquenchedspinandastrongcou-\npling (SC) low-energy FP1,29,30. Since the underlying stable\nFPs(USCandSC)associatedwiththesetworegimesarefun-\ndamentally distinct, a quantum phase transition (QPT) some -\nwherebetweenthetwomust thereforeoccur22.\nButwhatoftheotherisolateddotstates,encirclingthespi n-\n1 state as illustrated in fig. 2(a-c) (and all of which as noted\nabove are either spin singlets or doublets)? The salient poi nt\nhere is that, on coupling to the leads, allsuch give rise to\nFermiliquidstates: theirstablelow-energyFPsformacont in-\nuous line connecting the SC FP arising for the spin-1\n2Kondo\nmodelto the genericcase of the frozenimpurityFP31(asfol-\nlows from the original work of Krishnamurthy, Wilkins and\nWilson31on the asymmetric single-level Anderson model).\nNo phase transitions between these states can therefore oc-\ncur, the ‘transitions’ arising in the isolated dot limit (do tted\nlinesin fig.2(a-c))beingreplacedbycontinuouscrossover s.\nIn consequence, one expects the general structure of the\nphase diagram in the ( x,y)-plane to consist of a closed, con-\ntinuous line of QPTs separating an USC spin-1 phase from a\ncontinuously connected normal Fermi liquid phase. This is\nindeed as found from detailed NRG analysis, as will be seen\ninthefollowingsections. Atypicalresultantphasediagra mis\nshown in fig. 2(d) (for the same bare parameters as fig. 2(a),\nthe phase boundaryoccurringclose to the borderofthe (1,1)\nstate of the isolated dot as one might expect). It consists of a\nline of continuous QPTs; together with two first order level-\ncrossingQPTsontheline y=x(indicatedbydotsinfig.2(d)),\nwhich are equivalent to each other under the p-htransforma-\ntionx→−x.\nThe transitions will be discussed in detail below, but we\nadd here that the occurrence of first order transitions along\nthey=xline (ε2=ε1) is a general consequence of symme-\ntry. As noted in sec. IIA, for V2=V1the fullˆHtransforms5\nunder the ‘1-2’ transformation as ˆH(x,y)→ˆH(y,x), and is\nhence invariant on the line y=x. Along that line all states\nof the entire system thus have definite parity under the ‘1-2’\ntransformation,with the Hilbert space of ˆHstrictly separable\ninto disjoint parity sectors. A level-crossing transition must\nthus occur when the global many-body ground state changes\nparity(furtherdiscussionofit will begivenbelow).\nIII. PHASESAND THERMODYNAMICS\nDynamicsandtransportpropertieswillbediscussedinsec.\nIVff, but we begin with thermodynamics; in particular the\ntemperature ( T) dependence of two standard quantities1,34\nwhich provide clear signatures of the various FPs reached\nunder renormalization on decreasing the temperature/ener gy\nscale, namely the entropy Simp(T)and the uniform spin sus-\nceptibility χimp(T)=/an}bracketle{t(ˆSz)2/an}bracketri}htimp/T(whereˆSzreferstothespin\nof the entire system, and /an}bracketle{tˆΩ/an}bracketri}htimp=/an}bracketle{tˆΩ/an}bracketri}ht−/an}bracketle{tˆΩ/an}bracketri}ht0with/an}bracketle{tˆΩ/an}bracketri}ht0de-\nnotinga thermalaverageintheabsenceofthedot).\nWe also consider briefly the usual T=0 ‘excess impurity\ncharge’nimp,viz.the difference in charge of the entire sys-\ntemwithandwithoutthedotpresent( ˆΩ≡ˆN=∑k,σc��\nkσckσ+\n∑iˆniintheabove);andwhichinpracticecorrespondsclosely\nto the net dot charge, nimp≃ /an}bracketle{tˆn1+ˆn2/an}bracketri}ht, see also sec. IVA.\nProsaic though nimpis, we show later that it plays a key role\nin understanding the zero-bias conductance in boththe USC\nand FL phases, and relatedly the ‘Friedel-Luttingersum rul e’\nof sec. IVB. Under the p-hand 1-2 transformations of sec.\nIIA,nimp≡nimp(x,y)transformsrespectivelyas:\nnimp(x,y) =4−nimp(−x,−y) (14a)\n=nimp(y,x) (14b)\nResults shown are obtained using the full density ma-\ntrix formulation32,33of Wilson’s non-perturbativeNRG tech-\nnique29,30,31, employing a complete basis set of the Wilson\nchain; for a recent review see [34]. Calculationsare typica lly\nperformedforanNRGdiscretizationparameter Λ=3,retain-\ning the lowest 2000states per iteration. We here consider ex -\nplicitly the case V2=V1(sec. II), with the hybridization Γ\n(eqn. (9)) as the basic energy unit, choosing the lead band-\nwidthD/Γ=100 (≫1, such that results are independent of\nDforall practicalpurposes).\nFig. 3 shows the T/Γ-dependence of Simp(T)(top) and\nTχimp(T)(bottom),forfixed ˜U=20,˜U′=7and˜JH=2,tak-\ningaverticalcutthroughthe (x,y)-phasediagram: theenergy\nof level-1 is fixed at ˜ε1=−1\n2˜U−˜U′≡−17 (i.e. x=0), and\n˜ε2(orequivalently y) is progressivelydecreasedfromdeep in\nthe FL phase, towards and through the transition, and down\nto thep-hsymmetric point at the center of the USC phase;\nthe transition occurringat ˜ε2c=−6.536...(close to the value\n−6.5expectedfromtheisolateddotlimit).\nIn all cases the highest Tbehavior is naturally governed\nby the free orbital FP30,31, with all 42states of the two-level\ndot thermally accessible, hence Simp=ln16 (and Tχimp=\n2×1\n8). For case (a), ˜ε2= +5 is sufficiently large that0.00.51.01.52.02.53.0\n10-1210-1010-810-610-410-2100Simp\nT/Γln 2ln 3ln 16\na)b)c) d) e)h)\n0.00.10.20.30.40.50.6\n10-1210-1010-810-610-410-2100Tχimp\nT/Γa)b)c) d) e)f)g)h)1.01.21.41.61.82.0\n-15-10-5 0 5nimp\nε~2\nFIG. 3:T-dependence of the entropy Simp(T)(top panel) and spin\nsusceptibility Tχimp(T)(bottom);forfixedlevelenergy ˜ε1=−1\n2˜U−\n˜U′(i.e. x=0) on progressively decreasing ˜ε2from deep in the FL\nphase, across the QPT (at ˜ε2c=−6.536..), and through to the p-\nhsymmetric point atthe center of the USC phase. Shown for ˜U=\n20,˜U′=7 (i.e.˜ε1=−17) and˜JH=2, with:˜ε2=+5 (a),−5 (b),\n−6 (c),−6.3 (d),−6.43 (e),−7 (f),−10 (g) and −17 (h). [Labels\nf) and g) are omitted for clarity from the top panel, but are ea sily\nidentified from the bottom.] Inset, top panel : the corresponding\nT=0impuritycharge nimpvs˜ε2. ItchangescontinuouslyastheQPT\niscrossed at ˜ε2c(dashed vertical line),at whichpoint nimp≃1.4.\nlevel-2 is in essence irrelevant (provided T/Γ≪˜ε2−˜ε1),\nthe model thus reducing in effect to a single-level Anderson\nmodel30,31. Hence, on decreasing T,Simp(T)first flows to-\nwards the spin-1/2 local moment (LM) FP corresponding to\nSimp=ln2 (evident in this case as a relatively weak plateau\natT/Γ/lessorsimilar1, reflecting the modest minimum thermal excita-\ntion of∼ED(2,0)−ED(1,0) =ε1+U=3Γ). On further\ndecreasing T, the system then flows to the stable frozen im-\npurity (FI) FP symptomatic of the Fermi liquid ground state,\nwith vanishing entropy Simp(likewise Tχimp=0). A Kondo\nscaleTKmay be identified from the crossover between the\nmarginally unstable LM FP and the stable FI FP (we define\nit in practice via Simp(TK) =0.1). On further decreasing ˜ε2,\ncases (b-e) in fig. 3, the same essential behavior is found, th e\nFIFP remainingthestablelow- TFP.But the Simp=ln2 (and\nTχimp≃1\n4)LMplateauprogressivelylengthensandtheasso-\nciatedTKcorrespondinglydiminishes,vanishingasthetransi-6\n-20-16-12-8-4\n 0  2  4  6  8 10  12log(TK/Γ)\nε~2-ε~2,c-24-20-16-12-8-4 0\n 0 1 2 3 4 5 6 7 8log(TK/Γ)\n(ε~2-ε~2,c)-1\nFIG.4: EvolutionoftheKondoscaleintheFLphase ˜ε2>˜ε2cforthe\nsame parameters as fig. 3: log (TK/Γ)vs˜ε2−˜ε2c. Inset:TKvanishes\nexponentially as the QPT is approached, TK∝exp(−a/|˜ε2−˜ε2c|)\n(witha∼O(1)), characteristic of aKosterlitz-Thouless transition.\ntionisapproachedfromtheFermiliquidside (fig.4).\nThe behavior on the other side of the transition ˜ε2<˜ε2c\n(cases (f-h) in fig. 2) is qualitatively distinct. Here the T=0\nentropy is in all cases ln2 (with Tχimp=1\n4), characteristic\nof an unquenched doublet ground state. The stable FP is the\nspin-1/2LM FP – or equivalentlythe USC spin-1FP17, there\nbeingnodistinctionbetweenthemasFPs perse.\nTheQPT itself is ofKosterlitz-Thouless(KT)type. Thisis\nevident for examplefrom NRG flows, which indicate no sep-\narate, unstable critical FP, distinct from one of the stable FPs\nmentionedabove. Itisalsoevidentinthebehaviorofthesca le\nTK, whichas shownin fig. 4 (inset) vanishesexponentiallyin\n|˜ε2−˜ε2c|−1(rather than as a power-law) as the QPT is ap-\nproached from the Femi liquid side; and by the absence of a\nlow-energyscaleintheUSCphasewhichvanishesasthetran-\nsitionisapproachedfromthatside. Weaddthatthelatterdo es\nnotofcourseimplytheinherentabsenceofalow-energyscal e\nintheUSCphase. Fordeepinsidethisphase(where nimp≃2)\nthe effective low-energy model is spin-1 Kondo, as evident\ne.g.incase(h)offig.3fromboththeemergenceofanearfree\nspin-1 susceptibility with decreasing T(χimp∼S(S+1)/3T\nwithS=1), and from the intermediate Simp=ln3 plateau\nindicative of a spin-1 local moment FP; reached before the\ncrossover to the stable USC FP with Simp=ln2, and from\nwhichacharacteristicspin-1Kondoscale TS=1\nKmaybeidenti-\nfied(inparalleltothatabovefortheFermiliquidKondoscal e\nTK). ButthisscaleplaysnoroleintheQPT perse,andincon-\ntrast to the approach from the Fermi liquid phase, there is no\nvanishingscaleonapproachingtheQPT fromtheUSC side.\nThebehavioroutlinedaboveisnotconfinedtotheexample\nillustrated: all continuous transitions are found to be of K T\nform. Thisisinfacttobeexpected. HofstetterandSchoelle r19\nhave consider the model (with U′=U) in the regime where\nthe dot is doublyoccupiedby electrons, i.e. nimp≃2 – where\nfrom eqn. (14), nimp=2 arises by symmetry along the line\ny=−xin the phase-plane(or close enoughto it, in practice).\nAKTtransitionislikewise foundinthiscase19,andbyconti-\nnuityonewouldthusexpectthesamebehaviortoarisegener--15-10-5 0 5 10 15\n-15-10-5 0 5 10 15y/Γ\nx/Γa)b)c)\n-8-6-4-2 0 2 4 6 8\n-8-6-4-2 0 2 4 6 8y/Γ\nx/ΓUSCFL\nFIG. 5: Phase boundaries in the (x,y)-plane, separating the Fermi\nliquid phase (outer) from USC spin-1 (inner). Top: For˜U=20 and\n˜JH=5,varying ˜U′=0(a),7.5(b)and20(c). First-ordertransitions\nontheline y=xareindicatedbydots. Bottom: For˜U=10and˜U′=\n5.25, varying the exchange coupling ˜JH=5,1,0 and−0.5 (outside\ntoinside respectively). Notethe continued persistence of the USC\nspin-1phase, even forweak antiferromagnetic exchange (se e text).\nallyinthe (x,y)-plane.\nWe also note that the transition itself occurs generically i n\na mixed-valent regime of non-integral nimp; seee.g.fig. 3\n(top, inset) where nimpvaries continuously as the transition\niscrossed,with nimp≃1.4at thetransitionitself. Thisinturn\nmeans that even in a strongly correlated regime it is not in\ngeneral possible to construct, via a Schrieffer-Wolff38(SW)\ntransformationfrom the original Anderson-likemodel, an e f-\nfectivelow-energy spinmodelapplicableinthevicinityofthe\nQPT. An exception to this is the vicinity of the line y=−x\nalong which, as above, nimp=2 is guaranteed by symmetry.\nIn this case, as shown in [19], a SW transformationretaining\nsolely the 2-electron (1,1)triplet and (2,0)singlet states of\nthe isolated dot yields an effective two-spin, spin-1 /2 Kondo\nmodelknown20to exhibitaKT transition.\nPhase diagramsobtainedvia NRG areshownin fig. 5. The\ntop panel shows the effect of varying the inter-level intera c-\ntionU′forfixedUandJH,withbehaviorthatparallelsexpec-\ntations from the isolated dot limit (sec. IIB and fig. 2). The\nbottom panel by contrast shows the effect of varying the ex-\nchange coupling JHfor fixedU,U′, including ˜JH=0 and an\nantiferromagnetic(AF) ˜JH=−0.5. Notethatevenforweakly\nAF exchange the USC spin-1 phase still persists, as consid-7\nered further in sec. IIIB, reflecting a ferromagnetic effect ive\n(RKKY)spin-spininteractioninducedoncouplingtothelea d.\nA. First order transitionson y=xline\nWeturnnowtothefirstordertransitionspermittedbysym-\nmetry(secs.(IIA,IIB))ontheline y=x(˜ε2=˜ε1≡˜ε=ε/Γ).\nToillustratethis,fig.6showsthe T-dependenceof Simp(T)\nandTχimp(T), againfor fixed ˜U=20,˜U′=7 and˜JH=2 (cf\nfig. 3), as ˜εis varied and the transition is approached from\nbothsides: ˜ε=˜εc±10−nwithn=4,6,8,10( (b)-(e)respec-\ntively), as well as ˜ε=˜εc≃−4.7 itself ((f)). Shown for com-\nparison are the cases ((a)) ˜ε= +5 deep in the Fermi liquid\nregime,with ˜εheresufficientlylargethatthedegeneratelevels\narebarelyoccupied( nimp≃0.25,seetopinset);and ˜ε=−17\nat thep-hsymmetricpointdeepintheUSC phase((g)).\nThestablelow-temperatureFPsremainofcourseasbefore,\nviz.the FI FP for the Fermi liquid phase ˜ε>˜εcwhere the\nglobal ground state is a singlet; and the USC FP for ˜ε<˜εc,\nwith a doublet ground state. Close to the transition however\n– where the energy separation between these states is tend-\ning to zero (we denote its magnitude by T∗) – the singlet and\ndoublet states will appear effectivelydegeneratefor temp era-\nturesT/greaterorsimilarT∗;givingriseinconsequencetoanentropyplateau\nofSimp=ln3 seen clearly in fig. 6, with a corresponding\nplateau in the magnetic susceptibility of Tχimp=1\n6(readily\nunderstood as the mean /an}bracketle{t(Sz)2/an}bracketri}ht ≡1\n3(1×0+2×1\n4)for the\nquasidegenerate states). These are signatures of the ‘tran si-\ntion fixed point’ (TFP), characteristic of systems exhibiti nga\nlevel-crossing transition (see e.g.[39]). On further reducing\nTbelow∼T∗, the system is seen to cross over from the TFP\nto one or other of the stable FPs (which crossover in effect\ndefinesT∗40). Moreover, as the transition is approached, the\nlow-energy scale T∗vanishes – linearly in (˜ε−˜εc)as shown\nin fig. 6 (bottom inset), reflecting the level crossing charac -\nter of the QPT. And since T∗=0 precisely at the transition,\ntheTFPnaturallypersistsdownto T=039(wheretheground\nstate consists of precisely degenerate global singlet and d ou-\nbletstates), asevidentincase (f)offig.6.\nThe behaviorof the ( T=0) excessimpuritycharge nimpis\nalso shown in fig. 6 (top inset). In contrast to the continuous\nKT transitions, nimpis seen to change discontinuously as the\ntransitioniscrossed,commensuratewiththefirstordernat ure\nofthelevel-crossingtransition.\nA partial progenitor of the latter behavior is in fact appar-\nentinthetrivialnon-interactinglimit, U=0=U′=JH. Tak-\ning even ( e) and odd ( o) combinationsof the dot levels 1 and\n2,viz. deσ=(d1σ+d2σ)/√\n2 anddoσ=(d1σ−d2σ)/√\n2 (cf\neqn. (6) with Γii≡Γ, eqn. (9)), only the e-orbital tunnel cou-\nples directly to the lead and the non-interacting Hamiltoni an\nˆH0reducesto\nˆH0=1\n2(ε1+ε2)(ˆne+ˆno)+∑\nk,σ√\n2V(c†\nkσdeσ+h.c.)\n+∑\nσ1\n2(ε1−ε2)(d†\neσdoσ+h.c.)+ˆHL(15)0.00.51.01.52.02.53.0\n10-1210-1010-810-610-410-2100Simp\nT/Γln 2ln 3ln 16\na) b)c)d)e)f)\ng)\n0.00.10.20.30.40.50.60.7\n10-1210-1010-810-610-410-2100Tχimp\nT/Γ1/6a) b)c)d)e)f)g)0.00.51.01.52.0\n-15-10-5 0 5nimp\nε~\n0.00.51.01.5\n-1.0 0.0 1.0T∗ /10-5Γ\nε~-ε~c /10-4\nFIG. 6:T-dependence of the entropy Simp(T)(top panel) and spin\nsusceptibility Tχimp(T)(bottom) as the first-order transition on the\nline˜ε2=˜ε1≡˜εis approached and crossed from both sides of the\ntransition; for fixed ˜U=20,˜U′=7 and˜JH=2, the transition here\noccurring at ˜εc=−4.73864856302929. Solidlines refer tothe USC\nphase (˜ε<˜εc), dashed lines to the Fermi liquid (FL) phase. Shown\nfor (a)˜ε= +5 deep in the FL phase; (b-e) ˜ε=˜εc±10−nwith\nn=4,6,8,10 respectively; (f) ˜ε=˜εc; as well as for (g) ˜ε=−17\natthep-hsymmetricpointdeepintheUSCphase. ThetransitionFP\nhasacharacteristic Simp=ln3andTχimp=1\n6,asindicated; andper-\nsists down to T=0 precisely at the transition ((f)). Inset, top panel :\nT=0 impurity charge nimpvs˜ε1≡˜ε. It changes discontinuously\nas the transition is crossed (dashed vertical line), from nimp≃0.95\nto1.31.Inset,bottom panel : The low-energy scale T∗(see text)van-\nisheslinearlyin (˜ε−˜εc)asthetransitionisapproached,symptomatic\nof the level-crossing nature of the transition.\n(withˆHLthe leadHamiltonianeqn.(3)). Ingeneral,the eand\noorbitals are coupled by the penultimate term in eqn. (15).\nBut for the case ε2=ε1≡εof present interest the Hamilto-\nnianisseparable, ˆH0=ˆH0\ne+ˆH0\no,withˆH0\no=εˆnoafreeorbital\nwith energy ε. The transition in this case thus occurs triv-\niallyforε=0(thep-hsymmetricpointinthenon-interacting\nlimit)asthe o-orbital–whichisunoccupiedfor ε>0–moves\nacross the Fermi level, becoming singly occupied precisely\natε=0 and doubly occupied for all ε<0; such that nimp\nchangesdiscontinuouslyfrom1to3 as ε=0 iscrossed.\nWith interactions present the situation is of course much\nless simple. For although the o-orbital remains uncoupled\nfromthe lead when ε2=ε1, it is thencoupledto the e-orbital\nviathenon-trivialinteractiontermsin ˆHD(eqn.(1)),whichac-8\n-4-3-2-1 0 1 2 3 4\n-4-3-2-1 0 1 2 3 4y/Γ\nx/Γ\nFIG. 7: Evolution of phase boundaries in the ( x,y)-plane for antifer-\nromagnetic ˜JH<0. Shownforfixed ˜U=10and˜U′=5.25(cffig.5\nbottom), with: ˜JH=−0.55 (long dash line), −0.558 (solid), −0.56\n(short dash) and −0.6 (dotted). Interior regions in each case are the\nUSC phase, exterior regions the Fermi liquid. First-order t ransitions\nare indicated bydots, andoccur onthe lines y=x and y=−x.\nquire a rather complex (and physically unenlightening) for m\nwhen expressed in termsof eandooperators. We will return\nagaintothis case in sec. VB, fromtheperspectiveof dynam-\nicsandsingle-particle‘renormalizedlevels’.\nB. Weaklyantiferromagnetic JH\nAsnotedabove(fig.5),theUSCspin-1phasesurviveseven\nfor weak antiferromagnetic (AF) coupling JH<0, reflecting\naneffectiveferromagneticRKKYinteractioninducedoncou -\nplingtothelead. ForlargeenoughAF |JH|however,thesitua-\ntionisclearlydifferent. Onlythe singletstateofthedotinthe\n(1,1)sectoris relevanthere,andoncouplingtothe leadsone\nexpectsaglobalsingletgroundstatewithastableFIFPwhic h\niscontinuouslyconnectedtothatofthenormalFLphases: no\nphasetransitionsthenarise,andtheUSC phaseiseliminate d.\nSo how is the USC spin-1 phase destroyed as the strength\nof the AF coupling is progressively increased? This is il-\nlustrated in fig. 7, showing phase boundaries in the (x,y)-\nplane for fixed ˜U=10 and˜U′=5.25 (as in fig. 5) for AF\n˜JH=−0.55,−0.558,−0.56 and−0.6. For˜JH=−0.55, the\nphase boundary has the same form as in fig. 5, consisting of\ntheUSCphasecentredon( x,y)=(0,0),separatedfromtheex-\nteriorFermiliquidphasebyasingleboundarylineofKTtran -\nsitionsexceptontheline y=xwhereafirstorderQPTarises.\nOn decreasing ˜JHslightly to −0.558 however, the USC\nphase is seen to split into four distinct domains – symmetric\nas expected under both inversion, and reflection about y=x\n– with the p-hsymmetricpointin particularnowbeing in the\nFL phase. With a further slight decrease to ˜JH=−0.56, the\ntwo USC domains straddling y=xare now eliminated, leav-\ningtwoUSCregionsstraddlingtheline y=−x. Thisbehavior\npersistsonfurtherdecreasing ˜JH,theremainingUSCdomains\ndiminishinginextentuntilby ˜JH∼−1theytooevaporateand\ntheUSCphaseiseliminatedentirely.\nStrikingly, as indicated in fig. 7, one also sees that as theUSC phase fractionates, first order level-crossing transit ions\narise not only along y=x(as expected on general grounds\nanddiscussedin sec.IIIA);but alsoalongthe line y=−x.\nTo gain some insight into the above, note that the dif-\nference in energy (under ˆHD) between the (1 ,1) singlet and\ntriplet states of the isolated dot is |ES−ET|=|JH|. So for\n|JH|=−JH/lessorsimilarO(Γ), and at least close enough to p-hsym-\nmetry(x,y)=(0,0)(where nimp=2), oneexpectsit necessary\nto include both the (1 ,1) triplet andsinglet states in the low-\nenergydot manifold(higherdot states, such as (2 ,0),lie con-\nsiderably higher in energy provided U′is not within O(JH)\nofU). An effective low-energy model within this subspace\nmay then be constructed via a Schrieffer-Wolff transforma-\ntion38, the appropriatelocal unity operatorbeing ˆ1=ˆ1T+ˆ1S\nwithˆ1T=∑Sz|S=1,Sz/an}bracketri}ht/an}bracketle{tS=1,Sz|andˆ1S=|0,0/an}bracketri}ht/an}bracketle{t0,0|(with\n|S,Sz/an}bracketri}htreferring to the (1 ,1) triplet or singlet dot states). As\ndiscussedinAppendixA, theresultanteffectivemodelis\nˆHeff=J1ˆs1·ˆs0+J2ˆs2·ˆs0−Iˆs1·ˆs2+ˆHL(16)\nwhere as usual ˆs1andˆs2are the spin-1/2 operators for levels\n1and2,and ˆs0=∑σ,σ′f†\n0σσσσ′f0σ′isthespindensityofthe\nconductionchannelat the dot(with f†\n0σ=1√\nN∑kc†\nkσthe cre-\nation operator for the ‘0’-orbital of the Wilson chain30,31and\nNthe numberof k-states in the lead). Only exchangescatter-\ning contributions to eqn. (16) are shown explicitly, potent ial\nscattering being omitted for clarity. The effective exchan ge\ncouplings– viz.theJi>0couplingspin i=1or2tothelead,\nand the direct spin exchange I– are naturally functions of x\nandy; expressionsforthemaregivenin AppendixA.\nEqn. (16) is a two-spin Kondo model of the form studied\nin [20], so its physics is understood19,20. A QPT, occurring\nat a critical Ic, is obviously driven by the direct exchange I:\nfor ferromagnetic ΔI=I−Ic>0, spins 1 and 2 form a spin-\n1 which is underscreened on coupling to the lead, resulting\nin a residual free spin-1\n2; while for AF ΔI=I−Ic<0 by\ncontrast, the local singlet Fermi liquid phase naturally ar ises.\nTheresultantQPTisingeneralofKTform,withonepertinent\nexception20: ifJ2=J1in eqn. (16), then the Hamiltonian is\nseparable into distinct singlet and triplet sectors for the spin\nˆS=ˆs1+ˆs2, specifically\nˆHeff−ˆHL=ˆ1T/parenleftbig\nJ1ˆS·ˆs0−1\n4I/parenrightbigˆ1T+3\n4Iˆ1S:J2=J1(17)\n(on projecting eqn. (16) with ˆ1=ˆ1T+ˆ1S, and using ˆs1·ˆs2≡\n1\n2(ˆS2−3\n2)togetherwith ˆSˆ1S≡0 forall componentsof ˆS).\nThe resultant separability of the Hilbert space for J2=J1\nmeansofcoursethat afirst orderlevel-crossingtransition can\noccur in this case. As shown in Appendix A (eqn. (A14)),\nthis is precisely the situation arising for the present prob lem\nalong(andonlyalong)thelines y=xand y=−x;explaining\ntherebythelevel-crossingtransitionsfoundinfig. 741.\nIV. DYNAMICSAND TRANSPORT\nWe nowconsiderdynamicsandtransport,focussingon the\nT=0 zero-bias conductance and associated phase shift, δ.9\nA ‘Friedel-Luttingersum rule’ for δis derived, applicable to\nboth the FL and USC spin-1 phases (sec. IVB). A gener-\nalization of Luttinger’s integral theorem for the USC phase\nis deduced, and its significant implications for the zero-bi as\nconductancedetermined(sec.IVC).\nA. Single-particlepropagators\nWefirstsummarisebasicresultsforthelocalsingle-partic le\npropagators,embodiedinthe2 ×2matrixG(ω). Itselements\nGij(ω)are the retarded Green functions for the dot levels,\ni,j∈ {1,2}(as in sec. II), related to the corresponding non-\ninteractingpropagators G0(ω)bythe Dysonequation\n[G(ω)]−1= [G0(ω)]−1−Σ(ω) (18)\nwhereΣ(ω)is the 2×2 self-energy matrix (with elements\nΣij(ω)=ΣR\nij(ω)−iΣI\nij(ω)). Using equationof motion meth-\nods1,42theelementsof [G0(ω)]−1aregivenby\n/parenleftBig/bracketleftbig\nG0(ω)/bracketrightbig−1/parenrightBig\nij= (ω+−εi)δij−Γij(ω)(19)\nwhereω+=ω+i0+,andΓij(ω)isthehybridizationfunction\nΓij(ω) =∑\nkViVj\nω+−εk≡ΓR\nij(ω)−iΓI\nij(ω)(20)\nsuch that Γ2\n12(ω) =Γ11(ω)Γ22(ω)(for generality we allow\nhere for arbitrary V2,V1). For the standard1flat-band conduc-\ntion spectrum/lead considered (secs. (II,III)), the imagi nary\npart of the hybridization function ΓI\nij(ω) =Γij(eqns. (5,8))\nfor|ω|<Dand zero otherwise; and the corresponding real\npartΓR\nij(ω=0)=0at theFermilevel( ω=0).\nFrom eqns. (18,19) it follows that G(ω)has precisely the\nsame algebraic structure as G0(ω), but with Γij(ω)replaced\nby˜Γij(ω)definedby:\n˜Γij(ω)=Γij(ω)+Σij(ω) (21)\nUsingeqn.(19) thepropagators Gij(ω)thusfollowas\nG11(ω) =/parenleftbig\nω+−ε2−˜Γ22(ω)/parenrightbig\ndetG(ω)(22a)\nG22(ω) =/parenleftbig\nω+−ε1−˜Γ11(ω)/parenrightbig\ndetG(ω)(22b)\nG12(ω) =˜Γ12(ω)detG(ω) (=G21(ω))(22c)\nwiththe determinantgivenexplicitlyby:\ndetG(ω) =\n/bracketleftbig/parenleftbig\nω+−ε1−˜Γ11(ω)/parenrightbig/parenleftbig\nω+−ε2−˜Γ22(ω)/parenrightbig\n−˜Γ2\n12(ω)/bracketrightbig−1\n(23)\nTheseequationsenablethepropagatorsandtheirspectrald en-\nsitiesDij(ω) =−1\nπImGij(ω)to be determined; with self-\nenergies obtained in practice via a matrix generalization o f\nthestandardNRGmethod34,43,asoutlinedinAppendixBand\ndiscussedfurtherinsec. V.It is also convenient at this point to note an expres-\nsion for the ( T=0) excess impurity charge nimp; de-\nfined as the difference in charge of the entire system\nwith/without the dot, and hence nimp=2(−1)\nπIm/integraltext0\n−∞dω\n[∑kGkk(ω)+G11(ω)+G22(ω)−∑kGo\nkk(ω)]in terms of\nthe level propagators Gii(ω), the propagators Gkk(ω)for\nthe lead k-states, and their counterparts in the absence\nof the dot, Go\nkk(ω) = [ω+−εk]−1. Using equation of\nmotion methods it is simple to show that Gkk(ω) =\nGo\nkk(ω)+Go\nkk(ω)∑i,jViGij(ω)VjGo\nkk(ω),i.e.(via eqn. (20))\n∑k[Gkk(ω)−Go\nkk(ω)] =−∑i,j(∂Γij(ω)/∂ω)Gij(ω), and\nhence:\nnimp=2(−1)\nπIm/integraldisplay0\n−∞dω∑\ni,jGij(ω)/bracketleftbigg\nδij−∂Γij(ω)\n∂ω/bracketrightbigg\n(24)\nFor the commonly considered case1of an infinitely wide flat\nband/lead, where ∂Γij(ω)/∂ω=0 forallω,nimpreduces\ntonimp=2/integraltext0\n−∞dω[D11(ω)+D22(ω)]≡ /an}bracketle{tˆn1+ˆn2/an}bracketri}ht–i.e.to\nthe charge on the dot. For a finite lead bandwidth D(as\nconsidered here) nimpis in practice very close to the dot\ncharge,althoughdoesnotcoincideidenticallywith it.\nWe focus now on the T=0 zero-bias conductance, given\nfromeqn.(4) by\nGc(T=0)\nG0=2e2\nhπ(Γ11+Γ22)Dee(ω=0)(25)\nand determined by the Fermi level value of the ee-spectrum,\n(Γ11+Γ22)Dee(ω) =∑ijΓij(−1)\nπImGij(ω)(eqn. (7)); an ex-\nplicitexpressionforwhichcanbeobtainedusingeqns.(22, 23)\nand theω=0 behavior of the ˜Γij(ω). For both the normal\nFermi liquid phase andthe USC phase, the imaginarypart of\ntheself-energyvanishesat theFermilevel,\nΣI\nij(ω=0) =0. (26)\nFor the normal FL phase this is of course wholly familiar1.\nFor the USC phase, we have established it by direct NRG\ncalculation of the ΣI\nij(ω)(it is also consistent with purely\nelastic scattering at the Fermi level for a singular Fermi li q-\nuid28). Given eqn. (26), ˜Γij(0)follows from eqns. (20,21)\nas˜Γij(0) =−iΓij+ΣR\nij(0). Using this in eqns. (22,23), and\ndefiningrenormalizedsingle-particlelevelsin the usual w ay1\nby\nε∗\ni=εi+ΣR\nii(0), (27)\nasimple iftediouscalculationgives\nπ(Γ11+Γ22)Dee(0) =1\n1+/bracketleftbigg\nε∗\n1ε∗\n2−(ΣR\n12(0))2\nε∗\n1Γ22+ε∗\n2Γ11−2Γ12ΣR\n12(0)/bracketrightbigg2;\norequivalently\nπ(Γ11+Γ22)Dee(0) =sin2δ (28)\nwithδgivenexplicitlyby:\nδ=arctan/bracketleftBigg\nε∗\n1Γ22+ε∗\n2Γ11−2Γ12ΣR\n12(0)\nε∗\n1ε∗\n2−/parenleftbig\nΣR\n12(0)/parenrightbig2/bracketrightBigg\n.(29)10\nB. Friedel-Luttingersum rule\nThe quantity δappearing in eqns. (28,29) is simply the\nstatic scatteringphaseshift,givenalternativelyby\nδ=Imln(detG(0))≡Imln(detG(ω))/vextendsingle/vextendsingleω=0\nω=−∞(30)\n(the equivalence of eqns. (29,30) is readily shown using eqn .\n(23));therighthandsideofeqn.(30)alsousesarg [detG(ω=\n−∞)]=0,asfollowsfromeqn.(23)togetherwiththefactthat\nas|ω|→∞,ΣI(ω)vanisheswhilethe ΣR\nij(ω)tendtoconstants\n(theHartreecontributions).\nWe now point out a general result for the phase shift δ.\nFromeqns.(23,22),it followsthat\n∂\n∂ωln(detG(ω)) =−∑\ni,j/parenleftbigg\nδij−∂\n∂ω˜Γij(ω)/parenrightbigg\nGji(ω).\nIntegratingthisfrom ω=−∞to0(andnotingeqn.(21))then\ngivesdirectlyfromeqn.(30) that\nδ=π\n2nimp+IL (31)\nwherenimpisgivenbyeqn.(24),and(withTrdenotingatrace)\nIL=ImTr/integraldisplay0\n−∞dω∂Σ(ω)\n∂ωG(ω) (32)\nistheLuttingerintegral36,44(whichisdimensionless).\nWe emphasise that eqn. (31) is entirely general: applica-\nble to both the normal Fermi liquid and the USC phases (in-\ndeed its derivationdoesnot even requirea knowledgeof eqn.\n(26)). FortheparticularcaseoftheFLphase,Luttinger’st he-\noremgives IL=036,44;ILvanishingorderbyorderinperturba-\ntiontheoryaboutthenon-interactinglimit,reflectingadi abatic\ncontinuity to the non-interacting limit. In this case eqn. ( 31)\nreduces to the Friedel sum rule1,35,δ=π\n2nimp, relating the\nscattering phase shift to the excess (‘displaced’) charged in-\nducedon additionofthe dot/impurityto thesystem (andwith\nδ∈[0,2π]for a 2-level dot, since nimp∈[0,4]). More gener-\nally, however, δandnimpare related by eqn. (31), which we\nreferto asaFriedel-Luttingersumrule.\nThe Luttingerintegral for the normal Fermi liquid phase is\nan intrinsic characteristic of it; IL=0 holding independently\nof the underlying bare model parameters, provided only the\nsystemisaFL36,44. Assuch,theLuttingerintegralisthehall-\nmarkofthephasein aratherdeepsense.\nThe USC spin-1 phase by contrast is a singular Fermi liq-\nuid28. There is no reason here to suppose IL=0; and indeed\nit can be shown that the USC phaseis not perturbativelycon-\nnectedtothenon-interactinglimitofthemodel. Buttheobv i-\nousquestionarises: asfortheFL,doesananalogoussituati on\narise for the USC phase whereby the Luttinger integral has a\ncharacteristicvalueforthat phase?\nWe answer this question affirmatively, by direct numeri-\ncal calculation (and in several distinct ways). Since the se lf-\nenergiesΣ(ω)andGreenfunctions G(ω)arecalculablefrom\nNRG,wecancalculate ILdirectly(eqn.(32))asan ω-integral.Alternatively, nimpmaybeobtainedfromthermodynamiccal-\nculation (as in sec. III) and δfrom calculation of the ee-\nspectrum at the Fermi level alone (as in eqn. (28), or alter-\nnatively eqn. (29)); their difference then giving the Lutti nger\nintegral,IL=δ−π\n2nimpfrom eqn. (31). We have confirmed\nthat the same answer emerges in either way (and for the FL\nphase that IL=0 thereby results). Namely, for any region\nof the (x,y)-plane where the system is in the USC phase, the\nmagnitudeoftheLuttingerintegralisa constant,specifica lly:\n|IL|=π\n2:USC (33)\nWehaverepeatedthecalculationsformanydifferentvalues of\nthebareinteractionparameters ˜U,˜U′and˜JH. Thesameresult\nemerges;andwhile the numericsobviouslycannotamountto\naproof,weareconfidentin thevalidityofeqn.(33).\nAlthough the magnitude of ILis constant throughout the\nUSC phase, its sign is not. This is a natural consequence of\nsymmetry. By considering the symmetries of the propaga-\ntorsG(ω)≡G(ω;x,y)and self-energies Σ(ω;x,y)under a\nparticle-hole transformation (sec. IIA), it can be shown th at\ntheLuttingerintegral IL≡IL(x,y)isoddunderinversion,\nIL(x,y) =−IL(−x,−y). (34)\nInaddition,asappropriatetothecase V2=V1,thesymmetries\nofG,Σunder the 1-2 transformation (eqn. (13)) lead to the\nratherobviousinvarianceunderreflectionabouttheline y=x,\nIL(x,y) =IL(y,x). (35)\nWith|IL|=π\n2, eqn. (34) implies the existence of at least one\nbounding curve, of form yb=f(x)withf(x) =−f(−x),\nacrosswhich ILchangessignfrom +π\n2to−π\n2;whileeqn.(35)\nforthecase V2=V1impliesthatboundingcurvetobetheline\ny=−x. In practice(by directcalculation, as above)onlyone\nboundingline is found; and for the case V2=V1in particular\nwefind:\nIL(x,y) = +π\n2:y>−x (36a)\n=−π\n2:y<−x (36b)\nC. Zero-bias conductance\nAsabove, |IL|=π\n2isubiquitousthroughouttheUSCphase,\nas|IL|=0 is throughout the normal FL phase. This has im-\nmediateconsequencesforthebehaviorofthe T=0zero-bias\nconductance,givenfromeqns.(25,28,31) by\nGc(T=0)\nG0=2e2\nhsin2/parenleftBigπ\n2nimp+IL/parenrightBig\n.(37)\nSincesin2(π\n2[nimp±1])=cos2(π\n2nimp), it followsthat\nGc(T=0)\n2e2\nhG0=sin2/parenleftBigπ\n2nimp/parenrightBig\n: FL (38a)\n=cos2/parenleftBigπ\n2nimp/parenrightBig\n: USC (38b)11\nfor the FL and USC phases respectively. But as foundin sec.\nIII (seee.g.fig. 3),nimpvaries continuously on crossing the\nlineofKosterlitz-ThoulesstransitionsfromtheFLtotheU SC\nphase. Hencefromeqn.(38)it followsthatthezero-biascon -\nductancemustjumpdiscontinuouslyoncrossingtheQPT;the\ndiscontinuity on crossing from the FL to the USC phase be-\ning cos(πnimp), with its sign determined by the value of nimp\nat the QPT. From direct calculation of single-particle spec tra\nwe will verify explicitly in sec. V (see e.g.fig. 11) that eqns.\n(38a,38b) are satsified throughout the two phases. Eqn. (38)\nisofcourseequallyapplicabletothefirstorderlevel-cros sing\ntransitions(sec.IIIA),althoughinthiscase nimpitselfchanges\ndiscontinuouslyasthetransitioniscrossed(fig.6topinse t).\nV. SINGLE-PARTICLEDYNAMICS\nWeturnnowto ω-dependentsingle-particledynamics,here\nfocussing primarily on the Dee(ω)spectrum at T=0. The\nself-energies Σ(ω)are obtained from NRG via a generaliza-\ntion of the basic method34,43to the case of multilevelimpuri-\nties/dots, as outlined in Appendix B. Σis thereby calculated\nfrom\nΣ(ω) =F(ω) [G(ω)]−1(39)\nwherethe2 ×2matrixF(ω)haselements\nFij(ω) =/an}bracketle{t/an}bracketle{t[diσ,ˆHI];d†\njσ/an}bracketri}ht/an}bracketri}ht (40)\n(using conventional notation1,42for the ω-dependent\nFourier transform of a generic retarded correlation functi on\n/an}bracketle{t/an}bracketle{tˆA(t1);ˆB(t2)/an}bracketri}ht/an}bracketri}ht=−iθ(t1−t2)/an}bracketle{t{ˆA(t1),ˆB(t2)}/an}bracketri}ht); and where\nˆHIdenotes the interacting part of the dot Hamiltonian,\ngiven explicitly for the present problem by (eqn. (1))\nˆHI=U∑iˆni↑ˆni↓+U′ˆn1ˆn2−JHˆs1·ˆs2. Usingtheself-energies,\nthe fully interacting propagators are then obtained from th e\nDyson equation eqn. (18). As for single-level problems,\ncalculation of G(ω)in this way is numerically stable and\naccurate, and guarantees satisfaction of spectral sum rule s,/integraltext∞\n−∞dωDii(ω) =134,43(interleaved NRG/‘z-averaging’45\nalso beingusedforoptimalcalculationalaccuracy).\nAs in sec. III we consider explicitly V2=V1; for which\nΓij≡Γ(eqn. (9)), with the ee-spectrum thus given (eqn. (7))\nby:\nDee(ω) =1\n2[D11(ω)+D22(ω)+2D12(ω)](41)\nFig.8showsan‘allscales’overviewof2 πΓDee(ω)vs˜ω≡\nω/Γ,forfixed ˜U=20,˜U′=7and˜JH=2(asinfig.3forther-\nmodynamics),takingaverticalcutthroughthe (x,y)-phasedi-\nagram: theenergyoflevel-1isfixedat ˜ε1=−1\n2˜U−˜U′≡−17\n(i.e. x=0), and˜ε2is progressively decreased through the\nFermiliquidphasetowardsthetransition( ˜ε2c=−6.536...).\nThe most important spectral feature is of course the clear\nKondo resonance straddling the Fermi level. We consider it\nbelow,butfirst commentonthequalitativeoriginofthehigh -\nenergy spectral features, evident most clearly in the three ar-\nrowed peaks shown in fig. 8 for ˜ε2=+1. The corresponding0.000.250.500.751.00\n-25-20-15-10-5 0 5 10 15 202πΓDee(ω)\nω/Γ0.00.51.0\n-20 0 20\nFIG. 8: Single-particle spectrum 2 πΓDee(ω)vsω/Γin the Fermi\nliquid phase, for fixed level energy ˜ε1=−1\n2˜U−˜U′(i.e. x=0) on\nprogressively decreasing ˜ε2towards the QPT (occurring at ˜ε2c=\n−6.536..). For˜U=20,˜U′=7 (i.e.˜ε1=−17) and ˜JH=2, as in\nfig. 3; with ˜ε2=+1 (solid line), −3 (long dash), −6 (short dash).\nNote that all three cases contain a narrow Kondo resonance st rad-\ndling the Fermi level ω=0. For vertical arrows, see text. Inset:\nspectrumfor ˜ε2=−6.6onjustenteringtheUSCphase, showingthe\nabsence of a Kondo resonance.\nevolution of nimpvs˜ε2is shown in fig. 3 (inset), from which\nit is seen than nimp≃1.2 for˜ε2= +1 – sufficiently close to\nunity that we can interpret the high energy spectral feature s\nas removal or addition excitations from the singly-occupie d\n(n1,n2)=(1,0) state of the isolated dot. The removal exci-\ntation from dot level-1, contributing as such to the D11(ω)\nconstituentof Dee(ω)(eqn.(41)), thuscorrepondstrivially to\nED(1,0)−ED(0,0)=ε1(in the notationof sec. IIB), i.e.lies\nbelowtheFermi levelat ( ω/Γ=)˜ω=˜ε1≡−17here,gener-\natingthelower‘Hubbardsatellite’seenclearlyinfig.8;it spo-\nsition,dependentat thiscrudelevelof descriptiononlyon ˜ε1,\nvariesonlyslightlyonfurtherdecreasing ˜ε2intheFLregime.\nTwo addition excitations lying above the Fermi level are\nalso seen in fig. 8 for ˜ε2= +1 (arrowed). The lowest cor-\nresponds to electron addition to level-1, and hence shows up\n(again via D11(ω)) as an excitation at ED(2,0)−ED(1,0) =\nε1+U; thus lying at ˜ω=˜ω+=˜ε1+˜U= +3 as seen in the\nfigure. Thesecondexcitationcorrespondstoadditiontolev el-\n2, contributes as such to the D22(ω)constituent of Dee(ω),\nand thus corresponds to ED(1,1)−ED(1,0). Since there are\ntwo distinct (1 ,1) dot states – triplet and singlet – two such\nexcitations in principle arise; separated in energy by ˜JHand\noccurring at ˜ω=˜ωT=˜ε2+˜U′−1\n4˜JH(for triplet (1 ,1)) and\n˜ω=˜ωS=˜ε2+˜U′+3\n4˜JHfor the singlet. As evident from the\nfigure,couplingtotheleadsinpracticeblurstheseexcitat ions,\nsothat onlya singlespectralfeatureisseen.\nOn decreasing ˜ε2from+1 the˜ωS/Texcitations (which as\nabovedependon ˜ε2)decrease,andbecomecomparableinen-\nergy to the ˜ω+(=+3) excitation; so that as seen in fig. 8 the\nhigh-energy addition excitations in practice merge to a for m\na single peak, which on decreasing ˜ε2through the FL phase\nmoves towards – but does not reach – the Fermi level. In-\nstead, the single-particle spectrum in the immediate vicin ity\noftheFermilevel ω=0isnaturallydominatedbythenarrow12\n0.250.500.751.001.25\n-2.0-1.5-1.0-0.50.00.51.01.52.02.52πΓDee(ω)\nω / 10-4 Γ0.00.51.0\n-50-25 0 25 50\nω/TK\nFIG. 9: For same parameters as fig. 8, showing a close-up of the\nKondo resonance on approaching the transition (at ˜ε2c=−6.536..)\nfrom the FLside: 2 πΓDee(ω)vsω/Γfor˜ε2=−6.1(solidline)and\n−6.2(longdash). TheKondoresonancecollapses‘onthespot’as the\nQPT is approached and the Kondo scale TK→0. The short dashed\nline shows the spectrum for ˜ε2=−6.54 on just entering the USC\nphase; it is featureless on these scales, with no Kondo reson ance.\nInset: ScalingoftheKondoresonance onapproachingtheQPTfrom\ntheFLside. BothFLspectrainthemainfigurecollapsetoauni versal\nscaling resonance as a function of ω/TK(their individual TKs differ\nbymore thananorder ofmagnitude).\nlow-energyKondoresonanceevidentinfig.8.\nThe evolution of the Kondo resonance itself is shown in\nclose-upin fig. 9. As ˜ε2→˜ε2c+from the FL side, it narrows\nprogressively–reflectingtheincipientvanishingoftheKo ndo\nscaleTKknown from thermodynamics(sec. III, figs. (3,4)) –\nand collapses ‘on the spot’ at the transition itself, where TK\nvanishes. As a corollary, in the USC phase just on the other\nside of the transition the Kondo resonance is simply absent;\nas seen in fig. 9 (for ˜ε2=−6.54) where the USC spectrum\nis constant on the low ˜ω=ω/Γscales shown. The inset to\nfig. 8 also shows this USC spectrum on an ‘all scales’ level,\nshowing that while the Kondo resonance is absent here, the\nhigh-energyfeaturesdiscussedaboveevolveina smoothway\nfromthosearisingintheFL phase.\nSince the Kondo scale TKvanishes as the QPT is ap-\nproachedfrom the FL side, one expectsthe Kondoresonance\ntoexhibituniversalscalingintermsofit. Thatthisissois seen\ninfig. 9(inset),wherebothFL spectrashownin themainfig-\nurecollapsetoauniversalscalingformasafunctionof ω/TK.\nNote also that while we have scaled the spectra here in terms\nofTKobtainedfrom Simp(T)(asin sec. III),we couldequally\nhave defined TKspectrally – e.g.via the width of the Kondo\nresonance– andlikewise obtaineduniversalscalingbehavi or.\nThe essential point is simply that there is only one vanish-\ning low-energyscale as the QPT is approached,and different\npracticaldefinitionsofit areall fundamentallyequivalen t.\nThesubsequentevolutionofthespectrumintheUSCphase\nisshowninfig.10. NotfarintotheUSCphase( ˜ε2=−7),the\nspectrumlacksaKondoresonance,asabove. Howeveronfur-\nther decreasing ˜ε2, a second Kondo resonance straddling the\nFermi level is seen to arise. It is in fact well developed al-\nreadyby ˜ε2=−10,andnarrowsprogressivelyas ˜ε2decreases0.000.250.500.751.00\n-20-15-10-5 0 5 10 15 202πΓDee(ω)\nω/Γ\nFIG.10: USCphase. Forsameparametersasfigs.(8,9),2 πΓDee(ω)\nvsω/Γfor˜ε2=−7 (short dash line), −10 (dashed) and the center\nof the USC phase ˜ε2=−17 (p-hsymmetric point, solid line). The\nKondo resonance which develops here is that for a spin-1 Kond o\nmodel21;see alsosec. VA (fig.13).\ntowards the center of the USC phase at the p-h-symmetric\npoint˜ε2=−17. Theoriginofthisbehaviorisreadilyguessed\nfromnimpvs˜ε2(fig. 3 inset). For although the transition it-\nself corresponds to a ‘mixed valent’ nimp≈1.4, on entering\nthe USC phase nimpincreases quite rapidly; such that even\nby˜ε2=−10,nimpis close to 2. Here one expects the sys-\ntematlowenergiestobedescribedasymptoticallybyaspin- 1\nKondomodel,andhencethesecondKondoresonancetobeof\nthat ilk. Thisis indeed so; we discussit furtherin the conte xt\nof fig. 13 below. High-energyspectral featuresin this regim e\nare also naturallyinterpretablein terms of single-electr onex-\ncitations to/from the (1 ,1) triplet ground state of the isolated\ndot;e.g.at thep-hsymmetric point, all addition/removalex-\ncitations to/from both levels 1 and 2 have the same magni-\ntude,|˜ε1+˜U′−1\n4˜JH|, giving rise to the symmetrically dis-\nposedHubbardsatellites at |˜ω|≃10.5seenin fig.10.\nFinally and importantly, fig. 11 verifies the predictions of\nsec. IVC for the behavior of the zero-bias conductance on\ncrossingthe QPT. The Fermi levelspectrum2 πΓDee(0)vs˜ε2\nis shownin bothphases, andcomparedexplicitlyto eqn.(38)\nwithnimpobtainedfromanindependentthermodynamicNRG\ncalculation;theagreementbeingexcellent.\nA. Kondoantiresonances\nIntheexampleconsideredabovetheQPTisassociatedwith\nacollapsingKondo resonance intheFLphase;andhencenat-\nurally with a decrease in the zero-biasconductanceon cross-\ning into the USC phase. From eqn. (38), the latter behav-\nior is generic provided nimpat the transition lies in the in-\ntervalnimp∈[1\n2,3\n2](by symmetry eqn. (14) we can consider\nnimp∈[0,2]ratherthanthefullrange [0,4]). Ifhowever nimpat\ntheQPTliesintherange [3\n2,2]or[0,1\n2],theneqn.(38)predicts\ngenerically an increasein the conductance on crossing from\nthe FL to the USC phase. One might thus intuitively expect\nsuch behavior to be associated with a vanishing Kondo an-13\n0.00.20.40.60.81.0\n-15 -10 -5  0  5 102πΓDee(0)\nε~2USC FL\nFIG. 11: 2 πΓDee(ω=0)(equivalently the zero-bias conductance,\nseeeqns.(25,38)) vs˜ε2inbothphasesoneithersideoftheKostelitz-\nThoulesstransition(dashedverticallineat ˜ε2c=−6.536..,samebare\nparameters as figs. (8 - 10)). Crosses show the ω=0 spectra deter-\nmined from NRG,while solid lines show sin2(π\n2nimp)(in FLphase)\nand cos2(π\n2nimp)(in USCphase) with nimpobtained from a thermo-\ndynamic NRG calculation; verifyingthe predictions of eqn. (38). At\nthe QPT, nimp=1.43, whence the spectrum/conductance decreases\ndiscontinuously on crossing from the FL to the USC phase, nimpit-\nself evolvingcontinuously (fig.3inset).\n0.000.250.500.751.001.251.50\n-2.0-1.5-1.0-0.50.00.51.01.52.02πΓDee(ω)\nω/Γ0.00.51.0\n-20 0 200.00.51.0\n-50 0 50\nω/TK\nFIG. 12: 2 πΓDee(ω)vsω/Γalong the line y=−x(for same in-\nteraction parameters as figs. (8 - 11)), decreasing ˜ y=˜ε2+1\n2˜U+˜U′\nthrough theQPT(at ˜ yc=6.36..)fromtheFLside y>˜yc. Shownfor\n˜y=6.55 (short dash) and 6 .40 (long dash) in the FL phase, and ˜ y=\n6.30 (solid) just into the USC phase. A clear Kondo antiresonan ce\nat the Fermi level in the FL phase is seen (with 2 πΓDee(0) =0). It\ncollapses ‘on the spot’ as TK→0 and the QPT is approached; and\nexhibitsscalingasafunctionof ω/TKasitdoesso( rightinset ).Left\ninset: as mainfigure, on anexpanded ω/Γscale.\ntiresonance asthetransitionisapproachedfromtheFL side.\nThat this indeed arises19is illustrated in fig. 12, where dy-\nnamics on the line y=−xare considered; along which, by\nsymmetry(eqn.(14)), nimp=2 regardlessof phase(the spec-\ntraarelikewisereadilyshowntobesymmetricin ω). Forbare\ninteraction parameters ˜U=20,˜U′=7 and˜JH=2, we de-\ncrease(y/Γ≡) ˜y=˜ε2+1\n2˜U+˜U′(eqn.(11))acrossthetransi-\ntionoccuringatthe critical ˜ yc=6.36..,fromtheFL side( y>\n˜yc) to the USC phase. As shown in the main figure Dee(ω)indeed contains a Kondo antiresonance in the FL phase, here\nwith2πΓDee(ω=0)=0throughout. Thisantiresonancelike-\nwise vanishes on the spot as the transition is approached and\nthe Kondo scale TK→0; and as it does so exhibits scaling\nas a function of ω/TK(fig. 12 right inset), the low-frequency\nspectralbehaviorbeing2 πΓDee(ω)∝(ω/TK)2,symptomatic\nofanormalFermiliquid.\nNotethatthegeneralpredictionsofsec.IVareneatlyexem-\nplified by the above results: since nimp=2 everywherealong\nthey=−xline,eqns.(25,38)yield2 πΓDee(0)=1intheUSC\nphase and 0 in the FL phase (as confirmed by direct calcula-\ntion, figs. (12,13)); and hence that the zero-bias conductan ce\nGc(T=0)/G0inthiscase increasesbypreciselythe conduc-\ntancequantum2 e2/honcrossingtheQPTintotheUSCphase.\nFig. 13 continues fig. 12 into the USC phase, showing the\nee-spectra for ˜ y=5,3 and 0. As for its counterpart in fig.\n10, the Kondo resonance which develops in the USC phase\nis that for a spin-1 Kondo model21. As shown in Appendix\nA, its low-energy scale TS=1\nKvaries with the bare interaction\nparametersas(moduloanimmaterialprefactor)\nTS=1\nK∝exp/parenleftBigg\n−π\n8/bracketleftBigg\n(˜U+1\n2˜JH)2−˜y2\n˜U+1\n2˜JH/bracketrightBigg/parenrightBigg\n.(42)\nHence on decreasing ˜ ythrough the USC phase, the Kondo\nresonancebecomesincreasinglynarrowas TS=1\nKdecreasesto-\nwards its smallest (but non-zero) value occurring at the p-h\nsymmetric point ˜ y=0 (=˜x); and as shown in fig. 13, univer-\nsal spectralscalingasafunctionof ω/TS=1\nKtherebyarises.\nFig.13(inset)alsoshowstheclearcusp-likebehaviorofth e\nspin-1Kondoresonanceas |ω|→0,knownfromstudyofthe\nspin-1Kondomodelitself21(withspectrainferredfromthe t-\nmatrixoftheKondomodel). Thisbehaviorischaracteristic of\nthe singular Fermi liquid28nature of the underscreenedspin-\n1 phase; specifically the weak ferromagnetic coupling of the\nresidual spin-1\n2to the metallic lead, resulting in logarithmic\ncorrections to Fermi liquid behavior. As |ω|/TS=1\nK→0 we\nfind\n2πΓDee(ω=0)∼1−b\nln2(|ω|/TS=1\nK)(43)\n(withba constant), the leading logarithmic correction here\nstemming from the leading low- ωbehavior of the self-\nenergiesΣI\nij(ω)∼1/ln2(|ω|/TS=1\nK); and which form is in\nagreementwiththatof[21]forthespin-1Kondomodelitself .\nB. The y =xline\nAs considered in sec. IIIA in regard to thermodynamics,\nthe transition occurringalong the line ε2=ε1(i.e. y=x) is a\nfirstorderlevel-crossingtransition,aspermittedbysymm etry\nforV2=V1. Hereweconsiderthe ε2=ε1lineagain,fromthe\nperspective of dynamics, and the resultant channel separab il-\nityarisinginthe‘even/odd’representationasnowdiscuss ed.14\n0.000.250.500.751.00\n-20-15-10-5 0 5 10 15 202πΓDee(ω)\nω/Γ0.00.51.0\n-100 0 100\nω/TS=1\nK\nFIG. 13: Continuing fig. 12 into the USC phase along the line y=\n−x: 2πΓDee(ω)vsω/Γfor ˜y=5(shortdash),3(dash)andthe p-h\nsymmetricpoint ˜ y=0=˜x(solid). TheKondo resonance developing\nintheUSCphaseisthatforthespin-1Kondomodel21;and(inset)the\nthreespectrainthemainfigureshowlow-energyuniversalsc alingas\na function of ω/TS=1\nKwithTS=1\nKthe spin-1 Kondo scale (see text).\n1. Even/odd basis\nInprevioussectionstheelementsoftheGreenfunctionma-\ntrixG(ω)havebeenconsideredasthepropagatorsforthedot\nlevels,viz. Gij(ω)withi,j∈ {1,2}. Equally, one can take\neven/odd combinations of the dot levels, viz. deσ= (d1σ+\nd2σ)/√\n2 anddoσ=(d1σ−d2σ)/√\n2, and consider G(ω)in\nane/orepresentation;withelements Gαβ(ω)givenexplicitly\nbyGeeoo(ω)=1\n2[G11(ω)+G22(ω)±2G12(ω)],withGeo(ω)=\n1\n2[G11(ω)−G22(ω)](=Goe(ω)) for the off-diagonal ele-\nments. For ε2/ne}ationslash=ε1in general, there is no particular advan-\ntage in working with the e/orepresentation. However along\nthelineε2=ε1wherelevels1and2areequivalentbysymme-\ntry,G11(ω)=G22(ω)andhencetheoff-diagonal Geo(ω)=0.\nG(ω)inthee/orepresentationisthenpurelydiagonalfor all\nω,with elements:\nGeeoo(ω) =G11(ω)±G12(ω) (44)\nUsingeqns.(22,23,21),oneobtains\nGee(ω) =/bracketleftbig\nω+−ε−2Γ(ω)−Σee(ω)/bracketrightbig−1(45a)\nGoo(ω) =/bracketleftbig\nω+−ε−Σoo(ω)/bracketrightbig−1(45b)\nwhereε≡ε1=ε2denotes the common level energy, the hy-\nbridization function is Γ(ω)(≡Γij(ω), eqn. (20) with V2=\nV1),andthe ee/ooself-energiesaregivensimplyby:\nΣeeoo(ω) =Σ11(ω)±Σ12(ω) (46)\nNotice from eqn. (45b) that there is no direct hybridization\n(Γ(ω)) contribution to Goo(ω), reflecting the fact (sec. IIIA)\nthat forε2=ε1theo-orbital is not directly coupled to the\nlead. In the non-interacting limit the o-level is thus entirely\nfree,G0\noo(ω) =[ω+−ε]−1; but in general the o/elevels are\ncoupledviainteractions,asembodiedin Σoo(ω)/ne}ationslash=0.SinceG(ω)is diagonal in the e/orepresentation,\ndetG(ω) =Gee(ω)Goo(ω), and hence from eqn. (30) the\nstatic phaseshift δisseparableinto eandochannels,\nδ=δe+δo. (47)\nA short calculation using eqn. (30) (together with ΣI\nαα(ω=\n0)=0fromeqn.(26))thengives\nδe=arctan/parenleftbigg2Γ\nε∗e/parenrightbigg\n(48a)\nδo=arctan/parenleftbigg0+\nε∗o/parenrightbigg\n≡π θ(−ε∗\no)(48b)\nwhere each δα∈[0,π],Γ(=ΓI(ω=0)) is the usual hy-\nbridizationstrength (eqn. (9)), and θ(u)is the unit step func-\ntion. The ε∗\nαdenote the renormalized e/olevels, given by ( cf\neqn.(27))\nε∗\nα=ε+ΣR\nαα(0) (49)\nwithε∗\nα≡ε∗\nα(x)suchthat ε∗\nα(x)=−ε∗\nα(−x)(via ap-htrans-\nformation,sec.IIA). Likewise,considering ∂lnGαα(ω)/∂ω\nandrepeatingthecalculationleadingtoeqn.(31),gives\nδα=π\n2nimp,α+Iα\nL (50)\nwhere(cfeqn.(32))\nIα\nL=Im/integraldisplay0\n−∞dω∂Σαα(ω)\n∂ωGαα(ω)(51)\nisaLuttingerintegralforchannel α=eoro(withIα\nL≡Iα\nL(x)\nsuchthat Iα\nL(x)=−Iα\nL(−x)underinversion);and nimp,αisthe\nexcessimpuritychargeassociatedwith channel α,givenby\nnimp,e=2(−1)\nπIm/integraldisplay0\n−∞dωGee(ω)/bracketleftbigg\n1−2∂Γ(ω)\n∂ω/bracketrightbigg\n(52a)\nnimp,o=2(−1)\nπIm/integraldisplay0\n−∞dωGoo(ω) (52b)\nsuchthattheoverall nimp=nimp,e+nimp,o(eqn.(24)),andwith\nnimp,α(x) =2−nimp,α(−x). Since the behavior of relevant\nquantities under inversion x→ −xis as specified above, we\ncanfocuson x=ε+1\n2U+U′≥0;anddosointhefollowing.\n2. Results\nThe charges nimp,αmay be calculated directly from NRG.\nTheir evolution with ˜ x=x/Γis illustrated in fig. 14 (inset),\nondecreasing ˜ε=ε/Γwith˜U,˜U′,˜JHfixedatthesamevalues\nused in fig. 6 for thermodynamics; the level-crossing transi -\ntion here occurring at ˜ xc=12.26..(˜εc=−4.73..). On cross-\ningthetransitionfromthe FLside (˜ x>˜xc) totheUSCphase,\nnimp,oincreasesdiscontinuouslyfrom0–foundtobeitsvalue\nforall˜x>˜xc– tonimp,o=1, which constant value is like-\nwise found throughout the USC phase ˜ x<˜xc. For the e-\nchannelbycontrast nimp,eisnotfixedineitherphase;butittoo15\n-40-20 0 20 40 60 80 100\n-5 0 5 10 15 20 25 30εo*\nx~Γ0.00.51.0\n 0 5 10 15 20 25\nx~USCFL\nFIG.14: Renormalized odd level ε∗o/Γ(eqns. (49,54)) vs˜x=x/Γ=\n˜ε+1\n2˜U+˜U′(for fixed ˜U=20,˜U′=7 and˜JH=2). The level-\ncrossing transition at ˜ xc=12.26..is marked by an arrow; ε∗o∼\nx−xc≡ε−εcvanishes linearly as the transition is approached.\nInset: Behavior of nimp,o(solid line) and nimp,e(dashed) on cross-\ning from the FL phase (˜ x>˜xc) to the USC phase. Both jump dis-\ncontinuously on crossing the transition FL →USC,nimp,oupwards\nfrom 0 to 1 and nimp,edownwards as indicated; such that the overall\nnimp=nimp,o+nimp,eincreases as shown infig.6(topinset).\njumpsdiscontinuously, decreasing astheFL→USCtransition\nis crossed. The latter behavior is physically natural, sinc e\nthe piling of charge into the o-orbital which accompaniesthe\ntransition increases Coulomb repulsionswith electronsin the\ne-orbital, which the concomitant reduction in nimp,eacts to\noffset. The two effects do not however cancel, the overall\nnimp=nimp,o+nimp,e(showninfig.6,topinset)increasingas\nthe transition is crossed. The behavior just described is re do-\nlentof,butdistinctfrom,thatoccurringinthenon-intera cting\nlimit discussed in sec. IIIA; where at the transition, in tha t\ncaseoccuringfor ˜εc=0,nimp,ojumpsdiscontinuouslyfrom0\nto 2 but with no concomitant change in nimp,esince there are\nnointeractionspresent. Moreoversincethetransitionisg ener-\nically accompanied by occupancy of the o-orbital, one intu-\nitively expects the requisite critical ˜εcfor the transition with\ninteractions present to be reduced below its non-interacti ng\ncounterpart ˜εc=0, in order to offset the increased interac-\ntions;asindeedisfound. We alsoaddthatthebehaviorfound\nis not specfic to the interaction parameters used for illustr a-\ntion;inparticularthat\nnimp,o=0 : FL ,˜x>˜xc (53a)\n=1 : USC ,˜x<˜xc (53b)\nisfoundtooccurubiquitously.\nWe consider now the static renormalizedlevels, calculable\nfromeqn.(49); or,for ε∗\no,equivalentlyfrom\nε∗\no=ε\n1+FRoo(ω=0)(54)\nwhereFoo(ω)=F11(ω)−F12(ω)(withFR\noo(ω)its real part),\nand theFij(ω)are given by eqn. (40) and calculated directly\nvia NRG [eqn. (54) follows from eqn. (49) together witheqn. (B5) in the diagonal e/orepresentation]. The generic\n˜x-dependenceof ˜ε∗\no=ε∗\no/Γis illustrated in fig. 14. It evolves\ncontinuouslyforall ˜ x>0 (thedivergenceonapproachingthe\np-hsymmetric point ˜ x=0 at the center of the USC phase re-\nflects via eqn. (54) the fact that FR\noo(0)→ −1∓asx→0±).\nIn particular, in the USC phase 0 <˜x<˜xc, the renormalized\nlevelε∗\no<0,whilefor ˜ x>˜xcintheFLphase, ε∗\no>0;thelevel\nvanishinglinearlyastheQPTis crossed,\nε∗\nox→xc∼x−xc≡ε−εc. (55)\nAndforlargeenough ˜ε≫1,whereboththe eandolevelsare\nin practice empty and interaction effects embodiedin Σooare\nthusirrelevant, ε∗\no→ε(the‘bare’levelenergy,seeeqn.(49)).\nTheaboveresultsthenenablethe o-channelLuttingerinte-\ngralIo\nL(eqn. (51)) to be deduced. Since ε∗\no<0 [>0] in the\nUSC [FL] phase, eqn. (48b) gives a phase shift δo=πin the\nUSC phase 0 <˜x<˜xc, andδo=0 in the FL phase ˜ x>˜xc.\nCombining this with eqn. (53) for nimp,o, the Luttinger inte-\ngralIo\nL=δo−π\n2nimp,o(eqn.(50))followsdirectlyas\nIo\nL=0 : FL ,˜x>˜xc (56a)\n=π\n2: USC,0<˜x<˜xc (56b)\n– which result we havealso verified by direct computationof\nIo\nLitself, eqn.(51).\nFor thee-channelby contrast, directcalculationof Ie\nLgives\nIe\nL=0 inboththeFL phase andthe USCphase,\nIe\nL=0 : FL andUSC . (57)\nThe total Luttinger integral IL=Io\nL+Ie\nLthus vanishes as re-\nquired36,44throughouttheFLphase;whilefortheUSCphase\neqns.(56b,57) agree asthey must with the generalresult eqn .\n(36a)forIL(x,y)(whichisnotconfinedtothe y=xline). Note\nfurther,usingeqn.(57),thateqns.(48a,50)give\nε∗\ne=2Γtan/parenleftbigπ\n2/bracketleftbig\n1−nimp,e/bracketrightbig/parenrightbig\n: FL andUSC (58)\nindependentlyof the phase (as again verified by separate cal -\nculation of ε∗\neandnimp,e). From the ˜ x-dependence of nimp,e\nillustrated in fig. 14 (inset), eqn. (58) shows that ε∗\neprogres-\nsively decreases as ˜ xis decreased through the FL phase, in-\ncreasesdiscontinuouslyastheFL →USCtransitioniscrossed,\nand in the USC phase decreases monotonically as ˜ xis de-\ncreasedtowardsthe p-hsymmetricpoint ˜ x=0,wherenimp,e=\n1 and hence ε∗\ne=0 (and with ε∗\ne(x)for ˜x<0 following from\nthesymmetry ε∗\ne(x)=−ε∗\ne(−x)).\nAsabriefillustrationofsingle-particledynamicsalongt he\ny=xline, fig. 15 shows the evolution of the o-orbital spec-\ntrumDoo(ω)=−1\nπImGoo(ω)ondecreasing ˜ xthroughtheFL\nphase (main panel), across the transition and into the USC\nphase (right inset). In the vicinity of the QPT coming from\nthe FL side, a strong low-frequency spectral resonance (for\nω>0) isseen to develop;becominga poleat the Fermilevel\nprecisely at the transition, and crossing smoothly to ω<0\nin the USC phase. The position of the resonance tracks the\nvanishing renormalized level ε∗\no(eqn. (55)), the ω=0 pole\nat the transition reflecting ε∗\no=0 (from eqn. (45b), using16\n 0 1 2 3 4 5\n-4 -2  0  2  4ΓDoo(ω)\nω/Γ0.00.10.20.3\n-20-10 0 10 200.00.51.0\n-20-10 0 10 202πΓDee(ω)\nFIG.15:o-orbitalspectrum ΓDoo(ω)vsω/Γondecreasing ˜ x=x/Γ\nin the FL phase (for same interaction parameters as fig. 14, wi th the\ncritical ˜xc=12.26..). Shown for ˜ x=18,17,16,15,14,13 and 12 .5\n(fromrighttoleft). Rightinset : asmainfigure,butintheUSCphase\nfor ˜x=12 (dotted line), 7 (dashed) and at the p-hsymmetric point\n˜x=0 (solid). The o-spectrum evolves continuously as the transition\nis crossed, a pole occuring at the Fermi level precisely at th e transi-\ntionwhere ε∗ovanishes. Leftinset:e-orbitalspectrum2 πΓDee(ω)vs\nω/Γjust on either side of the QPT, for ˜ x=12.261351440 in the FL\nphase (solidline)and ˜ x=12.261351420 intheUSCphase (dashed).\nHeretheentirespectrumchanges abruptlyoncrossingthetr ansition.\nΣoo(ω=0)≡ΣR\noo(0)togetherwith eqn. (49), the Fermi level\nspectrum is given generally by Doo(ω=0)=δ(ε∗\no)). In the\nvicinityofthetransition,therenormalizedlevel ε∗\noisthecoun-\nterpart of the low-energy scale T∗introduced in sec. IIIA in\nrespect of thermodynamics (see e.g.fig. 6);T∗andε∗\noboth\nvanishinglinearlyasthetransitionisapproached,andcon trol-\nling the low-energy behavior of appropriate thermodynamic s\nandsingle-particledynamicsrespectively.\nWe also add that, as expected on physical grounds, the\nvanishing o-orbital renormalized level does not show up in\nthecorresponding e-channelspectrum Dee(ω),whichasillus-\ntratedin fig. 15 (leftinset) changesin a whollydiscontinuo us\nfashionon crossing the transition; commensuratewith the i n-\nherentlyfirst-ordernatureofthetransitionalongthe y=xline.\nVI. EXPERIMENT\nWe now consider the experiments of Kogan et al11on a\nGaAs-basedsingle-electrontransistorat lowtemperature (T),\nembodiedinthedifferentialconductanceasafunctionofga te\nvoltageΔVg(measured relative to a reference voltage), and\nalsothebias(orsource-drain)voltage Vsd. Onvaryingthegate\nvoltage,theresultantconductancemapsshown e.g.infig. 1of\n[11] (see also the theoretical fig. 16 below) show clear zero-\nbias Kondo peaks arising in the centersof adjacentCoulomb\nblockadevalleys;onevalleythusbeingassociatedwithano dd\nnumber of dot electrons and the other with an even number.\nThe former valley, which extends over a relatively wide ΔVg\nrange, is naturally interpreted11as the normal FL, or ‘singletphase’;whilethe latter,extendingovera narrower ΔVgrange,\nisinterpreted11asthe‘tripletphase’( i.e.theUSCphase).\nInconsideringtheoreticallytheconductance,\nGc(T,Vsd=0)\n(2e2/h)G0=/integraldisplay∞\n−∞dω−∂f(ω)\n∂ω2πΓDee(ω)(59)\nis exact37at zero-bias (as before we consider explicitly V2=\nV1), withDee(ω)the spectrum at the temperature of interest.\nAt finite bias by contrast, nothing exact can be said with the\nmethods at hand. To treat approximately Vsd/ne}ationslash=0 we neglect\nexplicitdependenceoftheself-energieson Vsd. Withthisstan-\ndardapproximation Gc≡Gc(T,Vsd)is readilyshownto be\nGc(T,Vsd)\n(2e2/h)G0≃1\n2/integraldisplay∞\n−∞dω/bracketleftbigg−∂fL(ω)\n∂ω+−∂fR(ω)\n∂ω/bracketrightbigg\n2πΓDee(ω)\n(60)\nwherefν(ω) =f(ω±1\n2eVsd)for leadν=R/Lrespectively\n(f(ω) = [eω/T+1]−1). ForVsd=0, eqn. (60) reduces cor-\nrectlytoeqn.(59);while for T=0 it yields\nGc(T=0,Vsd)\n(2e2/h)G0≃πΓ/bracketleftbig\nDee(ω=1\n2eVsd)+Dee(ω=−1\n2eVsd)/bracketrightbig\n(61)\nin terms of the T=0 spectra. In the above we have taken\na symmetric voltage split between the R/Lleads. From eqn.\n(60) this gives Gc(T,Vsd) =Gc(T,−Vsd), which symmetry is\nratherwell satisfied inexperiment(fig. 1,[11]).\nUnder application of a gate voltage, the level energy ε1∝\nΔVg, and one expectsthe level spacing Δε=ε2−ε1to be es-\nsentially fixed47. The experimental ‘trajectory’ in the ( ε1,ε2)\n(or(x,y))planeuponvarying ΔVgisthenasindicatedschemat-\nically in fig. 2(a), viz. y=x+Δε(i.e.ε2=ε1+Δε). In\nthis regard an interesting symmetry arises. Indicating exp lic-\nitly thex,ydependenceof Dee(ω)≡Dee(ω;x,y), it is readily\nshownthatunderthe p-hand1-2transformationsofsec.IIA,\nDee(ω;x,y)=Dee(−ω;−y,−x). Employingthisineqn.(60),\nnotingthat [∂fL(ω)/∂ω+∂fR(ω)/∂ω]isevenin ω,gives\nGc(T,Vsd;x,y) =Gc(T,Vsd;−y,−x)(62)\nfor the conductance Gc(T,Vsd)≡Gc(T,Vsd;x,y). That is,\nthe conductance is symmetric under reflection about the line\ny=−x. Now the ‘trajectory’ y=x+Δεis perpendicu-\nlar to the line y=−x, and intersects it for x=−1\n2Δε,i.e.\n(sincex=ε1+1\n2U+U′) forε1=−1\n2Δε−1\n2U−U′≡ε1,m.\nSince the phase boundaries are also symmetric under reflec-\ntion about the line y=−x(figs. (2,5,7)), this value ε1=ε1,m\n– and hence the corresponding ΔVg,m(ε1∝ ΔVg) – is thus\nthe midpoint of the triplet phase; such that the conductance\nshould be an even function of ε1−ε1,m, or equivalently of\nΔVg−ΔVg,m. This symmetry is quite well satisfied in ex-\nperiment (and is of course obeyed precisely in the theoreti-\ncal results). Fig. 16 (middle panel) shows the experimental\nzero-bias conductance11,46(crosses) as a function of ΔVg(in\nmV), together with corresponding theoretical results (as d e-\ntailed below). The midpointof the triplet (T) phase is readi ly\nidentifiedas ΔVg,m=−10mV, aboutwhichthe experimental\nconductance is indeed seen to be quite symmetric. And the17\n-6-4-2 0\n-20-10 0 10 20 30 40 50log (TK/Γ)\nΔVg /mV\n0.00.20.40.60.81.0\n-20-10 0 10 20 30 40 50Gc /2e2h-1\nΔVg /mVT SS\n-20-10 0 10 20 30 40 50\nΔVg /mV-0.50.00.5Vsd /mV\n0.20.30.40.50.60.70.8\nFIG. 16: Middle panel : Experimental zero-bias conductance11,46\n(crosses) vsΔVg(in mV). Theoretical results obtained as described\nintextarealsoshown,for T/Γ=0(solidline),5 ×10−3(longdash)\nand10−2(shortdash);for Γ=0.5meVthesecorrespondrespectively\ntoT=0, 30mK and 60mK. The positions of the theoretical phase\nboundaries ( T=0) between the USC triplet (T) and Fermi liquid\nsinglet (S) phases are as indicated. Upper panel : Kondo scales de-\ntermined as in sec. III, and shown as log (TK/Γ)vsΔVg(for the T\nphase,TK≡TS=1\nK).Bottom panel : Theoretical differential conduc-\ntance mapinthe ( Vsd,ΔVg)-plane,withthe grey-scale code indicated\n(in units of 2 e2/h). Shown for T=30mK, choosing Γ=0.5meV\n(see text). The experimental counterpart is shown infig.1of [11].\nexperimentalconductancemapshowninfig. 1of[11]( cffig.\n16bottompanel)isalsoclearlyrathersymmetricabout ΔVg,m.\nTo compare directly to experimentwe must specify the di-\nmensionlessinteractions ˜U,˜U′,˜JH(eqn.(10)), Δ˜ε=Δε/Γ,the\nrelationbetween ˜ε1=ε1/ΓandΔVg,andfinallythehybridiza-\ntionstrength Γ. Thisisobviouslyalargeparameterspace,and\nour intent here is simply to employ what we regard as a rea-\nsonableset of‘bare’parameters. Fora typicaldotthe relat ive\nhierarchy of energies satisfies3|˜JH| ≪Δ˜ε≪˜U; with which\nthe specific parameters we use here concur, |˜JH|=0.5,Δ˜ε=\n4.5 and˜U=12 (and with ˜UandΔ˜εin excess of unity, con-\nsistentwiththeoccurrenceofchargequantizationtowards the\ncenters of the Coulomb blockade valleys). No attempt to ex-plain experiment on the assumption ˜U′=˜Uwas found to be\nsuccessful,evenqualitatively,onvaryingthebareparame ters.\nThe main reason (as evident from inspection e.g.of fig. 2(c)\norfig.5(top,(c))isthattheresultantwidthoftheTphase(i n\n˜ε1orΔVg) is much too large compared to that of the singlet\n(S)phase,andassuchnotqualitativelyconsistentwithexp er-\niment11(fig. 16). For the results shown here we have used\n˜U′=6=˜U/2 (although tolerable variations from this value\ngivecomparableagreementwithexperiment).\nFrom the discussion above, the relation between ΔVgand\n˜ε1is of form ΔVg=c[˜ε1−˜ε1,m]+ΔVg,mwhere the propor-\ntionality constant cis to be determined (as above, ΔVg,m=\n−10mV and ˜ε1,m=−1\n2Δ˜ε−1\n2˜U−˜U′). For a chosen set of\n˜U,˜U′,˜JH,Δ˜ε, the theoretical zero-bias conductance at T=0\nis calculated from eqn. (59) as a function of ˜ε1. It is then\nscaled onto the experimental results shown in fig. 16 (mid-\ndle),overthe ΔVgrangeabove ∼35−40mV. We choosethis\nrangebecauseherethesystemisbeginningtheapproachtoth e\n‘emptyorbital’regimeof nimp≪1,whereonedoesnotexpect\nanyappreciable T-dependencetotheconductance[theexper-\nimentalTis not known with certainty46, for although the ex-\nperimentswereperformedattherefrigeratorbasetemperat ure\nof∼12mK, theelectrontemperature, T,was notdetermined;\nalthoughit is believedto be /lessorsimilar40mK46]. With this procedure\nwe determinethe constant c, which is then fixed and used for\nallΔVg(andT); as well as the dimensionless constant G0re-\nflecting (sec. II) the relative asymmetry in tunnel coupling to\ntheleads(fromscalingtheverticalaxisinfig.16,andleadi ng\ntoG0≃0.8 – as is obviously reasonable even from cursory\ninspectionoftheexperimentaldata).\nIn comparingto experiment,an obviouskey element is the\nrelative widths (in ˜ε1orΔVg) of the S and T phases, the for-\nmer being considerably wider than the latter in experiment.\nThiswenaturallyfindtobeinfluencedsignificantlybytheex-\nchange˜JH(and to a lesser extent by Δ˜ε), which is optimised\naccordingly. For the results shown here, we find ˜JH=−0.5\nto be optimal. Its magnitude is small, as expected, although\nits sign is antiferromagnetic. This is not however unreason -\nable, for on coupling to the leads as mentioned in sec. III,\nan AFbare˜JHstill generatesan effectiveferromagneticspin-\ncouplingvia an RKKY interaction(as evidentin the veryex-\nistence of the USC triplet phase for weakly AF bare ˜JH, and\nwhicheffectisinfactlargestforthecase V2=V1weconsider\nexplicitly).\nWith the above we calculate the T=0 zero-bias conduc-\ntance, shown in the middle panel (solid line) of fig. 16 for\n˜U=12,˜U′=6,Δ˜ε=4.5,˜JH=−0.5,with the two T/S phase\nboundaries indicated in the figure. The T phase, symmetri-\ncally disposed about the midpoint ΔVg,m=−10mV, occurs\nin the interval −13.45mV≤ΔVg≤ −6.55mV (correspond-\ning to−15.2≤˜ε1≤ −13.3); withnimpat the phase bound-\nariesofnimp=1.87(upperboundaryat ΔVg=−6.55mV)and\nnimp=4−1.87=2.13 (lower boundary),such that in accor-\ndance with eqn. (38) of sec. IVC the zero-bias conductance\nincreasesoncrossingfromtheS (FL)to theT(USC) phase.\nThe resultant Kondo scales as a function of ΔVgare shown\ninthetoppaneloffig.16(obtainedasspecifiedinsec.III,an d\nwithTK≡TS=1\nKfor the T phase). Since TKvanishes as the18\nQPT is approached from the S (FL) side, finite-temperature\neffects will obviously be most significant in the vicinity of\nthe transition. Fig. 16 thus shows the zero-bias conductanc e\nat two non-zero temperatures, T/Γ=0.005 and 0 .01. While\nthereisnotmuchnetdifferencebetweenthetwo,eachhasthe\neffectofsignificantlyincreasingtheconductanceinthevi cin-\nityofthetransition,andleadstowhatweregardasrathergo od\noverall agreement with experiment. Over the T-range shown\ntheconductance‘inside’theTphasedoesnoterodeasrapidl y\nasonemightlike,reflectingthefactthat TK≡TS=1\nKthereinis\ninexcessofthe T’sshown;althoughonecouldlikelyimprove\non this with a bare parameterset for which TS=1\nKinside the T\nphaseissomewhatsmaller. Thetemperaturerangeconsidere d\nhere is also entirely reasonable in relation to the experime n-\ntalTdiscussed above11,46; withΓ=0.5meV (as employed\nbelow)the temperaturesshowncorrespondto T=30mKand\n60mKrespectively.\nA. Conductancemaps\nThe bare parameters specified above are fixed. Using eqn.\n(60) the differential conductance, as a function of gate and\nbias voltages, may now be calculated and compared to ex-\nperiment. For this we must finally specify the hybridization\nstrengthΓ; in the followingwe take Γ=0.5meV(notingthat\ncomparison to experiment is not critically dependent on thi s\nchoice,with valuesin the range ∼0.3−0.6meVbeingfound\nquite acceptable). The resultant differentialconductanc emap\nforT=30mK is shown in fig. 16 (bottom panel), and is in\nrather good agreement with the experimental results report ed\ninfig. 1of[11].\nIn addition to the clear zero-bias Kondo ridges associated\nwith both the T and S phases, the conductance map shows\nother featuresnoted in experiment11. In particular,lookingat\nthe far left side of the conductance plot, one sees two dark\n‘ridges’ positioned symmetrically around Vsd=0. AsΔVgis\nincreased the two ridges move together, until they merge to\nform the zero-bias Kondo ridge associated with the T phase.\nThe latter persists for a range of ΔVg, and then the two ridges\nseparate again (the pattern being in other words symmetrica l\naboutΔVg,m=−10mV, for the reasons explained following\neqn. (62)). The obvious question arises as to the origin of\nthese‘ridges’,whichwenowconsider.\nAs seen in fig. 16, the T=0 zero-bias conductance in-\ncreasesonpassingfromtheS (FL)totheT(USC)phase;and\nin sec. VA we showed that such behavior was indicative of\na vanishing Kondo antiresonance in the single-particle spe c-\ntrum, as the transition is approached from the S side. This\nis the origin of the ridges seen in the conductance maps, as\nnowshownbyconsideringcutsthroughtheconductancemap\nof fig. 16, for a sequenceof differentfixed gate voltages ΔVg.\nThe top panel in fig. 17 accordingly shows the conductance\nvsbias voltage Vsd, for five different values of ΔVg:−10mV\n(at the midpoint of the T phase), and ΔVg=−6,−5,−3 and\n+2mVasonemovesintotheSphase(whichat T=0occurs\nforΔVg≥−6.5mV). For ΔVg=−10mVtheconductancenat-\nurally peaks at Vsd=0. But on entering the S phase a clear 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n-1.0 -0.5 0.0 0.5 1.0Gc /2e2h-1\nVsd /mV\n0.00.10.20.30.40.5\n-1.0 -0.5 0.0 0.5 1.0Gc /2e2h-1\nVsd /mV\nFIG. 17: Top panel :Vsd-dependence of conductance, taking cuts\nthrough the conductance map of fig. 16 (bottom) for a sequence\nof different fixed gate voltages: ΔVg=−10mV (midpoint of the T\nphase, sold line), and ΔVg/mV=−6 (long dash), −5 (short dash),\n−3 (point dash) and +2 (dotted), on moving into and through the S\nphase. Aclearantiresonance developsintheSphase,asdisc ussedin\ntext. Thepeaks intheconductance, symmetricalabout Vsd=0,lieat\nthe center of the ridges seen inthe conductance map.\nBottom panel : Focus on the ΔVg=−3mV case. Solid line: conduc-\ntance asintoppanel. Longdash: corresponding T=0conductance,\nproportionalto [Dee(ω=+1\n2eVsd)+Dee(ω=−1\n2eVsd)]. Shortdash:\ncontribution from Dee(ω= +1\n2eVsd)alone, showing a clear Kondo\nantiresonance inthe single-particle spectrum itself.\nantiresonance in the conductance is seen to develop, just se t-\nting in by ΔVg=−6mV and deepening progressively as ΔVg\nisincreasedintheSphasetowards −3mV(thennaturallydis-\nappearing as one gets considerably further into the S phase,\nas illustrated by the ΔVg= +2mV example). And the peaks\nin these conductance profiles, symmetrically disposed abou t\nVsd=0,lieatthecenteroftheridgesintheconductancemap.\nTo show that the above behavior indeed reflects a Kondo\nantiresonance in the single-particle spectrum itself, the bot-\ntom panel in fig. 17 focusses on the ΔVg=−3mV example.\nThe solid line again gives the conductance shown in the top\npanel;whilethelongdashlineshowsthecorresponding T=0\nconductanceobtained from eqn. (61), and thus being propor-\ntional to [Dee(ω= +1\n2eVsd)+Dee(ω=−1\n2eVsd)]. The lat-\nter clearly captures well the former (the differences natur ally\nbeing due to thermal smearing). Because the conductance is\nproportional to the symmetrized spectra at ω=±1\n2eVsd, it is\nnota prioriclear that the single-particle spectrum itself con-\ntains a Kondo antiresonance. That it does, however, is seen\nfromshort-dashline infig.17(bottom),whichshowsthe con-\ntributionfrom Dee(ω=+1\n2eVsd)itself,seentocontainaclear19\n-10 0 10 20 30 40 50\nΔVg /mV-0.50.00.5Vsd /mV\n0.00.10.20.30.40.50.60.70.8\nFIG. 18: As discussed in text, differential conductance map in the\n(Vsd,ΔVg)-plane; shown for T=30mK with the same bare param-\neters as figs. (16,17), except for a slight change in ˜JH. In this case\nno S/T transition occurs, the system remaining throughout i n the S\nphase. Thezero-biasKondoridgeassociatedwiththeTphase isthus\nabsent, and insteada zero-bias conductance antiresonance persists.\nKondoantiresonancecenteredontheFermilevel. Weaddtoo\nthat, since the peaks/ridgesin the conductancestem from th e\n‘peaks’ inherent to the Kondo antiresonance in Dee, they are\nobviouslynotinterpretablein termsofisolateddotstates .\nFinally, the zero-biasKondo ridge in the conductancemap\n– formed as described above on merging of the Vsd/ne}ationslash=0 con-\nductanceridges,and concomitantvanishingof a conductanc e\nantiresonance (as in fig. 17 top) – reflects of course the ex-\nistence of the T (USC) phase, and hence a transition to it\nfrom the S (FL) phase. However one can readily envisage\na situation where the underlying bare parameters of the sys-\ntem/device are slightly different, such that on ramping dow n\nthegatevoltagetheresultanttrajectory y=x+Δεcomesclose\nto but ‘misses’ the S/T transition; the system as such always\nremaininginaSphase(see e.g.fig.2(a)). Inthiscasenozero-\nbias Kondo ridge associated with the T phase can arise. In-\nstead,fromthediscussionabove,onemightintuitivelyexp ect\ncontinuedpersistenceoftheconductanceantiresonanceon de-\ncreasing ΔVg, with attendant finite-bias conductance ridges\nwhich never quite merge together. That this indeed occurs\nis illustrated in fig. 18, where the conductance map (here for\nT=30mK) is shown for the same bare parameters as figs.\n(16,17),exceptforaslightchangein ˜JHto−0.6(thesamebe-\nhavior arising also on changing e.g.Δ˜εrather than ˜JH). And\nthequalitativebehaviorseenhereisindeedsimilartothat ob-\nservedinaseconddevice,showninfig. 3of[11](althoughin\nthiscasewe havenotmadea quantitativecomparison).\nVII. CONCLUDING REMARKS\nAs exemplars of multilevel quantum dot systems, we have\nconsidered in this paper correlated two-level quantum dots ,\ncoupled in a 1-channel fashion to metallic leads. Thermo-\ndynamics, single-particle dynamics and electronic transp ort\nproperties show the physical behavior of the system to berich and varied; and our aim has been to obtain a unified\nunderstanding of the problem for essentially arbitrary dot\ncharge/occupancy. Exceptingpointsof highsymmetrywhere\nfirstorderlevel-crossingtransitionsarise,associatedq uantum\nphase transitionsare of Kosterlitz-Thoulesstype, eviden t in a\nvanishing Kondo scale as the transition to the underscreene d\nspin-1 phase is approached from the Fermi liquid side; and\nmanifestinparticularbyadiscontinuousjumpinthezero-b ias\nconductanceasthetransitioniscrossed,whichwehaveshow n\ncan be understoodhere from an underlyingFriedel-Luttinge r\nsum rule. We add in fact that an abrupt conductance change\nappearsto be a general signature of a KT transition, such be-\nhaviorarisinggenerallynotonlyinthepresentmodel,buta lso\nincapacitivelycoupled2-channelquantumdotswhichexhib it\na KT transitionfroma charge-KondoFermi liquidstate (with\na quenchedchargepseudospin)to a non-Fermiliquid,doubly\ndegenerate ‘charge-ordered’ phase48; and in the problem of\nspinless,capacitivelycoupledmetallicislands/largedo tsclose\ntoadegeneracypointbetween NandN+1electronstates,de-\nscribedbytwo Ising-coupledKondoimpurities49.\nSeveralissuesnaturallyremaintobeaddressed. Webelieve\nfor example that the generalization of Luttinger’s theorem to\nthesingularFermiliquidUSCphase(sec.IVB)issignificant ,\nand raises important basic questions (such as why, and what\nfundamentally does it reflect?). While we do not doubt its\nvalidity, we have however demonstrated it only numerically ;\nand a proper analytical understanding of the result is obvi-\nously desirable. In this work we have also considered the\nsystem in the absence of an applied magnetic field, B. Inter-\nesting physics arises also for B/ne}ationslash=0 (seee.g.[22]), where the\nunderlying quantum phase transitions are naturally smeare d\ninto crossovers. In fact, for the USC phase the limits of zero\nfield and B→0+are different for T=0, reflecting the to-\ntal polarizationof a free spin-1/2(asfor the USC fixedpoint )\non application of even an infinitesimal field. We will turn in\nsubsequentworktotheeffectsofmagneticfieldsuponsingle -\nparticledynamicsandtransportinthe model.\nAcknowledgments\nHelpfuldiscussionswithL.Borda,F.Essler,D.Goldhaber-\nGordon,A.Kogan,A.MitchellandM.Pustilnikaregratefull y\nacknowledged. Particular thanks are due to A. Kogan and D.\nGoldhaber-Gordon for kindly providing us with their exper-\nimental data from [11]. We thank the EPSRC for financial\nsupport,undergrantEP/D050952/1.\nAPPENDIXA:EFFECTIVELOW-ENERGYMODELS\nWe first sketch the derivation of the effective low-energy\nmodel considered in sec. IIIB, spanned by the (1 ,1) triplet\nand(1,1)singletstatesoftheisolateddot;withenergiesunder\nˆHDofET=ε1+ε2+U′−1\n4JHandES=ε1+ε2+U′+3\n4JH\nrespectively. Thelocalunityoperatorforthedotis\nˆ1=ˆ1T+ˆ1S (A1)20\nwithˆ1T=∑Sz|S=1,Sz/an}bracketri}ht/an}bracketle{tS=1,Sz|in an obvious notation,\nand likewise ˆ1S=|0,0/an}bracketri}ht/an}bracketle{t0,0|. These satisfy the following\nidentitiesin thelocalHilbertspace,\nˆ1T=ˆs1·ˆs2+3\n4ˆ1ˆ1S=−ˆs1·ˆs2+1\n4ˆ1,(A2)\nasfollowsusing ˆs1·ˆs2≡1\n2/parenleftbigˆS2−3\n2/parenrightbig\nwithˆS=ˆs1+ˆs2.\nOmittingforbrevitytheleadcontribution ˆHL(eqn.(3)),the\nlow-energymodelisgivenby ˆHeff=ESˆ1S+ETˆ1T+ˆH(2)\neff. The\nfirst two terms are simply the bare energies of the dot states;\nusing eqns. (A1,A2) they may be written as ESˆ1S+ETˆ1T=\n1\n4(ES+3ET)ˆ1+(ET−ES)ˆs1·ˆs2,orequivalentlyas −JHˆs1·ˆs2\nonomittingthefirst (constant/common)term:\nˆHeff=−JHˆs1·ˆs2+ˆH(2)\neff(A3)\nHereˆH(2)\neffis the leading ( O(V2)) contribution arising from\ntunnel coupling to the leads (eqn. (2) with V2=V1≡V, here\ndenotedas ˆH′),givenfromaSWtransformation38as\nˆH(2)\neff=1\n2∑\nα,βˆ1βˆH′/bracketleftBig/parenleftbig\nEα−ˆHD/parenrightbig−1+/parenleftbig\nEβ−ˆHD/parenrightbig−1/bracketrightBig\nˆH′ˆ1α\n(A4)\nwithα,β∈{S,T}(retardationeffectsareasusualneglected).\nInanalyzingeqn.(A4) oneencountersthe following‘natu-\nral’exchangecouplings Jα\ni>0,\nJα\ni=NV2/bracketleftbigg1\nΔEα\ni+1\nΔ˜Eα\ni/bracketrightbigg\n(A5)\n(withNthe number of k-states in the lead, such that NV2∼\nO(1)). Here the ΔEα\ni>0 denote electron removalexcitation\nenergies from level i=1 or 2, relative to the α=TorSdot\nground state, and the Δ˜Eα\ni>0 correspondingly denote elec-\ntronadditionenergies to level irelative to the α=TorS\nground state; e.g.ΔET\n1=ED(0,1)−ETorΔ˜ES\n2=ED(1,2)−\nESinthenotationofsec. IIB. Denoting\nλT=1\n2U+1\n4JHλS=1\n2U−3\n4JH(A6)\ntheseexcitationenergiesareeasily shownto begivenby\nΔEα\n1=λα−xΔ˜Eα\n1=λα+x(A7a)\nΔEα\n2=λα−yΔ˜Eα\n2=λα+y(A7b)\nwhere(eqn.(11)) x=ε1+1\n2U+U′,y=ε2+1\n2U+U′. Hence\nfromeqn.(A5),\nJα\n1≡Jα(x)Jα\n2≡Jα(y) (A8)\nwithJα(x)definedby:\nJα(x) =NV2/bracketleftbigg1\nλα−x+1\nλα+x/bracketrightbigg\n=Jα(−x)(A9)\nDirectanalysisofeqn.(A4)yields,afterastandardif labo -\nriouscalculation,theeffectivelow-energymodeleqn.(16 ),\nˆHeff=J1(x,y)ˆs1·ˆs0+J2(x,y)ˆs2·ˆs0−I(x,y)ˆs1·ˆs2(A10)where potential scattering contributions are omitted for c lar-\nity, andˆs0denotesthe spin density of the conductionchannel\nat the dot. The direct exchangecouplingbetween spins1 and\n2isfoundtobe\nI(x,y) =JH+1\n2/bracketleftbig\nJT(x)+JT(y)−JS(x)−JS(y)/bracketrightbig\n(A11)\nwhile the O(V2)antiferromagnetic exchange couplings be-\ntweenspins1 or2 andthe leadaregivenby\nJ1(x,y) =1\n2/bracketleftbig\n3JT(x)+JT(y)+/parenleftbig\nJS(x)−JS(y)/parenrightbig/bracketrightbig\n(A12a)\nJ2(x,y) =J1(y,x). (A12b)\nUsingeqn.(A9), theseexchangecouplingssatisfy\nJi(x,y) =Ji(−x,−y):i=1,2 (A13a)\nI(x,y) =I(−x,−y) =I(y,x) (A13b)\nby virtue of which ˆHeff≡ˆHeff(x,y)in eqn. (A10) satisfies\nˆHeff(x,y)≡ˆHeff(−x,−y)(reflecting its symmetry under a\np-htransformation); while from eqn. (A12b), ˆHeff(x,y)→\nˆHeff(y,x)underthe1-2transformation(eqn.(13)),asexpected\nongeneralgroundsfromsec.IIA. Thesesymmetries,andthe\nconsequent invariance of the phase boundaries in the ( x,y)-\nplane to both inversion and reflection about the line y=x,\nare also naturally satisfied when potential scattering term s,\nomittedexplicitlyfromeqn.(A10),areincluded. By contra st,\nthe apparent reflection symmetries ˆHeff(x,y) =ˆHeff(−x,y) =\nˆHeff(x,−y)which hold for eqn. (A10) itself (via eqn. (A9)),\narenotpreservedwhenpotentialscatteringisincluded;wh ich\nis why the phase boundaries in fig. 7 are not invariant to re-\nflectionaboutthelines x=0andy=0.\nAlongthelines y=±xinthe(x,y)-plane,itfollowsdirectly\nfromeqns.(A12b,A13a)that J2(x,±x)=J1(x,±x),andhence\nfromeqns.(A12a,A9)\nJ2(x,±x) =J1(x,±x) =2JT(x),(A14)\nwithI(x,±x)=JH+[JT(x)−JS(x)]followingsimilarlyfrom\neqn. (A11). In consequence,as discussed in sec. IIIB, ˆHeffis\nseparablealongthelines y=±x,andfirstorderlevel-crossing\ntransitions thus arise. Note incidentally that for given JH<\n0, bothJT(x)andI(x,±x)increase on increasing |x|from 0,\nboth of which act (see eqn. (17)) to favor the triplet phase;\nconsistent with the USC phase surviving for |x|>0 after it\nhasbeendestroyedfor x=0, asindeedfoundinfig.7.\nFinally, a byproduct of the above gives directly the pure\nspin-1 Kondo model appropriate for ferromagnetic JH>0\ndeep in the USC spin-1 phase15,16, where the (1 ,1) singlet\ndotstatesareenergeticallyirrelevantandonlythe (1,1)triplet\nstates need be retained. To this end simply project into the\ntriplet sector, ˆHK=ˆ1TˆHeffˆ1T. Writing J1ˆs1·ˆs0+J2ˆs2·ˆs0=\n1\n2(J1+J2)(ˆs1+ˆs2)·ˆs0+1\n2(J1−J2)(ˆs1−ˆs2)·ˆs0, recognising\nthatˆ1T(ˆs1−ˆs2)ˆ1T=0 and neglecting constants, eqn. (A10)\ngivesa spin-1Kondomodel\nˆHK=ˆ1TˆHeffˆ1T=JKˆS·ˆs0 (A15)21\nwhereˆS≡ˆ1T(ˆs1+ˆs2)ˆ1Tis a pure spin-1 operator, and JK=\n1\n2(J1+J2)isgivenfromeqn.(A12)by\nJK=JT(x)+JT(y) (A16)\nwithJT(x)given explicitly by eqn. (A9). Along the lines\ny=±xin particular, eqns. (A16,A9,A6) give JK=2JT(x);\ni.e.ρJK=8\nπΓ[U+1\n2JH]/([U+1\n2JH]2−x2)withρhere the\nlead density of states per conduction orbital (such that Γ≡\nπV2Nρ). From perturbative scaling1the spin-1 Kondo scale\nTS=1\nKfollows as TS=1\nK∼Dexp(−1/ρJK)(with the exponen-\ntial dependenceas usual of the essence, and the prefactorim -\nmaterial), and eqn. (42) for TS=1\nKthus results. From NRG\ncalculationswehaveconfirmedexplicitlythatthedependen ce\nofTS=1\nKon˜U+1\n2˜JHisindeedaspredictedbyeqn.(42).\nAPPENDIXB:SELF-ENERGIES\nThe key NRG method for calculating the self-energy43is\nreadily extended to multilevel dots/impurities. With ˆHIthe\ninteracting part of the dot Hamiltonian, equation of motion\ntechniques42are used to obtain the following basic equation\nfortheretardedpropagators {Gij(ω)}:\n∑\nl/parenleftbig\n(ω+−εi)δil−Γil(ω)/parenrightbig\nGlj(ω) =δij+/an}bracketle{t/an}bracketle{t[diσ,ˆHI];d†\njσ/an}bracketri}ht/an}bracketri}ht\n(B1)\nThe sum is over the dot levels ( l=1,2 here), and [,]denotes\na commutator. By definition, ˆHI≡0 in the non-interacting\nlimit; whenceeqn.(B1)is ofform\n/bracketleftbig\nG0(ω)/bracketrightbig−1G(ω) =1+F(ω) (B2)\nwithG0(ω)thenon-interactingpropagatormatrixandtheel-\nementsof F(ω)givenby:\nFij(ω) =/an}bracketle{t/an}bracketle{t[diσ,ˆHI];d†\njσ/an}bracketri}ht/an}bracketri}ht (B3)\nUsing the Dyson equation in the form [G0]−1= [G]−1+Σ,\neqn.(B2)givesdirectlyeqn.(39),\nΣ(ω) =F(ω)[G(ω)]−1(B4)from which the self-energies are calculated directly (sec. V).\nCombiningeqn.(B4)withtheDysonequationalsogives G=\nG0(1+F),so thateqn.(B4)maybewrittenalternativelyas\nΣ(ω) =F(ω)[1+F(ω)]−1/bracketleftbig\nG0(ω)/bracketrightbig−1(B5)\n(whichweexploittocalculatetherenormalizedlevel ε∗\nowhen\nconsideringdynamicsonthe y=xline,eqn.(54) sec.VB).\nˆHIis given explicitly for the present problem by the sep-\narable sum (eqn. (1)) ˆHI=U∑iˆni↑ˆni↓+U′ˆn1ˆn2−JHˆs1·ˆs2.\nThe elements eqn. (B3) of Fare thus linearly separable as\nFij=FU\nij+FU′\nij+FJ\nij(inobviousnotation),andarecalculated\nindividually. Since each such term is a retarded correlatio n\nfunction,they are Lehmannresolvable44, andin consequence\nsatisfy sum rules. Writing Fij(ω) =FR\nij(ω)−iFI\nij(ω)(with\nthereal/imaginarypartsrelatedbyHilberttransformatio n),the\ngeneralsumruleis\n/integraldisplay∞\n−∞dω\nπFI\nij(ω) =/an}bracketle{t/braceleftbig\n[diσ,ˆHI],d†\njσ/bracerightbig\n/an}bracketri}ht(B6)\nwhere{,}denotes an anticommutator. Specifically, for the\npresent problem in the absence of an applied magnetic field,\nit isreadilyshownthatthediagonal-elementsumruleis\n/integraldisplay∞\n−∞dω\nπFI\nii(ω) =1\n2U/an}bracketle{tˆni/an}bracketri}ht+U′/an}bracketle{tˆn¯i/an}bracketri}ht(B7)\n(where¯imeans the opposite level to i); while for the off-\ndiagonalelements:\n/integraldisplay∞\n−∞dω\nπFI\nij(ω)j/ne}ationslash=i=/parenleftbig\n−U′+3\n4JH/parenrightbig\n/an}bracketle{td†\njσdiσ/an}bracketri}ht (B8a)\n=/parenleftbig\n−U′+3\n4JH/parenrightbig/integraldisplay0\n−∞dωDij(ω)(B8b)\nThese sum rulesprovidea checkon the accuracyof the NRG\ncalculations,andinpracticeare wellsatisfied.\n1A. C. Hewson, The Kondo Problem to Heavy Fermions (Cam-\nbridge UniversityPress,Cambridge, 1993).\n2L. P. Kouwenhoven et al., in Mesoscopic Electron Transport ,\nedited byL.L.Sohn(Kluwer, Dordrecht, 1997).\n3M. Pustilnik and L. I. Glazman, J. Phys.: Condens. Matter 16,\nR513 (2004).\n4D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-\nMagder, U. Meirav,and M. A.Kastner, Nature 391, 156 (1998).\n5S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven,\nScience281, 540 (1998).\n6P.W.Anderson, Phys.Rev. 124, 41(1961).\n7P. W. Brouwer, Y. Oreg, and B. I. Halperin, Phys. Rev. B 60,\nR13977 (1999).8H. U. Baranger, D. Ullmo, and L. I. Glazman, Phys. Rev. B 61,\nR2425 (2000).\n9J. Schmid, J. Weis, K. Eberl, and K. v Klitzing, Phys. Rev. Let t.\n84, 5824 (2000).\n10W.G.vanderWiel,S.DeFranceschi,J.M.Elzerman,S.Taruch a,\nL. P. Kouwenhoven, J. Motohisa, F. Nakajima, and T. Fukui,\nPhys. Rev. Lett. 88, 126803 (2002).\n11A.Kogan,G.Granger,M.A.Kastner,D.Goldhaber-Gordon, an d\nH. Shtrikman,Phys. Rev. B 67, 113309 (2003).\n12C. H. L. Quay, J. Cumings, S. J. Gamble, R. de Picciotto,\nH. Kataura, and D. Goldhaber-Gordon, Phys. Rev. B 76, 245311\n(2007).\n13S.Sasaki, S.De Franceschi, J. M. Elzerman, W.G. van der Wiel ,22\nM. Eto, S. Tarucha, and L. P. Kouwenhoven, Nature 405, 764\n(2000).\n14N. Roch, S. Florens, V. Bouchiat, W. Wernsdorfer, and F. Bale -\nstro, Nature 453, 633 (2008).\n15M. Pustilnik and L. I. Glazman, Phys. Rev. Lett. 87, 216601\n(2001).\n16A. Posazhennikova, B.Bayani, and P.Coleman, Phys.Rev. B 75,\n245329 (2007).\n17P.Nozi` eres and A.Blandin, J. de Phys.(France) 41, 193 (1980).\n18K. Kikoinand Y. Avishai, Phys.Rev. Lett. 86, 2090 (2001).\n19W. Hofstetter and H. Schoeller, Phys. Rev. Lett. 88, 016803\n(2001).\n20M.Vojta,R.Bulla,andW.Hofstetter,Phys.Rev.B 65,140405(R)\n(2002).\n21W.Koller,A.C.Hewson,andD.Meyer,Phys.Rev.B 72,045117\n(2005).\n22M. PustilnikandL.Borda, Phys. Rev. B 73, 201301(R) (2006).\n23A. Posazhennikova and P. Coleman, Phys. Rev. Lett. 94, 036802\n(2005).\n24P.R.Bas and A.A.Aligia,cond-mat/09043236 (2009).\n25M.PustilnikandL.I.Glazman,Phys.Rev.Lett. 85,2993(2000).\n26M. Pustilnik, L. I. Glazman, and W. Hofstetter, Phys. Rev. B 68,\n161303(R) (2003).\n27W.Hofstetter and G.Zarand, Phys.Rev. B 69, 235301 (2004).\n28P.Mehta, N. Andrei, P.Coleman, L.Borda, andG. Zarand, Phys .\nRev. B72, 014430 (2005).\n29K. G.Wilson,Rev. Mod. Phys. 47, 773 (1975).\n30H.R.Krishnamurthy,J.W.Wilkins,andK.G.Wilson,Phys.Re v.\nB21, 1003 (1980).\n31H.R.Krishnamurthy,J.W.Wilkins,andK.G.Wilson,Phys.Re v.\nB21, 1044 (1980).\n32R.Peters,T.Pruschke,andF.B.Anders,Phys.Rev.B 74,245114\n(2006).\n33A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99, 076402\n(2007).34R. Bulla, T. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395\n(2008).\n35D. C.Langreth, Phys.Rev. 150, 516(1966).\n36J. M.Luttinger, Phys.Rev. 119, 1153 (1960).\n37Y. Meir andN. S.Wingreen, Phys.Rev. Lett. 68, 2512 (1992).\n38J. R.Schriefferand P.A.Wolff,Phys.Rev. 149, 491(1966).\n39A. K. Mitchell, T. F. Jarrold, and D. E. Logan, Phys. Rev. B 79,\n085124 (2009).\n40Inpracticewedefinethelow-energyscale T∗viaSimp(T∗)=0.85,\nsuitably betweenln2andln3.\n41The reader will notice from fig. 7 that the first order transiti ons\nalongy=−xoccur on the ‘inside’ intersection of this line with\nthephaseboundary, butnotontheouterintersection.Thisr eflects\nthe fact that eqn. (16) itself is not in fact applicable in the latter\nregion, constructed as it is retaining solely the (1 ,1) singlet and\ntriplet states of the dot. On progressively increasing yalong the\nliney=−x, the (2,0) (or (0,2)) singlet states of the dot must\nalsoberetainedinthelow-energymanifold,leadingtoadif ferent\neffective low-energy model witha non-separable Hilbertsp ace.\n42D. N.Zubarev, Sov. Phys.Usp. 3, 320 (1960).\n43R.Bulla,A.C.Hewson,andT.Pruschke,J.Phys.: Condens.Ma t-\nter10, 8365 (1998).\n44A. A. Abrikosov, L. P.Gorkov, and I. E. Dzyaloshinski, Methods\nofQuantumFieldTheoryinStatisticalPhysics (Dover,NewYork,\n1977).\n45W.C.OliveiraandL.N.Oliveira,Phys.Rev.B 49,11986(1994).\n46A. Koganand D.Goldhaber-Gordon, private communication.\n47Any variation of ΔεwithΔVgis a much weaker effect and thus\nusuallyneglected.Wecaninprincipleincludeit,butincom paring\ntoexperiment here donot finditnecessary todoso.\n48M. R. Galpin, D. E. Logan, and H. R. Krishnamurthy, Phys. Rev.\nLett.94, 186406 (2005).\n49M. Garst,S.Kehrein, T. Pruschke, A.Rosch, andM. Vojta,Phy s.\nRev. B69, 214413 (2004)."    },    {        "title": "0902.1230v1.Fermi_Velocity_Enhancement_in_Monolayer_and_Bilayer_Graphene.pdf",        "content": "arXiv:0902.1230v1  [cond-mat.str-el]  7 Feb 2009Fermi Velocity Enhancement in Monolayer and Bilayer Graphe ne\nGiovanni Borghi,1Marco Polini,2,∗Reza Asgari,3and A.H. MacDonald4\n1International School for Advanced Studies (SISSA), via Bei rut 2-4, I-34014 Trieste, Italy\n2NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy\n3School of Physics, Institute for research in fundamental sc iences, IPM 19395-5531 Tehran, Iran\n4Department of Physics, University of Texas at Austin, Austi n TX 78703, USA\nIn single-layer graphene sheets non-local interbandexcha nge leads toarenormalized Fermi-surface\neffective mass which vanishes in the low carrier-density lim it. We report on a comparative study of\nFermi surface effective mass renormalization in single-lay er and AB-stacked bilayer graphene. We\nexplain why the mass does not approach zero in the bilayer cas e, although its value is still strongly\nsuppressed.\nI. INTRODUCTION\nGraphene is an atomically thin two-dimensional elec-\ntron system composed of carbon atoms on a honeycomb\nlattice. Experimentershaverecently1madeprogresswith\ntechniques which enable isolation and study of systems\nwith one or a small number of graphene layers. Much\nof the fundamental physics interest in graphene sys-\ntems follows from the fact that its envelope-function low-\nenergy Schr¨ odingerequation is equivalent to the massless\nlimit of a two-dimensional Dirac equation. In the case\nof graphene the spinor structure in the Dirac equation\nrefers to honeycomb-sublattice and Brillouin-zone val-\nley, instead of spin and electron-positron, degrees of free-\ndom. Graphene therefore presents a new type of many-\nbody problem in which single-particle quantum mechan-\nics is relativisitc, but interactions are essentially non-\nrelativistic instantaneous Coulomb interactions.\nOne of the key consequences of electron-electron in-\nteractions in single-layer graphene is an enhancement of\nthe quasiparticle Fermi velocity2,3which diverges loga-\nrithmically as the carrier density goes to zero. For mod-\nerate interaction strengths4,5the velocity enhancement\n(Fermi liquid effective mass suppression) is dominated by\nnon-local interband exchange effects which are most eas-\nily explained using the pseudospin language3,6explained\nbelow. In this article we present a close comparison be-\ntween the physics of mass renormalization in single-layer\nand AB-stacked bilayer graphene, explaining why the ve-\nlocity renormalization is finite in the low-density limit in\nthe latter case.\nII. PSEUDOSPIN DESCRIPTION OF\nVELOCITY RENORMALIZATION IN\nSINGLE-LAYER GRAPHENE\nWe begin by recalling the continuum-model Hamilto-\nnian (¯h= 1) of massless Dirac fermions interacting with\nnon-relativistic Coulomb interactions. The Hamiltonian\nhas one-particle band-energy and two-particle Coulomb-\ninteraction contributions:\nˆH=v/summationdisplay\nk,α,βˆψ†\nk,α(σαβ·k)ˆψk,β+1\n2S/summationdisplay\nq/negationslash=0/summationdisplay\nk,k′/summationdisplay\nα,βvqˆψ†\nk−q,αˆψ†\nk′+q,βˆψk′,βˆψk,α,(1)\nwherevis the bare Fermi velocity, α,βare labels for the\ntwohoneycombsublatticeswhichwetreatasthelabelsof\na quantum spin-1 /2-like (pseudospin) degree-of-freedom,\nSis the area of the system, and vqis the 2D Fourier\ntransform of the interparticle potential. (Because our\nanalysis of Fermi velocity renormalization is based on\na screened exchange approximation, in which different\nspins and valleys are independent, we do not explicitly\naccount for these degrees of freedom.) From Eq. (1) we\nsee that in pseudospin language the band Hamiltonian\nconsists of a momentum-dependent pseudospin effective\nmagnetic field which acts in the direction of momentum\nk. The band eigenstates in the positive and negative\nenergy bands have their pseudospins either aligned or\nopposed to the direction of momentum.\nWhen interactions are treated in a mean-field approx-\nimation we obtain7\nˆHMF=−1\nS/summationdisplay\nk,k′/summationdisplay\nα,βvk−k′ραβ(k′)ˆψ†\nk,αˆψk,β.(2)\nwhere the density matrix ραβ(k′) =∝angbracketleftΨ0|ˆψ†\nk,βˆψk,α|Ψ0∝angbracketright\nand|Ψ0∝angbracketrightis the mean-field-theory ground state. (In both\nsingle-layer and bilayer cases we assume that transla-\ntionalsymmetryisnotbrokensothattheelectrondensity\nis constant and the Hartree contribution to the mean-\nfield Hamiltonian can be ignored.) We parametrize the\npseudospin-density matrix ραβ(k) in terms of charge and\npseudospin-density contributions:\nραβ(k) =n(0)\nk,++n(0)\nk,−\n2δαβ\n+n(0)\nk,+−n(0)\nk,−\n2ˆn(k)·σβα (3)\nwheren(0)\nk,sare noninteracting band occupation factors.\nAssumming that the valence band (label s=−1) is full\nand that spatial isotropy is not broken, the occupation\nfactors are completely fixed by the carrier density n. For\nexample, for a n-doped system at T= 0n(0)\nk,s= Θ(kF−k)2\nfors= +1 andn(0)\nk,s= 1 fors=−1, wherekF∝√nis\nthe Fermi wavevector and Θ( x) is a step function. The\nunit vector ˆn(k) specifies the pseudospin orientation of\nthe positive energy band at momentum k.\nUsing Eq. (3) in Eq. (2) and adding the band ki-\nnetic energy one can easily find that the total mean-field\nHamiltonian is\nˆHHF=/summationdisplay\nk,α,βˆψ†\nk,α[δαβB0(k)+σαβ·B(k)]ˆψk,β,(4)\nwhere the self-consistent Hartree-Fock fields are defined\nby\nB0(k) =−/integraldisplayd2k′\n(2π)2vk−k′[f+(k′)−1/2],(5)\nand\nB(k) =vk−/integraldisplayd2k′\n(2π)2vk−k′f−(k′)ˆn(k′).(6)\nIn Eq. (5) we have subtracted the neutral system value\nofB0, which is easily seen to be wavevectorindependent.\nThis subtraction is necessary in order to make B0finite\nand represents only a convenient choice for the zero of\nenergy. In Eq. (6) the first term is the band-structure\npseudospin magnetic field, and the second term is the\nexchange field. We also have introduced the short-hand\nnotation\nf±(k) =n(0)\nk,+±n(0)\nk,−\n2, (7)\nwhich recognizes that n(0)\nk,sdepends only on k=|k|.\nWe now demonstrate that the solution of the Hartree-\nFock equation is\nB(k) =vk−/integraldisplayd2k′\n(2π)2vk−k′f−(k′)ˆk′.(8)\nThis result is a consequence of the isotropic interaction\nvk−k′=v(|k−k′|) which can be decomposed in angular\nmomentum components using\nvk−k′=+∞/summationdisplay\nm=−∞Vm(k,k′)eim(ϕk−ϕk′)(9)\nwith\nVm(k,k′) =/integraldisplay2π\n0dθ\n2πe−imθ\n×v(/radicalbig\nk2+k′2−2kk′cos(θ)).(10)\nSubstituting this representation in Eq. (8), performing\nthe angular integration over ϕk′we find that\nBeq(k) =/bracketleftBigg\nvk−/integraldisplaykmax\n0dk′\n2πk′f−(k′)V1(k,k′)/bracketrightBigg\nˆk.(11)The ultraviolet cut-off kmaxis required8because the in-\ntegral is logarithmically divergent at large k′. We set\nkmaxto a value ∼1/a, wherea∼0.246 nm is graphene’s\nlattice constant, set by the energy scale over which the\nDirac model is valid. As expected Beq(k) is oriented\nalongˆkand|Beq(k)|=vk+ Σ(k) depends only on k.\nHere we have introduced the pseudospin-dependent part\nof the Hartree-Fock self-energy\nΣ(k) =−/integraldisplaykmax\n0dk′\n2πk′f−(k′)V1(k,k′).(12)\nNote that the fact that quasiparticles in monolayer\ngraphene have chirality J= 1 has selected the first mo-\nmentV1(k,k′) of the Coulomb interaction.\nFor convenience we assume a n-doped graphene layer\nand takeT= 0 so that\nf−(k) =1\n2[Θ(kF−k)−1] =−1\n2Θ(k−kF)\nf+(k)−1/2 =1\n2Θ(kF−k) (13)\nwherekFis the conduction-band Fermi radius. The\nrenormalized conduction s= + and valence s=−quasi-\nparticle energies are therefore given by\nEs(k) =svk−1\n4π/integraldisplaykF\n0dk′k′V0(k,k′)+sΣ(k).(14)\nwith\nΣ(k) =1\n4π/integraldisplaykmax\nkFdk′k′V1(k,k′). (15)\nThe first interaction correction in Eq. (14) is similar to\nthe familiar self-energy correction which occurs in an or-\ndinary two-dimensionalelectron gas. When the Coulomb\ninteraction is not screened it (famously and incorrectly)\npredicts a divergence of the quasiparticle velocity at the\nFermi energy. Our focus here is on the second self-energy\ncorrection, exhibited explicitly in Eq. (15), which is re-\nsponsibleforenhancedsplittingbetweenvalenceandcon-\nduction bands. To account partially for screening and\nto avoid the well-known Fermi velocity artifact of mean-\nfield-theory in systems with long-range interactions we\nhave used a screened interaction potential of the form\nvk=2πe2\nǫ(λqTF+k), (16)\nwhereqTF=αeekFis the Thomas-Fermi screening vec-\ntor andλis a control parameter which can be adjusted\nto represent the weaker screening at higher frequencies.\n(αee=e2/ǫ¯hvis the graphene continuum-model fine\nstructure constant.) We will mainly be interested in the\ncontribution to the self-energyat k∼kFdue to exchange\ninteractions with states at k′≫kF, for whichλplays lit-\ntle role.\nThe dependence of Σ( k) onk/kFforαee= 0.5 and\nΛ =kmax/kF= 10 is illustrated in Fig. 1. In this plot we3\nFIG. 1: (Color online) The pseudospin-dependent part of the\nHartree-Fock self-energy Σ( k) (in units of εF) for monolayer\ngraphene as a function of k/kF(for wave vectors up to the\ncut-off) for αee= 0.5 and Λ = 10. The vertical (black) line\nindicates the point k=kF. The data labeled by the dashed\n(red) line correspond to λ= 10−2in Eq. (16), while the data\nlabeled by the solid (blue) line correspond to λ= 1. Note the\nrapid change of Σ( k) around k=kFwhenλ= 10−2which is\nan artifact of theories which do not accout for screening. Th e\ndecrease in Σ for knearkmaxis an artifact of the ultraviolet\ncut-off, but plays little role in the region of interest k∼kF.\nhave shown Σ( k) for two different values of the control\nparameterλin Eq. (16): λ= 10−2, corresponding to an\nessentially unscreened Coulombpotential, and λ= 1 cor-\nresponding to the Thomas-Fermi screened potential. We\nsee that the unphysical λ= 0 anomalyat k=kFemerges\nonly at very small values of λ. For non-zero values of λ,\nΣ(k) is a smooth function of kand its contribution to\nthe quasiparticle velocity at k=kFcan be obtained by\nexpanding Σ( k) fork≪kmax. Forλ= 0 (unscreened\nCoulomb interactions) this expansion9can be performed\nanalytically. We find that for x>x′\nVm(xkF,x′kF) =2πe2\nǫkF¯Vm(x,x′) (17)\nwhere\n¯Vm(x,x′) =/integraldisplay∞\n0dt Jm(tx)Jm(tx′)\n=x′m\nxm+1Γ(m+1/2)\nΓ(m+1)Γ(1/2)\n×2F1(m+1/2,1/2,m+1,x′2/x2).(18)\nHereJm(x) is a Bessel function of order m, Γ(z) is the\nGamma function, and 2F1(a,b,c,z) is the Gauss hyper-\ngeometric function. (The x < x′result is obtained by\ninterchange x↔x′.) ForkF≪kmax, a requirement for\nthe utility of the Dirac continuum model of graphene,\nFIG. 2: (Color online) Pseudospin independent exchange en-\nergy contribution B0(k) (in units of εF) as calculated from\nEq. (5) as a function of k/kFforαee= 0.5. The data labeled\nby the dashed (red) line correspond to λ= 10−2in Eq. (16),\nwhile the data labeled by the solid (blue) line correspond to\nλ= 1. Top panel: Λ = 10. Bottom panel: Λ = 100. In both\npanels the vertical (black) line indicates the point k=kFand\nthe right axis scale refers to the λ= 1 data.\nthe leading term in the small x′expansion of Vmgoes\nlikex′m. As we explain below, this m-dependence is re-\nsponsible for the main qualitative difference between the\nphysics of quasiparticle mass renormalization in single-\nlayer and bilayer graphene.\nFor Λ = kmax/kFlarge and λnot too small,\nthe quasiparticle mass renormalization is due entirely\nto the pseudospin dependent part of the self-energy.\n(The pseudospin-independent contribution, illustrated in\nFig. 2, varies as k2at smallk.) Neglecting B0we find\nthat the renormalized velocity at k=kFis\nv⋆\nv= 1+1\nv∂Σ(k)\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=kF4\n= 1+1\n2αee∂fΛ(x)\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=1. (19)\nwhere\nfΛ(x) =/integraldisplayΛ\n1dx′x′¯V1(x,x′)\n=1\n2x2/integraldisplayx\n1dx′x′2\n2F1(3/2,1/2,2,x′2/x2)\n+x\n2/integraldisplayΛ\nxdx′1\nx′2F1(3/2,1/2,2,x2/x′2).\n(20)\nIt follows that\n∂fΛ(x)\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=1=1\n2/integraldisplayΛ\n1dx′1\nx′2F1(3/2,1/2,2,1/x′2)\n+3\n8/integraldisplayΛ\n1dx′1\nx′32F1(5/2,3/2,3,1/x′2).\n(21)\nBecause\nlim\nx′→∞2F1(3/2,1/2,2,1/x′2)\n= lim\nx′→∞2F1(5/2,3/2,3,1/x′2) = 1,(22)\nweimmediatelyseehowthefirstterminEq.(21)diverges\nlogarithmically for large Λ: we find, to leading order in\nΛ for Λ→ ∞\nlim\nΛ→∞∂fΛ(x)\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=1=1\n2ln(Λ), (23)\nand thus we recover the well-known2asymptotic result\nfor the velocity enhancement\nlim\nΛ→∞v⋆\nv= 1+αee\n4ln(Λ). (24)\nIII. VELOCITY RENORMALIZATION IN\nBILAYER GRAPHENE\nBilayer graphene has four atoms per unit cell and\nitsπ-electrons are therefore described by a model with\nfour-component spinors. In order to focus on the simil-\niarities and differences between single-layer and bilayer\nvelocity renormalization we use a commonly adopted\ntwo-component model for a bilayer which applies10at\nenergies below the inter-layer tunneling scale and ex-\nplains the bilayer’s unusual11quantum Hall effect. The\ntwo-component model for bilayer graphene has also\nbeen used to calculate the bilayer compressibility12and\nthe static noninteracting density-density linear-response\nfunction13.\nThe key differences between the two-band models of\nsingle layer and bilayer graphene are: i) the band disper-\nsionin the bilayercaseis quadraticwith an effectivemassm=t/(2v2) (wheretis the inter-layer tunneling and v\nis the Fermi velocity in an isolated monolayer), ii) the\nchirality is J= 2 rather than J= 1 for bilayer graphene\n(see below), and iii) the intra-layer [ V(S)\nk=vk] and inter-\nlayer [V(D)\nk=V(S)\nkexp(−kd)] Coulomb interactions are\ndifferent in the bilayer case. In our discussion of bilayer\ngraphene we use a Thomas-Fermi-like potential for vkas\nin Eq. (16), to avoid the well-known mean-field-theory\nartifacts in Coulombic systems. We also employ a cut-off\nkmaxfor the bilayer case, although we shall see that its\nrole is less essential. (The two-band continuum model ul-\ntraviolet cutoff scale in bilayer graphene is set by twhich\nis smaller than the π-band width scale ∼v/aappropri-\nate in the single-layer case.) Our calculations for bilayers\nhavescreening( λ) and cut-off ( kmax/kF≡Λ) parameters\nas in the single layer case, and are in addition depen-\ndent on the dimensionless inter-layer distance parameter\n¯d=dkmaxwhich has a value6≈0.2. The Thomas-Fermi\nscreening vector for bilayer graphene is given by qTF=\nme2/ǫso thatqTF/kF= [t/(2vkmax)]αeeΛ≡¯tαeeΛ. Us-\ningd= 3.35˚A andt= 0.3 eV, we find that\nqTF\nkF=¯tαeeΛ≃0.38αeeΛ. (25)\nThe mean-field theory calculations for the two-band\nmodel of bilayer graphene follow precisely the same lines\nas in the single-layer case. Eqs. (5) and (8) become:\nB0(k) =−/integraldisplayd2k′\n(2π)2V(S)\nk−k′f+(k′) (26)\nand\nBeq(k) =k2\n2mu2(k)\n−/integraldisplayd2k′\n(2π)2V(D)\nk−k′f−(k′)u2(k′).(27)\nIn Eq.(27) uJ(k) = (cos(Jϕk),sin(Jϕk)) specifies the\nk-dependence of the direction in pseudospin-space of\nthe band-structure contribution to the effective mag-\nnetic field. In the single-layer case the chirality Jhas\nthe valueJ= 1 and u1(k) =ˆk. It follows that only\nthe pseudospin-dependent part of the Hartree-Fock self-\nenergy differs between single-layer and bilayer cases:\nΣ(k) =−/integraldisplaykmax\n0dk′\n2πk′f−(k′)V(D)\n2(k,k′).(28)\nBecause the two low-energy sites which comprise the bi-\nlayer’s pseudospin are in different layers, the pseudospin-\ndependent self-energy is proportional to the interlayer\ninteraction V(D). The change from the m= 1 angular\ncomponent of the interaction in the single-layer case to\nthem= 2 component in the bilayer case follows imme-\ndiately from the J= 1 toJ= 2 band-chirality change.\nIn Eq. (28)\nV(D)\nm(k,k′) =/integraldisplay2π\n0dθ\n2πe−imθv(/radicalbig\nk2+k′2−2kk′cos(θ))\n×exp(−d/radicalbig\nk2+k′2−2kk′cos(θ)).(29)5\nUsing Eq. (13) we can rewrite Eq. (28) as\nΣ(k) =1\n4π/integraldisplaykmax\nkFdk′k′V(D)\n2(k,k′).(30)\nIndimensionlessunitswehave(scalingenergieswith vkF,\nwhich is notthe Fermi energy!)\nΣ(x)\nvkF=1\n2αee/integraldisplayΛ\n1dx′x′¯V(D)\n2(x,x′),(31)\nwhere, as in the single-layer case, all wavevectors have\nbeen rescaled with kF,i.e.x=k/kF,x′=k′/kF, and\nwhere we have introduced the dimensionless interaction\n¯V(D)\nm, which isV(D)\nmmeasured in units of 2 πe2/(ǫkF).\nThe dependence of Σ( k) onk/kFfor the bilayer case is\nillustrated in Fig. 3 for αee= 0.5 and Λ = 10 and 100. In\nthis plot we have shown Σ( k) for two different values of\nthe screening parameter λdefined in Eq. (16): λ= 10−2,\ncorresponding to an essentially unscreened Coulomb po-\ntential, and λ= 1 corresponding to the Thomas-Fermi\nscreened potential. The small kbehavior is not strongly\ninfluenced by screening and can be understood analyti-\ncally. For large Λ we can expand the exponential that\nenters in Eq. (29) in powers of ¯d/Λ:\nexp/parenleftbigg\n−¯d\nΛ/radicalbig\nx2+x′2−2xx′cos(θ)/parenrightbigg\n→1−¯d\nΛ/radicalbig\nx2+x′2−2xx′cos(θ)\n+1\n2/parenleftbigg¯d\nΛ/parenrightbigg2\n[x2+x′2−2xx′cos(θ)]+O((¯d/Λ)3).(32)\nFor unscreened Coulomb interactions\n¯V(D)\nm(x,x′) =/integraldisplay2π\n0dθ\n2πe−imθ 1/radicalbig\nx2+x′2−2xx′cos(θ)\n×exp/parenleftbigg\n−¯d\nΛ/radicalbig\nx2+x′2−2xx′cos(θ)/parenrightbigg\n.\n(33)\nThis implies that the expansion of ¯V(D)\nm(x,x′) starts at\nsecond order in ¯d/Λ form≥1: the term proportional\nto/radicalbig\nx2+x′2−2xx′cos(θ), which is first order in ¯d/Λ, is\ncanceledbythe denominatorin the integrandin Eq.(33).\nThus the angular average in Eq. (33) of this term would\ngive us a finite result only for m= 0.\nDisgarding this term, we obtain the following expan-\nsion of¯V(D)\nm(x,x′) in powers of ¯d/Λ valid for m≥1 :\n¯V(D)\nm(x,x′)→/bracketleftBigg\n1+1\n2/parenleftbigg¯d\nΛ/parenrightbigg2\n(x2+x′2)/bracketrightBigg\n¯Vm(x,x′)\n−1\n2xx′/parenleftbigg¯d\nΛ/parenrightbigg2/bracketleftbig¯Vm+1(x,x′)+¯Vm−1(x,x′)/bracketrightbig\n.\n(34)\nHere the coefficients ¯Vm(x,x′) are given by Eq. (18). Us-\ning this expression in Eq. (31) we find\nΣ(x)\nvkF=1\n2αee/integraldisplayΛ\n1dx′x′¯V2(x,x′)+1\n4αee/parenleftbigg¯d\nΛ/parenrightbigg2\nx2/integraldisplayΛ\n1dx′x′¯V2(x,x′)\n+1\n4αee/parenleftbigg¯d\nΛ/parenrightbigg2/integraldisplayΛ\n1dx′x′3¯V2(x,x′)−1\n4αee/parenleftbigg¯d\nΛ/parenrightbigg2\nx/integraldisplayΛ\n1dx′x′2[¯V1(x,x′)+¯V3(x,x′)]. (35)\nTo understand the limit x→0 of the bilayer Hartree-\nFock self-energy in Eq. (35) we use that\n¯Vm(x→0,x′) =xm\nx′m+11·3·5·...·(2m−1)\n2mm!.(36)\nInserting this expression in Eq. (35), carrying out the\nintegrations over x′, and neglecting terms that go to zero\nfaster than x2forx→0, we finally find\nΣ(x→0)\nvkF=3\n16αeex2Λ−1\nΛ\n−1\n32αeex2¯d2Λ−1\nΛ2+O(x3).(37)\nNote that the first term is finite in the limit Λ → ∞(zerodoping limit), while the second term which contains the\nlayer separation dependence goes to zero. As in the sin-\ngle layercase, the dominant contribution to the exchange\nself-energy at small wavevectors originates from interac-\ntions with states deep in the negative energy sea which\nare not sensitive to screening.\nThe renormalized mass m⋆in bilayer graphene is de-\nfinedvia\nkF\nm⋆≡∂\n∂k/bracketleftbiggk2\n2m+B0(k)+Σ(k)/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=kF\n=kF\nm+1\n2vαee∂ΓΛ(x)\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=1. (38)\nRestoring the pseudospin-independent self-energy contri-6\nFIG. 3: (Color online) The pseudospin-dependent part of the\nHartree-Fock self-energy Σ( k) (in units of vkF) for bilayer\ngraphene as a function of k/kF(for wavevectors up to the cut-\noff) forαee= 0.5. The data labeled by the dashed (red) line\ncorrespond to λ= 10−2in Eq. (16), while the data labeled by\nthe solid (blue) line correspond to λ= 1. Top panel: Λ = 10.\nThe vertical (black) line indicates the point k=kF. Bottom\npanel: Λ = 100.\nbution, which isidenticalin singlelayerand bilayercases,\nwe obtain that\nΓΛ(x)≡/integraldisplayΛ\n1dx′x′¯V(D)\n2(x,x′)−2/integraldisplay1\n0dx′x′¯V0(x,x′)\n−/integraldisplayΛ\n1dx′x′¯V0(x,x′). (39)\nIt follows that\nm⋆\nm=1\n1+1\n2αee¯tΛ∂ΓΛ(x)\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=1.(40)\nwhere¯t, defined aboveEq.(25), isthe samequantitythat\nFIG. 4: (Color online) The Hartree-Fock renormalized mass\nm⋆/mfor bilayer graphene [calculated from Eq. (40)] as a\nfunctionof αeeforλ= 10−2andvariousvaluesofΛ. Fromtop\nto bottom, Λ = 10 (squares), 20 (circles), 50 (triangles), an d\n100 (hexagons). Note how interactions suppress the quasipa r-\nticle effective mass.\nenters the Thomas-Fermi screening vector. We plot the\nratiom⋆/mobtained from this expression as a function\nofαeeforλ= 10−2in Fig. 4 while in Fig. 5 we plot\nm⋆/mas a function of density for various values of the\nscreening parameter λ. In this last plot we have taken\ninto account the spin and valley degeneracy factor g= 4\nof graphene by letting qTF→gqTF.\nIV. SUMMARY AND DISCUSSION\nIn summary, we have presented a comparative study\nof Fermi surface effective mass renormalization in single-\nlayer and AB-stacked bilayer graphene. Interband ex-\nchange adds a new twist to electron-electron-interaction\nvelocity renormalization physics compared to ordinary\ntwo-dimensional electron systems7,14, one which always\nleads to strongly enhanced velocities and correspond-\ninglysuppressed masses. We have shown analytically [see\nEq. (37)] and numerically (see Fig. 5, especially the in-\nset) that although this mechanism causes the mass to\napproach zero at low carrier densities (Λ → ∞) in single-\nlayer graphene, the renormalized bilayer mass remains\nfinite. The two cases are different because of the differ-\nence in the pseudospin-chirality of their quasiparticles,\nJ= 1 in the single layer case and J= 2 in the bilayer\ncase. Although the bilayer mass is finite, it is still esti-\nmated to be very strongly suppressed, depending on the\napproximation used for screening.\nThis behaviorshould be contrastedwith what happens\nin the case of conventional parabolic-band electrons liq-\nuids in Si-MOSFETs15,16or in GaAs17, where the quasi-7\nFIG. 5: (Color online) The Hartree-Fock renormalized mass\nm⋆/mfor bilayer graphene [calculated from Eq. (40)] as a\nfunction of density n(in units of 1012cm−2) forαee= 0.5.\nThe (blue) circles correspond to λ= 1, the (red) squares to\nλ= 10−2, while the(green)triangles to λ= 10−4. Intheinset\nwe have reported log10(m⋆/m) as a function of log10(n) for\nλ= 10−2andλ= 10−4. Note that the bilayer renormalized\nmass always saturates at a finite value for n→0.\nparticle velocity is generally suppressed by interactions\n(except in a narrowinterval of very high densities – small\nvalues of the Wigner-Seitz density parameter rs– which\ndepends in a complicated way on extrinsic effects such as\nthe quantum-well finite thickness14). The true value for\nm⋆as a function of density for graphene bilayers proba-\nbly occurs for a value of the screening parameter λ∼0.5,\nwhich represents a compromise between using Thomas-\nFermi theory and complete neglecting screening effects.\nIt should be emphasized however that a reliable numer-\nical estimate for m⋆as a function of density in bilayerslies outside the scope of this work and, as in the ordinary\ntwo-dimensional electron gas14,19,20, may ultimately re-\nquire elaborate calculations that are informed by quan-\ntum Monte Carlo numerical studies.\nWe conclude by commenting briefly on the possibil-\nity of broken symmetry states in single-layer and bi-\nlayer graphene systems. Enhanced quasiparticle veloc-\nities tend to increase the kinetic energy cost of mov-\ning the chemical potential further away from the Dirac\npoint. The same physics which leads to velocity en-\nhancement therefore also tends to suppress spontaneous\nspin-polarization, the type of instability most often con-\ntemplated7,18in ordinarylow-carrier-densityelectrongas\nsystems. Because of the pseudospin structure of single-\nlayer and bilayer graphene, interactions instead favor\nchiral-symmetry-breaking-type scenarios, as we have ex-\nplained elsewhere6. Chiral symmetry breaking, which\nwould lead to a spontaneous gap at the Dirac point and\nin the bilayer case to spontaneous charge transfer be-\ntween layers6, is most likely to occur in bilayer graphene.\nIf these instabilities occur, they are confined to a region\nof low carrier density6and the discussion of the current\npaper would apply only outside of this regime. In bi-\nlayer graphene the occurrence or absence of chiral sym-\nmetry breaking may depend on higher neighbor inter-\nlayer hopping processes not included in the model dis-\ncussed here. Recent lattice Monte Carlo calculations by\nDrut and L¨ ahde21suggest that these type of instabilities\neven occur for the J= 1 single-layer case.\nAcknowledgments\nM.P. was partly supported by the CNR-INFM “Seed\nProjects”. Work in Austin was supported by the NRI\nSWAN project, by the Welch Foundation, and by the\nNational Science Foundation under grant DMR-0606489.\n∗Electronic address: m.polini@sns.it\n1For recent reviews see A.H. Castro Neto, F. Guinea,\nN.M.R. Reres, K.S. Novoselov, and A.K. Geim, Rev. Mod.\nPhys.81, 109 (2009); A.K. Geim and A.H. MacDonald,\nPhys.Today 60, 35(2007); A.K.Geim andK.S.Novoselov,\nNature Mater. 6, 183 (2007); T. Ando, J. Phys. Soc. Jpn.\n74, 777 (2005).\n2J. Gonz´ ales, F. Guinea, and M.A. Vozmediano, Phys. Rev.\nB59, R2474 (1999); M.I. Katsnelson, Eur. Phys. J. B 52,\n151 (2006); O. Vafek, Phys. Rev. Lett. 98, 216401 (2007).\n3Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H.\nMacDonald, Phys. Rev. Lett. 98, 23660 (2007).\n4M. Polini, R. Asgari, Y. Barlas, T. Pereg-Barnea, and\nA.H. MacDonald, Solid State Commun. 143, 58 (2007); M.\nPolini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea,\nand A.H. MacDonald, Phys. Rev. B 77, 081411(R) (2008).\n5E.H. Hwang and S. Das Sarma, Phys. Rev. B 75205418(2007); S. Das Sarma, E.H. Hwang, and W.-K. Tse, Phys.\nRev. B75121406(R) (2007); E.H. Hwang, B. Y.-K. Hu,\nand S. Das Sarma, Phys. Rev. B 76115434 (2007).\n6H. Min, G. Borghi, M. Polini, and A.H. MacDonald, Phys.\nRev. B77, 041407(R) (2008).\n7G.F. Giuliani and G. Vignale, Quantum Theory of the\nElectron Liquid (Cambridge University Press, Cambridge,\n2005).\n8Whenkmaxis introduced, B0(k) is no longer independent\nofkin the undoped n→0 limit. This property is actu-\nally an artifact of the continuum model; the exchange self-\nenergy of an undoped system is local for a neutral lattice\nmodel because of particle-hole symmetry. For a discussion\nof this point see Y. Barlas and A.H. MacDonald, Proceed-\nings of HMF-18, to be published in Int. J. of Mod. Phys.\nB (2009). The cut-off related k-dependence of the neutral\nsystemB0(k) does not play an important role in velocity8\nrenormalization.\n9The artifacts associated with the neglect of screening do\nnot appear in the small kexpansion.\n10E. McCann and V.I. Fal’ko, Phys. Rev. Lett. 96, 086805\n(2006).\n11K.S. Novoselov, E. McCann, S.V. Morozov, V.I. Fal’ko,\nM.I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A.K.\nGeim, Nature Phys. 2, 177 (2006).\n12S. Viola Kusminskiy, J. Nilsson, D.K. Campbell, and A.H.\nCastro Neto, Phys. Rev. Lett. 100, 106805 (2008); S. Vi-\nola Kusminskiy, D.K. Campbell, and A.H. Castro Neto,\narXiv:0805.0305v1.\n13E.H. Hwang and S. Das Sarma, Phys. Rev. Lett. 101,\n156802 (2008).\n14R. Asgari, B. Davoudi, M. Polini, G.F. Giuliani, M.P. Tosi,\nand G. Vignale, Phys. Rev. B 71, 045323 (2005).\n15V.M. Pudalov, M.E. Gershenson, H. Kojima, N. Butch,\nE.M. Dizhur, G. Brunthaler, A.Prinz, andG. Bauer, Phys.Rev. Lett. 88, 196404 (2002).\n16A.A. Shashkin, S.V. Kravchenko, V.T. Dolgopolov, and\nT.M. Klapwijk, Phys. Rev. B 66, 073303 (2002); A.A.\nShashkin, M. Rahimi, S. Anissimova, S.V. Kravchenko,\nV.T. Dolgopolov, and T.M. Klapwijk, Phys. Rev. Lett. 91,\n046403 (2003).\n17Y.-W.Tan, J.Zhu,H.L.Stormer, L.N.Pfeiffer, K.W.Bald-\nwin, and K.W. West, Phys. Rev. Lett. 94, 016405 (2005).\n18J. Nilsson, A.H. Castro Neto, N.M.R. Peres, and F.\nGuinea, Phys. Rev. B 73, 214418 (2006).\n19R. Asgari and B. Tanatar, Phys. Rev. B 74, 075301 (2006).\n20For recent quantum Monte Carlo results on the effective\nmass in the ordinary two-dimensional electron gas see M.\nHolzmann, B. Bernu, V. Olevano, R.M. Martin, and D.M.\nCeperley, arXiv:0810.2450v1.\n21J.E. Drut and T.A. L¨ ahde, Phys. Rev. Lett. 102, 026802\n(2009)."    },    {        "title": "1401.8089v1.Interaction_between_graphene_and_copper_substrate__The_role_of_lattice_orientation.pdf",        "content": "1 \n Interaction between graphene and copper substrate: \nThe role of lattice orientation  \nOtakar Frank1,*, Jana Vejpravov a2, Vaclav Hol y3, Ladislav Kavan1, and Martin Kalbac1 \n1 J. Heyrovsky Institute of Physical Chemistry of the AS CR, v.v.i., Dolejskova 2155/3, CZ-182 \n23 Prague 8, Czech Republic  \n2 Institute of Physics of the AS CR, v.v.i., Na Slovance 1999/2, CZ -182 21 Prague 8, Czech \nRepublic  \n3 Faculty of Mathematics and Physics, Charles University  in Prague , Ke Karlovu 5, CZ -121 16 \nPrague 2, Czech Republic  \n \nAbstract  \nWe present a comprehensive  study of graphene grown by chemical vapor deposition on copper \nsingle crystals  with  exposed  (100), (110) and (111) faces . Direct examination of the as -grown \ngraphene by Raman spectroscopy using a range of visible excitati on energies and  microRaman \nmapping shows  distinct strain and doping levels for individual Cu surfaces . Comparison of \nresults from Raman mapping with X -ray diffraction techniques and A tomic F orce M icroscopy  \nshows it is neither the crystal quality nor the surface topography responsible for the specific strain \nand doping values, but it is the Cu lattice orientation itself. We also report an exceptionally narrow Raman 2D  band width caused by  the interaction between gr aphene and metallic substrate.  \n                                                           \n* Corresponding author. Tel. +420 266 053 446. email: otakar.frank@jh -inst.cas.cz  (Otakar Frank)  2 \n The appearance of this extremely narrow 2D band with full -width -at-half maximum (FWHM) as \nlow as 16 cm-1 is correlated with flat and undoped regions on the Cu(100) and (110) surfaces. \nThe generally compressed (~ 0.3% of strai n) and n- doped (Fermi level shift of ~250 meV) \ngraphene on Cu(111) shows the 2D band FWHM minimum of ~20 cm-1. In contrast, graphene \ngrown on Cu foil under the same conditions reflects the heterogeneity of the polycrystalline \nsurface and  its 2D band is  accordingly broader with FWHM > 24 cm-1. \n 1. Introduction  \nGraphene [ 1], a monoatomically thin membrane of sp\n2-hybridized covalently bonded carbon \natoms, attracts enormous research efforts due to its unique physical prope rties, such as high \ncarrier mobility, superior thermal conductivity, high optical transparency and extreme mechanical \nproperties  [2-5]. Many  applications exploiting these properties have been envisaged or already \ntested ranging from mechanical resonators  [6], integrated circuits  [7], solar cells  [8, 9], \ntransparent touch displays  [10] to terahertz detection  [11]. However, prospective application s of \nsuch devices  require large and high quality graphene layers . In this regard , chemical vapor \ndeposition (CVD) on metals is the most promising way of large -scale production of continuous \ngraphene layers  [12-14]. Even though general principles like the role of carbon solubility in the \ncatalyst, pressure or temperature in the CVD chamber are established  [15-18], a detailed \nunderstanding of the interplay between the surface structure of the metal and the resulting \ngraphene is still lacking. Copper, in the form of a thin polycrystalline foil, is the most widely \nemployed catalyst due to its low cost and ease- of-use for obtaining graphene monolayers with \nreasonable quality, with single -crystalline graphene domains in sizes of several millimetres  [19]. \nThe influence of the crystalline nature of Cu foils on the domain sizes and orientation, layer \nnumber and quality of the grown graphene has been shown recently  [20-24]. To avoid the 3 \n presence of Cu grain boundaries and transition between individual Cu lat tice orientations, both of \nwhich largely affect the resulting graphene , epitaxial growth on surfaces with defined long- range \ncrystalline order is particularly appealing  [15, 18, 25- 32]. The studies c onducted on single \ncrystals may provide direct information about the  effects of Cu lattice  on the graphene growth as \na function of pressure/temperature , without the perturbations from Cu grain boundaries.  \nRaman spectroscopy is  undoubtedly  one of the prominent methods  used for characterization of \ngraphene  [33, 34] . It can distinguish between single -layer, bilayer and multilayer graphene, it is \nhighly sensitive to doping  [35-37] and mechanical deformation  [38-41]. Moreover, the \ncapabilities o f Raman spectroscopy of CVD graphene are vastly  expanded by using precursors \nwith varying content of 12C and 13C isotopes  [42, 43]. The usual way of Raman characterization \nof CVD graphene transferred to Si/Si O2 suffers from the transfer process (ripples or even fissures, \ncontamination, etc.), and also the substrate can substantially modify both the electronic and \nphonon structure of the overlaying graphene. Hence the knowledge of Raman characteristics of as-grown graphene is of utmost importance  not only for a viable ex post  monitoring of the growth \nprocess but also from the fundamental point of view regarding the relationship between substrate and graphene. With a  few exceptions  [20, 27, 44, 45] , most of the Raman spectra are measur ed \non graphene transferred to Si/SiO\n2 substrate  instead of directly on Cu, due to a much larger \nRaman scattering intensity  and lower spectral background on Si/SiO 2. The differences in \ngraphene growth conditions (including substrate orientation), are in turn smeared and not \nresolved .   \nIn this study we focused on graphene grown on Cu(100), (110) and (111) single crystals by a \nlow pressure CVD from methane with either 12C or 13C isotope. Large -area microRaman \nmapping together with a thorough statistical ev aluation of the data (G and 2D band shifts, \nlinewidths and intensities, 2D band dispersion) are compared against the individual surfaces and 4 \n their qualities  and orientations  obtained by  X-ray diffraction ( Laue diffraction method, real and \nreciprocal space mapping)  and AFM topography . All three Cu surface  orientations show a \ndifferent impact on the overlying graphene concerning the amount and uniformity of strain and \ndoping. However, the most notable result consists in a remarkable and unprecedented narrowin g \nof the 2D band, especially for the graphene grown on Cu(100). The possible reasons are \nuniformity of the strain/doping fields and interaction between graphene and the substrate.  \n \n2. Experimental  \n2.1 Sample preparation \nThe  graphene samples were synthesized using CVD as reported previously [ 43]. In brief: The \nCu single crystals (MaTecK, purity 99.9999%, orientation accuracy < 0.1°) were heated to 1000 °C and annealed for 20 min under flowing H\n2 (50 standard cubic centimeter per minute \n(sccm)). Then the crystals were exposed either to  1 sccm  12CH 4 or 13CH 4 for 20 minutes leaving \nhydrogen gas on with the same flow rate  of 50 sccm . Finally the substrate was cooled down still \nunder H 2 flow by opening the furnace and removing the reaction tube from the heated zone (cool \ndown time to 500°C is ~  5 minutes) . The pressure was kept at 0.35 Torr during the whole growth. \nAll three Cu single crystals were subjected to the CVD at the same time, close to each other in \nthe furnace to avoid differences in the growth conditions.  \n 2.2 Sample characterization The copper single -crystals were checked by various X -ray diffraction techniques. First, the \ncrystallographic orientation was verified by the standard Laue method using a CHIRANA – Micrometa device with Cu Kα radiation. The single -crystals were mounted with the ( 100), (110) \nand ( 111) plane, respectively, p erpendicular  to the beam path and the diffraction pattern was 5 \n recorded in the back scattering geometry. The beam size was fixed to 3 mm in diameter. Quality \nof each sample was further investigated by mapping the characteristic diffractions (hkl) of the copper fcc  structure in the reciprocal space: ( 400), (200) for th e (100) face, ( 220), (222) for the \n(110) face , and (111), (222) for the [111] face. The reciprocal maps were recorded on the X’Pert \nPRO MRD XL in high -resolution set -up, the multiple symmetric 2Theta -Omega scans were \nrecorded. The data were analyzed using Matlab software considering a mosaic block model with a coherent crystal -truncation rod simulated by a two- dimensional PearsonVII function yielding \nthe characteristic vertical and lateral size of the mosaic blocks and their mean misorientation, respectivel y [46, 47] . The influence of multiple heating  cycles (equal to the conditions of the \nCVD) on the crystallinity was investigated by the reciprocal mapping. In order to probe the grain distribution within the ingot area, the mapping of the intensity of a single characteristic diffraction for each orien tation was performed in the real space. First, the orientation of each \nsingle -crystal was optimized to maximum intensity of the diffraction at a single point and then \nthe intensity of the diffraction was mapped within the sample area (10 x 10 mm\n2) with a s tep of 1 \nmm in the x and y directions, respectively. AFM images were recorded using a Dimension Icon \nMicroscope (Bruker) with ScanAsyst Air tips in the PeakForce tapping mode. \nThe Raman spectra were excited by He- Ne (1.96 eV) or Ar+-Kr+ (2.71, 2.54, 2.41, 2.34 or 2.18 \neV) lasers and acquired by a LabRam HR spectrometer (Horiba Jobin –Yvon) with a pixel -to-\npixel spectral resolution of approximately 1 to 2 cm-1 depending on the excitation energy (from \n1.96 to 2.71 eV). The spectrometer was interfaced to a micr oscope (Olympus, 100x objective) \nand the laser power was kept lower than 1 mW under the objective to avoid heating of the \nsample. Intensity response of the CCD detector was calibrated by tungsten halogen light source HL-2000- CAL (Ocean Optics) for each excitation. Raman peaks were fitted by Lorentzian line \nshapes for the analysis with a prior subtraction of a linear background in predefined spectral 6 \n ranges (1200 -1700 cm-1 for D and G band fitting, 2400- 2900 for 2D band fitting). The accuracy \nof peak position determination, as acquired by repeated measurements on one point is ~ ±0.6 cm-1 \nfor 1.96 eV excitation, which corresponds well to the pixel -to-pixel spectral resolution. The \naccuracy of FWHM determination was found to be also ±0.6 cm-1. Raman mapping was  \nconducted with lateral steps of 1 µm (both in X and Y directions) on rectangular areas with \nvarying sizes -  minimum number of 110 data points and 200 data points on average for each map. \nFor details on the map sizes, see Table S1 (Supplementary Information).  Accumulation time for \neach spectral window was 120 s. The samples were measured under ambient conditions, and \nstored in a vacuum desiccator.  \nDue the relatively low intensity of Raman  features of graphene on Cu, 13C was used to check \nwhether there are no interfering peaks from the background which might contribute to the analysed bands (D, G, and 2D). No such peaks were observed, apart from the vibration of \nmolecular oxygen at 1554 cm\n-1. The spectra acquired on 13C graphene were used in the same \nmanner as those from 12C, only the band positions were recalculated for the sake of direct \ncomparability according to the equation:  \n(ω0-ω)/ ω 0 = 1- [(12 + c 0)/(12+c)]1/2       (1) \nwhere ω0 is th e frequency of the particular Raman mode in the 12C sample, c  = 0.99 is the \nconcentration of 13C in the enriched sample, and c0 = 0.0107 is the natural abundance of 13C.  \nThe determination of electron concentration (for electron doping) from the G band position was \ndone by approximating the data from ref . [35] by a quadratic polynomial ∆𝑃𝑜𝑠(𝐺)=\n−0.986𝑛2+9.847𝑛 , where ∆𝑃𝑜𝑠(𝐺) is the shift from the “strain” line (see Results and \nDiscussion) and n is the electron concentration in 1013 cm-2. The corresponding Fermi level shift \nwas determined as  ∆𝐸𝐹=�𝜋𝑛𝜐𝐹2, where υ F is the Fermi level velocity.     7 \n 3. Results and Discussion  \nGraphene samples were synthesized using CVD as reported previously [ 48] on Cu (100), (110) \nand (111) single crystals and on a polycrystalline Cu foil for comparison. Cu surface quality was \nchecked by XRD real and reciprocal space mapping , and Laue diffraction  method. \nThe first simple check of the crystal orientation and quality was carried out by the Laue \nmethod ( typical Laue patterns are shown in Figure S1 , Supplementary Information). The patterns \nreflect  the characteristic Laue class symmetr y: m-3m expected for the (100), (110) and (111) \nplanes in the fcc structure  (space group no. 225) . The reciprocal maps for all orientations around \nthe two selected reflections are presented in Figures S2 -S4 (Supplement ary Information ). The fit \nto the simple mosaic model is also shown and the resulting values are summarized in Table 1. \nThe characteristic vertical size of the crystal block is identical for all directions within the \nexperimental error, while the lateral size varies between 400 – 800 nm. The mean misorie ntation \nangle is ca. 0.2°, 0.1°and 0.2° for the (100), (110) and (111) orientations, respectively. \nConsidering a typical size of a graphene grain (20 -  50 µm) [43, 48] , the mean block size is at \nleast one order of magnitude smaller. H ence, the fine structure variations of the copper crystal \n(within a single grain) should significantly influence neither the growth nor the interaction of the \ngraphene with the copper surface.  \nThe latter observation is also supported by the studies of the annealed samples. The reciprocal \nmaps for the ( 100) and ( 110) orientation before and aft er annealing (1000° C, H 2) are depicted in \nFigure S 5 (Supplementary Information) . For the ( 100)-oriented single -crystal, the diffuse \nscattering becomes significantly suppressed after annealing, which suggests improvement of the \ncrystallinity. For the ( 110)-crystal, the diffuse scattering after annealing is almost negligible, 8 \n which also points to a substantial improvement of the structure. The observed behavior is in \ncorrespondence with the well -know n effect of polygonization in copper upon annealing [ 49]. \n \nTable 1: Results of the XRD reciprocal map analysis using the simple mosaic model.  \n Vertical block size (nm)  Lateral block size (nm)  Misorientation (o) \n(100) 100 ± 20  400 ± 20  0.20  ± 0.05  \n(110) 100 ± 20  800 ± 20  0.10  ± 0.05  \n(111)  100 ± 20  600 ± 20  0.20  ± 0.05  \n Finally, the distribution of the macroscopic grains within the ingot was analyzed using the real \nspace mapping, the results are shown in Figure S 6 (Supplementary Information) . Typically, the \ningot contains several grains of few mm in diameter with misorien tation below 0.4\no. \nGraphene grown on the copper surfaces was studied in detail by microRaman spectroscopy. \nThe samples were measured freshly after graphene growth or stored under vacuum in the  \nmeantime. However, to check  for the effects of possible oxidation, e.g. during manipulation or \nmeasurement, i) we varied the order of measurement of samples in every experimental batch (see \nTable S1, Supplementary Information), and ii) we controlled the appearance of the samples under \noptical microscope. As an example, Figure S7 (Supplementary Information) shows the difference between graphene on Cu(111) freshly grown and after one week of storing under ambient \nconditions. The optical microphotograph of freshly grown graphene is esse ntially featureless due \nto high reflectivity of the Cu surface, with the exception of very faint and sparse darker spots \nbelonging to bilayer graphene patches. The area looks similarly  after 12 hours of Raman \nmapping. On the  other  hand, more features can be observed on the aged sample, where dark spots \nand thin lines appear due oxidation of the Cu surface, mainly on the defective grain boundaries  \nbut with an only slow progression into the grain interiors, in line with the previous observations 9 \n [50, 51] . Hence, the short (hours)  copper oxidation causes only marginal changes of  the Raman \nspectra. However, a partial and local influence cannot be ruled out, as proposed earlier [ 30]. \nFigure 1 shows typical Raman spectra of  the G and 2D (or G’) regions of graphene as -grown \non Cu(100) and (111) acquired using laser energies  from 1.96 eV (633 nm) to 2.71 eV (457 nm). \nThe measurements were taken on spots, which correspond to the most frequently appearing \nspectra (in terms of peak positions, widths and frequencies) for the particular Cu surface based on statistical evaluation o f Raman maps (see further). The spectra of graphene grown on Cu(110) are \ngenerally similar to those on Cu(100) and are shown in Fig. S 8 (Supplementary Information). \nThe D band region is not shown due to the absence  or very low intensity  of the  D peak in these \nspectra. It should be noted that the D band is present occasionally throughout the samples on the graphene grain boundaries or ad- layers [ 52]. Throughout the following text, the frequencies of G \nand 2D bands will be referred to as Pos(G) and Pos(2D), respectively, full -widths -at-half-maxima  \nas FWHM(G) and FWHM(2D), and I(G) or I(2D) denote  intensities of the bands expressed as \nintegrated areas. Lorentzian  lineshapes wer e used for fitting. If not stated otherwise, the results \nreferring to a certain Cu orientation are meant for graphene measur ed as -grown on the particular \nsurface.  10 \n  \nFigure 1. Raman spectra of graphene grown on Cu(100) and (111) single crystals. Excitation \nenergies are given for each spectrum, the spectra are normalized to the G band intensity for each \nCu surface, the corresponding zoom factors are labeled in the plots. \n Fig. 1  shows that  there are several differences in the spectra of graphene grown on the two \ndifferent Cu substrates. The I(2D)/I(G)  intensity ratio ( evaluated as integrated areas , 4 different \nmeasurements for each excitation ) is higher for Cu(100)  and Cu(110) , where it  amounts to 5.2 ± \n0.7 and 5.0 ± 0.7, respectively, with no statistically significant dependence on the laser excitation energy.  The I (2D)/I (G) for graphene on Cu(111) is 2.0 ± 0.4, again without a proportionality  to \nthe laser excitation  energy . Furthermore, the G band of graphene grown on Cu(111) is upshifted \nand narrowed compared to that grown on Cu(100), while the 2D band is upshifted but broadened \n11 \n for graphene grown on Cu(111). Selected relations between frequencies, FWHM  and I(2D)/I(G) \nratios are plotted in Figure 2 for all four Cu surfaces , as acquired from the Raman mapping using \nthe 1.96 eV laser excitation  energy . The Raman features of graphene on Cu(100) are grouped \nsimilarly to Cu(1 10) in terms of intensities and frequencies (panels a  and b in Fig.  2) and both \nhighly contrast with the features of graphene grown on Cu(111). The results on polycrystalline \nCu foil are usually in between the values obtained on Cu (100) and Cu(111)  surfaces . The \naverage d Raman spectra obtained from the mapping are shown in Figure S 9 (Supplementary \nInformation).  \n    \n 12 \n  \nFigure 2.  Correlations between the monitored parameters of the G and 2D peaks fitted by \nLorentzian lineshapes in Raman spectra of as -grown graphene on Cu(100) (blue), Cu(110) (red), \nCu(111) (green) and polycrystalline Cu foil (black). Each data point represents a m edian of all \nmeasurements (> 200 map points on average) on a given substrate from one CVD batch. The \nOffset(G) parameter is derived from Pos(G) by subtracting the G band value on the strain line for a constant Pos(2D), see text. The “error” bars represent the first (Q1) and third quartiles (Q3) of each dataset. Note: Raman shifts from CVD batch using \n13CH 4 are recalculated via Eq. 1 (see \nExperimental Section) to be comparable with samples produced from 12CH 4. Laser excitation \nenergy is 1.96 eV.  \n \nBesides  the extremely narrow FWHM(2D) (panels d and f) , which will be discussed later  in the \ntext, most of the differences shown in Fig. 2 can be understood in terms of  the current theories \n13 \n and previous experiments. The G band originates  from a conventional first  order Raman \nscattering process and corresponds to the in- plane, doubly degenerate phonon mode (transverse \n(TO) and longitudinal (LO) optical) with E 2g symmetry  at the Brillouin zone center  (Γ point)  \n[53]. Its frequency and width are influenced by doping  [35, 36]  and stress  [38], while its \nintegrated intensity was assumed to be dependent only on the excitation energy ( EL) as I(G) ∝ 𝐸𝐿4 \n[54, 55]  or I(G) ∝𝐸𝐿2 [56]. However, as has been shown recently, quantum interference effects \nplay a crucial role in the G band intensity, where blocking of excitation pathways dramatically \nincreases I(G) under strong doping [ 36, 57] . The increase in Pos(G) for both electron and hole \ndoped graphene is caused by  a non -adiabatic removal of the Kohn anomaly at Γ point  [37, 58] , \nand s imultaneous FWHM(G) decrease is caused by the Pauli blocking of phonon decay into \nelectron -hole pairs  [37]. For suspended graphene, the mean values of Pos(G) = 1580 cm-1 and \nFWHM(G) = 14 cm-1 can be regarded as experimental benchmarks for nearly undoped graphene \nwith the estimated upper bound of carrier density n of ~2 × 1011 cm-2 [59]. Mechanical s tress \ncauses G band downshift under tension and upshift under compression with a rate of 57 cm-1/% \nfor biaxial strain  [60]. Under uniaxial strain the G bands splits into two components G- and G+ \nwith the shift rates of 31 and 10 cm-1/%, respectively  [38, 40] . \nThe 2D mode come s from a second -order triple  resonant process between non- equivalent K \npoints in the Brillouin zone of graphene, involving two zone -boundary , TO-derived phonons  with \nopposite momenta q and –q [ 61-63]. Numerous theoretical and experimental works , e.g.  [61, 62, \n64-68], appeared in the last decade describing various aspects of the origin and properties of this \nintense, narrow and dispersive line, yet there are still several debatable issues. The 2D band has \nbeen regarded as a single Lorentzian with FWHM of ~ 24 cm-1 for graphene on Si/SiO 2 or \nbroader  depending on the substrate and/ or environment [ 34, 69] . As it turned out recently , free \nstanding graphene exhibits narrower (~ 22 cm-1) 2D band with a bimodal lineshape [ 59, 70, 71] . 14 \n Similar subfeatures were observed also for  the unstrained graphene on SU -8 photoresist  [72] and \nin uniaxially strained monola yers [ 72-74]. In a simplified one -dimensional picture  there can be \ntwo dominant directions of the phonon wavevectors contributing to the 2D mode – along K -Γ (so \ncalled inner) or K -M (outer) high symmetry lines. Earlier works usually assumed a preferential \ncontributi on of the latter to the 2D mode [ 66, 68, 69] , but recent works point to a larger \nimportance of the inner phonons  [62, 72, 73, 75] . However, the simple one -dimensional picture is \nnow being superseded by a full two- dimensional description of the electronic bands and phonon \ndispersion and matrix elements  [62], in which the notation of inner  or outer phonons is of a \nweaker relevance [ 67]. The 2D mode is dispersive, and its frequency changes with the excitation \nenergy with the slope ∂Pos(2D)/∂ EL of approx. 100 cm-1/eV [ 34]. The 2D band is also sensitive to \ndoping and mechanical stress but these effects  manifest themselves differently  from the case of \nthe G band. Strain causes the 2D band shift in the same directions as the G band, with the shift \nrates for  biaxial strain larger by a factor of  ~  2.2-2.5 [60, 76] . While broadening and splitting \naccompanies the shift for uniaxial strain [ 72], such effects are not observed for the biaxial one \n[60]. Charge injection  causes Pos(2D) increase for hole doping with a Pos(2D) /Pos(G) shift ratio \nof ~0.5- 0.7 [35, 36, 77] , whereas electron doping causes only a negligible Pos(2D) change for n ≤ \n2 × 1013 cm-2 followed by a non -linear Pos(2D) decrease for higher n -doping levels  [35, 36] . The \nFWHM(2D) increases and I(2D) decreases upon both p - and n- doping mainly due to electron-\nelectron interactions, and also electron -phonon coupling  strength  (expressed as a dimensionless \nparameter λ ), which increase s to a small extent upon doping [ 62, 65] . Additionally, the change in \nλ causes the 2D dispersio n slope to decrease upon doping [ 78]. \nWe first consider the Pos(G) in our samples (appearing in Fig. 2a and Fig s S10c,d, \nSupplementary Information). Graphene on Cu(100) exhibits unambiguously the lowest Pos(G), with medians from the three runs between 1586- 1587 cm\n-1 with the interquartile range (IQR = 15 \n Q3-Q1, see Fig.  2) between 2.2 and 3.3 cm-1. Graphene on Cu(110)  shows  the Pos(G) slightly \nhigher and with a larger variation, medians fall between 1585 and 1590 cm-1, with IQR of 3.0 – \n8.0 cm-1. Pos(G) for graphene on  Cu(111) is consistently the highest with medians between 1596 \nand 1597 cm-1 and IQR ~ 2.1 cm-1 in the latter two runs. The first run is rather exceptional for \nCu(111)  (see Fig. 3) , when  a second group of peaks at substantially lower frequencies (both G \nand 2D) appeared, occasionally also in the same spectra. The appearance of the lower frequency \nG and 2D or the simultaneous presence of the two G and 2D bands was spatially random in the maps and their origin is not clear at moment . We assume that these low frequency peaks belong \nto well defined areas with sizes even below the laser spot, where graphene relaxed from the \notherwise compressed state (see below ) with a simultaneous formation of wrinkles. The \nhypothesis is corroborated also by larger FWHM s of both G and 2D  bands  in the lower frequency \ngroup (13.2 vs. 9.5 cm\n-1 and 35.6 vs. 27.2 cm-1 for aver age FWHM(G) and FWHM(2D) , \nrespectively) , which is another indicator  of buckling  [38]. The median Pos(G) of graphene grown \non Cu foil is 1589 cm-1 with a large IQR of 9 cm-1, i.e. being the average of the oriented surfaces \nand spanning their whole  range. The correlation between Pos(2D) and Pos(G) is summarized in \nFig. 2 a and detailed in Fig. 3. The general trends for Pos(2D) are similar to Pos(G) : the lowest \nfrequencies are for Cu(100), the highest for Cu(111) with the foil being in between . From plots in \nFig. 3, we can see that most of the variations between individual points on each surface are driven \nby strain – the ∂Pos(2D)/∂Pos(G) is consistently ~ 2.5 for the linear correlations [ 39, 60, 77 ]. In \nseveral cases, the correlation is broadened or even a second set of points with an offset in the \nPos(G) direction appears, clearly evidencing additional variations in doping  [77]. The relative \nvariation of both the strain and doping levels between the individual surfaces can be estimated from Fig s 2 and S 11 (Supplementary Information ). Considering the “strain” line with slope of 2.5 \nin the Pos(2D)/Pos(G) plot delimiting the low frequency boundary of data points from Cu(100) 16 \n and Cu(110), we can estimate the change in charge carrier concentration in our samples.  Note \nthat the few datapoints of Cu(100) and Cu(110) with G band frequency downshifted from the \n“strain” line in Fig. S 11 have a large FWHM(2D) and low I(2D)/I(G), hence they probably come \nfrom small bilayer patches sometimes appearing i n CVD graphene on Cu [ 48, 79, 80] . For low \ncarrier concentrations, the Pos(2D) change with doping is negligible compared to ∂ Pos(G) [ 35, \n36]. The  separation between the “strain” line and Pos(G) median for Cu(111) is ~ 3 cm-1, \ncorresponding to a change in carrier concentration  n of ~ 3  × 1012 cm-2 or ~ 2 00 m eV shift of the \nFermi level  (|ΔEF|), see Experimental section for details of |ΔEF| estimation . Using the same \nmethod, the two distinct sets of data points within Cu(100) marked by arrows in Fig. S 11 exhibit  \nG band upshift from the “strain” line of 4 and 8 cm-1, corresponding to |Δ EF| of ~ 250 and 350 \nmeV, respectively . We should note  that it is impossible to derive the sign of doping at such low \nlevels from this approach  due to the aforementioned small ∂Pos(2D) . On the other hand, the \ndifference in strain can be easily estimated in our  case, as ∂Pos(2D) between Cu(100) and \nCu(111)  of ~ 23 cm-1 give s ~ 0.2% of biaxial compression [ 38, 40, 60] . \nThe variation in doping is also evidenced in Figs 2b, c and e, which display the mutual \ndependencies of I(2D)/I(G), FWHM(G) and the G band frequency. Since the role of strain cannot be directly discerned in these plots, we define the Offset(G) as the Raman shift difference between the measured Pos(G) and the “strain” line at the same value of Pos(2D) for the given point. We thus neglect the subtle slope of ∂Pos(2D)/∂Pos(G) caused by doping in the Offset(G) parameter (Fig. 2b and e), because ∂Pos(2D) /∂n is < 1 cm\n-1 per 1013 cm-2 for electron doping up \nto 2×1013 cm-2, hence  it is smaller than the measurement accuracy  in the encountered range of \ndoping levels . For completeness, Pos(G) instead of Offset(G) is shown in Fig. S 10c and d. \nGraphene on both Cu(100) and Cu(110) has median I(2D)/I(G) close to 4.5 with only a small \nvariation, caused to some extent by the aforementioned distinct doped areas in Cu(100), in 17 \n contrast to graphene on Cu(111), where the median I(2D)/I(G) is in the range of 1.3- 1.7 for the \nthree data sets . The decrease of I(2D) upon doping is in accord with previous experiments  [35, \n36] and theory  [62, 65] . \n \nFigure 3. Correlation between all measured Pos(2D) vs. Pos(G) for individual surfaces (rows) \nin each CVD batch (columns). Colors of the data points are scaled according to FWHM(2D) for \nthe particular point with the scale bar in the right bottom corner. The sketch wi th arrows in the \nbottom centre depicts the trends for strain and doping, derived from earlier works [ 35, 36, 38, 40, \n60, 77] . The numbers i n bottom right corner of each plot represent the median FWHM(2D) for \nthe particular run.  \n The state of graphene on Cu(100)  should be determined from its  Pos(G) and Pos(2D) , which \nexhibit the lowest frequencies among the studied samples. Assuming Pos(G) of 1580 cm\n-1 to be \n18 \n the benchma rk value for suspended graphene [ 59, 71] , median Pos(G) for Cu(100) is upshifted by \n~ 7 cm-1. Should the difference be caused solely by strain (~ 0.1% of biaxial compression), the \ncorresponding theoretical ∂Pos(2D) would amount for ~ 18 cm-1, which is still even lower than \nthe measured ∂Pos(2D). The higher -than-expected unstrained Pos(2D) can be partially explained \nby the smaller slope of 2D band frequency dispersion with E L. Note that doping would cause the \nrelative measured ∂Pos(2D) to drop compared to the theoretical “strained” one derived from  \n∂Pos(G). In addition, the main fraction (~ 60%) of measured points on Cu(100) exhibits \nFWHM(G) of 13 -14 cm-1 (Fig. S 12, Supplementary Information) , hence we might conclude that \nthe majority of graphene on Cu(100) is essentially undoped and slightly compre ssed (~ 0.1%). \nGraphene on Cu(110) shows a similar behavior, while graphene on Cu(111) manifests additional \n0.2% of compression, i.e. ~ 0.3% in total and > 200m eV |EF| shift.  \nFew data are available on the doping level of as -grown graphene on copper. Using angle -\nresolved photoemission spectroscopy (ARPES), Walter et al.  [30] reported on n- doping of \ngraphene on Cu(111) single crystals with E F shift of ~ - 300 m eV and additional ~ - 300 m eV for \ngraphene on Cu(100) presumed to originate from intercalation of oxygen from air  [30]. The \nresults closely match our Raman data, cf.  |EF| ~ 200 m eV for Cu(111) and three distinct doping \nlevels in Cu(100) with | EF| ~ 0, 250 and 350 m eV. Siegel et al.  [81] obtained varying levels of n-\ndoping  in graphene grown on Cu foil with the average value of  2 × 1013 cm-2 corresponding to \nΔEF ~ -500 m eV, likewise by means of ARPES  [81]. Fermi level shift smaller than 110 meV was \nfound by micro- spot ARPES on graphene grown on (100) facets of a copper foil by Wilson et al. \n[24]. Using  the first-principles density functional theory (DFT), Khomyakov et al. [ 82] calculated  \nthat the differences in work functions between graphene and Cu should cause n- doping in \ngraphene with Δ EF = -170 meV. 19 \n The strain in as -grown graphene on metal substrates is due to the difference between their  \nthermal expansion coefficients . Upon cooling, the metal lattice shrinks , while the graphene lattice \nexpands . Provided there is a good adhesion of graphene on the surface, this inevitably leads  to a \nbiaxial compression in graphene. Eventually, the strain can be partially released by the formation \nof wrinkles, as was documented by STM, e.g. in graphene on Cu(111)  [83] or Ir(111) [ 84]. \nAnother interpretation of strain in graphene on Cu surface is based on the mismatch between the lattice parameter of  Cu and equilibrium C -C bond length, when superstructures from graphene at  \nCu(111) and Cu(100) are formed [ 26]. Molecular dynamics simulation of the superstructures \nshowed inhom ogeneous c ompressive strain between 0.3 and 0.6% for graphene on Cu(111) and \nmore varying strain for graphene on Cu(100) ranging from 0.3% compression to 0.2% tension [26]. Our data show strain variation  of approx. 0.2% for data sets on Cu(100) and Cu(111), and \nup to ~ 0.4% for Cu(110), see Fig s 3 and S 11. In one case, the strain on Cu(111) was interpreted \nas relaxed by wrinkling in parts of the measured area, see above. It should be noted, howev er, \nthat the  molecular dynamics calculated strain  [26] is inhomogeneous on an atomic level and thus \nshould be manifested by broader Raman peaks with a narrow frequency distribution. We observe the opposite – narrow Raman peaks but with a wider frequency span, hence documenting strain \ninhomogeneities on a scal e closer to the size of the laser spot (~ 1 µm). On the other hand, given \nthe coincidence between the MD calculated  [26] and our measured strain values, the “lattice \nmismatch” interpretation cannot be ruled out. This could be especially valid for graphene on \nCu(111), which shows a   non-negligible charge transfer, and we assume the interaction there  to \nbe stronger than between graphene and Cu(100)  or Cu(110). The small doping level as well as \nsmall strain in the latter cases point towards relaxed graphene layers with larger distance  from the \nsubstrate and hence smaller interaction.  20 \n  \nFigure 4. Maps parameters obtained from fitted Raman spectra of graphene (3rd batch) grown on \nCu(100) – top row, (110) – middle row, and (111) – bottom row. The color scales are always the \nsame for a particular parameter across the Cu faces, as depicted below the respective columns. \nThe dimensions of all the Raman maps  are the same, the scale bar is 5 µm. The laser excitation \nenergy was 1.96 eV.  \n \nBy comparing Raman maps of the monitored G and 2D band parameters (Figure 4) and AFM \nimages taken in the same experimental batch (Figure S1 3, Supplementary Information), the \ninfluence of Cu topography can be discerned in some cases. Firstly, all three copper surfaces \nexhibit flat regions alternating with steep edges of varying height, originating presumably by evaporation of the metal during the low pressure CVD growth. The spatial extent of these \ntopographic features corresponds well with the spatial distribution of G and 2D band frequencies  \n(2 leftmost columns in Fig. 4). The height of these features derived from AFM profiles (Fig. S13) \n21 \n is then directly reflected in th e span of G and 2D band frequencies for the particular surface – the \nsmallest height as well as Pos(G), Pos(2D) differences are shown for Cu(100) and Cu(111) , \nwhereas the height profile of Cu(110) shows roughness one order of magnitude larger  with the \nPos(G) and Pos(2D) spanning over 20 and 50 cm-1, resp ectively . However, no other Raman \nfeature seems to be influenced by the microscale roughness of the Cu crystal. Furthermore, there \nis no correlation between neither nanoscale nor microscale roughness of the Cu surface and the Raman spectra of grown graphene, apart f rom the 2D mode width. As can be seen from Figure 4, \nthe narrowest FWHM(2D) is confined to areas which show rather uniform Pos(G) and Pos(2D) and correspond to the flat regions on Cu surfaces (regardless the lattice orientation).  \nThe XRD measurements show  only small differences in crystalline quality between the \noriented surfaces on all scales, which confirms that the obvious variations in strain and doping between graphene on the individual surface orientations are indeed driven by the particular alignmen t of graphene and Cu lattices.  The distinct orientations of graphene with respect to \nunderlying single crystalline Cu(100), Cu(110) and Cu(111) were also shown recently in ref. [32]. \nFinally, we turn our attention to the extremely narrow 2D  band with the FWHM  as low as 16 \ncm\n-1 for graphene on Cu(100) (see Fig. 5). There are again clear differences  between the tested \nCu surfaces.  The average FWHM(2D) increases from Cu(100) to Cu(110), further to Cu(111), \nwith graphene on the Cu foil exhibiting the broadest 2D comparable to the commonly measured values with the average of ~ 31 cm\n-1. The variations be tween the surfaces can be explained in \npart by different “ uniformity ” of graphene in terms of strain and doping at the scale of the laser \nspot (~  1 µm), which can be observed as the statistical distributions in Fig. 3. The broader the \ndistribution  of data points , both horizontal (doping) and skewed (strain), the broader the 2D band, 22 \n cf. the average FWHM(2D) in individual plots in Fig. 3. In other words, the FWHM(2D) reflects \nto some extent the microscopic spatial uniformity of graphene, which is pr ojected onto the \nsubmicro-  or nanoscale, i.e. below the laser spot size. This assumption is further corroborated by \nthe correlation between FWHM(2D) and FWHM(G)  (Fig s 2f and S14, Supplementary \nInformation ). Figure S1 4 shows two basic trends connecting the G and 2D linewidths. The first \ntrend  is connected with the increasing spatial homogeneity under the laser spot, when both G and \n2D bands are getting narrower. The second trend is visible when both FWHM(G) and \nFWHM(2D) reach their minima – for Cu(100). In that case , 2D band narrowing is ac companied \nby G band broadening caused by the lowered doping level in graphene down to | EF| ~ 0 eV, when \nFWHM(G) ~ 13 cm-1 and FWHM(2D) ~ 16 cm-1. On the other hand, maximum doping level \nreached in our samples (| EF| ~ 350 meV) is reflected in FWHM(G) drop down  to ~4.5 cm-1 and \nFWHM(2D) ~ 20 cm-1 for graphene on Cu(100) . It should be noted that in Fig. S14, all three \nmain groups of points on Cu(100) with distinct doping levels are clearly separated (along the yellow arrow).  23 \n  \nFigure 5. Histograms of FWHM(2D) accumulated from all measurements of as -grown \ngraphene on a particular Cu surface. Excitation energy is 1.96 eV. The vertical axis shows the \nnumber of occurrences (N) of the particular FWHM(2D).  \n The exact reason for the low absolute values of FWHM(2D), es pecially on Cu(100), is not fully \nunderstood  at the moment. However, one can expect to find the cause in the interaction between graphene and the metallic substrate. An exceptionally narrow 2D band was reported also in graphene on iridium  [85]. A recent ARPES study of graphene on Cu foil showed a substantial \nreduction of electron- electron interaction in graphene by the metallic screening of the substrate \nand thus provided a  more accurate analysis of electron -phonon interaction, which in turn showed \na bett er convergence to local density approximation calculations  [81]. We could therefore assume \nthat the current theoretical approaches to the calculations and interpretations of the Raman spectra of graphene might actually reflect the situation on copper, at least in terms of the electron-\n24 \n phonon coupling strength. A simple formula for the FWHM(2D) calculation was provided by \nBasko, though only conical bands were taken into account  [61]: \n𝐹𝑊𝐻𝑀 (2𝐷)=0.77𝜐𝑝ℎ\n𝜐𝐹8𝛾𝑒−𝑝ℎ     (2) \nWhere υ ph and υ F are the phonon and Fermi group velocities, respectively, and γ e-ph is a \nbroadening parameter due to electron -phonon scattering. The υ ph/ υF is reflected in the slope of \nthe 2D band dispersion with excitation energy, which is slightly higher for Cu(100) with the slope of 93.6 ± 0.6 cm\n-1/eV in contrast to 88.7 ± 1.4 cm-1/eV for Cu(111), caused by the higher \ndoping level [ 78] (see Figure 6 left  panel ). However, such slopes are still very much in the range \nof those reported also for graphene on Si/SiO 2 [34]. The dependence of γ e-ph scattering term on \nthe excitation energy was evaluated by Venezuela et al.  [62]: \n𝛾𝑒−𝑝ℎ=(18.88𝐸𝐿+6.802𝐸𝐿2) 𝑚𝑒𝑉     (3) \nwhere γe−ph   in Eq. 2 corresponds to γ e−ph /4 in Eq. 3. It should be noted that Basko [ 61] \nassumes the presence of two 2D components, each of them having the same FWHM calculated from  Eq. 2, with their separation being a function of electron -hole asymmetry.  The overall shape \nand width of the 2D peak stems  from a complex interplay involving both electronic and phononic \ntrigonal warping, which are opposite , i.e. they compensate  each other to a varying extent \n(depending on E\nL) [62], phonon self -energy corrections [ 86], polarization, instrumental \nbroadening, etc. As a consequence, a quantitative comparison between  the calculated and \nmeasured peak widths is usually avoided for doubly resonant scattering processes. In our case, a simple comparison of FWHM(2D) measured on graphene on Cu(111) and FWHM(2D) obtained from Eqs. 2 and 3 as a function of E\nL provides similar trends, i.e. the width increases with E L \n(Fig. 6 right panel). Nevertheless, the absolute values differ for the above mentioned reasons . On \nthe other hand, in graphene on Cu(1 00) and (110), the FWHM(2D) increase with E L is only very 25 \n small if present – the 2nd order polynomials can be fitted to the data with the minima at ~2.2 eV, \nbut with adjusted R2 of less than 0.85. ( Due to the small significance of the fits, we do not plot \nthem  in Fig. 6, in contrast to Cu(111) with adj. R2 > 0.99). The different scaling  of FWHM(2D)  \nwith EL might indicate a weaker dependence of electron -phonon coupling in graphene on \nCu(100) and Cu(110) and/or significant variations in the shapes of electronic a nd phononic bands \nresulting from the distinct orientation -dependent interaction strengths between graphene and \ncopper. Both these assumptions can be  justified by the data presented here as well as in  the \nliterature [ 24, 30, 45, 81, 82] , and  their separation requires further detailed experiments together  \nwith complex models em bracing a wider range of parameters.   \n \n \nFigure 6.  Dispersion of 2D band position (left) and FWHM (right) with laser excitation energy \nfor Cu(100) , Cu(110)  and Cu(111) plotted with blue , red and green, respectively. Dashed lines \nare linear and 2nd order polynomial weighted least -squares fits to experimental points in the left \nand right panels, resp.  Error bars represent standard deviation from 4 measurements. The grey \ndotted line in the right panel is a calculation according to Eq. 2, using the ex perimental 2D band \ndispersion for graphene on Cu(111) and γ e−ph from Eq. 3.  \n \n \n26 \n 4. Conclusions  \nWe have shown that  graphene grown by CVD on copper is  highly susceptible to the lattice \norientation of the  substrate, which influe nces the strain and doping in graphe ne – both locally and \ngenerally . As expected, the growth on Cu single  crystals, regardless of the orientation, provides  \nmore homogeneous graphene as compared to the commonly used polycrystalline Cu foil. \nGraphene on Cu(111) is mainly uniformly compre ssed with ~0.3% of biaxial strain and n- doped \nwith Fermi level shift of ~250 meV, though with minor occurrences of relaxed and wrinkled regions. Graphene on Cu(110) is compressed with a large range of values between 0.05 and 0.3% of biaxial strain and esse ntially undoped. Graphene on Cu(100) is slightly and uniformly \ncompressed (0.1%) and occurs in three states with distinct doping levels, where the majority (~60%) of the graphene is undoped and the other two parts, equal in appearance, show n -doping \nwith F ermi level shifts of ~ 250 and 350 meV, respectively. The data gathered for graphene on \nCu(100) and Cu(111) are in agreement with  the few previous studies by ARPES or STM and \nshow that Raman spectroscopy provides quantitative information when done on data sets large \nenough.  \nAn important result, which can have serious implications for the understanding  of the origin of \nthe double res onance 2D (or G’) band, is the  extremely narrow width of this band on all three \nsingle crystals, especially on Cu(100). Nevertheless,  more detailed studies  of the interaction \nbetween graphene and the metallic substrate are  necessary to fully describe the o bserved \nphenomenon.  \n    27 \n Acknowledgments  \nFinancial support was provided by the Grant Agency of the Czech Republic (Contract No. \nP204/10/1677) , FP7 -Energy -2010- FET project Molesol (Contract No. 256617), and Ministry of \nEducation, Youth and Sports ERC -CZ project (Contract No. LL1301).  \n \nAppendix A. Supplementary Data : Supporting data associated with this article (Laue images, \nreciprocal and real space XRD maps of Cu surfaces; additional  correlations of selected \nparamet ers from Raman spectra of graphene on C u single crystals, AFM images) , can be found in \nthe online version, at http://dx.doi.org/10.1016/*************. \n. \n \nREFERENCES  \n \n[1] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al. Electric \nfield effect in atomically  thin carbon films. Science 2004;306(5296):666- 9. \n[2] Balandin AA. Thermal properties of graphene and nanostructured carbon materials. Nat \nMater 2011;10(8):569- 81. \n[3] Du X, Skachko I, Barker A, Andrei EY. Approaching ballistic transport in suspended \ngraph ene. Nat Nanotechnol 2008;3(8):491- 5. \n[4] Geim AK. Graphene: Status and Prospects. Science 2009;324(5934):1530- 4. \n[5] Lee C, Wei XD, Kysar JW, Hone J. Measurement of the elastic properties and intrinsic \nstrength of monolayer graphene. Science 2008;321(5887):385- 8. \n[6] Bunch JS, van der Zande AM, Verbridge SS, Frank IW, Tanenbaum DM, Parpia JM, et \nal. Electromechanical Resonators from Graphene Sheets. Science 2007;315(5811):490- 3. 28 \n [7] Schwierz F. Graphene transistors. Nat Nanotechnol 2010;5(7):487- 96. \n[8] Kavan L, Yum JH, Gratzel M. Graphene Nanoplatelets Outperforming Platinum as the \nElectrocatalyst in Co -Bipyridine -Mediated Dye -Sensitized Solar Cells. Nano Lett \n2011;11(12):5501- 6. \n[9] Kavan L, Yum J -H, Graetzel M. Optically Transparent Cathode for Co(III/II) Mediated \nDye-Sensitized Solar Cells Based on Graphene Oxide. ACS Appl Mater Interfaces \n2012;4(12):6999- 7006.  \n[10] Bae S, Kim H, Lee Y, Xu XF, Park JS, Zheng Y, et al. Roll -to-roll production of 30- inch \ngraphene films for transparent electrod es. Nat Nanotechnol 2010;5(8):574- 8. \n[11] Vicarelli L, Vitiello MS, Coquillat D, Lombardo A, Ferrari AC, Knap W, et al. Graphene \nfield-effect transistors as room -temperature terahertz detectors. Nat Mater 2012;11(10):865- 71. \n[12] Kim KS, Zhao Y, Jang H, Le e SY, Kim JM, Kim KS, et al. Large -scale pattern growth of \ngraphene films for stretchable transparent electrodes. Nature 2009;457(7230):706- 10. \n[13] Li XS, Cai WW, An JH, Kim S, Nah J, Yang DX, et al. Large -Area Synthesis of High-\nQuality and Uniform Graphe ne Films on Copper Foils. Science 2009;324(5932):1312- 4. \n[14] Reina A, Jia X, Ho J, Nezich D, Son H, Bulovic V, et al. Large Area, Few -Layer \nGraphene Films on Arbitrary Substrates by Chemical Vapor Deposition. Nano Lett 2009;9(1):30 -\n5. \n[15] Ago H, Ogawa Y,  Tsuji M, Mizuno S, Hibino H. Catalytic Growth of Graphene: Toward \nLarge -Area Single -Crystalline Graphene. J Phys Chem Lett 2012;3(16):2228- 36. \n[16] Bhaviripudi S, Jia X, Dresselhaus MS, Kong J. Role of Kinetic Factors in Chemical \nVapor Deposition Synthesi s of Uniform Large Area Graphene Using Copper Catalyst. Nano Lett \n2010;10(10):4128- 33. 29 \n [17] Mattevi C, Kim H, Chhowalla M. A review of chemical vapour deposition of graphene on \ncopper. J Mater Chem 2011;21(10):3324- 34. \n[18] Bartelt NC, McCarty KF. Graphene growth on metal surfaces. MRS Bulletin \n2012;37(12):1158- 65. \n[19] Yan Z, Lin J, Peng Z, Sun Z, Zhu Y, Li L, et al. Toward the Synthesis of Wafer -Scale \nSingle -Crystal Graphene on Copper Foils. Acs Nano 2012;6(10):9110- 7. \n[20] Murdock AT, Koos A, Britton TB, Houben L, Batten T, Zhang T, et al. Controlling the \nOrientation, Edge Geometry, and Thickness of Chemical Vapor Deposition Graphene. Acs Nano \n2013;7(2):1351- 9. \n[21] Wofford JM, Nie S, McCarty KF, Bartelt NC, Dubon OD. Graphene Islands on Cu Foils: \nThe Int erplay between Shape, Orientation, and Defects. Nano Lett 2010;10(12):4890- 6. \n[22] Wood JD, Schmucker SW, Lyons AS, Pop E, Lyding JW. Effects of Polycrystalline Cu \nSubstrate on Graphene Growth by Chemical Vapor Deposition. Nano Lett 2011;11(11):4547- 54. \n[23] Yu Q, Jauregui LA, Wu W, Colby R, Tian J, Su Z, et al. Control and characterization of \nindividual grains and grain boundaries in graphene grown by chemical vapour deposition. Nat Mater 2011;10(6):443- 9. \n[24] Wilson N, Marsden A, Saghir M, Bromley C, Sch aub R, Costantini G, et al. Weak \nmismatch epitaxy and structural Feedback in graphene growth on copper foil. Nano Res 2013;6(2):99- 112. \n[25] Gao L, Guest JR, Guisinger NP. Epitaxial Graphene on Cu(111). Nano Lett \n2010;10(9):3512- 6. \n[26] He R, Zhao L, Petro ne N, Kim KS, Roth M, Hone J, et al. Large Physisorption Strain in \nChemical Vapor Deposition of Graphene on Copper Substrates. Nano Lett 2012;12(5):2408- 13. 30 \n [27] Hu B, Ago H, Ito Y, Kawahara K, Tsuji M, Magome E, et al. Epitaxial growth of large -\narea singl e-layer graphene over Cu(111)/sapphire by atmospheric pressure CVD. Carbon \n2012;50(1):57- 65. \n[28] Nie S, Wofford JM, Bartelt NC, Dubon OD, McCarty KF. Origin of the mosaicity in \ngraphene grown on Cu(111). Phys Rev B 2011;84(15):155425. \n[29] Orofeo CM, Hibi no H, Kawahara K, Ogawa Y, Tsuji M, Ikeda K -i, et al. Influence of Cu \nmetal on the domain structure and carrier mobility in single -layer graphene. Carbon \n2012;50(6):2189- 96. \n[30] Walter AL, Nie S, Bostwick A, Kim KS, Moreschini L, Chang YJ, et al. Electronic \nstructure of graphene on single -crystal copper substrates. Phys Rev B 2011;84(19):195443. \n[31] Zhao L, Rim KT, Zhou H, He R, Heinz TF, Pinczuk A, et al. Influence of copper crystal \nsurface on the CVD growth of large area monolayer graphene. Solid State Commun 2011;151(7):509- 13. \n[32] Jacobberger RM, Arnold MS. Graphene Growth Dynamics on Epitaxial Copper Thin \nFilms. Chem Mater 2013;25(6):871- 7. \n[33] Ferrari AC, Basko DM. Raman spectroscopy as a versatile tool for studying the properties \nof graphene. Nat Nanotechnol 2013;8(4):235- 46. \n[34] Malard LM, Pimenta MA, Dresselhaus G, Dresselhaus MS. Raman spectroscopy in \ngraphene. Phys Rep 2009;473(5- 6):51- 87. \n[35] Das A, Pisana S, Chakraborty B, Piscanec S, Saha SK, Waghmare UV, et al. Monitoring \ndopants by Raman scattering in an electrochemically top -gated graphene transistor. Nat \nNanotechnol 2008;3(4):210- 5. 31 \n [36] Kalbac M, Reina -Cecco A, Farhat H, Kong J, Kavan L, Dresselhaus MS. The Influence of \nStrong Electron and Hole Doping on the Raman Intensity of Chemical  Vapor -Deposition \nGraphene. Acs Nano 2010;4(10):6055- 63. \n[37] Pisana S, Lazzeri M, Casiraghi C, Novoselov KS, Geim AK, Ferrari AC, et al. \nBreakdown of the adiabatic Born- Oppenheimer approximation in graphene. Nat Mater \n2007;6(3):198- 201. \n[38] Frank O, Tsoukleri G, Parthenios J, Papagelis K, Riaz I, Jalil R, et al. Compression \nBehavior of Single -Layer Graphenes. Acs Nano 2010;4(6):3131- 8. \n[39] Frank O, Tsoukleri G, Riaz I, Papagelis K, Parthenios J, Ferrari AC, et al. Development of \na universal stress sensor  for graphene and carbon fibres. Nat Commun 2011;2:255, DOI: \n10.1038/ncomms1247. \n[40] Mohiuddin TMG, Lombardo A, Nair RR, Bonetti A, Savini G, Jalil R, et al. Uniaxial \nstrain in graphene by Raman spectroscopy: G peak splitting, Grueneisen parameters, and s ample \norientation. Phys Rev B 2009;79(20):205433- 8. \n[41] Young RJ, Kinloch IA, Gong L, Novoselov KS. The mechanics of graphene \nnanocomposites: A review. Compos Sci Technol 2012;72(12):1459- 76. \n[42] Kalbac M, Farhat H, Kong J, Janda P, Kavan L, Dresselhaus MS. Raman Spectroscopy \nand in Situ Raman Spectroelectrochemistry of Bilayer C -12/C -13 Graphene. Nano Lett \n2011;11(5):1957- 63. \n[43] Kalbac M, Frank O, Kavan L. Effects of Heat Treatment on Raman Spectra of Two -Layer \n12C/13C Graphene. Chem Eur J 2012;18(43): 13877- 84. \n[44] Costa SD, Righi A, Fantini C, Hao Y, Magnuson C, Colombo L, et al. Resonant Raman \nspectroscopy of graphene grown on copper substrates. Solid State Commun 2012;152(15):1317-20. 32 \n [45] Zhang L, Pollak E, Wang W -C, Jiang P, Glans P -A, Zhang Y, et  al. Electronic structure \nstudy of ordering and interfacial interaction in graphene/Cu composites. Carbon \n2012;50(14):5316- 22. \n[46] Bauer G, Li JH, Holy V. High resolution x -ray reciprocal space mapping. Acta Phys Pol \nA 1996;89(2):115- 27. \n[47] Fewster PF. Reciprocal space mapping. Crit Rev Solid State Mat Sci 1997;22(2):69- 110. \n[48] Kalbac M, Frank O, Kavan L. The control of graphene double -layer formation in copper -\ncatalyzed chemical vapor deposition. Carbon 2012;50(10):3682- 7. \n[49] Wei CT, Parthasarathi M N, Beck PA. X -Ray Study of Polygonization in Copper Single \nCrystals. J Appl Phys 1957;28(8):874- 7. \n[50] Fan Y, Yangqiao L, Wei W, Wei C, Lian G, Jing S. A facile method to observe graphene \ngrowth on copper foil. Nanotechnology 2012;23(47):475705. [51] Chen  S, Brown L, Levendorf M, Cai W, Ju S -Y, Edgeworth J, et al. Oxidation Resistance \nof Graphene -Coated Cu and Cu/Ni Alloy. Acs Nano 2011;5(2):1321- 7. \n[52] Frank O, Vejpravova J, Kavan L, Kalbac M. Raman spectroscopy investigation of defect \noccurrence in grap hene grown on copper single crystals. Phys Status Solidi B 2013, DOI: \n10.1002/pssb.201300105. [53] Tuinstra F, Koenig JL. Raman Spectrum of Graphite. J Chem Phys 1970;53(3):1126- 30. \n[54] Cançado LG, Jorio A, Pimenta MA. Measuring the absolute Raman cross s ection of \nnanographites as a function of laser energy and crystallite size. Phys Rev B 2007;76(6):064304. [55] Klar P, Lidorikis E, Eckmann A, Verzhbitskiy IA, Ferrari AC, Casiraghi C. Raman \nscattering efficiency of graphene. Phys Rev B 2013;87(20):205435. [56] Basko DM. Calculation of the Raman G peak intensity in monolayer graphene: role of \nWard identities. New J Phys 2009;11(9):095011.  33 \n [57] Chen C -F, Park C -H, Boudouris BW, Horng J, Geng B, Girit C, et al. Controlling inelastic \nlight scattering quantum p athways in graphene. Nature 2011;471(7340):617- 20. \n[58] Lazzeri M, Mauri F. Nonadiabatic Kohn anomaly in a doped graphene monolayer. Phys \nRev Lett 2006;97(26):266407. \n[59] Berciaud S, Ryu S, Brus LE, Heinz TF. Probing the Intrinsic Properties of Exfoliated \nGraphene: Raman Spectroscopy of Free -Standing Monolayers. Nano Lett 2009;9(1):346- 52. \n[60] Zabel J, Nair RR, Ott A, Georgiou T, Geim AK, Novoselov KS, et al. Raman \nSpectroscopy of Graphene and Bilayer under Biaxial Strain: Bubbles and Balloons. Nano Lett 2011;12(2):617- 21. \n[61] Basko DM. Theory of resonant multiphonon Raman scattering in graphene. Phys Rev B \n2008;78(12):125418. [62] Venezuela P, Lazzeri M, Mauri F. Theory of double -resonant Raman spectra in graphene: \nIntensity and line shape of defect -induced and two -phonon bands. Phys Rev B \n2011;84(3):035433. [63] Maultzsch J, Reich S, Thomsen C. Double -resonant Raman scattering in graphite: \nInterference effects, selection rules, and phonon dispersion. Phys Rev B 2004;70(15):155403. [64] Zólyomi  V, Koltai J, Kürti J. Resonance Raman spectroscopy of graphite and graphene. \nPhys Status Solidi B 2011;248(11):2435- 44. \n[65] Basko DM, Piscanec S, Ferrari AC. Electron -electron interactions and doping dependence \nof the two- phonon Raman intensity in graphe ne. Phys Rev B 2009;80(16):165413. \n[66] Kürti J, Zólyomi V, Grüneis A, Kuzmany H. Double resonant Raman phenomena \nenhanced by van Hove singularities in single -wall carbon nanotubes. Phys Rev B \n2002;65(16):165433. 34 \n [67] Narula R, Bonini N, Marzari N, Reich S . Dominant phonon wave vectors and strain-\ninduced splitting of the 2D Raman mode of graphene. Phys Rev B 2012;85(11):115451. \n[68] Thomsen C, Reich S. Double resonant Raman scattering in graphite. Phys Rev Lett \n2000;85(24):5214- 7. \n[69] Ferrari AC, Meyer JC,  Scardaci V, Casiraghi C, Lazzeri M, Mauri F, et al. Raman \nspectrum of graphene and graphene layers. Phys Rev Lett 2006;97(18):187401. [70] Luo Z, Cong C, Zhang J, Xiong Q, Yu T. Direct observation of inner and outer G' band \ndouble -resonance Raman scatteri ng in free standing graphene. Appl Phys Lett \n2012;100(24):243107. [71] Berciaud Sp, Li X, Htoon H, Brus LE, Doorn SK, Heinz TF. Intrinsic lineshape of the \nRaman 2D -mode in freestanding graphene monolayers. Nano Lett 2013;13(8):3517–23. \n[72] Frank O, Mohr M , Maultzsch J, Thomsen C, Riaz I, Jalil R, et al. Raman 2D -Band \nSplitting in Graphene: Theory and Experiment. Acs Nano 2011;5(3):2231- 9. \n[73] Huang M, Yan H, Heinz TF, Hone J. Probing Strain- Induced Electronic Structure Change \nin Graphene by Raman Spectros copy. Nano Lett 2010;10(10):4074- 9. \n[74] Yoon D, Son Y -W, Cheong H. Strain- Dependent Splitting of the Double -Resonance \nRaman Scattering Band in Graphene. Phys Rev Lett 2011;106(15):155502. [75] Popov VN, Lambin P. Theoretical Raman intensity of the G and 2D bands of strained \ngraphene. Carbon 2013;54(0):86- 93. \n[76] Filintoglou K, Papadopoulos N, Arvanitidis J, Christofilos D, Frank O, Kalbac M, et al. \nRaman spectroscopy of graphene at high pressure: Effects of the substrate and the pressure transmitting media. Phys Rev B 2013;88(4):045418. [77] Lee JE, Ahn G, Shim J, Lee YS, Ryu S. Optical separation of mechanical strain from \ncharge doping in graphene. Nat Commun 2012;3:1024. 35 \n [78] Attaccalite C, Wirtz L, Lazzeri M, Mauri F, Rubio A. Doped Graphene as Tunable \nElectron−Phonon Coupling Material. Nano Lett 2010;10(4):1172- 6. \n[79] Kim K, Coh S, Tan LZ, Regan W, Yuk JM, Chatterjee E, et al. Raman Spectroscopy \nStudy of Rotated Double -Layer Graphene: Misorientation -Angle Dependence of Electronic \nStructure. Phys Rev Le tt 2012;108(24):246103. \n[80] Kalbac M, Frank O, Kong J, Sanchez -Yamagishi J, Watanabe K, Taniguchi T, et al. Large \nVariations of the Raman Signal in the Spectra of Twisted Bilayer Graphene on a BN Substrate. J \nPhys Chem Lett 2012;3(6):796- 9. \n[81] Siegel DA , Hwang C, Fedorov AV, Lanzara A. Electron–phonon coupling and intrinsic \nbandgap in highly -screened graphene. New J Phys 2012;14(9):095006. \n[82] Khomyakov PA, Giovannetti G, Rusu PC, Brocks G, van den Brink J, Kelly PJ. First -\nprinciples study of the intera ction and charge transfer between graphene and metals. Phys Rev B \n2009;79(19):195425. \n[83] Tapaszto L, Dumitrica T, Kim SJ, Nemes -Incze P, Hwang C, Biro LP. Breakdown of \ncontinuum mechanics for nanometre -wavelength rippling of graphene. Nat Phys 2012;8(10) :739-\n42. [84] Hattab H, N’Diaye AT, Wall D, Klein C, Jnawali G, Coraux J, et al. Interplay of \nWrinkles, Strain, and Lattice Parameter in Graphene on Iridium. Nano Lett 2011;12(2):678- 82. \n[85] Vo-Van C, Kimouche A, Reserbat -Plantey A, Fruchart O, Bayle -Guillemaud P, Bendiab \nN, et al. Epitaxial graphene prepared by chemical vapor deposition on single crystal thin iridium films on sapphire. Appl Phys Lett 2011;98(18):181903- 3. \n[86] Araujo PT, Mafra DL, Sato K, Saito R, Kong J, Dresselhaus MS. Phonon Self -Energy \nCorrections to Nonzero Wave -Vector Phonon Modes in Single -Layer Graphene. Phys Rev Lett \n2012;109(4):046801. S1 \n Electronic Supplementary Information  \n \nInteraction between graphene and copper substrate: \nThe role of lattice orientation  \nOtakar Frank1,*, Jana Vejpravov a2, Vaclav Hol y3, Ladislav Kavan1, and Martin Kalbac1 \n1 J. Heyrovsky Institute of Physical Chemistry of the AS CR, v.v.i., Dolejskova 2155/3, CZ -182 \n23 Prague 8, Czech Republic  \n2 Institute of Physics of the AS CR, v.v.i., Na Slovance 1999/2, CZ -182 21 Prague 8, Czech \nRepublic  \n3 Faculty of Mathematics and Physics, Charles University  in Prague , Ke Karlovu 5, CZ -121 16 \nPrague 2, Czech Republic  \n \n* Corresponding author: otakar.frank@jh- inst.cas.cz  \n \n \n \n \n \n \n \n S2 \n Table S1. Samples of graphene grown on copper substrates for Raman mapping with laser \nexcitation of 1.96 eV.  All the samples in one batch were grown in the same time and measured in \nthe order as indicated. The samples in queue for characterization were stored under vacuum. The \norder of samples for measurement was varied amongst the batches to check if there is any \ninfluence of the waiting time on the results of Raman characterization. No such effects were observed. The numbers below sample orientation indicate the size of Raman maps as X x Y \n(lateral step size of 1 µm in both directions). Total measurement time f or one spectrum was ~ \n260 s (two spectral windows of 120 s each plus grating movement).  \nBatch #  Measured as  \n1st 2nd 3rd 4th \n1 Cu(100)  Cu(110)  Cu(111)  -- 16x16  17x16  18x17  \n2 Cu(111)  Cu(110)  Cu(100)  Cu foil  \n14x14  12x11  14x14  15x14  \n3 Cu(110)  Cu(100)  Cu(111)  -- 14x14  11x11  14x14  \n \n \n \n \n \nFigure S1.  Typical Laue images of the copper single- crystals recorded with the incident beam \nperpendicular to the (100), (110) and (111) planes. The patterns clearly reflect the expected \nminimum symmetry of the corresponding Laue class: m -3m. \n \nS3 \n \n \nFigure S2.  Reciprocal maps of the copper single -crystals for the (100) plane. The left panel \ncorresponds to the experimental data (top) and simulation (bottom) for the ( 200) and ( 400) \ndiffractions, the right panel shows the corresponding fits of a single scan along the Q x or Q z \ndirection in reciprocal space.  \n \n \n \nFigure S3.  Reciprocal maps of the copper single -crystals for the (110) plane. The left panel \ncorresponds to the experimental data (top) and simulation (bottom) for the ( 220) and (222) \ndiffractions, the right panel shows the corresponding fits of a single scan along the Q x or Q z \ndirection in reciprocal space.  \n S4 \n  \nFigure S4.  Reciprocal maps of the copper single -crystals for the (111) plane. The left panel \ncorresponds to the experimental data (top) and simulation (bottom) for the (111) and (222) \ndiffractions, the right panel shows the corresponding fits of a single scan along the Q x or Qz \ndirection in reciprocal space.  \n \n \n \nFigure S5.  Reciprocal maps of the cop per single -crystals for the (100) and ( 110) faces  recorded \non the samples before and after annealing, respectively.  \n \nS5 \n \n \nFigure S6. Real space maps of the copper single- crystals recorded perpendicular to the (100), \n(110) and (111) planes . The lateral dimensions of the maps is 10x10 m m2. \n \n \n \n \nFigure S7 . Optical microphotographs of Cu(111) surface with freshly grown graphene (left) and \nafter 1 week of storing at ambient conditions  (different area).  \n \nS6 \n  \nFigure S 8. Raman spectra of graphene grown on Cu(110) single crystal. Excitation energies are \ngiven for each spectrum, the spectra are normali zed to the G band intensity for each Cu surface, \nthe corresponding zoom factors are labeled in the plots.  \n \n \nS7 \n  \nFigure S 9. Average Raman spectra obtained by averaging all spectra of graphene acquired \nduring mapping of batch #2 on Cu(100) (blue), Cu(110) (red), Cu(111) (green), and Cu foil \n(black). Laser excitation energy is 1.96 eV.  Note that the peaks are much broader than wh at \nwould correspond to single spectra because of the span of strain (and doping) levels across the \nsampled areas. The peak at ~1555 cm-1 corresponds to the vibration of molecular oxygen.  \n \n \n \n \n \n \nS8 \n  \nFigure S10. Correlations between monitored parameters of the G peaks fitted by Lorentzian \nlineshapes in Raman spectra of as -grown graphene on Cu(100) (blue), Cu(110) (red), Cu(111) \n(green) and Cu foil (black). To show the difference between Pos(G) and Offset(G), panel s a and \nb are exact copies of Fig.  2e and b, respectively , with t he panels c and d show ing the same data, \nbut the G band frequencies displayed as  the “absolute” Raman shifts (i.e. Pos(G)) as opposed to \nthe Offset(G), which is the relative difference from t he shift caused purely by strain (for further \nexplanation, see main text).  \n \nS9 \n  \nFigure S1 1. Correlation between Pos(2D) and Pos(G) for all measured data points on the studied \nCu surfaces differentiated by colors. Laser excitation energy is 1.96 eV . The black line indicates \npure strain variation, with the slope of 2.5. Blue arrows label two data sets with doping levels \ndistinct from the majority of points on Cu(100).  \n \n \n \nS10 \n  \nFigure S 12. Histograms of FWHM(G) accumulated from all measurements of as -grow n \ngraphene on a particular Cu surface. Laser excitation energy is 1.96 eV. The vertical axis shows \nthe number of occurrences (N) of the particular FWHM(G).  \n \n \n \n \n \n \nS11 \n  \nFigure S1 3. Top:  AFM images (height  sensor ) of graphene on Cu(100), Cu(110) and Cu(111)  \nsingle crystals . Bottom: height profiles extracted along the dashed white lines from the respective \nAFM images. Note the different height scales (both in images and profiles) when comparing \nbetween the individual Cu orientations. \n \n  \n \n \n \n \n \nS12 \n  \nFigure S1 4. Correlation between FWHM(2D) and FWHM(G) for all measured data points on \nthe studied Cu surfaces differentiated by colors. Laser  excitation energy is 1.96 eV. The orange \nand yellow arrows depict the two main trends in 2D band narrowing: increasing the spa tial \nhomogeneity (orange) accompanied by G band narrowing and reducing the doping level (yellow) \naccompanied by G band broadening.  \n \n \n"    },    {        "title": "2012.05459v1.Evidence_of_non_trivial_Berry_phase_and_Kondo_physics_in_SmBi.pdf",        "content": "arXiv:2012.05459v1  [cond-mat.str-el]  10 Dec 2020Evidence of non-trivial Berry phase and Kondo physics in SmB i\nAnup Pradhan Sakhya, P. L. Paulose, A. Thamizhavel, and Kalobara n Maiti∗\nDepartment of Condensed Matter Physics and Materials Scien ce,\nTata Institute of Fundamental Research, Homi Bhabha Road, C olaba, Mumbai - 400 005, INDIA.\n(Dated: December 11, 2020)\nRealization of semimetals with non-trivial topologies suc h as Dirac and Weyl semimetals, have\nprovided a boost in the study of these quantum materials. Pre sence of electron correlation makes\nthe system even more exotic due to enhanced scattering of cha rge carriers, Kondo screening etc.\nHere, we studied the electronic properties of single crysta lline, SmBi employing varied state of the\nart bulk measurements. Magnetization data reveals two magn etic transitions; an antiferromagnetic\norder with a N´ eel temperature of ∼9 K and a second magnetic transition at a lower temperature (=\n7 K). The electrical resistivity data shows an upturn typica l of a Kondo system and the estimated\nKondo temperature is found to be close to the N´ eel temperatu re. High quality of the crystal enabled\nus to discover signature of quantum oscillation in the magne tization data even at low magnetic field.\nUsing a Landau level fan diagram analysis, a non-trivial Ber ry phase is identified for a Fermi pocket\nrevealing the topological character in this material. Thes e results demonstrate an unique example\nof the Fermiology in the antiferromagnetic state and opens u p a new paradigm to explore the Dirac\nfermion physics in correlated topological metal via interp lay of Kondo interaction, topological order\nand magnetism.\nI. INTRODUCTION\nTopological insulators (TIs) [1] having bulk band gaps\nand robust symmetry protected surface states have at-\ntracted enormous interest in the recent time due to their\nimportance in exploring rich fundamental science and\napplication in advanced technology. While impurity on\nthe surface of these materials often exhibit interesting\nevolutions [2, 3], breaking of suitable symmetry helped\nto realize semimetals with non-trivial topologies such\nas Dirac [4] and Weyl semimetals, [5]. The electronic\nstructure of Dirac or Weyl semimetals are gapless and\ncontain Dirac or Weyl points; the crossing of the lin-\nearly dispersive bands [4, 5]. These materials provide\nan wonderful opportunity to understand the physics of\nrelativistic particles in low-energy condensed matter sys-\ntems. Uniquebulk bandtopologyin thesesemimetalsex-\nhibit exoticphysicalpropertiessuchasextrememagneto-\nresistance (XMR), high charge carrier mobility, potential\ntopological superconductivity, and chiral-anomalydriven\nnegative magnetoresistance (MR) [6]. Till date most\nof the topological materials discovered experimentally\nare weakly correlated systems, where the effective single\nparticle descriptions adequately captures the electronic\nproperties.\nStrong spin-orbit coupling (SOC) in the correlated\nelectron systems may lead to topologically protected sur-\nface states in addition to other interesting properties\n[7, 8]. Interestingly, in the correlated topological insula-\ntors, the electron correlation enhances the effective elec-\ntron mass leading to insulating bulk ground state while\nthe surface states behave like Dirac Fermions despite the\npresence of strong electron correlation. For example, a\nKondo insulator, SmB 6, has been predicted to host topo-\nlogicalsurfacestateswithintheKondogap[9–11]. Subse-\nquent studies, however, showed anomalies such as quan-tum oscillations [9, 12] and finite Fermi surface [13–15].\nRSb/RBi (where R = rare earth) family of materials are\nalso proposed to be potential candidates for correlated\ntopologicalsemimetals [16–18]. Based on the band struc-\nture calculations LaX (X = P, As, Sb, Bi) have been\nclassified as Z2-topological metals [18]. Angle resolved\nphotoemission spectroscopy (ARPES) on LaSb indicate\nthat the material is topologically trivial and its proper-\nties can be well explained by electron-hole compensation\n[17]. On the other hand, LaBi is toplogically nontriv-\nial exhibiting three Dirac cones; one Dirac cone at the\nsurface Brillouin zone (SBZ) center and two are at the\nSBZ corner [19]. Another ARPES study contradicted\nthese results suggesting that the surface states in LaBi\nare highly unusual possessing the parabolic bottom band\nandlineartop band[20]. The originofsuchunusualmass\nacquisition is yet to be understood [20].\nIn this class of materials, SmBi exhibits band inversion\nbetween the Sm 5 dand Bi 6 pstates similar to LaBi [21]\nalthough the ARPES study by Li et al.could not find\nthe predictedsurfacestateatthe Γ pointpresumablydue\nto its strong hybridization with the bulk states. They\nfound two surface states at the high symmetry point, M,\nwhichinteractwitheachotherandyieldapeculiarpairof\ngapped trivial surface states [22]. ARPES and transport\nstudies of (Pr, Sm)Sb and (Pr, Sm)Bi revealed signa-\nture of extreme magnetoresistance (XMR) due to good\nelectron-hole compensation [23]. Evidently, presence of\nstrong electron correlation leads to unusual complexity\nin their properties. Multiple bands cross the Fermi level\nmaking the identification of the Fermi surfaces and its\nbehavior difficult.\nIn order to address the issues, we prepared very high\nqualitysinglecrystalsofSmBi and employeddeHaasvan\nAlphen (dHvA) measurements to derive the Fermiology,\nwhich relies on quantum oscillations as a function of ex-2\nternal magnetic field. The phase offset of the quantum\noscillation is related to Berry phase associated with the\ncyclotron orbits and provides a means to experimentally\naccess the signature of the Dirac cone [24]. We discover\npronounced quantum oscillations even at low magnetic\nfield; the analysis of the experimental data reveals pres-\nence of multiple Fermi pockets and a signature of a non-\ntrivial Berry phase. Resistivity upturn at around ∼12 K\ntypical of a Kondo behavior and the signature of Berry\nphase provide evidence of Dirac fermion physics for the\nelectronic properties of this correlated material.\nII. EXPERIMENTAL AND CRYSTAL\nSTRUCTURE\nHigh qualtiy single crystals of SmBi was grown fol-\nlowing flux method with high purity Sm (99.9%), Bi\n(99.998%) and In (99.9999%) [molar ratio of Sm:Bi:In\nwas 1:1:10]. The mixture was kept in a recrystallized\nalumina crucible and sealed in an evacuated quartz am-\npoule at a vacuum of 10−6Torr. The sealed ampoule\nheated to 1050oC at a rate of 60oC/hr, kept for 24\nhr and then the temperature was reduced very slowly\n(2oC/hr) to 700oC. The ampoule was centrifuged at\n700oC to remove the flux. We have verified the crystal\nstructure using powder x-raydiffraction method. Energy\ndispersive analysis of x-rays at various regions of the sin-\ngle crystal established homogeneity and stoichiometry of\nthe material. The orientation and single crystallinity of\nthe sample is determined using Laue diffraction. The\nelectrical resistivity was measured down to 5 K using a\nhomemadesetup. The dc magneticsusceptibility andthe\nmagnetization measurements of SmBi were performed in\nthe temperature range, 1.8 K to 300 K using a super-\nconducting quantum interference device (SQUID). The\ndHvA oscillations were probed through magnetization\nmeasurements in a vibrating sample magnetometer from\nQuantum Design. SmBi forms in rock salt structure as\nshown in Figure 1(a). We prepared single crystals of\nSmBi starting with highly pure ingredients and charac-\nterized the sample using varied methods. A typical Laue\ndiffraction pattern shown in Figure 1(b), exhibit clear\nfour-fold rotational symmetry pattern corresponding to\n(001) plane of the cubic structure with well defined spots\nthat demonstrates high quality of the crystals.\nIII. RESULTS AND DISCUSSION\nIn Figure 1(c), we show the electrical resistivity, ρas a\nfunctionoftemperatureforelectriccurrentparalleltothe\n[100] direction. A typical metallic behavior is observed\nwith the decrease in temperature from room tempera-\nture until around 12 K along with an upturn at lower\ntemperatures till 9 K and then there is a sharp decrease\nFIG. 1. (color online) (a) Crystal structure of SmBi. The red\nand blue colored solid spheres denote the Sm and Bi atoms\nrespectively. (b) Laue diffraction pattern of SmBi (001) sin -\ngle crystal. (c) Temperature dependent electrical resisti vity\nmeasured in the presence of a field of 0 Tesla and 2 Tesla. The\ndata show an upturn at about 12 K and then a sharp decrease\nbelow 9 K. (d) Enlarged view of the electrical resistivity in\nthe low temperature regime. (e) The low-temperature upturn\nin resistivity versus lnT. The solid line shows the fit over the\ntemperature range from 9.4 K to 18 K using equation 2.\nas shown in an expanded temperature scale in Figure\n1(d). A broad hump observed at about 50 K may be\ndue to the crystal field effect as found in other systems\n[25]. The resistivity upturn cannot be captured using\ntypical behavior due to disorder and/or other localiza-\ntion effects at low temperatures. In order to investigate\nif Kondo interactions plays a role here, the resistivity up-\nturn in this narrow temperature range is simulated using\nHamann’s model [26, 27], where the typical Kondo scat-\ntering is captured by the lnTdependence of resistivity.\nIn this model, the contribution due to the scattering by\nthe magnetic impurity, ρHcan be expressed as:\nρH=ρu\n2(1−ln(T/TK)/radicalbig\nln2(T/TK)+S(S+1)π2) (1)\nwhere,ρuis the resistivity at the unitarity limit, TKis\nthe Kondo temperature and Sis the spin moment of\nthe magnetic impurity. Above the resistivity minimum,\nthe major contribution to resistivity arises from electron-\nphonon scattering, which can be captured by fitting the\nresistivity, ρ(T) in the higher temperature range using\nan expression, a+bTn. From this fitting, we found b=\n1.6×10−7andn= 1.3. The resistivity in the low tem-3\nperature range (9.4 K to 18 K) is then fitted using an\nexpression that includes the Hamann term, along with\ntheelectron-phononcontribution bTnandatemperature-\nindependent term ρ0,\nρT=ρH+bTn+ρ0 (2)\nThe fit to the experimental data is shown in Figure\n1 (e) for the parameter values; ρu= 6.54×10−6Ω,\nρ0= 2.8×10−6Ω, S =0.02 and TK= 9.2 K. Good repre-\nsentation of the experimental data indicates importance\nof Kondo coupling of the Sm moments with the conduc-\ntionelectronsinitselectricaltransportpropertiesleading\nto resistivity upturn.\nFIG. 2. (color online) Zero field cooled and field cooled mag-\nnetic susceptibility, χ, vs temperature plot (left axis) and in-\nverse magnetic susceptibility, 1/ χ, vs temperature plot (right\naxis) measured under magnetic field of B= 40kOe. Inset\nshows the enlarged plot of χat T<18 K. The peaks labeled\nA and B in the inset shows signatures of the two magnetic\ntransitions.\nBelow 9 K, the resistivity shows a sharp decrease\nas often observed in various rare-earth intermetallics\ndue to antiferromagnetic ordering via Ruderman-Kittel-\nKasuya-Yosida (RKKY) interactions between the un-\ncompensated part of the magnetic moments [28]. This\nis verified by the magnetic susceptibility measurements.\nThe temperature-dependent molar magnetic susceptibil-\nityχ(T) =M/Hmeasured at the magnetic field, B=\n40kOe and it’s inverse, 1 /χ(T) are shown in Figure 2.\nA dome-shaped curve centered at about 25 K is evident\ninχ(T) plot akin to the observations in the Kondo in-\nsulator, SmB 6[29]. With the decrease in temperature,\nχ(T) increasesas expected in a paramagneticsystem and\nshows a sharp peak at around 9 K (labelled as B) [30].\nInterestingly, the experimental data reveal a second peak(labelled A) at 7 K (see the inset of the Figure 2) indi-\ncating presence of additional magnetic transition in the\nsystem. Further experimental studies, especially neutron\ndiffractionmeasurement arenecessaryto understand this\ncomplex behavior.\n/s48/s46/s53 /s48/s46/s52 /s48/s46/s51 /s48/s46/s50/s45/s50/s45/s49/s48/s49/s50\n/s48/s46/s50/s48 /s48/s46/s49/s56\n/s49 /s50 /s51 /s52/s50 /s51 /s52 /s53/s48/s46/s52/s48/s46/s56/s32\n/s32/s32/s32/s54/s32/s75\n/s32/s32/s55/s32/s75\n/s32/s32/s56/s32/s75\n/s32/s32/s56/s46/s53/s32/s75/s32/s49/s46/s56/s32/s75\n/s32/s50/s46/s53/s32/s75\n/s32/s32/s51/s32/s75\n/s32/s32/s52/s32/s75\n/s32/s32/s53/s32/s75\n/s32/s68 /s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s49/s47/s66/s32/s40/s84/s45/s49\n/s41/s40/s97/s41\n/s32\n/s32/s32/s55/s32/s75\n/s32/s56/s46/s53/s32/s75/s32/s32/s32/s49/s46/s56/s32/s75\n/s32/s32/s51/s32/s75\n/s32/s32/s53/s32/s75\n/s40/s98/s41\n/s70\n/s103\n/s70\n/s98/s32/s49/s46/s56/s32/s75\n/s32/s50/s46/s53/s32/s75\n/s32/s51/s32/s75\n/s32/s52/s32/s75\n/s32/s53/s32/s75\n/s32/s54/s32/s75/s32\n/s32\n/s32/s32/s70/s70/s84/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40 /s180 /s49/s48/s50\n/s32/s84/s41/s70\n/s97/s40/s99/s41\n/s32\n/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s70\n/s97\n/s32/s70\n/s98\n/s32/s70\n/s103\nFIG. 3. (color online) (a) dHVa oscillations obtained by sub -\ntracting polynomial backgroundfrom magnetic measurement ,\nplotted with inverse magnetic field (1/B) at different temper -\natures. (b) dHVa oscillations at some selected temperature s\nin an expanded x-scale. (c) Fast fourier transform of the\ndHVa oscillations exhibit three principal frequencies Fαat 37\nT,Fβat 105 T and Fγat 324 T. Inset in (c) - FFT ampli-\ntude of the dHVa oscillations as a function of temperature fo r\nthe frequencies Fα,FβandFγ(symbols). Solid lines are the\nfits using the thermal damping factor of the Lifshitz-Kosevi ch\nformula.\nWe have measured the magnetization as a function of\nmagnetic field at different temperatures. The magnetiza-\ntion isotherms of SmBi shows a paramagnetic type field\ndependence. In addition, we observe significant oscilla-\ntioninthemagnetizationasafunctionoftheappliedfield\nalthough the magnetic field applied is small compared\nto the kind of field required to get discernible oscilla-\ntions. Inordertoextracttheoscillations, thebackground\nmagnetization curve, M0(B) has been simulated using a\npolynomial; a fifth order polynomial replicates the back-\nground well. The subtracted magnetization, ∆ M(=\nM(T)−M0(B)) as a function of the applied magnetic\nfield,B(/bardblc) is shown in Figure 3(a) exhibiting clear\nsignature of de Haas van Alphen (dHvA) oscillations is\nobserved. The dHvA oscillations for some selected tem-4\nperatures in an expanded magnetic field scale are shown\nin Figure 3 (b) for clarity. Interestingly, the oscillations\nat 1.8 K can be observed down to a very low magnetic\nfield of about 1.1 T. With the increase in temperature,\nthe oscillations dies down drastically and almost disap-\npears above 8.5 K.\nThe frequency spectra obtained by fast Fourier trans-\nform (FFT) of the dHvA oscillations is shown in Figure\n3(c). There are three principal frequencies Fαat 37 T,\nFβat 105 T and Fγat 324 T indicating presence of three\nFermi pockets for B/bardblcin SmBi. Signatures of higher\nharmonics are also observed, which are not shown here\nfor clarity. With the increase in temperature, the am-\nplitude of all the frequencies decreases fast due to the\nthermal broadening of the Landau levels.\nFIG. 4. (color online) (a) Oscillatory part of magnetizatio n as\na function of 1/B at 1.8 K (the black line is the experimental\ndata and the red line is the sum of the three filtered frequen-\ncies). (b) The three filtered oscillatory parts of magnetiza tion\ncorresponding to Fα(blue line), Fβ(thick red line) and Fγ\n(black line).\nThese three pockets can be filtered from the total os-\ncillation data by using a Band-pass filter. The filtered\noscillatory parts for the Fα(blue line), Fβ(red line) and\nFγ(black line) branch are shown in Figure 4 (b). The\nsum of the three filtered frequencies (red line) in Figure\n4(a) matches very well both in amplitudes and phases\nwith the experimental data (black line) providing confi-\ndence on the analysis.\nThe above results suggest that there are three Fermi\npockets having finite cross section perpendicular to the c\naxis. The frequency (in inverse field) of the oscillationsTABLE I. Fermi surface parameters for SmBi obtained from\nthe dHvA oscillations data.\nF(T)AF(10−2˚A−2)kF(10−2˚A−1)m∗/mevF(105m/sec)\n37 0.354 3 0.216 1.6\n105 1.01 5.65 0.304 2.1\n324 3.1 9.9 0.287 3.98\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48/s46/s50/s48 /s48/s46/s50/s52/s49/s50\n/s110 /s32/s47/s32 /s110\n/s48/s46/s50/s50\n/s32/s49/s47/s66/s32/s40/s84/s45/s49\n/s41/s32/s70\n/s97\n/s32/s70\n/s98 \n/s32/s70\n/s103\n/s76/s97/s110/s100/s97/s117/s32/s108/s101/s118/s101/s108/s32/s105/s110/s100/s101/s120/s40/s98/s41/s32/s82 /s32/s47/s32 /s82\n/s48/s46/s50/s50/s68/s105/s110/s103/s108/s101/s32/s112/s108/s111/s116/s40/s97/s41\nFIG. 5. (color online) (a) The Dingle plots for the frequenci es\nFα(open circle), Fβ(closed circle) and Fγ(open triangle)\nalong with the fits using Lifshitz-Kosevich formula. In the\ny-axis,R=ln[∆MB1/2sinh(14.69m∗T/B)]. The magnitude\nofRis normalized by it’s value at 1 /B= 0.22 T−1[1.045,\n0.1667 and 1.65 for Fα,FβandFγ, respectively]. (b) Landau\nfan diagram extracted from the quantum oscillation data; th e\nnormalized Landau band index, n/n0.22vs the reciprocal of\nthe field, 1 /B[n= Landau band index and n0.22is the value\nofnat 1/B= 0.22 T−1, which are 8.5, 22.5 and 71 for Fα,\nFβandFγ, respectively]. Universality in field dependence for\nall the Fermi pockets.\nis related to the k-space cross-sectional area of the Fermi\nsurfaceAFby the Onsager relation, F= (φ0/2π2)AF\n(magnetic flux quantum, φ0=h/2e). Using Onsager re-\nlation, the Fermi surface cross-section ( AF), correspond-\ning Fermi momentum ( kF), and Fermi velocity ( vF) are\ncalculated and listed in the Table I. Temperature depen-\ndence of the oscillation amplitudes are shown in the inset\nof Fig. 3(c), which is calculated using the thermal damp-\ning factor in the Lifshitz-Kosevich (LK) formula [31],\nRT= (2π2kBT/β)/sinh(2π2kBT/β) (3)\nwhereβ=e/planckover2pi1B/m∗,m∗is the cyclotron effective mass.\nDue to the presence of multiple branches in the quantum\noscillation, the fitting parameter Bhas been defined as\nthe average inverse field of the FFT interval, and the\namplitudes of the FFT peaks are used for fitting. The\nbest fit is shown by solid line in the inset of Fig. 3(c).\nUsing the fitting parameters, we have also calculated the\ncyclotron mass, m∗for theFα,Fβand the Fγpockets\nand are listed in Table I.\nThe Dingle temperature has been calculated by fitting5\nthe dHvA oscillation amplitudes with inverse magnetic\nfield to the Dingle damping term,\nRD= exp(−2π2kBm∗TD//planckover2pi1eB) (4)\nTo improve accuracy in fittings, we have separated the\noscillation components of Fα,FβandFγbands by fil-\ntering out irrelevant oscillations, and then extracted the\noscillation amplitude of 1.8 K as a function of 1 /B. The\nbest fit using the LK equation is shown in Fig. 5 (a) and\nthe estimated Dingle temperatures, TDare 3.9 K, 1.5 K\nand 2.5 K for Fα,Fβand the Fγ, respectively. The\nsmall Dingle temperature indicates little influence from\ndisorder due to impurities. Using the estimated Dingle\ntemperatures, the mobility, µq= (e/planckover2pi1/2πkBm∗TD) of the\ncharge carriers corresponding to the three Fermi surfaces\nhas been found to be 2510cm2/Vsec, 4650cm2/Vsec and\n2954 cm2/Vsec, respectively.\nWe now investigate the possible topological nature of\nthe bands by constructing the Landau fan diagram. The\nLandau level assignment is done using the peak positions\nof the oscillation component of each frequency shown in\nFig. 3 (c). When a closed cyclotron orbit is quantized\nunder an external magnetic field B, then according to\nthe Lifshitz-Onsager (LO) quantization rule,\nAn/planckover2pi1\neB= 2π(n+γ−δ) (5)\nHere,Anis the Fermi surface cross section of the Lan-\ndau level (LL) n,γ=1\n2−φB\n2πis the phase factor ( φB\nis the Berry phase) and δis an additional phase shift\n(δ= 0 in two dimensions, = ±1/8 in three dimensions)\n[32, 33]. Clearly, a vanishingly small γmakesφBclose to\nπ, thereby providea signatureof non-trivialBerry phase.\nThe Landau fan diagram, a plot of Landau index num-\nbern(y-axis) vs 1 /B(x-axis), is shown in Fig. 5(b).\nHere, the Landau level index, n(n+ 1/2) is assigned to\neach ∆M(B) minimum(maximum) [32] and is calculated\nusing LO formula as shown in the figure by lines super-\nimposed over the data points (symbols). Interestingly, a\nnormalization of nby its values at 1 /B= 2.2 puts all\nthe data on a single straight line indicating an universal\nbehavior of the charge carriers in all the Fermi pockets.\nThe magnitude of |γ−δ|for theFα,FβandFγpockets\narefoundtobe0.07,0.34and0.4,respectively. Thesmall\nvalue of|γ−δ|for theFαpocket(within ±1/8)revealsan\nevidence of a non-trivial Berry phase of πand indicates\nits correspondence to Dirac fermions [33, 34].\nIV. CONCLUSION\nFrom the above results it is clear that very high qual-\nitysinglecrystalofSmBi helpedtounveilseveralpuzzles.\nTransport measurements exhibit a resistivity upturn at\nabout12Kwitha lnTdependencetypicalofaKondolat-\ntice system. The estimated Kondo temperature is foundto be 9.2 K. Magnetization data reveals signature of two\nmagnetic transitions; one at 9 K and the other at 7 K.\nThe first transition at 9 K correspond to antiferromag-\nnetic ordering, signature of which is also seen in the re-\nsistivity data via a sharp decrease in resistivity as often\nobserved in various strongly correlated systems [8]. The\nN´ eel temperature of 9 K is close to the estimated Kondo\ntemperature suggesting a delicate competition between\nthe Kondo effect and the indirect exchange interaction.\nClearlythe compensationofmagneticmomentisfarfrom\ncomplete providing a possible case of spin-density wave\nquantum criticality [35–37]. The origin of second transi-\ntionisnot clearyet. However,it’s strikingsimilaritywith\nother Kondo lattice systems such as CeB 6is interesting.\nWhile doing the magnetization studies at low temper-\natures, we discover beautiful quantum oscillations at a\nfield as low as 1.1 Tesla. Prominent dHvA quantum os-\ncillations are observed at temperatures upto about 8.5 K\nwherethesystemisinantiferromagneticphase. Fromthe\nanalysis of the magnetization data, three intriguing oscil-\nlation branches, Fα,Fβ, andFγcorresponding to three\nFermi surfaces are observed. The Berry phase was un-\nambiguouslycharacterizedby extrapolatingthe observed\ndHvA oscillations to the zero-energy Landau level using\nLandau fan diagram. Signature of a non-trivial Berry\nphase is discovered for the FαFermi-pocket, suggesting\nthe nontrivial topological character. Ironically, although\nthe system possesses Kondo-type interactions, the effec-\ntive mass for all the Fermi pockets found is very small\nalong with high mobility. These results thus provide an\nunique example of a system exhibiting Kondo interac-\ntions along with highly diminished effective mass and an\nevidence for a nontrivial Berry phase, which provides an\nopportunity for studying the interplay between electron\ncorrelations, topology and magnetism.\nACKNOWLEDGMENTS\nAuthors acknowledges financial support from the De-\npartment of Atomic Energy, Govt. of India under the\nproject no. 12-R&D-TFR-5.10-0100. KM acknowledges\nfinancial support from DAE, Govt. of India under the\nDAE-SRC-OI Award program, and Department of Sci-\nence and Technology, Govt. of India under J. C. Bose\nFellowship program.\n∗Corresponding author: kbmaiti@tifr.res.in\n[1] M. Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[2] D. Biswas, S. Thakur, K. Ali, G. Balakrishnan, and K.\nMaiti, Sci. Rep. 5, 10260 (2015).\n[3] D. Biswas, and K. Maiti, J. Electron Spectrosc. Relat.\nPhenom. 208, 90 (2016).6\n[4] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,\nD. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai\nZ Hussain, and Y. L. Chen, Science 343, 864 (2014).\n[5] A. A. Burkov, and L. Balents, Phys. Rev. Lett. 107,\n127205 (2011) .\n[6] B.Yan, andC. Felser, Annu.Rev.Condens.MatterPhys.\n8, 337 (2017).\n[7] S. Patil, V. R. R. Medicherla, R. S. Singh, E. V. Sam-\npathkumaran, and K. Maiti, Europhys. Lett. 97, 17004\n(2012).\n[8] S. Patil, K. K. Iyer, K. Maiti, and E. V. Sampathku-\nmaran, Phys. Rev. B 77, 094443 (2008).\n[9] G. Li, Z. Xiang, F. Yu, T. Asaba, B. Lawson, P. Cai, C.\nTinsman, A. Berkley, S. Wolgast, Y. S. Eo, D.-J. Kim, C.\nKurdak, J. W. Allen, K. sun, X. H. Chen, Y. Y. Wang,\nZ. Fisk, and Lu Li, Science 346, 1208 (2014).\n[10] M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys.\nRev. Lett. 104, 106408 (2010).\n[11] D. J. Kim, S. Thomas, T. Grant, J. Botimer, Z. Fisk,\nand Jing Xia, Sci. Rep. 3, 3150 (2013).\n[12] B. S. Tan, Y.-T. Hsu, B. Zeng, M. C. Hatnean, N. Har-\nrison, Z. Zhu, M. Hartstein, M. Kiourlappou, A. Srivas-\ntava, M. D. Johannes, T. P. Murphy, J.-H. Park, L. Bal-\nicas, G. G. Lonzarich, G. Balakrishnan, and S. E. Sebas-\ntian, Science 349, 287 (2015).\n[13] E. Frantzeskakis, N. de Jong, B. Zwartsenberg, Y. K.\nHuang, Y. Pan, X. Zhang, J. X. Zhang, F. X. Zhang, L.\nH. Bao, O. Tegus, A. Varykhalov, A. de Visser, and M.\nS. Golden, Phys. Rev. X 3, 041024 (2013).\n[14] A. P. Sakhya and K. Maiti, AIP Conf. Proc. 2115,\n030338 (2019).\n[15] A. P. Sakhya and K. Maiti, Sci. Rep. 10, 1262 (2020).\n[16] F. F. Tafti, Q. D. Gibson, S. K. Kushwaha, N. Hal-\ndolaarachchige, and R. J. Cava, Nat. Phys. 12, 272-277\n(2015).\n[17] L.-K. Zeng, R. Lou, D.-S. Wu, Q. N. Xu, P.-J. Guo, L.-\nY. Kong, Y.-G. Zhong, J.-Z. Ma, B.-B. Fu, P. Richard,\nP. Wang, G. T. Liu, L. Lu, Y.-B. Huang, C. Fang, S.-S.\nSun, Q. Wang, L. Wang, Y.-G. Shi, H. M. Weng, H.-C.\nLei, K. Liu, S.-C. Wang, T. Qian, J.-L. Luo, and H. Ding,\nPhys. Rev. Lett. 117, 127204 (2016)\n[18] M. Zeng, C. Fang, G. Chang, Y.-A. Chen, T. Hsieh, A.\nBansil, H. Lin, and L. Fu, arXiv:1504.03492.\n[19] J. Nayak, S.-C. Wu, N. Kumar, C. Shekhar, S. Singh, J.\nFink, E. E. D. Rienks, G. H. Fecher, S. S. P. Parkin, B.Yan, and C. Felser, Nat. Commun. 8, 13942 (2017).\n[20] Y. Wu, T. Kong, L.-L. Wang, D. D. Johnson, D. Mou,\nL. Huang, B. Schrunk, S. L. Bud’ko, P. C. Canfield, and\nA. Kaminski, Phys. Rev. B 94, 081108(R) (2016).\n[21] X. Duan, F. Wu, J. Chen, P. Zhang, Y. Liu, H. Yuan,\nand C. Cao, Commun. Phys. 1, 71 (2018).\n[22] P. Li, Z. Z. Wu, F. Wu, C. Cao, C. Y. Guo, Y. Wu,\nY. Liu, Z. Sun, C.-M. Cheng, D.-S. Lin, Frank Steglich,\nHuiqiu Yuan, Tai-Chang Chiang, and Yang Liu, Phys.\nRev. B98, 085103 (2018).\n[23] Z. Z. Wu, F. Wu, P. Li, C. Y. Guo, Y. Liu, Z. Sun, C.-M.\nCheng, T.-C. Chiang, C. Cao, Huiqiu Yuan, and Yang\nLiu, Phys. Rev. B 99, 035158 (2019)\n[24] A. R. Wright, and R. H. McKenzie, Phys. Rev. B 87,\n085411 (2013).\n[25] P. K. Das, N. Kumar, R. Kulkarni, and A. Thamizhavel,\nPhys. Rev. B 83, 134416 (2011).\n[26] D. R. Hamann, Phys. Rev. 158, 570-580 (1967).\n[27] V. K. C. Liang and C. C. Tsuei, Phys. Rev. B 7, 3215-\n3225 (1973).\n[28] J. K¨ astner and E. F. Wassermann, J. Low Temp. Phys.\n29, 411-429 (1977).\n[29] J. Stankiewicz, M. Evangelisti, P. F. S. Rosa, P.\nSchlottmann, and Z. Fisk, Phys. Rev. B 99, 045138\n(2019).\n[30] F. Hulliger, B. Natterer, and K. R¨ uegg, Z. f¨ ur Phys. B\nCondens. Matter 32, 37-42 (1978).\n[31] D. Shoenberg, Magnetic Oscillations in Metals (Cam-\nbridge University, Cambridge, England, 2009)\n[32] J. Cao, S. Liang, C. Zhang, Y. Liu, J. Huang, Z. Jin,\nZ.-G. Chen, Z. Wang, Q. Wang, J. Zhao, Shiyan Li, Xi\nDai, Jin Zou, Zhengcai Xia, Liang Li, and Faxian Xiu,\nNat. Commun. 6, 7779 (2015)\n[33] Y. P. Li, C. C. Xu, M. S. Shen, J. H. Wang, X. H. Yang,\nX. J. Yang, Z. W. Zhu, C. Cao, and Z. A. Xu, Phys. Rev.\nB98, 115145 (2018).\n[34] K. S. Noveselov, A. K. Geim, S. V. Morozov, D. Jiang,\nM. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and\nA. A. Firsov, Nature (London) 438, 197 (2005).\n[35] A. J. Millis, Phys. Rev. B 48, 7183 (1993).\n[36] S. Patil, V. R. R. Medicherla, R. S. Singh, S. K. Pandey,\nE. V. Sampathkumaran, and K. Maiti, Phys. Rev. B 82,\n104428 (2010).\n[37] P. Gegenwart, Q. Si, and F. Steglich, Nat. Phys. 4, 186\n(2008)."    },    {        "title": "1702.02458v1.A_magnetic_phase_transition_graphene_transistor_with_tunable_spin_polarization.pdf",        "content": "A magnetic phase -transition graphene transistor  with tunable spin \npolarization  \n \nPéter Vancs ó1*, Imre Hagymási2* and Levente  Tapasztó1 \n1. Centre for Energy Research,  Institute of Technical  Physics and Materials Science, 2D \nNanoelectronics „ Lend ület” Research Group, Budapest, Hungary  \n2.  Wigner Research Centre for Physics, Institute of S olid State Physics and Optics,  \nStrongly  Correlated Systems ‘‘Lendület’’ Research Group, Budapest, Hungary  \n* These authors contributed equally to this work.  \n \nAbstract  \nGraphene nanoribbons (GNRs) have been proposed as potential building blocks for field \neffect transistor (FET) devices due to their quantum confinement bandgap. Here, we \npropose a novel GNR device concept, enabling the control of both charge and spin \nsigna ls, integrated within the simplest three -terminal device configuration. In a \nconventional FET device, a gate electrode is employed to tune the Fermi level of the \nsystem in and out of a static bandgap. By contrast, in the switching mechanism proposed \nhere, the applied gate voltage can dynamically open and close an interaction gap, with \nonly a minor shift of the Fermi level. Furthermore, the strong interplay of the band \nstructure and edge spin configuration in zigzag ribbons enables such transistors to carry \nspin polarized current without employing an external magnetic field or ferromagnetic \ncontacts. Using an experimentally validated theoretical model, we show that such \ntransistors can switch at low voltages and high speed, and the spin polarization of the \ncurrent can be tuned fro m 0% to 50% by using the same back gate electrode. \nFurthermore, such devices are expected to be robust against edge irregularities and can \noperate at room temperature. Controlling both charge and spin signal within the \nsimplest FET de vice configuration could open up new routes in data processing with \ngraphene based devices.  \n  Introduction  \nThe ability to manipulate electronic and magnetic sign als is at the heart of our information \nand communication technology. Today the information proc essing relies on our ability to \nprocess  electric signals , while data storage is built on manipulating  magnetic moments . \nDevices that can provide control over both charge and spin signal s could open up new \nprospects for future electronics. Graphene has emerged as a material with strong potential for \nboth electronic and spintronic applications1,2. The presence of a band  gap in the electronic \nband structure is crucial in order to enable digital  electronic applications, con ventiona lly \nbased on devices such as field -effect transistors (FETs). Due to the absence of a  band  gap in \n2D graphene  sheets , several strategies have been proposed to enable the complete  depletion of \ncharge carriers , and therefore a leakage free OFF state of the transistor. Among these \napproaches  the scientific attention quickly  turned towards  graphene nanoribbons (GNRs) , \nnarrow stripes of graphene, as promising candidate s for low-dimensional FETs  with increased \nIon/Ioff ratio3,4,5. The operation of early GNR FET devices fabricated by e -beam lithography \nwas dominated by edge roughness and disorder, which considerably reduces their operation \nspeed, as also evidenced by their low mobility3. However, with the rise  of more precise \nnanofabrication techniques6,7,8,9, as well as bottom -up synthesis10,11,12, such limitations can in \nprinciple be surmounted . Nevertheless , the realization of high mobility electronic devices \nbased on semiconductor GNRs have proven particularly  challenging. Classically, armchair \ngraphene nanoribbons have been proposed as building blocks of FET devices , as ribbons of \nthis orientat ion possess the largest quantum confinement band gap . However, for armchair \nnanoribbons even  a single  atomic row  variation  in their width ca n drastically alter the \nbandgap13. This implies that such devices are particularly sensitive to structural irregulari ties \nand disorder  along their length . We propose here that by using zigzag -edge GNRs (ZGNRs) \nthis limitation can be overcome.  \nThe semiconducting nature  of ZGNRs is related to interaction effects and their spin \npolarized edge states14,15. The semiconducting  state of ZGNRs at half filling, with oppositely \nspin-polarized edges , is robustly  predicted by  several th eoretical models , including density \nfunctional  theory (DFT)15,16,17, the mean -field approximation of the Hubbard model14,18, exact \ndiagonalization and qu antum Monte Carl o simulation19, as well as  density -matrix \nrenormal ization group algorithm (DMRG)20,21. Furthermore the spin polarized edge states, in \nprinciple also enable us to gain access to the spin degree of freedom in such ribbons. It has \nalready  been shown that the magnetic configuration  of ZGNRs can be tuned by using an external magnetic field22,23,24,25 or ferro magnetic electrodes26,27,28,29, which can be  exploited in \nvarious  device applications such as spin valve s, spin diode s, spin transistor and GMR devices . \nHowever, controlling the device by means of external magnetic fields  is technologically  \ncumbersome , as compared to a more desirable electrical control.  In this direction, Son et al.30 \nreported th e concept of a half -metallic ZGNR  transistor s that can carry fully spin polarized \ncurrent, without the application of a magnetic field , only  by using  an in-plane external electric \nfield across the ribbon . In this case  due to the spatially separated  edge states , the transverse \nelectric field closes  the gap for one spin  orientation and widens it for the  other , resulting in \nhalf-metallicity . Several calculations were performed using transverse electric field s for \npossible spintronic applications31,32,33,34. However,  from the device fabrication point of view  \nthe application of a homogeneous  in-plane electric field across a few nanometer wide ribbon \nis technologically quite challenging  and implies a  large lateral device size, which is more \ndifficult to integrate. It would be a huge technological advantage  to control  the spin signal in \nZGNRs by conventional back/top gate electrode s, involving  a perpendicular electric field . \nWe propose that this can be achieved  by exploiting  a novel switching mechanism  in \nnarrow zigzag GNRs enabled by doping the ribbons  (moving away from the half -filling ). It is \nknown that s lightly above the antiferromagnetic  (AFM) ground state energy a ferromagnetic \n(FM)  configuration  exists with parallel spin orientation on both edges16. In contrast  to the \nhalf-filled system, where  the AF M state is a lways more stable, in a doped Z GNR, both AF M \nand FM sates can become energetically favorable. Most importantly, there is an int riguing \ninterconnection of the  spin configuration and the electronic band structure that renders  the \nribbons in the AFM spin configuration insulating15, while FM nanoribbons are metallic17. We \nhave recently shown experimentally that a  sharp transition from semiconducting  (AFM) to \nmetallic (FM)  state can indeed occur  as a function  of the ribbon width35. Howe ver, varying \nthe ribbon width is not a practical parameter for device applications . In this study, by using \nthe same  experimentally validated the oretical model, we show that the sharp semiconductor  \n(AFM)-metal  (FM)  transit ion can also be induced through  changi ng the carrier density  (𝑛2𝐷) \nof the Z GNR18,36,37. Our calculations, based on the Hubbard model and the Landauer transport \nformalism, reveal that the novel switching mechanism based on magnetic phase transition \nenables  excellent FET characteristics, and the device can also carry spin -polarized current. \nThis is enabled by  the fact that in contrast  to the conventional  FETs, where upon gating the \nelectr onic structure remains unchanged, only the charge density is modulated,  in an \ninteracting system the density of states depends strongly on the electron density . The main \nadvantage of our device concept  is that the carrier density  in ZGNR  can be easily tuned using a back gate electrode , therefore  neither external magneti c field nor transverse electric field is \nrequired  to control both the charge  and spin currents  in the device . \n \nResults  and discussion  \nFigure 1 shows the schematic  idea of the proposed  device  concept , comprising  two \nnonmagnetic metal contacts and a back -gate electrode . A long, narrow , zigzag graphene \nnanoribbon  of the width w is placed on a gate insulator  which has a thickness d and a relative \ndielectric constant  𝜀𝑟.  \nFirst, we calculate t he electronic and magnetic properties of the Z GNR constituting the \nchannel material  of the device using  the mean -field approximation of the Hubbard -model \naccounting for  both the doping of the ribbon and the effect of finite temperature:  \nℋ=−∑𝑡𝑖𝑗𝑐̂𝑖𝜎†\n〈𝑖,𝑗〉𝑐̂𝑗𝜎+𝑈∑𝑛̂𝑗↑𝑛̂𝑗↓\n𝑗−𝜇𝑁̂. \nThe fi rst term is the nearest -neighbor tight -binding Hamiltonian , while the second term  \naccounts for the onsite Coulomb repulsion.  The effects of temperature and doping  have been \nincluded in the third term through  the g rand canoni cal ensemble , where 𝜇 is the chemical \npotential, and 𝑁̂ the particle number operator. The electron densit y and the chemical potential \nwere  determined self -consistently  with the interaction parameter U = 3.24 eV estimated from \nexperiments35 (see Supplementary Information) . In the calculations we  considered hydrogen -\nsaturated and relaxed edges similar to first -principles DFTcalculations15. A nanoribbon is a \nquasi -one-dimensional system, where quantum fluctuations can be significant;  hence , it is not \nstraightforward that mean -field theory can  give an accurate description  of the system . \nHowever, for wide enough ribbons – like in our case – it has been shown based on QMC \nresults38 that mean -field theory gives quantitatively good description in the weakly interacting \nlimit, U < 5.4 eV.  \nUsing the above  theoretical model, in a previous work, we were able to quantitatively \ninterpret the sharp semiconductor -metal transition experimentally observed as a function of \nribbon width in zigzag graphene nanoribbons defined by STM Lithograp hy35. Now, using the \nsame theoretical model , we foun d that the semiconductor (AFM) -metal (FM)  phase transition \ncan also occur as a function o f charge carrier concentration.  This finding is of huge  practical \nimportance , as the charge carrier concentration can be easily tuned, in contrast to the ribbon \nwidth, enabling us to exploit the opening of an interaction gap associated with the phase \ntransit ion in device applications . Figure 2 a shows the phase diagram of ZGNRs as a function of ribbon width and charge carrier density , as calculated from the Hubbard Hamiltonian \nabove . Due to the fact that our lattices are bipartite, pa rticle -hole symmetry is present;  \ntherefore , our results are valid both for the n-type or p-type doping. The most  important  \nfinding is that for realistic ribbon widths of orders of nanometers, the phase transition occurs \nin the low  doping regime ( 𝑛2𝐷 ~ 1012 cm-2) which  can be ea sily accessed experimentally  by \nsimple electrostatic gating . Concerning the magnetic properties of such ZGNRs, in the  low \ndoping regime  we have found only two types of magnetic phases, AFM and FM, which are \nthe result of the competition between band and interaction energies . The ef fect of doping can \nbe understood  by considering that the extra electrons start occupying states of the conduction \nband minima . However, in the AFM configu ration, due to the presence of a significant  band \ngap, this is energetical ly quite unfavorable. Consequently, when the doping reaches a critical \nvalue, it is more favorable for the system to switch to a metallic state and gain band energy, \nby changing the spin configuration on the edges from AFM to FM . As can be seen fro m the \nphase diagram in Fig 2a , the critical doping value depends on the ribbon width and the \ntemperature. By decreasing the width of the ribbons the AFM/FM switching occurs at higher \ndoping level s. Regarding the temperature, the AFM -FM phase transition is shifted towards the \nhigher doping values as the temperature is increased. For example in a 20-ZGNRs (ZGNR \nthat is twenty carbon zigzag lines wide, w= 4.2 nm) the critical doping value is 1.6×1012 cm-2 \nat 𝑇=60K compared to 2.6×1012 cm-2 at 𝑇=300K, which means that the AFM ground state is \nrealized in a wider parameter range at higher temperatures . In Figure 2b we plot the band gap \nvalues  (𝐸𝑔) and the total magnetic moment  (𝑀= 1/2(𝑛↑−𝑛↓) geµB) per unit length ( 𝑎0=0.246 \nnm) as a function of the doping for a 20-ZGNR at 𝑇=300K. In the AFM configuration the \nband gaps were found to be about  𝐸𝑔 ~ 0.25 𝑒𝑉 with zero ( 𝑀=0) total magnetic moment \nresulting from the opposite spin directions at the ed ges. By contrast,  in the FM config uration \nthe emerging net magnetic moment is about 𝑀 ~ 0.6 µ B/ao (Bohr magnetron per unit length ), \nwith no band gap ( 𝐸𝑔=0 𝑒𝑉). The band structures shown in the insets o f Fig 2b correspond to \nthe 𝑛2𝐷=1×1012 cm-2 (AFM) and 𝑛2𝐷=4×1012 cm-2 (FM)  case, respectively. We found that in \nthe low doping regime the band gaps and magnetic moments display only a small variation as \na function of carrier density, except near the transition point. It is also worth mentioning that \nin the high er doping regime  further magnetic phases appear18 until the edge -state bands are \nfilled and the ribbon  become s paramagnetic (near 1014 cm-2 doping values)18,39. However, \nsuch doping levels are difficult to achieve in a solid gate device configuration . To calculate  the device characteristics  besides the properties of the ZGNRs, the gate \ncapacitance  (𝐶=𝑒𝜕𝑛2𝐷𝜕𝑉𝐺⁄ ), which is the relationship between the applied gate \nvoltage (𝑉𝐺) and the induced carrier density  (𝑛2𝐷) at room temperature , has also to be \naccurately determined . Our d etailed calculations reveal that besides the classical capacitance  \n(𝐶𝐶), the quantum contribution  (𝐶𝑄) is also significant , as it is  often the case in nanoscale \ndevices40 (for details see Supplementary Information).  \nThe quantum capacitance of the ZGNRs was numerically derived from our Hubbard \ncalculations, while the classical capacitance of a w width  ZGNR  was calculated from  classical \nelectrostatics41,42  leading to  the following expression:  \n𝐶𝐶=𝜀0𝜀𝑟𝜋[2𝑑arctan (𝑤\n4𝑑)+𝑤\n4ln(1+16𝑑2\n𝑤2)]−1\n, \nwhere d and 𝜀𝑟 are the thickness and the relative dielectric constant  of the insulator. This \nanalytical expression for 𝐶𝐶 is valid as long as the surface charge density of the ZGNR  is \nroughly constant,  which is fulfilled in our low doping regime calculations. In addition, this \nexpression describes quite accurate ly the exp erimental results of back gated graphene \nnanoribbons43. Figure 3  shows t he calculated 𝑛2𝐷 as a function of 𝑉𝐺 for different oxide s SiO 2 \n(𝜀𝑟=3.9) and HfO 2 (𝜀𝑟=47)44 in the case of 20 -ZGNR  at 𝑇=300K. Due to our device geometry, \nwhere the ZGNR is interfaced by the gate  insul ator and  air, we used an effective dielectric \nconstant in the calculations, which is the average value of the dielectric constants of substrate \nand air . For a SiO 2 insulating layer with  d=300 𝑛𝑚 and d =30 𝑛𝑚 we found that the  AFM -FM \ntransition  (𝑛2𝐷=2.6×1012 cm-2) occurs at  𝑉𝐺=0.96 𝑉and 𝑉𝐺=0.67 𝑉, respectively . As for both \nlayer thicknesses  𝐶𝐶≪𝐶𝑄, a linear behavior is obtained as a function of  𝑛2𝐷(𝑉𝐺). In contrast, \n𝐶𝐶≫𝐶𝑄 for a HfO 2 insulating layer with  d=30 𝑛𝑚, therefore the nonlinear behavior of 𝑉𝑄 \n(see Figure S1 ) dominates in  the 𝑛2𝐷(𝑉𝐺) function. The phase transition for HfO 2 layer was \nfound to occur at 𝑉𝐺=0.17  𝑉. One can also note  that the values of the 𝑛2𝐷 jump at the critical \ngate voltage  value where the first order magnetic phase transition takes place. From practical \npoint of view, the most important finding is that such devices can already switch at very low \ngate voltages. Furthermore, the critical value of the gate voltage, at which th e AFM -FM \ntransition occurs, can be further decreased by reducing the thickness of the oxide -layer or \nincreasing the dielectric constant. By changing the width of the ZGNR we found that the \ncritical value of the gate voltage further decreases when w is incr eased  as also apparent f rom \nFigure 2a , where wider ribbons are characterized by  lower critical doping values . Overall, we can conclude that ZGNR transistor s can switch  in the range of 𝑉𝐺≈ 0.1-1 𝑉, depending on the \nwidth of the ZGNR and the prop erties of the gate insulator layer.  \nTo explore  the transport characteristics  of the proposed novel device, featuring  a magnetic \nphase transition and interaction band gap , the interaction effects need  to be incorporated  into \nthe transport calculations. This  is not straightforward , as the widely used Landauer formalism \nwas developed for non -interacting Hamiltonians . However, by using the  mean -field \napproximation , the original many -body problem can be  mapped to an effective single -particle \nHamiltonian to which  the Landauer’s method can be applied45. We have  include d into our \ncalculations the effe ct of e lectron interactions by  using the spin dependent Hamiltonian matrix \nelements from our Hubbard calculations . The conductance calculations  between the source \nand drain contacts  have been p erformed based on the Landauer formalism using the standard \nrecursive Green’s function technique : \n𝐺𝜎=−𝑒2\nℎ∫𝑑𝐸∑𝑇𝑖𝑗𝜎(𝐸)𝜕𝑓(𝐸−𝐸𝐹)\n𝜕𝐸\n𝑖𝑗, \nwhere 𝜎=↑,↓ is the spin index, 𝑇𝑖𝑗(𝐸) is the transmission coefficient fro m sub -band j in the \nsource to sub-band i in the drain electrode at energy E, while f is the Fermi function.  Figure 4 \nshow s the spin dependent conductance 𝐺𝜎 as a function of charge carrier density  𝑛2𝐷 for the \ndefect -free 20 -ZGNR at 𝑇=0K and 𝑇=300K.  \nAt T=0K (Figure 4a) below the phase transition carrier concentration the conductance is \nzero according to the presence of the band gap in the AFM configuration (Figure 4c). \nReaching the critical value of doping  (𝑛2𝐷=1.6×1012 cm-2) the ZGNR become s metallic (FM \nconfiguration) due to the open conduction channel s for both up and down  spins at the Fermi \nenergy (Figure 4d). The equal number of conduction channels for the different spin directions \nimplies that the current is not spin polarized even though the ribbon itself  is spin polarized \n(𝑛↑>𝑛↓). However , the current can become spin  polarized by further increasing the charge \ncarrier concentration (gate voltage) to 𝑛2𝐷=8.7 ×1012 cm-2 where additional conductance \nchannels open for spin down while the number of conductance channel for spin up state \nremains the same. This asymmetry of t he band structure for  up and down spins  in the FM \nconfiguration (Figure 4e) enables our device to carry spin polarized current. Consequently, \nour device layout  does not require the use of an external magnetic field , ferromagnetic \ncontacts  or transvers electric fields to achieve spin polarized current , which is a huge \ntechnological advantage . Another intriguing property of our device is that the charge and the \nspin currents can be independently tuned by the charge carrier density (sweeping the gate volta ge). This is due to the somewhat counter -intuitive fact that after switching the ribbon into \nthe FM phase, although the spin polarization (net magnetic moment) of the ribbon becomes \nnonzero, the spin polarization of the current still remains zero. This can  be understood in the \nLandauer formalism, where  the product of electron velocity ( v~𝜕𝐸𝑘 /𝜕𝑘) and density \n(𝑛~𝜕𝑘/𝜕𝐸) in the current  (𝐼=𝑒𝑛𝑣) cancels  the 𝜕𝐸/𝜕𝑘 term, thus the conductance is \ndetermined only by the number of the open conductance  channels , independently of the shape \nof the dispersion curve.  \nIn Figure 4b we plot the conductance of the same 20 -ZGNR at room temperature \n(T=300K). The effect of the temperature has two important effects: (1) i t shift s the position of \nthe AFM -FM (semicon ductor -metal) transition, and (2) it modifies the shape of the \nconductance plateaus due to the broadened energy distributions of the incident electrons \ndescribed by the Fermi -function. In contrast to the step -like transition of both charge and spin \ncurrent  for the T=0  K case, at room temperature  both transitions become gradual, due to \ncombined effect of the temperature and doping. In contrast to the sharp sw itching of spin \npolarization fro m 0% to 50% at low temperatures (0K), at room temperature the spin \npolarization degree of the current can be continuously tuned between 0 and 50% by sweeping \nthe back -gate voltage. This provides us a fully electrical control through a simple back gate \nelectrode over the spin polarization degree of the current, which could be exploited in future \nspintronic devices. In principle, such a ribbon could act as spin source replacing \nferromagnetic electrodes, which would be a huge technologic al advantage and on the long run \nenabling  all-graphene spintronic circuits2. \nBeside  the tunable spin polarization, the FET characteristics of the device are also \ninteresting as it  employs a  novel switching mechanism.  Instead of tuning the Fermi level in \nand out of the static bandgap of a channel material,  we can dynamically open a nd close an \ninteraction gap, with only a  minor shift of the Fermi leve l. The calculated I on/Ioff ratio of a 20-\nZGNR  device  is about ~105 at T=150K, but decreases  to around ~102 at room temperature. \nHowever, this value at room temperature can be significantly increased by  using narrower \nZGNR characterized by larger  band gap s. For instance a lready a  10-ZGNR ( w = 2.1 nm ) has \nan Ion/Ioff ratio of about 104 at room temperature , while this value reaches 107 at T=150K. To \ncharacterize the switching speed of such a device, w e have also calculated its subthreshold \nswing 𝑆=𝑑𝑉𝐺/(𝑑(log 𝐼𝑑)). The 𝑆 value  is defined to be the amount of change in gate voltage \n(𝑉𝑔) required to produce a 10x change in drain  current ( 𝐼𝑑) in the low 𝑉𝐺 region , where the \ndrain current has an exponential dependence. From the calculated gate voltage and \nconductance values we can estimate the subthreshold swing of our 20 -ZGNR device to 𝑆=81 𝑚𝑣/𝑑𝑒𝑐 in the case of 30 𝑛𝑚 HfO 2 layer at room temperature . Choosing  narrower ribbons \n(10-ZGNR ) this value is further decreased to  𝑆=71 𝑚𝑣/𝑑𝑒𝑐. The low 𝑆 values are  \ncomparable with recent ly reported MOSFETs  (𝑆=77 𝑚𝑣/𝑑𝑒𝑐)46 and single layer MoS 2 \ntransistors (𝑆=74 𝑚𝑣/𝑑𝑒𝑐)47 approaching the fundamental thermionic limitation of the \nsubthreshold swing48 (𝑆𝑚𝑖𝑛 =60 𝑚𝑣/𝑑𝑒𝑐) in conventional FETs at room temperature.  Finally , \nwe note that the large I on/Ioff ratio and low subthreshold swing values of our device are similar \ncompared to previously reported GN R FETs based on more complex  structures including both \narmchair - and zigzag -edge segments49. \nThe above result s clearly show that both the electronic and spintronic properties of such \ndevices are predicted to be exceptional . However, from practical consideration it is very \nimportant to  assess how robust such properties are against deviations from the ideal \nconditions. So far our calculations have been performed on ZGNRs with perfect edges. In \nreality , several types of deviation s from the ideal zigzag edge structure can occur50,51,52. Since  \nthe operation  mechanism of our device is based on the edge magnetism, edge defects and \nreconstruct ions that can completely suppress  the spins at the edges , in principle, render our \ndevice inoperable. Nevertheless, the experimental observation of the AFM/FM transition35 \nevidences that such defects are not commonly encountered.  STM lithography7,35 is an \neffective nanofabrication technique  to define graphene nanoribbons with nanometer precision \nand well -defined crystallographic edge orientations . Therefore we have considered the most \nlikely edge irregularities  resulting from  this process, namely, atomic scale step -like defects  \n(Figure 5a inset), along an  overall zigzag edge orientation.  We define the defect concentration \n(𝑥𝑑⁄) as the ratio of the length of the protrusion  (x=2∙𝑎0) and the length of the unit cell \n(d=n∙𝑎0), thus Figure 5a inset shows the geometry of 20 -ZNGR with 40%  edge  defect \nconcentration (n=5). We found that up to  such high edge defect concentra tions the ribbon \nedges  still become spin polarized  in the half -filled case , although  the magnitude of magnetic \nmoment at the edges significantly decreases (Medge=0.069 µ B/ao) as compared to the defect -\nfree 20 -ZGNR ( Medge=0.28 2 µ B/ao) (Fig 5a).  We note that a highly similar decrease of the \nedge magnetic moments was previously predicted  for simple edge vacancies and impurities in \nGNRs53. The b and structure (Fig 5b) and t ransport calculations on  the highly defective 20 -\nZGNR at room temperature confirmed the  persistence  of the  semiconductor -to-metal \ntransition  as a function of  carrier density , and the current flowing through the device becomes \nspin polarized in a similar manner as for the defect -free ZGNRs. Consequently, the spin -\nrelated device properties are predicted to be particularly robust against atomic scale edge \ndisorder, as they persist  up to extremely high edge defect concentrations of 40%.  It is also very important to note, that the change in the ribbon width introduced by the edge defects \nonly slightly changes the bandgap of such a ZGNR. This is in striking contrast with the case \nof armchair ribbons, where one atomic wide edge steps can induce an order of magnitude \nchange in the band gap along the ribbon13. Consequently , not on ly the spin but also  the \nelectronic transport properties of ZGNRs are expected to be much more robust against edge \ndisorder as compared to their armchair counterparts.  \n \n \nConclusion  \nIn summary, we have  identified a novel switching mechanism in FET devices based on \nzigzag graphene nanoribbons, and  demonstrated its feasibility to control both charge and spin \ncurrents by using  a simple gate electrode. We have shown that such devices can switch a low \ngate voltages (0.1 V - 1 V), and high speed (S = 71 𝑚𝑣/𝑑𝑒𝑐.), while the spin polarization of \nthe current can be tuned between 0 and 50% by the same  gate electrode. We have also shown \nthat the proposed electronic  and spintronic device characteristics are  particularly robust \nagainst edge dis order rendering their operation feasible under realistic conditions even at  \nroom temperature. We believe that the device concept presented here allows  the simple  \nrealizati on of  graphene circuits, enabling the integrated control  of both charge and spin \nsignal s within the same device that could open new rout es for novel data processing \ntechniques.  \n \nAcknowledgements: The work has been  supported by  the ERC  StG and the  “Lend ület” \nprogramm e of the  Hungarian  Academy  of Sciences grants LP2014 -14, as well as  the Korea \nHungary Joint Laboratory for Nanosciences . L.T. acknowledges  OTKA  grant  K10875 . I.H. \nwas supported in part by the  Hungarian Resear ch Fund (OTKA) through Grant K120569 . \nAuthor contributions:  L.T. proposed the device concept. I.H. and P.V. performed the \ntheoretical calculations. V.P. and L.T. wrote the paper. All authors discussed the results and \ncommented on the manuscript.  \n  Figures  \n \n \n \nFigure 1. Schematic represent ation of the three states  of the proposed ZGNR  FET: (a) an \ninsulating OFF state with antiparallel spin orientation on opposite ribbon edges in the absence \nof a gate bias, (b) t he conducting ON  state characterized by parallel spin orientation on ribbon \nedge s but no net spin polarization of the current  induced by an applied gate voltage, and (c) \nspin polarized ON state  achieved by further increasing the gate potential . (VG values \ncorrespond to 30 𝑛𝑚 HfO 2 oxide layer at  room temperature .) \n(a) (b) \n(c)      \nFigure 2 . (a) Phase diagram of ZGNRs as a function of charge carrier concentration  (𝑛2𝐷) \nand ribbon width ( w) at two different temperatures  (T=60K and T=300K ). The AFM ground \nstate is the only stable without external doping; however,  the FM configuration can become \nfavorable above  a critical  doping  value and ribbon width . (b) A sharp transition in the b and \ngap and magnetic moment  of a ZGNR  (w=4.2 nm) occurs as a function of charge carrier \ndensity at room temperature . Inset s correspond to the band structures  for one spin  orientation  \nof the 20-ZGNR at 𝑛2𝐷=1×1012 cm-2 (AFM state) and at 𝑛2𝐷=4×1012 cm-2 (FM state) . \n \n \nFigure 3 . Carrier density  (𝑛2𝐷) in 20-ZGNR  (w=4.2 nm) as a function of gate voltage  (𝑉𝐺) for \ndifferent oxide thicknesses  at room temperature . Such a  device can switch at a critical charge \ndensity marked by dashed line, corresponding to very low gate voltages.  \n \n(a) (b)     \n \nFigure 4 . Spin dependent c onductance  (𝐺𝜎) as a function of charge carrier density  (𝑛2𝐷) for a \n20-ZGNR  at (a) T=0K  and (b) T=300K. The AFM -FM transition  (dashed line)  dynamically \ncloses the bandgap of the ribbon switching the transistor into a conducting state. Further \nincreasing the car rier concentration opens additional spin down channels leading to th e spin \npolarization of the current . (c-e) Spin resolved band structure in the three operation regimes  \nfor: 0 cm-2, 3×1012 cm-2, and 1013 cm-2 at T=0K, respectively.  \n      \nFigure 5  (a) Edge m agnetic moment s (Medge) as a function of edge  defect concentration  for \n20-ZGNR . Inset shows the geometry of 20 -ZNGR with 40% edge defect concentration. \nDashed line corresponds to edge magnetic moment of the defect -free 20 -ZGNR. (b) Band gap \nand magnetic moment  of highly defective (40%) 20 -ZGNR  at room temperature as a function \nof doping . Inset s correspond to the band structures for one spin orientation  in the AFM \n(𝑛2𝐷=5×1012 cm-2) and FM ( 𝑛2𝐷=9×1012 cm-2) state, respectively.  \n \n(a) (b) \n(a) (b) (c) (d) (e) References  \n                                                           \n1 K. S. Novoselov, V. I. Fal'ko, L. Colombo, P. R. Gellert, M. G. Schwab, K. Kim, A roadmap for graphene. \nNature 490, 192 –200 (2012).  \n \n2 W. Han, R. K. Kawakami, M. Gmitra , J. Fabian, Graphene spintronics. Nat. Nanotechnol. 9, 794 –807 (2014).  \n \n3 Z. Chen, Y. -M. Lin, M. Rooks, P. Avouris, Graphene nano -ribbon electronics. Physica E 40, 228 -232 (2007).  \n \n4 M. Y. Han, B. Ö zyilmaz, Y. Zhang, P. Kim, Energy Band -Gap Engineering of  Graphene Nanoribbons. Phys. \nRev. Lett. 98, 206805 (2007).  \n \n5 X. Wang, Y. Ouyang, X. Li, H. Wang, J. Guo, H. Dai, Room -Temperature All -Semiconducting Sub -10-nm \nGraphene Nanoribbon Field -Effect Transistors. Phys. Rev. Lett. 100, 206803 (2008).  \n \n6 X. Li, X. Wang, L. Zhang, S. Lee, H. Dai, Chemically derived, ultrasmooth graphene nanoribbon \nsemiconductors. Science 319, 1229 –1232 (2008).  \n \n7 L. Tapasztó, G. Dobrik, Ph. Lambin, L. P. Biró, Tailoring the atomic structure of graphene nanoribbons by \nscanning tunnell ing microscope lithography. Nat. Nanotechnol. 3, 397 –401 (2008).  \n \n8 L. Jiao, L. Zhang, X. Wang, G. Diankov, H. Dai, Narrow graphene nanoribbons from carbon nanotubes. Nature \n458, 877 (2009).  \n \n9 X. Wang, H. Dai, Etching and narrowing of graphene from the edges. Nature Chem. 2, 661 –665 (2010).  \n \n10 J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. \nFeng, K. Müllen, R. Fasel, Atomically precise bottom -up fabrication of graphene nanoribbons. Nature 466, 470 –\n473 (2010).  \n \n11 Y.-C. Chen, D. G. de Oteyza, Z. Pedramrazi, C. Chen, F. R. Fischer, M. F. Crommie, Tuning the band gap of \ngraphene nanoribbons synthesized from molecular precursors. ACS Nano 7, 6123 –6128 (2013).  \n \n12 H. Sakaguchi, Y. Kawagoe, Y. Hirano, T. Iruka, M. Yano, T. Nakae , Width -controlled sub -nanometer \ngraphene nanoribbon films synthesized by radical -polymerized chemical vapor deposition. Adv. Mater. 26, \n4134 –4138 (2014).  \n \n13 K. Nakada , M. Fujita, G. Dresselhaus, M. S. Dresselhaus, Edge state in graphene ribbons: Nanometer size \neffect and edge shape dependence. Phys. Rev. B 54, 17954 -17961 (1996).  \n \n14 M. Fujita , K. Wakabayashi, K. Nakada,  K. Kusakabe, Peculiar Localized State at Zigzag G raphite Edge. J. \nPhys. Soc. Jpn. 65, 1920 -1923  (1996).  \n \n15 Y.-W. Son, M. L. Cohen, S. G. Louie, Energy Gaps in Graphene Nanoribbons. Phys. Rev. Lett.  97, 216803 \n(2006).  \n \n16 H. Lee, Y. -W. Son, N. Park, S. Han, J. Yu, Magnetic ordering at the edges of graphitic fragments: Magnetic \ntail interactions between the edge -localized states. Phys. Rev. B 72, 174431 (2005).  \n \n17 L. Pisani, J. A. Chan, B. Montanari, N. M. Harrison, Electro nic structure and magnetic properties of graphitic \nribbons. Phys. Rev. B 75, 064418 (2007).  \n \n18 J. Jung, A. H. MacDonald, Carrier density and magnetism in graphene zigzag nanoribbons. Phys. Rev. B 79, \n235433 (2009).  \n \n19 H. Feldner, Z. Y. Meng, A. Honecker , D. Cabra, S. Wessel, F. F. Assaad, Magnetism of finite graphene \nsamples: Mean -field theory compared with exact diagonalization and quantum monte carlo simulations. Phys. \nRev. B 81, 115416 (2010).                                                                                                                                                                                       \n \n20 T. Hikihara, X. Hu, H. -H. Lin, C. -Y. Mou, Ground -state properties of nanographite systems with zigzag edges. \nPhys. Rev. B  68, 035432 (2003).  \n \n21 I. Hagymási, Ö. Legeza, Entanglement, excitations and correlation effects in narrow zigzag graphene \nnanoribbons. Phys. Rev. B 94, 165147 (2016) . \n \n22 F. Mu ñoz-Rojas, J. Fernández -Rossier, J. J. Palacios, Giant Magnetoresistance in Ultrasmall Graphene Based \nDevices. Phys. Rev. Lett. 102, 136810 (2009).  \n \n23 Y.-T. Zhang, H. Jiang, Q. Sun, X. C. Xie, Spin polarization and giant magnetoresistance effect induced by \nmagnetization  in zigzag graphene nanoribbons . Phys. Rev. B 81, 165404 (2010).  \n \n24 R. Farghadan, E. Saievar -Iranizad, Spin -polarized transport in zigzag -edge graphene nanoribbon junctions. J. \nAppl. Phys. 111, 014304 (2012).  \n \n25 Z. F. Wang, F. Liu, Giant magnetoresistance in zigzag graphene nanoribbon. Appl. Phys. Lett. 99, 042110 \n(2011).  \n \n26 W. Y. Kim, K. S. Kim, Prediction of very large values of magnetoresistance in a graphene nanoribbon device. \nNat. Nanotechnol. 3, 408 –412 (2008).  \n \n27 M. Zeng, L. Shen, M. Zhou, C. Zhang, Y. Feng, Graphene -based bipolar spin diode and spin transistor: \nRectification and amplification of spin -polarized current. Phys. Rev. B 83, 115427 (2011).  \n \n28 M. Zeng, L. Shen, H. Su, C. Zhang, Y. Feng, Graphene -based s pin logic gates. Appl. Phys. Lett. 98, 092110 \n(2011).  \n \n29 J. Cao, C. P. Tang, S. -J. Xiong, A Datta -Das transistor and conductance switch based on a zigzag graphene \nnanoribbon. J. Appl. Phys. 113, 054304 (2013).  \n \n30 Y.-W. Son, M. L. Cohen, S. G. Louie, Hal f-metallic graphene nanoribbons.  Nature 444, 347 -349 (2006).  \n \n31 J. Guo, D. Gunlycke, C. T. White, Field effect on spin -polarized transport in graphene nanoribbons. Appl. \nPhys. Lett. 92, 163109 (2008).  \n \n32 J. Jung, A. H. MacDonald, Magnetoelectric  coupling in zigzag graphene nanoribbons. Phys. Rev. B 81, \n195408 (2010).  \n \n33 G. P. Tang, J.  C. Zhou, Z.  H. Zhang, X.  Q. Deng, Z.  Q. Fan, A theoretical investigation on the possible \nimprovement of spin -filter effects by an electric field for a zigzag graphe ne nanoribbon with a line defect. \nCarbon 60, 94–101 (2013).  \n \n34 F. J. Culchac, R. B. Capaz, A. T. Costa, A. Latgé, Magnetic response of zigzag nanoribbons under electric \nfields, J. Phys. Condens. Matter 26, 216002 (2014).  \n \n35 G. Z. Magda, X. Jin, I. Hagymási , P. Vancsó, Z. Osváth, P. Nemes -Incze, C. Hwang, L. P. Biró,  L. Tapasztó, \nRoom -temperature magnetic order on zigzag edges of narrow graphene nanoribbons. Nature 514, 608 -611 \n(2014).  \n \n36 K. Swada, F. Ishii, M. Saito, S. Okada, T. Kawai, Phase control of gra phene nanoribbon by carrier doping: \nappearance of noncollinear magnetism. Nano Lett. 9, 269 –272 (2009).  \n \n37 S. Dutta, K. Wakabayashi, Tuning Charge and Spin Excitations in Zigzag Edge Nanographene Ribbons. Sci. \nRep. 2, 519 (2012).  \n \n38 H. Feldner, Z. Y. Meng,  T. C. Lang, F. F. Assaad, S. Wessel, A. Honecker, Dynamical Signatures of Edge -\nState Magnetism on Graphene Nanoribbons. Phys. Rev. Lett. 106, 226401 (2011).  \n                                                                                                                                                                                      \n39 F. Wu, E.  Kan, H. Xiang, S. -H. Wei, M. -H. Whangbo , J. Yang, Magnetic states of zigzag graphene \nnanoribbons from first principles. Appl. Phys. Lett. 94, 223105 (2009).  \n \n40 D. L. John, L. C. Castro, D. L. Pulfrey, Quantum capacitance in nanoscale device modeling. J. Appl. Phys. 96, \n5180 (2004).  \n \n41 A. A. Shy lau, J. W. Kłos, I. V. Zozoulenko, Capacitance of graphene nanoribbons. Phys. Rev. B 80, 205402 \n(2009).  \n \n42 R. Reiter, U. Derra, S. Birner, B. Terrés, F. Libisch, J. Burgdörfer, C. Stampfer , Negative quantum capacitance \nin graphene nanoribbons with lateral gates. Phys. Rev. B 89, 115406 (2014).  \n \n43 Y.-M. Lin, V. Perebeinos, Z. Chen, P. Avouris, Electrical observation of subband formation in graphene \nnanoribbons. Phys. Rev. B 78, 161409(R) (20 08). \n \n44 P. K. Parka,  S.-W. Kangb, Enhancement of dielectric constant in HfO 2 thin films by the addition of Al 2O3. \nAppl. Phys. Lett . 89, 192905 (2006).  \n \n45 S. Ihnatsenka, G. Kirczenow , Effect of electron -electron interactions on the electronic structure and \nconductance of graphene nanoconstrictions. Phys. Rev. B 86, 075448 (2012).  \n \n46 V. Chobpattana, C. -Y. Huang, B. J. Thibeault, W. Mitchell, S. Stemmer, A. C. Gossard, M. J. W. Rodwell , \nRecord Ion (0.50 mA/µm at VDD = 0.5 V and Ioff = 100 nA/µm) 25 nm -gate-length ZrO2/InAs/InAlAs \nMOSFETs. 2014 Symposium on VLSI Technology (VLSI -Technology): Digest of Technical Papers. \ndoi:10.1109/VLSIT.2014.6894363  \n \n47 B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, A. Kis, Single -layer MoS2 transistors. Nat. \nNanotechnol. 6, 147 –150 (2011).  \n \n48 P. M. Solomon, Inability of single carrier tunneling barriers to give subthermal subthreshold swings in  \nMOSFETs. IEEE Electron Device Lett. 31, 618 –620 (2010).  \n \n49 Q. Yan , B. Huang, J. Yu, F. Zheng, J. Zang, J. Wu, B. -L. Gu, F. Liu, W. Duan, Intrinsic Current -Voltage \nCharacteristics of Graphene Nanoribbon Transistors and Effect of Edge Doping. Nano Lett. 7, 1469 -1473 \n(2007).  \n \n50 T. Wassmann , A. P. Seitsonen , A. M. Saitta, M. Lazzeri, F. Mauri, Structure, stability, edge states, and \naromaticity of graphene ribbons. Phys. Rev. Lett. 101, 096402 (2008).  \n \n51 A. P. Seitsonen, A. M. Saitta, T. Wassmann, M. Lazzeri , F. Mauri, Structure and stability of graphene \nnanoribbons in oxygen, carbon dioxide, water, and ammonia. Phys. Rev. B 82, 115425 (2010).  \n \n52 K. Kim, S. Coh, C. Kisielowski, M. F. Crommie, S. G. Louie, M. L. Cohen, A. Zettl , Atomically perfect torn \ngraphene edges and their reversible reconstruction. Nature Commun. 4, 2723 (2013).  \n \n53 B. Huang, F. Liu, J. Wu, B. -L. Gu, W. Duan, Suppression of spin polarization in graphene nanoribbons by \nedge defects and impurities. Phys. Rev. B 77, 153411 (2008).  "    },    {        "title": "0710.5717v1.Low_momentum_ring_diagrams_of_neutron_matter_at_and_near_the_unitary_limit.pdf",        "content": "arXiv:0710.5717v1  [nucl-th]  30 Oct 2007Low-momentum ring diagrams\nof neutron matter at and near the unitary limit\nL.-W. Siu, T. T. S. Kuo\nDepartment of Physics and Astronomy, Stony Brook Universit y, NY 11794-3800, USA\nR. Machleidt\nDepartment of Physics and Astronomy, University of Idaho, M oscow, ID 83844, USA\nWe study neutron matter at and near the unitary limit using a l ow-momentum ring diagram ap-\nproach. By slightly tuningthe meson-exchange CD-Bonn pote ntial, neutron-neutronpotentials with\nvarious1S0scattering lengths such as as=−12070fmand +21 fmare constructed. Such potentials\nare renormalized with rigorous procedures to give the corre sponding as-equivalent low-momentum\npotentials Vlow−k, with which the low-momentum particle-particle hole-hole ring diagrams are\nsummed up to all orders, giving the ground state energy E0of neutron matter for various scattering\nlengths. At the limit of as→ ±∞, our calculated ratio of E0to that of the non-interacting case\nis found remarkably close to a constant of 0.44 over a wide ran ge of Fermi-momenta. This result\nreveals an universality that is well consistent with the rec ent experimental and Monte-Carlo compu-\ntational study on low-density cold Fermi gas at the unitary l imit. The overall behavior of this ratio\nobtained with various scattering lengths is presented and d iscussed. Ring-diagram results obtained\nwithVlow−kand those with G-matrix interactions are compared.\nPACS numbers: pacs\nI. INTRODUCTION\nBack in 1999, Bertsch[1] formulated a many-body\nproblem, asking: what are the ground state properties\nof a two-species fermion system that has a zero-range in-\nteractionandaninfinitescatteringlength? Suchproblem\nwas originally set up as a parameter-free model for a fic-\ntitious neutron matter. Recently, as the experiments on\ntrapped cold alkali gas undergo huge breakthroughs, de-\ngenerate Fermi gas with a tunable scattering length (in-\ncluding±∞) becomes accessible in laboratories[2]. Since\nthen cold Fermi systems have aroused growing attention.\nThe term ‘unitary limit’ has been used by many au-\nthors to refer to the special scenarioin a low-density two-\nspecies many-body system where the scattering length\nbetween particles approaches infinity. More specifically,\nat the unitary limit, the scattering length as, the Fermi\nmomentum kF, and the range of the interaction rintsat-\nisfy|as|>> k−1\nF>> rint. Under such condition, atoms\nare ‘strongly interacting’, and a full theoretical descrip-\ntionoftheirpropertiesisachallengingtaskinmany-body\ntheory. Universal behavior is expected to show up in\nvarious aspects, including ground state properties as dis-\ncussedbelow,collectiveexcitations[3, 4, 5, 6, 7, 8, 9], and\nthermodynamic properties [10, 11, 12, 13, 24]. Such uni-\nversality can be naively understood as the ‘dropping’ of\nthe scattering length asout of the problem, leaving kFas\nthe only relevant length scale. In particular, the ground\nstate energy E0, is expected to be proportional to that of\nthe non-interacting gas Efree\n0[14], that is E0/Efree\n0=ξ\n, or equivalently\nE0\nA=3\n5k2\nF\n2ξ (1)(/planckover2pi1=m= 1),Abeing the number of particles. The uni-\nversal constant ξis of great interest and many attempts\nhave been made to derive it analytically or determine it\nexperimentally.\nTheoretical calculations suggest that ξis between 0.3\nto 0.7. For example, an early work based on dif-\nferent Pad´ e approximations gives ξ= 0.326,0.568[14].\nDiagrammatic approach gives 0.326 with Galitskii\nresummation[15], 0.7 with ladder approximation[15],\nand 0.455 with a diagrammatic BCS-BEC crossover\ntheory[17]. Other theoretical approaches have also been\nused, including ǫexpansion, which gives ξ=0.475 in [18]\nand [20], and variational formalism, which gives 0.360 in\n[19]. The four most recent experimental measurements\nare listed in Table I. Though the experimental results\nare consistent with each other, the experimentally deter-\nmined value of ξstill falls between relatively large error\nbars(∼10%). By far the best estimate on ξis considered\nto be that from Quantum Monte-Carlo methods, giving\nξ= 0.44(1)[21] and 0.42(1)[22].\nξAuthors Ref.\n0.36(15) Bourdel el.al[23]\n0.51(4) Kinast et.al. [24]\n0.46(5) Partridge et.al.[25]\n0.46+0.05\n−0.12Stewart et.al.[26]\nTABLE I: Comparison of recent experimental values on ξ.\nCold and dilute neutron matter is a special class of\ncold Fermi system with greatimportance in astrophysics.\nIts properties at resonance has attracted much inter-\nest recently [27, 28]. In this work we report results\nfrom low-momentum ring diagram calculations on the\nground-state energy of neutron matter at and near the2\nunitary limit. As is well-known, the1S0channel of\nneutron matter has a fairly large scattering length as\n(−18.97fm), nonetheless, it is still finite. Here, by ad-\njusting the interaction parameters of the CD-Bonn po-\ntential [29], we construct ‘tuned’ neutron interactions\nwith different as’s such as −9.83fm,−12070fmand\n+21fm(which possesses a bound state). For a wide\nrange of neutron density, the case of as=−12070fm\ncan be considered the same as the unitary limit, namely\nas→ −∞. We shall compute the ground state en-\nergy of neutron matter, with inter-neutron potentials\nbeing these ‘tuned’ CD-Bonn’s, by two steps: renor-\nmalization followed by ring summation. We first renor-\nmalize neutron interactions with a T-matrix equivalence\nrenormalization method [30, 31, 32, 33, 34, 36], where\nthe high-momentum components beyond a decimation\nscale Λ are integrated out. This gives the corresponding\nlow-momentum interactions Vlow−k’s with the scattering\nlengths being preserved. Then, we calculate the ground\nstate energy by summing the particle-particle-hole-hole\n(pphh) ring diagrams[37] to all orders. In such ring sum-\nmation, we employ a model space approach, namely, the\nsummation is carried out within a model space charac-\nterized by {k≤Λ}.\nWe shall closely examine how our results differ from\nsimilar calculations with a different renormalized inter-\naction - the Brueckner G-matrix on which the Brueckner\nHartree-Fock(BHF) method is based. The BHF method\nhas been widely used for treating the stronglyinteracting\nnuclear many body problems [38, 39]. However, BHF is a\nlowest-order reaction matrix ( G-matrix) theory and may\nbe improved in several aspects. To take care of the short\nrange correlations, the ladder diagrams of two particles\ninteracting with the bare interaction are summed to all\norders in BHF. However, this method does not include\ndiagrams representing hole-hole correlations such as dia-\ngram (iii) of Fig.1. Note that this diagram has repeated\n(pphh) interactions as well as self-energy insertions to\nboth hole and particle lines. Another aspect of the tra-\nditional BHF is that it employs a discontinuous single-\nparticle (s.p.) spectrum which has a gap at the Fermi\nsurfacekF. To improve upon these drawbacks, Song et\nal. [37]haveformulateda G-matrixring-diagrammethod\nfor nuclear matter, with which the pphhring diagrams\nsuch as diagrams (i) to (iii) of Fig.1 are summed to all\norders. This ring-diagram method has been applied to\nnuclear matter and given satisfactory result [37]. The\nVlow−kring diagram method used in this work is highly\nsimilar to [37]’s , except for one significant difference: the\ninteraction used in the G-matrix ring diagram method is\nenergy dependent. (The Brueckner G-matrix is energy\ndependent, as we shall later discuss.) This complicates\nthe calculation a lot. Vlow−kprovides a cleaner and sim-\npler implementation on such all-order ring summation.\nWe shall first provide an outline of the ring-diagram\napproach in section II. The derivation details of the\nlow-momentum interaction from the CD-Bonn potentials\nshall be followed in section III. Our major results fromtheVlow−kring diagram method are in section IV. There\nwe shall present our results for the ground-state energy\nand ratio E0/Efree\n0obtained with potentials of various\nscattering lengths. A fixed-point criterion for determin-\ning the decimation scale Λ will be discussed. There one\ncan also find a comparison of data on the ground state\nenergy obtained with two different methods-the Vlow−k\nand theG-matrix ring diagram methods. We shall sum-\nmarize and discuss our work in the last section.\nII. LOW-MOMENTUM RING DIAGRAMS\nIn this section we describe how we calculate the ring\ndiagrams for the ground state energy shift ∆ E0, which\nis defined as the difference ( E0−Efree\n0) whereE0is the\ntrue ground-state energy and Efree\n0is the corresponding\nquantity for the non-interacting system. In the present\nwork, we consider the pphhring diagrams as shown in\nFig. 1. We shall calculate the all-order sum, denoted\nas ∆Epp\n0, of such diagrams. Our calculation is carried\nout within a low-momentum model space {k≤Λ}and\neach vertex of the diagrams is the renormalized effec-\ntive interaction corresponding to this model space. Two\ntypes of such interactions will be employed, one being\nthe energy-independent Vlow−kand the other being the\nenergy-dependent G-matrix interaction. Let us consider\nfirst the former. In this case, ∆ Epp\n0can be written [37]\nas\n∆Epp\n0=−1\n2πi/integraldisplay∞\n−∞dωeiω0+tr<Λ[F(ω)Vlow−k\n+1\n2(F(ω)Vlow−k)2+1\n3(F(ω)Vlow−k)3+···] (2)\nwhereFis the free pphhpropagator\nFab(ω) =¯na¯nb\nω−(ǫa+ǫb)+i0+−nanb\nω−(ǫa+ǫb)−i0+(3)\nwithna= 1, a≤kF; = 0, k > k Fand ¯na= (1−na).\nWe now introduce a strength parameter λand aλ-\ndependent Green function Gpp(ω,λ) defined by\nGpp(ω,λ) =F(ω)+λF(ω)Vlow−kGpp(ω,λ).(4)\nThe energy shift then takes the following simple form\nwhen expressed in terms of Gpp, namely\n∆Epp\n0=−1\n2πi/integraldisplay1\n0dλ/integraldisplay∞\n−∞eiω0+tr<Λ[Gpp(ω,λ)Vlow−k]\n(5)\nUsing Lehmann’s representation for Gpp, one can show\nthat\n∆Epp\n0=/integraldisplay1\n0dλΣmΣijkl<ΛYm(ij,λ)Y∗\nm(kl,λ)/an}bracketle{tij|Vlow−k|kl/an}bracketri}ht,\n(6)3\nwhere the transition amplitudes Yare given by the fol-\nlowing RPA equation:\n/summationdisplay\nef[(ǫi+ǫj)δij,ef+λ(1−ni−nj)/an}bracketle{tij|Vlow−k|ef/an}bracketri}ht]\n×Ym(ef,λ) =ωmYm(ij,λ); (i,j,e,f)<Λ.(7)\nThe index mdenotes states dominated by hole-hole com-\nponents, namely, states that satisfy /an}bracketle{tYm|1\nQ|Ym/an}bracketri}ht=−1\nandQ(i,j) = (1−ni−nj). We have used the HF s.p.\nspectrum given by Vlow−k, namely\nǫk=/planckover2pi12k2/2m+/summationdisplay\nh<kF/an}bracketle{tkh|Vlow−k|kh/an}bracketri}ht(8)\nfor both holes and particles with k≤Λ. Thus the propa-\ngators of the diagrams as shown in Fig. 1 all include HF\ninsertions to all orders. The above spectrum is continu-\nous up to Λ.\nThe above ring-diagram method is a renormalization\ngroup approach for a momentum model space defined by\namomentumboundaryΛ,andthespacewithmomentum\ngreater than Λ is integrated out. The resulting effective\ninteraction for the model space is Vlow−kwhich is en-\nergy independent. This renormalization procedure can,\nhowever, also lead to a model-space effective interaction\nwhich is energy dependent. The G-matrix ring-diagram\nmethod of [37] is of the latter approach. Formally, these\ntwo approaches should be the same. In the present work\nwe shall carry out ring-diagram calculations using both\napproaches; it would be ofinterest to comparethe results\nof these two different approaches.\nIn the following, let us briefly describe the G-matrix\nring diagram method [37]. Here each vertex of Fig. 1 is a\nmodel-space G-matrix interaction, to be denoted as GM.\nIt is defined by\nGM\nijkl(ω) =Vijkl+/summationdisplay\nrsVijrsQM(rs)\nω−k2r−k2s+i0+GM\nrskl(ω)\n(9)\nwherek2\nrstands for the kinetic energy /planckover2pi12k2\nr/2mand sim-\nilarly for k2\ns. The Pauli projection operator QMis to\nassure the intermediate states being outside Λ and kF,\nnamely it is defined by\nQM(rs) = 1,if max(kr,ks)>Λand min (kr,ks)< kF\n= 0,otherwise. (10)\nIn the above kF<Λ. In Ref.[37] Λ is chosen to be\n∼3fm−1. Note that the above GMis energy dependent,\nnamely it is dependent on the energy variable ω. How-\never,ωis not a free parameter; it is to be determined\nin a self-consistent way. For example, the model-space\ns.p. spectrum is given by the following self-consistent\nequations:\nǫa=/planckover2pi12k2\na\n2m+/an}bracketle{ta|U|a/an}bracketri}ht; (11)/an}bracketle{ta|U|a/an}bracketri}ht=/summationdisplay\nh≤kF/an}bracketle{ta,h|GM(ω=ǫa+ǫh)|a,h/an}bracketri}ht, a <Λ\n= 0, otherwise. (12)\nIntheabove Uisthes.p. potentialand ǫthemodel-space\ns.p. energywhich is determined self-consistentlywith the\nenergy variable of GM. Note that this s.p. spectrum\ndoes not have a gap at kF; it is a continuous one up to\nΛ. When choosing Λ= kFthe above is the same as the\nself-consistent BHF s.p. spectrum.\nWhen calculating the ring diagrams using GM, its\nenergy variable is also determined self-consistently. In\ntermsof GM, the all-ordersumofthe pphhringdiagrams\nis [37]\n∆Epp\n0=/integraldisplay1\n0dλ/summationdisplay\nm/summationdisplay\nijkl(<Λ)Ym(ij,λ)Y∗\nm(kl,λ)GM\nkl,ij(ω−\nm)\n(13)\nwhere the transition amplitudes Ymand eigenvalues ω−\nm\nare given by the following self-consistent RPA equation:\n/summationdisplay\nef[(ǫi+ǫj)δij,ef+λ(1−ni−nj)Lij,ef(ω)]Ym(ef,λ)\n=µm(ω,λ)Ym(ij,λ); (i,j,e,f)<Λ. (14)\nThe index mdenotes states dominated by hole-hole com-\nponents. The vertex function Lis obtained from 2- and\n1-body diagrams first order in GM[37]. The above equa-\ntion is solved with the self-consistent condition that the\nenergy variable of Lis equal to the eigenvalue, namely\nω=µm(ω,λ)≡ω−\nm(λ). (15)\nComparing with the Vlow−kring diagram calcula-\ntion described earlier, the above G-matrix calculation is\nclearly more complicated. Because of the energy depen-\ndence of the interaction GM, the above equations have to\nbe solvedself-consistently both for the s.p. spectrum and\nfor the RPA equations. To attain this self consistency,\nit is necessary to use iteration methods and this pro-\ncedure is often numerically involved. In contrast, ring-\ndiagram calculation using the energy-independent inter-\nactionVlow−kis indeed much simpler. As mentioned ear-\nlier, we shall carry out ring-diagram calculations using\nboth methods.\nIII.Vlow−kWITH INFINITE SCATTERING\nLENGTH\nTo carry out the above ring-diagram calculation, we\nneed the low-momentum potential Vlow−k. Since we are\ninterestedatneutronmatteratandneartheunitarylimit\n(infinite scattering length), we should have Vlow−k’s of\ndefinite scattering lengths, including ±∞, so that the\ndependence of our results on scattering lengths can be\ninvestigated. In the present work, we have chosen a two-\nstepproceduretoconstructsuchpotentialssothatthere-\nsultingpotentialsarecloseto realisticneutronpotentials.4\n+ + Δ E0=\n+......\n......\n(iii)(i) (ii)\nFIG. 1: pphhring-diagram summation in the calculation of\nthe ground state energy shift.\nWe first construct bare potentials Vabased on a realis-\ntic nucleon-nucleon potential; these potentials are tuned\nso that they have definite scattering lengths. Renormal-\nized low-momentum potentials Va\nlow−kare then obtained\nfromVausing a renormalization procedure which pre-\nserves the scattering length.\nWe start from the high-precision CD-Bonn [29]\nnucleon-nucleon potential. For this potential, the scat-\ntering length of the1S0channel is already fairly large\n(-18.97fm), and it is found to depend rather sensitively\non the interaction parameters. Thus by slightly tuning\nthe interaction parameters of the CD-Bonn potential, we\nhave obtained a family of1S0neutron potentials of def-\ninite scattering lengths. We shall denote them as Va.\nOur tuning procedure will be discussed in section IV(A).\nRecently there have been a number of studies on\nthe low-momentum nucleon-nucleon potential Vlow−k\n[30, 31, 32, 33, 34, 36]. Vlow−kis obtained from a bare\nnucleon-nucleon potential by integrating out the high-\nmomentum components, under the restriction that the\ndeuteron binding energy and the low-energy phase-shifts\nare preserved. The Vlow−kobtained from different re-\nalistic potentials ( CD-Bonn [29], Argonne [40] , Ni-\njmegen [41] and Idaho [42]) all flow to a unique potential\nwhen the cut-offmomentum isloweredto around2 fm−1.\nThe above Vlow−kis obtained using a T-matrix equiva-\nlence renormalization procedure [30, 31, 32, 33, 34, 36].\nSince thisprocedurepreservesthe half-on-shellT-matrix,\nit of course preserves the scattering length. Thus this\nprocedure is suitable for constructing Va\nlow−k, the low-\nmomentum interaction with definite scattering length.\nUsing this procedure, we start from the T-matrix equa-\ntion\nT(k′,k,k2) =Va(k′,k)+/integraldisplay∞\n0q2dqVa(k′,q)T(q,k,k2)\nk2−q2+i0+,\n(16)\nwhereVais a modified CD-Bonn potential of scattering\nlengtha. Notice that in the above the intermediate state\nmomentum qis integrated from 0 to ∞. We then definean effective low-momentum T-matrix by\nTlow−k(p′,p,p2) =Va\nlow−k(p′,p)\n+/integraldisplayΛ\n0q2dqVa\nlow−k(p′,q)Tlow−k(q,p,p2)\np2−q2+i0+,\n(17)\nwhere the intermediate state momentum is integrated\nfrom 0 to Λ, the momentum space cut-off. We require\nthe above T-matrices to satisfy the condition\nT(p′,p,p2) =Tlow−k(p′,p,p2); (p′,p)≤Λ.(18)\nThe above equations define the effective low momentum\ninteraction Va\nlow−k. The iteration method of Lee-Suzuki-\nAndreozzi [43, 44] has been used in calculating Va\nlow−k\nfrom the above T-matrix equivalence equations. From\nnow on, we shall denote Va\nlow−ksimply as Vlow−k.\nIV. RESULTS\nA. Low-momentum interactions and scattering\nlengths\nTo study neutron matter at the unitary limit, we first\nneed a realistic neutron-neutron interaction that would\nlead to a huge1S0scattering length as, and a small ef-\nfective range re. We obtain such interaction by ‘tun-\ning’ the meson mass mσin the usual CD-Bonn potential.\nThe exchange of a lighter meson generates a stronger at-\ntraction, therefore making the scattering length asmore\nnegative until a bound state is formed. As one ‘tunes’\nacross the bound state, aswill pass from −∞to +∞,\neventually become less and less positive. In this work,\nthismσ‘tuning’ is taken as a manual adjustment in the\nstrength of the neutron-neutron potential. Of great in-\nterest is that this ‘tuning’ may naturally come from the\ndensity-dependence of the nucleon-nucleon potential via\nthe mechanism of Brown-Rho (BR) scaling[45, 46, 47],\nwhich suggests the in-medium meson masses should de-\ncrease.\nAt normal nuclear matter density, the meson masses\nofρ,ωandσare all expected to decrease by about 15%\n[47]comparedtotheirmassesinfreespace. Thisdecrease\nwill enhance not only the attraction from σbut also the\nrepulsion from ρandω. As a preliminary study, we shall\ntune only mσin the present work. To compensate for the\nrepulsive effect from ρandω(which are not tuned in the\npresent work), we shall only tune mσslightly, namely a\nfew percent. We shall consider that the aboveBR scaling\nis compatible with neutron matter of moderate density\n(kF∼1fm−1). In a future publication, we plan to carry\nout further studies, including the tuning of ρ- andω-\nmeson masses.\nVarious ‘tuned’ CD-Bonn potentials are listed in Table\nII. Fromthereonecanseethesensitivityofthescattering\nlength to the changein mσ. Atmσ≈442MeV, namely a5\nname mσ(MeV)as(fm)re(fm)\noriginal CD-Bonn 452 -18.97 2.82\nCD-Bonn-10 460 -9.827 3.11\nCD-Bonn-42 447 -42.52 2.66\nCD-Bonn- ∞442.85 -12070.00 2.54\nCD-Bonn+ ∞442.80 +5121.00 2.54\nCD-Bonn+21 434 +21.01 2.31\nTABLE II: mσin the original CD-Bonn potential is tuned\nto give neutron-neutron potentials with different scatteri ng\nlengths.\n2.4% decrease from the original, as≈ −12000fm. Notice\nthat the effective ranges for the CD-Bonn potentials are\nlargerthantheactualrangesofthem. Forexample, refor\nthe original CD-Bonn potential is 2 .82fm, considerably\nlarger than the range of one-pion exchange. Within the\nrange of Fermi momenta from 0 .8fm−1to 1.5fm−1that\nwe use in our computation below, as≈ −12000fmis\nobviously enormous compared to any length scale in the\nsystem, thus we expect the neutron matter to be at the\nunitarylimit, i.e., nodifferent fromthelimiting case as=\n−∞. Forconvenience, wenamesuchpotentialCD-Bonn-\n∞.\nFollowing the renormalization procedures as already\ndescribed in Section III, we obtain the low-momentum\npotential Vlow−k’s for several CD-Bonn potentials listed\nabove. A comparison of the diagonal matrix elements\nin theVlow−k’s (with a fixed cut-off momentum Λ) is\nshown in Figure 2. It is of interest that the strength of\nVlow−konly changes weakly with the scattering length.\nFor example, it changes by merely about 10% from as=\n−18.97fmto−12070fm.\n0 0.5 1 1.5 2−1.4−1.2−1−0.8−0.6−0.4−0.200.2\nk / fm−1V(k,k) / fm\nas = −9.83 fm\nas = −18.97 fm\nas = −12070 fm\nas = +21.01 fm\nFIG. 2: Diagonal matrix elements of Vlow−kconstructed from\nCD-Bonn potentials with different scattering lengths. Λ =\n2.4fm−1is used in all cases.\nB. Ground-state energy andthe universal constant ξ\nHere we shall present our major results, namely the\nground state energies E0of neutron matter at and\nclose to the unitary limit from the summation of low-\nmomentum ring diagrams to all orders. Following the\npotential renormalization procedure described in section\nIII, we first calculate Vlow−kfor certain chosen values\nfor the decimation scale Λ. Then the all-order sum of\nthepphhring diagrams are calculated using the above\nVlow−k. As introduced in Section II, the calculation de-\ntails in the summation of pphhring diagrams can be\nfound in Ref.[37]. How to choose the decimation scale\nΛ is clearly an important step in our calculation, and in\nthe present work we shall use a stable-point, or ‘fixed-\npoint’, criterion in deciding Λ. Before discussing this\ncriterion, let us first present some of our results for the\nground-state energy per particle ( E0/A). In Fig. 3 we\npresent such results for four asvalues, calculated with\nΛs determined by the above criterion. (The details of\nthis determination will be described a little later.) As\nshown by the figure, we see that E0/Adoes not change\nstrongly with as. The ratios ξ=E0/Efree\n0are then\nreadily obtained, as shown in Fig.4. It is of interest that\nthe ratios for the four ascases are all weakly dependent\nonkF. To help understand this behavior, we plot in\nFig.5 the potential energy per particle PE/A(namely\n∆Epp\n0/Aof Eq.(6)) versus k2\nF, for the same four ascases.\nIt is rather impressive that they all appear to be straight\nlines. We have fitted the ‘lines’ in the figure to the equa-\ntionPE/A= (/planckover2pi12/m)/parenleftbig\nβk2\nF+γ/parenrightbig\n: We have found ( β,γ)\n=(-0.1370, 0.0002), (-0.1498, -0.0008), (-0.1649, -0.0035)\nand (-0.1797, -0.0082) respectively for as=−9.87fm,\n−18.97fm,−12070fmand +21 .0fm. The rms devia-\ntion for the above fitting are all very small (all less than\n0.0013), confirming that they are indeed very close to\nstraight lines. The above results are of interest, and are\nconsistent with those shown in Fig. 4. In fact the ratios\nof Fig.4 are determined by the ‘slopes’ of these ‘lines’.\nBeforefurtherdiscussingourresults,letusnowaddress\nthe question of how to determine the decimation scale\nΛ. There are basically two considerations: The first one\nconcerns the experimental NN scattering phase shifts on\nwhich realistic NN potentials are based. The second is\nabout the dependence of our results on Λ. Realistic NN\npotentials [29, 40, 41, 42] are constructed to reproduce\nthe experimental NN phase shifts up to Elab≈300MeV.\nThissuggeststhatΛisabout2 fm−1, asbeyondthisscale\nNN potential models are not experimentally constrained\nand are thus rather uncertain (model dependent) [34].\nWe now turn to the dependence ofour results on Λ. As\ndescribed in Section II, Vlow−kis used in the determina-\ntion of the H.F. single particle spectrum (see Eq.8), the\ntransition amplitudes Yin the RPA equation (see Eq.7),\nand finally, the ground state energy E0(see Eq. 6). In-\ntuitively, E0should exhibit a non-trivial Λ-dependence.\nFor various Fermi-momenta, this dependence is studied\nand is found to be remarkably mild.6\n0.8 0.9 1 1.1 1.2 1.3 1.4 1.5246810121416\nkF / fm−1 E0/ A\nas = −9.87 fm\nas = −18.97 fm\nas = −12070 fm\nas = +21.01 fm\nFIG. 3: Ground state energy per particle, E0/A, of neutron\nmatter with various tuned CD-Bonn potentials, computed\nfrom the summation of low-momentum pphhring diagrams.\nOnly1S0contribution is included.\n0.80.9 11.11.21.31.41.50.350.40.450.50.55\nkF / fm−1E0 / E0free\nas = + 21.01 fmas = − 12070 fmas = −18.97 fmas = −9.87 fm\nFIG. 4: The ratio E0/Efree\n0as a function of Fermi momen-\ntumkFfor the various CD-Bonn potentials listed in Table II.\nThe data with CD-Boon- ∞(as=−12070fm) indicates that\nE0/Efree\n0is a constant of 0 .443±0.006 over the range of kF\nas shown.\nAs an example, let us present in Fig. 6 our re-\nsults obtained with the potential CD-Bonn- ∞. For\nΛ = (2.0−2.6)fm−1, it is seen that ξvaries actually\nby a rather small amount (note that the range of our\nplot is from 0.438 to 0.444). Furthermore the Λ depen-\ndence of ξshows up as a curve with a minimum. The\nfinal choice of Λ is based on the criterion that E0should\nbe stable against changes in Λ. As shown in the fig-\nure, an obvious stable-point, or fixed-point, defined by\ndE0(Λ)/dΛ = 0, is found at about 2 .3fm−1. Thus we\nhave used Λ = 2 .3fm−1for CD-Bonn- ∞. We found that\nthe position of the fixed point is almost the same for the\ndifferent Fermi-momenta in the range (0 .8−1.5)fm−1.\nThe same procedure is done on the original CD-Bonn,0.60.811.21.41.61.822.2−18−16−14−12−10−8−6−4−2\nkF2 / fm−2 PE / A (MeV)as = −9.87 fm\nas = −18.97 fm\nas = −12070 fm\nas = +21.01 fm\nFIG. 5: Potential energy per particle, PE/A, of neutron mat-\nter with various tuned CD-Bonn potentials, computed from\nthe summation of low-momentum pphhring diagrams. Only\n1S0contribution is included.\nandothertunedpotentials. Thefixedpoints,alsowithan\nnegligible dependence on kF, are found to be 2 .15fm−1,\n2.25fm−1and 2.4fm−1respectively for CD-Bonn poten-\ntials of scattering lengths −9.8fm,−18.9fm(the orig-\ninal CD-Bonn), and +21 .01fm. The above fixed-point\nΛ’s have been used for the results presented in Figs. 3-5.\n22.1 2.2 2.3 2.4 2.5 2.60.4380.4390.440.4410.4420.4430.4440.445\nΛ / fm−1E0 / E0freekF = 1.3 fm−1\nkF = 1.2 fm−1\nkF = 1.1 fm−1\nkF = 1.05 fm−1\nFIG. 6: Determination of the fixed point where dE0/dΛ = 0\nfor CD-Bonn- ∞.\nOf great significance is the ratio of the ground state\nenergy to that of the non-interacting case, namely\nE0/Efree\n0. At the unitary limit, it is expected to be an\nuniversal constant, named ξ. This constant is of great\nimportance as it determines the equation of state of all\nlow-densitycoldFermigas. Attheunitarylimit, ourdata\nonE0/Efree\n0all lie within a narrowwindow from0.437to\n0.448. Such result confirms a universality over Fermion\ndensity in a wide range (1 .73−11.40)×10−2fm−3. Most\nimportantly, the numerical value of ξis remarkably close7\nto that from Monte Carlo methods, which by far is be-\nlieved to be the best estimate. Astra et. al.obtained\n0.42(1)based on a square well potential and particle den-\nsitynR3\n0= 10−6(whereR0is the potential range). Carl-\nsonet. al.obtained 0.44(1) based on a ‘cosh potential’,\nand particle density nµ−3= 0.020 (where 2 /µis the ef-\nfective range). In our case, nΛ−3= (1.4−9.4)×10−3\n(where Λ = 2 .3fm−1is the decimation scale in the renor-\nmalization). These works, including ours, employ very\ndifferent interactions and various particle densities. Still,\nthe value of ξagrees incredibly well.\nIn Figure 4 we contrast the data from CD-Bonn- ∞\nwith that from the original CD-Bonn and other tuned\npotentials. Even though the1S0scattering length in\nthe original CD-Bonn is already fairly large ( as=\n−18.97fm), still the equation of state, as predicted from\nthe ratio E0/Efree\n0, has significant difference from the\nunitary limit. As seen in our data with CD-Bonn- ∞po-\ntential, at the unitary limit the ratio E0/Efree\n0= 0.44 is\npractically independent of the underlying neutron den-\nsityn.\nC. Comparison with G-matrix results\nAs discussed in section II, our ring-diagram calcula-\ntions are based on a model space framework. A model-\nspace is defined by momentum {k≤Λ}where Λ is the\ndecimationscale. Thespacewith k >Λisintegratedout,\nresulting in a model-space effective interaction Veff. We\nhave used so far the energy-independent Vlow−kforVeff.\nAlternatively, on can also use the energy-dependent GM-\nmatrix (of section II) as Veff. These two approaches are\nformally equivalent. We have carried out calculations to\ncheck this equivalence.\nWe have repeated the ring diagram summation with\nthe energy-independent Vlow−kreplaced by the energy-\ndependent model-spaceBrueckner GM-matrix, and carry\noutafullyself-consistentcomputationinsummingupthe\npphhring diagrams. The exact procedures in Ref.[37]\nare followed (section II). Ring diagrams within a model\nspace up to a cut-off momentum Λ is summed to all or-\nders. We found that the ground state energy is rather\ninsensitive to the choice of Λ. See Figure 7 for the\ndata of CD-Bonn- ∞and CD-Bonn(-18.97), done with\nΛ = 2.3fm−1,2.25fm−1respectively. As illustrated,\nthe two methods, namely, ring diagram summation with\nVlow−kand that with GM-matrix, are fully consistent.\nThis is a remarkable and reassuring result, as the cal-\nculational procedures of them are vastly different. For\ntheGMcase, the s.p. spectrum, the RPA amplitudes Y\nand energies ω−\nmare all calculated self-consistently, while\nfor theVlow−kcase no such self-consistent procedures are\nneeded. Clearly the Vlow−kring-diagrammethod is more\ndesirable.0.80.9 11.11.21.31.41.50.30.350.40.450.50.55\nkF / fm−1E0 / E0free\nas = −18.97 fm, Ring w/ Vlow−k\nas = −18.97 fm, Ring w/ G−matrix\nas = −12070 fm, Ring w/ Vlow−k\nas = −12070 fm, Ring w/ G−matrix\nFIG. 7: The ratio E0/Efree\n0for the potentials CD-Bonn- ∞\nand CD-Bonn(-19.87) computed with two methods. Ring\nw/G-mat:pphhring diagrams summation with Brueckner\nGM-matrix. Λ = 2 .3fm−1is used, computation is fully self-\nconsistent. Ring w/ Vlow−k:pphhring diagrams summation\nwithVlow−k, fixed point is at Λ = 2 .3.\nD. Schematic effective interaction at unitary limit\nAt the unitary limit, the simple equation of state\nE0=ξEfree\n0in neutron matter suggests a very counter-\nintuitive nature in the underlying system: strongly in-\nteracting fermions essentially can be described by a non-\ninteracting picture with an effective mass. This unex-\npected ‘simplicity’ can best be captured by a schematic\ninteraction. To illustrate this, let us consider neutron\nmatter confined in a closed Fermi sea |Φ0(kF)/an}bracketri}ht. In other\nwords, we consider neutron matter in a one-dimensional\nmodel space. We denote the effective interaction for this\nmodel space as VFS. Then the potential energy per par-\nticle is\nPE\nA=/an}bracketle{tΦ0(kF)|VFS|Φ0(kf)/an}bracketri}ht/A\n=8\nπ/integraldisplaykF\n0/parenleftbigg\n1−3k\n2kF+k3\n2k3\nF/parenrightbigg\n/an}bracketle{tk|VFS|k/an}bracketri}htk2dk(19)\nwherekis the relative momentum.\nSuppose we take VFSas a contact effective interaction\nVFS=1\nS\nas−2\nπkF(20)\n(/planckover2pi1=m= 1) where Sis a positive parameter with S <<\n|as|. WhenS=1 andkFreplaced by Λ, VFSis the same\nthe effective interaction for the pion-less effective field\ntheory [34, 35]. Substituting the above into Eq.(19) gives\nξ= 1+5\n91\nπ\n2S\naskF−1. (21)8\nAt the unitary limit (infinite as), the above gives\nξ=4/9, independent of kF, which is practically the same\nas the result for ξ(-12070) of Fig. 4. The above also\ngivesξfor finite as. At the unitary limit, we expect\nVFSto be unique. For finite as(away from the unitary\nlimit), it is not expected to be unique and the param-\neterSis expected to depend on the underlying poten-\ntial. As shown in Fig. 4, we have calculated ξusing the\nCD-Bonn potentials of finite scattering lengths. These\nresults can also be qualitatively described by the above\nequation. For instance, for S= 1.25 andkF= 1.0, the\nabove equation gives ξ= 0.54, 0.50 and 0.39 respectively\nforas=−9.87fm,−18.97fmand +21 .01fm. In short,\ncertain main features of our results obtained from ring-\ndiagram calculations with the CD-Bonn potentials can\nbe qualitatively reproduced by the above simple contact\neffective interaction.\nV. SUMMARY\nIn conclusion, we have carried out a detailed study on\nneutron matter at and close to the unitary limit with a\nlow-momentum ring diagram approach. By slightly tun-\ning the realistic CD-Bonn potential, we have obtained\n1S0neutron potentials of specific scattering lengths, in\nparticular the CD-Bonn- ∞one with asof−12070fm.\nBy integrating out their momentum components be-\nyond a decimation scale Λ, we obtain renormalized low-momentum interactions Vlow−kof the same specific scat-\ntering lengths. The ground state energy E0of neutron\nmatter are then calculated by summing up the pphhring\ndiagramstoallorderswithinthemodelspace {k <Λ}. A\nfixed-point criterion is used to determine the decimation\nscale Λ. We have carried out ring-diagram calculations\nusing two types of renormalized interactions, the energy-\nindependent Vlow−kand the energy-dependent G-matrix,\nwith results given by them being nearly identical. The\nVlow−kring-diagrammethodhasasimplerformalismand\nis also more suitable for numerical calculation. For the\nCD-Bonn- ∞potential, the ratio E0/Efree\n0is found to be\nvery near a universal constant of 0.44 over the neutron\ndensity range (1 .73−11.40)×10−2fm−3. Our result\nagrees well with the recent experimental measurement\nand Monte-Carlo computation on cold Fermi gas at the\nunitary limit.\nAcknowledgement We thank G.E. Brown, E.\nShuryak, T. Bergmann and A. Schwenk for many helpful\ndiscussions. This work is supported in part by U.S. De-\npartment of Energy under grant DF-FG02-88ER40388,\nand by the U.S. National Science Foundation under\nGrant PHY-0099444.\n[1] R. F. Bishop, Int. J. Mod. Phys. B 15, iii (2001), Many-\nBody Challenge Problem by G. F. Bertsch.\n[2] C. A. Regal and D. S. Jin, Phys. Rev. Lett. 90, 230404\n(2003); S. Jochim, M. Bartenstein, G. Hendl, J. Hecker\nDenschlag, R. Grimm, A. Mosk, M.Weidem¨ uller, Phys.\nRev. Lett. 89, 273202 (2002); C. H. Schunck, M. W.\nZwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle, A.\nSimoni, E. Tiesinga, C. J. Williams, and P. S. Julienne,\nPhys. Rev. A. 71, 045601 (2005).\n[3] S. Stringari, Europhys. Lett. 65, 749 (2004).\n[4] A.Bulgac and G. F. Bertsch, Phys.Rev.Lett. 94, 070401\n(2005).\n[5] H. Heiselberg, Phys. Rev. Lett. 93, 040402 (2004).\n[6] J. Kinast, S.L. Hemmer, M.E. Gehm, A. Turlapov, and\nJ.E. Thomas, Phys. Rev. Lett. 92, 150402 (2004).\n[7] J. Kinast, A. Turlapov, and J. E. Thomas, Phys. Rev. A\n70, 051401(R) (2004).\n[8] A. Altmeyer, S. Riedl, C. Kohstall, M. J. Wright, R.\nGeursen, M. Bartenstein, C. Chin, J. Hecker Denschlag,\nand R. Grimm, Phys. Rev. Lett. 98, 040401 (2007).\n[9] M.J. Wright, S. Riedl, A. Altmeyer, C. Kohstall, E.R.\nS´ anchez Guajardo, J. Hecker Denschlag, and R. Grimm,\nPhys. Rev. Lett. 99, 150403 (2007).\n[10] A. Bulgac, Phys. Rev. Lett. 95, 140403 (2005).\n[11] A. Bulgac, Joaqu´ ın E. Drut, and Piotr Magierski, Phys.\nRev. Lett. 96, 090404 (2006).\n[12] A. Bulgac, Joaqu´ ın E. Drut, and Piotr Magierski, Phys.Rev. Lett. 99, 120401 (2007).\n[13] J. E. Thomas, J. Kinast, and A. Turlapov, Phys. Rev.\nLett.95, 120402 (2005).\n[14] G. A. Baker, Jr., Phys. Rev. C 60, 054311 (1999).\n[15] H. Heiselberg, Phys. Rev. A 63, 043606 (2001).\n[16] G. M. Bruun, Phys. Rev. A 70, 053602 (2004).\n[17] A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett.\n93, 100404 (2004).\n[18] Y. Nishida, and D. T. Son, Phys. Rev. Lett. 97, 050403\n(2006).\n[19] R. Haussmann, W. Rantner, S. Cerrito, and W. Zwerger,\nPhys. Rev. A 75, 023610 (2007).\n[20] J.-W. Chen, and E. Nakano, Phys. Rev. A 75, 043620\n(2007).\n[21] J. Carlson, S.-Y. Chang, V. R. Pandharipande, and K.\nE. Schmidt, Phys. Rev. Lett. 91, 050401 (2003).\n[22] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S.\nGiorgini, Phys. Rev. Lett. 93, 200404 (2004).\n[23] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F.\nChevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkel-\nmans, and C. Salomon, Phys. Rev. Lett. 93, 050401\n(2004).\n[24] J. Kinast, A. Turlapov, J. E. Thomas, Q. Chen, J. Stajic,\nand K. Levin, Science 307, 1296 (2005).\n[25] G. B. Partridge, W. Li, R. I. Kamar, Y.-A. Liao, and R.\nG. Hulet, Science 311, 503 (2006).\n[26] J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin,9\nPhys. Rev. Lett 97220406 (2006).\n[27] A. Schwenk and C. J. Pethick, Phys. Rev. Lett. 95,\n160401 (2005).\n[28] D. Lee and T. Sch¨ afer, Phys. Rev. C 73, 015202 (2006).\n[29] R. Machleidt, Phys. Rev. C 63, 024001 (2001).\n[30] S.K. Bogner, T.T.S. Kuo and L. Coraggio, Nucl. Phys.\nA684, 432 (2001).\n[31] S.K. Bogner, T.T.S. Kuo, L. Coraggio, A. Covello and\nN. Itaco, Phys. Rev. C 65, 051301R (2002).\n[32] L. Coraggio, A. Covello, A. Gargano, N. Itako, T.T.S.\nKuo, D.R. Entem and R. Machleidt, Phys. Rev. C 66,\n021303(R) (2002).\n[33] A. Schwenk, G.E. Brown and B. Friman, Nucl. Phys.\nA703, 745 (2002).\n[34] S.K. Bogner, T.T.S. Kuo and A. Schwenk, Phys. Rep.\n386, 1 (2003).\n[35] T. Sch¨ afer, C.-W. Kao, S.R. Cotanch, Nucl. Phys. A762,\n82 (2005).\n[36] J.D. Holt, T.T.S. Kuo and G.E. Brown, Phys. Rev. C\n69, 034329 (2004).[37] H.Q. Song, S.D. Yang and T.T.S. Kuo, Nucl. Phys.\nA462, 491 (1987).\n[38] H. A. Bethe, Annu. Rev. Nucl. Sci. 21, 93 (1971).\n[39] J.W. Holt and G.E. Brown, p.239 in Hans Bethe and\nHis Physics (World Scientific, July 2006, editted by G.E.\nBrown and C.-H. Lee).\n[40] R.B.Wiringa, V.G.J. StoksandR.Schiavilla, Phys.Rev .\nC51, 38 (1995).\n[41] V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen and J.J.\nde Swart, Phys. Rev. C 49, 2950 (1994).\n[42] D.R. Entem, R. Machleidt, Phys. Rev. C 68, 041001\n(2003).\n[43] K. Suzuki and S. Y. Lee, Prog. Theor. Phys. 64, 2091\n(1980).\n[44] F. Andreozzi, Phys. Rev. C 54, 684 (1996).\n[45] G.E. Brown, M. Rho, Phys. Rev. Lett. 66, 2720 (1991).\n[46] G.E. Brown, M. Rho, Phys. Rept. 396, 1 (2004).\n[47] R. Rapp, R. Machleidt, J.W. Durso and G.E. Brown,\nPhys. Rev. Lett. 82, 1827 (1999)."    },    {        "title": "1204.4863v1.Scanning_Tunneling_Microscopy_Spectroscopy_of_Vortices_in_LiFeAs.pdf",        "content": "arXiv:1204.4863v1  [cond-mat.supr-con]  22 Apr 2012Scanning Tunneling Microscopy/Spectroscopy of Vortices i n LiFeAs\nT. Hanaguri,1,2K. Kitagawa,3,∗K. Matsubayashi,2,3Y. Mazaki,3,4Y. Uwatoko,2,3and H. Takagi1,2,5\n1RIKEN Advanced Science Institute, Wako, Saitama 351-0198, Japan\n2TRiP, Japan Science and Technology Agency, Chiyoda, Tokyo 1 02-0075, Japan\n3Institute for Solid State Physics, University of Tokyo, Kas hiwa, Chiba 277-8581 Japan\n4Department of Physics, Nihon University, Chiyoda, Tokyo 10 1-8308, Japan\n5Department of Physics, University of Tokyo, Bunkyo, Tokyo 1 13-0033 Japan\n(Dated: July 29, 2021)\nWe investigate vortices in LiFeAs using scanning tunneling microscopy/spectroscopy. Zero-field\ntunneling spectra show two superconducting gaps without de tectable spectral weight near the Fermi\nenergy, evidencing fully-gapped multi-band superconduct ivity. We image vortices in a wide field\nrange from 0.1 T to 11 T by mapping the tunneling conductance a t the Fermi energy. A quasi-\nhexagonal vortexlattice at low field contains domain bounda ries which consist of alternating vortices\nwith unusual coordination numbers of 5 and 7. With increasin g field, the domain boundaries become\nill-defined, resulting in a uniformly disordered vortex mat ter. Tunneling spectra taken at the vortex\ncenter are characterized by a sharp peak just below the Fermi energy, apparently violating particle-\nhole symmetry. The image of each vortex shows energy-depend ent 4-fold anisotropy which may\nbe associated with the anisotropy of the Fermi surface. The v ortex radius shrinks with decreasing\ntemperature and becomes smaller than the coherence length e stimated from the upper critical field.\nThis is direct evidence of the Kramer-Pesch effect expected i n a clean superconductor.\nPACS numbers: 74.70.Xa, 74.55.+v, 74.25.Uv, 74.20.Rp\nI. INTRODUCTION\nVortices in type-II superconductors have attracted\nmuch attention. Since superconductivity is destroyed\nat the vortex center, the vortex core can be regarded\nas a potential well surrounded by the superconducting\ngap (SG). Therefore, quasi-particle states in the vortex\ncore provide us with useful information of the SG. In\nthis respect, it is particularly interesting to study vor-\ntices in iron-based superconductors [1–4]. Characteris-\ntic features of iron-based superconductors, namely high\ntransition temperature, multi-band nature, [5] small car-\nrier density and possible sign-reversal in the SG, [6–14]\nshould bring about unusual quasi-particle states in and\naround the vortex core. Here we investigate vortices of\nan iron-based superconductor using scanning tunneling\nmicroscopy/spectroscopy (STM/STS).\nIn general, the excitation spectrum in the vortex core\nconsists of discrete bound states separated by ∼∆2/EF,\nwhere ∆ is the SG and EFis the Fermi energy. [15] In\nconventionalsuperconductors, EFismanyordersofmag-\nnitude larger than ∆ and the excitation spectrum can be\nregarded as continuous. Depending on the quasi-particle\nlifetimeτassociated with impurity scattering, the vortex\ncore can be classified into either dirty or clean regimes.\nIf the lifetime broadening /planckover2pi1/τis greater than ∆ (equiv-\nalent to ξ > l, whereξis the coherence length and lis\nthe mean-free path), quasi-particles are frequently scat-\ntered in the core and the core is well approximated as a\n∗Presentaddress: Graduate School ofIntegrated Artsand Sci ences,\nKochi University, Kochi 780-8520, Japannormal metal. The influence of SG on the core states is\nmasked in this dirty case. On the other hand, if /planckover2pi1/τ <∆\norξ < l, quasi-particles can travel from one side of the\nvortex core to the other without being scattered. The\nmost prominent feature of such a clean vortex core is\nthat local density-of-states (LDOS) at the vortex center\nexhibits a peak at EFbecause low-energy bound states\nare localized near the center. This LDOS peak was first\nobserved by Hess et al.in NbSe 2[16] and subsequent\ntheoretical analysis revealed that the energy and spatial\nevolution of the LDOS peak reflect the nature of SG as\nwell as normal-state band structure [17–20].\nIf the ratio EF/∆, which roughly corresponds to the\nnumber of bound states, becomes small, energy separa-\ntion between the bound states is large and the excita-\ntion spectrum is no longer continuous, provided /planckover2pi1/τ <\n∆2/EF. Unfortunately, this so-called quantum-limit\nregime may not be reachable in conventional phonon-\nmediated superconductors. However, in iron-based su-\nperconductors, because of small carrier density (which\nleads small EF) and high transition temperature (which\nmeans large ∆), the quantum-limit regime may be an-\nticipated. Theoretical analysis of the quantum-limit vor-\ntex predicts strong particle-hole asymmetry in an LDOS\nspectrum at the vortex center and a Friedel-like oscilla-\ntion in the spatial variation of LDOS. [21]\nSo far, vortices are successfully observed by STM/STS\nin some iron-based supercondcutors. [22–24] The LDOS\npeak associated with the bound states has been detected\nin the tunneling spectra at the vortex centers of FeSe\nand Ba 0.6K0.4Fe2As2. [23, 24] In the latter compound,\npossible quantum-limit behavior is discussed based on\nthe asymmetry of the spectrum. [23] However, detailed\nelectronicstructure of the vortexcore and effects of multi2\nband are still elusive.\nIn addition to the quasi-particle states in the vortex\ncore, properties of many vortices, vortex matter, have\nprovided a fertile area of research. In general, vor-\ntices tend to form a triangular lattice (Abrikosov lattice)\ndue to the repulsive interaction between them. How-\never, the Abrikosov lattice may qualitatively be altered\nby quenched disorders and/or thermal agitation. [25]\nThecompetitionamongassociatedenergyscales, namely,\nelastic energy of the Abrikosov lattice, pinning energies\nof quenched disorders and thermal energy, brings about\nvariousvortexphases: hexaticAbrikosov-likephase(elas-\ntic energy dominant), vortex glass or amorphous vortices\n(pinningenergydominant)andvortexliquid(thermalen-\nergy dominant). The effect of thermal agitation and con-\nsequent vortex-lattice melting transition has extensively\nbeen studied in cuprate high-temperature superconduc-\ntors. However, competition between elastic and pinning\nenergies is still to be clarified.\nThe stability of the Abrikosov lattice is determined\nby its shear modulus C66which is roughly given by\nB2\nc2b(1−b)2whereBc2is the upper critical field and bis\nthe magnetic field Bnormalized by Bc2. [26] At low and\nhighbwhereC66is small, the Abrikosov lattice should\nbe destroyed by the quenched disorders and the amor-\nphous vortex matter may emerge. The vortex lattice\npartially retains its order in the intermediate brange but\nshould contain dislocations. [27] Numerical simulation of\nthe vortexconfigurationshowsthatvorticeswith unusual\ncoordination numbers of 5 and 7 are aligned alternately,\nforming domain boundaries. [28] A crossoverto the high-\nfield amorphous phase occurs through roughening of the\nboundaries.\nIn this article, we report results of comprehensive\nSTM/STS experiments on vortices in an iron-based su-\nperconductor LiFeAs. LiFeAs is one of the very few sto-\nichiometric iron-based superconductors and thus clean\nsingle crystals are available. In addition to this, cleaved\nsurfaces studied by STM/STS are electronically neutral,\nwhich is highly advantageous to minimize the surface-\nrelated extrinsic effects. Tunneling spectrum exhibits\nclear two-gap features indicating that the SG amplitude\nis different depending on the Fermi-surface cylinder. In\ncontrast to other vortex-imaging techniques such as Bit-\nterdecoration,STM/STScanimagevorticesevenathigh\nmagnetic field where Bis almost uniform. By taking this\nadvantage, we study the structure of vortex matter over\nwideBrange and find a signature of high-field crossover\ntotheamorphousphase. Wealsoexaminedtheelectronic\nstate of the vortex cores. Tunneling spectra at the vor-\ntex center are strongly particle-hole asymmetric with a\nsharp LDOS peak just below EF. We compare the spec-\ntra and their spatial evolution with those expected in the\nquantum-limit vortex core and show that the LDOS fea-\ntures associated with the larger SG are consistent with\nthe quantum-limit behavior, but those for the smallerSG\nincluding the sharp peak just below EFare not compat-\nible with the quantum-limit vortex-core state. We alsofound that vortex radius shrinks with decreasing tem-\nperature. This is a direct evidence of so-called Kramer-\nPescheffect[29,30]associatedwiththelow-energybound\nstates in the vortex core.\nII. EXPERIMENTAL METHOD\nLiFeAs single crystals were grown by a LiAs self-flux\ntechnique. [31] A resistivity ρratioρ(300 K)/ρ(20 K) of\nas high as 45 and no detectable residual specific heat\ncoefficient are observed. These data demonstrate the\nhigh quality of the crystals. Superconducting transition\ntemperature Tcdefined at zero-resistance is about 18 K.\nSince LiFeAs degrades in an ambient atmosphere, all of\nthe sample handling procedures were carried out inside\nof glove boxes filled with purified Ar or N 2gases. All\nSTM/STS experiments were performed in a constant-\ncurrentmode using acommercialultra-high-vacuumlow-\ntemperatureSTMsystemmodifiedbyourselves(Unisoku\nUSM-1300 with Nanonis controller). [32] Tunneling con-\nductance, which is proportional to the LDOS at given\nbiasvoltage, wasmeasuredby standardlock-intechnique\nusing the built-in software lock-in detector of Nanonis\ncontroller. Clean and flat surfaces perpendicular to the\n[001] axis were prepared by in situcleaving at liquid N 2\ntemperature. Immediately after the cleaving, samples\nwere transferred to the STM unit which was precooled\nbelow 10 K. Electrochemically etched W wires, which\nwere cleaned, sharpened and characterized by a field-ion\nmicroscope, were used for scanning tips. Magnetic fields\nwere applied along the [001] axis. Whenever magnetic\nfield was changed, the sample was heated up above Tcin\nadvance and then it was field-cooled to the measurement\ntemperatures in order to ensure the homogeneous field\ndistribution inside the sample.\nIII. RESULTS AND DISCUSSION\nA. Topography and spectra in zero magnetic field\nWe show a typical STM topography in Fig. 1(a). Al-\nthough various types of natural defects exist, most of the\ncleaved surface is clean and flat enough to image a reg-\nular square atomic lattice as shown in the inset. The\nlattice constant is determined to be 0.39 nm, indicating\nthat the imaged atomic site is either the As site or Li\nsite.\nFigure 1(b) represents the tunneling spectrum mea-\nsured in the clean region. The spectrum is skewed V-\nshaped bottomed at EFwith a bump at about 40 meV.\nWe confirmed that the spectrum does not exhibit obvi-\nous spatial variations except in the vicinity of defects.\nResults of spectroscopic studies near defects will be pub-\nlished everywhere.\nAt low energy, SG structure is clearly identified. As\nshown in Fig. 1(c), spectral weight over a finite energy3\nFIG. 1: (color online). (a) STM topograph of cleaved (001) su rface of LiFeAs at 0.54 K. The set-up conditions for imaging w ere\nsample-bias voltage Vs= +20 mV and tunneling current It= 10 pA. The inset shows a magnified topography with atomic\nresolution ( Vs= +20 mV, It= 100 pA). (b) Tunneling spectrum taken away from defects. Ti p stabilization condition was\nVs= +98.9 mV and It= 100 pA. Measurement was done with a bias modulation amplitu deVmod= 1mV rms. Data were taken\nat 1.5 K. (c) Low-energy tunneling spectra at different tempe ratures. Two gaps are identified in the spectra and both gaps\ndisappear above 18 K which corresponds to bulk Tc. Tip was stabilized at Vs= +25 mV and It= 125 pA. Vmod= 0.1 mVrms.\nrange near EFis completely removed at low tempera-\ntures. This is a direct evidence that the SG is finite\neverywhere on the Fermi surface. Two pairs of peaks\ncan be clearly identified, implying that the SG consists\nof at least two components. By defining the SG am-\nplitude 2∆ using the peak-to-peak separation, the SG\nratio 2∆/kBTc(kBdenotes Boltzmann constant) for the\nsmaller SG (∆ ∼2.9 meV) is estimated to be 3.7, which\nis close to the value expected from the weak-coupling\nBardeen-Cooper-Schrieffer theory. The SG ratio for the\nlarger SG (∆ ∼6.0 meV) takes a very large value of\n7.7. In both SGs, the rising edges at lower energy\nhave finite width which is wider than our energy reso-\nlution (∼0.1 meV). This result is consistent with the\nanisotropic SG suggested by recent angle-resolved pho-\ntoemission spectroscopy (ARPES) experiments [33–35]\nand quasi-particle interference effect. [36]\nThe observed SG amplitudes are quantitatively consis-\ntent with the ARPES data. [33–35] We identify that the\nsmallerSGopens onthe outer-mosthole cylinderandthe\nlarger SG is associated with the inner hole cylinders. A\ncontribution from the electron cylinders, where ARPES\nobserved ∆ ∼4 meV, [33, 34] is not clear in the tunnel-\ning spectrum. Possible reason for this apparent absence\nmay be that the gap-edge peak for the electron cylinders\noverlaps with those for the hole cylinders, or electron\ncylinders are ”masked” by the tunneling matrix element.\nAt elevated temperatures, both gaps are smeared out\nand disappear at about bulk Tcof 18 K, indicating that\nthey faithfully represent bulk superconductivity.\nB. Imaging vortex matter\nSince superconductivity is destroyed at the vortex cen-\nter, LDOS below the SG energy should be enhanced\naround the vortices. Therefore, vortices can be imagedby spectroscopic imaging technique in which tunneling\nconductance is measured at every pixel of STM image.\nFigures 2(a)-(c) are conductance maps at EFat rep-\nresentative B’s, showing vortices as bright spots. We\nrepeated the same experiment over wide Brange from\n0.1 T to 11 T. As shown in Fig. 3(a), Bdependence of\nvortex density perfectly follows the trend expected in the\ncase that single vortex carries the single flux quantum of\n2.07×10−15Wb.\nInterestingly, vortices do not form into the ordered\ntriangular lattice. Two-dimensional auto-correlation\n(Figs. 2(g)-(i)) and Fourier-transform (Figs. 2(j)-(l)) im-\nages are characterized by B-dependent ring-like struc-\ntures around the origin. At 0.5 T, weak 6-fold spots are\nidentified along the ring in the auto-correlation image\n(Fig. 2(g)), indicating that hexatic correlation is devel-\noping. At higher B, the ring-like structures are isotropic\nin magnitude but the rings in the Fourier-transform im-\nagesaredeformed into rectangularshapeelongatedalong\nthe As-As direction. The emergence of 4-fold anisotropy\nsuggests that the symmetry of underlying crystal lattice\nand/or anisotropy of the SG affect the configuration of\nvortices but they are not strong enough to arrange vor-\ntices into regular square lattice which was observed in\nYNi2B2C, for example. [37, 38]. Nevertheless, the ob-\nserved ring-like structures indicate that there is a well-\ndeveloped distance correlation between vortices but the\norientational correlation is almost absent.\nSuch a highly disordered vortex matter in LiFeAs is\nin sharp contrast to the vortices in Ba 0.6K0.4Fe2As2\nwhere well-ordered triangular vortex lattice has been ob-\nserved. [23] The difference between these two materials\ncan be ascribed to the difference in the shear modu-\nlusC66associated with the difference in Bc2. [26] In\nLiFeAs, Bc2along the [001] axis is about 15 T, [39]\nwhile that of (Ba,K)Fe 2As2is estimated to be more\nthan 60 T. [40, 41] This difference gives rise to an4\nFIG. 2: (color online). (a)-(c) Images of vortices at 1.5 K ob tained by mapping tunneling conductance at EF. Tip was\nstabilized at Vs= +20 mV and It= 100 pA. Vmod= 0.7 mV rms. Arrows denote the nearest Fe-Fe direction. (d)-(f) Delaun ay\ntriangulation diagrams obtained from vortex images shown i n (a)-(c). Vertices with symbols denote vortices with coord ination\nnumbers different from conventional value of 6. (Blue triang le: 5, red square: 7, green star: others) Dashed black square is\ndrawn separately from the edge of the field of view (black soli d square) by the mean value of inter-vortex distance at each fi eld.\nDefect fraction (Fig. 3(b)) was calculated using vortices w ithin this dashed black square to avoid the effect of the edge. (g)-(i)\nTwo-dimensional autocorrelation-function calculated fr om vortex images shown in (a)-(c). (j)-(l) Two-dimensional Fourier\ntransformations from vortex images shown in (a)-(c).\norder-of-magnitude smaller C66in LiFeAs than that in\n(Ba,K)Fe 2As2. Therefore, vortex lattice in LiFeAs is\nmuch more susceptible to quenched disorders than that\nin (Ba,K)Fe 2As2. In other words, LiFeAs can be a play-\nground to investigate how the structure of vortex matter\nsubject to quenched disorder changes as a function of B.\nInordertoanalyzethelocalstructureofvortexmatter,\nweperformedDelaunaytriangulationanalysisassumma-\nrized in Figs. 2(d)-(f). Each vertex denotes the position\nof vortex and vertices with symbols correspond to defect\nvortices whose coordination numbers are different from\n6. We plot defect-vortex density as a function of Bin\nFig. 3(b). In the low field region, defect density increases\nrapidly and reaches more than 40 %.\nAtB= 0.5T,adomainboundary, whichconsistsofal-\nternating defect vortices with coordination numbers of 5\nand 7, has been observed. (Fig. 2(d)) More grain bound-aries are observed at 2 T but each grain boundary still\nkeeps its alternating structure. (Fig. 2(e)) At 8 T, de-\nfects tend to cluster and defect vortices with extraordi-\nnary coordinationnumbers of 4 and 8 emerge. (Fig. 2(f))\nIn order to examine the defect structure within the clus-\nter at 8T, we compare local vortex configurations at 2 T\nand 8 T in the fields of view which contain equal num-\nber of vortices. (Figs. 3(c),(d)) In contrast to the domain\nboundary at 2 T, well-defined alternating line-like struc-\nture is no longer observed at 8T. Namely, even though\ndefect density saturates above ∼2 T, local vortex config-\nuration still changes with B.\nThese observations suggest the following picture of\norder-disorder crossover in the vortex matter. In the\nBregion where the elastic energy dominates, number of\ndefect vortices is small. Even if there is a defect, its co-\nordination number should be 5 or 7, which is close to5\nFIG. 3: (color online). (a) Applied-field dependence of vort ex\ndensity in LiFeAs. Solid black line denotes the behavior ex-\npected in the case that single vortex carries the single vort ex\nquantum of 2.07 ×10−15Tm2. (b) Applied-field dependence\nof fraction of defect vortices with coordination numbers di f-\nferent from conventional value of 6. (c) and (d) Delaunay\ntriangulation diagrams at 2 T and 8 T in the fields of view\nwhich contain equal number of vortices. Symbols have the\nsame meanings as in Figs. 2(d)-(f).\nthe regular value of 6. Such defect vortices are aligned\nalternately forming a domain boundary. This situation\ndoes not change so much as long as the elastic energy\nis relevant. At high enough B, however, C66diminishes.\nAs a result, the domain boundary becomes rough and\nthevortexmatterapproachestotheuniformlydisordered\namorphous phase which contains vortices with extraor-\ndinary coordination numbers less than 5 and more than\n7. These behaviors are totally consistent with the result\nof numerical simulation. [28]\nC. Shape of a single vortex\nWe now focus on spectroscopic features of a single vor-\ntex. All of the data shown below are taken at B= 0.5 T\nand the reproducibility of the results has been checked\nin a different tip-sample combination. Figures 4(a)-(d)\nshow conductance maps at different energies around a\nvortex. Vortex-core state at EFexhibits a 4-fold star\nshape with high LDOS tails along the nearest As-As\ndirections. Each tail splits in parallel with increasing\nenergy. These behaviors closely resemble the energy-\ndependent vortex-core shape of NbSe 2, [16] except that\nin NbSe 2vortex core has a 6-fold anisotropy rather than\n4-fold.Theoretical models exist which account for the energy-\ndependent vortex-coreshape include effects of vortex lat-\ntice [17, 18], SG anisotropy [19, 20] and crystal-lattice\npotential which determines the shape of the Fermi sur-\nface.[17, 18, 42] In the caseofLiFeAs, the effect ofvortex\nlattice can be safely excluded because vortices at 0.5 T\nform into a distorted hexatic lattice (Fig. 2(a)), which\ndoes not fit with the observed 4-fold anisotropy of the\nvortex core.\nAt low enough energies, LDOS distribution around a\nvortex is primarily governed by the smallest SG and the\nFermi surface where it opens. In LiFeAs, ARPES ob-\nserved the smallest SG in the outer-most hole cylinder\nand the SG exhibits 4-fold anisotropy. [33–35] The SG is\nsmaller along the Fe-Fe direction than along the As-As\ndirection, which would give rise to the vortex-core shape\nwith the LDOS tails along the Fe-Fe direction. This is in\ncontradiction to the observed LDOS distribution at EF\nshown in Fig. 4(a), indicating that the SG anisotropy\ndoes not play a major role.\nOn the other hand, the shape of the Fermi sur-\nface is consistent with the observed anisotropy of the\nvortex-core state. According to ARPES, the outer-most\nhole cylinder has a rounded square cross section with\nflat region perpendicular to the As-As direction. [43]\nThis anisotropy should result in a larger number of\nquasi-particles traveling along the As-As direction, form-\ning high LDOS tails around a vortex. Indeed, Wang,\nHirschfeld and Vekhter analyzed the LDOS distribution\naround a single vortex in LiFeAs with the assumptions of\nvarious SG anisotropy and concluded that Fermi-surface\nanisotropy in LiFeAs is strong enough to dominate the\nvarious types of SG anisotropies. [44]\nD. LDOS spectrum and its spatial evolution\nIn order to get more insight into the vortex core state,\nwe study the individual tunneling spectrum and its de-\ntailed spatial evolution. Figure 4(e) shows the spectrum\ntaken at the vortex center. We also plot the spectrum\naway from vortices where the two-gap feature observed\nin the absence of Bis well maintained. The spectrum at\nthe center exhibits a pronounced peak just below EF(-\n0.9 meV) as well as some other weaker peaks. False color\nplots of spatial evolutions of tunneling spectrum along\nFe-Fe and As-As directions are shown in Fig. 4(f) and\n(g), respectively. The LDOS peak at ∼-0.9 mV shifts\nfurther away from EFwith increasing distance from the\nvortex center and smoothly approaches the smaller SG.\nThe counter branch is also observed at symmetric en-\nergies above EF, although the intensity is much weaker\nnear the vortex center. There is an energy-gap-like fea-\nture apparent at EFeven at the vortex center. Spectral\nweight at low energies is higher along the As-As direc-\ntion than along the Fe-Fe direction, giving rise to the\nanisotropic shape of the vortex.\nLDOS features associated with the larger SG show6\nFIG. 4: (color online). (a)-(d) Tunneling-conductance ima ges\naroundasinglevortexatdifferentenergies. Arrowsdenotet he\nnearest Fe-Fe direction. Tip was stabilized at Vs= +25 mV\nandIt= 250 pA. Vmod= 0.2 mV rms. Data were taken at\n0.54 K. (e) Tunneling spectra taken at the center of vortex\n(red) and away from vortices (blue). Vmod= 0.10 mV rms.\nLine profiles of tunneling conductance across the vortex cen -\nter. (f) Along the nearest Fe-Fe direction. (g) Along the\nnearest As-As direction. Vmod= 0.15 mV rms.\nmore complicated behavior. A broad LDOS peak at ∼-\n2.3 mV at the vortex center exhibits jumps in its energy\nwith increasing distance from the center and finally de-\nvelops as the peak at the larger-SG energy. Above EF,\nthe gap-edge peak evolves from the peak at ∼+4.2 mV\nat the center. Namely, the quasi-particle branches asso-\nciated with the larger SG are asymmetric between filled\nand empty states not only in their intensity but also in\ntheir energy. These features are in sharp contrast to the\nsmooth particle-hole symmetric evolutions observed for\nthe quasi-particle branches associated with the smaller\nSG.The observed vortex core states in LiFeAs are appar-\nently different from those of conventional superconduc-\ntors, e.g., NbSe 2. In NbSe 2where the vortex-core ex-\ncitation spectrum in almost continuous, LDOS peak is\nlocated just at EF. [16] With increasing distance from\nthe vortex center, the peak splits along smooth branches\nwhich are symmetric in both intensity and energy be-\ntweenfilledandemptystates.[16]Bycontrast,inLiFeAs,\nspectral weight in the vicinity of vortex center is strongly\nparticle-hole asymmetric, there is a gap-like feature even\nat the vortex center, and evolution of quasi-particle\nbranches for larger SG is neither smooth nor symmetric.\nThese observations indicate that factors which are lack-\ning in the vortex of conventional superconductor play a\nvital role.\nProximity to the quantum limit is a plausible candi-\ndate ofthe observedasymmetry, as hasbeen suggestedin\nBa0.6K0.4Fe2As2. [23] In the quantum-limit vortex, the-\noretical calculation of LDOS spectrum predicts strong\nparticle-hole asymmetry at the vortex center which is as-\nsociated with the lowest bound state. [21] LDOS peak\nfor the lowest bound state appears only on empty (filled)\nstate if the carriers are electron (hole). [21, 23] This is\napparently consistent with the observed asymmetry in\nLiFeAs; the LDOS peaks are located below EFand the\nrelevant SGs are opening in the hole cylinders.\nIn this scenario, energy of the LDOS peak |Ep|at\nthe vortex center represents the energy of the lowest\nbound state given by ∆2/2EF. For the larger SG, we\nobserved that Ep∼ −2.3 meV and ∆ ∼6.0 meV, lead-\ningEF∼7.8 meV. This EFvalue is very small, in gen-\neral sense, but does not contradict with the ARPES re-\nsults [33–35, 43] where the top of the inner hole band\nis very close to EF. The particle-hole asymmetric spa-\ntial evolution of the LDOS peak, especially the step-like\nchange in its energy in the filled state, may be due to the\nFriedel-like oscillation. [21] Namely, it is possible that\nthe observed asymmetric quasi-particle states associated\nwith the largerSG reflect the quantum nature of the vor-\ntex electronic states in LiFeAs.\nHowever, the quantum-limit picture may not apply\nfor the smaller SG because calculated EF∼4.7 meV\nfromEp∼ −0.9 meV and ∆ ∼2.9 meV is much\n(at least order-of-magnitude) smaller than the value ex-\npected from ARPES dispersion. [33–35, 43] In addition,\nthe quasi-particle branch approaching to the smaller SG\nis observed to be very smooth without any trace of\nFriedel-like oscillation. Such a smooth evolution may\nsuggest that quantum nature is absent in the vortex elec-\ntronic states associated with the smaller SG. It is an im-\nportant open issue to identify what gives rise to the pro-\nnounced particle-hole asymmetry in the spectral weight\nand opens an energy gap at the vortex center.7\nFIG. 5: (color online). Temperature dependence of the tun-\nneling spectra at the vortex center (a) and away from vortice s\n(b). Tip was stabilized at Vs= +25 mV and It= 250 pA.\nVmod= 0.1 mV rms. Each curve (except at 7.5 K) is shifted\nvertically for clarity. Thin black lines denote the convolu ted\n0.54 K spectra by Fermi-Dirac distribution function.\nE. Temperature dependence: Kramer-Pesch effect\nNext we examine temperature Tdependence of the\nvortex-corestates. Figure 5(a) shows the tunneling spec-\ntra taken at the vortex center at different T. The LDOS\npeak at -0.9 meV diminishes rapidly with increasing T.\nWe compare the observed spectrum at each Twith the\nspectrum at 0.54 K convoluted with the Fermi-Dirac dis-\ntribution function at T. It is clear that the observed\nLDOS peak is broader than that expected from the ther-\nmal smearing effect. This is in stark contrast to the T\ndependence of the spectrum away from vortices where\neach spectrum is well reproducedby the convoluted spec-\ntrum as shown in Fig. 5(b). This contrasting behavior\nstrongly suggests that the temperature not only smears\nthe quasi-particle states but also changes the excitation\nspectrum in the vortex core.\nThe most plausible origin of this unusual temperature\neffect is the so-called Kramer-Pesch (KP) effect [29, 30].\nIn clean superconductors where /planckover2pi1/τ <∆, quasi-particle\ndistribution below ∆ depends on T. The lower-energy\nbound states are more and more populated at lower T.\nThisT-dependent quasi-particle distribution gives rise\nto theTdependence of spatial variation of SG around\na vortex, and vice versa. Kramer and Pesch first ana-\nlyzed this process using quasi-classical Eilenberger equa-\ntion and pointed out that, with lowering T, the spatial\nvariation of SG deviates from ∼tanh(r/ξ) behavior ex-\npected from Ginzburg-Landau theory. [29] Here, rde-\nnotes distance from the vortex center. In this case, it is\nconvenient to introduce a new length scale ξ1defined as\nξ1= limr→0(r/∆(r))∆(r→ ∞) to measure the vortex-\ncore size. Kramer and Pesch found that ξ1shrinks lin-\near inT, provided kBT >∆2/EF[29]. Later on, Gygi\nand Schl¨ uter analyzed a clean vortex core based on the\nquantum Bogoliubov-de Gennes formalism. As well as\nreproducing the shrinkage of vortex core, they clarified\nFIG. 6: (color online). Tunneling-conductance images at EF\nat different temperatures. Tip was stabilized at Vs= +25 mV\nandIt= 250 pA. Vmod= 0.2 mV rms. Note that color scale is\ndifferent from that for Fig.4(a)-(d).\ntheTdependence of vortex bound states. [18]\nFor the sake of examining the relevance of the KP ef-\nfect, the analysis of the Tdependence of tunneling spec-\ntrum would not be appropriatebecause individual bound\nstates are not resolved in Fig. 5(a). Instead, the Tde-\npendence of the vortex-core size can be a direct evidence\nof the KP effect. Figure 6 shows the conductance maps\natEFtaken at various T. It is clear that apparent vortex\nsizeincreaseswith increasing T, as expected from the KP\neffect. In order to quantify this process, we take conduc-\ntance profiles along Fe-Fe and As-As directions at each\nT. (Figs. 7(a) and 7(b)) We fit each profile with Gaus-\nsian function to obtain the half-width-at-half-maximum\n(HWHM), which can be a good measure of the vortex\nsize. As shown in Fig. 7(c), HWHM decreases rapidly\nwith decreasing Tand tends to saturate below about\n2 K. We note that Tdependence of ξestimated from\nBc2[39] is very weak in the Trange studied and does\nnot account for the observed strong Tdependence. In\naddition, HWHM at our lowest T= 0.54 K (1.9 nm and\n2.5 nm along Fe-Fe and As-As directions, respectively)\nis much shorter than ξ(T= 0)∼4.64 nm. [39] These\nobservations strongly suggest that the KP effect plays an8\nFIG. 7: (color online). Line profiles of tunneling-conducta nce atEFat different temperatures. (a) Along the nearest Fe-Fe\ndirection. (b) Along the nearest As-As direction. Solid lin es denote results of Gaussian fitting. (c) Temperature depen dence of\nhalf-width-at-half-maximum of the fitted Gaussian peak, wh ich represent the vortex-core size.\nimportant role in LiFeAs.\nThe importance of the observation of KP effect in\nLiFeAs is twofold. Firstly, to our knowledge, the KP\neffect has only been detected by muon spin rotation tech-\nnique. [45] Muon spin rotation primarily senses the field\nor current distribution inside the sample and the vortex-\ncore size is defined as the distance from the vortex cen-\nter at which the current density reaches the maximum\nvalue. [45] In contrast to this rather indirect way which\nrequires a specific model, STM/STS gives a direct unam-\nbiguous evidence of KP effect.\nSecondly, the observation of KP effect and its satu-\nration at low Tallows us to estimate the energy of the\nlowest bound state. In the quantum-limit vortex, vortex\ncore stops shrinking at T <∆2/kBEF, because no more\nlow-energystates are available. [21, 29] In the presence of\nimpurity scattering, the saturation occurs even at higher\nT. [30] The observed saturation at T∼2 K sets the up-\nper bound of the lowest bound-state energy associated\nwith the smaller SG to be about 0.2 meV.\nIV. CONCLUSIONS\nComprehensiveSTM/STS experimentson LiFeAs over\nwideBandTranges provide new insights into the prop-\nerties of vortex matter and the electronic states of indi-\nvidual vortices.\nZero-field tunneling spectra contain two superconduct-\ning gaps. By comparing with ARPES results, [33–35] we\nidentify the larger (∆ ∼6.0 meV) and smaller (∆ ∼\n2.9 meV) gaps as opening on inner and outer hole Fermi\nsurface cylinders, respectively.\nVortices in LiFeAs do not form into a well-ordered lat-\ntice while there is a well-developed distance correlation\nbetween vortices. We found that localhexaticcorrelation\nretains at low Bbut the vortex matter contains domain\nboundaries which consist of alternating vortices with un-\nusual coordination numbers of 5 and 7. With increasingB, the vortex matter approaches to uniformly disordered\namorphous state through increasing number of domain\nboundaries and roughening of each boundary.\nLDOS distribution at EFaround a single vortex ex-\nhibits tails along the nearest As-As directions and each\ntail splits in a parallel fashion at elevated energies. This\n4-fold anisotropic shape of the vortex core is associated\nwith the square shape of the outer hole Fermi-surface\ncylinder.\nThe multi-gap character of LiFeAs manifests itself in\nthe spatial evolution of LDOS spectrum around a vor-\ntex as two quasi-particle branches. Tunneling spec-\ntrum at the vortex center is strongly particle-hole asym-\nmetric, being reminiscent of the behavior expected in\nthe quantum-limit vortex core where the discrete bound\nstatesinsidethe corearesmearedoutneitherbyimpurity\nscattering nor thermal broadening. The quasi-particle\nbranch associated with the larger superconducting gap\nexhibitstep-likeenergychangeasapproachingtothevor-\ntex center. This is consistent with the Friedel-like oscilla-\ntion expected in the quantum-limit vortex. [21] However,\nquasi-particle states associated with the smaller super-\nconducting gap are not compatible with the quantum-\nlimit behavior, although energy-gap like feature is ob-\nservedeven at the vortexcenter. ObservationofKramer-\nPescheffect, strong Tdependence in the vortex-coresize,\nalsosuggeststhat energy-separationbetween the discrete\nbound states associated with the smaller gap is not large\nenough to bring about detectable energy gap at the vor-\ntex center. The mechanism which opens the energy gap\nat the vortex center is an interesting open question. Nev-\nertheless, we anticipate that observed rich spectroscopic\nfeatures exposed in the vortex core of LiFeAs serve as a\ntouchstone of various theoretical models for the super-\nconducting state in iron-based materials.\nNote added.- During the completion of the manuscript,\nwe became aware that Chi et al.also observed two su-\nperconducting gaps in LiFeAs by STM/STS [46].9\nAcknowledgments\nThe authors thank M. P. Allan, A. F. Bangura, T. -\nM. Chuang, J. C. Davis, N. Hayashi, P. J. Hirschfeld, Y.\nKato, E. -A. Kim, K. Machida, Y. Nagai, K. Okazaki, A.Rost, M. Takigawa, I. Vekhter, Y. Wang and H. -H. Wen\nfor valuable discussions and comments. This work has\nbeen supported by a Grant-in-Aid for Scientific Research\nfrom the Ministry of Education, Culture, Sports, Science\nand Technology of Japan.\n[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,\nJ. Am. Chem. Soc. 130, 3296 (2008).\n[2] J. Paglione and R. L. Greene, Nat. Phys. 6, 645 (2010).\n[3] P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep.\nProg. Phys. 74, 124508 (2011).\n[4] G. R. Stewart, Rev. Mod. Phys. 83, 1589 (2011).\n[5] D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003\n(2008).\n[6] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du,\nPhys. Rev. Lett. 101, 057003 (2008).\n[7] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H.\nKontani, and H. Aoki, Phys. Rev. Lett. 101, 087004\n(2008).\n[8] K. Seo, B. A. Bernevig, and J. Hu, Phys. Rev. Lett. 101,\n206404 (2008).\n[9] F. Wang, H. Zhai, Y. Ran, A. Vishwanath, and D.-H.\nLee, Phys. Rev. Lett. 102, 047005 (2009).\n[10] A. V. Chubukov, M. G. Vavilov, and A. B. Vorontsov,\nPhys. Rev. B 80, 140515 (2009).\n[11] V. Cvetkovic and Z. Tesanovic, EPL 85, 37002 (2009).\n[12] A. D. Christianson, E. A. Goremychkin, R. Osborn,\nS. Rosenkranz, M. D. Lumsden, C. D. Malliakas, I. S.\nTodorov, H. Claus, D. Y. Chung, M. G. Kanatzidis, R.\nI. Bewley, and T. Guidi, Nature 456, 930 (2008).\n[13] C.-T. Chen, C. C. Tsuei, M. B. Ketchen, Z.-A. Ren, and\nZ. X. Zhao, Nat. Phys. 6, 260 (2010).\n[14] T. Hanaguri, S. Niitaka, K. Kuroki, and H. Takagi, Sci-\nence328, 474 (2010).\n[15] C. Caroli, P. G. De Gennes, and M. J., Phys. Lett. 9, 307\n(1964).\n[16] H. F. Hess, R. B. Robinson, and J. V. Waszczak, Phys.\nRev. Lett. 64, 2711 (1990).\n[17] F. Gygi and M. Schl¨ uter, Phys. Rev. Lett. 65, 1820\n(1990).\n[18] F. Gygi and M. Schl¨ uter, Phys. Rev. B 43, 7609 (1991).\n[19] N.Hayashi, M. Ichioka, andK.Machida, Phys.Rev.Lett.\n77, 4074 (1996).\n[20] N. Hayashi, M. Ichioka, and K. Machida, Phys. Rev. B\n56, 9052 (1997).\n[21] N. Hayashi, T. Isoshima, M. Ichioka, and K. Machida,\nPhys. Rev. Lett. 80, 2921 (1998).\n[22] Y. Yin, M. Zech, T. L. Williams, X. F. Wang, G. Wu,\nX. H. Chen, and J. E. Hoffman, Phys. Rev. Lett. 102,\n097002 (2009).\n[23] L. Shan, Y.-L. Wang, B. Shen, B. Zeng, Y. Huang, A.\nLi, D. Wang, H. Yang, C. Ren, Q.-H. Wang, S. -H. Pan,\nand H. -H. Wen, Nat. Phys. 7, 325 (2011).\n[24] C.-L. Song, Y.-L. Wang, P. Cheng, Y.-P. Jiang, W. Li,\nT. Zhang, Z. Li, K. He, L. Wang, J.-F. Jia, H. -H. Hung,\nC. Wu, X. Ma, X. Chen, and Q. -K. Xue, Science 332,\n1410 (2011).\n[25] G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I.Larkin, and V. M. Vinokur, Rev. Mod. Phys. 66, 1125\n(1994).\n[26] E. H. Brandt, Phys. Rev. B 34, 6514 (1986).\n[27] A. Larkin and Y. Ovchinnikov, J. Low Temp. Phys. 34,\n409 (1979).\n[28] M. Chandran, R. T. Scalettar, and G. T. Zim´ anyi, Phys.\nRev. B69, 024526 (2004).\n[29] L. Kramer and W. Pesch, Z. Phys. 59, 269 (1974).\n[30] N. Hayashi, Y.Kato, andM. Sigrist, J. Low. Temp. Phys.\n139, 79 (2005), and references therein.\n[31] Y. Imai, H. Takahashi, K. Kitagawa, K. Matsubayashi,\nN. Nakai, Y. Nagai, Y. Uwatoko, M. Machida, and A.\nMaeda, J. Phys. Soc. Jpn. 80, 013704 (2011).\n[32] T. Hanaguri, J. Phys.: Conference Series 51, 514 (2006).\n[33] K. Umezawa, Y. Li, H. Miao, K. Nakayama, Z.-H. Liu,\nP. Richard, T. Sato, J. B. He, D.-M. Wang, G. F. Chen,\nH. Ding, T. Takahashi, and S.-C. Wang, Phys. Rev. Lett.\n108, 037002 (2012).\n[34] S. V. Borisenko, V. B. Zabolotnyy, A. A. Kordyuk, D. V.\nEvtushinsky, T. K. Kim, I. V. Morozov, R. Follath, and\nB. B¨ uchner, Symmetry 4, 251 (2012).\n[35] K. Okazaki, private communication.\n[36] M. Allan, A. Rost, T.-M. Chuang, and J. C. Davis, pri-\nvate communication.\n[37] H. Sakata, M. Oosawa, K. Matsuba, N. Nishida, H.\nTakeya, and K. Hirata, Phys. Rev. Lett. 84, 1583 (2000).\n[38] H. Nishimori, K. Uchiyama, S.-i. kaneko, A. Tokura, H.\nTakeya, K. Hirata, and N. Nishida, J. Phys. Soc. Jpn.\n73, 3247 (2004).\n[39] N. Kurita, K. Kitagawa, M. Kazuyuki, A. Kismara-\nhardja, E.-S. Choi, J. S. Brooks, Y. Uwatoko, S. Uji,\nand T. Terashima, J. Phys. Soc. Jpn. 80, 013706 (2011).\n[40] M. M. Altarawneh, K. Collar, C. H. Mielke, N. Ni, S.\nL. Bud’ko, and P. C. Canfield, Phys. Rev. B 78, 220505\n(2008).\n[41] H. Q. Yuan, J. Singleton, F. F. Balakirev, S. A. Baily,\nG. F. Chen, J. L. Luo, and N. L. Wang, Nature 457, 565\n(2009).\n[42] Y.-D. Zhu, F. C. Zhang, and M. Sigrist, Phys. Rev. B\n51, 1105 (1995).\n[43] S. V. Borisenko, V. B. Zabolotnyy, D. V. Evtushinsky, T.\nK. Kim, I. V. Morozov, A. N. Yaresko, A. A. Kordyuk,\nG. Behr, A. Vasiliev, R. Follath, et al., Phys. Rev. Lett.\n105, 067002 (2010).\n[44] Y. Wang, P. J. Hirschfeld, and I. Vekhter, Phys. Rev. B\n85, 020506(R) (2012).\n[45] J. E. Sonier, J. Phys.: Condensed Matter 16, S4499\n(2004).\n[46] S. Chi, S. Grothe, R. Liang, W. N. Hardy, S. A. Burke,\nD. A. Bonn, and Y. Pennec, arXiv:1204.0273v1."    },    {        "title": "0908.2343v3.Optical_conductivity_of_metal_nanofilms_and_nanowires__The_rectangular_box_model.pdf",        "content": "arXiv:0908.2343v3  [cond-mat.mes-hall]  24 Feb 2010Optical conductivity of metal nanofilms and nanowires: The r ectangular-box model\nValery P. Kurbatsky and Valentin V. Pogosov∗\nDepartment of Micro- and Nanoelectronics, Zaporozhye Natio nal\nTechnical University, Zhukovsky Str. 64, Zaporozhye 69063, U kraine\n(Dated: June 18, 2018)\nThe conductivity tensor is introduced for the low-dimensio nal electron systems. Within the\nparticle-in-a-box model and the diagonal response approxi mation, components of the conductivity\ntensor for a quasi-homogeneous ultrathin metal film and wire are calculated under the assumption\nd∼=λF(wheredis the characteristic small dimension of the system, λFis the Fermi wavelength\nfor bulk metal). We find the transmittance of ultrathin films a nd compare these results with\navailable experimental data. The analytical estimations f or the size dependence of the Fermi level\nare presented, and the oscillations of the Fermi energy in ul trathin films and wires are computed.\nOur results demonstrate the strong size and frequency depen dences of the real and imaginary parts\nof the conductivity components in the infrared range. A shar p distinction of the results for Au and\nPb is observed and explained by the difference in the relaxati on time of these metals.\nPACS numbers: 73.60.Ag, 73.40.-c, 72.20.Dp\nKeywords:\nI. INTRODUCTION\nThin-film materials, in particular, metal films are\nused widely in modern technologies including electron-\nics. As a rule, ultrathin films are fragmentized (the is-\nland films) and consist of flat islands connected with the\nthin threads-channels [1–3].\nExperimental techniques allow the optical characteris-\ntics in the infraredrangetobe measurednotonly forthin\nfilms (see, for example, [4–15]), but also for the specifi-\ncally grown nanorods-antennas of the micrometer length\n[16–18].\nIn [7], the authors for the first time measured the in-\nfrared conductivity of Pb ultrathin films. A decrease in\nthe conductivity of the films was explained by their gran-\nular structure. Subsequently, Tu et al. [9] measured the\noptical characteristics of metal films at a temperature of\n10 K and revealed an anomalous optical transparency in\nthe far-IR range. Pucci et al. [14] were the first to study\nthe quantum size effects in the transmission spectra of\nlead thin films by IR spectroscopy. It should be noted\nthat results of measurements have been usually inter-\npretedbyexperimentersintheframeworkofthemodified\nDrude theory. A theoretical analysis of optical proper-\ntiesofultrathinfilmsandwiresisnecessary,inparticular,\nfor the diagnostics of the nanostructure materials [19] in\norder to use them in micro- and nanoelectronics [20].\nThe important feature of the metal 1D- and 2D sys-\ntems, films and wires, is an anisotropy of their electrical\nand optical properties caused by the size quantization.\nFor this reason, the conductivity of the low-dimensional\nsystems is represented by a tensor σαβ(q,ω) which, in\nparticular, determines the optical absorption. The dissi-\npation of energy of the plane monochromaticelectromag-\n∗Corresponding author: vpogosov@zntu.edu.uanetic wave with the frequency ωand the wave vector q\nin unit volume per unit time for a nonmagnetic material\nis\nQ(q,ω) =1\n4/summationdisplay\nα,β/braceleftbig\nσ∗\nαβ(q,ω)+σβα(q,ω)/bracerightbig\nEαE∗\nβ,\nwhereEα,βare the components of the electric field [21].\nHowever, the only value directly measurable for an ultra-\nthin film in IR range is the transmittance.\nThe purpose of this work is to calculate components of\nthe conductivity tensor for quasi-homogeneous ultrathin\nmetal films and wires provided the condition d∼=λFis\nsatisfied. We use the Wood and Ashcroft approach [22]\nadapted to this case. The main advancement is the pro-\ncedure of the accurate determination of the Fermi level\nfor a film and a wire of such a thickness with taking into\naccount the size oscillations. The transmittance of the\nultrathin films is also calculated in order to compare the-\noretical results with experimental data [4, 5].\nTo the best of our knowledge, the oscillatory behavior\nof the Fermi energy in confined (2D) electron gas was\nstudied for the first time by Sandomirskii [23]. Later\ncalculations were performed for thin films, spheres and\nwires [24–34]. In the present paper, in order to describe\nthe Fermi energy behavior in low-dimensional metallic\nsystems we use an elementary one-particle analytical ap-\nproach [32].\nII. CONDUCTIVITY TENSOR\nA film of thickness L(or a wire of radius ρ0) compara-\nble in magnitude to the Fermi wavelength of an electron\nin an infinite metal ( λ0\nF≈0.5 nm) will be referred to as\nthe ultrathin film or wire (see Fig. 1). The longitudinal\nsizes of the sample are assumed to be considerably larger\nthan the film thickness: L≪a,b(orρ0≪ Lfor wire),2\nwhich leads to the pronounced quantization of the trans-\nverse component of the electron momentum. This re-\nsults in the formation of subbands, i.e., groups of energy\nlevels corresponding to the same value of the transverse\nmomentum component.A response of an electron gas to the electromagnetic\nfieldE=E0exp[i0(qr−ωt)] may be determined in a\nlinear approximation by the density matrix technique.\nFor the induced current one can obtain [22]:\nˆj(k,ω) =i0e2\nΩmeω/braceleftigg\nE0/summationdisplay\nifi∝an}bracketle{ti|ei0(q−k)r|i∝an}bracketri}ht\n+1\nme/summationdisplay\nijfi−fj\nεij−/planckover2pi1w/parenleftbigg\n∝an}bracketle{tj|e−i0krˆp|i∝an}bracketri}ht−1\n2/planckover2pi1k∝an}bracketle{tj|e−i0kr|i∝an}bracketri}ht/parenrightbigg/parenleftbigg\n∝an}bracketle{ti|ei0qrE0ˆp|j∝an}bracketri}ht+1\n2/planckover2pi1qE0∝an}bracketle{ti|ei0qr|j∝an}bracketri}ht/parenrightbigg\n\n,(1)\nFigure 1: Choice of coordinates.\nwherei0=√−1;|i∝an}bracketri}ht,|j∝an}bracketri}htare the wave functions of the\ninitial and final electron states corresponding to energies\nεiandεj;εij=εi−εj;fjandfjare occupation factors;\nΩ is the volume of sample, meis the electron mass, −e\nis the electron charge, ˆpis the momentum operator.\nWe describe conductivity electrons in metal films\nand wires within the framework of the particle-in-a-box\nmodel. In many cases, this model turns out to be quite\nproductive for metals with high conductivity [35]. In the\ncase of the low-dimensional electron systems, a potential\nbox model includes the key feature of these systems –\na confinement of electrons inside a region with certain\nsmall dimension. The model remains applicable up to acertain critical size when conductivity vanishes. As for\nthe depth of a box, this parameter remains important as\nlong as electron emitting is permitted.\nIn general, the model of electron gas in a rectangular\npotential box is a good initial approximation, while var-\nious details (particularities of structure, impurities etc.)\ncan be taken into account by introducing corresponding\ncorrections.\nAn electron system in low-dimensional structures is\nanisotropic, and its characteristics can be expressed in\ntensor form. The tensor origin of the conductivity be-\ncomes obvious after converting the expression (1) into\nthe form\njα(k,ω) =/summationdisplay\nβσαβ(k,q,ω)Eβ(q,ω),\nwhereα,β=x,y,zandσαβis the conductivity tensor.\nIt is not difficult to demonstrate that the conduc-\ntivity tensor is proportional to δαβδk,q, whereδαβor\nδk,q={1,k=q; 0,k∝ne}ationslash=q}is Kronecher’s symbol,\nfor macroscopic samples with the wave functions of the\nkindΩ−1/2exp(−pr//planckover2pi1). ThisimpliesthatalltheFourier\ncomponents of the current, except the one with k=q,\nare equal to zero. Of course, this is not the case for ul-\ntrathin films and wires, but the component with k=q\nis still dominating. At the first step, known as the di-\nagonal response approximation, this component only is\ntaken into account. We then find\nσαβ(q,q,ω) =i0e2N\nmeωΩδαβ+i0e2\nm2eωΩ/summationdisplay\ni,jfi−fj\nεij−/planckover2pi1ω/parenleftbigg\n∝an}bracketle{tj|e−i0qrˆpα|i∝an}bracketri}ht−1\n2/planckover2pi1qα∝an}bracketle{tj|e−i0qr|i∝an}bracketri}ht/parenrightbigg\n×/parenleftbigg\n∝an}bracketle{ti|ei0qrˆpβ|j∝an}bracketri}ht+1\n2/planckover2pi1qβ∝an}bracketle{ti|ei0qr|j∝an}bracketri}ht/parenrightbigg\n≡σαβ(q,ω).(2)\nHere the relation/summationtext\nifi=Nis used with Nequal to the number of the conductivity electrons.3\nOver infrared region, the condition qL, qρ 0≪1 is sat-\nisfied allowing us to express the conductivity tensor in\nterms of the according small value.\nIII. FILM\nIt is assumed that the conduction electrons of the film\nare located in a rectangular potential box V(r) with a\ndepthU0<0, so that the box shape reproduces the film\nshape (see Fig. 1), and\n|U0|=ε0\nF+W0, ε0\nF=/planckover2pi12\n2m(3π2¯n)2/3.(3)\nHereW0, ε0\nF, and ¯nare the electron work function, the\nFermi energy and the electron concentration for a bulk\nmetal, respectively.\nThe unperturbed states of the film are described by\nthe wave functions\nΨmnp(x,y,z) =1√\nabψm(x)e2πni0y/ae2πpi0z/b,(4)\nwheren, p=±1,±2,...andm= +1,+2,.... The sub-\nscriptmnumbers the subbands. The wave functions\nψm(x) are represented in the following form:\nfor even values of m,\nψm(x) =\n\nCmsinkxmx,−L/2<x<L/ 2,\n(−1)(m/2)+1Bme−κmx, x>L/ 2,\n(−1)m/2Bmeκmx, x< −L/2,(5)\nand for odd values of m,\nψm(x) =\n\nCmcoskxmx,−L/2<x<L/ 2,\n(−1)(m−1)/2Bme−κmx, x>L/ 2,\n(−1)(m−1)/2Bmeκmx, x<−L/2,(6)\nCm=/radicalbigg\n2κm\n2+κmL, Bm=Cmkxm\nk0eκmL/2.\nHere,Cmis the normalization factor, kxmare the roots\nof the equation\nkxmL=−2arcsin(kxm/k0)+πm, (7)\nwhereκm=/radicalbig\nk2\n0−k2xmand/planckover2pi1k0=/radicalbig\n2me|U0|(see Ref.\n[32]).\nIn this section, we focus on optical transitions between\nsubbands accompanied by changing the transverse com-\nponentoftheelectronwavevector kxm. Thesetransitions\ncontribute to the σxxcomponent of the conductivity ten-\nsor. SinceqL≪1, we have in zero approximation\nσxx=i0e2\nmeωΩ\n×\nN+1\nme/summationdisplay\ni,jfi−fj\nεij−/planckover2pi1ω/vextendsingle/vextendsingle/vextendsingle∝an}bracketle{tj|e−i0(qyy+qzz)ˆpx|i∝an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle2\n.\n(8)\nDividing by εij−/planckover2pi1ωin the sum and interchanging iand\njfor the second term appeared after this dividing, ex-pression (8) can be transformed into\nσxx=i0e2\nmeωΩ\n×\nN+2\nme/summationdisplay\ni,jfiεij\nε2\nij−/planckover2pi12ω2/vextendsingle/vextendsingle/vextendsingle∝an}bracketle{tj|e−i0(qyy+qzz)ˆpx|i∝an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle2\n.\n(9)\nSince\n∝an}bracketle{tj|e−i0(qyy+qzz)ˆpx|i∝an}bracketri}ht=∝an}bracketle{tm′|ˆpx|m∝an}bracketri}htδqy,kyn−kyn′δqz,kzp−kzp′,\nand in view of the fact that |kxm−kxm′| ≫q, further\nsimplifications are possible:\nσxx≈i0e2\nmeωΩ\n×\nN+2\nme/summationdisplay\nm,m′\nn,pfmnpεmm′\nε2\nmm′−/planckover2pi12ω2|∝an}bracketle{tm′|ˆpx|m∝an}bracketri}ht|2\n.(10)\nHere the occupation factor is approximated by the step\nfunctionfmnp=θ(εF−εmnp), whereεFis the Fermi\nenergy for nanofilm, εmm′=/planckover2pi12(k2\nxm−k2\nxm′)/2me. Using\nThomas-Reiche-Kuhnsumrule(seeRef. [22]), werewrite\n(10) as\nσxx=2i0e2/planckover2pi12ω\nm2eΩ/summationdisplay\nm,m′\nn,pfmnp|∝an}bracketle{tm′|ˆpx|m∝an}bracketri}ht|2\nεmm′(ε2\nmm′−/planckover2pi12ω2),(11)\nand then one can obtain corresponding component of the\ndielectric tensor\nǫxx= 1+4πi0\nωσxx. (12)\nThe matrix elements of the momentum projection op-\nerator ˆpx=i/planckover2pi1∂/∂xfrom (4) – (6) are\n|∝an}bracketle{tm′|ˆpx|m∝an}bracketri}ht|2=/braceleftig\n1−(−1)m+m′/bracerightig\n×8/planckover2pi12k2\nxmk2\nxm′κmκm′\n(k2\nxm′−k2xm)2(2+κmL)(2+κm′L).(13)\nThe broadening is introduced in a manner proposed\nby Mermin [36]. As a result of this procedure the tensor\ncomponents σxxandǫxxget both real and imaginary\nparts:\nReσxx=/parenleftig4\nL/parenrightig3a0γ2\nπ/parenleftig1\n/planckover2pi1e2\n2a0/parenrightig\nH(+),(14)\nImσxx=−/parenleftig4\nL/parenrightig3a0k2\nω\nπ/parenleftig1\n/planckover2pi1e2\n2a0/parenrightig\nH(−),(15)\nReǫxx= 1+/parenleftig4\nL/parenrightig4L\na0H(−),Imǫxx=/parenleftig4\nL/parenrightig4Lγ2\na0k2ωH(+),\n(16)4\nwhere\nH(∓)=mF/summationdisplay\nm=1mmax/summationdisplay\nm′=1{1−(−1)m+m′}L2κmκm′k2\nxmk2\nxm′(k2\nF−k2\nxm){(k2\nxm′−k2\nxm)2∓k4\nω∓γ4}\n(2+κmL)(2+κm′L)(k2\nxm′−k2xm)3{{(k2\nxm′−k2xm)2−k4ω+γ4}2+4k4ωγ4}.(17)\nHereγ=/radicalbig\n2me//planckover2pi1τ,τis the relaxation time, /planckover2pi1kω=√2me/planckover2pi1ω,a0is the Bohr radius and\nmF=/bracketleftbiggLkF\nπ+2\nπarcsin/parenleftigkF\nk0/parenrightig/bracketrightbigg\n, mmax=/bracketleftbiggLk0\nπ/bracketrightbigg\n+1.\n(18)\nSquarebracketsin (18) and in the text belowindicate the\ninteger number. Instead of the summation over nandp\nin (11), (12) we perform integration.\nIn order to use Eqs. (14) – (18) in calculations, it is\nnecessary to supplement them by the relation determin-\ning the Fermi energy of film [32]\nk2\nF=1\nmF/parenleftigg\n2π¯nL+mF/summationdisplay\nm=1k2\nxm/parenrightigg\n. (19)\nThe relation (19) together with Eqs. (7) and (18) de-\nscribes the size-dependent Fermi level in ultrathin films.\nIn the caseof a film, transmittance is a quantity, which\nis directly measurable\nTR =I/I0, (20)\nwhereI0andIare intensities of a wave at surfaces x=\n−L/2 andx=L/2, respectively.\nFor a film of thickness Lthe transmittance may be\nestimated as:\nTR = exp {−η(ω,L)L}, (21)\nwhere the absorption coefficient ηshould be calculated\nby using (16) and the formula\nη=2ω\ncIm/radicalbig\nǫ(ω,L). (22)\nIV. WIRE\nThe simplest model for an ultrathin wire (see Fig. 1)\nis to consider it as a cylindrical potential well V(ρ,z)\nof infinite depth. The length of the well Lis assumed\nto be much larger than its radius ρ0. The conductivity\nelectrons are described by the wave functions of the kind\nψmnp(ρ,ϕ,z) =Rmn(ρ)Φm(ϕ)Zp(z).(23)\nThe function\nZp(z) =1√\nLei0kzpz(24)corresponds to the longitudinal motion of an electron.\nThe subscript pnumbers values of z−component of its\nwave vector. The angle part of the wave function\nΦm(ϕ) =1√\n2πei0mϕ(25)\nhas to satisfy the periodicity condition\nΦm(ϕ+2π) = Φm(ϕ), (26)\nfrom which follows the eigenvalues spectrum m= 0,±1,\n±2,....\nThe radial dependence of the wave function is de-\nscribed by the Bessel functions of an integer order\nRmn(ρ) =CmnIm(kmnρ), (27)\nwhere\nCmn=√\n2\nρ0|I′m(kmnρ0)|. (28)\nHerekmn=amn/ρ0, whereamnare positive roots of the\nBessel function of the m-th orderIm(ξ),n= 1,2,....\nThe prime marks a derivative with respect to ξ.\nIn the next Section, for the case of a wire, we obtain\nthe relation similar to Eq. (19).\nA. The Fermi energy\nWe start from the expression for the energy of an elec-\ntron\nεmnp=/planckover2pi12\n2me/parenleftbig\nk2\nmn+k2\nzp/parenrightbig\n, (29)\nwherekmnandkzpare the eigenvalues of transverse and\nlongitudinal components of the electron wave vector, re-\nspectively [20]. The electron states in a wire correspond\nto points (kmn,kzp) on thek⊥kzphalf-plane ( k⊥>0).\nSince the spectrum kzpis quasicontinuous ( L ≫ρ0),\nthese points form a system of straight lines k⊥=kmn.\nThe occupied states distribute on intercepts cut off by\nthe semicircle of radius kF(see Fig. 2). Density of the\nelectron states at the intercepts is equal to L/π.\nThe total number of the occupied states (equal to the\nnumber of the conduction electrons in a wire) is\nN= 2L\nπ/summationdisplay\nm,n/radicalig\nk2\nF−k2mn.5\nFigure 2: Geometrical diagram of electron-state filling in\nquantum wire.\nTaking into account that N= ¯nΩ, we then obtain the\nequation for the computation of the Fermi level kFin an\nultrathin wire\n¯n=2\nπ2ρ2\n0/summationdisplay\nm,n/radicalig\nk2\nF−k2mn. (30)\nThe electron concentration ¯ nis assumed to be the same\nin a wire and in a bulk metal. The summation should\nbe performed over all numbers mandnsatisfying the\ncondition\nkmn≤kF. (31)\nThe size dependence of the Fermi level in ultrathin\nfilms and wires has an ”oscillatory” form. In order to\ndetermine a magnitude of the variations, let us evaluate\nthe averaged (smoothed) size dependence. In this case,\nan averaging means a replacement of summation in Eqs.\n(19) and (30) by integration.\nForafilm, weusethe Euler-MacLaurinsummationfor-\nmula [37], in which it is enough to take into account the\nfirst two terms. Allowing mFto take any value (not only\ninteger,mF≈LkF/π) and neglecting the corrections for\na finite depth of a potential box, we obtain\nkF/k0\nF≈1+π/(4k0\nFL), (32)\nwherek0\nFis the Fermi wave number for a bulk metal.\nIn the case of a wire, it is hard to estimate directly the\nsizedependenceoftheFermilevelbecauseitisimpossible\nto express explicitly the roots of the Bessel functions.\nHowever, the averaged size dependence kF(ρ0) can be\nobtained in an indirect way.\nLet us rewrite (30) as\n(k0\nFρ0)3= 6/summationdisplay\nm,n/radicalig\n(k0\nFρ0)2−a2mn.\nHerek0\nF= (3π2¯n)1/3, and the relation\nkmn=amn/ρ0 (33)\nwas used.amnare positive roots of the Bessel function\nof orderm= 0,±1,±2,...andn= 1,2,.... Assumingtheamn=a(m,n) function to be continuous, we turn to\nthe integration:\n(k0\nFρ0)3= 12/integraldisplay /integraldisplay/radicalbig\n(kFρ0)2−a2(m,n)dmdn (34)\n(m≥0 now). Limits of the integration are determined\nby the condition a(m,n)≤kFρ0.\nThe left-hand side of Eq. (34) tends to zero with ρ0→\n0. At the same time, the right-hand side tends to zero\nonly ifkFρ0→a01. Hence, the averaged size dependence\nis\nkF(ρ0)≈a01/ρ0\nfor the small values ρ0. For the large values ρ0, the in-\ntegrated expression is kFρ0and the area of the region\nof integration/integraltext/integraltext\ndmdnis proportional to k2\nFρ2\n0. Then,\ncomparing the left-hand and right-hand sides, we find\nthatkF(ρ0)→const with ρ0→ ∞. Accepting the con-\nstant to be k0\nF, we finally obtain\nkF/k0\nF= 1+a01/(k0\nFρ0),\nwherea01≈2.4048.\nIn the next subsection, to calculate conductivity com-\nponents, we use the size-dependent Fermi energy εF\nfound from the exact expression (30).\nB. Components of the conductivity tensor\nLet us consider the case when a wave is directed nor-\nmally to the axis of a wire (see Fig. 1). The wave vector\nisthen locatedin the x−yplane, i.e.,qz= 0. Orientating\nthex−axis along the wave propagation, we get qy= 0,\nqr=qxx≃ρ0/λ≪1, ande±i0qr≈1±i0qxx.\nIn zero order of the expansion σαβin terms of ρ0/λ,\nthe expression (2) takes the form\nσαβ=i0e2N\nmeωΩδαβ+i0e2\nm2eωΩ/summationdisplay\ni,jfi−fj\nεij−/planckover2pi1ω∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht∝an}bracketle{ti|ˆpβ|j∝an}bracketri}ht.\n(35)\nFollowingthe procedure, whichled ustoEq. (9), wehave\nσαβ=i0e2N\nmeωΩδαβ+i0e2\nm2eωΩ/summationdisplay\ni,jfi\n×/parenleftigg\n∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht∝an}bracketle{ti|ˆpβ|j∝an}bracketri}ht\nεij−/planckover2pi1ω+∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht∗∝an}bracketle{ti|ˆpβ|j∝an}bracketri}ht∗\nεij+/planckover2pi1ω/parenrightigg\n.(36)\nUsing (24) – (28), after rather cumbersome transfor-\nmations(seeAppendixA),thematrixelementsofvarious\nprojections of the momentum operator can be written as\n∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht=\n\n/planckover2pi1kzpδij, α=z;\n−i0/planckover2pi1\n2δpp′kmnCmnG(−), α=x;\n/planckover2pi1\n2δpp′kmnCmnG(+), α=y;(37)6\nG(∓)=δm−1,m′J(−)∓δm+1,m′J(+),(38)\nJ(∓)=Cm∓1,n′ρ0/integraldisplay\n0Im∓1(km∓1,n′ρ)Im∓1(kmnρ)ρdρ.\nBecause of a specific form of ∝an}bracketle{tj|ˆpz|i∝an}bracketri}htthe sum in Eq. (35)\nbecomes zero if α=zorβ=z. Hence,\nσxz,zx,yz,zy = 0, σzz=i0e2¯n\nmeω. (39)For other diagonal components, the expression (36) can\nbe easily transformed into\nσαα=i0e2¯n\nmeω+2i0e2\nm2eωΩ/summationdisplay\ni,jfiεij\nε2\nij−/planckover2pi12ω2|∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht|2,(40)\nwhere the subscript α=x,y. After a substitution of the\nmatrix elements (37) into (40), we find\nσxx,yy=i0e2¯n\nmeω+i0e2\nmeωΩ/summationdisplay\nm,n\np,n′fmnpk2\nmnC2\nmn/braceleftigg\n(k2\nmn−k2\nm−1,n′)J2\n(−)\n(k2mn−k2\nm−1,n′)2−k4ω+(k2\nmn−k2\nm+1,n′)J2\n(+)\n(k2mn−k2\nm+1,n′)2−k4ω/bracerightigg\n,(41)\nwhere\nfmnp=/braceleftbigg1, k2\nmn+k2\nzp<k2\nF,\n0, k2\nmn+k2\nzp>k2\nF.\nAn expression for the non-diagonal components σxy\nandσyxfollows from (36)\nσαβ=i0e2\nm2eωΩ/summationdisplay\ni,jfi\n×/parenleftbigg∝an}bracketle{tj|ˆpx|i∝an}bracketri}ht∝an}bracketle{ti|ˆpy|j∝an}bracketri}ht\nεij∓/planckover2pi1ω+∝an}bracketle{tj|ˆpx|i∝an}bracketri}ht∗∝an}bracketle{ti|ˆpy|j∝an}bracketri}ht∗\nεij±/planckover2pi1ω/parenrightbigg\n.(42)\nThe upper sign corresponds to α=x,β=y, and the\nlower one to α=y,β=x.\nThe axis symmetry of the problem is reflected by the\nfact that in (35) and (36) the summation is performed\nover positive mandm′as well as negative ones but the\nsame in absolute value. The analysis of the expressions\n(37) and (38) based on the properties of the Bessel func-\ntions [38]\nk(−m)n=kmn, I−m(ξ) = (−1)mIm(ξ)\nreveals a different behavior of the matrix elements when\nchanging together m→ −mandm′→ −m′,\n∝an}bracketle{tj|ˆpx|i∝an}bracketri}ht → −∝an}bracketle{tj|ˆpx|i∝an}bracketri}ht,∝an}bracketle{tj|ˆpy|i∝an}bracketri}ht → ∝an}bracketle{tj|ˆpy|i∝an}bracketri}ht.(43)\nThis causes the terms in (42) to cancel pairwise, and we\nthen find\nσxy=σyx= 0. (44)\nThus, all non-diagonal components of the conductiv-\nity tensor vanish in zero approximation of the expansion\nin terms of ρ0/λ. However, in linear approximation the\nresult is different. We take account that terms, whichcontainδij, lead to the vanishing of the sum. Therefore,\nin this approximation, components σzxandσzyhave a\nform\nσzβ=qxe2\nm2eωΩ/summationdisplay\ni,jfi\n×/parenleftigg\n∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∝an}bracketle{ti|ˆpβ|j∝an}bracketri}ht\nεij−/planckover2pi1ω+∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∗∝an}bracketle{ti|ˆpβ|j∝an}bracketri}ht∗\nεij+/planckover2pi1ω/parenrightigg\n,(45)\nwhereβ=x,yandforthematrixelementsseeAppendix.\nAn analysis, similar to the one, which resulted in Eq.\n(43), gives\n∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht → −∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht,\n∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∝an}bracketle{ti|ˆpy|j∝an}bracketri}ht → −∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∝an}bracketle{ti|ˆpy|j∝an}bracketri}ht,\n∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∝an}bracketle{ti|ˆpx|j∝an}bracketri}ht → ∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∝an}bracketle{ti|ˆpx|j∝an}bracketri}ht.(46)\nHence, to linear order in ρ0/λwe haveσzy= 0 butσzx∝ne}ationslash=\n0. Using Eq. (38), the relation\n∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∗∝an}bracketle{ti|ˆpx|j∝an}bracketri}ht∗=−∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht∝an}bracketle{ti|ˆpx|j∝an}bracketri}ht\nand Eq. (A3), we derive\nσzx=2i0qxe2\n/planckover2pi1Ω/summationdisplay\nn,n′\nm,pfmnpkzpC2\nmn(F(−)−F(+)),(47)\nwhere\nF(∓)=J(∓)Cm∓1,n′ρ0/integraltext\n0Im∓1(km∓1,n′ρ)Im(kmnρ)ρ2dρ\n(k2mn−k2\nm∓1,n′)2−k4ω.\nDissipation is introduced by the substitution ω→\nω+i0/τin expression for conductivity. When τ= 0,7\nthe diagonal components of conductivity are imaginary.\nSince the remaining components of the tensor vanish in\nzero approximation, dissipation is also absent ( Q= 0).\nIn general, dissipation is small for optical frequencies in\nwhich we are interested ( ω≫1/τ).\nSubstituting ω→ω+i0/τin (39), after straightfor-\nward transformations we obtain the Drude formula [39]\nσzz(ω) =σ(0)1+i0ωτ\n1+ω2τ2, (48)\nwhereσ(0)≡e2¯nτ/meis the static conductivity. Thus,\nthe component σzz(ω) is associated with the classical\nconductivity. Other diagonal components (40) can be\nrepresented as\nσαα=σzz{1+S(ω,ρ0,L)}, (49)\nwhere\nS≡2\nNme/summationdisplay\ni,jfiεij/parenleftbig\nε2\nij−/planckover2pi12ω2+2/planckover2pi12ωi0/τ/parenrightbig\n(ε2\nij−/planckover2pi12ω2)2+4/planckover2pi14ω2/τ2|∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht|2\n(50)\nandα=x,y.\nAfter interchanging subscripts iandj, terms of the\nsum (50) reverse their sign. As a result,\n/summationdisplay\ni,j\nεi,εj<εFfiεij/parenleftbig\nε2\nij−/planckover2pi12ω2+2/planckover2pi12ωi0/τ/parenrightbig\n(ε2\nij−/planckover2pi12ω2)2+4/planckover2pi14ω2/τ2|∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht|2= 0,\nand\nS=2\nNme/summationdisplay\ni,j\nεi<εFεj>εFεij/parenleftbig\nε2\nij−/planckover2pi12ω2+2/planckover2pi12ωi0/τ/parenrightbig\n(ε2\nij−/planckover2pi12ω2)2+4/planckover2pi14ω2/τ2|∝an}bracketle{tj|ˆpα|i∝an}bracketri}ht|2.\n(51)\nHereεij<0, i.e. only transitions coupled with absorp-\ntion participate in the conductivity. It is important to\nremark that Im S <0 for any frequency. Since in the\noptical region the real part of the component σzzcan be\nignored and its imaginary part is positive, it follows from\n(49) that Re σxx,yy>0 andQ>0 over the whole region.\nLet us compare in magnitude components of the con-\nductivity tensor. For Au, the frequency /planckover2pi1ω= 1 eV, dis-\nsipation /planckover2pi1/τ= 0.02 eV we find σ(0) = 4.6×1017s−1,\n|σzz| ≈σ(0)/ωτ≈1016s−1. We use below the value\ne2/2a0/planckover2pi1= 2.0×1016s−1as a unit of the conductivity.\nThen|σzz| ≈0.5.\nWe can now estimate, for example, height of peaks in\nReσxx. We use relationships\nReσxx=−|σzz|ImS\nand\nImS≈ −τ\n/planckover2pi1Nme|∝an}bracketle{tm+1,n′|ˆpx|mn∝an}bracketri}ht|2/summationdisplay\np1(which may be obtained from (49) and (50) under con-\ndition that the peaks are well separated). Taking into\naccount that\n/summationdisplay\np1 =2L\nπ/radicalig\nk2\nF−k2mn∼=2L\nπk0\nF,(52)\n|∝an}bracketle{tm+1,n′|ˆpx|mn∝an}bracketri}ht|2∝k2\nmn∼=1\n4/planckover2pi12k0\nF2(53)\nand, using (3), we have\nImS∼=−3/planckover2pi1τ\n2meρ2\n0.\nForτ= 2.1×10−14s−1(Au),d= 2ρ0= 2 nm, we find\nImS∼=−1, Reσxx∼=1. In macroscopic limit ρ0→ ∞we\nfind that Re σxx= 0 and Im σxx= Imσzz, as we have\nexpected.\nComparing (36) with (45), one can obtain |σzx/σxx|∼=\nqxρ0. Forλ= 103nm,d= 2 nm we have |σzx/σxx|∼=\n10−2.\nV. RESULTS AND DISCUSSION\nThe difference between our approach and the theory\n[22] is associated with peculiarities of a electronic levels\ndistribution in films/wiresofnanometersthickness, when\nthed∼=λFcondition is satisfied.\nIn this case, a number of subbands, formed as a result\nof the size quantization, is small, while the contribution\nof each of them in the sum (17) is significant. Opposite\nassumptions are made in [22]: a characteristic size dis so\nlarge that the number of subbands is much larger than\n1. Then, the separation between neighboring subbands\n(with numbers mandm+1) is small, while contributions\nof individual items in the sum [22] are not significant\nanymore, so the summation can be substituted by inte-\ngration, as usually done in the case of a quasi-continuous\ndistribution. Discreteness, coming from the size quan-\ntization, manifests itself only weakly. The Fermi level\nin films and wires with small thickness noticeably differs\nfrom the Fermi level of a bulk metal (30% difference for\na wire of 1 nm diameter, see Fig. 3). In [22], the authors\nuse the Fermi level of a bulk metal to find a number of\nsubbands, while wetakeintoaccountthe sizedependence\nof the Fermi level, when determining a number of sub-\nbands. For a few nanometers thickness, these numbers\nare found to be different and, because they are small, a\nnoticeable divergence in results is revealed.\nFinally, theory [22] was developed in order to apply it\nfor an isotropic composite medium. Therefore, from the\nvery beginning, the direction of an applied field was con-\nsideredasa preferredone. In ourapproach,anisotropyof\nthe metallic 1Dand 2Dsystems(wiresand films) is taken\ninto consideration, their conductivity and dielectric func-\ntion are assumed to be tensors that allows a response of8\nFigure 3: Reduced size dependences of the Fermi energy of\nwires and films vs diameter d= 2ρ0and thickness L, respec-\ntively.\nwires and films to be determined for any orientation in\nan external field.\nAnisotropyaswellasdiscretenessmanifestsitselfmuch\nstronger in systems with the small characteristic size d∼=\nλF. Under the condition d≫λF, both our theory and\ntheory [22] lead to the same results.\nA. The Fermi energy\nFig. 3 demonstrates the size dependence of the Fermi\nenergy for films and wires of Au and Al computed from\nEqs. (19) and (30). The size dependences have an ”os-\ncillatory” form. In contrast to the Fermi energy of a\nfilm [32], the size variation of the Fermi energy of a wire\nseems to be random. Input parameters for calculations\nwere taken from Ref. [32].\nIn the case of a film, cusps on the size dependence\n(i.e., the jumps of the derivative dεF/dL) are distributed\nnearly regular with the approximately constant period\n∆L≈π/k0\nF. The cusp on the size dependence of a wire\nappears each time when the increasing radius ρ0reaches\nthe valueρ0(m′n′)for which the condition (31) is satisfied\nby one more pair ( m′,n′):\nam′n′=kFρ0(m′n′).\nDistance between the neighboring cusps is\n∆d≈2(am′n′−amn)/k0\nF,whereamnis the root of the Bessel function closest to\nam′n′in value. Roots of the Bessel functions of different\norders mix up so that ∆ dvaries, at first sight, randomly\nwith change in size.\nThe oscillationsof the Fermi energyin a wire of diame-\nterdand in afilm ofthickness Lare similarin magnitude\nifd∼=L. Asinthecaseofafilm, the”period”∆ dandthe\namplitude of the oscillations tend to zero with increasing\ndiameter.\nCharacteristic properties of the size dependence of the\nFermi energy for various metal wires (and various metal\nfilms too) may be explained exclusively by different value\nk0\nF. As compared to the Au wire, for the Al wire, the\nscale ∆dof the oscillations is finer, the amplitude of the\noscillations and the averaged value εF/ε0\nFare smaller.\nB. Film\nThe specific feature of the optical characteristics of\nthin films is the presence of peaks associated with the\noptical transitions between the subbands. The size effect\nmanifests itself in a change of the number of peaks, their\nposition, and the spacing between them.\nThe positions of the peaks is determined by the ap-\nproximate expression /planckover2pi1ωmm′≈/planckover2pi1ω0|m′2−m2|, where\nmandm′are the numbers of subbands between which\nthe transition occurs and /planckover2pi1ω0≡π2/planckover2pi12/(2meL2) = 0.34\n[eV]/L2[nm2]. The frequency range under consideration\nlies in the infrared and visible spectral ranges. The lower\nlimit of the range ( /planckover2pi1ω12) corresponds to the beginning of\nthe optical transitions between the subbands. The upper\nlimit of the frequency range is the electron work function\nWof the film. The estimates can be made with the work\nfunctionW0for infinite metals Au and Ag.\nThe calculated real and imaginaryparts of the conduc-\ntivity component σxxfor the Au films 2 and 6 nm thick\nare presented in Fig. 4. For ultrathin films, the number\nofsubbands completelyorpartiallyoccupied byelectrons\nis small:mF≈2L/λ0\nF. Therefore, the number of peaks\nis small as well. For the film of the thickness L= 2 nm,\nthe peak at /planckover2pi1ω12≈0.25 eV corresponding to the lower\nlimit of the frequency range is clearly seen. The peaks\nthat represent the transitions between the neighboring\nsubbands with the numbers mandm′=m+ 1 are lo-\ncated to the left of the maximum height peak observed\nat the frequency /planckover2pi1ωmax≃/planckover2pi1ω0(2mF+1). This frequency\ncorresponds to the transition between the subbands with\nthe numbers m=mFandm′=mF+1. The spacing be-\ntween any two neighboring peaks is identical and approx-\nimately equal to 2 /planckover2pi1ωc. As the film thickness Lincreases,\nall peaks shift toward the left, the spacing between peaks\ndecreases, and they begin to merge together.\nThe overlappingofthe peaks becomes significant when\nthe spacing between them is equal to their width. The\npeak width is determined by the dissipation mechanisms\nand is approximately equal to 2 /planckover2pi1/τ. Peaks for the film of\nthe thickness L= 2 nm are clearly distinguishable (see9\nFigure 4: Frequency dependences of the real and imaginary\nparts of the film conductivity component (in e2/2a0/planckover2pi1units)\ncalculated by Eqs. (14) and (15) (solid lines). Dashed curve s\ncorrespond to the results of calculations by formulas (68) f rom\nwork [22].\nFig. 4), but for the thickness L= 6 nm the peaks disap-\npear completely. (It should be noted that the results of\nourcalculationsappeartobeweaklysensitivetoachange\nin the relaxation time τwithin one order of magnitude.)\nIt can be seen from Fig. 4 that, as the film thickness\ndecreases, the discrepancy between the results of calcu-\nlations by Eqs. (14), (15) and by (68) from Ref. [22]\nincreases and becomes substantial. This discrepancy is\nassociated with the fact that relationships (14) and (15)\nwere derived with allowance made for the dependence of\nthe Fermi energy on the film thickness kF(L) and the ex-\nact calculation of the number mFof occupied subbands.\nIn Ref. [22], the number mFwas calculated by the pro-\ncedure which gives an error ±1 for films with thickness\nL≃λF. This is an essential error because the number of\noccupied subbands for such thickness is small and ranges\nfrom 2 to 6.\nThe calculated frequency dependences of the transmit-\ntance forAu and Agthin films ofdifferentthicknessesare\ncompared with the experimental data in Fig. 5. When\ncomparingtheresultsofourcalculationswithexperimen-\ntal data, it is necessary to take into consideration that\nour definition of the transmittance is different from that\nwhich is normally used by experimentalists.\nIn the transmittance (20) the value Iis the same in\nboth cases: This is an intensity of radiation, which comes\nout from the film through the surface x=L/2. Experi-()\nTR\nTRexp\nTRTRTR\nTRexp\nTRexp\nTRexp\nFigure 5: Frequency dependences of the film transmittance\ncalculated by Eqs. (14) – (22) using ǫxx(solid lines) and ǫzz\n(dashed lines, left hand scale). Opened triangles [5], circ les\n[4], and square [9] indicate the experimental data (right ha nd\nscale) for the Au and Ag films. Solid triangles and circles\nindicate the recalculated experimental data (left hand sca le)\nin according to Eq. (54).\nmentalists take I0as intensity of radiation incident onto\nthe surface of the film x=−L/2. Of course, certain\nfraction of the radiation penetrates inside, while the re-\nmaining part is reflected. We don’t consider reflection\nand assume I0to be intensity of radiation, which comes\ninto the film through the surface x=−L/2. To make a\ncomparison with the theory, experimental values of the\ntransmittance TR expare recalculated by using the for-\nmula\nTR =TRexp\n1−R, (54)\nwhere R is the value of reflection coefficient obtained by10\nFigure 6: Frequency dependences of the real and imaginary\nparts of the film dielectric function calculated by Eqs. (16) .\nmeasuring under the same conditions as TR exp[4, 5].\nThe results of the recalculation are also presented in Fig.\n5.\nThe absorbance ηis determined by the functions\nImǫ(/planckover2pi1ω) and Reǫ(/planckover2pi1ω) according to expression (22). The\nfrequency dependences of these functions exhibit a dif-\nferent behavior (Fig. 6).\nUnlikethefunctionIm ǫ(/planckover2pi1ω), thefunctionRe ǫ(/planckover2pi1ω)has\nnot only pronounced resonance maxima but also minima\nshown as inverted peaks. The height of both peaks in-\ncreaseswith an increasein the frequency, sothat, eventu-\nally, oneofthe invertedpeaksintersectstheabscissaaxis,\nand the function Re ǫ(/planckover2pi1ω) becomes negative (in contrast\nto the function Im ǫ(/planckover2pi1ω) that is always positive in sign).\nThe minimum transmittance should be identified with\nthe minimum of the function Re ǫ(/planckover2pi1ω), which is located\nin the vicinity of the frequency /planckover2pi1ωmax=/planckover2pi1ω0(2mF+1).\nAt frequencies /planckover2pi1ω >/planckover2pi1ωmax, the absorbance is deter-\nmined only by the real part of the dielectric function:\nη≈(2ω/c)/radicalbig\n|Reǫ|. It is easy to check that the ab-\nsorbance tends to a specific constant value with an in-\ncrease in the frequency. The transmittance TR (21)\nis characterized by the same tendency. This tendency\ncan be clearly seen in Fig. 5. The peaks associated\nwith the transitions between far subbands areclearly dis-\ntinguished against the background of the monotonic in-\ncrease in the transmittance. In particular, the transition\nmF−3→mFmanifests itself at /planckover2pi1ω≈1.8 eV (L= 4\nnm).To the zero order in L/λ, diagonal components of the\ndielectric tensor only are not equal to zero. Solid lines in\nFig. 5 represent transmittance computed by Eqs. (21)\nand (22) using ǫxx. The transmittanceindicates achange\nin a normal to surface component of an electric field of\na wave passed through a film. It is this component that\ncauses optical transitions between subbands formed by\nthe size quantization. Dashed lines represent transmit-\ntance calculated with using ǫzz(ǫzz=ǫyy) which shows a\nweakening of a parallel to surface component of an elec-\ntric field. Such a transmittance is observed at normal\nincidence of radiation onto a film.\nIn the region /planckover2pi1ω>/planckover2pi1ωmax, a mechanism of dissipation\n(i.e. value of the relaxation time τ) affects the trans-\nmittance weakly. Dissipation manifests itself noticeably\nonly in vicinity of the minimum of transmittance. Thus,\ndiscrepancies between theory and experiment at these\nfrequencies could not be explained by either a mecha-\nnism of dissipation or an orientation of a film in field. A\nremarkable feature, as seen from Fig. 5, is a noticeable\nexceeding of measured transmittance over computed one\n(except of the last section, where there is a good agree-\nment). This implies that the discrepancy between the\ntheory and experiment can be attributed to a large non-\nhomogeneity in thickness and especially an absence of\ncontinuity, i.e. it can be explained by the presence of\nregions of a substrate without coating.\nLet’s name the ratio of coating area to substrate area\nby the coating coefficient pand denote coating thickness\nasL′. The mean thickness which is usually determined\nby experimentalistsfrom the massofa film and substrate\narea isL=pL′. Transmittance of a “holey” film of\nthicknessLis\nTR(L) = 1−p+pTR(L′) = 1−p+pTR(L/p).\nReducingp, it is possible to increase transmittance up\nto 1. For example, transmittance of a film with thickness\nL= 4 nm and coating coefficient p= 0.45 at the fre-\nquency/planckover2pi1ω= 1 eV is equal to 0.75, i.e. discrepancy with\nthe experimental value is twice lowered (see Fig. 5).\nIn the frequency region /planckover2pi1ω<0.5 eV, there are experi-\nmental data for transmittance of thin films of Au [9] and\nPb [9, 14]. Unfortunately data for reflection are absent\nhence the recalculation of experimental results like that\nrepresented above is impossible. However, taking into\naccount that this recalculation leads to an increase of\ntransmittance value, we guess experimental data for Au\n[9] to be in agreement with our calculations under as-\nsumption of a normal incidence of radiation onto a film\n(Fig. 5). Moreover, the recalculation can change a type\nof dependence on frequency for transmittance, i.e. a ris-\ning can be replaced by a falling after the recalculation\n(see Fig. 5 for Au, L= 5.8 nm). This could explain\nwhy within frequency interval (0 .2,0.5) eV dependence\nof transmittance on frequency is decreasing or absent at\nall according to our calculations whereas an increasing\nis observed in experiment [14]. As to value of transmit-\ntance for Pb films, it is difficult to compare calculations11\nwith experimental data because results of different ex-\nperiments vary essentially. Thus transmittance value for\nPb film of 4 nm thickness is given in [9] as 0.79 at 0.05\neV but it is 0.12 only at the close frequency 0.12 eV fol-\nlowing [14] (in addition, transmittance diverges in value\n1.5 times for different technologies of film coating [14]).\nC. Wire\nThe frequency dependences of Re σxxand Imσxxfor\nthe Au wire of diameter 1.6 nm are presented in Fig.\n7. For such a small diameter, the peaks corresponding\nto the transitions between levels of the size quantization\n(subbands) manifest itself clearly. In spite of the rather\ncomplete spectrum kmn, position of the peaks is well pre-\ndicted.\nLet us find, for example, the position of the peak in\nReσxxwhich hasthe maximum height. The height ofthe\npeaks is proportional to |∝an}bracketle{tm+1,n′|ˆpx|mn∝an}bracketri}ht|2/summationtext\np1. The\nmatrix element has the maximum magnitude at n′=n\nbecause under this condition the integral in (38) takes on\nthe maximum value. Further, by using (52) and (53), it\nis easy to determine that the maximum height is realized\natm= 0,n′=n=nF. For the diameter d= 1.6 nm,\nthe number nF≈k0\nFρ0/πis equal to 3. As a result we\nhave\n/planckover2pi1ωmax=/planckover2pi12/parenleftbig\nk2\n1,3−k2\n0,3/parenrightbig\n2me=/planckover2pi12/parenleftbig\na2\n1,3−a2\n0,3/parenrightbig\n2meρ2\n0= 1.40 eV.\nThis is in good agreement with numerical calculations\npresented in Fig. 7. The height of the peaks in Fig. 7\nalso confirms our estimation.\nFig. 7 demonstrates the important fact Re σxx>0\nover all frequency range. In contrast to this, Im σxxis a\nvariable in sign function of the frequency.\nThe frequency dependences of Re σzxand Im��zxare\nalso presented in Fig. 7. As it was expected, the position\nof the peaks is identical both for Re σzxand Imσzxbut\nthe height is one order smaller in the first case.\nComparing the upper and lower parts of Fig. 7, we\ncan trace the size dependence of the conductivity for ul-\ntrathin metal wires. When dincreases, the peaks shift\nto the left with displacement equal to ∆ ω=ω′−ω=\nω(ρ2\n0/ρ′\n02−1). More distant peaks (with lager value ω)\nhave lager displacement, so that the interval occupied by\nthe peaks contracts. At the same time, new peaks ap-\npear within this interval, because with enlarging ρ0the\nnumber of levels and the number of the possible tran-\nsitions between them increase. Distance between peaks\ndecreases, and when it approaches /planckover2pi1/τ, the peaks begin\nto merge together.\nIt is interesting to compare results of the study for the\noptical conductivity of ultrathin metal wires with anal-\nogous results for ultrathin films. Divergences are asso-\nciated with the different dimensionality of the systems.\nThis is reflected in an essential difference in the energeticFigure 7: Calculated frequency dependences of the real and\nimaginary parts of σαβ(ine2/2a0/planckover2pi1units) for Au wires of\nvarious diameter d.\nspectra and also in the fact that after calculation of qua-\nsicontinuous states, in the case of a wire, the summation\nover two numbers m,nremains, while in the case of a\nfilm, it remains over one number only (which numerates\nvalues of the x−component of the electron momentum).\nIt isthis fact that explainsapproximatelyone orderlower\nheight of the maximum in the frequency dependence of\nthe conductivity of a wire compared to the case of a film.\nIndeed,\nReσwire\nxx\nReσfilmxx∼=Ω−1\nwire/summationtext\np1\nΩ−1\nfilm/summationtext\np,n1=2L\nπ/radicalig\n(kwire\nF)2−k2\n0nF\nπρ2\n0L\n×/braceleftbiggab\nπ2π{(kfilm\nF)2−k2\nmF}\nabL/bracerightbigg−1\n∼=10−1\n/radicalbig\nρ0[nm]L2\nρ2\n012\nFigure 8: Calculated frequency dependences of the real part\nofσxx(ine2/2a0/planckover2pi1units) for Al and Pb wires.\nbecause\n(kwire\nF)2−k2\n0nF∼=2πk0\nF/ρ0,(kfilm\nF)2−k2\nmF∼=2πk0\nF/L.\nForL,ρ0∼=1 nm we obtain Re σwire\nxx/Reσfilm\nxx∼=10−1.\nAs to the different position of the peaks, this may be\ncompletely explainedby characteristicpropertiesofspec-\ntra of the 1D and 2D systems.\nThe frequency dependences of Re σxxfor the Al and\nPb wires of diameter 1.6 nm are presented in Fig. 8. It\nissurprisingthatpeaksin theconductivityofthe Pbwire\nare absolutely absent. The reason of this is a small value\nof the relaxation time for Pb equal to τ= 1.4×10−15s,\nso that width of the peaks /planckover2pi1/τ= 0.44 eV. In this respect,\nAl, with /planckover2pi1/τ= 0.08 eV, holds an intermediate position\nbetween Au and Pb. For calculations we use values of\nthe relaxation time for bulk metals taken from [35].\nIn spite of an absence of peaks in the frequency depen-\ndence of the conductivity for the Pb wire, its maximum\nmay be found in such a way as the position of the max-\nimum height peak in the conductivity of the Au wire\nwas determined above, with the difference that this time\nnF= 4:\n/planckover2pi1ωmax=/planckover2pi12/parenleftbig\na2\n1,4−a2\n0,4/parenrightbig\n2meρ2\n0= 2.1 eV.\nThis value agrees well with Fig. 8 taking into account\nthe large width of the peaks.\nSurprisingly, the difference in the obtained results for\nAl and Pb (due to the different values of τ) indicate size-\nfrequency dependences, which can be expected for films\nand wires, inhomogeneous in thickness considered, for\nexample, in Refs. [40–44]. If fluctuations of sizes in 1D-\nand 2D-systems lead to a strong effective reduction of\nτ, experimental size dependences of conductivity are no-\nticeably smoothed irrespective of a metal kind.\nWewilldevoteselectpublicationtothetheoryoftrans-\nport in films and wires with rough surface.\nVI. CONCLUSIONS\nThe conductivity tensor is introduced for the low-\ndimensional electron systems. Components of the con-ductivity tensor for a quasi-homogeneousultrathin metal\nfilm and wire are calculated within the particle-in-a-box\nmodel on the assumption that the component of the in-\nduced current with the wave vector equal to the wave\nvector of the electromagnetic field is dominating.\nOver infrared region the condition qL, qρ 0≪1 is sat-\nisfied allowing us to express components of the conduc-\ntivity tensor in terms of the according small value. All\nnon-diagonal components of the conductivity tensor are\nequal to zero in zero order of the expansion. They ap-\npear in linear approximation. The important fact that\nthe real part of the diagonal components is non-negative\nover all frequency range, with the guarantee Q >0 for\nthe dissipation of energy, is proved.\nAs a result of comparing the according components of\nthe conductivity tensor for a film and a wire of the same\nthickness of order 1 nm, one order smaller value for a\nwire is obtained. In such a manner di��erent density of\nstates near the Fermi level manifests itself (it is greater\nfor a film). It is found that the discrepancy between\nour results and the theory [22] increases and becomes\nsubstantial, as the characteristic small dimension of the\nsystem decreases. This discrepancy is associated with\nthe strong dependence of the Fermi level on this dimen-\nsion for small values of order of the Fermi wavelength.\nThis size dependence of the Fermi level has an “oscil-\nlatory” form. Transmittance is calculated for a simple,\nwell-defined model without fitting parameters.\nAcknowledgments\nWe are grateful to A. V. Babich and A. V. Korotun\nfor help in the calculations and to W. V. Pogosov for\nreading the manuscript. This work was supported by the\nMinistry of Education and Science of Ukraine.\nAppendix A: The matrix elements\nThe expressionsfor momentum projections in cylindri-\ncal coordinates have a form\nˆpz=−i0/planckover2pi1∂\n∂z,\nˆpx=−i0/planckover2pi1/braceleftbigg\ncosϕ∂\n∂ρ−sinϕ\nρ∂\n∂ϕ/bracerightbigg\n,(A1)\nˆpy=−i0/planckover2pi1/braceleftbigg\nsinϕ∂\n∂ρ+cosϕ\nρ∂\n∂ϕ/bracerightbigg\n.\nUsing (23) – (28) and (A1), we have13\n∝an}bracketle{tj|ˆpx|i∝an}bracketri}ht=−i0/planckover2pi1\n2δpp′\n\n1\n2π2π/integraldisplay\n0/parenleftig\ne−im′ϕei(m−1)ϕ+e−im′ϕei(m+1)ϕ/parenrightig\ndϕρ0/integraldisplay\n0Rm′n′dRmn\ndρρdρ\n+m\n2π2π/integraldisplay\n0/parenleftig\ne−im′ϕei(m−1)ϕ−e−im′ϕei(m+1)ϕ/parenrightig\ndϕρ0/integraldisplay\n0Rm′n′Rmndρ\n\n\n=−i0/planckover2pi1\n2δpp′\n\nδm−1,m′\nρ0/integraldisplay\n0Rm′n′dRmn\ndρρdρ+mρ0/integraldisplay\n0Rm′n′Rmndρ\n\n+δm+1,m′\nρ0/integraldisplay\n0Rm′n′dRmn\ndρρdρ−mρ0/integraldisplay\n0Rm′n′Rmndρ\n\n\n; (A2)\nThen using relation I′\nm(x) =±mIm(x)/x∓Im±1(x) from\n[38], we obtain Eqs. (37) – (38).In a similar way we find\n∝an}bracketle{tj|xˆpz|i∝an}bracketri}ht=−i0/planckover2pi1/integraldisplay /integraldisplay /integraldisplay/braceleftbigg\nRm′n′Φ∗\nm′Z∗\np′(ρcosϕ)RmnΦmdZp\ndz/bracerightbigg\nρdρdϕdz\n=/planckover2pi1kzpδpp′2π/integraldisplay\n0Φ∗\nm′Φmcosϕdϕρ0/integraldisplay\n0Rm′n′Rmnρ2dρ=1\n2/planckover2pi1kzpδpp′(δm−1,m′+δm+1,m′)ρ0/integraldisplay\n0Rm′n′Rmnρ2dρ\n=1\n2/planckover2pi1kzpδpp′\nδm−1,m′ρ0/integraldisplay\n0Rm−1,n′Rmnρ2dρ+δm+1,m′ρ0/integraldisplay\n0Rm+1,n′Rmnρ2dρ\n.(A3)\n[1] R. D. Fedorovich, A. G. Naumovets, and P. M. Tomchuk,\nPhys. Rep. 328, 73 (2000).\n[2] R. Otero, A. L. Vazquez de Parga, and R. Miranda, Phys.\nRev. B66, 115401 (2002).\n[3] A. Gloskovskii, D. A. Valdaitsev, M. Cinchetti, S. A.\nNepijko, J. Lange, M. Aeschlimann, M. Bauer, M.\nKlimenkov, L. V. Viduta, P. M. Tomchuk, and G.\nSch¨ onhense, Phys. Rev. B 77, 195427 (2008).\n[4] J. P. Gasparini and R. Fraisse, Thin Solid Films 30, 11\n(1975).\n[5] J. Dryzek and A. Czapla, Phys. Rev. Lett. 58, 721 (1987).\n[6] D. A. Evans, M. Alonso, R. Cimino, and K. Horn, Phys.\nRev. Lett. 70, 3483 (1993).\n[7] P. F. Henning, C. C. Homes, S. Maslov, G. L. Carr, D.\nN. Basov, B. Nicoli´ c, and M. Strongin, Phys. Rev. Lett.\n83, 4880 (1999).\n[8] G. Fahsold, A. Bartel, O. Krauth, N. Magg, and A. Pucci,\nPhys. Rev. 61, 14108 (2000).\n[9] J. J. Tu, C. C. Homes, and M. Strongin, Phys. Rev. Lett.\n90, 017402 (2003).\n[10] G. Fahsold, A. Priebe, N. Magg, and A. Pucci, Thin SolidFilms428, 107 (2003).\n[11] N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere,\nOpt. Express 11, 482 (2003).\n[12] I. Vilfan and H. Phur, Eur. Phys. J. B 36, 281 (2003).\n[13] S. Ding, X. Wang, D. J. Chen, and Q. Q. Wang, Opt.\nExpress 14, 1541 (2006).\n[14] A. Pucci, F. Kost, G. Fahsold, and M. Ja/suppress lochowski, Phys .\nRev.74, 25428 (2006).\n[15] M. C. Tringides, M. Ja/suppress lochowski, and E. Bauer, Phys.\nToday4, 50 (2007).\n[16] F. Neubrech, T. Kolb, R. Lovrincic, G. Fahsold, A. Pucci ,\nJ. Aizpurua, T. W. Cornelius, M. E. Toimil-Molares, R.\nNeumann, and S. Karim, Appl Phys. Lett. 89, 253104\n(2006).\n[17] O. L. Muskens, V. Giannini, J.A. Sanchez-Gil, J. Gomez\nRivas, Nano Lett. 7 (9) , 2871 (2007).\n[18] M. Klevenz, F. Neubrech, R. Lovrincic, M. Ja/suppress lochowski ,\nA. Pucci, Appl. Phys. Lett. 92, 133116 (2008).\n[19] V. V. Pogosov, Introduction to the Physics of Charge and\nSize Effects: Surface, Clusters, and Low-Dimensional\nSystems (Fizmatlit, Moscow, 2006) [in Russian].14\n[20] V. V. Pogosov and E. V. Vasyutin, Nanotechnology 17,\n3366 (2006).\n[21] L. D. Landau and E. M. Lifshits, Elecrodynamics of con-\ntinuum media (Fizmatlit, Moscow, 2001) [in Russian].\n[22] D. M. Wood and N. W. Ashcroft, Phys. Rev. B 25, 6255\n(1982).\n[23] V. B. Sandomirskii, Sov. JETF 25101 (1967).\n[24] F. K. Schulte, Surf. Sci. 55427 (1976).\n[25] W. Ekardt, Phys. Rev. B 291558 (1984).\n[26] P. J. Feibelman, and D. R. Hamann, Phys. Rev. B 29\n6463 (1984).\n[27] J. C. Boettger, Phys. Rev. B 5313133 (1996).\n[28] A. Kiejna, J. Peisert, and P. Scharoch, Surf. Sci. 54432\n(1999).\n[29] N. Zabala, M. J. Puska, and R. M. Nieminen, Phys. Rev.\nB5912652 (1999).\n[30] I. Sarria, C. Henriques, C. Fiolhais, and J. M. Pitarke,\nPhys. Rev. B 621699 (2000).\n[31] E. Ogano, N. Zabala, and M. J. Puska, Nanotechnology\n13363 (2002).\n[32] V. V. Pogosov, V. P. Kurbatsky, and E. V. Vasyutin,\nPhys. Rev. B 71, 195410 (2005).\n[33] C. M. Horowitz, C. R. Proetto, and J. M. Pitarke, Phys.\nRev. B78, 085126 (2008).[34] Y. Han, D.-J. Liu, Phys. Rev. B 80, 155404 (2009).\n[35] N. Ashcroft and N. Mermin, Solid State Physics (Holt,\nRinehart, and Winston, New York, 1976).\n[36] N. D. Mermin, Phys. Rev. B 1, 2362 (1970).\n[37] G. A. Korn and T. M. Korn, Mathematical Handbook\nfor Scientists and Engineers , 2nd ed. (Dover, New York,\n2000).\n[38] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,\nSeries and Products , 5th ed. (Academic Press, San Diego,\n1994).\n[39] J. M. Ziman, Principles of the Theory of Solids (Cam-\nbridge University Press, New York, 1972).\n[40] Z. Te˘ sanovi´ c, M. V. Jari´ c, and S. Maekawa, Phys. Rev.\nLett.57, 2760 (1986).\n[41] N. Trivedi and N. W. Ashcroft, Phys. Rev. 38, 12298\n(1988).\n[42] G. Fishman and D. Calecki, Phys. Rev. Lett. 62, 1302\n(1989); Phys. Rev. 43, 11581 (1991).\n[43] A. E. Meyerovich and I. V. Ponomarev, Phys. Rev. B 65,\n155413 (2002).\n[44] Y. Cheng and A. E. Meyerovich, Phys. Rev. B 73, 085404\n(2006)."    },    {        "title": "2008.09769v1.Twistronics_in_graphene_based_van_der_Waals_structures.pdf",        "content": "Twistronics in graphene -based van der Waals structures  \n \nYa-Ning Ren§, Yu Zhang§, Yi-Wen Liu§, and Lin He† \nCenter for Advanced Quantum Studies, Department of Physics, Beijing Normal \nUniversity, Beijing, 100875, People’s Republic of China  \n§These authors contributed equally to this work.  \n†Correspondence and requests for materials should be addressed to L.H. (e -mail: \nhelin@bnu.edu.cn ). \nThe electronic properties of van der Waals (vdW) structures can be \nsubstantially modified by the moiré  superlattice potential, which strongly depends \non the twist angle among the compounds. In twisted bilayer graphene (TBG), two \nlow-energy Van Hove singula rities (VHSs) move closer with decreasing twist \nangles and finally become highly non -dispersive flat bands at the magic angle (~ \n1.1º). When the Fermi level lies within the flat bands of the TBG near the magic \nangle, Coulomb interaction is supposed to exce ed the kinetic energy of the \nelectrons, which can drive the system into various strongly correlated phases. \nMoreover, the strongly correlated states  of flat bands are also realized  in other \ngraphene -based vdW structures with an interlayer twist . In this ar ticle, we mainly \nreview the recent experimental advances on the strongly correlated physics  of the \nmagic -angle TBG (MATBG) and the small -angle twisted multilayer graphene . \nLastly we will give out a perspective of this field.  \n \n  1. Introduction  \nThe discovery of graphene has generated an extensive investigation on two -\ndimensional van der Waals (vdW) materials, among which the bilayer graphene is \nconsidered as an ideal model system and has attracted much attention in the past few \nyears. Both experi ments and theoretical calculations reveal that the interlayer vdW \ninteractions and band structures of bilayer graphene can be dramatically modulated  by \nthe twist between two adjacent graphene layers [1-29]. For the large twist angles of θ > \n5.5°, the two g raphene layers are usually electronically decoupled and each layer of \nthem behaves as a monolayer graphene  [21, 28 -32], except for a small set of angles which \nyield commensurate structures  [2, 16 -18, 33]. With decreasing the twist angle, low -energy \nVan Hove singularities (VHSs) of twisted bilayer graphene (TBG) gradually move \ncloser, accompanied by a pronounced suppression of Fermi velocity due to the strongly \ninterlayer coupling  [4-15, 21, 25, 26, 34]. When the twist angle approaches the magic angle (~ \n1.1°), the Fermi velocity almost vanishes, resulting in two highly non -dispersive flat \nmoiré  bands closely flanking the charge neutrality point  [6-11, 25, 35 -58]. In the past two \nyears, the TBG near the magic angle has opened a fascinating new chapter in t he field \nof strongly correlated quantum matter  [49-85]. An exceptionally wide range of correlated \nphysics, such as Mott insulator, superconductivity, ferromagnetism, and topology, are \nexperimentally observed in the TBG near the magic angle  [51, 52, 63 -82]. In this article, we \nfirst summarize the electronic properties of the TBG  within a broad range of the twist \nangles from 0 °  to 60°. And then, we mainly focus on the strongly correlated phases in \nthe TBG near the magic angle, following by the compar ison of flat -band physics in the \nsmall -angle twisted multilayer graphene. After that, the moiré  periodic potential \ninduced exotic physics in graphene/hexagonal boron nitride (hBN) heterostructure and \nthe proximity effect induced properties in graphene/transiti on metal dichalcogenides \n(TMDs) heterostructures are briefly introduced. At last, we present the outlook and \nchallenges in this field .  \n 2. Electronic properties of TBG  \nConsidering that the complex geometry of the TBG can dramatically affect its \nelectronic  properties [1-29], we first devote some attention to the atomic  structures. \nStrictly, the periodic moiré superlattice  of the TBG  only occurs at specific so -called \ncommensurate angles that satisfy  [2-7, 14 -16]: \n𝜃\n=cos−1(3𝑝2+3𝑝𝑞+𝑞22⁄\n3𝑝2+3𝑝𝑞+𝑞2)                                                (1) \nwhere p and q are coprime positive integers . As shown in Fig. 1, the commensurate \nsuperlattice with a long -ranged moiré period in real space is easily formed for the twist \nangles in the vicinity of 0°  and 60°. In contrast, the incommensurate superlattice with \nrotational symmetry but not a long -ranged translational symmetry is more common for \nthe twist angles around 30° [87-96]. Usually,  study  on the TBG mainly focuses on the \ncommensurate superlattices because the strength of interlayer coupling is expected to \nintroduce many exotic phenomena, which are quite different from that in the \ncomponents  [51-85, 97 -115]. Among all, the TBG near the magi c angle of approximately \n1.1° has attracted enormous interest. Many strongly correlated physics, such as Mott \ninsulating state, superconductivity, strange metal (non -Fermi liquid), nematicity, \ntopology, and magnetism for the partially filled moiré flat minibands have all been \nreported in recent experiments  [51-85] (Fig. 1).  \nWith the reduction of the twist angle, the interplay between the vdW interaction \nenergy and the elastic energy of the TBG is expected to result in obviously structural \nrelaxation and  atomic reconstruction [115-119]. Specifically, the atomic registry varies \ncontinuously across the moiré period between AA, AB, and BA stacking configurations, \nwith the growth of the energy -favorable AB and BA stacking regions and the reduction \nof the AA stacking region (In the AA stacking region, each top -layer atom locates \ndirectly onto a bottom -layer atom. In the AB/BA stacking region, each A/B -sublattice \natom of the top layer locates onto a B/A -sublattice atom of the bottom layer, while the \nB/A-sublatt ice atoms of the top layer have no partner in the bottom layer)  [115-119]. Once the inverse symmetry of the AB/BA stacking is broken by an external displacement \nfield or the substrate -induced staggered potential, it is expected to open  a band gap  [120-\n125], accompanied by the different band topology of the AB and BA stacking  regions  \n[126-132]. In this case, the topologically protected one -dimensional  network of valley Hall \nstates and two pairs of chiral edge modes at the AB -BA domains are expected to appear \n[97-113, 115], as shown in the bottom left panel of Fig. 1. Moreover, the AB -BA domain \nand its band structure can be tuned by electrostatic force  near the magic angle [105]. The \nlarge coherence length of the topological network modes provides a platform to \naccommodate various interference oscillations. Very recently, the Aharonov -Bohm \noscillations under perpendicular magnetic fields are experimental ly observed in \nmarginally TBG due to the existence of a triangular network of one -dimensional states  \n[101-103] (top left panel of Fig. 1) .  \nFor the large twist angles  around 30°, the atomic structure of the TBG is usually \nincommensurate with only quasi -periodicity symmetry  [87-96]. On the one hand, the \ninterlayer interaction of the large -angle TBG was assumed to be negligible due to the \nlacking of phase coherence, which was thought of as a pair of decoupled monolayer \ngraphene sheets [21, 28 -32]. On the other hand, the conventional moiré  effective theory, \nwhich bases on the interference of the lattice periods  [1-19], cannot describe the electronic \nstructures of such an incommensurate stacking. The large -angle TBG has attracted \nincreasing attention on ly recently when the 30 ° TBG was successfully synthesized on \nthe 4H -SiC (0001)  [88, 133, 134], Pt (111)  [89], Cu-Ni(111)  [135], and Cu(111)  [92, 94, 136] \nsubstrates. The 30 ° TBG, a typical incommensurate configuration with a 12-fold \nrotational symmetry (Fig. 1), is the first experimental realization of the two -\ndimensional quasicrystals based on graphene. Contradicting the belief, recent angle -\nresolved photoelectron spectroscopy (ARPES)  [88, 89] and scanning tunneling \nmicroscop y (STM) [92] experiments show the signatures of extra mirrored Dirac points \ninside the Brillouin zone of each graphene layer, together with a gap opening at the \nzone boundary (Fig. 1). This result demonstrates the strongly interlayer  coupling  with \nquasi -periodicity in the 30 ° TBG. Moreover, the well -defined Landau quantization of \nmassless Dirac fermions is observed in the scanning tunneling spectroscopy (STS) spectra, highlighting the relativistic Dirac fermion quasicrystal of the 30 ° TBG  [92].  \nAround the 30°, there still are a series of discrete large twist angles of the TBG \nwith the commensurate configuration  [2, 16 -18, 33], as shown in Fig. 1. The angular -\ndependence conductivity measurement also highlights the commensurate crystalline \nsuperstructure of t he TBG at the twist angles of 21.8°  and 38.2°  [33], arising from the \ncoherent two -dimensional electronic interface state  [2, 16]. Theoretically, the TBG with \nlarge commensurate angles (such as  𝜃= 21.8° ) are predicted to be a generic higher -\norder topological insulator, which is characterized by the topologically protected corner \nstates [114]. Moreover,  for the  TBG  with a twist angle that is  sufficiently close to the \ncommensurate 38.2° , the alternative commensurate configurations, i.e. the gapped \nexchange symmetry even (SEE)  and the gapless exchange symmetry odd (SEO)  [2], \ngeometrically frustrate the network of topologically protected counterpropagating \nchiral modes [137]. The emergent geo metric frustration of the system gives rise to flat \nbands and a Kagome lattice [137], which may be an exciting platform for exploring the \nspin-liquid physics and other exotic states  [138-140] \nAs we summarized in Fig. 1, the TBG exhibits a variety of exotic electronic \nproperties, which strongly depend on the interlayer twist angle. In the following part, \nwe mainly focus on strongly correlated physics in the magic -angle TBG (MATBG) . \n3. MATBG  \n3.1 Symmetry breaking in the MATBG  \nIn the TBG,  a finite interlayer coupling between the two graphene layers leads to the \nemergence of two saddle points appearing at the intersections of the two Dirac cones, \nwhich consequently generate two low -energy VHSs in th e density of states (DOS) \n(inset of Fig. 2). Theoretically, the energy separation of the two VHSs can be roughly \nestimated as  [4, 21 -26, 28]:  \n∆𝐸𝑉𝐻𝑆 ≈ℏ𝑣𝐹∆𝐾−\n2𝑡𝜃                                                 (2)                         where ∆𝐾=2|𝐾|sin (𝜃/2) is the relative shift of the Dirac points of the adjacent \ngraphene bilayers with the reciprocal -lattice vector 𝐾 in reciprocal space , 𝑡𝜃 is the \ninterlayer hopping,  ℏ is the reduced Planck constant, and 𝑣𝐹 is the Fermi velocity. \nAccording to the Eq. (2), the two VHSs move closer to each other with decreasing the \ntwist angles. Here we summarize the energy separations of the two VHSs ∆𝐸𝑉𝐻𝑆 vs \n𝜃 of the TBG acquired from the STS measurements  [20, 22 -28, 55 -60, 86], as shown in Fig. \n2. Consistent with the Eq. (2), the ∆𝐸𝑉𝐻𝑆 generally decreases with decreasing the twist \nangles. However, there are a wide distribution of the experimental data, which may \nresult from the variation of the 𝑡𝜃 in different TBG samples.  \nWhen the twist angle is close to the magic angle (~ 1.1 °), the quantum interference \nof electrons in the superlattice potential gives rise to extremely narrow low -energy \nbands, i.e. , the flat bands, and isolates them from higher -energy dispersive bands  [6-11, \n35-48], as shown in Fig. 3(a). There are three important parameters in the single -particle \nband structure of the TBG near the magic angle : (i) the energy separations of the two \nVHSs or the two flat bands (labeled as ∆𝐸𝑉𝐻𝑆), (ii) the bandwidth of eac h VHS or each \nflat band, and (iii) the band gap between the flat bands and the upper dispersive bands \n(labeled as ∆𝑔𝑎𝑝). These parameters are systematically measured by different groups \nin the past few years [20-25, 50 -69, 115]. In Fig. 3(b) -(e), we summarized the ∆𝐸𝑉𝐻𝑆, the full-\nwidth at half -maximum (FWHM) of the VHSs, and the ∆𝑔𝑎𝑝 acquired from STS and \ntransport measurements in the TBG near the magic angle, respectively. Obviously, the \nmeasured data vary from sample to s ample and  exhibit large differences,  sometimes \neven by an order of magnitude in the TBG with almost the same twist angle. This result \nhighlights the complexity and lack of reproducibility even in the single -particle band \nstructure  of the TBG near the magic angle. . \nWhen the Fermi level lies within the flat bands in the MATBG, the Coulomb \ninteractions between electrons, which can be estimated by 𝑈~𝑒2𝜖𝑙𝑚⁄   (𝜖  is the \neffective dielectric constant, and 𝑙𝑚  is the moiré period), bec ome important with \nrespect to the kinetic energy set by the bandwidth W of the flat bands  [143-145, 187 -191]. \nTherefore, the system is expected to exhibit various strongly correlated phases. Indeed, \nrecent STS measurements have shown obvious signatures of many -body correlations in the TBG near the magic angle  [52-60, 62, 84, 85]. When the two flat bands in the MATBG \nare fully occupied or empty, the STS spectra show sharp peaks  associated with the two \nnearly flat conduction and valence bands, which are consistent with the single-particle \npictures  [52-58]. When one of the flat bands is partially occupied (not just at \ncommensurate fillings), the spectra show pronounced  bands  splitting together with the \nbroadening of  both the two flat bands  [52-58], indicating the importance of interaction -\ninduced symmetry breaking states.  \nOne of the major differences in the STS spectra extracted from different research \ngroups is worth highlighting here. For the devices re ported by Kerelsky  (1.15°) et al. \n[57] Xie (1.01°) et al. [55] Choi (1.04°) et al. [56] and Jiang (1.07°)  et al. [58], the partially \noccupied flat bands split into two peaks with their energies locating on either side of \nthe Fermi level in the spectra (as schematically shown in Fig. 4(a)). Other groups, such \nas Ren et al. [59], reported that the partially filled flat band spontaneously splits into four \npeaks in a 1.49º TBG, as shown in Fig. 4(b). Because of the fourfold spin –valley flavour \ndegeneracies of each flat band, the strong electron –electron interactions are supposed \nto drive the degenerate bands splitting into sub -bands in the framework of Hartree –\nFock approximation  [143, 176, 186]. The behaviours of the spontaneous symmetry -breaking \nstates of electrons in the MATBG are analogous to the quantum Hall ferromagnets in \nLandau levels (LLs) of  graphene monolayer  [201, 202 ], in which the isospin SU(4) \nsymmetry of each LL is lifted due to the strong electron –electron interactions [194-200]. \nAnd simultaneously, the partially occupied LL splits into subset at all the integer fillings \n[201-210]. Therefore, the splitting flat bands in the MATBG can be naturally attributed to \nthe correlation -induced valley and/or spin polarized states, suggesting the abundant \nmany -body ground states in the MATBG.  \n3.2 Correlated insulator and superconductivity in the MATBG  \nA central question in t he TBG near the magic angle is the nature of the observed \nmany -body ground states, which have been widely studied in the past two years  [51-85]. \nTo date, many groups have examined the longitudinal resistance (conductance) in the TBG near the magic angle as  a function of carrier densities, temperatures, magnetic \nfields, and pressure via the transport measuremen ts [63-81], and a zoo of strongly \ncorrelated phases, such as correlated insulat ing states and superconductivity, are clearly \ncaptured  [63-81]. Similarly, the doping -dependent STS intensities at the Fermi level can \nalso mimics the conductance measured with the transport where only charge carriers \nnear the Fermi level have contributions  [53-56, 58]. Figure 5 summarized representative \nresults obtain ed in transport measuremen ts [68, 70 -72, 78] and STM measurements [54-56, 58] \nin the TBG near the magic angle. A common signature of insulating states appears at \nmost or all integer occupancies of the fourfold degenerate flat conduction and valence \nbands, i.e., at moiré band filling factors ν, where ν = 0, ±1, ±2, ±3 indicate the number \nof charge carriers per moiré superlattice unit cell. Close to the certain insulating states, \nsuperconducting domes are observed below critical temperatures via the transport  \nmeasurements, as illustrated schematically in Fig. 5  [68, 70 -72, 78]. Importantly, there are \nsignificant differences in the acquired phase  diagrams in different experiments, as we \nwill discuss subsequently.  \nFirst, some groups show the existence of correla ted insulat ing states at all integer \nmoiré band fillings [72] while other groups find semimetallic or metallic behavior at the \nsame integer moiré band fillings [63-71, 73 -81]. The reasons for this difference might be \nattributed to the unavoidable strains  [55-57, 104, 211], inhomogeneous twist angles [77, 79, 82, \n212], the aligned or unaligned devices of the MATBG with respect to the underlying \nsubstrate  [177-183, 213 -215], and the thickness of a dielectric spacer layer (typically hBN) [70, \n71, 75, 216]. On the one hand, the electronic properties of the MATBG is very sensitive to \nthe lattice relaxation and heterostrain [115, 117, 211], and the strain values of the devices \nduring the sample preparation processes are indeed different according to several STM \nexperiments  [52-60, 62, 84, 85]. On the other hand, the device setups employed by different \ngroups show great differences. For instance, Cao et al. encapsulated the MATBG by \nhBN flakes with the thicknesses of about 10 -30 nm  [65, 68]. In contrast, the thickn esses \nof hBN in the fabricated devices are 30 -50 nm, 25 -50 nm, and 10 nm for the \nexperiments in  Saito et al [70], Polshyn et al [63], and Lu et al [72], respectively. Recent \nexperiments show that the dielectric environment of the MATBG, i.e. thickness of a dielectric hBN layer, has a strong influence on the screened electron -electron \ninteractions, which in turn have significant influence on the obtained phase diagram [70, \n71, 216]. In addition, a recent superconducting quantum interference device fabricated on \nthe apex of a sharp pipette (SQUID -on-tip) measurement also highlights the \nunavoidable varying degree of twist -angle homogeneity in the TBG samples  [77]. The \nlarge variations in the obtained correlation phase diagrams in the TBG near the magic \nangle are reasonable and expected with considering the lack of reproducibility even in \nthe single -particle band structure in this system .   \nSecond, superconduc ting signatures are not always found at the same doping levels \nin the TBG with almost the same twist angle in the transport experiments  [63-81] and no \nsignature of superconductivity is observed in the STM measurements [52-58] up to now \n(Fig. 5(b)). In the transport experiments, the superconductivity is usually deduced from  \na (near) zero -resistance state that is rapidly suppressed with increasing temperatures \nand magnetic fields, accompanied with a field -dependent critical current that is \nconsistent with Ber ezinskii –Kosterlitz –Thouless (BKT) theory  [68, 69, 71, 72, 76]. At present, \nthe superconductivity is lack of reproducibility in the transport measurement and only \na few samples exhibit the zero -resistance state. It is even more abnormal that all the \nSTS spectra show no signature of superconducting gap in the MATBG up till now [52-\n58]. The local superconducting signals should be much more accessible in the STM \nstudies since the spatial resolved STM and STS measurements can efficiently avoid the \nimpurities and inhomogeneous of the sample [217-220]. This reminds us to tre at the (near) \nzero-resistance phenomena in transport measurements more critically, since other \nmechanisms in the transport measurements such as ballistic transport may also lead to \nsimilar zero -resistance phenomena [221]. In addition, recent studies pointe d out that the \nsimilar sudden drop in resistivity and nonlinear I–V characteristic were also observed \nin twisted double -bilayer graphene , and these phenomena  may possibility originate \nfrom the spin -polarized correlated state  [222]. Further  measurements  should be carefully \ncarried out in the near future to rule out any other possibilities .   \nOn the other hand, if the (near) zero -resistance feature indeed comes from the \nsuperconductivity, the origin and mechanism of the superconductivity in the MATBG is another key issue for researchers to address. So far, there are mainly two different \nopinions on the relationship between superconductivity and insulating behavior in the \nMATBG. One is that the superconductivity and insulating behavior are connecting  [66-\n68, 72-78], the other is that they are competing [63, 69 -71]. For the research groups such as \nCao et al. [67, 68, 76], their experimental data show that both of the superconductivity and \ninsulating state arise from strongly electron -electron interactions and  are closely \nconnecting in the MATBG. Their experiments indicate a tantalizing resemblance to that \nof cuprate superconductors for the following three reasons: (1) The phase diagrams \nshow that the superconducting domes are always in close proximity to the i nsulating \nstates [68, 152, 171, 223 -225]; (2) A broad regime of temperature -linear resistivity is observed \nfor a range of fillings near the correlated insulator, and the ratio of critical temperature \nto the Fermi energy is relatively large  [67, 226 -231]; (3) A strongly anisotropic phase \nappears in a wedge above the under -doped region of the superconducting dome, and \nthe superconductivity exhibits an anisotropic response to the direction of in -plane \nmagnetic fields, which reveal a nematic pairing state acro ss the entire superconducting \ndome  [76, 173, 185, 232 -241]. However, for the results obtained by the research groups such \nas Saito et al. [70] and Stepanov et al. [71], the superconductivity and correlated insulator \nare competing with each other in the MAT BG: the insulating phase arises from electron -\nelectron interactions while superconductivity arise s from conventional phonon -\nmediated pairing [165-170]. In their experiments, the correlated insulating states at ν = ±2 \nare substituted to metallic states by decreasing the thickness of dielectric environment \nto independently suppress the Coulomb energy  [70, 71]. However, the superconductivity \nstill persists in the absence of the correlated insulators and even takes over the phase \nspace vacated by the correlate d insulators  [69-71]. Moreover, the superconductivity can \nsurvive to a larger detuning of the twist angles  [69-71]. In this framework, the electric \npairing between weakly interacting band electrons, since any tendency of the electrons \nto form an insulating  phase would suppress superconductivity because fewer electrons \nwould be available to pair. In addition, conventional phonon -mediated pairing in \ngraphene -based systems with a high DOS at the Fermi level is a realistic possibility, \nsuch as in an intercalate d graphite [242-246]. Despite much theoretical work [40-42, 141 -173, 187], the microscopic mechanism of the pairing interaction and the symmetry of the \nsupercondu cting order parameter still have not been reached consensus, which needs to \nbe continuously ex plored.  \n3.3 Nematicity  in the MATBG  \nIn the single -particle theoretical picture, the moiré periodic potential in the TBG \nleads to the spatial variation of the DOS, which are expected to exhibit the same or \ninverted contrast as atomic topographies  in the STM images  [1-3, 7-9]. The Coulomb \ninteractions can efficiently break the symmetry of electronic wavefunctions  [173, 185, 239 -\n241], resulting in the spatial redistribution of the DOS within each moiré superlattice. \nExperimentally, the spatial -resol ved DOS can be directly visualized via the \nconductance maps in the STM studies [53, 56 -60, 192]. Figure 6(a) shows a schematical \natomic structure of two misoriented graphene sheets with the twist angle θ, which \nexhibits the C 6 rotational symmetry. When the  twist angle is far from the magic angle, \nthe conductance maps at energies around the VHS peaks show the same features as \ntopographies and maintain the C 6 symmetry [58, 60, 192]. Similar C 6-symmetry \nconductance maps are also observed in the MATBG when the  two flat bands are fully \nfilled or empty (Fig. 6(b)) [58]. However, when the Fermi level locates at the charge \nneutrality point (CNP) of the MATBG, the low -energy conductance maps show an \nobvious correlation -induced charge redistribution in each moiré, whi ch suppresses the \nDOS at AA regions and highlights the domain wall areas [56]. Particularly, the \nconductance maps at the energies of two flat bands show pronounced anisotropy, \nreducing the symmetry from the initial C 6 to C 2 with the symmetry axes perpendicular \nto each other  [53, 56, 6 0, 175].   \nWhen one of the flat bands (VHS peaks) in the MATBG is partially filled, the \nwavefunctions for both the flat bands change dramatically, resulting in different \nsymmetry breaking phases  for different moiré band fillings  [51-59, 62 -84, 182 -188]. For the \npartially filled flat band, the DOS for ν = 1 are no longer localized at the AA regions. \nIn contrast, the DOS for ν = 2 are still localized at AA regions, accompanied by the broken of C 6 rotational symmetry [53] (as schematically shown in Fig. 6(c)). The stark \ndifferences between ν = 1 and 2 fillings indicate that they are intrinsically different \ncorrelated phases  [53]. These results are well consistent with the fact that the ground \nstate of ν = 1 is both spin and valley polarized, while the ground state of ν = 2 is either \nspin or valley polarized [176-178, 180 -183]. Moreover, the charge redistributions in partially \nfilled flat band prefer to result in the stripe charge order  [53, 57 -59], suggesting a high \nsusceptibility towards nematicity [173, 185, 239 -241]. This result is similar to the cuprate \nsuperconductors, where the superconductivity is intimately connected to the nematic \nphase and pseudo -gap state [232-237]. However, we should still be much careful about this \nviewpoint. On the one hand, only a small amount of strain, which is almost unavoidable \nin the TBG, can introduce additional broken of symmetries  [211, 239, 24 7]. Although the \nstrain -induced trivial symmetry -breaking state is supposed to be uniform with different \nenergies and fillings, there still exists the possibility that the coexistence of both strain \nand unaccounted electronic correlations stabilize the sys tem as the observed ground \nstate  [175, 248]. On the other hand, the domains of stripe charge order with different \norientations are expected to appear in the nematic phase, which is still experimentally \nlacking in the MATBG up till now. In addition, theoret ical calculations have predicted \nabundant competing quantum states and correlated ground phases in the MATBG, \nwhich strongly depend on the subtle atomic structures and sample preparation processes  \n[170-191]. Therefore, the intrinsic ground states and fundamental effects in the MATBG \nneed to be further carefully studied.  \n3.4 Topology and Magnetism in the MATBG  \nRecent experiments on ν = 3 filling of the conduction band in the TBG near the magic \nangle reveal quite distinct correlation -induced quantum stat es, such as metallic phase  \n[69-71], thermally activated trivial insulator  [63, 72 -74, 76 -78], and (quantized) anomalous Hall \neffect [64, 81]. These differences highlight the extreme sensitivity of symmetry -breaking \nphases to the fine atomic structures of t he TBG and the surrounding atmosphere. \nAmong all, the (quantized) anomalous Hall effect, which is supposed to be energy -favored when the MATBG is closely aligned to the hBN layers  [64, 81, 178 -183, 213 -215], is of \ngreat interest and attracts much attention . Sharpe  et al. observed a giant anomalous Hall \n(AH) effect at around ν = 3 of the TBG near the magic angle (1.17º) that is on top of \nhBN layers with the corresponding angle of 0.81º, and the Hall resistance 𝑅𝑥𝑦 \ndisplayed hysteresis loops under external magnetic fields  [81]. Moreover, they acquired \nthe chiral edge currents via the nonlocal transport measurements, which are the \ncharacteristic of topological phases  [81]. Very recently, Serlin  et al . not only well \nrepeated the hysteresis loops, but also realized the switch of the magnetization via a \nsmall charge current  [64]. More importantly, they observed a well -quantized  resistance  \n𝑅𝑥𝑦=ℎ𝑒2⁄ under zero magnetic field at ν = 3 of the hBN -aligned TBG near the magi c \nangle (1.15º), demonstrating the realization of a quantized anomalous Hall (QAH) state \nin this system  [64] (Fig. 7).   \nThe observation of hysteresis loops at ν = 3 filling of the hBN -aligned TBG near the \nmagic angle demonstrates a clear experimental evid ence of the ferromagnetic state  [64, \n81, 82]. Due to the extremely weak spin -orbit coupling and the absence of magnetic atoms \nin graphene system, the ferromagnetism in the TBG near the magic angle is attributed \nto the purely orbital magnetism, with an easy -axis anisotropy arising from the graphene \nbands  [177-183, 213, 215]. Recent STM and superconducting quantum interference device \n(SQUID) measurements also demonstrate the existence of a large orbital magnetic \nmoment of several 𝜇𝐵  per electron per moiré unit cell in the MATBG [62, 82, 85]. \nMoreover, the coexistence of (quantized) anomalous Hall effects and ferromagnetic \ninsulating states is reminiscent of the Chern insulator where the longitudinal resistivity \n𝑅𝑥𝑥 approaches to z ero and the Hall resistivity is quantized to 𝑅𝑥𝑦=ℎ𝐶𝑒2⁄  (C is the \nChern number arising from the Berry curvature of the filled bands) [64, 81, 179 -181]. \nTheoretically, the achievement of a Chern insulator should open up a topologically \nnontrivial gap, i.e. the moiré miniband carries a nonzero valley -contrast Chern number  \n[179-181, 214]. In the TBG near the magic angle, the atomic relaxation leads to the low -\nenergy flat band isolating from the higher -energy dispersive bands by an energy gap [45-\n48, 115, 11 7]. Although the combination of C2z symmetry (a twofold rotational symmetry around the out -of-plane axis) and time -reversal symmetry in pristine TBG interchanges \nthe two valleys and forbids the Berry curvature of the low -energy flat bands, the \nunderlying n ear-aligned hBN can open up a band gap at the Dirac point of graphene \nand efficiently break the C2z symmetry [64, 81, 178 -183, 213 -215]. Moreover, for ν = 3 filling \nflat band in TBG near the magic angle, the strongly e-e interactions are expected to \nsponta neously break both the spin and valley symmetries, resulting in the electrons \npolarizing into a spin - and valley -resolved moiré miniband  [176-178, 180 -182]. Therefore, a \nvalley Chern number of C = 1 is generated, resulting in the energy -favored QAH state \nat ν = 3 filling flat band of the hBN -aligned MATBG [64]. Very recently, several groups \nalso observed the field -driven phase transition of valley -projected bands from zero \nChern numbers to nonzero Chern numbers at ν = ±1 and -2 fillings of the hBN -non-\nalign ed MATBG  [51, 52, 71 -74, 80], highlighting the possible competition between trivial \ninsulator and Chern insulator. However, the primary cause for the diversities still \nremains to be determined.   \nThe discovery of the QAH state in the MATBG provides a new insight to search for \nhigh-quality QAH materials. In traditional QAH materials such as (Bi,Sb) 2Te3, the \nbroken of time -reversal symmetry and the realization of topologically nontrivial Chern \nbands are attributed to the external magnetic dopants [249-254]. Hence, the critical \ntemperature of QAH is much lower than the magnetic ordering temperature, because it \nis strongly limited by the inhomogeneous structures  [249-254]. In contrast, the intrinsic \naligned MATBG -hBN processes the necessary ingredients for the QAH state, offering \na less disordered system to realize a more robust quantization [64].  \n4. Multilayer graphene  \nAs we shown above, the ratio between Coulomb interactions and kinetic energy \n(bandwidth) is a key parameter to determine the strength of electronic correlations  [143-\n145, 187 -191]. One of the new perspectives for extending the correlated behaviors is to seek \nfor the graphene -based superlattices with a tunable bandwidth. Twisted double -bilayer \ngraphene (TDBG), which consists of two sheets of  untwisted Bernal -stacked bilayer graphene stacked together with a twist angle θ, is an idea candidate for investigating \nthe novel correlated physics. On the one hand, the Bernal -stacked bilayer graphene \nshows parabolic band dispersion  [255-257], and the b and gap opens up at the charge neutral \npoint with the increasing perpendicular electrical displacement field  [120-125], resulting \nin the TDBG inheriting this electrically tunable flat bands [222, 258 -268]. On the other hand, \nthe flat bands in TDBG emerge o ver a wide range of twist angles and are much narrow \nowing to the gap [222, 258 -269]. This  distinguishes from the TBG that the flat bands only \nemerge in the vicinity of the magic angles  [6-11, 25, 35 -58]. Therefore, the TDBG is \nsupposed to exhibit a much richer phase diagrams that are highly sensitive to both the \ntwist angle and the application of an electric displacement field (Fig. 8).   \nRecent t ransport experiments highlighted the correlated insulating behaviour for \nTDBG with the twist angle θ≲1.3° at ν = 1, 2, 3  fillings of the moiré  unit cell . The \ncorrelated insulating behaviour can be  sensitively switched  on and off by the \ndisplacement field [222, 258 -262]. Among all, the ν = 2 correlated insulating state is \nparticular. In stark contr ast to the MATBG,  the ν = 2 insulating gap of the TDBG shows \na linear response to the parallel magnetic fields and yields the g -factor that is close to g \n= 2, suggesting the signature of the spin -polarized ground state [222, 259, 260] (Fig. 8). With \nthe twist angle decreas ing to 0.84°, multiple correlated states appear at the half filling \nof each high -energy flat bands, all of which are tunable by the displacement field as \nwell [259]. Similar to the MA TBG, Liu  et al. [222], Cao et al.  [259] and Shen et al.  [260] \nobserve  half- and quarter -filling correlated state s in the TDBG system. However, the \nspin-polarized ground states at half -filling , the drops in resistivity of metallic states  at \nlow temperature and their nonlinear I -V characteristics  indicate that it exhibits mor e \ncomplex and richer correlated phases than the MATBG  system [222, 259, 260]. Untill now, \nnot many explicit zero -resistance experiments have been presented or repeated . Liu et \nal. conjecture tha t the critical transition is a result of Cooper -pair formation, but the \npossible superconductivity is developed only in the 1.26° device  [222], and Lee et al. \nargue that it is a spin-triplet superconductivity from pairing between opposite valleys  \n[265]. However , because of the absence of well -defined critical current , no saturation to \na normal state and the nonlinear I -V outside the halo , He et al. conjecture  that the larger nonlinear I -V may be  formed by  Joule heating , and the reversals in the sign of the Hall \ncoefficient  indicates  that spontaneous symmetry breaking is the origin of resistivity \ndrops  [261]. Therefore, more experiments should be taken to reveal the underlying \nmechanism in the TDBG system. Moreover, when the twist angle is only about  0.1°, it \nis energetically favorable for commensurate regions to maximize in area, leading to  a \nsignificant structural reconstruction of the triangular domains that alternate with \nABAB - and ABCA -stacked regions  [270-272]. This result is similar to the tiny -angle TBG  \n[115-119]. It’s worth noting that the ABCA -stacked regions are a good region to study the \ncorrelated physics of itself due to the band structure 𝐸(𝑘)∝𝑘4 [273-277].  \nABC -stacked trilayer graphene (ABC -TLG) is  also a great candidate for achieving \nthe flat -band  physics due to the low -energy cubic dispersion relation  of 𝐸(𝑘)∝𝑘3 at \nthe K point in the first Brillouin [278-282]. When placing the ABC -TLG  onto the \nhexagonal boron nitride (hBN) with a large moiré superlattice, the moiré periodic \npotential  can lead to the extremely narrow electronic minibands, which are expected to \nexhibit abundant strongly correlated physics  [56, 178, 283 -290]. Recent transport experiments \nshow the signatures of the Mott insulating states at 1/4 and 1/2 fillings of the hole \nminibands  in the ABC -TLG/hBN heterostructures at a large vertical displacement field \n|D| [284]. In the slightly electron and hole -doped 1/4 -filling Mott insulating state, the \nsuperconducting domes emerge at the temperature below 1 kelvin [56]. Moreover, the \ntopolog y of the ABC -TLG/hBN moiré miniband show s greatly displacement -field-\ndependent. For the displacement field D > 0, the hole miniband is topological with a \nnonzero Chern number of C = 2 at the 1/4 filling. In contrast, reversing the displacement \nfield to D <  0 can lead to a topological trivial moiré miniband (C = 0) [285]. These \nexperimental results have been followed by a number of theoretical studies on the exotic  \nproperties of the flat bands in the ABC -TLG/hBN heterostructures [178, 286 -290]. However, \nthe possible mechanism of the superconductivity is still full of debate up till now.  In \naddition to the intrinsic flat band of th is system,  the magnetic field can condense the \nenergy band into more flat ten LLs, and the interactions  of electron can be revealed by \nthe broken -symmetry states of L Ls, which indicates that it is an ideal system to study \nthe novel correlated phases in the quantum Hall regime  [278]. More recently, the twisted TLG  is also considered as a perfect  system to st udy the \nemergent electronic correlations of the engineered flat bands[291-303]. In the twisted \nmonolayer graphene -bilayer graphene (tMBG), i.e. the monolayer graphene rotates \nwith respect to the Bernal -stacked bilayer graphene, experimental results show th e \ncorrelated insulating states at fillings ν = 1, 2, 3 electrons per moiré unit cell when \n1.2≲θ≲1.4° [292-294]. The correlated insulating states in the tMBG can be switched  \non and off by a displacement field D with an asymmetric response to D, highlighting  \nthe electron -hole asymmetry in this system [292-294]. In the vicinity of ν = 1 correlated \ninsulating state, there is the signature of superconductivity [294]. Moreover, the quantized \nanomalous Hall effect with 𝑅𝑥𝑦=ℎ2𝑒2⁄   is observed at ν = 1 and 3 fillings, \nsuggesting the spontaneous valley -polarized bands with the Chern number of C = 2 [293]. \nThese results have been theoretically verified recently  [295-298]. In addition, abundant \nstrongly correlated physics such as super conducting, topological, magnetic, insulating, \nand metallic states and their interactions are also widely predicted in the twisted TLG \nwhere one of the three layers is twisted by a small angle relative to perfect AAA, ABA, \nor ABC stacking [298, 299] and ev en in twisted multilayer graphene [300-306], which need \nfurther research.  \n5. Graphene with other 2D materials  \nThe lattice mismatch between graphene and the underlying substrates gives rise to \nthe moiré periodic potential, which can significantly modify the  electronic features of \npristine graphene. Especially, the graphene/hBN heterostructure has attracted much \ninterest. On the one hand, hBN can be used as an ideal dielectric substrate, which not \nonly couples weakly to the graphene , but also significantly su ppress es the charge \ninhomogeneities and greatly improve s the mobility of charge carriers in graphene [307-\n311]. Therefore, the graphene/hBN heterostructure enables the experimental probing of \nrich intrinsic physics in graphene such as the relativistic quan tum Hall effect (QHE), \nfractional QHE, and more exotic correlated states [207, 312 -318]. On the other hand, hBN \ncan act as an active component of the vdW heterostructures and realize a moiré superlattice structure together with graphene due to the existenc e of lattice mismatch \nand related rotation between them [319]. In this case, the electronic properties of  \ngraphene can be strongly modified and the new superlattice Dirac points arise, which \nmay be accompanied by the energy gap opening at the charge neutra lity [320-322]. \nMoreover, the long -wavelength graphene/hBN moiré pattern provides a unique \nplatform to experimentally realize the stunning self -similar recursive energy spectrum, \ni.e. Hofstadter butterfly [323-325]. The plasmon and phonon polaritons with exceptional \nproperties are also expected to be achieved in these regimes [326, 327]. Moreover, the \ntwisted hBN/graphene/hBN heterostructure can host a second -order moiré superlattice \nwith an even larger periodicity, which may result in a highly reconstruct ed graphene \nband structure featuring multiple secondary Dirac points [328]. Recent experiments \nreport the tunable on/off bandgap at the original and secondary Dirac points by rotating \nthe partial component of the heterostructure, suggesting a tunable trans ition between \nthe absence or presence of inversion symmetry in the twisted hBN/graphene/hBN \nheterostructure [329]. This result provides an insight for engineering the band structure \nof the vdW system.  \nRecent advances in fabrication techniques have made it possible to realize the vdW \nheterostructures with the components of graphene and TMDs materials [193, 330, 331]. The \n(twisted) vdW heterostructures offer a unique platform for addressing many \nfundamental physics that the individual constituent layers may n ot have [332-334]. One \nsignificant feature that all TMDs have in common is the strong intrinsic spin -orbit \ncoupling (SOC), with the order of 100 meV in the valence band and 10 meV in the \nconduction band [335-340]. As for the graphene/TMDs heterostructures,  the proximity -\nenhanced SOC in graphene with several meV is expected to achieve active graphene \nspintronics such as the spin Hall effect and the inverse spin galvanic effect even at the \nroom temperature [341-350]. Moreover, the large proximate SOC can effi ciently generate \nthe band inversion of bilayer graphene (BLG) in the BLG/TMDs heterostructures [350], \nand can also result in the gate -tunable transitions between topological and trivial \ninsulating states in the multilayer graphene (MLG)/TMDs heterostructur es [351]. \nAnother distinctive electronic property of typical TMDs such as NbSe 2, NbS 2, TaSe 2, and TaS 2 is the superconductivity at low temperature [352-355]. The strong SOC induced \nspin splittings cause the superconducting pairing to be of the Ising type, which is  \nextremely robust to external in -plane magnetic fields [352, 356, 357]. Previous ab initio \ncalculations show a large superconducting pairing can be induced into the graphene \nlayer despite the large mismatch between the lattice constants of graphene  and NbSe 2, \nand the proximity -induced superconductivity in graphene can still maintain within a \nwide range of twist angles due to the large size of the Fermi pockets in NbSe 2 [358]. \nExperimentally, the proximity -induced superconducting gap in BLG of the BL G / \nNbSe 2 heterostructure and the Andreev reflection at their junction are clearly observed \n[359]. Moreover, recent STM experiment captures the periodically modulated pseudo -\nmagnetic field in graphene/ NbSe 2 moiré superlattices, which reminds us a new way to \ndesign the flat electronic bands in graphene [360]. In the future , the proximity -induced \nvarious electronic properties should be carefully studied relying on available degrees \nof freedom such as the twist  angle and strength of interlayer coupling.  \n6. Conclusions and perspectives  \nIn summary, the TBG is expected to exhibit an exceptionally wide range of physical \nphenomena, which dramatically depends on the  twist angles. Especially in th e TBG \nnear the magic a ngle, many outstanding issues in condensed matter physics such as \nMott insulating state, superconductivity, strange metal, nematicity, topology, and \nmagnetism are totally achieved for the partially filled moiré flat minibands. Moreover, \nmuch richer strongl y correlated phase diagram is realized in twisted multilayer \ngraphene due to the additional tunable degree of freedom of the electrical displacement \nfield. However, there still exist many fundamental mysteries hidden in the graphene -\nbased twisted materials . One of the key questions is whether there is any relation \nbetween the zero -resistance feature in transport measurement and superconductivity. If \nyes, what is the origin and mechanism of the superconductivity? Moreover, the ground \nstates of these material s at different fillings of the moiré flat band and their phase \ntransitions under external magnetic fields are key question s for researchers to address. Corresponding theor etical calculations  need to be carried out  to capture the intrinsic \nphysics and predi ct new phases in this regime.  In addition, th e result s in the MATGB  \ncan also extend to other flat -band system , such as the strain ed graphene with flat bands.  \n \nReferences  \n[1] Landgraf W, Shallcross S, Türschmann K, Weckbecker D and Pankratov O 2013 \nPhys. Rev. B  87 075433  \n[2] Mele E J 2010 Phys. Rev. B  81 161405  \n[3] Shallcross S, Sharma S and Pankratov O 2013 Phys. Rev. B  87 245403  \n[4] Lopes dos Santos J M B, Peres N M R and Castro Neto A H 2007 Phys. Rev. Lett.  \n99 256802  \n[5] Shallcross S, Sharma S, Kan delaki E and Pankratov O A 2010 Phys. Rev. B  81 \n165105  \n[6] Trambly de Laissardière G, Mayou D and Magaud L 2012 Phys. Rev. B  86 125413  \n[7] Trambly de Laissardière G, Mayou D and Magaud L 2010 Nano Lett.  10 804  \n[8] Bistritzer R and MacDonald A H 2011 Proc. Natl. Acad. Sci. U.S.A.  108 12233  \n[9] San -Jose P, Gonzalez J and Guinea F 2012 Phys. Rev. Lett.  108 216802  \n[10] Suárez Morell E, Correa J D, Vargas P, Pacheco M and Barticevic Z 2010 Phys. \nRev. B  82 121407(R)  \n[11] Uchida K, Furuya S, Iwata J -I and  Oshiyama A 2014 Phys. Rev. B  90 155451  \n[12] Wang Z F, Liu F and Chou M Y 2012 Nano Lett.  12 3833  \n[13] Weckbecker D, Shallcross S, Fleischmann M, Ray N, Sharma S and Pankratov O \n2016 Phys. Rev. B  93 035452  \n[14] Lopes dos Santos J M B, Peres N M R and Ca stro Neto A H 2012 Phys. Rev. B  86 \n155449  \n[15] Sboychakov A O, Rakhmanov A L, Rozhkov A V and Nori F 2015 Phys. Rev. B  92 \n075402  \n[16] Shallcross S, Sharma S and Pankratov O A 2008 Phys. Rev. Lett.  101 056803  [17] Chu Z -D, He W -Y and He L 2013 Phys. Rev.  B 87 155419  \n[18] Bistritzer R and MacDonald A H 2010 Phys. Rev. B  81 245412  \n[19] Moon P and Koshino M 2012 Phys. Rev. B  85 195458  \n[20] Cherkez V , de Laissardière G T, Mallet P and Veuillen J Y 2015 Phys. Rev. B  91 \n155428  \n[21] Luican A, Li G, Reina A, Kong J, Nair R R, Novoselov K S, Geim A K and Andrei \nE Y 2011 Phys. Rev. Lett.  106 126802  \n[22] Yan W, Liu M, Dou R -F, Meng L, Feng L, Chu Z -D, Zhang Y , Liu Z, Nie J -C and \nHe L 2012 Phys. Rev. Lett.  109 126801  \n[23] Brihuega I, Mallet P, González -Herrero H , Trambly de Laissardière G, Ugeda M \nM, Magaud L, Gómez -Rodríguez J M, Ynduráin F and Veuillen J Y 2012 Phys. Rev. \nLett. 109 196802  \n[24] Li G, Luican A, Lopes dos Santos J M B, Castro Neto A H, Reina A, Kong J and \nAndrei E Y 2010 Nat. Phys.  6 109  \n[25] Yi n L-J, Qiao J -B, Zuo W -J, Li W -T and He L 2015 Phys. Rev. B  92 081406(R)  \n[26] Yin L -J, Qiao J -B, Wang W -X, Zuo W -J, Yan W, Xu R, Dou R -F, Nie J -C and He \nL 2015 Phys. Rev. B  92 201408(R)  \n[27] Wong D, Wang Y , Jung J, Pezzini S, DaSilva A M, Tsai H -Z, Jung H S, Khajeh R, \nKim Y and Lee J 2015 Phys. Rev. B  92 155409  \n[28] Yan W, Meng L, Liu M, Qiao J -B, Chu Z -D, Dou R -F, Liu Z, Nie J -C, Naugle D \nG and He L 2014 Phys. Rev. B  90 115402  \n[29] Kim Y , Yun H, Nam S -G, Son M, Lee D S, Kim D C, Seo S, Choi H C, Lee H -J, \nLee S W and Kim J S 2013 Phys. Rev. Lett.  110 096602  \n[30] Miller D L, Kubista K D, Rutter G M, Ruan M, de Heer W A, First P N and Stroscio \nJ A 2009 Science  324 924  \n[31] Hass J, Varchon F, Millán -Otoya J E, Sprinkle M, Sharma N, de Heer W A, Berger \nC, First P N, Magaud L and Conrad E H 2008 Phys. Rev. Lett.  100 125504  \n[32] Sprinkle M, Siegel D, Hu Y , Hicks J, Tejeda A, Taleb -Ibrahimi A, Le Fèvre P, \nBertran F, Vizzini S, Enriquez H, Chiang S, Soukiassian P, Berger C, de Heer W A, \nLanzara A and Conrad E H 2009 Phys. Rev. Lett.  103 226803  [33] Koren E, Leven I, Lörtscher E, Knoll A, Hod O and Duerig U 2016 Nat. Nanotech.  \n11 752  \n[34] Ni Z, Wang Y , Yu T, You Y and Shen Z 2008 Phys. Rev. B  77 235403  \n[35] Angeli M, Mandelli D, Valli A, Amaricci A, Capone M,  Tosatti E and Fabrizio M \n2018 Phys. Rev. B  98 235137  \n[36] Lin X and Tománek D 2018 Phys. Rev. B  98 081410  \n[37] Carr S, Fang S, Po H C, Vishwanath A and Kaxiras E 2019 Phys. Rev. Research  1 \n033072  \n[38] Goodwin Z A H, Corsetti F, Mostofi A A and Lischner J 2019 Phys. Rev. B  100 \n235424  \n[39] Po H C, Zou L, Senthil T and Vishwanath A 2019 Phys. Rev. B  99 195455  \n[40] Koshino M, Yuan N F Q, Koretsune T, Ochi M, Kuroki K and Fu L 2018 Phys. \nRev. X  8 031087  \n[41] Su Y and Lin S -Z 2018 Phys. Rev. B  98 195101  \n[42] Tarnopolsky G, Kruchkov A J and Vishwanath A 2019 Phys. Rev. Lett.  122 106405  \n[43] Julku A, Peltonen T J, Liang L, Heikkilä T T and Törmä P 2020 Phys. Rev. B  101 \n060505  \n[44] Ledwith P J, Ta rnopolsky G, Khalaf E and Vishwanath A 2020 Phys. Rev. Research  \n2 023237  \n[45] Carr S, Fang S, Zhu Z and Kaxiras E 2019 Phys. Rev. Research  1 013001  \n[46] Lucignano P, Alfè D, Cataudella V , Ninno D and Cantele G 2019 Phys. Rev. B  99 \n195419  \n[47] Fang S, Ca rr S, Zhu Z, Massatt D and Kaxiras E 2019 arXiv:1908.00058   \n[48] Leconte N, Javvaji S, An J and Jung J 2019 arXiv:1910.12805   \n[49] Utama M I B, Koch R J, Lee K, Leconte N, Li H, Zhao S, Jiang L, Zhu J, Watanabe \nK, Taniguchi T, Ashby P D, Weber -Bargioni A, Zettl A, Jozwiak C, Jung J, Rotenberg \nE, Bostwick A and Wang F 2020 Nat. Phys.   \n[50] Lisi S, Lu X, Benschop T, de Jong T A, Stepanov P, Duran J R, Margot F, Cucchi \nI, Cappelli E and Hunter A 2020 arXiv:2002.02289   \n[51] Tomarken S L, Cao Y , Demir A, Watanab e K, Taniguchi T, Jarillo -Herrero P and Ashoori R C 2019 Phys. Rev. Lett.  123 046601  \n[52] Nuckolls K P, Oh M, Wong D, Lian B, Watanabe K, Taniguchi T, Bernevig B A \nand Yazdani A 2020 arXiv:2007.03810   \n[53] Zhang Z, Myers R, Watanabe K, Taniguchi T and LeR oy B J 2020 \narXiv:2003.09482   \n[54] Wong D, Nuckolls K P, Oh M, Lian B, Xie Y , Jeon S, Watanabe K, Taniguchi T, \nBernevig B A and Yazdani A 2020 Nature  582 198  \n[55] Xie Y , Lian B, Jäck B, Liu X, Chiu C -L, Watanabe K, Taniguchi T, Bernevig B A \nand Yazdani A 2019 Nature  572 101  \n[56] Choi Y , Kemmer J, Peng Y , Thomson A, Arora H, Polski R, Zhang Y , Ren H, Alicea \nJ, Refael G, von Oppen F, Watanabe K, Taniguchi T and Nadj -Perge S 2019 Nat. \nPhys.  15 1174  \n[57] Kerelsky A, McGilly L J, Kennes D M, Xian L, Yankowitz  M, Chen S, Watanabe \nK, Taniguchi T, Hone J, Dean C, Rubio A and Pasupathy A N 2019 Nature  572 95  \n[58] Jiang Y , Lai X, Watanabe K, Taniguchi T, Haule K, Mao J and Andrei E Y 2019 \nNature  573 91  \n[59] Ren Y -N, Lu C, Zhang Y , Li S -Y , Liu Y -W, Yan C, Guo Z -H, Liu C -C, Yang F and \nHe L 2019 arXiv:1912.07229   \n[60] Li S -Y , Liu K -Q, Yin L -J, Wang W -X, Yan W, Yang X -Q, Yang J -K, Liu H, Jiang \nH and He L 2017 Phys. Rev. B  96 155416  \n[61] Kim K, DaSilva A, Huang S, Fallahazad B, Larentis S, Taniguchi T, Watanabe K, \nLeRoy B J, MacDonald A H and Tutuc E 2017 Proc. Natl. Acad. Sci. U.S.A.  114 \n3364  \n[62] Li S -Y , Zhang Y , Ren Y -N, Liu J, Dai X and He L 2019 arXiv: 1912.13133   \n[63] Polshyn H, Yankowitz M, Chen S, Zhang Y , Watanabe K, Taniguchi T, Dean C R \nand Young A F 2019 Nat. Phys.  15 1011  \n[64] Serlin M, Tschirhart C L, Polshyn H, Zhang Y , Zhu J, Watanabe K, Taniguchi T, \nBalents L and Young A F 2020 Science  367 900  \n[65] Cao Y , Fatemi V , Demir A, Fang S, Tomarken S L, Luo J Y , Sanchez -Yamagishi J \nD, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C and Jarillo -Herrero P 2018 Nature  556 80  \n[66] Codecido E, Wang Q, Koester R, Che S, Tian H, Lv R, Tran S, Watanabe K, \nTaniguchi T, Zhang F, Bockrath M and Lau C N 2019 Sci. Adv.  5 eaaw9770  \n[67] Cao Y , Chowdhury D, Rodan -Legrain D, Rubies -Bigorda O, Watanabe K, \nTaniguchi T, Senthil T and Jarillo -Herrero P 2020 Phys. Rev. Lett.  124 076801  \n[68] Cao Y , Fatemi V , Fang S, Watanabe K, Taniguchi T, Kaxiras E and Jarillo -Herrero \nP 2018 Nature  556 43  \n[69] Arora H S, Polski R, Zhang Y , Thomson A, Choi Y , Kim H, Lin Z, Wilson I Z, Xu \nX, Chu J -H, Watanabe K , Taniguchi T, Alicea J and Nadj -Perge S 2020 Nature  583 \n379  \n[70] Saito Y , Ge J, Watanabe K, Taniguchi T and Young A F 2020 Nat. Phys.   \n[71] Stepanov P, Das I, Lu X, Fahimniya A, Watanabe K, Taniguchi T, Koppens F H L, \nLischner J, Levitov L and Efetov D K  2020 Nature  583 375  \n[72] Lu X, Stepanov P, Yang W, Xie M, Aamir M A, Das I, Urgell C, Watanabe K, \nTaniguchi T, Zhang G, Bachtold A, MacDonald A H and Efetov D K 2019 Nature  \n574 653  \n[73] Saito Y , Ge J, Rademaker L, Watanabe K, Taniguchi T, Abanin D A and  Young A \nF 2020 arXiv:2007.06115   \n[74] Wu S, Zhang Z, Watanabe K, Taniguchi T and Andrei E Y 2020 arXiv:2007.03735   \n[75] Liu X, Wang Z, Watanabe K, Taniguchi T, Vafek O and Li J I A 2020 \narXiv:2003.11072   \n[76] Cao Y , Rodan -Legrain D, Park J M, Yuan F N, Wa tanabe K, Taniguchi T, Fernandes \nR M, Fu L and Jarillo -Herrero P 2020 arXiv:2004.04148   \n[77] Uri A, Grover S, Cao Y , Crosse J A, Bagani K, Rodan -Legrain D, Myasoedov Y , \nWatanabe K, Taniguchi T, Moon P, Koshino M, Jarillo -Herrero P and Zeldov E 2020 \nNature  581 47  \n[78] Yankowitz M, Chen S, Polshyn H, Zhang Y , Watanabe K, Taniguchi T, Graf D, \nYoung A F and Dean C R 2019 Science  363 1059  \n[79] Zondiner U, Rozen A, Rodan -Legrain D, Cao Y , Queiroz R, Taniguchi T, Watanabe \nK, Oreg Y , von Oppen F, Stern A, Berg E,  Jarillo -Herrero P and Ilani S 2020 Nature  582 203  \n[80] Das I, Lu X, Herzog -Arbeitman J, Song Z -D, Watanabe K, Taniguchi T, Bernevig \nB A and Efetov D K 2020 arXiv:2007.13390   \n[81] Sharpe A L, Fox E J, Barnard A W, Finney J, Watanabe K, Taniguchi T, Kastne r M \nA and Goldhaber -Gordon D 2019 Science  365 605  \n[82] Tschirhart C L, Serlin M, Polshyn H, Shragai A, Xia Z, Zhu J, Zhang Y, Watanabe \nK, Taniguchi T, Huber M E and F Y A 2020 arXiv:2006.08053   \n[83] Balents L, Dean C R, Efetov D K and Young A F 2020 Nat. Phys.  16 725 \n[84] Liu Y -W, Qiao J -B, Yan C, Zhang Y , Li S -Y and He L 2019 Phys. Rev. B  99 201408  \n[85] Zhang Y , Hou Z, Zhao Y -X, Guo Z -H, Liu Y -W, Li S -Y , Ren Y -N, Sun Q -F and He \nL 2020 arXiv: 2002.10073   \n[86] Wang W -X, Jiang H, Zhang Y , Li S -Y , Liu H, Li X, Wu X and He L 2017 Phys. \nRev. B  96 115434  \n[87] Koren E and Duerig U 2016 Phys. Rev. B  93 201404(R)  \n[88] Ahn S J, Moon P, Kim T -H, Kim H -W, Shin H -C, Kim E H, Cha H W, Kahng S -J, \nKim P, Koshino M, Son Y -W, Yang C -W and Ahn J R 2018 Science  361 782  \n[89] Yao W, Wang E, Bao C, Zhang Y , Zhang K, Bao K, Chan C K, Chen C, Avila J, \nAsensio M C, Zhu J and Zhou S 2018 Proc. Natl. Acad. Sci. U.S.A.  115 6928  \n[90] Moon P, Koshino M and Son Y -W 2019 Phys. Rev. B  99 165430  \n[91] Park M J, Kim H S and Lee S 2019 Phys. Rev. B  99 245401  \n[92] Yan C, Ma D -L, Qiao J -B, Zhong H -Y , Yang L, Li S -Y , Fu Z -Q, Zhang Y and He \nL 2019 2D Mater.  6 045041  \n[93] Yu G, Wu Z, Zhan Z, Katsnelson M I and Yuan S 2019 NPJ Comput. Mater.  5 1  \n[94] Deng B, Wang B, Li N, Li R, Wang Y , Tang J, Fu Q, Tian Z, Gao P, Xue J and Peng \nH 2020 ACS Nano  14 1656  \n[95] Hua C -B, Chen R, Zhou B and Xu D -H 2020 arXiv: 2001.07551   \n[96] Yu G, Katsnelson M I and Yuan S 2020 Phys. Rev. B  102 045113  \n[97] San -Jose P  and Prada E 2013 Phys. Rev. B  88 121408(R)  \n[98] Anđelković M, Covaci L and Peeters F M 2018 Phys. Rev. Mater.  2 034004   \n[99] Efimkin D K and MacDonald A H 2018 Phys. Rev. B  98 035404  [100] Huang S, Kim K, Efimkin D K, Lovorn T, Taniguchi T, Watanabe K, MacDonald \nA H, Tutuc E and LeRoy B J 2018 Phys. Rev. Lett.  121 037702  \n[101] Rickhaus P, Wallbank J, Slizovskiy S, Pisoni R, Overweg H, Lee Y , Eich M, Liu \nM-H, Watanabe K, Taniguchi T, Ihn T and Ensslin K 2018 Nano Lett.  18 6725  \n[102] Xu S G, Berdyugin A I, Kumaravadivel P, Guinea F, Krishna Kumar R, Bandurin \nD A, Morozov S V , Kuang W, Tsim B, Liu S, Edgar J H, Grigorieva I V , Fal’ko V I, \nKim M and Geim A K 2019 Nat. Commun.  10 4008  \n[103] De Beule C, Dominguez F and Recher P 2020 arXiv:2005.05352   \n[104] Q iao J-B, Yin L -J and He L 2018 Phys. Rev. B  98 235402  \n[105] Liu Y -W, Su Y , Zhou X -F, Yin L -J, Yan C, Li S -Y , Yan W, Han S, Fu Z -Q, Zhang \nY , Yang Q, Ren Y -N R and He L 2020 arXiv:2007.01993   \n[106] Ramires A and Lado J L 2018 Phys. Rev. Lett.  121 146801  \n[107] Tsim B, Nam N N T and Koshino M 2020 Phys. Rev. B  101 125409  \n[108] Walet N R and Guinea F 2019 2D Mater.  7 015023  \n[109] Fleischmann M, Gupta R, Wullschläger F, Theil S, Weckbecker D, Meded V , \nSharma S, Meyer B and Shallcross S 2020 Nano Lett.  20 971  \n[110] De Beule C, Dominguez F and Recher P 2020 arXiv:2003.08987   \n[111] Hou T, Ren Y , Quan Y , Jung J, Ren W and Qiao Z 2020 Phys. Rev. B  101 201403  \n[112] Chou Y -Z, Wu F and Sarma S D 2020 arXiv:2004.15022   \n[113] Sunku S S, Ni G X, Jiang B Y , Yoo H, Stern bach A, McLeod A S, Stauber T, \nXiong L, Taniguchi T, Watanabe K, Kim P, Fogler M M and Basov D N 2018 Science  \n362 1153  \n[114] Park M J, Kim Y , Cho G Y and Lee S 2019 Phys. Rev. Lett.  123 216803  \n[115] Yoo H, Engelke R, Carr S, Fang S, Zhang K, Cazeaux P, S ung S H, Hovden R, \nTsen A W and Taniguchi T 2019 Nat. Mater.  18 448  \n[116] Wijk M M v, Schuring A, Katsnelson M I and Fasolino A 2015 2D Mater.  2 \n034010  \n[117] Nam N N T and Koshino M 2017 Phys. Rev. B  96 075311  \n[118] Gargiulo F and Yazyev O V 2017 2D Mater.  5 015019  \n[119] Jain S K, Juričić V and Barkema G T 2016 2D Mater.  4 015018   [120] McCann E 2006 Phys. Rev. B  74 161403(R)  \n[121] Oostinga J B, Heersche H B, Liu X, Morpurgo A F and Vandersypen L M K 2008 \nNat. Mater.  7 151  \n[122] Li J, Wang K, McFaul K J, Zern Z, Ren Y , Watanabe K, Taniguchi T, Qiao Z and \nZhu J 2016 Nat. Nanotech.  11 1060  \n[123] Zhang Y , Tang T -T, Girit C, Hao Z, Martin M C, Zettl A, Crommie M F, Shen Y \nR and Wang F 2009 Nature  459 820  \n[124] Castro E V , Novo selov K S, Morozov S V , Peres N M R, dos Santos J M B L, \nNilsson J, Guinea F, Geim A K and Neto A H C 2007 Phys. Rev. Lett.  99 216802  \n[125] Mak K F, Lui C H, Shan J and Heinz T F 2009 Phys. Rev. Lett.  102 256405  \n[126] Vaezi A, Liang Y , Ngai D H, Yang L a nd Kim E -A 2013 Phys. Rev. X  3 021018  \n[127] Zhang F, MacDonald A H and Mele E J J P o t N A o S 2013 Proc. Natl. Acad. \nSci. 110 10546  \n[128] Pelc M, Jaskólski W, Ayuela A and Chico L 2015 Phys. Rev. B  92 085433  \n[129] Lee C, Kim G, Jung J and Min H 2016 Phys. Rev. B  94 125438  \n[130] Alden J S, Tsen A W, Huang P Y , Hovden R, Brown L, Park J, Muller D A and \nMcEuen P L 2013 Proc. Natl. Acad. Sci.  110 11256  \n[131] Ju L, Shi Z, Nair N, Lv Y , Jin C, Velasco J, Ojeda -Aristizabal C, Bechtel H A, \nMartin M C, Zettl A, Analytis J and Wang F 2015 Nature  520 650  \n[132] Yin L -J, Jiang H, Qiao J -B and He L 2016 Nat. Commun.  7 11760  \n[133] Suzuki T, Iimori T, Ahn S J, Zhao Y , Watanabe M, Xu J, Fujisawa M, Kanai T, \nIshii N, Itatani J, Suwa K, Fukidome H, Tanaka S, Ahn J R, Okazaki K, Shin S, \nKomori F and Matsuda I 2019 ACS Nano  13 11981  \n[134] Lin Y -R, Samiseresht N, Franke M, Parhizkar S, Soubatch S, Amorim B, Lee T -\nL, Kumpf C, Tautz F and Bocquet F 2018 arXiv:1809.07958   \n[135] Takesaki Y , Kawahara K, Hibino H, Okada S, Tsu ji M and Ago H 2016 Chem. \nMater.  28 4583  \n[136] Pezzini S, Mišeikis V , Piccinini G, Forti S, Pace S, Engelke R, Rossella F, \nWatanabe K, Taniguchi T, Kim P and Coletti C 2020 Nano Lett.  20 3313  \n[137] Pal H K, Spitz S and Kindermann M 2019 Phys. Rev. Lett.  123 186402  [138] Anderson P W 1987 Science  235 1196  \n[139] Balents L 2010 Nature  464 199  \n[140] Norman M R 2016 Rev. Mod. Phys.  88 041002  \n[141] Fidrysiak M, Zegrodnik M and Spałek J 2018 Phys. Rev. B  98 085436   \n[142] Guo H, Zhu X, Feng S and Scalettar R T 2018 Phys. Rev. B  97 235453  \n[143] Guinea F and Walet N R 2018 Proc. Natl. Acad. Sci. U.S.A.  115 13174  \n[144] Isobe H, Yuan N F Q and Fu L 2018 Phys. Rev. X  8 041041  \n[145] Kennes D M, Lischner J and Karrasch C 2018 Phys. Rev. B  98 241407  \n[146] Kang J and Vafek O 2018 Phys. Rev. X  8 031088  \n[147] Liu C -C, Zhang L -D, Chen W -Q and Yang F 2018 Phys. Rev. Lett.  121 217001  \n[148] Padhi B, Setty C and Phillips P W 2018 Nano Lett.  18 6175  \n[149] Rademaker L and Mellado P 2018 Phys. Rev. B  98 235158  \n[150] Venderbos J W F and Fernandes R M 2018 Phys. Rev. B  98 245103  \n[151] Xu C and Balents L 2018 Phys. Rev. Lett.  121 087001  \n[152] Yuan N F Q and Fu L 2018 Phys. Rev. B  98 045103  \n[153] Alidou st M, Willatzen M and Jauho A -P 2019 Phys. Rev. B  99 155413  \n[154] Chou Y -Z, Lin Y -P, Das Sarma S and Nandkishore R M 2019 Phys. Rev. B  100 \n115128  \n[155] González J and Stauber T 2019 Phys. Rev. Lett.  122 026801  \n[156] Huang T, Zhang L and Ma T 2019 Sci. B ull. 64 310  \n[157] Kumar P, Vanhala T I and Törmä P 2019 Phys. Rev. B  100 125141  \n[158] Lin Y -P and Nandkishore R M 2019 Phys. Rev. B  100 085136  \n[159] Ray S, Jung J and Das T 2019 Phys. Rev. B  99 134515  \n[160] Roy B and Juričić V 2019 Phys. Rev. B  99 1214 07  \n[161] Tang Q -K, Yang L, Wang D, Zhang F -C and Wang Q -H 2019 Phys. Rev. B  99 \n094521  \n[162] Wu F and Sarma S D 2019 Phys. Rev. B  99 220507  \n[163] Wu X -C, Keselman A, Jian C -M, Pawlak K A and Xu C 2019 Phys. Rev. B  100 \n024421  \n[164] You Y -Z and Vishwanath A 2019 npj Quantum Materials  4 16  [165] Choi Y W and Choi H J 2018 Phys. Rev. B  98 241412(R)  \n[166] Peltonen T J, Ojajärvi R and Heikkilä T T 2018 Phys. Rev. B  98 220504  \n[167] Wu F, MacDonald A H and Martin I 2018 Phys. Rev. Lett. 121 257001  \n[168] Lian B, Wang Z and Bernevig B A 2019 Phys. Rev. Lett.  122 257002  \n[169] Wu F, Hwang E and Sarma S D 2019 Phys. Rev. B  99 165112  \n[170] Angeli M, Tosatti E and Fabrizio M 2019 Phys. Rev. X  9 041010  \n[171] Po H C, Zou L, Vishwanath A and Senthil T 2018 Phys. Rev. X  8 031089  \n[172] Seo K, Kotov V N and Uchoa B 2019 Phys. Rev. Lett.  122 246402  \n[173] Dodaro J F, Kivelson S A, Schattner Y , Sun X Q and Wang C 2018 Phys. Rev. B  \n98 075154  \n[174] Ochi M, Koshino M and Kuroki K 2018 Phys. Rev.  B 98 081102(R)  \n[175] Liu S, Khalaf E, Lee J Y and Vishwanath A 2019 arXiv:1905.07409   \n[176] Xie M and MacDonald A H 2020 Physical Review Letters  124 097601  \n[177] Alavirad Y and Sau J D 2019 arXiv:1907.13633   \n[178] Repellin C, Dong Z, Zhang Y -H and Senthil T 2020 Phys. Rev. Lett.  124 187601  \n[179] Wu F and Sarma S D 2020 Phys. Rev. Lett.  124 046403  \n[180] Zhang Y -H, Mao D and Senthil T 2019 Phys. Rev. Research  1 033126  \n[181] Bultinck N, Chatterjee S and Zalete l M P 2020 Phys. Rev. Lett.  124 166601  \n[182] Liu J and Dai X 2019 arXiv:1911.03760   \n[183] Chatterjee S, Bultinck N and Zaletel M P 2020 Phys. Rev. B  101 165141  \n[184] Bultinck N, Khalaf E, Liu S, Chatterjee S, Vishwanath A and Zaletel M P 2019 \narXiv:1911. 02045   \n[185] Chichinadze D V , Classen L and Chubukov A V 2020 arXiv:2007.00871   \n[186] Zhang Y , Jiang K, Wang Z and Zhang F 2020 Phys. Rev. B  102 035136  \n[187] Kang J and Vafek O 2019 Phys. Rev. Lett.  122 246401  \n[188] Kang J and Vafek O 2020 Phys. Rev. B  102 035161  \n[189] Da Liao Y , Meng Z Y and Xu X Y 2019 Phys. Rev. Lett.  123 157601  \n[190] Sherkunov Y and Betouras J J 2018 Phys. Rev. B  98 205151  \n[191] Cea T and Guinea F 2020 Phys. Rev. B  102 045107  \n[192] Yao Q, van Bremen R, Slotman G J, Zhang L, Haart sen S, Sotthewes K, Bampoulis P, de Boeij P L, van Houselt A, Yuan S and Zandvliet H J W 2017 Phys. \nRev. B  95 245116  \n[193] Cao Y , Luo J Y , Fatemi V , Fang S, Sanchez -Yamagishi J D, Watanabe K, \nTaniguchi T, Kaxiras E and Jarillo -Herrero P 2016 Phys. Rev. Le tt. 117 116804  \n[194] Goerbig M O 2011 Rev. Mod. Phys.  83 1193  \n[195] Nomura K and MacDonald A H 2006 Phys. Rev. Lett.  96 256602  \n[196] Yang K, Das Sarma S and MacDonald A H 2006 Phys. Rev. B  74 075423  \n[197] Sheng L, Sheng D N, Haldane F D M and Balents L  2007 Phys. Rev. Lett.  99 \n196802  \n[198] Nomura K, Ryu S and Lee D -H 2009 Phys. Rev. Lett.  103 216801  \n[199] Lian Y , Rosch A and Goerbig M O 2016 Phys. Rev. Lett.  117 056806  \n[200] Alicea J and Fisher M P A 2006 Phys. Rev. B  74 075422  \n[201] Song Y J, Otte A F, Kuk Y , Hu Y , Torrance D B, First P N, De Heer W A, Min H, \nAdam S and Stiles M D 2010 Nature  467 185  \n[202] Li S -Y , Zhang Y , Yin L -J and He L 2019 Phys. Rev. B  100 085437  \n[203] Zhang Y , Jiang Z, Small J, Purewal M, Tan Y -W, Fazlollahi M, Chudow J, \nJaszczak J, Stormer H and Kim P 2006 Phys. Rev. Lett.  96 136806  \n[204] Jiang Z, Zhang Y , Stormer H L and Kim P 2007 Phys. Rev. Lett.  99 106802  \n[205] Bolotin K I, Ghahari F, Shulman M D, Stormer H L and Kim P 2009 Nature  462 \n196  \n[206] Du X, Skachko I, Duerr F, Luican A and Andrei E Y 2009 Nature  462 192  \n[207] Dean C R, Young A F, Cadden -Zimansky P, Wang L, Ren H, Watanabe K, \nTaniguchi T, Kim P, Hone J and Shepard K L 2011 Nat. Phys.  7 693  \n[208] Feldman B E, Krauss B, Smet J H an d Yacoby A 2012 Science  337 1196  \n[209] Young A F, Dean C R, Wang L, Ren H, Cadden -Zimansky P, Watanabe K, \nTaniguchi T, Hone J, Shepard K L and Kim P 2012 Nat. Phys.  8 550  \n[210] Yu G L, Jalil R, Belle B, Mayorov A S, Blake P, Schedin F, Morozov S V , \nPonom arenko L A, Chiappini F, Wiedmann S, Zeitler U, Katsnelson M I, Geim A K, \nNovoselov K S and Elias D C 2013 Proc. Natl. Acad. Sci.  110 3282  \n[211] Bi Z, Yuan N F Q and Fu L 2019 Phys. Rev. B  100 035448  [212] Wilson J H, Fu Y , Das Sarma S and Pixley J H 2020 Phys. Rev. Research  2 023325  \n[213] He W -Y , Goldhaber -Gordon D and Law K T 2020 Nat. Commun.  11 1650  \n[214] Zhu J, Su J -J and MacDonald A H 2020 arXiv:2001.05084   \n[215] Liu J and Dai X 2020 NPJ Comput . Mater.  6 57  \n[216] Kim M, Xu S G, Berdyugin A I, Principi A, Slizovskiy S, Xin N, Kumaravadivel \nP, Kuang W, Hamer M, Krishna Kumar R, Gorbachev R V , Watanabe K, Taniguchi \nT, Grigorieva I V , Fal’ko V I, Polini M and Geim A K 2020 Nature Communications  \n11 2339  \n[217] Wang Q -Y , Li Z, Zhang W -H, et al. 2012 Chinese Phys. Lett.  29 037402  \n[218] Natterer F D, Ha J, Baek H, Zhang D, Cullen W G, Zhitenev N B, Kuk Y and \nStroscio J A 2016 Phys. Rev. B  93 045406  \n[219] Tonnoir C, Kimouche A, Coraux J, Magaud L, Dels ol B, Gilles B and Chapelier \nC 2013 Phys. Rev. Lett.  111 246805  \n[220] Qiao J -B, Gong Y , Zuo W -J, Wei Y -C, Ma D -L, Yang H, Yang N, Qiao K -Y , Shi \nJ-A, Gu L and He L 2017 Phys. Rev. B  95 201403  \n[221] Mayorov A S, Gorbachev R V , Morozov S V , Britnell L, Jali l R, Ponomarenko L \nA, Blake P, Novoselov K S, Watanabe K, Taniguchi T and Geim A K 2011 Nano Lett.  \n11 2396  \n[222] Liu X, Hao Z, Khalaf E, Lee J Y , Ronen Y , Yoo H, Haei Najafabadi D, Watanabe \nK, Taniguchi T, Vishwanath A and Kim P 2020 Nature  583 221  \n[223]  Lee P A, Nagaosa N and Wen X -G 2006 Rev. Mod. Phys.  78 17  \n[224] Keimer B, Kivelson S A, Norman M R, Uchida S and Zaanen J 2015 Nature  518 \n179  \n[225] Scalapino D J 2012 Rev. Mod. Phys.  84 1383  \n[226] Gurvitch M and Fiory A T 1987 Phys. Rev. Lett.  59 1337  \n[227] Takagi H, Batlogg B, Kao H L, Kwo J, Cava R J, Krajewski J J and Peck W F \n1992 Phys. Rev. Lett.  69 2975  \n[228] Cooper R A, Wang Y , Vignolle B, Lipscombe O J, Hayden S M, Tanabe Y , Adachi \nT, Koike Y , Nohara M, Takagi H, Proust C and Hussey N E 2009 Science  323 603  \n[229] Jin K, Butch N P, Kirshenbaum K, Paglione J and Greene R L 2011 Nature  476 73  \n[230] Greene R L, Mandal P R, Poniatowski N R and Sarkar T 2020 Annu. Rev. Condens. \nMatter Phys.  11 213  \n[231] González J and Stauber T 2020 Phys. Rev. Let t. 124 186801  \n[232] Lawler M J, Fujita K, Lee J, Schmidt A R, Kohsaka Y , Kim C K, Eisaki H, Uchida \nS, Davis J C, Sethna J P and Kim E -A 2010 Nature  466 347  \n[233] Daou R, Chang J, LeBoeuf D, Cyr -Choinière O, Laliberté F, Doiron -Leyraud N, \nRamshaw B J, Liang R, Bonn D A, Hardy W N and Taillefer L 2010 Nature  463 519  \n[234] da Silva Neto E H, Aynajian P, Frano A, Comin R, Schierle E, Weschke E, Gyenis \nA, Wen J, Schneeloch J, Xu Z, Ono S, Gu G, Le Tacon M and Yazdani A 2014 \nScience  343 393  \n[235] Sato Y , Kasahara S, Murayama H, Kasahara Y , Moon E G, Nishizaki T, Loew T, \nPorras J, Keimer B, Shibauchi T and Matsuda Y 2017 Nat. Phys.  13 1074  \n[236] Hanaguri T, Lupien C, Kohsaka Y , Lee D H, Azuma M, Takano M, Takagi H and \nDavis J C 2004 Nature  430 1001  \n[237] Kohsaka Y , Taylor C, Fujita K, Schmidt A, Lupien C, Hanaguri T, Azuma M, \nTakano M, Eisaki H, Takagi H, Uchida S and Davis J C 2007 Science  315 1380  \n[238] Fradkin E, Kivelson S A, Lawler M J, Eisenstein J P and Mackenzie A P 2010 \nAnnu. Rev. Condens. Matter Phys.  1 153  \n[239] Fernandes R M and Venderbos J W F 2019 Sci. Adv.  6 eaba8834  \n[240] Chichinadze D V , Classen L and Chubukov A V 2020 Phys. Rev. B  101 224513  \n[241] Kozii V , Isobe H, Venderbos J W F and Fu L 2019 Phys. Rev. B  99 144507  \n[242] Yang S L, Sobota J A, Howard C A, Pickard C J, Hashimoto M, Lu D H, Mo S \nK, Kirchmann P S and Shen Z X 2014 Nat. Commun.  5 3493  \n[243] Ichinokura S, Sugawara K, Takayama A, Takahashi T and Hasegawa S 2016 ACS \nNano  10 2761  \n[244] Chapman J, Su Y , Howard C A, Kundys D, Grigorenko A N, Guinea F, Geim A \nK, Grigorieva I V and Nair R R 2016 Sci. Rep.  6 23254  \n[245] Weller T E, Ellerby M, Saxena S S, Smith R P and Skipper N T 2005 Nat. Phys.  \n1 39  [246] Margine E R, Lambert H and Giustino F 2016 Sci. Rep.  6 21414  \n[247] Wu F and Das Sarma S 2019 Phys. Rev. B  99 220507(R)  \n[248] Wu F 2019 Phys. Rev. B  99 195114  \n[249] Chang C -Z, Zhang J, Feng X, et al. 2013 Science 340 167  \n[250] Kou X, Guo S -T, Fan Y , Pan L, Lang M, Jiang Y , Shao Q, Nie T, Murata K, Tang \nJ, Wang Y , He L, Lee T -K, Lee W -L and Wang K L 2014 Phys. Rev. Lett.  113 137201  \n[251] Chang C -Z, Zhao W, Kim D Y , Zhang H, Assaf B A, Heiman D, Zhang S -C, Liu  \nC, Chan M H W and Moodera J S 2015 Nat. Mater.  14 473  \n[252] Lee I, Kim C K, Lee J, Billinge S J L, Zhong R, Schneeloch J A, Liu T, Valla T, \nTranquada J M, Gu G and Davis J C S 2015 Proc. Natl. Acad. Sci.  112 1316  \n[253] Mogi M, Yoshimi R, Tsukazaki A, Ya suda K, Kozuka Y , Takahashi K S, Kawasaki \nM and Tokura Y 2015 Appl. Phys. Lett.  107 182401  \n[254] Yasuda K, Mogi M, Yoshimi R, Tsukazaki A, Takahashi K S, Kawasaki M, \nKagawa F and Tokura Y 2017 Science  358 1311  \n[255] Rozhkov A, Sboychakov A, Rakhmanov A a nd Nori F 2016 Phys. Rep.  648 1  \n[256] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 \nRev. Mod. Phys.  81 109  \n[257] McCann E and Koshino M 2013 Rep. Prog. Phys.  76 056503  \n[258] Burg G W, Zhu J, Taniguchi T, Watanabe K, MacDonald A H and Tutuc E 2019 \nPhys. Rev. Lett.  123 197702  \n[259] Cao Y , Rodan -Legrain D, Rubies -Bigorda O, Park J M, Watanabe K, Taniguchi \nT and Jarillo -Herrero P 2020 Nature  583 215  \n[260] Shen C, Chu Y , Wu Q, Li N, Wang S, Zhao Y , Tang J, Liu J, Tian J, Watanabe K, \nTaniguchi T, Yang R, Meng Z Y , Shi D, Yazyev O V and Zhang G 2020 Nat. Phys.  \n16 520  \n[261] He M, Li Y , Cai J, Liu Y , Watanabe K, Taniguchi T, Xu X and Yankowitz M 2020 \narXiv:2002.08904   \n[262] Adak P C, Sinha S, Ghorai U, Sangani L D V , Watanabe K, Taniguchi T, Sensarma \nR and Deshmukh M M 2020 Phys. Rev. B  101 125428  \n[263] Chebrolu N R, Chittari B L and Jung J 2019 Phys. Rev. B  99 235417  [264] Choi Y W and Choi H J 2019 Phys. Rev. B  100 201402(R)  \n[265] Lee J Y , Khalaf E, Liu S, Liu X, Hao Z, Kim P and Vishwanath A 2019 Nat. \nCommun.  10 5333  \n[266] Wu F and Das Sarma S 2020 Phys. Rev. B  101 155149  \n[267] Culchac F J, Del Grande R, Capaz R B, Chico L and Morell E S 2020 Nanosc ale \n12 5014  \n[268] Samajdar R and Scheurer M S 2020 Phys. Rev. B  102 064501  \n[269] Haddadi F, Wu Q, Kruchkov A J and Yazyev O V 2020 Nano Lett.  20 2410  \n[270] Kerelsky A, Rubio -Verdú C, Xian L, Kennes D M, Halbertal D, Finney N, Song \nL, Turkel S, Wang L an d Watanabe K 2019 arXiv:1911.00007   \n[271] Lee K, Utama M, Kahn S, Samudrala A, Leconte N, Yang B, Wang S, Watanabe \nK, Taniguchi T, Zhang G, Weber -Bargioni A, Crommie M, Ashby P D, Jung J, Wang \nF and Zettl A 2020 arXiv:2006.04000   \n[272] Halbertal D, Finney N R, Sunku S S, Kerelsky A, Rubio -Verdú C, Shabani S, \nXian L, Carr S, Chen S and Zhang C 2020 arXiv:2008.04835   \n[273] Min H and MacDonald A H 2008 Prog. Theor. Phys. Supp.  176 227  \n[274] Latil S and Henrard L 2006 Phys. Rev. Lett.  97 036803  \n[275] Mak K F,  Shan J and Heinz T F 2010 Phys. Rev. Lett.  104 176404  \n[276] Yan J -A, Ruan W Y and Chou M Y 2011 Phys. Rev. B  83 245418  \n[277] Wang W, Shi Y , Zakharov A A, Syväjärvi M, Yakimova R, Uhrberg R I G and \nSun J 2018 Nano Lett.  18 5862  \n[278] Yin L -J, Shi L -J, Li S-Y , Zhang Y , Guo Z -H and He L 2019 Phys. Rev. Lett.  122 \n146802  \n[279] Bao W, Jing L, Velasco J, Lee Y , Liu G, Tran D, Standley B, Aykol M, Cronin S \nB, Smirnov D, Koshino M, McCann E, Bockrath M and Lau C N 2011 Nat. Phys.  7 \n948  \n[280] Lui C H, Li Z, Mak  K F, Cappelluti E and Heinz T F 2011 Nat. Phys.  7 944  \n[281] Scherer M M, Uebelacker S, Scherer D D and Honerkamp C 2012 Phys. Rev. B  \n86 155415  \n[282] Yin L -J, Wang W -X, Zhang Y , Ou Y -Y , Zhang H -T, Shen C -Y and He L 2017 Phys. Rev. B  95 081402(R)  \n[283] Lee Y , Che S, Jr. J V , Tran D, Baima J, Mauri F, Calandra M, Bockrath M and \nLau C N 2019 arXiv:1911.04450   \n[284] Chen G, Jiang L, Wu S, Lyu B, Li H, Chittari B L, Watanabe K, Taniguchi T, Shi \nZ, Jung J, Zhang Y and Wang F 2019 Nat. Phys.  15 237  \n[285] Chen G, Sharpe A L, Fox E J, Zhang Y -H, Wang S, Jiang L, Lyu B, Li H, \nWatanabe K, Taniguchi T, Shi Z, Senthil T, Goldhaber -Gordon D, Zhang Y and Wang \nF 2020 Nature  579 56  \n[286] Chittari B L, Chen G, Zhang Y , Wang F and Jung J 2019 Phys. Rev. Lett.  122 \n016401  \n[287] Zhang Y -H and Senthil T 2019 Phys. Rev. B  99 205150  \n[288] Pantaleon P A, Cea T, Brown R, Walet N R and Guinea F 2020 arXiv:2003.05050   \n[289] Scheurer M S and Samajdar R 2020 Phys. Rev. Research  2 033062  \n[290] Zhang Y -H and Senthil T 2020 arXiv:2003.13702   \n[291] Qiao J -B and He L 2014 Phys. Rev. B  90 075410  \n[292] Chen S, He M, Zhang Y -H, Hsieh V , Fei Z, Watanabe K, Taniguchi T, Cobden D \nH, Xu X and Dean C R 2020 arXiv:2004.11340   \n[293] Polshyn H, Zhu J, Kumar M A, Zhang Y , Yang F, Tschirha rt C L, Serlin M, \nWatanabe K, Taniguchi T, MacDonald A H and Young A F 2020 arXiv:2004.11353   \n[294] Shi Y , Xu S, Ezzi M M A, Balakrishnan N, Garcia -Ruiz A, Tsim B, Mullan C, \nBarrier J, Xin N, Piot B A, Taniguchi T, Watanabe K, Carvalho A, Mishchenko A, \nGeim A K, Fal'ko V I, Adam S, Neto A H C and Novoselov K S 2020 \narXiv:2004.12414   \n[295] Ma Z, Li S, Zheng Y -W, Xiao M -M, Jiang H, Gao J -H and Xie X C 2019 \narXiv:1905.00622   \n[296] Park Y , Chittari B L and Jung J 2020 arXiv:2005.01258   \n[297] Rademaker L, Protop opov I and Abanin D 2020 arXiv:2004.14964   \n[298] Li X, Wu F and MacDonald A H 2019 arXiv:1907.12338   \n[299] Carr S, Li C, Zhu Z, Kaxiras E, Sachdev S and Kruchkov A 2020 Nano Lett.  20 \n3030  [300] Mora C, Regnault N and Bernevig B A 2019 Phys. Rev. Lett.  123 026402  \n[301] Tsai K -T, Zhang X, Zhu Z, Luo Y , Carr S, Luskin M, Kaxiras E and Wang K 2019 \narXiv:1912.03375   \n[302] Zhu Z, Carr S, Massatt D, Luskin M and Kaxiras E 2020 arXiv:2006.00399   \n[303] Zuo W -J, Qiao J -B, Ma D -L, Yin L -J, Sun G, Zhang J -Y , Guan L -Y and He L \n2018 Phys. Rev. B  97 035440  \n[304] Liu J, Ma Z, Gao J and Dai X 2019 Phys. Rev. X  9 031021  \n[305] Zhang Y -H, Mao D, Cao Y , Jarillo -Herrero P and Senthil T 2019 Phys. Rev. B  99 \n075127  \n[306] Ma Z, Li S, Xiao M -M, Zheng Y -W, Lu M, Liu H, Gao J -H an d Xie X 2020 \narXiv:2001.07995   \n[307] Decker R, Wang Y , Brar V W, Regan W, Tsai H -Z, Wu Q, Gannett W, Zettl A and \nCrommie M F 2011 Nano Lett.  11 2291  \n[308] Xue J, Sanchez -Yamagishi J, Bulmash D, Jacquod P, Deshpande A, Watanabe K, \nTaniguchi T, Jarillo -Herrero P and LeRoy B J 2011 Nat. Mater.  10 282  \n[309] Yankowitz M, Xue J and LeRoy B J 2014 J. Phys.: Condens. Matter  26 303201  \n[310] Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, \nTaniguchi T, Kim P, Shepard K L and Hone J 2010 Nat. Nanotech.  5 722  \n[311] Yankowitz M, Ma Q, Jarillo -Herrero P and LeRoy B J 2019 Nat. Rev. Phys.  1 112  \n[312] Kim Y , Balram A C, Taniguchi T, Watanabe K, Jain J K and Smet J H 2019 Nat. \nPhys.  15 154  \n[313] Zhou H, Polshyn H, Taniguchi T, Watanabe K and Young A F 2020 Nat. Phys.  16 \n154  \n[314] Yin J, Slizovskiy S, Cao Y , Hu S, Yang Y , Lobanova I, Piot B A, Son S -K, \nOzdemir S, Taniguchi T, Watanabe K, Novoselov K S, Guinea F, Geim A K, Fal’ko \nV and Mishch enko A 2019 Nat. Phys.  15 437  \n[315] Amet F, Bestwick A J, Williams J R, Balicas L, Watanabe K, Taniguchi T and \nGoldhaber -Gordon D 2015 Nat. Commun.  6 5838  \n[316] Chiappini F, Wiedmann S, Novoselov K, Mishchenko A, Geim A K, Maan J C \nand Zeitler U 2015 Phys. Rev. B  92 201412(R)  [317] Young A F, Sanchez -Yamagishi J D, Hunt B, Choi S H, Watanabe K, Taniguchi \nT, Ashoori R C and Jarillo -Herrero P 2014 Nature  505 528  \n[318] Lee D S, Skákalová V , Weitz R T, von Klitzing K and Smet J H 2012 Phys. Rev. \nLett. 109 056602  \n[319] Yankowitz M, Xue J, Cormode D, Sanchez -Yamagishi J D, Watanabe K, \nTaniguchi T, Jarillo -Herrero P, Jacquod P and LeRoy B J 2012 Nat. Phys.  8 382  \n[320] Gorbachev R V , Song J C W, Yu G L, Kretinin A V , Withers F, Cao Y , Mishchenko \nA, Grigorie va I V , Novoselov K S, Levitov L S and Geim A K 2014 Science  346 448  \n[321] Wang E, Lu X, Ding S, Yao W, Yan M, Wan G, Deng K, Wang S, Chen G, Ma L, \nJung J, Fedorov A V , Zhang Y , Zhang G and Zhou S 2016 Nat. Phys.  12 1111  \n[322] Chen Z -G, Shi Z, Yang W, Lu  X, Lai Y , Yan H, Wang F, Zhang G and Li Z 2014 \nNat. Commun.  5 4461  \n[323] Hunt B, Sanchez -Yamagishi J D, Young A F, Yankowitz M, LeRoy B J, Watanabe \nK, Taniguchi T, Moon P, Koshino M, Jarillo -Herrero P and Ashoori R C 2013 Science  \n340 1427  \n[324] Ponomare nko L A, Gorbachev R V , Yu G L, Elias D C, Jalil R, Patel A A, \nMishchenko A, Mayorov A S, Woods C R, Wallbank J R, Mucha -Kruczynski M, Piot \nB A, Potemski M, Grigorieva I V , Novoselov K S, Guinea F, Fal’ko V I and Geim A \nK 2013 Nature  497 594  \n[325] Dean C R, Wang L, Maher P, Forsythe C, Ghahari F, Gao Y , Katoch J, Ishigami \nM, Moon P, Koshino M, Taniguchi T, Watanabe K, Shepard K L, Hone J and Kim P \n2013 Nature  497 598  \n[326] Ni G X, Wang H, Wu J S, Fei Z, Goldflam M D, Keilmann F, Özyilmaz B, Castro \nNeto A H, Xie X M, Fogler M M and Basov D N 2015 Nat. Mater.  14 1217  \n[327] Dai S, Ma Q, Liu M K, Andersen T, Fei Z, Goldflam M D, Wagner M, Watanabe \nK, Taniguchi T, Thiemens M, Keilmann F, Janssen G C A M, Zhu S E, Jarillo -\nHerrero P, Fogler M M and Basov D N 201 5 Nat. Nanotech.  10 682  \n[328] Wang L, Zihlmann S, Liu M -H, Makk P, Watanabe K, Taniguchi T, Baumgartner \nA and Sch önenberger C 2019 Nano Lett.  19 2371  \n[329] Finney N R, Yankowitz M, Muraleetharan L, Watanabe K, Taniguchi T, Dean C R and Hone J 2019 Nat. N anotech.  14 1029  \n[330] Kim K, Yankowitz M, Fallahazad B, Kang S, Movva H C P, Huang S, Larentis S, \nCorbet C M, Taniguchi T and Watanabe K 2016 Nano Lett.  16 1989  \n[331] Ribeiro -Palau R, Zhang C, Watanabe K, Taniguchi T, Hone J and Dean C R 2018 \nScience  361 690  \n[332] Yu L, Lee Y -H, Ling X, Santos E J G, Shin Y C, Lin Y , Dubey M, Kaxiras E, \nKong J, Wang H and Palacios T 2014 Nano Lett.  14 3055  \n[333] Massicotte M, Schmidt P, Vialla F, Schädler K G, Reserbat -Plantey A, Watanabe \nK, Taniguchi T, Tielrooij K J and Koppens F H L 2016 Nat. Nanotech.  11 42  \n[334] Roy K, Padmanabhan M, Goswami S, Sai T P, Ramalingam G, Raghavan S and \nGhosh A 2013 Nat. Nanotech.  8 826  \n[335] Kośmider K, González J W and Fernández -Rossier J 2013 Phys. Rev. B  88 245436   \n[336] Liu G -B, Shan W -Y , Yao Y , Yao W and Xiao D 2013 Phys. Rev. B  88 085433  \n[337] Yuan H, Bahramy M S, Morimoto K, Wu S, Nomura K, Yang B -J, Shimotani H, \nSuzuki R, Toh M, Kloc C, Xu X, Arita R, Nagaosa N and Iwasa Y 2013 Nat. Phys.  \n9 563  \n[338] Zeng H, L iu G-B, Dai J, Yan Y , Zhu B, He R, Xie L, Xu S, Chen X, Yao W and \nCui X 2013 Sci. Rep.  3 1608  \n[339] Zhang Y , Chang T -R, Zhou B, Cui Y -T, Yan H, Liu Z, Schmitt F, Lee J, Moore R, \nChen Y , Lin H, Jeng H -T, Mo S -K, Hussain Z, Bansil A and Shen Z -X 2014 Nat. \nNanotech.  9 111  \n[340] Zhu Z Y , Cheng Y C and Schwingenschlögl U 2011 Phys. Rev. B  84 153402  \n[341] Yang B, Tu M -F, Kim J, Wu Y , Wang H, Alicea J, Wu R, Bockrath M and Shi J \n2016 2D Mater.  3 031012  \n[342] Avsar A, Tan J Y , Taychatanapat T, Balakrishnan J, K oon G K W, Yeo Y , Lahiri J, \nCarvalho A, Rodin A S, O’Farrell E C T, Eda G, Castro Neto A H and Özyilmaz B \n2014 Nat. Commun.  5 4875  \n[343] Yang B, Lohmann M, Barroso D, Liao I, Lin Z, Liu Y , Bartels L, Watanabe K, \nTaniguchi T and Shi J 2017 Phys. Rev. B  96 041409  \n[344] Antonio Benítez L, Sierra J F, Savero Torres W, Arrighi A, Bonell F, Costache M V and Valenzuela S O 2018 Nat. Phys.  14 303  \n[345] Ghiasi T S, Ingla -Aynés J, Kaverzin A A and van Wees B J 2017 Nano Lett.  17 \n7528  \n[346] Omar S, Madhushankar B N and van Wees B J 2019 Phys. Rev. B  100 155415  \n[347] Wang Z, Ki D -K, Khoo J Y , Mauro D, Berger H, Levitov L S and Morpurgo A F \n2016 Phys. Rev. X  6 041020  \n[348] Zihlmann S, Cummings A W, Garcia J H, Kedves M, Watanabe K, Tani guchi T, \nSchönenberger C and Makk P 2018 Phys. Rev. B  97 075434  \n[349] Wakamura T, Reale F, Palczynski P, Guéron S, Mattevi C and Bouchiat H 2018 \nPhys. Rev. Lett.  120 106802  \n[350] Island J O, Cui X, Lewandowski C, Khoo J Y , Spanton E M, Zhou H, Rhodes D, \nHone J C, Taniguchi T, Watanabe K, Levitov L S, Zaletel M P and Young A F 2019 \nNature  571 85  \n[351] Zaletel M P and Khoo J Y 2019 arXiv:1901.01294   \n[352] Xi X, Wang Z, Zhao W, Park J -H, Law K T, Berger H, Forró L, Shan J and Mak \nK F 2016 Nat. Phys.  12 139  \n[353] Guillamón I, Suderow H, Vieira S, Cario L, Diener P and Rodière P 2008 Phys. \nRev. Lett.  101 166407  \n[354] Galvis J A, Rodière P, Guillamon I, Osorio M R, Rodrigo J G, Cario L, Navarro -\nMoratalla E, Coronado E, Vieira S and Suderow H 2013 Phys. Rev. B  87 094502  \n[355] Sipos B, Kusmartseva A F, Akrap A, Berger H, Forró L and Tutiš E 2008 Nat. \nMater.  7 960  \n[356] Lu J M, Zheliuk O, Leermakers I, Yuan N F Q, Zeitler U, Law K T and Ye J T \n2015 Science  350 1353  \n[357] de la Barrera S C, Sinko M R, Gopalan D  P, Sivadas N, Seyler K L, Watanabe K, \nTaniguchi T, Tsen A W, Xu X, Xiao D and Hunt B M 2018 Nat. Commun.  9 1427  \n[358] Gani Y S, Steinberg H and Rossi E 2019 Phys. Rev. B  99 235404  \n[359] Li J, Leng H -B, Fu H, Watanabe K, Taniguchi T, Liu X, Liu C -X and Zhu J 2020 \nPhys. Rev. B  101 195405  \n[360] Jiang Y , Anđelković M, Milovanović S P, Covaci L, Lai X, Cao Y , Watanabe K, Taniguchi T, Peeters F M and Geim A K 2019 arXiv: 1904.10147   \n \n \nFig. 1 Abundant physical phenomena in TBG.  On the logarithmic axis of twist angle, \nthe blue regions correspond to the  commensurate  angle, and the red regions correspond \nto the incommensurate  angle. The arrow points out several typical physical phenomena, \nthe angles from small to large correspond to: Aharonov –Bohm oscillations along the \ntriangular network of AB/BA domain walls (θ ~ 0.1° ) [102], topologically protected \nhelical edge states  on the domain wall network ( θ ~ 0.245° ) [100], strongly correlated \nphase s for the partially filled moiré flat minibands on MATBG systems ( θ ~ 1.1° ) [67, \n143], the higher -order topological insulator with topological corner states ( θ ~ 21.78° ) \n[114], the emergence of mirrored Dirac cones in  graphene quasicrystal ( θ ~ 30° )  [88, 89], \nflat bands caused by the electrons confined in a geometrically frustrated network of \ntopologically protected modes  (θ ~ 38.21° ) [137]. \n \n \n \n \nFig. 2 VHSs separation as a function of twist angle 𝜽. Solid colorful circles are the \nexperimental data measured in different TBG from different references  [20, 23 -28, 55 -60, 86]. \nError bars in energy represent the peak position change caused by slight doping change. \nThe inset shows the electronic band struct ure of TBG. Left: The first Brillouin zone of \nTBG with twist angle 𝜃 . K 1 and K 2 are the Dirac points of top and bottom layers. \nMiddle: Energy dispersion relation of the overlap of the two Dirac cones, giving rise to \ntwo VHSs which generate peaks in the DO S. Right: Diagram of the energy dependence \nof DOS near the Fermi level.  \n \n \n \n \n \n \nFig. 3 The sample differences of TBG in different references.  (a) A schematic \ndiagram of the MATBG band structure and the corresponding DOS showing VHS peaks. \nThe colored arrows point out three typical parameters: the band gap between the flat \nand the upper dispersive band ( ∆𝑔𝑎𝑝 marked by blue),  the full-width at hal f-maximum \nof the VHSs  (FWHM  marked by red), the energy separations of two VHSs ( ∆𝐸𝑉𝐻𝑆 \nmarked by green ). (b) VHSs separation ∆𝐸𝑉𝐻𝑆 as a function of twist angle 𝜃 near \nMA-TBG. Solid colorful circles  [23, 24,  53, 55-60] are the experimental data measured in \ndifferent TBG, and the chemical potential is in between the two VHSs. The open stars  \nsymbols  [20, 25, 53, 55, 56, 58, 62] represent data for the band becomes fully filled or fully \nunfilled. (c) The FWHM of the VHSs  as a function of twist angle 𝜃, each of VHS is \nfully occupied or unoccupied . [23-25, 55 -60]. Solid colorful circles represent the FWHM of \nconduction VHS. Open colorful circles represent FWHM of valence VHS. (d) and (e) \nThe energy gap ∆𝑔𝑎𝑝 between the flat bands and the higher -energy dispersive bands  \nas a function of twist angle 𝜃. (d) represents data for STM experiments  [55, 56]. (e) \nrepresents data for transport experiments  [61, 63 -65, 69, 115, 193]. Solid colorful circles \nrepresent the gap between the electron flat band and the upper dispersive bands. Open \ncolorful circles represent the gap between the hole flat band and the lower dispersive \nbands. Error bars in (b-e) originate from the slight dopin g change.  \n \n \nFig. 4 The STS spectra of TBG  when one of the flat bands is partially filled.  (a) \nThe schematic diagram of that the partially filled DOS peak in MATBG splits into two \npeaks.  (b) The partially filled flat band splits into four DOS peaks in a no n-magic -angle \nTBG with θ ~ 1.49º [59]. \n \n \n  \n \nFig. 5 Correlated insulator and superconductivity in the MATBG. Top panel:  \nConductance at the Fermi level as a function of filling factors ν in STM experiments.   \nGrey areas correspond to fully occupied or unoccupied flat bands. Green areas \ncorrespond to the CNP ( ν = 0). Twist angles include 1.05°  [54], 1.07°  [58], 0.99°  [56] and \n1.01°  [55]. Bottom panel:  Superconducting phase diagrams acquired from the transport \nmeasurements. Solid superconducting domes indicate the coexistence of \nsuperconductivity and insulating states at  ν = ±2. Twist angles include 1.14°  [78], 1.10°  \n[72], 1.05°  [68], 1.16°  [68], 1.27°  [78]. Dotted superconducting domes indicate that the \nsuperconductivity persists in the absence of the correlated insulators at  ν = ±2. Twist \nangles include 1.04°  [71], 1.15°  [71] and 1.18°  [70]. \n \n \n \nFig. 6  Spatially resolved conductance maps at the energies of the flat bands (VHSs) \nin MATBG. (a) Schematic atomic structure of the TBG with a twist angle θ. The AA \nstacking configurations are marked in the panel, exhibiting the C 6 rotational symmetry. \n(b) The conductance maps at the energies around the VHS peaks show the same features \nas topographies and maintain the C 6 symmetry when the two VHSs are fully fil led or \nempty. (c) The conductance maps at the energies around the VHS peaks show a \npronounced anisotropy in each moiré when the Fermi level lies in one of the VHSs, \nreducing the symmetry from the initial C 6 to C 2. \n \n \n \n \nFig. 7  Quantized anomalous Hall effec t in TBG (θ ~ 1.15º) at at ν =3.  (a) \nLongitudinal resistance 𝑅𝑥𝑥 and Hall resistance 𝑅𝑥𝑦 measured at B =150 mT and T \n= 1.6 K as a function of carrier density over the entire flat bands.  Near the ν =3, 𝑅𝑥𝑦 \napproaches ℎ𝑒2⁄ and 𝑅𝑥𝑥  reaches a deep minimum, illustrating the quantized \nanomalous Hall state.  (b) Magnetic -field-dependent  𝑅𝑥𝑥  and 𝑅𝑥𝑦  at ν =3. The \nsweep directions are indicated by the arrows, showing an obvious hysteresis  [64]. \n \nFig. 8  Schematic of the electrical -displacement -field -dependent low energy moiré \nbands and DOS in the TDBG. The two parabolic dispersive bands are originated from \nthe Bernal -stacked BLG. The two low -energy flat bands are separated from the high -\nenergy dispersing bands by the moiré gaps. The bandwidth of the flat bands W and the \nband gap Δ at the charge neutral point can be easily tuned via the perpendicular \nelectrical displacement field D. In a specific range of D, the correlated insulating states \nemerge when the isolated flat band is half filled, exhibiting the spin -polarized ordering.  \n"    },    {        "title": "1212.1591v1.Environmental_sensitivity_of_n_i_n_and_undoped_single_GaN_nanowire_photodetectors.pdf",        "content": " 1Environmental sensitivity of n-i-n  and undoped single GaN \nnanowire photodetectors  \nF González-Posada1,*, R Songmuang2, M Den Hertog2, and E Monroy1  \n1 CEA-CNRS Group “Nanophysique et Semi-conduc teurs”, INAC-SP2M, CEA-Grenoble, 17 rue \ndes Martyrs, 38054 Grenoble Cedex 9, France. \n2Institut Néel-CNRS, BP 166, 25 rue des Martyrs, 38042 Grenoble Cedex 9, France. \n*Presently at Applied Physics Department, Chal mers University of Technology, SE 412 96, \nGothenburg, Sweden. \nE-mail: fernando.gonzal ez.posada@gmail.com \n \nReceived data / Published / Online : \n \nAbstract \nIn this work, we compare the photodetector  performance of single defect-free undoped and n-i-\nn GaN nanowires (NWs). In vacuum, undoped NWs present a responsivity increment, \nnonlinearities and persistent photoconductivity eff ects (~ 100 s). Their unpinned Fermi level at \nthe m-plane NW sidewalls enhances the surface st ates role in the photodetection dynamics. Air \nadsorbed oxygen accelerates the carrier dynamics at  the price of reducing the photoresponse. In \ncontrast, in n-i-n  NWs, the Fermi level pinning at the contact regions limits the photoinduced \nsweep of the surface band bending, and hence re duces the environment sen sitivity and prevents \npersistent effects even in vacuum. \nKEYWORDS: Nanowire (62.23 Hj), GaN (77.84 Bw),  Photoconductivity (73.50 Pz), Ultraviolet  \nphotodetectors (42.79 Pw,85.60 Gz).  2Photodetectors based on semiconductor nanowires (NWs) picture new technological solutions \nfor quantum electronics,1 energy generation,2 optoelectronics,3 and sensorics.4,5 The photodetector \ncapabilities of single NWs are under study for a variety of materials,6 including GaN.7−10 For this \napplication, III-nitride NWs present advantages in terms of physical and chemical robustness and \nbio-compatibility,11  in addition to the photodetecting capabilities of a material family with a \ndirect bandgap tunable in the ultraviolet (UV)  / visible spectral range. We have previously \nreported that single n-i-n  NWs display photocurrent gain in the range of 105−108, with the \nphotocurrent scaling sublinearly with the excitation power.10 Their spectral response, relatively \nflat for excitation above the GaN bandgap (< 360 nm  wavelength), presents a visible rejection of \nmore than six orders of magnitude, and their photocurrent time response is in the millisecond \nrange. Although these results confirm that the photor esponse is dominated by the redistribution of \ncharge at surface levels, these n-i-n  GaN NWs are insensitive to the measuring environment. \nGiven the large surface-to-volume ratio that ch aracterizes the NW geometry, understanding the \nsurface dynamics becomes a fundamental issue. From photoluminescence (PL) measurements in \nGaN-based NWs,12,13 it is generally accepted that adsorbed oxygen reduces the nonradiative \nrecombination time, resulting in a faster quenching of the NW emission in the air in comparison to experiments in vacuum, as also observed in GaN 2D layers\n14 and ZnO NWs.15 However, \nresults become contradictory if we look at th e photodetection properties: Some authors have \nstated that the photocurrent from GaN NWs is  insensitive to the air/vacuum atmosphere,10,16 \nwhereas other groups report variations of th e photocurrent in different gas environments.8,17  \nIn this work, we compare the photodetector  performance of single defect-free undoped and n-i-\nn GaN NWs paying particular attention to the effe ct of the measuring environment (in vacuum or \nin the air). We conclude that the environment insensitivity of n-i-n  NWs with ohmic contacts is \nassociated to the pinning of the Fermi level at the NW edges. In contrast, undoped NWs with Schottky contacts are significantly sensitive to the measurement atmo sphere: In the air, adsorbed \noxygen accelerates the surface processes, whereas in vacuum, due to the longer recombination \ntime associated to the surface states, the behavi our of single NW photoconductors approaches the \none of two-dimensional (2D) GaN photodetectors.  3GaN NWs were grown by plasma-assisted molecular-beam epitaxy (PAMBE) on Si (111) \nunder nitrogen-rich conditions, at a substrate temperature TS ~ 790 ºC.18 The NWs are defect free, \nthey have a length of 1.2–1.5 μm and a diameter of 50 −80 nm, and they exhibit N-polarity. The \nNWs under study are either undoped or n-i-n  structures, the latter consisting of two Si-doped n-\ntype edges and an undoped middle section with a nominal length of 400 nm. The residual doping \nin the PAMBE chamber is in the range of ~ 1×1017 cm-3 for N-polar GaN. For n-type doping, the \nSi cell temperature was tuned to the value th at yields a Hall electron concentration of \n~ 2 × 1019 cm-3 in 2D GaN layers. \nDue to the large surface-to-volume ratio, the fr ee carrier distribution in NWs is strongly \naffected by surface states. Applying the abrupt depletion approximation to a cylindrical \nstructure,19−21 total depletion of the NW occurs for a diameter below \ndcNed2016 Φ=εε \nwhere Φ is the surface potential (conducti on band edge - Fermi level) and Nd the doping density. \nTaking into account the diameter of the NWs under study, the typical values of surface potential \nin the literature (0.3 eV < Φ < 0.6 eV),22−24 and the residual doping in our system, the undoped \nsections of the NWs should be fully depleted.  \nTo perform the characterization of single obj ects, NWs were mechanically dispersed on \nheavily-doped Si substrates capped with a 280 nm thick SiO 2 layer. Single NWs were contacted \nusing e-beam lithography, e-beam metallization and lift-off tec hnique. The electrodes consisted \nof Ti/Al/Ti/Au (5/25/15/100 nm) for n-i-n  NWs and Ni/Au (10/100 nm) for undoped NWs, in \norder to fabricate the ohmic and Schottky contacts,  respectively. The right side of Figure 1 shows \ntop-view scanning electron microscopy (SEM) images of n-i-n (top) and undoped (bottom) \ncontacted NWs. \nCurrent-voltage ( I−V) characteristics were recorded us ing an Agilent 4155C semiconductor \nparameter analyzer. Photocurrent meas urements were performed under bias ( Vb) with the n-i-n   4NWs connected to a load resistance ( RL = 100 – 180 k Ω),10 and the undoped NWs connected to a \nPDA200C current amplifier. Unless indicated , measurements were performed under chopped \nexcitation using a TDS2022C oscilloscope or to an SR830 lock-in amplifier (input resistance = \n10 M Ω). The spectral response was characterized using a 450 W Xe-arc lamp coupled to a \nGemini 180 monochromator. At the exit, the light was directed with an optic fiber and focused \nonto the NW device. The variation of the photocurrent as a function of the optical power was \nanalyzed using an Ar+ laser (wavelength λ = 244 nm, 488 nm) as excitation source. All the \nmeasurements presented in this work were performed at room temperature. \nFigure 1 (left) compares typical I−V characteristics of single n-i-n  and undoped NWs in the \ndark (measured in the air and in vacuum) and under UV illumination (in the air). Regarding the \ndark current, we observe a strong sensitivity to the atmosphere: Measurements in the air render \ncurrent levels one order of magnitude lower than in vacuum, in agreement with previous reports8 \nand with results in CdS NWs.25 In the case of n-i-n  NWs, the dark current displays the  I ∼ V α \n(α ≥ 2) behavior characteristic of space-charge-lim ited transport, as expected due to the surface-\ninduced depletion of the non-intentionally-dope d NW sections. In the literature, space-charge \nlimited current is observed for Ga N NWs with diameters below 200 nm,7,21 as also reported for \nCdS,25 GaAs,26 or ZnO NWs.6 On the other hand, undoped NWs present dark current densities at \nleast two orders of magnitude lower than n-i-n  devices. Their approximately linear I−V \ncharacteristic is explained by the trap filling pr ocess characteristic of insulators prior to the \napplication of the voltage.27 Under illumination, undoped NWs display larger on/off current \ncontrast in comparison to n-i-n  NWs. However, the overall photocurrent level in undoped NWs is \nsmaller than the one of n-i-n  NWs by at least one order of magnitude. \n Figure 2 presents on the left side the variation of the photocurrent as a function of the laser \nexcitation power for an undoped NW m easured in vacuum (pressure < 10\n-4 mbar) and in the air. \nThe right side of figure 2 describes the photodetector gain28 as a function of the measurement  5frequency for n-i-n  and undoped NWs in vacuum and in the air. In comparison to n-i-n  devices, \nundoped NWs display lower photocurrent gain. In both cases, the photocurrent scales sublinearly \nwith the optical power following an I ~ P β law with β < 1.10 Statistically, the sublinear trend is \nless marked in the case of undoped NWs, with higher values of β (0.4 − 0.6 in undoped NWs and \n0.2 − 0.5 in n-i-n  NWs).29 Furthermore, undoped NWs present a high sensitivity to the \nmeasurement atmosphere: Whereas for n-i-n  NWs photocurrent measurements in vacuum and in \nthe air are in the same range (±5 %), for undoped NWs the photocurrent measured in vacuum can \nbe more than one order of magnitude higher than  in the air. Despite the enhancement of the \nphotocurrent in vacuum, its sublinear behavior with the optical power remains unchanged, i.e. β \nremains constant within an error bar ±5 %.  A similar enhancement of the photoresponse in \nvacuum has been reported in ZnO NWs, 30 and in GaN NWs grown along the m-axis.17 \nFigure 3 compares the spectral response of an undoped NW photodetector in the air and in \nvacuum. In the air, the spectral response of n-i-n and undoped NWs is similar, approximately \nconstant for wavelengths below 360 nm and displaying a visible rejection of more than three \norders of magnitude, in agreement with the data presented of Bertness et al. ,8 although some \nprevious reports show distinct photoc urrent features below the GaN band gap.31,32 To verify the \nvisible blindness, the NWs were exposed to 1 W of a 488 nm Ar+ laser without observing any \nphotoresponse, whatever the NW or the measuring atmosphere. These results are in contradiction \nwith a previous report which shows a degrada tion of the GaN NW UV/visible contrast under \nvacuum,8 although the presence and role of persistent effects in that measurement were not \nindicated. In our case, the enhancement of th e photoresponse of undoped NWs in vacuum is \naccompanied by a shape modification of the spectral response at short wavelengths (energies \nabove the GaN bandgap) with the appearance of a second peak around 280 nm. This difference is \nunderstood by taking into account the high abso rption coefficient at short wavelengths, which \nrestricts the light penetration to a few nanometers,  thus increasing the sens itivity to surface states \nand adsorbed impurities.  6 The variation of the gain with the chopping frequency displayed in Figure 2(right) points out \nthe presence of slow phenomena . To clarify this issue, time response measurements were \nperformed under 3 V bias exciting with a Xe arc lamp ( λ = 350 nm). Figure 4 presents \nphotocurrent decays recorded after ~2 min of c ontinuous excitation. As previously reported, n-i-n  \nNWs present nonexponential dynamics with an initial decay constant around 4-10 ms.10 The \ndecay becomes faster and increasingly nonexpo nential for higher excitation power, but no \nsignificant difference is observed as a function of  the NW measuring environment, either in \nvacuum or in the air. In the case of undoped NWs, their behavior is similar to n-i-n  devices when \nthey are operated in the air. However, persis tent photoconductivity effects are activated under \nvacuum in the range of tens-hundreds of seconds. These persistent effects approach the behavior \nof 2D GaN photoconductors.33 \nIn summary, comparing undoped and n-i-n  GaN NW photodetectors, in both devices we find a \ndecrease of the dark current in the air. Under illu mination, they display a similar spectral response \nwith large UV/visible contrast. However, undope d NWs present a dark current three orders of \nmagnitude lower than n-i-n  structures, a gain of about one order of magnitude lower, and a strong \ndependence of the measurement environment: In vacuum, undoped NWs react with an \nenhancement of the photoresponse, accompanie d by stronger nonlinearities, and persistent \nphotoconductivity effects in the ra nge of tens-hundreds of seconds.  \nTo understand these results, we must keep in mind that the behavior of GaN NWs is dominated \nby the m-face sidewalls. Theoretical and experimental studies show that the Fermi level is \nunpinned in clean m-plane GaN surfaces, and their more stable  atomic configuration is Ga-\nterminated surface [34,35]. Therefore, the Ga N NW sidewalls show three Ga dangling bonds, \nfrom its inner core (4s2p1), that do not find the subsequent N layer to finish the sp3 hybridization. \nIn the air, adsorbed oxygen passivates the surf ace reducing the amount of charges available for \nconduction and modifying the radial potential prof ile in the NW. Furthermore, it is generally  7accepted that adsorbed oxygen results  in a decrease of the carrier lifetime that can be measured by \nPL.12,13 \nUnder illumination, we observe a variation of the NW conductivity Δσ. Keeping in mind that \nthe conductivity is given by σ = en μ, where e is the electronic charge, n is the carrier density, and \nμ is the carrier mobility, a change in conductivity, Δσ = e(μΔn+nΔμ), can occur either due to \nchange in the carrier concentration, Δn, or to a change in the carrier mobility, Δμ . The carrier \nconcentration should scale linearly with the excitation, in contrast with the observed nonlinear \nbehaviour of the photocurrent. Moreover, the carri er lifetime measured by PL in the ns range12 is \nin contradiction with the photocurrent decay times described above (millisecond times for n-i-n  \nNWs and significantly longer for undoped NWs). We can hence conclude that the photodetector \nresponse is dominated by Δμ, which is affected by the surface  band bending and by the scattering \nassociated to surface states. When light is switche d off, the photogenerated carriers recombine at \na rate that is accelerated in presence of oxygen. However, the current recovery involves also a \nrearrangement of the surface charge, which is at the origin of the slow photocurrent components. \nThe different photoresponse of n-i-n  NWs and undoped NWs is assigned to the different \nlocation and behavior of the Fermi level at th e surface. Surface states play a major role on the \nundoped NW photocurrent dynamics due to th e unpinned Fermi level. In the case of n-i-n  NWs, \nthe pinning of the Fermi level close to the conduction band, due to the n-regions, and the \nreduction of the m-plane surface potential, due to residual silicon on the sidewalls in the undoped \nregions, reduce the photoinduced sweep of the Ferm i level, preventing persistent effects and \nreducing the environment sensitivity. Therefore, an important projection of the present work is \nthat the doping profile of the NW is a critical pa rameter to determine not only its performance as \na photodetector, but also its functionalization capabilities for chemical sensor applications. \n \nIn conclusion, single GaN NWs present high phot ocurrent gain, with th e photocurrent scaling \nsublinearly with the excitation power, and with a spectral selectivity larger than six orders of  8magnitude. In the case of undoped NWs with Schot tky contacts, the unpinned Fermi level at the \nNW sidewalls enhances the role of surface stat es in the photodetection process. In vacuum, the \ndesorption of oxygen from the sidewalls slows dow n the recovery process, which results in higher \nresponsivity associated to persistent photoconduc tivity effects. In contrast, in the case of n-i-n  \nNWs, the pinning of the Fermi level at the contact regions limits the photoinduced sweep of the \nsurface band bending, thus reducing the sensitivity to the environment and preventing persistent \neffects even in vacuum.  \nWe thank J. Dussaud, Y. Curé, and Y. Genuist for their technical support.  Partial financial \nsupport from ANR-2011-NANO-027 “UVLamp” and EU  ERC-StG « TERAGAN » (#278428) is \nacknowledged. F. G-P thanks Prof. P. Lefebvre for sharing the ref. 13. \n  9 \n0.1 110-1210-1010-810-6\nVoltage (V)            n −i−n  Undoped\nDark in air\nDark vacuum  \nLight in airCurrent (A)I~V 2\nI~V\n \n \n \n \n \nFigure 1. Left: I−V characteristics of single n-i-n  and undoped NWs measured in the dark (both in \nvacuum and in the air) and under UV illumination (in air). Black symbols correspond to the dark \ncurrent; red symbols correspond to illumination with ~ 0.2 W/cm2 of UV light ( λ = 244 nm). \nRight: Top-view SEM image of an n-i-n  (top) and undoped (bottom) contacted single NW.  10 \n0.1 1 10 10010-1010-910-810-7\n1 10 100 1000101102103104105106 Photocurrent (A)Undoped NW\nin vacuum\nin the air\nPower (mW/ cm2) \nGain Vac.   Air\nn-i-n NWs\nUndoped NWs\nFrequency (Hz)  \n \n \nFigure 2. Left: Photocurrent variation as a f unction of the excitation power for a single undoped \nNW in vacuum and in the air. Measurements we re made at 3 V bias and 11 Hz frequency. Right: \nVariation of the gain as a function of the measuring frequency of single n-i-n  and undoped NWs \nunder 3 V. The excitation power was 8 mW/cm2. Dotted lines are guides for the eye. \n  11 \n250 300 350 400Undoped NW\n In vacuum\n In the air\nWavelength (nm) Intensity (arb. units)\n  \n \nFigure 3. Spectral response of a single undoped NW  in vacuum and in the air. Measurements \nwere performed under 3 V bias, with a chopping  frequency of 11 Hz, and scanning from long to \nshort wavelengths.  12 \n100 150 200 2500.010.11 Undoped NW in vacuum\n n-i-n NW in vacuum\n Undoped NW in the air\n Normalized Photocurrent\nTime (s) \n \n \nFigure 4. Photocurrent decays from single n-i-n  and undoped NWs under 3 V bias after \nillumination with a Xe arc lamp ( λ = 350 nm).  \n  13 \nReferences \n1 Y. Li, F. Qian, J. Xiand and C. M. Lieber, Materials Today  9, 18 (2006). \n2 P. Hiralal, H. E. Unalan and G. A. J. Amaratunga, Nanotechnology 23, 194002 (2012). \n3 G.-C. Yi (Editor) Semiconductor nanostructur es for optoelectronic devices, NanoScience and \nTechnology, DOI 10.1007/978-3-642-22480-5 11 (S pringer-Verlag Berlin: Heidelberg)  pp 1 −37 \n(2012). \n4 G. S. Aluri, A. Motayed, A. V. Davydov, V. P. Oleshko, K. A. Bertness, N. A. Sanford and R. \nV. Mulpuri, Nanotechnology 23, 175501 (2012). \n5 R.-S. Chen, A. Ganguly, L.-C. Chen and K. -H. Chen, InGaN and ZnO-based materials and \ndevices (Springer Series in Materials Science; Springer-Verlarg Berlin: Heidelberg) Ed. S. \nPearton, pp 295 −315 (2012). \n6 C. Soci, A. Zhang, X.-Y. Bao, H. Kim, Y. Lo and D. Wang, J. Nanosci. Nanotechnol. 10, 1 \n(2010). \n7 R. Calarco, T. Stoica, O. Brandt and L. Geelhaar, J. Mater. Res. 26, 2157 (2011). \n8 K.A. Bertness, N. A. Sanford and A. Davydov,  IEEE J. Quant. Electr . 17, 847 (2011). \n9 R.-S. Chen, H.Y Chen, C.-Y. Lu, K.-H. Chen, C.-P. Chen, L.-C. Chen and Y.-J. Yang, Appl. \nPhys Lett . 91, 223106 (2007). \n10 F. González-Posada, R. Songmuang, M. Den Hertog and E. Monroy, Nano Lett. 12 172 (2012). \n11 S.A. Jewett, S. Makowski, B. Andrews, M. J.  Manfra and A. Ivaniseivc Acta Biomaterialia 8, \n728 (2012).  1412 C. Pfüller, O. Brandt, F. Grosse, T. Flissi kowski, C. Chèze, V. Consonni, L. Geelhaar, H. \nGrahn and T. H. Riechert, Phys. Rev. B 82, 0453320 (2010). \n13 P. Lefebvre, S. Albert, J. Risti ć, S. Fernández-Garrido, J. Gran dal, M.-A. Sánchez-García and \nE. Calleja, Superlattices and Microstructures 52 165 (2012). \n14 M. Foussekis, A. A. Baski and M. A. Reshchikov, Appl. Phys. Lett.  94, 162116 (2009). \n15 M. A. Reschikov, A. Behrends, A. Bakin, A. Waag, J. Vac. Sci. Technol B 27, 1688 (2009). \n16 R. Calarco, M. Marso, T. Ritcher, A. I. Ayka nat, R. Mejiers, A. Hart, T. Stoica and H. Luth, \nNano Lett. 5, 982 (2005). \n17 R. S. Chen, C. Y. Lu, K. H. Chen and C. L. Chen, Appl. Phys. Lett. 95, 233119 (2009). \n18 R. Songmuang, O. Landré and B. Daudin, Appl. Phys. Lett. 91, 251902 (2007). \n19 J. H. Luscombe and C. L. Frenzen, Solid-State Electronics 46, 885 (2002). \n20 V. Dobrokhotov, D. N. McIlroy, M. G. Norton, A. Abuzir, W. J. Yeh, I. Stevenson, R. Pouy, J. \nBochenek, M. Cartwright, L. Wang, J. Dawson, M. Beaux and C. Berven, J. Appl. Phys.  99, \n104302 (2006). \n21 N. A. Sanford, P. T. Blanchard, K. A. Bertness,  L. Mansfield, J. B. Schlager, A. W. Sanders, A. \nRoshko, B. B. Burton and S. M. George, J. Appl. Phys. 107, 034318 (2010). \n22 M. Kočan, A. Rizzi, H. Lüth, S. Keller and U. K. Mishra, Phys. Stat. Sol. 234, 773 (2002). \n23 K. M. Tracy, W. J. Mecouch, R. F. Davis, R. J. Nemanich, J. Appl. Phys. 94, 3163 (2003). \n24 S. Chevtchenko, X. Ni, N. Q. Fan, A. A. Baski and H. Morkoç, Appl. Phys. Lett. 88, 122104 \n(2006). \n25 Y. Gu and J. Lauhon,  Appl Phys Letts 89, 143102 (2006).  1526 A. D. Schrieker, F. M. Davidson III, R. J. Wiacek and B. A. Korgel, Nanotech.  28, 2681 \n(2006). \n27 M. A. Lampert, Phys. Rev. 103, 1648 (1956). \n28 The gain, g, is calculated from the me asurement of photocurrent, Iph, using the equation: g = \n(Iph/Popt)·(hc/ λq) where h is Planck’s constant, c is the speed of light, λ is the excitation \nwavelength and q is the charge of the electron. For si mplicity, we assume the external quantum \nefficiency is unity. \n29 average values after measuring eight devices. \n30 C. Soci, A. Zhang, X.-Y. Bao, H. Kim, Y. Lo and D. Wang, J. Nanosci. Nanotechnol. 10, 1 \n(2010). \n31 A. Armstrong, G. T. Wang and A. A. Talin,  J. Electron. Mat. 38, 484 (2009). \n32 A. Cavallini, L. Polenta, M. Rossi, T. Richter,  M. Marso, R. Meijers, R. Calarco and H. Lüth, \nNano Lett. 6, 1548 (2006). \n33 E. Monroy, F. Omnès and F. Calle, Semicond. Sci. Technol  18, R33 (2003). \n34 M. Bertelli, P. Löptien, M. Wnderoth, A. Rizzi, R. G. Ulbrich, M. C. Righi, A. Ferretti, L. \nMartin-Samos, C. M. Bertoni and A. Catellani, Phys. Rev. B 80, 115324 (2009). \n35 D. J. Carter and C. Stampfl, Phys. Rev B 79, 195302 (2009). "    },    {        "title": "1312.6346v1.Rydberg_dressing__Understanding_of_collective_many_body_effects_and_implications_for_experiments.pdf",        "content": "arXiv:1312.6346v1  [cond-mat.quant-gas]  22 Dec 2013Rydberg dressing: Understanding of collective\nmany-body effects and implications for experiments\nJ. B. Balewski, A. T. Krupp, A. Gaj, S. Hofferberth, R. L¨ ow\nand T. Pfau\nE-mail:j.balewski@physik.uni-stuttgart.de\n5. Physikalisches Institut, Universit¨ at Stuttgart, Pfaffenwaldr ing 57, 70569 Stuttgart,\nGermany.\nAbstract. The strong interaction between Rydberg atoms can be used to con trol\nthe strength and character of the interatomic interaction in ultra cold gases by weakly\ndressing the atoms with a Rydberg state. Elaborate theoretical p roposals for the\nrealization of various complex phases and applications in quantum simu lation exist.\nAlso a simple model has been already developed that describes the ba sic idea of\nRydberg dressing in a two-atom basis. However, an experimental r ealization has\nbeen elusive so far. We present a model describing the ground stat e of a Bose-\nEinstein condensate dressed with a Rydberg level based on the Ryd berg blockade.\nThis approach provides an intuitive understanding of the transition from pure two-\nbody interaction to a regime of collective interactions. Furthermor e it enables us to\ncalculate the deformation of a three-dimensional sample under rea listic experimental\nconditions in mean-field approximation. We compare full three-dimen sional numerical\ncalculations of the ground state to an analytic expression obtained within Thomas-\nFermi approximation. Finally we discuss limitations and problems arising in an\nexperimental realization of Rydberg dressing based on our experim ental results. Our\nwork enables the reader to straight forwardly estimate the exper imental feasibility of\nRydberg dressing in realistic three-dimensional atomic samples.\nPACS numbers: 32.80.Rm,03.75.Hh,34.20.Cf,05.30.RtRydberg dressing 2\n1. Introduction\nUltracold atoms provide an ideal system for the simulation of complex many-body\nsystems encountered for example in condensed matter physics [ 1]. Prime examples\nare the superfluid to Mott insulator transition [ 2], studies of the BEC-BCS crossover\n[3] or the Ising model [ 4]. So far in such approaches the atoms have mostly been used\nas hard spheres in various trapping potentials. Long-range intera ctions [5] have been\nrealized by dipolar atomic species [ 6] and cold polar molecules [ 7,8]. A controllable\ninteratomic interaction beyond short-range isotropic character would greatly enrich the\navailable toolbox for quantum simulation and quantum computation. F or ground state\natoms, theisotropics-waveinteractionpotentialcanbetunedus ingFeshbachresonances\n[9]. Atomic species with a magnetic dipole moment show a combination of an isotropic\ns-wave interaction potential and a long-range dipolar part [ 10]. Combining both, the\noverall character of the interaction can be tuned by changing the strength of the s-wave\npart [11].\nRydberg atoms in contrast show a very strong van-der-Waals typ e interaction typically\nmore than ten orders stronger than the interaction between gro und state atoms [ 12]. At\nshortdistancesorclosetoaF¨ orsterresonancethereisatrans itiontoalongrangedipolar\ninteraction which can exhibit different angular dependencies [ 13,14]. Furthermore such\na F¨ orster resonance allows to easily tune the interaction strengt h [15].\nThemostobviousprobleminusingRydbergexcitationinquantumgase sisthemismatch\nin timescales; the lifetime of Rydberg atoms in low l states is on the ord er of tens of\nmicroseconds whereas it typically takes three orders of magnitude longer for a typical\nexperimental system to equilibrate. This problem can be overcome b y only weakly\ndressing [ 16,17] the atomic ground state with a small fraction fof the Rydberg state\nenhancing theoverall lifetimeofthesystemby1 /f. Rydbergdressing hasbeenproposed\nto realize a number of interesting phases in ultracold gases, such as rotons, solitons or\nsupersolids [ 16,17,18,19,20,21,22,23].\nThe strong interaction between Rydberg atoms can result in collect ive effects in the\nexcitation process, namely the suppression of further excitation in a volume around\nthe first Rydberg atom, the so-called Rydberg-blockade effect [ 24,25]. The blockade\neffect has to be included in the calculation of the effective dressing po tential as has been\nalready shown by a very simple two-atom model [ 26]. At typical densities in quantum\ndegenerate atomic gases, the number of atoms inside a blockaded v olume can become\nvery large. How this affects the dressing potential has been shown by Honer et al.\n[27]. By numerically solving an effective spin Hamiltonian they found that th ere is a\ntransition from the pure two-body interaction to a collective N-bod y regime with the\neffective interaction vanishing at large densities.\nIn this paper, we develop an analytic N-atom model for the dressing potential that fully\naccounts for blockade effects. We show that our model gives the s ame result as the\nnumerical approach in [ 27]. We use our model to study the effect of Rydberg dressing\non a three-dimensional BEC, using realistic experimental paramete rs. We find that evenRydberg dressing 3\nfar from the fully blockaded regime collective effects significantly red uce the interaction\ninduced by Rydberg dressing. This is preventing an observation in cu rrent experiments.\nWe provide evidence for this by showing experimental data that also demonstrates\nfurther practical problems arising. We discuss the limits of Rydberg dressing in current\nexperiments and point out possible solutions for future approache s.\n2. N-atom model of Rydberg dressing\nFollowing the argumentationof [ 26], we beginwith the simple case of two atoms, dressed\nwith a Rydberg state by a coupling field with Rabi frequency Ω and lase r detuning ∆.\nFor simplicity we assume the interaction U(R) between atoms in the particular Rydberg\nstate to be of purely repulsive van-der-Waals type. In fact it turn s out, that the actual\nshape of the Rydberg interaction plays a minor role, effectively redu ced to the sign of\nthe potential and the value of the blockade radius rB; one can account for attractive\ninteraction by simply changing the sign of the detuning. The Hamiltonia nin the dressed\nstates basis |gg∝angb∇acket∇ight, 1/√\n2(|gr∝angb∇acket∇ight+|rg∝angb∇acket∇ight) and|rr∝angb∇acket∇ight(g: ground state, r: Rydberg state) reads:\nH=h\n0 Ω/√\n2 0\nΩ/√\n2 ∆ Ω /√\n2\n0 Ω/√\n2 2∆+U(R)\n (1)\nHere the asymmetric singly excited state 1 /√\n2(|gr∝angb∇acket∇ight−|rg∝angb∇acket∇ight) has been left out, as it is\nnot coupled by Ω. It is straight forward to obtain the new ground st ate of the system by\ndiagonalizing ( 1) depending on the interatomic distance R. As can be seen in Figure 1a\nfor the combination of blue laser detuning and repulsive potentials th is ground state\nshows a steep anticrossing at around the blockade radius rBwhereas for red detunings\nthere is a smooth step. The blockade radius rBis defined as the interatomic distance,\nwherethepower broadeningΩ eff=/radicalbig\nΩ2+(2∆)2≈2|∆|(forweak dressing Ω2/∆2≪1)\nequates the absolute value of the Rydberg-Rydberg interaction U(rB). For a repulsive\nvan-der-Waals interaction U(R) =C6/R6, as it is present for most Rydberg states far\nfrom possible resonances, one obtains:\nrB=/parenleftigg\nC6\nh/radicalbig\nΩ2+(2∆)2/parenrightigg1/6\n≈/parenleftbiggC6\nh|2∆|/parenrightbigg1/6\n(2)\nIn the case of red detuning and repulsive potentials, the potential (up to an offset)\nis given as the Rydberg interaction potential, weighted with the prob ability to find\nboth atoms in the Rydberg state, which is at large distances Rapproximately f2=\n(Ω2/(2(Ω2+∆2+∆√\nΩ2+∆2)))2≈Ω4\n16∆4. AtrBthedetuning ofthedoubly excited state\nincreases due to the Rydberg-Rydberg interaction, tuning this st ate out of resonance.\nThis is the well known phenomenon of Rydberg blockade [ 12]. Therefore the probability\nto find both atoms in the Rydberg state decreases dramatically and the potential\nsaturates at a constant value. For blue detuning the interaction e nergy saturates on the\nsame asymptotic values, but with the values for small and large inter atomic distances RRydberg dressing 4\n0 1 2 3U1(a)Interaction energy E ( / ) Ω Δ2\nInteratomic distance R (r )Bred detuning\nblue detuning (Δ<0)\nf U(R)2\nR>rBR<rB( >0) Δ\nattractiveDetuning Δ0repulsiveInteraction dU/dR\nU2(b)\nrepulsive\nattractiverB\nFigure 1. Schematic of binary Rydberg dressing potentials. In (a) the intera ction\npotential of a pair of atoms dressed with a repulsive Rydberg state (dU\ndR>0) by\na coupling laser field with Rabi frequency Ω and detuning ∆ (∆ >0 red, ∆ <0\nblue). In both cases the interaction potential saturates for ver y large and very small\ndistances onto a constant value U1/2which is only depending on the laser parameters\nΩ and ∆. In case of red detuning and repulsive interaction (or blue de tuning and\nattractive interaction) the potential converges to the Rydberg -Rydberg interaction\npotential U(R), weighted with the Rydberg fraction fsquared (dashed line, corrected\nfor offset U1). Note the scaling of the energy axis with the sign of the detuning ∆.\nFor blue detuning ∆ <0 the potential effectively becomes repulsive. The parameter\nregimes where the four possible potential shapes occur are furth er illustrated in (b).\nFor red detuning the asymptotic form of the potential is repulsive; for blue detuning\nit is attractive. Anticrossings appear for the combination of repuls ive interaction and\nblue detuning as well as for attractive interaction and red detuning .\ninterchanged. It is important to note that the difference between the asymptotic values\nof the potential energy and therefore also the overall magnitude of the dressing effect\nis determined only by the laser parameters Ω and ∆ [ 28,23]. These values have been\nalready given by Johnson et al. [ 26]:\nU1=h∆/parenleftigg\n1−/radicalbigg\nΩ2\n∆2+1/parenrightigg\n(3)\nU2=h∆\n2/parenleftigg\n1−/radicalbigg\n2Ω2\n∆2+1/parenrightigg\n(4)\nThe overall energy scale is given by the difference of equations ( 3) and (4), which can be\napproximated as hΩ4\n8∆3for large detunings ∆ ≫Ω. The actual shape and strength of the\nRydberg interaction determines mainly thevalue rBof theblockade radiusandtherefore\nthe transition between the two regimes. For attractive Rydberg in teraction potentials\nthe same potential shapes appear (see Figure 1b). Only the sign of the laser detuning\n∆ has to be inverted in Figure 1a to account for attractive interaction potentials U.\nThe two-atom result has a very simple explanation: If the two atoms are separated by\na distance much larger than the blockade radius rB, the system can be described as two\nindependent atoms in a laser field with Rabi frequency Ω and detuning ∆ with respectRydberg dressing 5\nto the transition to the Rydberg level. Therefore both atoms individ ually exhibit an AC\nStark shift U1/2. If the two atoms approach each other, at the distance of the b lockade\nradiusrB(equation 2) the Rydberg-Rydberg interaction potential becomes so strong\nthat the state with both atoms excited is tuned out of resonance. The explanation of\nthe behaviour of the system in the blockaded regime becomes obviou s from the following\ndiscussion of the N-atomic case.\nWestudynowthedensitydependent energyofasystem consisting ofNRydbergdressed\natoms. At very low densities the interatomic distances Rare much larger than the\nblockade radius rB. The atoms are therefore independent and the energy of the sys tem\nisNtimes the light shift U1/2 of one single atom in equation ( 3):\nU1(N) =Nh∆\n2/parenleftigg\n1−/radicalbigg\nΩ2\n∆2+1/parenrightigg\n(5)\nIn the high density limit all atoms are situated within one blockade sphe re of volume\n4\n3πr3\nB. In this case, the dimensionality of the Hilbert space can be dramatic ally reduced\nby the fact that all states with more than one Rydberg excitation a re completely tuned\noutofresonanceduetotheRydberg blockade. Theresulting Hamilt onianinthe( N+1)-\ndimensional basis |gg...g∝angb∇acket∇ight,|rg...g∝angb∇acket∇ight,|gr...g∝angb∇acket∇ight,... ,|gg...r∝angb∇acket∇ightreads:\nHN=h\n0Ω\n2Ω\n2...Ω\n2\nΩ\n2∆ 0...0\nΩ\n20.........\n.........∆ 0\nΩ\n20...0 ∆\n(6)\nIt can be shown, e.g. by mathematical induction, that the polynomia l determining the\neigenenergies E is given as:\ndet(HN−1E) =/bracketleftbigg\nE2−h∆E−h2NΩ2\n4/bracketrightbigg\n(h∆−E)N−1(7)\nThe energy of the ground state of the fully blockaded N-atomic sta te therefore is\nU2(N) =h∆\n2/parenleftigg\n1−/radicalbigg\nNΩ2\n∆2+1/parenrightigg\n(8)\nComparing this result to equations ( 3) and (4), the explanation of the energy in the\nlimit of full blockade becomes obvious. Due to the Rydberg blockade a ll N atoms share\none Rydberg excitation forming a collective state1√\nN/summationtextN\ni=1|g1,g2,...,ri,...,gN∝angb∇acket∇ight. This\nstate, a so called super atom [ 29], is coupled by the light field to the ground state with a\ncollectively enhanced Rabi frequency√\nNΩ. The collective Rabi frequency then also has\nto be used in the calculation of the AC Stark effect. The energy of th e fully blockaded\nN-atomic system (equation 8) is therefore the light shift of a single super atom with a\ncollective Rabi frequency√\nNΩ and detuning ∆. In the special case N= 2 the result of\nequation ( 4) is recovered. The interaction potential induced by Rydberg dres sing shown\nin Figure 1a can therefore be viewed as the gradual transition between a colle ctive lightRydberg dressing 6\nshift of one super atom and the individual light shift of two independe nt atoms in the\nvicinity of the blockade radius rB.\nTo study the consequences of high densities for the effect of Rydb erg dressing, we\ncalculate now the additional interaction energy Edressper atom. The number of atoms\nNwithin one blockade sphere of radius rBis determined by the ground state atom\ndensityρ:\nN=4\n3πr3\nBρ=ρ\nfρB(9)\nwhereρBdenotes the critical density, where according to [ 27] blockade phenomena start\nto play a role. This is when the average number of Rydberg atoms in a b lockade sphere\nf·ρ·4/3πr3\nBapproaches unity:\nρB=3\nπr3\nB∆2\nΩ2=3√\n2h\nπ√C6|∆|5/2\nΩ2(10)\nThe difference of light shift between one super atom U2(N) (equation 8) and N\nindependent atoms U1(N) (equation 5) is shared among all Natoms, so that the\ncontribution per atom reads:\nEdress(ρ) =h∆\n2/parenleftigg\nΩ2\n4∆2ρB\nρ/parenleftbigg\n1−/radicalbigg\n4ρ\nρB+1/parenrightbigg\n−/parenleftigg\n1−/radicalbigg\nΩ2\n∆2+1/parenrightigg/parenrightigg\n(11)\nNote that this expression is only valid for ρ/ρB≥Ω2/(4∆2); at lower densities the\nblockade does not play a role any more and the mean interaction ener gy per particle\nis given as f2U(ρ−1/3)≪1. The relevant quantity for calculating the impact on an\nultracoldsample (see section 3) isthevariationoftheenergy density Eeff(ρ) =ρEdress(ρ)\nwith density:\n∂ρEeff(ρ) =h∆\n2\n/radicalbigg\nΩ2\n∆2+1−1−Ω2\n2∆21/radicalig\n4ρ\nρB+1\n (12)\nThe density dependence of this quantity is shown in Figure 2. For low densities ρand\nweak dressing Ω2/∆2≪1, there is a linear increase with slope\ngeff=hπ\n6r3\nBΩ4\n∆3(13)\nThis slope deviates from the value in [ 27] only by a constant factor of π/2. At higher\ndensities, the last part of equation ( 12) vanishes and the energy functional ∂ρEeffquickly\nsaturates on a constant value\nµsat=h∆\n2/parenleftigg/radicalbigg\nΩ2\n∆2+1−1/parenrightigg\n≈hΩ4\n2∆3(14)\nThis expression agrees with the result in [ 27]. The energy Edress(ρ) from equation ( 11)\nsaturates also exactly on the same value. This is the AC Stark shift o f a single free\nRydberg dressed atom, as the contribution of the dressed super atom to the energy of\na single atom becomes negligible.Rydberg dressing 7\n0 5 10 15 20050100150200250\nDensity (10 cm ) ρ12 -3Energy functional E (Hz)effgeffρμsat\nρBρC 0 0.01 0.02 0.03 0.04\n02.55\nFigure 2. Rydberg dressing induced energy functional ∂ρEeff(ρ) versus ground state\natom density ρ. The dashed lines show the asymptotic behaviour at high and low\ndensities. The inset shows a zoom at very low densities. The values ar e calculated\nfor a Rabi frequency of Ω = 10kHz, a red detuning of ∆ = 100kHz and a purely\nrepulsive van-der-Waals interaction with C6= 1.89·10−28Hzm6, corresponding to the\n35S Rydberg state [ 30] and resulting in a blockade radius of rB= 3.1µm. The critical\ndensities ρBandρCare explained in the text.\nAs can be seen in Figure 2the energy functional ∂ρEeffdeviates from the initial linear\nslope already at densities well below ρB. The simple explanation is that collective effects\nalreadystarttoplayaroleassoonastherearemorethantwo atom swithinoneblockade\nvolume, which is the case at densities above\nρC=3\n2πr3\nB=3\n2π/radicaligg\n2|∆|\nC6(15)\nAs a consequence, collective effects beyond two-body interaction have to be considered\nalso at relatively low densities, where they already start to reduce a ny influence of\nRydberg dressing.\n3. Rydberg dressing of Bose-Einstein condensates\nWe now apply our results to calculate the modification of the wavefun ction of a Bose-\nEinsteincondensateunderRydberg-dressing. Ourmodelprovide sananalyticexpression\nof the steady state density distribution of a BEC dressed with a hom ogeneous coupling\nfield Ω even in case of a cylindrically symmetric harmonic trapping poten tial. This not\nonly allows us to predict effects in realistic experimental situations bu t also to extract\nthe scaling of the deformation with different parameters.\nWe consider a BEC in a three dimensional harmonic trap. The evolution of theRydberg dressing 8\n0 5 10 15 200510152025\nRadial size ( m) μDensity (10 cm ) ρ12 30\n1\n23\n4\n5\nN=105\n0246810Density (10 cm ) ρ12 3ΩΔ\nΔc0.072 Δc\n0\n10\n2 345\nN=1040/1\n2\n3\n4\n5\n0 5 10 15 20\nRadial size ( m) μ(a) (b)\nFigure 3. Reproduction of Figure3 of [ 27] with the super atom model: Density profile\nρ(r) of a Bose-Einstein condensate consisting of N= 104(a) andN= 105(b) atoms\n[33] in a radially symmetric trap with trap frequency ω= 2π·15.9Hz. The inset in\n(a) shows the parameters of the laser field coupling to the Rydberg level in the plane\nof Rabi frequency Ω and detuning ∆ in units of the critical detuning ∆ c= 107kHz.\nFor larger condensates and therefore higher densities ρthe effect of Rydberg dressing\nis strongly reduced.\ncondensate wavefunction ψ(/vector r,t) is described by the Gross-Pitaevskii equation:\ni/planckover2pi1∂ψ\n∂t=/bracketleftigg\n−h2∇2\n2m+3/summationdisplay\ni=1mω2\nix2\ni\n2+gs|ψ|2+∂ρEeff(|ψ|2)/bracketrightigg\nψ (16)\nThe first two terms are the usual kinetic energy contribution and t he harmonic trapping\npotential, characterized by the three oscillation frequencies ωi. The third term is the\nmean-field contribution of the contact interaction between the at oms, while the fourth\nterm∂ρEeff(ρ) from equation ( 12) describes the effect of Rydberg dressing on the BEC.\nBesides mean-field approximation this incorporates two further ap proximations: First\nof all, local density approximation requires the sample to be much larg er thanthe length\nscaleoftheinteractiongivenbytheblockaderadius rB(equation 2). Secondly theactual\nshape of the Rydberg interaction potential is neglected by assumin g a step between the\ntwo asymptotic values of the dressing potential at the blockade ra diusrB.\nEvenwithoutsolvingtheequationsomeimportantconclusionscanbe drawn. Inthevery\nlow density regime the interaction energy is linear in the density |ψ|2and can therefore\nbe described with an effective s-wave scattering lengthmgeff\n4π/planckover2pi12(see equation 13). For high\ndensities, the energy functional ∂ρEeff(ρ) becomes constant and thus only contributes as\na constant offset to the chemical potential µ. In this case no effect of Rydberg dressing\non the density distribution can be expected.\nThegroundstatestationarysolution ψ(/vector r,t) =ψ(/vector r)e−iµ//planckover2pi1tofequation 16canbeobtained\nnumerically e.g. using the split step Fourier method [ 31]. For condensates with large\natom numbers the trapping potential and the interaction energy a re large [ 32]. In\nThomas-Fermi approximation then the kinetic energy term can be n eglected which isRydberg dressing 9\n0 1 2 3 4 5012345\nRadial coordinate ( m) μDensity (10 cm ) ρ13 -3\n0 4 8 12 16 20012345\nAxial coordinate ( m) μDensity (10 cm ) ρ13 -3no dressing (TF)\nrepulsive (TF)\nattractive (TF)no dressing (TF)\nrepulsive (TF)\nattractive (TF)\nDensity (10 cm ) ρ13 -3\n0 1 2 3 4 5012345\nno dressing (num.)\nrepulsive (num.)\nattractive (num.)\n0 4 8 12 16 20012345\nno dressing (num.)\nrepulsive (num.)\nattractive (num.)\nAxial coordinate ( m) μRadial coordinate ( m) μDensity (10 cm ) ρ13 -3\nFigure 4. Radial and axial density profiles of a BEC consisting of 104 87Rb atoms in\nan axially symmetric trap with trap frequencies ωr= 2π·80Hz and ωz= 2π·20Hz.\nThe analytic result using Thomas-Fermi approximation (left) is comp ared to the full\nnumerical solution of the Gross-Pitaevskii equation in three dimens ions (right). The\nunperturbed density distribution as well as two different Rydberg d ressed situations\nare shown. The repulsive case has been calculated for the same par ameters as Figure 2\n(35S, Ω = 10kHz, ∆ = 100kHz). For the attractive case the sign of t he detuning has\nbeen inverted.\nusually a good approximation except for the low density wings of the c ondensate [ 34].\nWithout the additional energy functional ∂ρEeffthe equation becomes linear in |ψ|2\nmakingtheanalyticsolutionparticularlysimple. Thewellknownresult is aparaboloidal\ndensity distribution ρ=|ψ|2where the value of the chemical potential µis determined\nby the normalization to the total atom number N. With the contribution from Rydberg\ndressing (equation 12) the calculation of the modified ground state density ρbecomes\nonly slightly more involved. For red detunings ∆ >0 equation ( 16) can be rewritten as\na cubic equation in the density ρthat can be efficiently solved analytically. Only the\nchemical potential µhas to be calculated numerically under the constraint that the tota l\nBEC atom number remains constant. In Figure 3, we show the resulting condensate\ndensity distributions when using the same parameters as in [ 27]. Our analytic model is\nin very good agreement with the numerical results by Honer et al. als o obtained within\nThomas-Fermi approximation, but there is a finite deviation. Never theless our results\nshould be morethanenough precise to allowpredictions about overa ll scaling andorders\nof magnitude. However, since the additional term ∂ρEeffof Rydberg dressing is expected\nto be small, it is not obvious that Thomas-Fermi approximation is valid h ere. As an\nexemplary check, we calculate the density distribution for typical p arameters using bothRydberg dressing 10\nmethods as shown side by side in Figure 4. As expected a repulsive dressing potential\nleads to an expansion of the condensate whereas an attractive po tential makes the cloud\nbecome smaller and denser. Furthermore the full numerical solutio n and the result of\nThomas-Fermi approximation agree very well in the centre of the c ondensate, whereas\nthe deviation at the outer parts becomes significant. It turns out that in these regions\nthe kinetic energy term is dominating the density distribution so that here the effect of\nRydberg dressing is even less visible than in the centre of the BEC. Th is is in contrast\nto what one could have expected from the scaling of the energy fun ctional∂ρEeff(ρ) with\ndensity alone (see Figure 2). However, the fact that the impact of Rydberg dressing\nis vanishing at higher densities can be seen comparing the attractive and repulsive\ndressing. The effect of repulsive Rydberg dressing is slightly strong er since it becomes\nself-amplifying.\nSince the modification of the density distribution by the Rydberg dre ssing is the\nstrongest in the centre of the condensate we use in the following th e relative change\nof peak density to quantify the response of a three-dimensional a symmetric BEC. As\nthe behaviour of this quantity is well reproduced using Thomas-Fer mi approximation,\nwe can make use of the significant speed up provided by the analytic c alculation within\nThomas-Fermi approximation. Furthermore, the residual deviat ion from the more exact\nnumerical simulation further decreases at higher atom numbers in t he condensate.\n4. Systematic study of parameter space\nWe can now compute the deformation of a BEC depending on different experimentally\naccessible parameters. These are essentially the laser parameter s, the Rabi frequency Ω\nand detuning to the Rydberg state ∆, the initial peak density ρ0of the condensate and\nthe blockade radius rBwhich within the model of equation ( 2) can be controlled via the\nC6coefficient of the Rydberg state.\nFrom Figure 2one can conclude that at high atomic densities the effect of Rydberg\ndressing is vanishing. The Rydberg blockade here limits the achievable interaction\nstrength. In a not fully blockaded sample, at densities below ρB, a higher density\nincreases the collective Rabi frequency√\nNΩ leading to more Rydberg excitation in\nthe system. Above ρB, however, no further Rydberg excitation is possible. The\ntotal interaction energy, determined by the number of Rydberg e xcitations in the\nsystem, is then saturated on a constant value that is shared amon g more atoms. The\ncontribution for each individual atom is thus effectively vanishing at h igh densities. At\nvery low densities, however, the interatomic distance increases th ereby also reducing\nthe interaction strength. The maximum effect is therefore expect ed in the intermediate\nregime where the energy difference per atom between the blockade d system and the non\nblockaded system is the highest. We estimate this region to be at the point where the\nsystem just starts to become fully blockaded, at densities around ρ=ρB.\nThe laser parameters Ω and ∆ mainly determine the fraction of Rydbe rg excitations\nf≈Ω2/4∆2. Large Rabi frequencies Ω and small detunings ∆ to the Rydberg st ate areRydberg dressing 11\n0 5 10 15 2002004006008001000\nRabi frequency (kHz) ΩDetuning (kHz) Δ\n-25-20-15-10-50\nRel. change of peak density / (%)Δρ□ρ0\n-2\n-10-5\n-20-1\n-15\nFigure 5. Dependency of Rydberg dressing of a BEC on laser parameters Ω an d ∆.\nTherelativechangeofpeakdensity∆ ρ/ρ0in steadystateiscalculatedforacondensate\nconsisting of N= 2·104atoms in a cylindrically symmetric trap with trap frequencies\nωr= 2π·20Hzand ωz= 2π·80Hz. The condensateisdressedwith arepulsiveRydberg\nstate with C6= 6.10·10−29Hzm6, corresponding to the 32S Rydberg state.\ntherefore increasing the effect of Rydberg dressing. However, t here is an upper bound\nof the tolerable Rydberg fraction fgiven by the decay of the Rydberg state. Large\nRydberg fractions flead to a strongly reduced lifetime ∝1/fof the dressed state that\nprevents the observation of any mechanical effect. This constra int is further discussed\nin part7.\nIn order to quantify the effect of Rydberg dressing onto a BEC we c alculate the relative\nchange ∆ρ/ρ0of the peak density of the condensate. The dependency of this qu antity\non the Rabi frequency Ω and the laser detuning ∆ to the Rydberg st ate are shown in\nFigure5. The range of Rabi frequencies is given by technical constraints a s discussed in\npart6. As expected, the effect increases towards higher Rabi frequen cies Ω and lower\ndetunings ∆. At very small detunings ∆ ≤Ω, however, there is some deviation from\nthis trend since here the blockade radius rBaccording to equation ( 2) becomes large.\nAnyway in this parameter region the approximation of weak dressing f≪1 is not\nfulfilled any more.\nSo far we assumed that the finite blockade radius rBis given by power broadening\naccording to equation ( 2) neglecting technical sources of laser broadening. In Figure 6\nwe study also the influence of the blockade radius rBas an independent quantity. The\neffect increases monotonously with Rabi frequency Ω and decreas es with detuning ∆.\nFor each value of Ω and ∆, however, there is a clear maximum in the bloc kade radius\nrB. This can be explained by the fact that the blockade radius rBdetermines the length\nscale of the system. The density ρis rescaled by the blockade radius rBaccording to\nequation ( 9). As discussed before, the maximum effect of Rydberg dressing is e xpected\nwhen the system becomes fully blockaded at ρ=ρB. Keeping the density ρfixed weRydberg dressing 12\n0 200 400 600 800 10000246810\nDetuning (kHz) ΔBlockade radius r ( m)Bμ\n-70-60-50-40-30-20-100\nRel. change of peak density / (%)Δρ□ρ0(b)\n-1-2\n-3\n-4\n-6\n-20-10\n-40\n0 5 10 15 200246810\nRabi frequency (kHz) ΩBlockade radius r ( m)Bμ\n-40-30-20-100\nRel. change of peakd ensity / (%)Δρ□ρ0(a)\n-1\n-2\n-4\n-6\n-10\n-20-15\n-30-3\n-40\nFigure 6. Dependency of Rydberg dressing of a BEC on the Rabi frequency Ω (a),\nlaser detuning ∆ (b) and the blockade radius rBas an independent quantity. The\nrelative change of peak density ∆ ρ/ρ0in steady state is calculated for a condensate\nconsisting of N= 2·104atoms in a cylindrically symmetric trap with trap frequencies\nωr= 2π·20Hz and ωz= 2π·80Hz. The condensate is dressed detuned by\n∆ = 100kHz (a) with Rabi frequency Ω = 10kHz (b) to a repulsive Rydb erg state\nwithC6= 6.10·10−29Hzm6, corresponding to the 32S Rydberg state. The black dots\nshow the blockade radii with maximum effect at fixed Ω and ∆ respectiv ely. The black\nsolid line shows the value of rB,m(equation 17) when the sample on average starts to\nbecome fully blockaded. This is where the maximum effect is expected ( see text).\ncan calculate the optimal blockade radius rB,mfrom equation ( 10):\nrB,m=3/radicaligg\n3\nπρ∆2\nΩ2(17)\nItturnsoutthatthisevenquantitativelyreproducesthedepend encyobservedinFigure 6\nif we use 3/4 times the mean density of the dressed condensate. For small def ormation\nof the condensate also the mean density ρ= 2/5ρ0of the initial Thomas-Fermi density\ndistribution with peak density ρ0is a good approximation.\n5. Dipolar interaction\nWithin the model presented here, the shape of the interaction pot ential only determines\nthe value of the blockade radius rBand the sign of the Rydberg dressing. Therefore\none would not expect that a Rydberg-Rydberg interaction of dipola r type could lead\nto fundamentally new effects. However, it seems worthwhile to cons ider the dipolar\ninteraction of Rydberg states close to a F¨ orster resonance. So me Rydberg states\ncan be tuned into resonance by applying microwaves [ 35,36] or small electric fields\n[37,38,39,40,41,42]. First of all, the interaction between two Rydberg atoms close to\nsuch a resonance shows an angular dependency [ 13,14] that is expected to translate\ndirectly into the Rydberg dressed interaction. Secondly and more im portantly the\nstrength and sign of the interaction close to the F¨ orster resona nce can be tuned via\nthe F¨ orster defect ∆ F[15]. This would not only allow to easily realize repulsive andRydberg dressing 13\n0\nFörster defect ΔFDetuning Δ\nΔ=-2FΔ\nFigure 7. Scheme of different potentials of a pair of atoms dressed with a Rydb erg\nstate close to a F¨ orster resonance. In the plane of detuning ∆ an d F¨ orster defect ∆ F\nfour different regimes are identified. The regime of a smooth potent ial is narrowed\ndown compared to Figure 1b to small absolute detunings |∆|<|∆F|/2.\nattractive Rydberg dressing while keeping other parameters of th e system like e.g. the\nRabi frequency Ω constant, but also enable to control the blocka de radiusrBof the\nsystem by changing the interaction strength. As the discussion ab ove showed, this is a\nkey parameter that rescales the important density scale of the pr oblem.\nYet in case of a F¨ orster resonance the level scheme becomes sligh tly more involved and\nthe two-atom Hamiltonian ( 1) has to be extended by the pair state |r′r′′∝angb∇acket∇ightwhich on\nF¨ orster resonance (∆ F= 0) becomes degenerate with the doubly excited Rydberg state\n|rr∝angb∇acket∇ight:\nH=h\n0 Ω/√\n2 0 0\nΩ/√\n2 ∆ Ω /√\n2 0\n0 Ω/√\n2 2∆ Udd(R)\n0 0 Udd(R) 2∆+∆ F\n(18)\nwhereUdd(R) =C3/R3is the coupling of the two doubly excited Rydberg states.\nNegative F¨ orster defects ∆ F<0 lead to repulsive interaction, attractive interaction\ncan be realized by choosing positive F¨ orster defects. It turns ou t that similar potential\ncurves as in Figure 1a can be obtained. In particular the asymptotic values are the\nsame as for the Hamiltonian ( 1). However, the smooth dressing potential (red curve\nin Figure 1a) is only present in the regime of small absolute detunings |∆|<|∆F|/2.\nFor larger detunings ∆ there is an additional anticrossing occurring . The regimes of\ndifferent two body potentials in the plane of detuning ∆ and F¨ orster defect ∆ Fare\nsketched in Figure 7. However, we note that the anticrossing is likely to play a role only\nin very cold atomic samples since fast atoms follow the potential curv es diabatically\n[43]. Then again only the asymptotic values of the potential and the bloc kade radius\nrBare important. In this case, the sign of the Rydberg dressed pote ntial can be fullyRydberg dressing 14\ncontrolled with the sign of the laser detuning ∆ alone, also in the absen ce of a F¨ orster\nresonance.\n6. Current experimental situation\nIn our present experimental apparatus described in [ 44], we can realize Bose-Einstein\ncondensates in a cylindrically symmetric trap with a radial trapping fr equencyωr=\n2π·22Hz and axial trapping frequencies down to ωz= 2π·82Hz. Typical atom numbers\nvary around 105corresponding to a peak density of 1014cm−3. Atom numbers down\nto 2·104can be realized in order to reduce the peak density to 5 ·1013cm−3at the\nexpense of increasing atom number fluctuations observed. We exc ite Rydberg S and D-\nstatesviaatwo-photontransitiondetunedby500MHzfromtheint ermediate5 P3/2state\nusing continuous wave diode lasers at wavelengths of 780nm and 480 nm. Depending\non the Rydberg state, the Rabi frequency of the upper transitio n is limited by technical\nconstraintstoseveral tensofMHz. Thefocused480nmlaserissw itched onadiabatically\nduring 400ms since it creates a dipole potential for the ground stat e atoms [ 12]. As it is\ncommon for all Rydberg excitation schemes based on a two-photon transition [ 44,45],\nthe Rabi frequency of the lower transition is limited by off-resonant scattering from the\nintermediate state [ 46] as in our case, or two-photon ionization [ 47]. The first effect can\nbe estimated if reabsorption of the emitted photons [ 48] is taken into account. In the\nexperiment we choose the red laser power such that the total ato m loss over the length\nof the laser pulse is largely negligible. For experiments lasting 100ms we can apply\neffective Rabi frequencies on the order of few kHz. This sequence length corresponds\nto about twice the inverse axial trap frequency and was chosen to allow for the BEC\ndensity distribution to reach its equilibrium. We switch off the Rydberg lasers and the\ntrap simultaneously. After a time of flight lasting 50ms, an absorptio n image is taken.\nFrom these images we extract the change of the BEC atom number a nd the aspect ratio\nsimilar to the procedure in [ 49] in order to compare them to reference measurements\nwith the 780nm laser off.\nThe results for two different Rydberg S-states and one Rydberg D -state close to a\nF¨ orster resonance are shown in Figure 8in comparison to reference Rydberg spectra\nmeasured by field ionization and ion detection in a thermal sample as de scribed in\n[49,44]. For the S-states and the Rabi frequencies given in Figure 8, the onset of full\nblockade according to equations ( 2) and (17) using the mean density ρ= 2/5ρ0is taking\nplace at detunings ∆ just below 100kHz. This is the regime where the la rgest effect\ncan be expected (see part 4). Here obviously the decay from the Rydberg state, visible\nas a loss feature in Figure 8c, is the limiting factor; the decay leads to strong heating\nthat destroys nearly the whole condensate at small detunings |∆|<200kHz. Outside\nthis regime the expected effect of Rydberg dressing according to F igure5is very small.\nConsequently we are not able to detect any significant deformation of the condensate\nwithin the experimental error (see Figure 8e). Furthermore, from the measurement on\nthe 35S state another problem becomes obvious. In the referenc e spectrum taken inRydberg dressing 15\n0246810Ion signal [10 arb.u.]3\n020406080100Rel. atom number [%]\n2 1 0 -1 -29095100105110\nDetuning [MHz]Rel. aspect ratio [%](a)\n(c)\n(e)32S\n35S\n0123(b)Ion signal [10 arb.u.]3\n020406080Rel. atom number [%]\n95100105110115120Rel. aspect ratio [%]4\n(d)\n(f)44D ( ∆=-6.7MHz)F\n44D ( ∆= 2.2MHz)F\n10 5 0 -5 -10\nDetuning [MHz]\nFigure 8. Experimental results on dressing different Rydberg states to a Bo se-\nEinstein condensate: The condensate consisting of 2 ·104(S-states, left panels) and\n8·104(D-state, right panels) atoms is dressed for 100ms at fixed Rabi f requencies\n(32S1/2, mJ= 1/2: Ω = 3 .4kHz, 35S 1/2, mJ= 1/2: Ω = 2 .3kHz, 44D 5/2, mJ= 5/2:\nΩ = 2.6kHz) and variable detuning ∆ to the Rydberg state respectively. T he relative\nchange of BEC atom number (c, d) and aspect ratio (radial/axial wid th, e, f) was\nextracted from absorption images taken after a time of flight of 50 ms. In (a, b) the\nion signal from reference measurements in a thermal sample is show n. The parameters\nwere chosen to make tiny signals from molecular states visible (S-sta tes (a): excitation\npulse length 100 µs, Rabi frequencies Ω = 17 .5kHz/2.3kHz; D-state (b): excitation\npulse length 5 µs, Rabi frequency Ω = 8 .5kHz). Therefore the signal on resonance is\nhighly saturated. The solid lines in (c-f) are a moving average propor tional to the\nexcitation linewidth (width 0 .3MHz, c, e; 1MHz, d, f) as guide to the eye. Error bars,\n±1 s.d. from eight independent measurements.\na thermal sample at 1012cm3, there is a small additional line red detuned by about\n0.85MHz to the atomic line (see Figure 8a). This line can be assigned to a molecular\nbound state of a Rydberg atom and a ground state atom, so called u ltralong-range\nRydberg molecules [ 50,51]. In the BEC, this leads to loss features that are more\npronounced for two reasons. First, the scaling of the Franck-Co ndon factor for the\nphotoassociation of molecules with the density of ground state ato ms increases the\nexcitation probability ina BEC. And second, these bound states sho w a reduced lifetime\nat higher densities [ 52] thereby causing stronger atom losses.\nThe44D 5/2, mJ= 5/2statewasstudied attwo different F¨ orsterdefectstothereso nance\nwith the 46P 3/2, mJ= 3/2 and 42F 7/2, mJ= 7/2 pair state [ 53,15]. Since the sign ofRydberg dressing 16\nthe F¨ orster defect is inverted, we would expect here a dressing e ffect of opposite sign\n(see part 5). These experiments were preformed with condensates at a four times higher\natom number and therefore next to twice the peak density compar ed to the ones on the\nS-states. The better stability of the experiment at these parame ters leads to a lower\nnoise level; however, the expected effect of Rydberg dressing is ev en further decreased\nbecause of the higher atomic density and therefore again below the experimental noise\nlevel. In particular, we observe no significant difference between th e two measurements\nat different F¨ orster defects. Instead, for the Rydberg D-sta tes, we observe a band of\nmolecular states [ 54] in the spectrum (see Figure 8b). In the same range of detunings,\nthere is a strong and several MHz broad loss feature in the measur ements with the BEC,\nas can be seen from Figure 8d. Furthermore this loss seems to be connected to a change\nin aspect ratio (Figure 8f). This does not necessarily imply an asymmetric deformation\nof the condensate. Calculations within Thomas-Fermi approximatio n [55] indicate that\neven an isotropic deformation of the BEC in the trap can lead to a cha nge in aspect\nratio at finite time of flight. However, it also seems unlikely that this de formation is\ncaused by uniform losses alone, since it does not fully coincide with the observed BEC\natomlosses. Especially there is no such pronounced effect observe d inthe measurements\nof the Rydberg S-states (see Figure 8e) although there are equally strong atom losses\npresent close to resonance.\nIn any case ultralong-range Rydberg molecules can possibly inhibit th e observation of\nRydberg dressing at red detuning for several Rydberg states. T his effect becomes even\nmore important at Rydberg states with higher principal quantum nu mbern. As the\nbinding energy is decreasing proportional to n−6, the molecular states come closer to\nthe atomic Rydberg state. Then polyatomic bound states [ 51,56] start to play a role,\neventually leading to a density dependent shift of the Rydberg line [ 57,49].\n7. General challenges in realizing Rydberg dressing\nSeveral technical constraints are likely to impede an experimental observation of\nRydberg dressing in our current setup. One is the unfavourable sc aling of the effect with\nthe density of ground state atoms. A low temperature sample is req uired to observe\nthe small effects expected. Such samples, like Bose-Einstein conde nsates typically\nfeature a high atomic density. As discussed before, the impact of R ydberg dressing\nis greatly reduced in strongly blockaded samples. Elements with large r background s-\nwave scattering length, such as cesium, allow the preparation of co ndensates at lower\npeak density [ 58]. In this case, the modification of the density distribution, however ,\nwould be even smaller due to the large mean-field interaction between the atoms.\nFurthermore, ultracold samples of cesium can only be prepared in op tical dipole traps.\nCommon red detuned dipole traps create differential light shifts bet ween the ground\nand Rydberg state that can lead to an inhomogeneous laser detunin g ∆ to the Rydberg\nstate. Therefore, a magic wavelength trap [ 59] would be required. Instead of reducing\nthe atomic density one possibility would be to reduce the blockade rad iusrBof theRydberg dressing 17\n0 1 2 3 4 5 6 7 8 910102030405060708090100\nOffset density (10 cm ) ρOffs12 -3Initial peak density (10 cm ) ρ012 -3\n-50-40-30-20-100\nRel. change of peak density / (%)Δρ□ρ0\n-4\n-5\n-6\n-10-8\n-15\n-20\n-30\n-40\nFigure 9. Dependency of Rydberg dressing of a BEC on the initial peak density ρ0\nandan offset density ρOffs. The relativechangeofpeak density∆ ρ/ρ0in steady stateis\ncalculated for a condensate consisting of N= 2·104atoms in a cylindrically symmetric\ntrap with trap frequencies ωr= 2π·20Hz and ωz= 2π·80Hz. The condensate is\ndressed detuned by ∆ = 100kHz with Rabi frequency Ω = 10kHz to a r epulsive\nRydberg state with C6= 6.10·10−29Hzm6, corresponding to the 32S Rydberg state.\nsample in order to tune the system just to the onset of saturation . To this end either a\nRydberg state at low principal quantum number can be chosen since theC6-coefficient\nis scaling as n11[30]. Alternatively also the excitation linewidth ∆ fcould in principle\nbe increased artificially. The drawback of the first possibility is that t he lifetime of the\nRydberg state at the same time decreases ∝n3, thereby reducing the tolerable Rydberg\nfraction. The latter is not practicable due to the unfavourable sca lingrB∝(∆f)1/6and\nthe fact that large excitation linewidths preclude realization of small laser detunings ∆.\nAnother possibility would involve a F¨ orster resonance to tune the R ydberg interaction\nstrength as discussed in part 5.\nAnother principal problem is the long timescale required for experime nts studying\nmechanical effects. The maximum achievable Rydberg fraction f≈Ω2/4∆2is limited\nby the product of pulse length and the decay rate from the dresse d state. This can be\nsimply estimated for the experimental parameters from part 6, in particular the 32S\nRydberg state and a pulse length of t= 100ms. If we restrict the atom losses to an\narbitrary value of 50%, we obtain the condition f <1/(2Γrt)≈10−4using the decay\nrate Γ r= 50.6kHz of the 32S Rydberg state including blackbody radiation at 298K\n[41]. At a detuning of ∆ = 100kHz, corresponding to the optimal detunin g for the\ncurrent density (see part 6) therefore the Rabi frequency is limited to Ω = 2kHz. As\ncan be seen from Figure 5the expected density change due to Rydberg dressing at these\nparameters is on the order of only a percent.\nA rather technical problem finally is related to the preparation of th e condensate. A\nvery pure BEC is required since any non condensed atoms take part in the collective\nlight shift but do not lead to a deformation of the condensate. We ca n account for a\nresidual thermal cloud by simply introducing an offset density ρOffsin our calculations.Rydberg dressing 18\nAs can be seen from Figure 9the reduction of the effect of Rydberg dressing onto the\ncondensate is sizeable even at small offset densities ρOffssince the main effect according\nto Figure 2is expected at low densities.\nEven though at present it seems unlikely to observe Rydberg dress ing at the parameters\ndiscussed above, we see two main paths towards a possible experime ntal realization.\nOne would involve samples at reduced dimensionality as optical lattices [23] thereby\nreducing the effective density. Adhering to three-dimensional sam ples instead would\nrequire a significant increase in Rabi frequency. The effect of Rydb erg dressing scales\nroughly as ∼Ω4/∆3(equations 13and14) while the decay rate of the dressed state is\nonly scaling as the Rydberg fraction ∼Ω2/∆2. By increasing the laser detuning ∆ by\nthe same amount as the Rabi frequency Ω one can therefore expe ct a stronger impact\nat the same loss rate. For the parameters discussed above, a mod ification of the peak\ndensity on the order of ten percent would therefore require a mor e than hundred times\nlarger laser power.\n8. Conclusion\nInsummary, wehavepresented asimplemodelofaRydbergdresse d system. Thismodel\nprovides an easy qualitative and quantitative description of the colle ctive effects arising.\nWe studied the impact of Rydberg dressing onto a three-dimensiona l axially symmetric\nBose-Einstein condensate, a situation that is experimentally releva nt, depending on\ndifferent parameters. Our model allows to intuitively understand th e dependency of\nthe maximum effect on various experimental parameters. The limit of Thomas-Fermi\napproximation in the case of Rydberg dressing was identified by a com parison with a\nfull numerical solution of the Gross-Pitaevskii equation. We propo sed the extension\nto Rydberg states close to a F¨ orster resonance. Here further work is required in\norder to understand possible collective effects arising in the F¨ orst er coupling. Finally\nwe described our current experimental situation in view of a possible realization of\nRydberg dressing. Principle as well as practical challenges were dem onstrated and, as\na conclusion, possible solutions were discussed.\nAcknowledgments\nWe thank H. P. B¨ uchler and his group for fruitful discussions and s upport. This work is\nfundedbytheDeutscheForschungsgemeinschaft (DFG)withinth eSFB/TRR21andthe\nproject PF 381/4-2. We also acknowledge support by the ERC unde r contract number\n267100 and A.G. acknowledges support from E.U. Marie Curie progra m ITN-Coherence\n265031.\nReferences\n[1] Immanuel Bloch, Jean Dalibard, and Sylvain Nascimb` ene. Quantu m simulations with ultracold\nquantum gases. Nature Physics , 8:267–276, 2012.Rydberg dressing 19\n[2] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. H¨ ansc h, and Immanuel Bloch.\nQuantum phase transition from a superfluid to a Mott insulator in a ga s of ultracold atoms.\nNature, 415(6867):39–44, January 2002.\n[3] N. Navon, S. Nascimb` ene, F. Chevy, and C. Salomon. The equat ion of state of a low-temperature\nFermi gas with tunable interactions. Science, 328(5979):729–732, 2010.\n[4] Jonathan Simon, Waseem S. Bakr, Ruichao Ma, M. Eric Tai, Philipp M. Preiss, and Markus\nGreiner. Quantum simulation of antiferromagnetic spin chains in an op tical lattice. Nature,\n472:307–312, 2011.\n[5] Bo Yan, Steven A. Moses, Bryce Gadway, Jacob P. Covey, Kade n R. A. Hazzard, Ana Maria Rey,\nand Jun Ye Deborah S. Jin. Observation of dipolar spin-exchange int eractions with lattice-\nconfined polar molecules. Nature, 501:521–525, 2013.\n[6] Thierry Lahaye, Tobias Koch, Bernd Fr¨ ohlich, Marco Fattori, J onas Metz, Axel Griesmaier,\nStefano Giovanazzi, and Tilman Pfau. Strong dipolar effects in a quan tum ferrofluid. Nature,\n284:672–675, 2007.\n[7] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis , J. J. Zirbel, S. Kotochigova,\nP. S. Julienne, D. S. Jin, and J. Ye. A high phase-space-density gas of polar molecules. Science,\n322(5899):231–235, 2008.\n[8] S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis , G. Qu´ em´ ener, P. S.\nJulienne, J. L. Bohn, D. S. Jin, and J. Ye. Quantum-state controlle d chemical reactions of\nultracold potassium-rubidium molecules. Science327(5967):853–857, 2010.\n[9] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshba ch resonances in ultracold\ngases.Rev. Mod. Phys. , 82:1225–1286, Apr 2010.\n[10] T Lahaye, C Menotti, L Santos, M Lewenstein, and T Pfau. The p hysics of dipolar bosonic\nquantum gases. Reports on Progress in Physics , 72(12):126401, 2009.\n[11] Stefano Giovanazzi, Axel G¨ orlitz, and Tilman Pfau. Tuning the d ipolar interaction in quantum\ngases.Phys. Rev. Lett. , 89:130401, Sep 2002.\n[12] M. Saffman, T. G. Walker, and K. Mølmer. Quantum information wit h Rydberg atoms. Rev.\nMod. Phys. , 82(3):2313–2363, August 2010.\n[13] Thomas J. Carroll, Katharine Claringbould, Anne Goodsell, M. J. L im, and Michael W. Noel.\nAngular dependence of the dipole-dipole interaction in a nearly one-d imensional sample of\nRydberg atoms. Phys. Rev. Lett. , 93:153001, Oct 2004.\n[14] A. Reinhard, T. Cubel Liebisch, B. Knuffman, and G. Raithel. Lev el shifts of rubidium Rydberg\nstates due to binary interactions. Phys. Rev. A , 75:032712, Mar 2007.\n[15] J. Nipper, J. B. Balewski, A. T. Krupp, S. Hofferberth, R. L¨ ow , and T. Pfau. Atomic pair-state\ninterferometer: Controlling and measuring an interaction-induced phase shift in Rydberg-atom\npairs.Phys. Rev. X , 2:031011, Aug 2012.\n[16] N. Henkel, R. Nath, and T. Pohl. Three-dimensional roton excit ations and supersolid formation\nin Rydberg-excited Bose-Einstein condensates. Phys. Rev. Lett. , 104:195302, May 2010.\n[17] G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller. Strongly correlated gases of\nRydberg-dressed atoms: Quantum and classical dynamics. Phys. Rev. Lett. , 104:223002, Jun\n2010.\n[18] F. Cinti, P. Jain, M. Boninsegni, A. Micheli, P. Zoller, and G. Pupillo. S upersolid droplet crystal\nin a dipole-blockaded gas. Phys. Rev. Lett. , 105:135301, Sep 2010.\n[19] F. Maucher, N. Henkel, M. Saffman, W. Kr´ olikowski, S. Skupin, a nd T. Pohl. Rydberg-induced\nsolitons: Three-dimensional self-trapping of matter waves. Phys. Rev. Lett. , 106:170401, Apr\n2011.\n[20] N.Henkel, F.Cinti, P.Jain,G.Pupillo, andT.Pohl. Supersolidvort excrystalsinRydberg-dressed\nBose-Einstein condensates. Phys. Rev. Lett. , 108:265301, Jun 2012.\n[21] S. M¨ obius, M. Genkin, A. Eisfeld, S. W¨ uster, and J. M. Rost. E ntangling distant atom clouds\nthrough Rydberg dressing. Phys. Rev. A , 87:051602, May 2013.\n[22] Marco Mattioli, Marcello Dalmonte, Wolfgang Lechner, and Guido P upillo. Cluster LuttingerRydberg dressing 20\nliquids of Rydberg-dressed atoms in optical lattices. Phys. Rev. Lett. , 111:165302, Oct 2013.\n[23] T. Macri and T. Pohl. Rydberg-dressing of atoms in optical latt ices.arXiv:1308.5562 , 2013.\n[24] I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti, P. Nosb aum, V. M. Akulin, and P. Pillet.\nMany-body effects in a frozen Rydberg gas. Phys. Rev. Lett. , 80:253–256, Jan 1998.\n[25] D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhan g, R. Cˆ ot´ e, E. E. Eyler, and P. L.\nGould. Local blockade of Rydberg excitation in an ultracold gas. Phys. Rev. Lett. , 93:063001,\nAug 2004.\n[26] J. E. Johnson and S. L. Rolston. Interactions between Rydbe rg-dressed atoms. Phys. Rev. A ,\n82:033412, Sep 2010.\n[27] Jens Honer, Hendrik Weimer, Tilman Pfau, and Hans Peter B¨ uch ler. Collective many-body\ninteraction in Rydberg dressed atoms. Phys. Rev. Lett. , 105:160404, Oct 2010.\n[28] Isabelle Bouchoule and Klaus Mølmer. Spin squeezing of atoms by t he dipole interaction in\nvirtually excited Rydberg states. Phys. Rev. A , 65:041803, Apr 2002.\n[29] Vladan Vuletic. Quantum networks: When superatoms talk phot ons.Nature Physics , 2:801–802,\n2006.\n[30] Kilian Singer, Jovica Stanojevic, Matthias Weidem¨ uller, and Robin Cˆ ot´ e. Long-rangeinteractions\nbetween alkali Rydberg atom pairs correlated to the ns-np,np-np andnd-nd asymptotes. J.\nPhys. B: At. Mol. Opt. Phys. , 38(2):S295–S307, 2005.\n[31] Juha Javanainen. Noncondensate atoms in a trapped Bose gas .Phys. Rev. A , 54:R3722–R3725,\nNov 1996.\n[32] W. Ketterle, D. S. Durfee, and D.M. Stamper-Kurn. Making, pr obing and understanding Bose-\nEinsteincondensates. Proceedingsofthe InternationalSchool ofPhysics”EnricoFermi”, Course\nCXL, Amsterdam, 1999. IOS Press.\n[33] There is a discrepancy between legend and graph in Figure 3 of Ph ys. Rev. Lett. 105, 160404\n(2010): The atom number is misquoted one order of magnitude too h igh.\n[34] Lene Vestergaard Hau, B. D. Busch, Chien Liu, Zachary Dutto n, Michael M. Burns, and J. A.\nGolovchenko. Near-resonant spatial images of confined Bose-Ein stein condensates in a 4-Dee\nmagnetic bottle. Phys. Rev. A , 58:R54–R57, Jul 1998.\n[35] P.Bohlouli-Zanjani,J.A.Petrus,andJ.D. D.Martin. Enhance mentofRydbergatominteractions\nusing ac Stark shifts. Phys. Rev. Lett. , 98:203005, May 2007.\n[36] Atreju Tauschinsky, C. S. E. van Ditzhuijzen, L. D. Noordam, and H. B. van Linden van den\nHeuvell. Radio-frequency-drivendipole-dipoleinteractionsinspatia llyseparatedvolumes. Phys.\nRev. A, 78:063409, Dec 2008.\n[37] W. R. Anderson, M. P. Robinson, J. D. D. Martin, and T. F. Gallag her. Dephasing of resonant\nenergy transfer in a cold Rydberg gas. Phys. Rev. A , 65:063404, Jun 2002.\n[38] K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. F edorov, and J. D. D. Martin.\nSpectroscopic observation of resonant electric dipole-dipole inter actions between cold Rydberg\natoms.Phys. Rev. Lett. , 93:233001, Nov 2004.\n[39] K. Afrousheh, P. Bohlouli-Zanjani, J. A. Petrus, and J. D. D. M artin. Determination of the\n85Rbng-series quantum defect by electric-field-induced resonant energ y transfer between cold\nRydberg atoms. Phys. Rev. A , 74:062712, Dec 2006.\n[40] Thomas J. Carroll, Shubha Sunder, and Michael W. Noel. Many-b ody interactions in a sample of\nultracold Rydberg atoms with varying dimensions and densities. Phys. Rev. A , 73:032725, Mar\n2006.\n[41] I. I. Ryabtsev, D. B. Tretyakov, I. I. Beterov, and V. M. En tin. Observation of the Stark-tuned\nF¨ orster resonance between two Rydberg atoms. Phys. Rev. Lett. , 104:073003, Feb 2010.\n[42] A. Reinhard, K. C. Younge, T. Cubel Liebisch, B. Knuffman, P. R . Berman, and G. Raithel.\nDouble-resonance spectroscopy of interacting Rydberg-atom s ystems. Phys. Rev. Lett. ,\n100:233201, Jun 2008.\n[43] Curt Wittig. The Landau-Zener formula. The Journal of Physical Chemistry B , 109(17):8428–\n8430, 2005.Rydberg dressing 21\n[44] RobertL¨ ow, HendrikWeimer, JohannesNipper, JonathanBBa lewski, Bj¨ ornButscher, HansPeter\nB¨ uchler, and Tilman Pfau. An experimental and theoretical guide t o strongly interacting\nRydberg gases. Journal of Physics B: Atomic, Molecular and Optical Physics , 45(11):113001,\n2012.\n[45] M. Viteau, J. Radogostowicz,M. G. Bason, N. Malossi, D. Ciampin i, O.Morsch, and E. Arimondo.\nRydberg spectroscopyof a Rb MOT in the presence of applied or ion c reated electric fields. Opt.\nExpress, 19(7):6007–6019, Mar 2011.\n[46] Rudolf Grimm, Matthias Weidem¨ uller, and Yurii B. Ovchinnikov. Op tical dipole traps for neutral\natoms.Adv. At. Mol. Opt. Phys. , 42:95–170, 2000.\n[47] Marco Anderlini, Emmanuel Courtade, Donatella Ciampini, J¨ org H . M¨ uller, Oliver Morsch,\nand Ennio Arimondo. Two-photon ionization of cold rubidium atoms. J. Opt. Soc. Am. B ,\n21(3):480–485, Mar 2004.\n[48] Y. Castin, J. I. Cirac, and M. Lewenstein. Reabsorption of light by trapped atoms. Phys. Rev.\nLett., 80:5305–5308, Jun 1998.\n[49] Jonathan B. Balewski, Alexander T. Krupp, Anita Gaj, David Pet er, Hans Peter B¨ uchler, Robert\nL¨ ow, Sebastian Hofferberth, and Tilman Pfau. Coupling a single elect ron to a Bose-Einstein\ncondensate. Nature, 502:664–667, 2013.\n[50] V. Bendkowsky, B. Butscher, J. Nipper, J. P. Shaffer, R. L¨ o w, and T. Pfau. Observation of\nultralong-range Rydberg molecules. Nature, 458:1005–1008, 2009.\n[51] V. Bendkowsky, B. Butscher, J. Nipper, J. B. Balewski, J. P. S haffer, R. L¨ ow, T. Pfau, W. Li,\nJ. Stanojevic, T. Pohl, and J. M. Rost. Rydberg trimers and excite d dimers bound by internal\nquantum reflection. Phys. Rev. Lett. , 105:163201, Oct 2010.\n[52] Bj¨ orn Butscher, Vera Bendkowsky, Johannes Nipper, Jona than B Balewski, Ludmila Kukota,\nRobert L¨ ow, Tilman Pfau, Weibin Li, Thomas Pohl, and Jan Michael Ros t. Lifetimes of\nultralong-rangeRydberg molecules in vibrational ground and excite d states. Journal of Physics\nB: Atomic, Molecular and Optical Physics , 44(18):184004, 2011.\n[53] J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. L¨ ow, a nd T. Pfau. Highly resolved\nmeasurements of Stark-tuned F¨ orster resonances between R ydberg atoms. Phys. Rev. Lett. ,\n108:113001, Mar 2012.\n[54] Alexander T. Krupp et al. in preparation.\n[55] Y. Castin and R. Dum. Bose-Einstein condensates in time depend ent traps. Phys. Rev. Lett. ,\n77:5315–5319, Dec 1996.\n[56] Anita Gaj et al. in preparation.\n[57] E. Amaldi and E. Segr` e. Effect of pressure on high terms of alk aline spectra. Nature, 133:141,\n1934.\n[58] Tino Weber, Jens Herbig, Michael Mark, Hanns-Christoph N¨ ag erl, and Rudolf Grimm. Bose-\nEinstein condensation of cesium. Science, 299(5604):232–235, 2003.\n[59] S. Zhang, F. Robicheaux, and M. Saffman. Magic-wavelength op tical traps for Rydberg atoms.\nPhys. Rev. A , 84:043408, Oct 2011."    },    {        "title": "1305.3864v1.Thermally_induced_spin_polarization_of_a_two_dimensional_electron_gas.pdf",        "content": "arXiv:1305.3864v1  [cond-mat.mes-hall]  16 May 2013Thermally-induced spin polarization of a two dimensional e lectron gas\nA. Dyrda/suppress l1, M. Inglot2, V. K. Dugaev2,3, and J. Barna´ s1,4\n1Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Pozna´ n, Poland\n2Department of Physics, Rzesz´ ow University of Technology,\nal. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland\n3Department of Physics, Instituto Superior T´ ecnico,\nTU Lisbon, av. Rovisco Pais, 1049-001, Lisbon, Portugal\n4Institute of Molecular Physics, Polish Academy of Sciences ,\nul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland\n(Dated: June 24, 2021)\nSpin polarization of a two-dimensional electron gas with Ra shba spin-orbit interaction, induced\nby a thermo-current, is considered theoretically. It is sho wn that a temperature gradient gives rise\nto an in-plane spin polarization of the electron gas, which i s normal to the temperature gradient.\nThe low-temperature spin polarization changes sign when th e Fermi level crosses bottom edge of\nthe upper electronic subband. We also compare the results wi th spin polarization induced by an\nexternal electric field (current).\nPACS numbers: 71.70.Ej, 72.25.Pn, 85.75.-d\nIntroduction – Spin-orbit interaction is responsible for\nmixing of orbital and spin degrees of freedom. This mix-\ning, in turn, gives rise to a variety of interesting phe-\nnomena which are observable experimentally1–5. For in-\nstance, it is well known that insulating materials hav-\ning sufficiently low symmetry can reveal linear magneto-\nelectric phenomena6, which follow from direct coupling\nof magnetic and electric degrees of freedom. Such ma-\nterials can be magnetized electrically, and the induced\nmagnetic moment is linear in electric field. In turn, mag-\nnetic field leads then to a linear electric polarization. The\nsituation is more complex in conducting materials, where\nelectric field is inevitably associated with a charge cur-\nrent. It was predicted long time ago that the electric\ncurrent in a system with spin-orbit (SO) interaction can\ninduce a spin polarization of conduction electrons7, with\nthe polarization vector perpendicular to the direction of\ncurrent and electric field. This phenomenon was studied\nlater in various systems exhibiting SO interaction8–13.\nEdelstein8has considered the spin polarization induced\nby electric field (current) in a two-dimensional (2D) elec-\ntron gas with Rashba spin-orbit interaction, and found\nthat the spin polarization is in the plane of the system\nand is normal to the electric field, as shown schemati-\ncally in Fig.1 (left part). The current-induced spin po-\nlarization in a 2D electron gas for a general case of SO\ninteraction including both Rashba and Dresselhaus terms\nhas been studied theoretically in a recent paper12. The\ncurrent-induced spin polarization has also been proved\nexperimentally1–3,14–17.\nOn the other hand, it is well known that electric cur-\nrent can be also driven by a temperature gradient, like\nfor instance in the Seebeck effect. Thus, combining the\nabove two phenomena, one can expect that spin polar-\nization of a system can be induced by the gradient of\ntemperature (by the thermo-current) as well, as shown\nschematically in Fig.1 (right part). Physical origin of\nspin polarization, however, is now different, as the elec-tric field and the temperature gradient a��ect the car-\nrier distribution in quite different ways. As the applied\nelectric field shifts the Fermi surface of electrons in the\nmomentum space, the temperature gradient makes dif-\nferent ”smearing” of the Fermi surface at different tem-\nperatures.\nFIG. 1. Schematic illustration of the electric-field-induc ed\nspin polarization (left side) and of the thermally-induced spin\npolarization (right side).\nSpin polarization induced by a temperature gradient is\nin the plane of the 2D electron gas. Additionally, the spin\npolarization in the low temperature regime changes sign\nwhen the Fermi level crosses bottom edge of the upper\nsubband. As the spin polarization for the Fermi level\nabove this edge is linear in the chemical potential and\nspin-orbit parameter, this linearity does not hold when\nthe Fermi level is below the edge.\nGeneral considerations – To describe energy transport,\na fictitious ’gravitational’ potential has been introduced\nlong time ago by Luttinger18. Gradient of this potential\nis a driving force for the energy current – analogously\nas gradient of electrostatic potential is a driving force\nfor charge current. It was shown that the transport co-\nefficients corresponding to the ’gravitational’ potential\ncoincide with those appropriate for the temperature gra-\ndient18. The concept of gravitational potential was ex-\nplored in a couple of papers, see e.g. Refs 19 and 20.\nIn this paper we use another approach. Instead of the2\n’gravitational’ potential we introduce an auxiliary (’grav-\nitational’) vector field, which may be considered as an\nanalog of the vector potential in electromagnetism.\nThus, we consider a system that can be described by\nthe Hamiltonian H=H0+V, whereH0includes kinetic\nand spin-orbit interaction terms, while Vis a perturba-\ntion induced by the ’gravitational’ field, which we write\nin the form\nV=−ˆjQ·A. (1)\nHere,Ais the ’gravitational’ vector field, which can be\nalso understood as an auxiliary field, response to which\ngives the heat current, ˆjQ=−∂H/∂A. In turn, the heat\ncurrent operator ˆjQin Eq.(1) is defined as\nˆjQ=1\n2[(H0−µ),v]+, (2)\nwhere [A,B]+=AB+BAandvis the velocity operator.\nApplying the formalism to the well know results on the\nSeebeck effect, we may identify Ai(ω) asAi(ω) =i\nω∇iT\nT.\nThus, applying the imaginary-time (Matsubara) Green\nfunction formalism, one can write the thermally-induced\nspin polarization of the system, linear in ∇T/T, in the\nform\nSj(ωm) =−1\n2∇iT\nTiT\nωmTr/integraldisplayd2k\n(2π)2/summationdisplay\nnσj\n×G0\nk(εn+ωm) [(H0−µ),vi]+G0\nk(εn), (3)\nwhereσjare the Pauli matrices, G0\nk(εn) is the temper-\nature Green’s function corresponding to the Hamilto-\nnianH0, the sum goes over imaginary discrete Matsub-\nara energies εn= (2n+ 1)iπT whileωm= 2miπT for\n(m,n∈Z). Upon calculating the sum over Matsubara\nenergies in Eq.(3) and making analytical continuation to\nthe whole complex plane, one needs to take the limit\nω→0 to find the static spin polarization.21\nSystem and solution – Below we apply this approach to\nthe thermally-induced spin polarization of a 2D electron\ngas with Rashba SO interaction. We will use the units\nwith/planckover2pi1= 1. Hamiltonian H0of the system can be written\nin the form\nH0=εk+α(kyσx−kxσy), (4)\nwhereαis the Rashba SO coupling parameter, and εk=\nk2/2m. We assume, that a temperature gradient is along\nthe axis yand calculate the spin polarization along the\naxisx(the other components vanish). Let us consider\nfirst the case of a finite temperature, when both subbands\nof the electron states described by the Hamiltonian (4)\nare populated for arbitrary chemical potential µ. The\nspin polarization can be calculated from Eq.(3), with the\nGreen function G0\nkgiven by\nG0\nk(εn) =εn−εk+µ+α(kyσx−kxσy)\n(ε−E1k+µ) (εn−E2k+µ). (5)HereE1,2k=εk±αkare the dispersion relations of the\ntwo (upper and lower) electron subbands corresponding\nto the Hamiltonian H0.\nUpon inserting Eq.(5) into Eq.(3), calculating the trace\nand then calculating the sum over Matsubara energies ε\n(by integrating over an appropriate Contour in the Com-\nplex plane), we make analytical continuation to the com-\nplex plane, and then take the limit ω→0. As a result,\none finds the dominant contribution to the static spin\ndensitySxin the form\nSx=∇T\nT/integraldisplaydk\n2πτkεk(εk−µ) [f′(E1k)−f′(E2k)]\n+α\n2∇T\nT/integraldisplaydk\n2πτkk(2εk−µ) [f′(E1k) +f′(E2k)],(6)\nwheref′(ε) is the first derivative of the distribution func-\ntion. In order to take into account relaxation processes,\nwe have also replaced δ→1/2τk, withτkbeing the rele-\nvant relaxation time and δan infinitesimally small num-\nber which emerges from integration over the Contour in\nthe complex plane.21Sinceαis rather small, the deriva-\ntive of the distribution functions can be expanded for\nα≪T, which leads to the following formula for spin\ndensity,\nSx= 2α∇T\nT/integraldisplaydk\n2πτkkεk(εk−µ)f′′(εk)\n+α∇T\nT/integraldisplaydk\n2πτkk(2εk−µ)f′(εk). (7)\nThis formula clearly shows that the leading term in spin\npolarization for α≪Tis linear in α, and that the spin\npolarization vanishes for α= 0.\nThe above expansion, however, is not valid in the limit\nof low temperatures, where the derivatives of the distri-\nbution functions are sharp. We derive now some approxi-\nmate formula for low- Tspin polarization by replacing the\nderivatives in Eq.(6) by the corresponding delta-Dirac\nfunctions. We will distinguish the cases of positive and\nnegative chemical potentials, µ > 0 andµ < 0. In the\nformer case both electron subbands are then occupied\nwith electrons while in the later one only the lower sub-\nband,E2k, is populated. Let us consider first the case of\nµ >0. From Eq.(6) one finds the dominant contribution\n(linear in α) in the form\nSx=αµ∇T\nTm\n2πτkF. (8)\nHere,τkFis the relaxation time at the Fermi wavevec-\ntorkFcorresponding to zero spin-orbit coupling, kF=√2mµ. In turn, when µ < 0, only the lower band,\nE2k, is populated, but there are two Fermi vavectors,\nk±\nF2=αm±/radicalbig\nα2m2+ 2mµ. ¿From Eq.(6) one finds\nthen the dominant contribution in the form\nSx=αµ∇T\nTm\n2πn∗\nnτ0, (9)\nwhereτ0=τkF=0andµ≥ −α2m/2. The latter con-\ndition follows from the position of the lower band edge.3\nApart from this, nis the electron concentration corre-\nsponding to the Fermi level µandn∗is the correspond-\ning electron density when µ= 0. For instance, assuming\nparameters typical of GaAs-based quantum wells, i.e.,\nα= 2×10−9eV·cm,kF= 107cm−1,τ= 10−11s,\nT= 6.5K, and ∆ T= 5K at the sample of length 0.1 cm,\none can estimate spin polarization Sxto be of the order\nofSx≃2×109cm−2.\nNumerical results – In the low temperature regime we\nhave found above some approximate analytical solutions.\nFor higher temperatures, however, we need to find the in-\ntegrals in Eq.(6) or Eq.(7). To do this, we need to know\nthe explicit form of the relaxation time τk. We will con-\nsider a special case, where τkis constant, τk=τ. Such a\nsituation takes place for instance in the case non-ionized\nimpurities22, where one finds 1 /τ= 4π2e4Nim/ǫ2\n0κ2\n0.\nHere,Niis the impurity concentration, ǫ0is the dielectric\nconstant, and κ0is the Thomas-Fermi momentum. The\nlow-temperature approximate solutions are then given by\nEqs (8) and (9) with τkF=τandτ0=τ, respectively.\nIn the finite temperature regime, the integrals in Eqs\n(6) or (7) can be easily calculated by changing the inte-\ngration variable from k to εk. The dependence on tem-\nperature is now more complex as additionally the dis-\ntribution functions contribute to this dependence. Nu-\nmerical results on the spin polarization induced by tem-\nperature gradient are shown in Fig.2. For convenience,\nwe normalized there the spin polarization to τ(∇T/T)\nand the normalized spin polarization is presented as a\nfunction of the chemical potential µfor different tem-\nperatures T. Consider first the upper part of this figure\n(a), which describes the range of positive µ,µ > 0, i.e.\nthe range which is most relevant experimentally. Accord-\ning to Eq.(8), the spin polarization for low temperatures\ngrows linearly with increasing µ, i.e. with increasing\nelectron concentration (at constant ∇T/T). This lin-\near behavior does not hold at higher temperatures (see\nFig.2(a)). Apart from this, keeping constant tempera-\nture gradient one can conclude that the spin polarization\ndecreases with increasing temperature. This is clearly\nshown in the inset in Fig.2(a), where to emphasize the\ntemperature dependence, Sxis normalized to τ∇Tin-\nstead of τ(∇T/T).\nPart (b) of Fig.2, in turn, shows the normalized spin\npolarization in the range of low chemical potentials –\ndown to values below the bottom of the lower subband\n(band edge). We remind the readers, that the band edge\nof the upper subband E2kis shifted by spin-orbit in-\nteraction to a negative energy, indicated in Fig.2(b) as\nthe band edge. The approximate solution for the low-\ntemperature spin polarization, given by Eqs.(8) and (9),\nis also shown in Fig.2(b). As one can easily note, the for-\nmulas (8) and (9) describe very well the low-temperature\nspin polarization, except for chemical potentials very\nclose to the band edge. Apart from this, spin polarization\nat low temperatures changes sign at µ= 0, and becomes\nnegative for negative µ. Whenµis below the band edge,\nboth subbands are empty at T= 0 (electrons are lo-\nFIG. 2. Spin polarization induced by temperature gradient,\nnormalized to τ(∇T/T) and shown as a function of chemi-\ncal potential µfor indicated values of temperature. Part (a)\npresents the situation for µ >0, while part (b) shows spin po-\nlarization for low chemical potentials, down to values belo w\nthe band edge. The inset in (a) shows the temperature de-\npendence of the spin polarization, while the inset in (b) cor re-\nspondsto T= 1K.Theotherparametersare: α= 2×10−11eV\nm,m= 0.05m0, wherem0is free electron mass.\ncalized at the donor states), but for a nonzero Tsome\nelectrons are excited to the 2D subbands and a tail in\nthe spin polarization appears for µbelow the band edge\n(see Fig.2(b)).\nThe above numerical results clearly show that spin po-\nlarization can be induced by a temperature gradient. As\nmentioned already in the introductory part, the spin po-\nlarization can also be induced by a charge current (elec-\ntric field), as shown already by Edelstein8. Below we cal-\nculate the current-induced spin polarization in the whole\ntemperature range and for arbitrary chemical potential.\nThis will allow us to compare the results on spin po-\nlarization induced by a temperature gradient with those\nobtained with an electric field. Moreover, this will also\nprove the theoretical method used in this paper.\nSpin polarization induced by electric field – Now we\nderive some formula for spin polarization due to electric\ncurrent flowing through the system. Instead of a tem-\nperature gradient, however, there is now an electric field\nE= (0,Ey,0). Accordingly, Hamiltonian (1) is now re-\nplaced with V=−ev·A, whereAis the vector potential4\nFIG. 3. Spin polarization induced by electric field, normal-\nized toτeEyand shown as a function of chemical potential\nfor indicated temperatures. Part (a) corresponds to positi ve\nchemical potentials µ >0, while part (b) shows spin polar-\nization for low chemical potentials, down to values below th e\nband edge. The other parameters are as in Fig.2.\nfor electromagnetic field. Following the same methodol-\nogy as above, the xcomponent of spin polarization can\nbe written as\nSx(ωm) =−eEyiT\nωmTr/integraldisplayd2k\n(2π)2/summationdisplay\nnσxG0\nk(εn+ωm)vyG0\nk(εn).\n(10)\nFor finite temperatures and constant relaxation time\nτ, the above formula leads to the following expression for\nstaticSx:\nSx=eEyτ/integraldisplaydk\n2πεk[f′(E1k)−f′(E2k)]\n+αeEy/integraldisplaydk\n2πk[f′E1k) +f′(E2k)], (11)\nwhich can be used for numerical calculations. In turn, the\nspin polarization at T= 0, can be obtained in a similar\nway as the analytical formulas (8) and (9) in the case ofa temperature gradient. For positive chemical potentials\none arrives then at the following formula:\nSx=αeEym\n2πτ, (12)\nwhich coincides with the known expression for spin polar-\nization induced by electric current8,11,13. In turn, when\nµ <0 the corresponding formula reads\nSx=αeEym\n2πn\nn∗τ. (13)\nNumerical results for current-induced spin polariza-\ntion are shown in Fig.3, where the part (a) corresponds\nto positive µ, while part (b) presents spin polarization\nfor low chemical potentials, down to values below the\nband edge when the subbands are populated for nonzero\ntemperatures. As before, the spin polarization increases\nwith increasing µ. There is however no sign change of\nthe spin polarization, contrary to the case of thermally\ninduced spin polarization. Furthermore, spin polariza-\ntion decreases with increasing temperature, except for\nthe chemical potentials in the vicinity of the band edge.\nSummary – We have calculated spin polarization of\na 2D electron gas with Rashba spin-orbit coupling, in-\nduced by a temperature gradient. We have shown that\nthe thermo-current can effectively induce spin polariza-\ntion in the plane of the electron gas and normal to the\ndirection of the temperature gradient. The method we\napplied is based on the concept of ’gravitational’ poten-\ntial, but we used the Green function formalism requiring\nrather an auxiliary ’gravitational’ vector potential. We\nhave shown, that the thermally induced spin polarization\nchanges sign when the Fermi level crosses the band edge\nof the upper subband. The results also show that the\nthermally induced spin polarization decreases with in-\ncreasing temperature. Additionally, we have calculated\nthe electric-field-induced spin polarization in the whole\nrange of chemical potentials and for arbitrary tempera-\nture. Contrary to the case of thermally-induced spin po-\nlarization, there is now no change of the spin polarization\nsign. We note that the phenomenon studied in this pa-\nper is different from the effects of thermally-induced spin\nvoltage (spin thermopower or spin Seebeck effect)23–25.\nThis work was supported by the National Science Cen-\nter in Poland as Projects Nos. DEC-2012/04/A/ST3/\n00372 and partly as the Grant N N202 236540.\nA.D. also acknowledges support by the Adam Mick-\niewicz University Foundation and by the European\nUnion under European Social Fund Operational Pro-\ngramme ”Human Capital” (POKL.04.01.01-00-133/09-\n00). M.I. acknowledges support through the project\nDEC-2011/01/N/ST3/00394. Two of us (A.D. and\nV.K.D.) acknowledge discussion with J. Berakdar and\nhis kind hospitality.\n1Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Phys. Rev. Lett. 93, 176601 (2004).2A. Yu. Silov, P. A. Blaynov, J. H. Wolter, R. Hey, K. H.5\nPloog, and N. S. Averkiev, Appl. Phys. Lett. 85, 5929\n(2004).\n3V.Sih, R.C.Myers, Y.K.Kato, W.H.Lau, A.C. Gossard,\nand D. D. Awschalom, Nature Phys. 1, 31 (2005).\n4A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y.\nLyanda-Geller, and L. P. Rokhinson, Nature Phys. 5, 656\n(2009).\n5T. Jungwirth, J. Wunderlich, and K. Olejnk, Nature Ma-\nterials11, 382 (2012).\n6M. Fiebig, J. Phys. D: Appl. Phys. 38, R123 (2005).\n7M. I. Dyakonov and V. I. Perel, Phys. Letters A 35, 459\n(1971).\n8V. M. Edelstein, Sol. State Communs. 73, 233 (1990).\n9A. G. Aronov and Y. B. Lynda-Geller, JETP Lett 50, 431\n(1989).\n10M.-H. Liu, S.-H. Chen, and C.-R. Chang, Phys. Rev. B 78,\n165316 (2008).\n11C. Gorini, P. Schwab, and M. Dzierzawa, Phys. Rev. B 78,\n125327 (2008).\n12C. M. Wang, H. T. Cui, and Q. Lin, Phys. Status Solidi B\n246, 2301 (2009).\n13P. Schwab, R. Raimondi and C. Gorini, Europhysics Lett.\n90, 67004 (2010).\n14C. L. Yang, H. T. He, L. Ding, L. J. Cui, Y. P. Zeng, J. N.\nWang, and W. K. Ge, Phys. Rev. Lett. 96, 186605 (2006).15N. P. Stern, S. Ghosh, G. Xiang, M. Zhu, N. Samarth, and\nD. D. Awschalom, Phys. Rev. Lett. 97, 126603 (2006).\n16W. F. Koehl, M. H. Wong, C. Poblenc, B. Swenson, U.\nK. Mishra, J. S. Speck, and D. D. Awshalom, Appl. Phys.\nLett.95, 072110 (2009).\n17S. Kuhlen, K. Schmalbuch, M. Hagedorn, P. Schlammes,\nM. Patt, M. Lepsa, G. G¨ untherodt, and B. Beschoten,\nPhys. Rev. Lett. 109, 146603 (2012).\n18J. M. Luttinger, Phys. Rev. B 6, A1505 (1964).\n19L. Smrcka and P. Streda, J. Phys. C: Solid State Phys. 10,\n2153 (1977).\n20N. R. Cooper, B. I. Halperin, and I. M. Ruzin, Phys. Rev.\nB55, 2344 (1997).\n21A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,\nMethods of Quantum Field Theory in Statistical Physics\n(Dover, New York, 1963).\n22T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54,\n437 (1982).\n23K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature (London)\n455, 778 (2008).\n24T.S. Nunner and F. von Open, Phys. Rev. B 84,\n0250405(R) (2011).\n25J. Borge, C. Gorini, and R. Raimondi, Phys. Rev. B 87,\n085309 (2013)."    },    {        "title": "0708.1432v1.Low_temperature_heat_capacity_of_Fe__1_x_Ga__x__alloys_with_large_magneostriction.pdf",        "content": "arXiv:0708.1432v1  [cond-mat.mtrl-sci]  10 Aug 2007Low temperature heat capacity of Fe 1−xGaxalloys with large\nmagnetostriction\nJ. M. Hill\nInstitute for Physical Research and Technology,\nIowa State University, Ames, IA 50011\nR. J. McQueeney∗\nDepartment of Physics and Astronomy and Ames Laboratory,\nIowa State University, Ames, IA 50011\nRuqian Wu\nDepartment of Physics and Astronomy,\nUniversity of California, Irvine, CA 92697\nK. Dennis, R. W. McCallum, M. Huang, and T. A. Lograsso\nAmes Laboratory, Ames, IA 50011\n(Date: August 9, 2007)\nAbstract\nThe low temperature heat capacity Cpof Fe1−xGaxalloys with large magnetostriction has been\ninvestigated. The data were analyzed in the standard way usi ng electron ( γT) and phonon ( βT3)\ncontributions. The Debye temperature Θ Ddecreases approximately linearly with increasing Ga\nconcentration, consistent with previous resonant ultraso und measurements and measured phonon\ndispersion curves. Calculations of Θ Dfrom lattice dynamical models and from measured elastic\nconstants C11,C12andC44are in agreement with the measured data. The linear coefficien t of\nelectronic specific heat γremains relatively constant as the Ga concentration increa ses, despite the\nfact that the magnetoelastic coupling increases. Band stru cture calculations show that this is due\nto the compensation of majority and minority spin states at t he Fermi level.\nPACS numbers: 75.80.+q, 65.40.Ba, 62.20.Dc, 71.15.Mb\n1I. INTRODUCTION\nFe1−xGaxalloys areknown fortheir largemagnetostriction. Values ofthetet ragonalmag-\nnetostriction coefficient,3\n2λ100, can reach values as high as ∼400ppm for certain alloy com-\npositions and heat treatments.1,2,3The sharp rise in the magnetostriction near 19at.%Ga\ncomposition can be explained by a simultaneously increasing magnetoe lastic coupling, b1,\nand decreasing tetragonal shear modulus, C′.1The decrease in C′has been independently\nmeasured via resonant ultrasound techniques1,4,5,6and neutron scattering7. However, the\nnature and characterization of the large increase in magnetoelast ic coupling with composi-\ntion has been difficult to determine. It has been suggested that the increase in b1is related\nto short-range ordered clustering of the Ga atoms prior to the fo rmation of long-range\nordered structures near 19at.%Ga.8Below∼19%Ga, Fe 1−xGaxalloys are disordered and\ncrystallize in a body-centered-cubic ( bcc)α-Fe (A2) structure. Above this composition, two\nordered phases are possible; D0 3and B 2. For itinerant magnetic alloys without significant\nshort-range ordering (local) effects, b1depends on the spin-orbit coupling of electrons near\nthe Fermi level. In strongly ferromagnetic alloys, the compositiona l dependence of the mi-\nnority spin electronic density-of-states (DOS) at the Fermi level, n↓(εF), can be related to\nmagnetostriction.9Measurements of the linear coefficient of the electronic specific hea t at\nlow temperatures, γ, (also called the Sommerfeld constant) are directly proportional t o the\ntotal electronic DOS at the Fermi level, n(εF), and can be used to characterize the ori-\ngins of magnetostriction. This is most clearly demonstrated in Ni-Fe a lloys, where the zero\nmagnetostriction composition (permalloy) corresponds to a full ma jority spin band and a\nminimum in n↓(εF) leading to a minimum in γ, as predicted in the ”split-band model” of\nBergeret al.(Ref.9). We have undertaken a study of the low tempeature heat capacit y\nof Fe-Ga alloys to determine if variations in γare present that can be correlated with the\nlarge increase in magnetostriction. In addition, the lattice contribu tion to the specific heat,\ncharacterized by the Debye temperature Θ D, indicates strong lattice softening with added\nGa in agreement with ultrasound and neutron scattering data.\n2II. SAMPLE PREPARATION\nSingle crystal alloys of Fe 1−xGaxwere grown by the Bridgman technique (see Ref.2for\nmore details of sample preparation). Gallium (99.999% pure) and elect rolytic iron (99.99%\npure) were cleaned and arc melted together several times under a n argon atmosphere. To\npreparesinglecrystalsamples, theas-castingotwasplacedinana luminacrucibleandheated\nunder a vacuum to 1500 C. After reaching 1500 C, the growth cham ber was backfilled with\nultra high purity argon to a pressure of 2.76 x 105Pa. Following pressurization, heating\nwas continued until the ingot reached a temperature of 1600 C and held for 1 hour before\nbeing withdrawn from the furnace at a rate of 5mm/hr. Following cry stal growth, the\ningot was annealed at 1000 C for 168 hours (using heating and cooling rates of 10 degrees\nper minute). Small parallelepipeds (2mm ×1mm×0.5mm) were cut from the ingot by\nwire electrical discharge machining and cleaned by acid etching and po lished on one side.\nSamples were sealed in a quartz tube and annealed at 1000 C for 4 hou rs and furnace cooled\ndown to room temperature. Composition measurements were done by energy-dispersive\nspectrometers (EDS) in a JEOL 840A Scanning Electron Microscope (SEM).\nIII. MEASUREMENTS\nHeat capacity measurements were carried out using a Quantum Des ign Physical Prop-\nerty Measurement System. The addenda were measured separat ely immediately before the\nsample measurement and subsequently subtracted. Heat capacit yCpdata for slow-cooled\nFe1−xGaxalloys are shown in Fig. 1.\nIV. DATA ANALYSIS\nAt lowT, the heat capacity for a soft ferromagnet is given by the formula\nCp=γT+βT3+αT3\n2, (1)\nwhere the terms represent the electronic, phonon, and spin-wav e contributions, respectively.\nThe spin wave contribution has been measured at very low temperat ures and is estimated\nto be about a factor of fifty times smaller than β.10,11,12Our own analysis of the pure α-Fe\n(x= 0) data demonstrates that the heat capacity is not sensitive to t he small spin wave\n3/s48 /s49/s48 /s50/s48 /s51/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48\n/s32/s67\n/s112/s32/s40/s109 /s74/s32/s75/s45/s49\n/s32/s109 /s111/s108/s45/s49\n/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s49/s57/s46/s52/s32/s37\n/s49/s55/s46/s53/s32/s37\n/s49/s53/s46/s52/s32/s37\n/s57/s46/s48/s32/s37\n/s48/s32/s37/s70/s101\n/s49/s45/s120 /s71/s97\n/s120 \nFIG. 1: Heat capacity Cpversus temperature Tdata for slow-cooled Fe 1−xGaxalloys.\nTABLE I: Parameters obtained from fits by Eq. (2) to slow-cool edCp/Tdata.\nat.% Ga γ(mJ\nmolK2)β(mJ\nmolK4) ΘD(K)\n0.0 4.89(1) 0.0187(1) 470.2(8)\n9.0 5.03(1) 0.0276(2) 413(3)\n15.4 4.46(1) 0.0332(1) 388(3)\n17.5 4.69(1) 0.0582(3) 322\n19.4 5.21(1) 0.0521(2) 334(4)\nterm proportional to αbelow 10 K. In this limit, the spin wave term is ignored and the heat\ncapacity can be written in the following form\nCp\nT=γ+βT2. (2)\nThe plot of the data as Cp/Tvs.T2is shown in Fig. (2). The Sommerfeld constant\n(y-intercept) and lattice specific heat coefficient (slope) can then be obtained by a linear\nleast-squares fit to the plot of Cp/Tvs.T2. The parameters obtained from the fits are shown\nin Table I. The Debye temperature, Θ Dcan be derived from βin the procedure described\nbelow and is also shown in Table I. The value obtained for γand Θ Dfor pure α-Fe are\nconsistent with literature values.13,14,15,16,17,18\n4/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s48/s50/s48/s52/s48/s54/s48/s32\n/s32/s49/s57/s46/s52/s32/s37\n/s49/s55/s46/s53/s32/s37\n/s49/s53/s46/s52/s32/s37\n/s57/s46/s48/s32/s37\n/s48/s32/s37/s67\n/s112/s47/s84/s32/s40/s109 /s74/s32/s75/s45/s50\n/s32/s109 /s111/s108/s45/s49\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41/s70/s101\n/s49/s45/s120 /s71/s97\n/s120 \nFIG. 2: Heat capacity divided by temperature Cp/TversusT2for slow-cooled Fe 1−xGaxalloys.\nA. Electronic heat capacity\nFigure 3 shows that the electronic coefficient γremains essentially constant as the Ga\nconcentration increases. At low temperatures, the electronic he at capacity is proportional\nto the DOS at the Fermi level n(εF), according to the formula γ=π2\n3R2n(εF), where Ris\nthe universal gas constant.\nTo better understand our experimental results, we also perform ed density functional\ncalculations for Fe 1−xGaxalloys, using the highly precise full potential linearized augment\nplane wave (FLAPW) method.19No shape approximation is assumed for the charge, poten-\ntial, and wave function expansions in the entire space. We used the g eneralized gradient\napproximation20for the description of the exchange correlation interaction. The c onver-\ngence against parameters such as the number of k-points and energy cutoff was carefully\nmonitored. We used a (2 ×2×2) supercell that comprises 16 atoms throughout the cal-\nculations and we studied cases with x= 0.0 (pure bcc α−Fe),x= 0.0625 (Fe 15Ga1),x=\n0.125 (Fe 14Ga2) andx= 0.1875 (Fe 13Ga3). For the Fe 14Ga2and Fe 13Ga3cells, there are\nseveral different ways to arrange Ga atoms on the bcc lattice sites are related to different\nshort-range ordered structures. Results reported below corr espond to their minimum energy\nconfigurations. Fig. 4 shows the calculated electronic DOSfor Fe 1−xGaxalloys with x= 0.0,\n0.0625, 0.125 and 0.1875, respectively. As xincreases, n(εF) changes oppositely in the two\nspin channels. In the majority spin part, the Fe d-holes are gradually purged and n↑(εF)\n5/s48/s49/s50/s51/s52/s53/s54\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s53/s49/s48/s40/s97/s41\n/s32/s32\n/s32/s32/s40/s109 /s74/s32/s75/s45/s50\n/s32/s109 /s111/s108/s45/s49\n/s41\n/s40/s98/s41\n/s32/s110/s40\n/s70/s41/s32/s40/s115/s116/s97/s116/s101/s115/s32/s74/s45/s49\n/s32/s115/s112/s105/s110/s45/s49\n/s41\n/s120/s44/s32/s65/s116/s111/s109/s105/s99/s32/s112/s101/s114/s99/s101/s110/s116/s32/s71/s97/s32/s40/s37/s41\nFIG. 3: (a) The electronic coefficient of the heat capacity ( γ) of slow-cooled Fe 1−xGaxalloys\nplotted as a function of Ga concentration, x; experimental data (empty circles) and calculated\n(filled circles). In experimental data, the statistical err or bars are smaller than the symbol size.\n(b) Thespin-projected electronic density of states of majo rity (uptriangles, solid line) andminority\n(down triangles, dotted line) at the Fermi level n(εF).\nfalls monotonically. For x=0.15, the Fe- dband in the majority spin channel is completely\nfilled. Meanwhile, the number of non-bonding states aroundthe Fer mi level grows steadily in\nthe minority spin channel. The trend of the n(εF) versusxcurve reasonably matches with\nthe experimental data of γ, as shown in Fig.3. In the calculation, the small dip around x=\n0.15 mainly stems from the elimination of the Fe majority spin d-holes due to the presence\nof Ga atoms. This dip feature is seen also in the experimental data, h owever it is a small\neffect that is on the limit of the sensitivity of the technique.\n6/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48\n/s32/s32\n/s32\n/s32/s32/s110/s40 /s41/s44/s32/s69/s108/s101/s99/s116/s114/s111/s110/s105/s99/s32/s68/s79 /s83/s32/s40/s115/s116/s97/s116/s101/s32/s101/s86/s45/s49\n/s32/s115/s112/s105/s110/s45/s49\n/s41\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\nFIG. 4: The calculated electronic density-of-states for Fe 1−xGaxalloys with black solid lines for\nx= 0 (i.e., pure bcc Fe), green dashed-dotted lines for x= 0.0625, red dashed lines for x= 0.125,\nand blue dotted line for x= 0.1875. The positive side is for the majority spin channel whil e the\nnegative side is for the minority spin channel. Zero energy i s for the position of the Fermi level.\nB. Lattice heat capacity\nFor a single, isotropic phonon mode with sound velocity c, the contribution to the low-\ntemperature molar heat capacity in the Debye model is\nCV=2π2R\n5k3\nBV\n/planckover2pi13c3T3, (3)\nwhereVis the volume of the primitive cell, Ris the gas constant, and /planckover2pi1is Planck’s con-\nstant divided by 2 π. For a general cubic crystal with elastic anisotropy and three pho non\npolarizations, the form of the heat capacity is the same as Eq. (3), withcreplaced by the\neffective sound velocity ¯ c, which is obtained by averaging the inverse-cubed sound velocities\nover all possible propagation directions and modes\n1\n¯c3=/summationdisplay\ni/integraldisplay1\nν3\ni(θ,φ)dΩ. (4)\n7/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48/s52/s53/s48/s53/s48/s48\n/s32/s32/s68/s44/s32/s68/s101/s98/s121/s101/s32/s84/s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s120/s44/s32/s65/s116/s111/s109/s105/s99/s32/s112/s101/s114/s99/s101/s110/s116/s32/s71/s97/s32/s40/s37/s41/s70/s101\n/s49/s45/s120 /s71/s97\n/s120 \nFIG. 5: The Debye temperature Θ Dplotted as a function of Ga concentration xas determined\nfrom the data are shown as solid points ( •). The solid line is a linear fit to the slow-cooled data.\nVarious calculated Θ Dvalues are indicated by the open circles ( ◦), as referenced in the text.\nwhereνi(θ,ϕ) is the sound velocity in a crystalline direction given by θ,ϕ. The Debye\ntemperature is defined as\nΘD=/planckover2pi1¯c\nkB/parenleftbigg6π2\nV/parenrightbigg1\n3\n, (5)\nsuch that the low-temperature heat capacity can be written\nCV=12π4R\n5T3\nΘ3\nD=βT3(6)\nand the slope of CV/TversusT2equalsβ= (1943.9J mol−1K−1)Θ−3\nD. Experimentally de-\ntermined values of Θ Dare given in Table I and are also shown in Fig. 5. Θ Ddecreases\napproximately linearly with increasing Ga concentration, consistent with previous measure-\nments of the tetragonal shear modulus C′via resonant ultrasound5and neutron scattering7.\nIn an effort to affirm the validity of the measured Debye temperatur es, we numerically\ncalculated Θ Dvia three methods outlined below. These various estimates of Θ Dare also\nshown in fig. 5.\n(1). Integral: The simplest way to calculate the Debye temperature from a lattice\ndynamical model is to calculate the heat capacity by integration. Giv en the model lattice\ndynamical parameters (obtained from fits to inelastic neutron sca ttering data, for example),\n8/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s48 /s53/s48/s48 /s49/s48/s48/s48/s48/s49/s48/s50/s48/s51/s48\n/s32/s32/s103/s40/s69/s41/s44/s32/s68/s79 /s83/s32/s40/s109 /s101/s86/s45/s49\n/s41\n/s45/s70/s101/s97/s41\n/s32/s32/s103/s40/s69/s41/s47/s69/s50\n/s32/s40/s49/s48/s45/s53\n/s32/s109 /s101/s86/s45/s51\n/s41\n/s69/s44/s32/s80/s104/s111/s110/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s52/s46/s55/s120/s49/s48/s45/s53\n/s32/s109/s101/s86/s45/s51/s98/s41\n/s68/s61/s52/s54/s52/s32/s75\n/s32/s32/s67\n/s86/s47/s84/s32/s40/s109 /s74/s32/s75/s45/s50\n/s32/s109 /s111/s108/s45/s49\n/s41\n/s84/s50\n/s32/s40/s75/s50\n/s41/s61/s49/s46/s57/s52/s120/s49/s48/s45/s53\n/s32/s74/s32/s32/s75/s45/s52\n/s32/s109/s111/s108/s45/s49/s99/s41\nFIG. 6: (a) Phonon density-of-states g(E) versus energy Eforα-Fe as determined from a force\nconstant model. (b) g(E)/E2versusEshowing the limiting value (dashed line) as Eapproaches\n0meV. (c) CV/TversusTfor the data (empty circles), and as calculated from Eqn. (7) (solid\nline). The best linear fit to the calculation (dashed line) gi ves a Debye temperature of 464 K.\nthe phonon density-of-states, g(E), can be calculated. The phonon DOS can then be used\nto calculate the heat capacity according to\nCV= 3R/integraldisplay∞\n0g(E)/parenleftbiggE\nkBT/parenrightbigg2eE\nkBT\n/bracketleftBig\neE\nkBT−1/bracketrightBig2dE. (7)\nAt low temperatures, the result of this calculation can be fit to the lim iting form in Eq. (6)\n9to obtain Θ Din the same way as it was obtained from the measured data above. Fig . 6(a)\nshowsthecalculatedDOSfor α-Fewheretheforceconstantsareobtainedbyfittingtheroom\ntemperature phonon dispersion curves measured by inelastic neut ron scattering.21Using this\nDOS, we can calculate the heat capacity at low temperatures via Eq. (7) and, subsequently,\nfit the curve up to 10K as shown in Fig. 6(c). The fitted slope, 1.94 ×10−5Jmol−1K−4, can\nthen be used to obtain Θ D= 464K via Eq. (6).\n(2). Extrapolation: One can obtain Θ Ddirectly from the DOS without calculating the\nheat capacity. The T3dependence arises from the low energy quadratic energy depende nce\nof the DOS (resulting from a linear dispersion relation). Plotting lim E→0g(E)/E2gives the\ncoefficient of the low energy quadratic form of the DOS, which leads d irectly to the average\n¯c:\n¯c=/bracketleftBigg\nV\n2π2/parenleftbigg\nlim\nE→0g(E)\nE2/parenrightbigg−1/bracketrightBigg1\n3\n. (8)\nThen, Eq. (8) can be used to calculate Θ D. Fig. 6(b) shows g(E)/E2as calculated for\nα-Fe. The (average) zero frequency limit of this function has a value of approximately\n4.7×10−5meV−3. Using Eqs. (5) and (8), Θ Dis calculated to be 463K.\n(3) Elastic constants: Additionally, the Debye temperature can be calculated directly\nfrom the elastic constants. Any cubic system can be described by t hree independent elastic\nconstants; C11,C12andC44. Generally these are experimentally measured as two transverse\nmodes,C44(rhombohedral shear) and C′= 1/2(C11−C12) (tetragonal shear), and one\nlongitudinal mode, K= 1/3(C11+ 2C12) (bulk modulus). Given these elastic constants,\nthe sound velocities νi(θ,φ) can be calculated for any propagation direction ( θ,φ) using the\nGreen-Cristoffel equations:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(C11−C44)q2\nx+C44q2−ρω2(C12+C44)qxqy (C12+C44)qxqz\n(C12+C44)qyqx(C11−C44)q2\ny+C44q2−ρω2(C12+C44)qyqz\n(C12+C44)qzqx (C12+C44)qzqy(C11−C44)q2\nz+C44q2−ρω2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.\n(9)\nSolutions of Eqs. (9) yield the sound velocities νj(q) =ωj(q)/qfor each phonon branch\nalong the crystal direction given by wave vector q. The average sound velocity can then\nbe evaluated for an anisotropic cubic crystal by numerical averag ing as shown in Eq. (4).\nSubsequently, Eq. (5) can be used to calculate Θ D.\n10Forα-Fe, the zero-Kelvin elastic constants areextrapolated to be C11= 243.1GPa, C12=\n138.1GPa and C44= 121.9GPa.11Calculation of the Debye temperature based on the above\nmethod gives Θ D= 478K which compares favorably to the value determined from fits t o\nthe low temperature calorimetry data, as shown in Fig. 5. This can als o be compared to the\nresult obtained via the de Launay formula; a general semi-analytic f unction to determine\nΘD(T) for cubic metals from the elastic constants.22,23,24Given the elastic constants listed\nabove, this formula yields Θ D(T= 0K) = 477K.11\nThe elastic constants for quenched Fe 81.3Ga18.7were independently measured by Clark\net al. (Ref.1) as a function of temperature. These were extrapolated to 0K an d substi-\ntuted into the de Launay formula yielding Θ D(T= 0K) = 386K. They were also used via\nEqs. (9, 4 and5) to calculate Θ D= 396K. Ascan beseen inFig. 5, these values aregenerally\nconsistent with trends in the data.\nV. DISCUSSION\nWe find only a weak correlation of γandλ100with composition. Band structure calcu-\nlations within density functional theory show that the small change inγis caused by the\ncancellation of two effects; a simultaneous depletion of holes in the ma jority spin band and\nthe expected increase in n↓(εF) in the minority band. Thus, while the heat capacity is not a\nsensitive probe of the relevant electronic states for magnetostr iction in the case of Fe 1−xGax,\nthe agreement with band structure calculations is a positive step. T he increase in n(εF)\nafterx >0.15 manifests the presence of non-bonding states around the Fe rmi level in the\nminority spin channel. As revealed in our previous studies,8these states play a key role for\nthe increase of λ001and the role of the effect of short-range ordering is an open quest ion.\nOf course, more analysis on the details of wave functions, such as t heir magnetic quantum\nnumbers, are needed for correct prediction. Results concerning the theoretical prediction of\nλ001will be published elsewhere.\nIn addition to the electronic behavior, the low temperature heat ca pacity can indepen-\ndently probe the lattice softening by determination of the Debye te mperature. We find that\nthe decrease of the Debye temperature agrees with previous est imates of lattice softening.\n11VI. SUMMARY\nWe have measured the heat capacity as a function of temperature for Fe 1−xGax, 0.0\n< x <0.194 solid solutions crystallizing in the bccstructure. The Debye temperatures\nfollow a linearly decreasing trend with increasing Ga concentration co nsistent with known\nlattice softening. The electronic coefficient of the specific heat rem ains relatively constant,\nin agreement with band structure calculations.\nVII. ACKNOWLEDGMENTS\nWe thank S. Bud’ko and R. Rink for performing additional heat capac ity measurements.\nAmes Laboratory is supported by the U.S. Department of Energy u nder Contract No. DE-\nAC02-07CH11358. This project is partially supported by the Office o f Naval Research under\nONR grant No. N000140610530.\n∗Electronic address: mcqueeney@ameslab.gov\n1A. E. Clark, K. B. Hathaway, M. Wun-Fogle, J. B. Restorff, T. A. L ograsso, V. M. Keppens,\nG. Petculescu, and R. A. Taylor, J. Appl. Phys. 93, 8621 (2003).\n2A. E. Clark, M. Wun-Fogle, J. B. Restorff, T. A. Lograsso, and J. R. Cullen, IEEE Trans. Magn.\n37, 2678 (2001).\n3A. E. Clark, J. B. Restorff, M. Wun-Fogle, T. A. Lograsso, and D. L. Schlagel, IEEE Trans.\nMagn.36, 3238 (2000).\n4G. Petculescu, K. B. Hathaway, T. A. Lograsso, M. Wun-Fogle, and A. E. Clark, J. Appl. Phys.\n97, 10M315 (2005).\n5M. Wuttig, L. Dai, and J. Cullen, Appl. Phys. Lett. 80, 1135 (2002).\n6P. Mungsantisuk and S. Guruswamy, in EPD Congress 2005, Proceedings of Sessions and Sym-\nposia held during the TMS Annual Meeting , edited by M. E. Schlesinger, The Minerals, Metals\n& Materials Society (TMS, Warrendale, PA, 2005), pp. 193–19 8.\n7J. L. Zarestky, V. O. Garlea, T. A. Lograsso, D. L. Schlagel, a nd C. Stassis, Phys. Rev. B 72,\n180408(R) (2005).\n8R. Wu, J. Appl. Phys. 91, 7358 (2002).\n129L. Berger, Physica B+C 91, 31 (1977).\n10M. Hatherly, K. Hirakawa, R. D. Lowe, J. F. Mallett, M. W. Stri ngfellow, and B. H. Torrie, J.\nAppl. Phys. 35, 802 (1964).\n11J. A. Rayne and B. S. Chandrasekhar, Phys. Rev. 122, 1714 (1961).\n12P. S. Mahesh and B. Dayal, Phys. Rev. 143, 443 (1966), and cited references.\n13G. Duyckaerts, Physica 6, 401 (1939).\n14W. H. Keesom and B. Kurrelmeyer, Physica 6, 633 (1939).\n15M. Dixon, F. E. Hoare, T. M. Holden, and D. E. Moody, Proc. Roy. Soc. London A 285, 561\n(1965).\n16S. S. Shinozaki and A. Arrott, Phys. Rev. 152, 611 (1966).\n17M. B. Stearns, in Landolt-B¨ ornstein, Numerical Data and Functional Relati onships in Science\nand Technology, New Series , edited by K.-H. Hellwege and O. Madelung (Springer-Verlag ,\nBerlin, 1986), vol. 19a of III, chap. 1, p. 118.\n18D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill Book Company, New\nYork, 1972), chap. 4, p. 115, 3rd ed.\n19E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Phys. R ev. B24, 864 (1981).\n20J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996), copyright (C)\n2007 The American Physical Society Please report any proble ms to prola@aps.org PRL.\n21V. J. Minkiewicz, G. Shirane, and R. Nathans, Phys. Rev. 162, 528 (1967).\n22J. de Launay, Solid State Physics 2, 219 (1956).\n23J. de Launay, J. Phys. Chem. 21, 1975 (1953).\n24J. de Launay, J. Phys. Chem. 22, 1676 (1954).\n13"    },    {        "title": "2311.09534v1.Dissipation_Enhanced_Unidirectional_Transport_in_Topological_Systems.pdf",        "content": "Dissipation Enhanced Unidirectional Transport in Topological Systems\nMing Lu,1Xue-Zhu Liu,2Hailong Li,3Zhi-Qiang Zhang,4, 5,∗Jie Liu,6,†and X.C. Xie3, 7, 8\n1Beijing Academy of Quantum Information Sciences, Beijing 100193, China\n2Center for Advanced Quantum Studies, Department of Physics,\nBeijing Normal University, Beijing 100875, China\n3International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China\n4School of Physical Science and Technology, Soochow University, Suzhou 215006, China\n5Institute for Advanced Study, Soochow University, Suzhou 215006, China\n6Department of Applied Physics, School of Science, Xian Jiaotong University, Xian 710049, China\n7Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China\n8Hefei National Laboratory, Hefei 230088, China\n(Dated: November 17, 2023)\nDissipation is a common occurrence in real-world systems and is generally considered to be detri-\nmental to transport. In this study, we examine the transport properties of a narrow quantum\nanomalous Hall system with dissipation applied on one edge. When the Fermi level resides within\nthe hybridization gap, we find that while transport is suppressed on one edge, it is significantly en-\nhanced on the other. We reveal that this enhancement arises from dissipation-induced gap closure,\nwhich is deeply rooted in the point gap topology of the system, resulting in a reduction of the de-\ncaying coefficient. When the dissipation is very large, we find that the low-energy physics is nearly\nindistinguishable from a narrower system, whose dissipation amplitude is inversely proportional\nto that of the original one. To get more physical intuition, we demonstrate that the low-energy\nphysics can be well captured by a pair of coupled counter-propagating chiral edge states, one of\nwhich has a modified group velocity and an effective dissipation. We also briefly discuss the possible\nexperimental realizations of this enhanced unidirectional transport.\nI. INTRODUCTION\nNon-Hermitian (NH) physics has drawn intensive re-\nsearch interests recently[1–4]. Many unique features\nhave been uncovered which has no Hermitian coun-\nterparts, including exceptional degeneracies[5–21], ex-\ntended topological classifications[22–27], and NH skin\neffect (NHSE)[28–30]. Among them, NHSE is of par-\nticular importance. It leads to the failure of conven-\ntional bulk-boundary correspondence and motivates the\ncrucial concepts such as generalized Brillouin zone and\npoint gap topology[31–42]. Macroscopic states are ag-\ngregated near the boundary of a system in NHSE, orig-\ninating from the point gap topology that exists only in\nNH Hamiltonians[43, 44]. Meanwhile, the protection of\nthe real line gap may also lead to the localization of states\nnear the system boundary, namely the topologically pro-\ntected edge states[45–50]. A growing trend is the study\nof the interplay between these two kinds of topology and\ntheir corresponding boundary phenomena[51–68]. As a\nnontrivial example, when the NHSE is manifested only on\nthe topological edge modes but not on the bulk, higher\norder skin effect becomes possible. It shows higher di-\nmensional localization without the requirement of high-\norder topology. To realize this, several studies have been\nput forward[51–60]. A rather universal and convenient\nway is proposed in Ref. [60], in which a constant dissi-\npation term is applied on one of the ( d−1)-dimensional\nsurfaces of a d-dimensional topological material. As we\nknow, the topological edge states play an essential role in\nthe transport properties of quantum materials, whereasthe study of the interplay between dissipation and the\ntopological edge states is still in its infancy[69–75]. Pre-\nvious works have shown that dissipation at the end of a\none-dimensional Majarana chain can stabilize Majarana\nzero mode, improving its potential for topological quan-\ntum computing[76–78]. Yet, transport studies on the ef-\nfects of dissipation on topological edge states in higher\ndimensional topological materials are still rare.\nIn this work, we investigate the transport properties\nof a narrow quantum anomalous Hall system with dissi-\npation applied upon one of its edges. When the Fermi\nlevel resides inside the hybridization gap due to finite\nsize effect[79], we find that with appropriate dissipation\nstrength, transport can be significantly enhanced on the\nopposite edge. The reason for this enhancement is due to\nthe closing of the real energy gap by dissipation, which\nresults in a reduction of the decay rate for the current\ndensity. As we increase the dissipation amplitude from\nzero, we find that the real part of the band gap first\ncloses and then reopens, while the imaginary part first\nreopens and then closes, resulting in two separate excep-\ntional transitions. When the dissipation becomes very\nlarge, we find the low-energy physics is almost identical\nto that of a narrower system, with very small dissipation\namplitude that is inversely proportional to the original\none. To gain more physical insight, we show that low-\nenergy physics can be effectively described by a pair of\ncounter-propagating chiral edge states that are coupled\ntogether. One of these states has a modified group ve-\nlocity and an effective dissipation. Experimentally, we\nbriefly discuss some possible realizations of this enhancedarXiv:2311.09534v1  [cond-mat.mes-hall]  16 Nov 20232\nunidirectional transport phenomenon.\nII. MODEL SYSTEM WITH EDGE\nDISSIPATION\nWe consider a quantum anomalous Hall model pro-\nposed by Qi, Wu and Zhang (QWZ) [80]\nH= sin kxσx+ sin kyσy+ (m+ cos kx+ cos ky)σz.(1)\nThis model is topological as long as |m|<2. When\na constant dissipation is added at the y= 0 edge, the\nHamiltonian for a ribbon with Nysites in the ydirection\nreads\nH(kx) =Ny−1X\nn=0[sinkxσx+ (m+ cos kx)σz]c†\nncn\n+Ny−2X\nn=0\u0014\n(i\n2σy+1\n2σz)c†\nn+1cn+ h.c.\u0015\n−iγσ0c†\n0c0.\n(2)\nWe fix the parameters m=−1.85 and Ny= 16 in the\nfollowing unless otherwise stated.\nAs shown in Fig. 1(a), in the absence of dissipation,\nthe hybridization of the counter-propagating chiral edge\nstates occurs due to finite system width, opening up a\nsmall band gap around zero energy. As the dissipation\nis applied upon the y= 0 edge, the gap decreases and\nthe energy bands develop a gapless imaginary part, as\nshown in Fig. 1(b). When γ=γc1[Fig. 1(c)], both the\nreal and the imaginary band gaps close and the system\nhas an exceptional degeneracy. Increasing the dissipa-\ntion further, the imaginary part of the bands develops\na gap and the real part remains gapless, see Fig. 1(d)\nand Fig. 1(e). This behavior holds until arriving at the\nsecond exceptional point γ=γc2[Fig. 1(f)], where both\nthe real and imaginary parts of band gap become gapless\nagain. Beyond γc2, the real part of the gap opens and\nthe imaginary part keeps gapless, as shown in Fig. 1(g).\nFrom Fig. 1(b) to Fig. 1(g), we can clearly see that the\nmagnitude of the imaginary parts of the right-moving\nstates are smaller than that of the left-moving ones, ex-\ncept at the rare places where exceptional degeneracies\noccur. This is simply because the dissipation is added\naty= 0 sites, which is far from the right-moving chi-\nral edge states. The dissipation is transferred from the\nbottom to the top by the hybridization effect, giving the\nright-moving chiral edge state an effective dissipation.\nTherefore, the effective dissipation for the right-moving\nstate is unavoidable given the existence of the coupling\nof the counter-propagating edge state and is expected to\nrapidly decrease with the increasing of the system width.\nThe evolution of the energy gap as a function of dis-\nsipation strength can be more clearly seen in Fig. 1(h).\nAs shown in Fig. 1(h1), the real part of the energy gaprapidly decreases to zero from γ= 0 to γc1, keeping at\nzero between γc1andγc2, and then increases from zero\nforγ > γ c2and gradually saturates to a new gap value,\nwhich is slightly larger than the gap without dissipation.\nThe closing of the hybridization gap is a result of the large\nmismatch of the imaginary part of the energies between\nthe counter-propagating chiral edge states[79, 81, 82], as\nwell as the point gap topology dictated by the extended\nNielsen-Ninomiya theorem[83]. For large enough dissipa-\ntion, the system behaves like a finite width ribbon with\none slice of sites narrower and with very small dissipation,\nleading to the increased coupling between edge states and\nthus a larger hybridization gap (see also Sec. V). The\nimaginary parts of chiral edge states at kx= 0 are shown\nin Fig. 1(h2), which bifurcate at γc1and then merge\natγc2, and finally slowly decrease towards zero for very\nlarge dissipation. The relatively small imaginary part for\nthe left-moving edge states when γis very small or very\nlarge indicates the effective dissipation is small in these\nregimes; while the imaginary part of it is large between\nthe two critical points, indicating the effective dissipation\nis large.\nThe low-energy physics can be well captured by a\ncontinuum model, which describes two coupled counter-\npropagating chiral edge states with effective dissipation\napplied on one of them. The effective model reads\nHeff(kx) =\u0012vkx ∆\n∆−αvkx−iγeff\u0013\n, (3)\nwhere vandαvare the group velocities of the right and\nleft-moving edge states, respectively; and ∆ is the cou-\npling strength of them. γeffis the effective dissipation,\nwhich differs from the bare dissipation γapplied on the\nlattice edge. The band structure can be easily obtained\nE±(kx) =−iγeff+ (α−1)vkx\n2±s\u0012(α+ 1)vkx+ iγeff\n2\u00132\n+ ∆2.\n(4)\nNote that ∆, αandγeffdepend on γ, and can be de-\ntermined in the following sense. At kx= 0, E±=\n−iγeff/2±p\n∆2−γ2\neff/4, giving γeff=−Im (E++E−)\nand ∆ =p\n(E+−E−)2+γ2\neff/4. We fix v= 1 from the\ninspection of the Hermitian limit and determine αby\nthe slope of E+(kx) +E−(kx). The dependence of these\nparameters on γis given in Fig. 2. We can see the cou-\npling strength ∆ first increases, reaching the peak value\nat about γ≈1, and then approaches a new larger value.\nThe effective dissipation also peaks at about γ≈1 and\ngradually decreases to zero for larger γ. Notably, we have\nequal group velocities for the small and large γ, while\nthey become rather unbalanced in the intermediate range\n(see also Appendix A). With these parameter values, the\nband structures calculated from the lattice model can be\nwell fitted by Eq. (4), as shown by the black crosses in\nFig. 1.3\n0.1\n 0.0 0.10.1\n0.00.1Ek\n(a1)\n0.1\n 0.0 0.10.1\n0.00.1ReEk\n(b1)\n0.1\n 0.0 0.10.1\n0.00.1\n(c1)\n0.1\n 0.0 0.10.1\n0.00.1\n(d1)\n0.1\n 0.0 0.10.1\n0.00.1\n(e1)\n0.1\n 0.0 0.10.1\n0.00.1\n(f1)\n0.1\n 0.0 0.10.1\n0.00.1\n(g1)\n0 200.000.04ReEgap(h1)\n1\n0 1\nkx1\n01Ek(a2)\n0.1\n 0.0 0.1\nkx0.00\n0.03\nImEk\n(b2)\n0.1\n 0.0 0.1\nkx0.00\n0.02\n0.04\n(c2)\n0.1\n 0.0 0.1\nkx0.00\n0.02\n0.04\n(d2)\n0.1\n 0.0 0.1\nkx0.00\n0.02\n0.04\n(e2)\n0.1\n 0.0 0.1\nkx0.00\n0.02\n0.04\n(f2)\n0.1\n 0.0 0.1\nkx0.00\n0.02\n0.04\n(g2)\n0 20\n0.00.10.2ImE0(h2)\n|ImEN1|\n|ImEN|00.20.04\n00.20.00.1\nFIG. 1. Band structures of the ribbon with width Ny= 16 and m=−1.85 and different dissipation magnitudes, where in (a)\nγ= 0; (b) γ= 0.13; (c) γ=γc1= 0.14894; (d) γ= 0.17; (e) γ= 5.5; (f) γ=γc2= 5.76628; (g) γ= 6.0. The continuously\nchanging color represents the magnitude of the imaginary part, and the black crosses represent the fit from Eq. 4. The gap\nof the real energy band (h1) and imaginary parts of two edge states with kx= 0 (h2) as a function of γ. The insets are the\nenlarged view for the small γvalues.\n0 2 4 6 8\n0.00.10.2\neff\n(a)\n0 2 4 6 8\n0.0200.0250.030\n(b)\n1.01.31.61.9\nFIG. 2. Fitted parameters for the continuum model Eq. (3).\n(a)γeffas a function of γ. (b) ∆ (green) and α(orange) as a\nfunction of γ.\nIII. DISSIPATION ENHANCED\nUNIDIRECTIONAL TRANSPORT\nWe have shown that edge dissipation can effectively\nmanipulate the low-energy band structures, which mo-\ntivates us to study its impact on transport properties.\nWe consider a two terminal device with two semi-infinite\nmetal leads connected to a finite width system described\nby Eq. (2). The two terminal linear conductance is\ncalculated by Gpq(E) = Tr[Γ pGrΓqGa], where Γ p=\ni(Σr\np−(Σr\np)†) is the line width function of lead p(=L, R),\nand Σr\npis the self-energy due to the coupling of the lead\npwith the central region[84, 85]. The retarded Green’s\nfunction is defined as Gr(E) = ( E−Hc−P\npΣr\np)−1,\nwith Hcrepresent the finite size lattice Hamiltonian of\nthe central region. We can also calculate the local current\nbetween neighboring site iandjusing the formula[85, 86]\nJij=2e2V\nhImX\nα,βHiα,jβGp\njβ,iα, (5)\nwhere the sum is over the internal degree of freedom α, β\nof the lattice site. Gp=GrΓpGais the electron correla-\n104\n102\n100102104\n0.00.20.40.6 G(e2/h)(a)\nGRL\nGLR\n104\n102\n100102104\n0.00.51.0(b)\nGRL\nGLR\n010y(c)\n010y(d)\n010y(e)\n0 20 40 60 80 100 x010y(f)\n0.10.20.3\n0.10.2\n0.10.20.3\n0.00.10.2FIG. 3. Two terminal conductance GRLandGLRforEf=\n0.01 (a) which is inside the hybridization gap and Ef= 0.045\n(b) which is outside the hybridization gap as a function of\nγ. Current density distribution for γ= 0 (c) and γ= 0.8\n(d) when VL> VR, and for γ= 0 (e) and γ= 0.8 (f) when\nVL< VR.\ntion function and Vis the voltage difference between the\nleft and right leads.\nIn Fig. 3, we show the linear conductance and local\ncurrent distribution for a system with Nx= 100. When\nthe Fermi level is in the gap, as shown in Fig. 3(a) with\nEf= 0.01, we can clearly see that the conductance from4\nleft to right and from right to left are distinct for a wide\nrange of dissipation values. For some parameter regimes,\nthe unidirectional transport is found where GLRis al-\nmost zero while GRLis finite, which is a manifestation of\nNHSE and point gap topology in the context of quantum\ntransport[87, 88]. Quite amazingly, dissipation can even\nenhance the transport from left to right, where GRLrises\nas dissipation increases from zero and reaches the peak\nwhen γ≈0.8. In the Hermitian limit where there is no\ndissipation, we have GLR=GRL̸= 0. The nonzero\nvalue of the conductance is the result of finite length\nwhich allow part of the current tunneling through, see\nalso Fig. 3(c) and Fig. 3(e). When γis very large, we\ncan see GRLandGLRbecome identical but smaller than\nthat when γ= 0, indicating for very large dissipation, the\nsystem behaves like a Hermitian system with a slightly\nlarger gap value. When Fermi energy Ef= 0.045 is out of\nthe gap as shown in Fig. 3(b), we also find the dissipation\ninduced unidirectional transport for the intermediate dis-\nsipation regime. For very large or very small dissipation,\nthe system is in the Hermitian limit and the transmission\nprobability is large and tends to unity.\nThe enhanced transport can be more clearly seen in\nthe local current distribution in Fig. 3(c) with γ= 0\nand Fig. 3(d) with γ= 0.8, where the left lead has the\nhigher voltage. As shown in in Fig. 3(c), most of the\ncurrent coming from left lead on top of the sample is\nreflected back to the left-moving channel on the bottom,\nand only a small fraction is managed to transmit through\nthe sample to the right lead. On the other hand, when\nthe dissipation is added, the left-moving channel in the\nbottom has a strong dissipation and thus is unable to\ncarry the otherwise reflected current. Therefore, the re-\nflection is suppressed and the transmission is enhanced,\nas clearly shown in Fig. 3(d). The unidirectional trans-\nport is also clearly seen by comparison between Fig. 3(f)\nand Fig. 3(d). As mentioned, the large effective dissipa-\ntion destroys the bottom left-moving edge state, so that\nthere is no channel to carry the current from right to left.\nIV. SUPPRESSION OF THE DECAY RATE BY\nDISSIPATION\nThe enhancement of the transport by dissipation can\nalso be explained by the decreasing of the imaginary\npart of the wave vector. As shown in Fig. 4(a), we\ncalculate GRLby changing the system length Nxfor\nseveral dissipation amplitudes with fixed Fermi energy\nEf= 0.01. As clearly shown, GRLdecays exponentially\nwith respect to the system length. The decay rate varies\nnon-monotonically when increasing γ: for γsmaller than\n0.8, the slope decreases; while for γlarger than 0 .8, the\nslope increases. Fig. 4(b) shows the decay rate for dif-\nferent Fermi energies with respect to γ, which agrees\nwith the observation in Fig. 4(a). For Efresiding in-\n200 400 600 800\nNx1010\n107\n104\n101\nGRL(e2/h)\n(a)\n=0.000\n=0.025\n=0.050\n=0.800\n=2.000\n=4.000\n=8.000\n0 2 4 6 8\n0.000.010.020.03\n(b)\nEf=0.010\nEf=0.020\nEf=0.030\nEf=0.045FIG. 4. (a) GRLas a function of Nxfor different γ’s when\nEf= 0.01. The dots are the numerical results from transport\ncalculations and the lines are the exponential fit Ae−βNx. (b)\nDecay rate βas a function of γfor different Fermi energies.\nThe dots are the exponential fitted values and the lines are\nthe analytical results from Eq. (6).\nside the hybridization gap, the decay rate starts from\na nonzero value and decreases rapidly with γand then\nslowly increases again. On the other hand, when Efis\noutside the gap, the decay rate is zero at the beginning,\nrapidly increasing when γis small, and then varies non-\nmonotonically with the dissipation amplitude.\nThe decay rate can be extracted from the continuum\nmodel Eq. (3). To do this, we make the substitution kx→\n−i∂xand assume the right-moving state with energy E\nhas the form ψE(x) =eiqxψE(0). We can then get the\nexpression for the wave vector qby solving the Sh¨ ordinger\nequation Heff(x)ψE(x) =EψE(x). We obtain\nq±=1\n2αvh\n(α−1)E−iγeff±p\n[(α+ 1)E+ iγeff]2−4α∆2i\n.\n(6)\nClearly, q+is the corresponding wave vector for the state\nmoving along the positive xdirection. The current car-\nried by state ψE(x) isJ(x) =eℏ\nmIm[ψ∗(x)∂xψ(x)]∝\ne−2Imq+x. Therefore, the decay rate of the GRLwith\nrespect to the system length is simply β(E, γ) =\n2Imq+(E, γ), where we have implicitly use the fact that\nparameters such as γeff, αand ∆ are functions of γ. We\nplot the decay rate in Fig. 4(b) using the analytical ex-\npression Eq. (6), which nicely agrees with the data points\nfrom the linear conductance calculations. From the above\nanalysis, we can see that the enhancement of the trans-\nport from left to right is a result of dissipation enabled de-\ncreasing of the imaginary wave vector of the right-moving\nedge states.\nV. THE LARGE DISSIPATION LIMIT\nIn this section, we explicitly show that in the large\ndissipation limit, the low-energy physics of the original\nsystem is almost identical to a system that is one slice\nnarrower with very small dissipation applied on the edge.\nIn Fig. 5(a) and Fig. 5(b), we plot the band structure of\nthe system with Ny= 16 and γ= 20 using the continu-5\n0.1\n 0.0 0.1\nkx0.1\n0.00.1ReEk(a)\n0.1\n 0.0 0.1\nkx0.0\n0.5\n1.0\n1.5\nImEk1e2\n (b)\n101102103104\n(Ny=16)\n104\n103\n102\n101\n(Ny=15)\n(c)\n051015y(d)\n0 20 40 60 80 100x051015y(e)\n0.10.20.3\n0.10.20.3\nFIG. 5. Real (a) and imaginary (b) part of the low-energy\nbands. The continuous changing color lines (black circles)\nrepresent the lattice model with Ny= 16 and γ= 20 ( Ny=\n15 and γ= 0.05). (c) The dissipation magnitudes for system\nwith Ny= 16 and the corresponding values for system with\nNy= 15. The current density distribution for system with\nγ= 20 when VL> VR(d) and VL< VR(e).\nously changing color line. Upon it, we also plot several\neigenvalues of the system with Ny= 15 and γ= 0.05\nmarked as black circles. All other parameters are identi-\ncal. As we can see, both the real part and the imaginary\npart of the bands of these two systems matches quite ac-\ncurately. To see the equivalency more directly, we plot\nthe current distribution for the system with Fermi energy\nEf= 0.01 and γ= 20. As clearly shown in Fig. 5(d)\nand Fig. 5(e), the current density avoids the sites with\nvery large dissipation and flows one site above it, behav-\ning like a system with one slice narrower with stronger\nedge state coupling. The effective small dissipation for\nthis narrower system is manifested from the asymmetric\ntransport, which shows that the edge current from right\nto left decays slightly faster than that from left to right.\nIn Fig. 5(c), we compared more dissipation amplitudes\nforNy= 15 and Ny= 16 systems. Quite amazingly,\nit shows that the effective dissipation for the narrower\nsystem is precisely inverse proportional to that of the\noriginal system. In Appendix B, we give a formal proof\nfor this observation. Thus as previously mentioned, the\nsystem will be Hermitian-like with one slice narrower in\nthe very large dissipation limit.\nVI. SUMMARY AND DISCUSSION\nWe find that edge dissipation can significantly en-\nhance the unidirectional transport of the narrow quan-\ntum anomalous Hall system when the Fermi level re-\nsides inside the hybridization gap. We reveal that theenhancement is due to the gap closing by the dissipa-\ntion, which in turn decreases the decay rate of the cur-\nrent density. When varying the dissipation strength, we\nfind two exceptional transitions, where both the real and\nimaginary part of the band gap close. As the dissipa-\ntion becomes very large, we show that the low-energy\nphysics for the system is almost identical to a one slice\nnarrower system whose dissipation is inversely propor-\ntional small. Using an effective continuum model, which\ndescribes two coupled counter-propagating chiral edge\nstates with different group velocities and dissipation, we\nare able to capture most of the low-energy physics from\nthe lattice calculations. Experimentally, the quantum\nanomalous Hall effect has been realized in several physi-\ncal systems where the dissipation can be implemented in\nprinciple, including real quantum materials[89, 90], cold\natoms systems[91, 92], acoustic systems[93], and pho-\ntonic crystal systems[94–96]. In Appendix C, we explic-\nitly show that this enhanced unidirectional transport can\nbe realized in a quantum anomalous Hall material with\na coupled metallic lead acts as the source of dissipation.\nAt last, we believe that the edge dissipation effect dis-\ncussed in this work is quite general and can potentially\nbe extended to other topological quantum systems such\nas quantum spin Hall insulators and topological super-\nconductors.\nVII. ACKNOWLEDGEMENT\nWe are grateful for the valuable discussions with Qing\nYan, Ming Gong, Haiwen Liu and Hua Jiang. This work\nis supported by the Innovation Program for Quantum\nScience and Technology (Grants No. 2021ZD0302400)\nand National Natural Science Foundation of China\n(Grants No. 12204044, No. 12304052, No. 11974271,\nNo. 92265103). Hailong Li is also funded bv China Post-\ndoctoral Science Foundation (Grant No. BX20220005).\nAppendix A: UNBALANCED GROUP\nVELOCITIES IN THE INTERMEDIATE\nDISSIPATION REGIME\nIn this part, we show the band structures for the in-\ntermediate γregime. As previously shown in Fig. 2, the\neffective dissipation γeff, the coupling strength ∆, and the\ngroup velocity αfor the left-moving state varies dramati-\ncally in this range. In Fig. 6, we plot the low-energy band\nstructures and the corresponding continuum model fits of\nfour different γvalues. As clearly shown, the group veloc-\nity for the left-moving edge state deviates from that of the\nright-moving state quite apparently, making the real part\nof the band structures looks like a tilted Dirac cone. The\ncontinuum model fit (black circles) for the real part of\nthe band structure is quite accurate. However, the lower6\nimaginary band structure is not well fitted by the con-\ntinuum model. Nevertheless, the upper imaginary band\nis well fitted, which represents the imaginary part of the\nright-moving edge states. Therefore, the wave vectors of\nthe right-moving states can still be well captured by the\ncontinuum model, as we have already shown in Fig. 4(b).\nThe reason for the group velocity change by dissipation\nis not clear at this moment, and the low-energy model\nfor a better fit of the lower imaginary energy band also\nneed further investigation in the future.\n0.1\n 0.0 0.10.1\n0.00.1ReEk(a1)\n0.1\n 0.0 0.1\nkx0.0\n0.1\nImEk(a2)\n0.1\n 0.0 0.10.2\n0.00.2(b1)\n0.1\n 0.0 0.1\nkx0.0\n0.2\n(b2)\n0.1\n 0.0 0.10.15\n0.000.15(c1)\n0.1\n 0.0 0.1\nkx0.0\n0.1\n(c2)\n0.1\n 0.0 0.10.1\n0.00.1(d1)\n0.1\n 0.0 0.1\nkx0.0\n0.1\n(d2)\nFIG. 6. Band structures for intermediate γvalues, where in\n(a)γ= 0.5, (b) γ= 1, (c) γ= 2, and (d) γ= 3. The lines are\nfrom numerical calculations of lattice model, and the black\ncircles are from the continuum model result Eq. (4).\nAppendix B: CORRESPONDENCE IN THE\nLARGE DISSIPATION LIMIT\nIn this part, we show that the low-energy spectrum of\nH(Ny) and ˜H(Ny−1) are nearly identical when γis very\nlarge, where\nH(Ny) =\nh00−iγσ0h010··· 0\nh10 h00h01......\n0.........0\n......h10h00h01\n0 ··· 0h10h00\n,\n˜H(Ny−1) =\nh00−i1\nγσ0h010··· 0\nh10 h00h01......\n0.........0\n......h10h00h01\n0 ··· 0h10h00\n.\n(B1)\nIn the above, h00= sin kxσx+ (m+ cos kx)σz, h10=\n1\n2σz+i\n2σx, h01=h†\n10. The Sch¨ odinger equation forH(Ny) is H(Ny)ψ=Eminψ, where Eminand ψ=\u0000ψ1, ψ2,···, ψNy\u0001Tare the energy and wave vector\nfor the edge state, respectively. Explicitly,\n(h00−iγσ0)ψ1+h01ψ2=Eminψ1, (B2)\nh10ψ1+h00ψ2+h01ψ3=Eminψ2, (B3)\n... (B4)\nh10ψNy−1+h00ψNy=EminψNy. (B5)\nIn the large dissipation limit,\nψ1= (Emin−h00+iγσ0)−1h01ψ2≈ −i1\nγh01ψ2.(B6)\nSubstituting Eq.(B6) into Eq.(B3), we have\n\u0012\nh00−i1\nγh10h01\u0013\nψ2+h01ψ3=Eminψ2. (B7)\nTherefore, the eigen-equation H(Ny)ψ=Eminψtrans-\nforms to ¯H(Ny−1)¯ψ=Emin¯ψ, where\n¯H(Ny−1) =\nh00−i1\nγh10h01h010··· 0\nh10 h00h01......\n0.........0\n......h10h00h01\n0 ··· 0h10h00\n\n(B8)\nand¯ψ=\u0000ψ2, ψ3,···, ψNy\u0001T.\nNow we show the low-energy spectrum ˜Eminof˜H(Ny−\n1) can be approximated to Eminusing perturbation the-\nory. The Hamiltonian of the ( Ny−1)-wide QWZ ribbon\nwithout dissipation reads\nH0=\nh00h010··· 0\nh10h00h01......\n0.........0\n......h10h00h01\n0··· 0h10h00\n, (B9)\nIts eigen-equation is written as H0ψ(0)=E(0)\nminψ(0),\nwhere ψ(0)=\u0010\nψ(0)\n1, ψ(0)\n2,···, ψ(0)\nNy−1\u0011T\n. To the first\norder perturbation, we have\nEmin=E(0)\nmin−i1\nγψ(0)†\n1h10h01ψ(0)\n1, (B10)\n˜Emin=E(0)\nmin−i1\nγψ(0)†\n1σ0ψ(0)\n1. (B11)\nwhere ψ(0)\n1satisfies\nh00ψ(0)\n1+h01ψ(0)\n2=E(0)\nminψ(0)\n1. (B12)7\nBy a unitary transformation, we have\n˜h00ϕ1+σ−ϕ2=E(0)\nminϕ1, (B13)\nwhere σ−=\u00120 0\n1 0\u0013\n,˜h00= (m+cos kx)σx−sinkxσzand\nψ(0)\n1,2=uxϕ1,2with ux=\u0012\n1 1\n−1 1\u0013\n/√\n2. From Eq. (B13),\nϕ1can be written as\nϕ1= 1\n−\u0010\nE(0)\nmin+sinkx\u0011\nM!\nϕ11, (B14)\nwhere M=−(m+ cos kx). Accordingly,\nEmin=E(0)\nmin−i1\nγ|ϕ11|2, (B15)\n˜Emin=E(0)\nmin−i1\nγ\n \nE(0)\nmin+ sin kx\nM!2\n+ 1\n|ϕ11|2.\n(B16)\nFor small kx,M≫E(0)\nmin+ sin kx, therefore we have\n˜Emin≈Emin.\nAppendix C: METALLIC LEAD AS THE SOURCE\nOF DISSIPATION\nIn the following, we explicitly show that the aforemen-\ntioned dissipation enhanced transport can be realized for\na quantum anomalous hall material with a metallic lead\nconnected to its edge. The lead act as the source of dissi-\npation, into which the electrons from the edge states can\nflow. From the aspect of Green’s function, the coupling\nof the metallic lead with the central region provides the\nlater a self-energy. This self-energy is nonzero only on\nthe edge sites and its magnitude can be experimentally\ntuned by the coupling strength, which in principle can\nbe adjusted by applying gate voltage on the coupling re-\ngion. In Fig. 7, we show the linear conductance GRL\nandGLRas a function of the coupling strength t. The\ndissipation enhanced unidirectional transport and the re-\ncovery of the hermiticity in the large coupling limit are\nobserved, which resembles the features shown in Fig. 3.\nTo obtain the GRLin experiment, we set the voltage of\nleft lead VL>0, and connect right lead and bottom lead\nto the ground. GRLis calculated as the ratio between\nthe current flow into the right lead IRand the voltage of\nleft lead VL. Similarly, GLRcan be obtained by IL/VR\nwhen left and bottom leads are grounded.\n∗zhangzhiqiangphy@163.com\n†jieliuphy@xjtu.edu.cn\n102\n100102\nt0.00.40.8G(e2/h)(a)\nGRL\nGLR\n102\n100102\nt0.00.51.0(b)\nGRL\nGLRFIG. 7. Three terminal linear conductance GRLandGLRfor\nEf= 0.01 (a) and Ef= 0.045 (b) as a function of coupling\nstrength twith the third metallic lead.\n[1] C. M. Bender and S. Boettcher, Physical Review Letters\n80, 5243 (1998).\n[2] C. M. Bender, D. C. Brody, and H. F. Jones, Physical\nReview Letters 89, 270401 (2002).\n[3] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H.\nMusslimani, S. Rotter, and D. N. Christodoulides, Na-\nture Physics 14, 11 (2018).\n[4] Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics\n69, 249 (2020).\n[5] C. Dembowski, H.-D. Gr¨ af, H. L. Harney, A. Heine,\nW. D. Heiss, H. Rehfeld, and A. Richter, Physical Re-\nview Letters 86, 787 (2001).\n[6] C. Dembowski, B. Dietz, H.-D. Gr¨ af, H. L. Harney,\nA. Heine, W. D. Heiss, and A. Richter, Physical Review\nE69, 056216 (2004).\n[7] M. Berry, Czechoslovak Journal of Physics 54, 1039\n(2004).\n[8] W. D. Heiss, Journal of Physics A: Mathematical and\nTheoretical 45, 444016 (2012).\n[9] J. Wiersig, Physical Review Letters 112, 203901 (2014).\n[10] B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer,\nA. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇ ci´ c,\nNature 525, 354 (2015).\n[11] Y. Xu, S.-T. Wang, and L.-M. Duan, Physical Review\nLetters 118, 045701 (2017).\n[12] W. Chen, S ¸. K. ¨Ozdemir, G. Zhao, J. Wiersig, and\nL. Yang, Nature 548, 192 (2017).\n[13] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia,\nR. El-Ganainy, D. N. Christodoulides, and M. Kha-\njavikhan, Nature 548, 187 (2017).\n[14] A. Cerjan, M. Xiao, L. Yuan, and S. Fan, Physical Review\nB97, 075128 (2018).\n[15] A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong,\nand M. C. Rechtsman, Nature Photonics 13, 623 (2019).\n[16] M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. Stone,\nand L. Jiang, Physical Review Letters 123, 180501\n(2019).\n[17] M.-A. Miri and A. Al` u, Science 363, 10.1126/sci-\nence.aar7709 (2019).\n[18] J. Wiersig, Photonics Research 8, 1457 (2020).\n[19] J. Wiersig, Nature Communications 11, 10.1038/s41467-\n020-16373-8 (2020).\n[20] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews\nof Modern Physics 93, 015005 (2021).\n[21] K. Ding, C. Fang, and G. Ma, Nature Reviews Physics\n4, 745 (2022).\n[22] H. Shen, B. Zhen, and L. Fu, Physical Review Letters\n120, 146402 (2018).8\n[23] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-\ngashikawa, and M. Ueda, Physical Review X 8, 031079\n(2018).\n[24] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys-\nical Review X 9, 041015 (2019).\n[25] K. Kawabata, S. Higashikawa, Z. Gong, Y. Ashida, and\nM. Ueda, Nature Communications 10, 10.1038/s41467-\n018-08254-y (2019).\n[26] L. E. F. F. Torres, Journal of Physics: Materials 3,\n014002 (2019).\n[27] A. Ghatak and T. Das, Journal of Physics: Condensed\nMatter 31, 263001 (2019).\n[28] X. Zhang, T. Zhang, M.-H. Lu, and Y.-F. Chen, Ad-\nvances in Physics: X 7, 10.1080/23746149.2022.2109431\n(2022).\n[29] N. Okuma and M. Sato, Annual Review of Condensed\nMatter Physics 14, 83 (2023).\n[30] R. Lin, T. Tai, L. Li, and C. H. Lee, Frontiers of Physics\n18, 10.1007/s11467-023-1309-z (2023).\n[31] T. E. Lee, Physical Review Letters 116, 133903 (2016).\n[32] V. M. M. Alvarez, J. E. B. Vargas, and L. E. F. F. Torres,\nPhysical Review B 97, 121401 (2018).\n[33] S. Yao and Z. Wang, Physical Review Letters 121,\n086803 (2018).\n[34] Y. Xiong, Journal of Physics Communications 2, 035043\n(2018).\n[35] C. H. Lee and R. Thomale, Physical Review B 99, 201103\n(2019).\n[36] S. Yao, F. Song, and Z. Wang, Physical Review Letters\n121, 136802 (2018).\n[37] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J.\nBergholtz, Physical Review Letters 121, 026808 (2018).\n[38] K. Yokomizo and S. Murakami, Physical Review Letters\n123, 066404 (2019).\n[39] K. Kawabata, N. Okuma, and M. Sato, Physical Review\nB101, 195147 (2020).\n[40] Z. Yang, K. Zhang, C. Fang, and J. Hu, Physical Review\nLetters 125, 226402 (2020).\n[41] Z.-Q. Zhang, H. Liu, H. Liu, H. Jiang, and X. Xie, Science\nBulletin 68, 157 (2023).\n[42] H. Liu, M. Lu, Z.-Q. Zhang, and H. Jiang, Physical Re-\nview B 107, 144204 (2023).\n[43] K. Zhang, Z. Yang, and C. Fang, Physical Review Letters\n125, 126402 (2020).\n[44] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato,\nPhysical Review Letters 124, 086801 (2020).\n[45] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics\n82, 3045 (2010).\n[46] X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics\n83, 1057 (2011).\n[47] C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, Re-\nviews of Modern Physics 88, 035005 (2016).\n[48] A. Bansil, H. Lin, and T. Das, Reviews of Modern Physics\n88, 021004 (2016).\n[49] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang,\nS. S. P. Parkin, B. A. Bernevig, and T. Neupert, Science\nAdvances 4, 10.1126/sciadv.aat0346 (2018).\n[50] B. Xie, H.-X. Wang, X. Zhang, P. Zhan, J.-H. Jiang,\nM. Lu, and Y. Chen, Nature Reviews Physics 3, 520\n(2021).\n[51] C. H. Lee, L. Li, and J. Gong, Physical Review Letters\n123, 016805 (2019).\n[52] L. Li, C. H. Lee, and J. Gong, Physical Review Letters\n124, 250402 (2020).[53] D. Zou, T. Chen, W. He, J. Bao, C. H. Lee, H. Sun, and\nX. Zhang, Nature Communications 12, 10.1038/s41467-\n021-26414-5 (2021).\n[54] X. Zhang, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F.\nChen, Nature Communications 12, 10.1038/s41467-021-\n25716-y (2021).\n[55] Y. Li, C. Liang, C. Wang, C. Lu, and Y.-C. Liu, Physical\nReview Letters 128, 223903 (2022).\n[56] W. Zhu and J. Gong, Physical Review B 106, 035425\n(2022).\n[57] W. Zhu and J. Gong, Physical Review B 108, 035406\n(2023).\n[58] X. Ma, K. Cao, X. Wang, Z. Wei, and S. Kou, Arxiv\n(2023), arXiv:2304.01422 [quant-ph].\n[59] F. Schindler, K. Gu, B. Lian, and K. Kawabata, PRX\nQuantum 4, 030315 (2023).\n[60] D. Nakamura, K. Inaka, N. Okuma, and M. Sato, ArXiv\n(2023), arXiv:2304.08110 [cond-mat.mes-hall].\n[61] X.-W. Luo and C. Zhang, Physical Review Letters 123,\n073601 (2019).\n[62] Z. Zhang, M. R. L´ opez, Y. Cheng, X. Liu, and J. Chris-\ntensen, Physical Review Letters 122, 195501 (2019).\n[63] E. Edvardsson, F. K. Kunst, and E. J. Bergholtz, Physi-\ncal Review B 99, 081302 (2019).\n[64] T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata,\nM. Ueda, and F. Nori, Physical Review Letters 122,\n076801 (2019).\n[65] R. Okugawa, R. Takahashi, and K. Yokomizo, Physical\nReview B 102, 241202 (2020).\n[66] K. Kawabata, M. Sato, and K. Shiozaki, Physical Review\nB102, 205118 (2020).\n[67] M. Lu, X.-X. Zhang, and M. Franz, Physical Review Let-\nters127, 256402 (2021).\n[68] Y. Fu, J. Hu, and S. Wan, Physical Review B 103, 045420\n(2021).\n[69] C. Wang and X. R. Wang, arXiv (2019),\narXiv:1901.06982 [cond-mat.mes-hall].\n[70] A. Marguerite, J. Birkbeck, A. Aharon-Steinberg, D. Hal-\nbertal, K. Bagani, I. Marcus, Y. Myasoedov, A. K. Geim,\nD. J. Perello, and E. Zeldov, Nature 575, 628 (2019).\n[71] S. A. Sato, P. Tang, M. A. Sentef, U. D. Giovannini,\nH. H¨ ubener, and A. Rubio, New Journal of Physics 21,\n093005 (2019).\n[72] G. Shavit and M. Goldstein, Physical Review B 101,\n125412 (2020).\n[73] Z. Fedorova, H. Qiu, S. Linden, and J. Kroha, Nature\nCommunications 11, 10.1038/s41467-020-17510-z (2020).\n[74] Y.-W. Huang, P.-Y. Yang, and W.-M. Zhang, Physical\nReview B 102, 165116 (2020).\n[75] W. Lin, Y. Feng, Y. Wang, Z. Lian, H. Li, Y. Wu, C. Liu,\nY. Wang, J. Zhang, Y. Wang, X. Zhou, and J. Shen,\nPhysical Review B 105, 165411 (2022).\n[76] P. San-Jose, J. Cayao, E. Prada, and R. Aguado, Scien-\ntific Reports 6, 10.1038/srep21427 (2016).\n[77] J. Avila, F. Pe˜ naranda, E. Prada, P. San-Jose, and\nR. Aguado, Communications Physics 2, 10.1038/s42005-\n019-0231-8 (2019).\n[78] H. Liu, M. Lu, Y. Wu, J. Liu, and X. C. Xie, Physical\nReview B 106, 064505 (2022).\n[79] B. Zhou, H.-Z. Lu, R.-L. Chu, S.-Q. Shen, and Q. Niu,\nPhysical Review Letters 101, 246807 (2008).\n[80] X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Physical Review B\n74, 085308 (2006).9\n[81] Z. Wang, J. Song, H. Liu, H. Jiang, and X. C. Xie, New\nJournal of Physics 17, 113040 (2015).\n[82] M. Shafiei, F. Fazileh, F. M. Peeters, and M. V.\nMiloˇ sevi´ c, Physical Review B 106, 035119 (2022).\n[83] T. Bessho and M. Sato, Physical Review Letters 127,\n196404 (2021).\n[84] S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge University Press, 1995).\n[85] H. Jiang, L. Wang, Q. feng Sun, and X. C. Xie, Physical\nReview B 80, 165316 (2009).\n[86] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Physical Re-\nview B 50, 5528 (1994).\n[87] F. Song, S. Yao, and Z. Wang, Physical Review Letters\n123, 170401 (2019).\n[88] Y. Yi and Z. Yang, Physical Review Letters 125, 186802\n(2020).\n[89] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang,\nM. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji,Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C.\nZhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue,\nScience 340, 167 (2013).\n[90] C.-Z. Chang, C.-X. Liu, and A. H. MacDonald, Reviews\nof Modern Physics 95, 011002 (2023).\n[91] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,\nT. Uehlinger, D. Greif, and T. Esslinger, Nature 515,\n237 (2014).\n[92] M.-C. Liang, Y.-D. Wei, L. Zhang, X.-J. Wang, H. Zhang,\nW.-W. Wang, W. Qi, X.-J. Liu, and X. Zhang, Physical\nReview Research 5, l012006 (2023).\n[93] M. Xiao, W.-J. Chen, W.-Y. He, and C. T. Chan, Nature\nPhysics 11, 920 (2015).\n[94] Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljaˇ ci´ c,\nNature 461, 772 (2009).\n[95] L. Lu, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Nature Pho-\ntonics 8, 821 (2014).\n[96] J. Chen and Z.-Y. Li, Physical Review Letters 128,\n257401 (2022)."    },    {        "title": "1503.07913v2.Surface_Fermi_arcs_in__Z_2__Weyl_semimetals__A_3_Bi___A_Na__K__Rb_.pdf",        "content": "arXiv:1503.07913v2  [cond-mat.str-el]  22 Jun 2015Surface Fermi arcs in Z2Weyl semimetals A 3Bi (A=Na, K, Rb)\nE. V. Gorbar,1,2V. A. Miransky,3,4I. A. Shovkovy,5and P. O. Sukhachov1\n1Department of Physics, Taras Shevchenko National Kiev Univ ersity, Kiev, 03680, Ukraine\n2Bogolyubov Institute for Theoretical Physics, Kiev, 03680 , Ukraine\n3Department of Applied Mathematics, Western University, Lo ndon, Ontario N6A 5B7, Canada\n4Department of Physics and Astronomy, Western University, L ondon, Ontario N6A 3K7, Canada\n5College of Letters and Sciences, Arizona State University, Mesa, Arizona 85212, USA\n(Dated: March 12, 2018)\nThe surface Fermi arc states in Z2Weyl semimetals A 3Bi (A = Na, K, Rb) are studied by\nemploying a continuum low-energy effective model. It is show n that the surface Fermi arc states\ncan be classified with respect to the ud-parity symmetry. Bec ause of the symmetry, the arcs come\nin mirror symmetric pairs. The effects of symmetry breaking t erms on the structure of the Fermi\narcs are also studied. Among other results, we find at least tw o qualitatively different types of the\nsurface Fermi arcs. The arcs of the first type link disconnected sheets of the bulk Fermi surface,\nwhile arcs of the second type link different points of the samebulk Fermi surface sheet.\nPACS numbers: 73.20.At, 71.10.-w, 03.65.Vf\nI. INTRODUCTION\n3D Dirac semimetals are 3D analogs of graphene [1]. Their conduction a nd valence bands touch only at discrete\n(Dirac) points in the Brillouin zone with the electron states described by the 3D massless Dirac equation. Each Dirac\npoint in momentum space is composed of two superimposed Weyl node s of opposite chirality. Such points are usually\nobtained by fine tuning of certain physical parameters (e.g., the sp in-orbit coupling strength or chemical composition)\nand are difficult to control. Additionally, they are often unstable with respect to mixing of Weyl modes and opening\na gap.\nAn important idea was proposed in Refs. [2, 3], where it was shown th at an appropriate crystal symmetry can\nprotect and stabilize the gapless 3D Dirac points. Indeed, if a pair of crossing bands belong to different irreducible\nrepresentations of the discrete (rotational) crystal symmetry and if this symmetry is not broken dynamically, then the\nmass term for the corresponding Dirac fermions will be prohibited. T heab initio calculations in Ref. [2] showed that\nβ-cristobalite BiO 2exhibits three Dirac points at the Fermi level. Unfortunately, this m aterial is metastable. By using\nthe first-principles calculations and an effective model analysis, the compounds A 3Bi (A=Na, K, Rb) and Cd 3As2\nwere identified in Refs. [4, 5] as possible 3D Dirac semimetals protecte d by crystal symmetry. Giant diamagnetism,\nlinear quantum magnetoresistance, and the quantum spin Hall effec t are expected in these materials. Furthermore,\nvarious topologically distinct phases can be realized in these compoun ds by breaking the time-reversal and inversion\nsymmetries. By using angle-resolved photoemission spectroscopy , the Dirac semimetal band structure was indeed\nobserved [6–8] in Cd 3As2and Na 3Bi opening the path toward experimental investigations of the pro perties of 3D\nDirac semimetals.\nWeyl semimetals is another group of materials that is closely related t o 3D Dirac semimetals and have already\nattracted a lot of theoretical interest (for reviews, see Refs. [ 9–11]). They are characterized by topologically non-\ntrivial Weyl nodes in reciprocal space. Weyl nodes are monopoles o f the Berry flux and, therefore, can appear\nor annihilate only in pairs. Weyl semimetals were proposed to be realize d in pyrochlore iridates [12], topological\nheterostructures[13], magneticallydoped topological insulators[1 4], and nonmagnetic materials such as TaAs [15, 16].\nRecently, first experimental studies of Weyl semimetal candidate TaAs were reported in Refs. [17–20]. The authors\nobservedunusual transport properties and surface states th at are characteristicofthe Weyl semimetal phase. Another\ninteresting realization of the Weyl points in the context of photonic crystals has been recently reported in Ref. [21].\nSince the magnetic field breaks the time-reversal symmetry, a Dira c (semi-)metal in a magnetic field may transform\ninto a Weyl one with Weyl nodes separated in momentum space by a no nzero chiral shift [22]. Experimentally, the\ntransition from a Dirac metal to a Weyl one in a magnetic field might hav e been observed in Bi 1−xSbxforx≈0.03\n[23]. In moderately strong magnetic fields, a negative magnetoresis tivity is observed and interpreted as a fingerprint\n[24–26] of a Weyl/Dirac metal phase.\nThe surface Fermi arcs [12, 27–29], which connect Weyl nodes of o pposite chirality, are related to the non-trivial\ntopology of Weyl semimetals. In equilibrium, the presence of such su rface states ensures that the chemical potentials\nat different Weyl points are identical [27]. Although Fermi arcs alwa ys connect Weyl nodes of opposite chirality,\ntheir shapes depend on the boundary conditions and, as shown in Re f. [30], Fermi arcs of an arbitrary form can\nbe engineered. The Fermi arcs on the opposite surfaces of a semim etal sample together with the Fermi surfaces of2\nbulk states form a closed Fermi surface. In an external magnetic field, the nontrivial structure of the corresponding\nFermi surface gives rise to closed magnetic orbits involving the surf ace Fermi arcs [31]. These orbits produce periodic\nquantum oscillations of the density of states in a magnetic field leading to unconventionalFermiology of surface states.\nIt was argued in Ref. [32] that the interaction effects can change t he separation between Weyl nodes in momentum\nspace and the length of the Fermi arcs in the reciprocal space and , thus, affect these magnetic orbits. As a result,\nwe found that the period of oscillations of the density of states rela ted to closed magnetic orbits involving Fermi arcs\nhas a non-trivial dependence on the orientation of the magnetic fie ld projection in the plane of the semimetal surface\n[32]. If experimentally observed, such a dependence would provide a n important clue to the effects of interactions in\nWeyl semimetals.\nNormally, one would not expect any surface Fermi arcs in 3D Dirac se mimetals because the Dirac point has\nno topological charge and the associated Berry flux vanishes. In R efs. [4, 5], however, it was shown that the 3D\nDirac semimetals A 3Bi (A=Na, K, Rb) and Cd 3As2possess non-trivial surface Fermi arcs. This finding suggests a\ntopologically nontrivial nature of the corresponding Dirac materials . Recently we showed [33] that this is indeed the\ncase for Dirac semimetals A 3Bi (A = Na ,K,Rb). The physical reason for their nontrivial topological propert ies is\nconnected with a discrete symmetry of the low-energy effective Ha miltonian. The symmetry classification allows one\nto split all electron states into two separate sectors, each descr ibing a Weyl semimetal with a pair of Weyl nodes\nand broken time-reversal symmetry. The time-reversal symmet ry is preserved in the complete theory because its\ntransformation interchanges states from the two different sect ors. The nontrivial topological structure of each sector\nwas supported by explicit calculations of the Berry curvature, whic h revealed a pair of monopoles of the Berry flux at\nthe positions of Weyl nodes in each of the two sectors of these sem imetals [33]. In essence, these results demonstrated\nthat Dirac semimetals A 3Bi (A = Na ,K,Rb) are, in fact, Z2Weyl semimetals.\nIn Refs. [4, 5], the surface Fermi arcs in 3D Dirac semimetals were ob tained in a tight-binding model by using an\niterative method that produces the surface Green’s function of t he semi-infinite system [34]. The imaginary part of\nthe surface Green’s function makes possible to determine the local density of states at the surface. While such a\ntechnique is very powerful, it is essentially a “black box”. In contras t, in the present paper, we study analytically the\nsurface Fermi arc states by employing the continuum low-energye ffective model with appropriateboundary conditions\nat the surface. We hope that such a consideration will provide a dee per understanding of the physical properties and\ncharacteristics of the surface Fermi arcs, as well as shed more lig ht on the nontrivial topological properties of the\nA3Bi compounds.\nThe paper is organizedas follows. In Sec. II, we introduce the low-e nergyeffective model and discuss its symmetries.\nThe recently revealed Z2Weyl semimetal structure of A 3Bi (A = Na, K, Rb) is emphasized. In order to clarify the\norigin and the structure of the surface Fermi arcs, we study in Se c. III the corresponding states in a simplified model\nthat contains a single Weyl semimetal sector. In Sec. IV, we prese nt the rigorous analysis of the surface Fermi\narc states in a realistic low-energy model of semimetals A 3Bi (A = Na, K, Rb). The effects of several possible\nsymmetry breaking terms on the structure of the surface Fermi arc states are investigated in Sec. V. The discussion\nand the summary of the main results are given in Sec. VI. Technical d etails regarding the symmetry properties and\nclassification of the Fermi arc states are presented in Appendices A and B.\nFor convenience, throughout the paper, we set /planckover2pi1= 1 andc= 1.\nII. MODEL\nA. Low-energy effective Hamiltonian\nThe low-energy Hamiltonian derived in Ref. [4] for A 3Bi (A = Na ,K,Rb) has the form\nH(k) =ǫ0(k)+H4×4, (1)\nwhereǫ0(k) =C0+C1k2\nz+C2(k2\nx+k2\ny) and\nH4×4=\nM(k)Ak+0B∗(k)\nAk−−M(k)B∗(k) 0\n0B(k)M(k)−Ak−\nB(k) 0 −Ak+−M(k)\n. (2)\nWhile the diagonal elements of H4×4are given in terms of a single function, M(k) =M0−M1k2\nz−M2(k2\nx+k2\ny), the\noff-diagonal elements are determined by functions Ak±andB(k) =αkzk2\n+, wherek±=kx±iky.3\nBy fitting the energy spectrum of the effective Hamiltonian with the ab initio calculations, the numerical values of\nparameters in the effective model were determined in Ref. [4]. They are\nC0=−0.06382 eV, C 1= 8.7536 eV˚A2, C 2=−8.4008 eV˚A2,\nM0=−0.08686 eV, M 1=−10.6424 eV˚A2, M 2=−10.3610 eV˚A2,\nA= 2.4598 eV˚A, a = 5.448˚A, c = 9.655˚A,(3)\nwhere we also included the lattice constants aandc. Since no specific value for αwas quoted in Ref. [4], we will treat\nit as a free parameter below.\nThe energy eigenvalues of the low-energy Hamiltonian (1) are given b y the following explicit expression:\nE(k) =ǫ0(k)±/radicalbig\nM2(k)+A2k+k−+|B(k)|2. (4)\nIt is easy to check that the term with the square root vanishes at t he two Dirac points, k±\n0= (0,0,±√m), where√m≡/radicalbig\nM0/M1. With the choice of the low-energy parameters in Eq. (3), we find th at√m≈0.09034˚A−1. The\nfunctionB(k) plays the role of a momentum dependent mass (gap) function that vanishes at the Dirac points.\nIt is instructive to show that linearizing M(k) in the vicinity of the Dirac points k±\n0, Hamiltonian (2) takes the\nform of a 3D massive Dirac Hamiltonian. In the vicinity of k−\n0, expanding M(k) to the linear order in δk=k−k−\n0,\nwe obtain\nHlin\n4×4=/parenleftbigg\nA(˜kxσx−˜kyσy−˜kzσz)B∗(k)σx\nB(k)σx −A˜k·σ/parenrightbigg\n, (5)\nwhereσarePaulimatricesand ˜k= (kx,ky,2δkz√M0M1/A). Furthermore, byperformingthe unitarytransformation,\n˜Hlin\n4×4≡U+\nxHlin\n4×4Ux, whereUx= diag(σx,I2) andI2is the 2×2 unit matrix, we find that the Hamiltonian takes the\nstandard form of the Dirac Hamiltonian in the chiral representation ,\n˜Hlin\n4×4=/parenleftbigg\nA˜k·σB∗(k)\nB(k)−A˜k·σ/parenrightbigg\n. (6)\nTaking into account that the mass term B(k) vanishes at the Dirac point, we conclude that the upper and lower\n2×2 blocks describe quasiparticle states of opposite chiralities. Also, s ince the leading order nonzero corrections to\nthe mass function are quadratic in momentum, the chirality remains a good quantum number in a sufficiently small\nvicinity of the Dirac point. Hamiltonian (6), describing two subsets of the opposite chirality states near a single Dirac\npoint, does not appear to have any interesting topological proper ties. Also, by itself, it is unlikely to give rise to any\nFermi arcs states. It is easy to check, however, that Hamiltonian (2) linearized near k+\n0has a similar structure and\ndescribes two additional subsets of the opposite chirality states. As we argue below, the superposition of the two\nsectors of the theory is nontrivial and gives rise to an interesting t opological structure [33].\nB. Symmetries\nLet us briefly review the symmetry properties of the low-energy Ha miltonian following Ref. [33]. We start by\npointing out that, as expected, the Hamiltonian (1) is invariant unde r the time-reversal and inversion symmetries,\ni.e.,\nΘH−kΘ−1=Hk,(time-reversal symmetry) (7)\nPH−kP−1=Hk,(inversion symmetry) (8)\nwhere Θ = TK(Kis complex conjugation) and\nT=\n0 0 1 0\n0 0 0 1\n−1 0 0 0\n0−1 0 0\n, P=\n1 0 0 0\n0−1 0 0\n0 0 1 0\n0 0 0 −1\n. (9)\nHaving both, the time-reversal and inversion symmetries, sugges ts that the corresponding compounds are not Weyl\nsemimetals. This is not the whole story, however.4\nAs shown in Ref. [33], the low-energy Hamiltonian in Eq. (1) possesses a new discrete symmetry, the so-called\nup-down parity (ud-parity), that protects its topological natur e. In order to understand the corresponding symmetry,\nit is instructive to start from the approximate Hamiltonian without th e mass function B(k) (or, alternatively, α= 0).\nIn this case, the 4 ×4 Hamiltonian takes a block diagonal form: H4×4(α= 0)≡H+\n2×2⊕H−\n2×2. The explicit form of\nthe upper block is given by\nH+\n2×2=/parenleftbiggM0−M1k2\nz−M2(k2\nx+k2\ny) A(kx+iky)\nA(kx−iky) −/bracketleftbig\nM0−M1k2\nz−M2(k2\nx+k2\ny)/bracketrightbig/parenrightbigg\n. (10)\nThis block Hamiltonian defines a Weyl semimetal with two Weyl nodes loc ated atk±\n0. (The lower block H−\n2×2has a\nsimilar form, except that kxis replaced by −kx.) It is well known [29, 31] that such a Weyl semimetal has the surfac e\nFermi arc connecting the Weyl nodes of opposite chirality at k+\n0andk−\n0. Because of the sign difference, kx→ −kx,\nthe chiralities of the states near the Weyl nodes at k±\n0are opposite for the upper and lower block Hamiltonians. Thus,\nthe complete 4 ×4 block diagonal Hamiltonian H4×4(α= 0) describes two superimposed copies of Weyl semimetal\nwith two pairs of overlapping nodes. Since the opposite chirality Weyl nodes coincide exactly in momentum space,\nthey effectively give rise to a pair of Dirac points at k±\n0. At the same time, because the opposite chirality nodes\ncome from two different Weyl copies, they cannot annihilate and can not form topologically trivial Dirac points. In\nfact, the corresponding approximate model describes a Z2Weyl semimetal [33]. The nontrivial topological properties,\nassociated with the underlying Z2Weyl semimetal structure, ensure that the resulting Dirac semime tal possesses\nsurface Fermi arcs.\nIt is easy to show that the existence of the Z2Weyl semimetal structure in the absence of B(k) is connected with\nthe continuous symmetry U +(1)×U−(1) of the approximate Hamiltonian H4×4(α= 0). This symmetry describes\nindependent phase transformations of the spinors that corresp ond to the up- and down-block Hamiltonians, H+\n2×2and\nH−\n2×2, respectively.\nForB(k)/negationslash= 0, the continuous symmetry U +(1)×U−(1) is broken down to its diagonal subgroup U em(1) that\ndescribes the usual charge conservation. However, the low-ene rgy Hamiltonian (1) with the momentum dependent\nmass function B(k) =αkzk2\n+possesses a ud-parity, defined by the following transformation [3 3]:\nUH−kzU−1=Hkz,(ud-parity), (11)\nwhere matrix Uhas the following block diagonal form: U≡diag(I2,−I2) andI2is the 2×2 unit matrix. For the\nHamiltonian to be symmetric under the ud-parity, it is crucial that th e mass function B(k) changes its sign when\nkz→ −kz[while the functions ǫ0(k) andM(k) in the diagonal elements do not change their signs]. In the special\ncase of a momentum independent mass function, such a discrete sy mmetry does not exist.\nAs was argued in Ref. [33], the existence of the noncommuting time- reversal and ud-parity symmetries implies that\nthe A3Bi semimetal is, in fact, a Z2Weyl semimetal. In such a semimetal, all quasiparticle states can be s plit into\ntwo separate groups, labeled by the eigenvalues χ=±1 ofUχ=UΠkz, where Π kzis the operator that changes the\nsign of the zcomponent of momentum, kz→ −kz. Effectively, each group of states defines a Weyl semimetal with a\nbroken time-reversal symmetry. The corresponding symmetry is preserved in the complete theory, in which the two\ncopies of Weyl semimetals are superimposed.\nTheZ2Weyl semimetal structure of A 3Bi (A = Na ,K,Rb) is also supported by the explicit calculation of the Berry\nconnection and the Berry curvature in each Weyl sector describe d [33]. In particular, the corresponding results for\nthe curvature in the momentum space reveal a clear dipole structu re. It is natural, that each Weyl sector, described\nby quasiparticle states with a fixed eigenvalue of Uχ, should give rise to Fermi arcs connecting the pairs of Weyl nodes\natk±\n0. Moreover, such arcs should be topologically protected and could n ot be removed by small perturbations of\nmodel parameters.\nIn our discussion of Fermi arcs below, it will be also useful to take int o account that there exists yet another discrete\nsymmetry defined by the following transformation:\n˜UH−kx˜U−1=Hkx, (12)\nwhere\n˜U=\n0 0 1 0\n0 0 0 1\n1 0 0 0\n0 1 0 0\n. (13)\nIt is interesting to note that the product of the Uχand˜UΠkxtransformations Uχ˜UΠkx=TΠkxΠkzis also a symmetry\nof the low-energy Hamiltonian (1). The symmetry TΠkxΠkzis related to the time-reversal symmetry. This follows5\nfrom the fact that KΠkyis also the symmetry of the low-energy Hamiltonian (1). Together th e operators Uχ,˜UΠkx,\nandTΠkxΠkzform a non-commutative discrete group.\nHamiltonian (1) is rather complicated, therefore, the correspond ing analytic calculations of its surface Fermi states\nare quite involved and not much revealing. Therefore, our general strategy in analyzing these states will be to start\nfrom a simplified model and then move forward to the realistic model b y adding step-by-step the necessary missing\npieces.\nIII. SURFACE FERMI ARCS IN SIMPLIFIED 2×2MODEL\nIn order to get an insight into the structure of the surface Fermi arcs in the low-energy model described by\nHamiltonian (1), it is instructive to first study the surface Fermi ar cs in a simplified 2 ×2 model, given by one of the\ndiagonal blocks, e.g., H+\n2×2in Eq. (10). (The solutions for the other block Hamiltonian, H−\n2×2, can be obtained simply\nby changing kx→ −kx.) For completeness, we will also include the term ǫ0(k) proportional to the unit matrix, which\nis present in the low-energy Hamiltonian. Thus, our model 2 ×2 Hamiltonian reads\nH2×2=ǫ0(k)+H+\n2×2=ǫ0(k)+/parenleftbiggM0−M1k2\nz−M2(k2\nx+k2\ny) A(kx+iky)\nA(kx−iky) −/bracketleftbig\nM0−M1k2\nz−M2(k2\nx+k2\ny)/bracketrightbig/parenrightbigg\n.(14)\nBefore proceeding to the analysis, it is convenient to perform a unit ary transformation, ˜H2×2≡U−1\nyH2×2Uy, where\nUy=1√\n2(I2+iσy). The transformed Hamiltonian has the following explicit form:\n˜H2×2=ǫ0(k)+/bracketleftbig\nγ/parenleftbig\nk2\nz−m/parenrightbig\n−M2(k2\nx+k2\ny)/bracketrightbig\nσx−vkxσz−vkyσy, (15)\nwhere we introduced the notations similar to those in Ref. [29]: v=Aandγ=−M1.\nTo study the surface Fermi arcs, we will assume that the surface of a semimetal is at y= 0. The semimetal itself\nis in the upper y >0 (lowery <0) half-space when we describe the surface arc states on the bot tom (top) surface.\n(Of course, in the absence of any effects that break the inversion symmetryky→ −kyexplicitly, the two cases will\nbe related by a simple symmetry transformation.) Without loss of gen erality, we will concentrate primarily on the\nbottom surface states. The boundary condition on the semimetal surface will be imposed by replacing the parameter\nmwith the −˜mon the vacuum side of the boundary and taking the limit ˜ m→ ∞[29]. From a physics viewpoint,\nsuch a replacement is the simplest way to prevent quasiparticle from escaping into the vacuum.\nTaking into account that the Fermi arc states should be localized at they= 0 boundary, let us rewrite Hamiltonian\n(15) in the following form:\n˜H2×2=/parenleftbiggC0+C1k2\nz+C2(k2\nx−∂2\ny)−vkxγ/parenleftbig\nk2\nz−m/parenrightbig\n−M2(k2\nx−∂2\ny)+v∂y\nγ/parenleftbig\nk2\nz−m/parenrightbig\n−M2(k2\nx−∂2\ny)−v∂yC0+C1k2\nz+C2(k2\nx−∂2\ny)+vkx/parenrightbigg\n, (16)\nwhere, for the convenience of further derivations, we replaced ky≡ −i∂y.\nA. Simplified model with C2=M2= 0\nWewill seein whatfollowsthatthe presenceofthetermswith thesec ondderivativewithrespectto yinHamiltonian\n(16) leads to many technical complications and makes the analysis ra ther involved. Therefore, to set up the stage, in\nthis subsection we start our analysis in an even more simplified model, d escribed by Hamiltonian (16) with C2and\nM2set to zero. Then, by introducing the two-component spinor Ψ = ( ψ1,ψ2)T, we see that the eigenvalue problem\n(˜H2×2−E)Ψ = 0 is equivalent to the following system of equations:\n/parenleftbig\n−vkx+C1k2\nz+C0/parenrightbig\nψ1+/bracketleftbig\nv���y+γk2\nz−γm(y)/bracketrightbig\nψ2=Eψ1, (17)\n/parenleftbig\nvkx+C1k2\nz+C0/parenrightbig\nψ2+/bracketleftbig\n−v∂y+γk2\nz−γm(y)/bracketrightbig\nψ1=Eψ2. (18)\nHerem(y) =mθ(y)−˜mθ(−y), whereθ(y) is the step function. Recall that, by assumption, the boundary c ondition\naty= 0 is enforced by taking the limit ˜ m→ ∞on the vacuum side ( y <0). Formally, Eqs. (17) and (18) have the\nfollowing surface state solutions:\nΨ1(y) =/parenleftbigg\nN1eγ\nv/integraltextydy′[k2\nz−m(y′)]\n0/parenrightbigg\n,Ψ2(y) =/parenleftbigg0\nN2e−γ\nv/integraltextydy′[k2\nz−m(y′)]/parenrightbigg\n, (19)6\nIn the region occupied by the semimetal ( y >0), the solution Ψ 1(y) is normalizable only for k2\nz−m <0, while the\nsolution Ψ 2(y) is normalizable only for k2\nz−m>0. However, on the vacuum side ( y<0), only Ψ 1(y) is normalizable.\nThe dispersion relation for this normalizable surface state solution f ollows from Eq. (17). It is given by\nE=−vkx+C1k2\nz+C0. (20)\nBy making use of this relation, we derive the equation for the bottom surface Fermi arc in the transverse kxkzplane,\nkx=−E−C1k2\nz−C0\nv. (21)\nIt is instructive to compare this surface Fermi arc with that in the m odel of Ref. [29], where C1= 0. While the surface\nFermi arcs run between kz=−√mandkz=√min both models, the arcs in the model of Ref. [29] do not depend\non the momentum kz. This is in contrast to the surface Fermi arc in Eq. (21), for which kxis a quadratic function\nofkz. Thus, we see that the presence of the quadratic in kzterm in the diagonal component of Hamiltonian (16)\nproduces a nonzero curvature of the surface Fermi arcs in momentum space. For illustration, sev eral surface Fermi\narcs for different values of the Fermi energy are shown in Fig. 1. Th e arcs have parabolic shapes. The corresponding\narcs in the model of Ref. [29] would be given by straight lines.\nFIG. 1: (Color online) The bottom surface Fermi arcs for seve ral different values of the Fermi energy in a simplified two-\ncomponent model, described by Hamiltonian (16) with C2=M2= 0. The analytical form of the arcs is given in Eq. (21).\nBefore concluding this section, let us note that the solution Ψ 2(y) in Eq. (19) describes Fermi arcs on the top\nsurface. We find from Eq. (18) that the corresponding dispersion relation is given by E=vkx+C1k2\nz+C0. Let us\nalso note in passing that there exists another set of the (top and b ottom) Fermi arcs for the lower block Hamiltonian,\nH−\n2×2. The corresponding arcs are obtained from the solutions for the u pper block Hamiltonian, H+\n2×2, by making the\nreplacement kx→ −kx.\nB. The case with C2/negationslash= 0andM2/negationslash= 0\nLet us now consider the general case with C2/negationslash= 0 andM2/negationslash= 0. By noting that the Hamiltonian in Eq. (16) contains\nsecond derivatives with respect to y, the eigenvalue problem ( ˜H2×2−E)Ψ = 0 becomes more complicated. In the\nsemimetal ( y>0), it is equivalent to the following system of coupled equations:\n/bracketleftbig\nC2(k2\nx−∂2\ny)−vkx+C1k2\nz+C0/bracketrightbig\nψ1+/bracketleftbig\n−M2(k2\nx−∂2\ny)+v∂y+γk2\nz−γm/bracketrightbig\nψ2=Eψ1, (22)\n/bracketleftbig\nC2(k2\nx−∂2\ny)+vkx+C1k2\nz+C0/bracketrightbig\nψ2+/bracketleftbig\n−M2(k2\nx−∂2\ny)−v∂y+γk2\nz−γm/bracketrightbig\nψ1=Eψ2. (23)7\nFIG. 2: (Color online) The bottom surface Fermi arcs (25) for several values of the Fermi energy in a two-component model,\ndescribed by Hamiltonian (16) with C2/negationslash= 0 and M2/negationslash= 0.\nOn the vacuum side ( y <0), the corresponding set of equations has the same form, but wit hmreplaced by −˜m.\nAt the vacuum-semimetal interface ( y= 0), the wave functions and their derivatives should satisfy the co nditions of\ncontinuity, see Eqs. (A1) through (A4) in Appendix A1.\nThe keydetails ofthe derivation ofthe surface Fermi arcsolutions arepresented in Appendix A1. On the semimetal\nside, the spinor structure of the solution takes the following form:\nΨy>0(y) =2/summationdisplay\ni=1/parenleftbigg\nai\nbi/parenrightbigg\ne−piy, (24)\nwhere the explicit expressions for the exponents are given in Eq. (A 9). Note that the exponents take real values in\nthe case of surface Fermi arc states. The condition of existence of nontrivial surface Fermi arc solutions is given by\n−C2(p2\n1−k2\nx)+C1k2\nz+C0−E−vkx\n−M2(p2\n1−k2x)−γ(k2z−m)+vp1=−C2(p2\n2−k2\nx)+C1k2\nz+C0−E−vkx\n−M2(p2\n2−k2x)−γ(k2z−m)+vp2. (25)\nThis equation defines the functional dependence kz(kx) for the possible surface Fermi arc states. A numerical study\nshows that nontrivial solutions exist only in a finite range of energies , i.e.,−0.168 eV/lessorsimilarE/lessorsimilar0.373 eV. Several\nsolutions for different values of the energy are shown in Fig. 2. The r esults of the numerical analysis show that the\nfollowing condition is satisfied: b1/a1=b2/a2= 0.5115 for all solutions. It is worth noting that the E= 0 surface\nFermi arc in Fig. 2 appears to be almost identical to the correspond ing arc, obtained by a very different method in\nRef. [4], see Fig. 3c in that paper.\nSo far, we considered the arc states only for one of the two-comp onent block Hamiltonians, defined in Eq. (14).\nSimilar solutions also exist for the lower two-component block Hamilton ian,ǫ0(k) +H−\n2×2. It is straightforward to\nshow that the solutions to the eigenvalue problem for the lower block are the same as for the upper one, after one\nmakes the replacement kx→ −kx. Graphically, these solutions are mirror images of the arcs in Fig. 2.\nBefore concluding this subsection, let us also note that the descrip tion of the Fermi arc states on the top surface is\nsimilar to the bottom ones. By assuming that Weyl semimetal is at y<0 and the vacuum is at y>0, the appropriate\nboundary conditions are implemented by using the y-dependent parameter m(y) =mθ(−y)−˜mθ(y) and taking the\nlimit ˜m→ ∞at the end. Up to a reflection kx→ −kx, the corresponding final results for the Fermi arcs on the top\nsurface look similar to those on the bottom surface, shown in Fig. 2.8\nC. Effective Hamiltonian for surface Fermi arc states\nFollowing the usual approach in the studies of topological insulator [ 35], it may be natural to derive an effective\nHamiltonian for the surface Fermi arcstates. The block Hamiltonian s in the simplified model at hand can be naturally\nseparatedintotwoparts,i.e., ˜H±\n2×2=H0+H±\n1, wherethezerothorderpart H0correspondstotheoriginalHamiltonian\natkx=kz= 0, i.e.,\nH0=/parenleftbiggC0−C2∂2\ny−γm+M2∂2\ny+v∂y\n−γm+M2∂2\ny−v∂yC0−C2∂2\ny/parenrightbigg\n. (26)\nwhileH1contains all the terms with nontrivial dependence on kxandkz, i.e.,\nH±\n1=/parenleftbigg\nC1k2\nz+C2k2\nx∓vkxγk2\nz−M2k2\nx\nγk2\nz−M2k2\nxC1k2\nz+C2k2\nx±vkx/parenrightbigg\n. (27)\nAs in the previous analysis, we used ky≡ −i∂y. To start with, we have to solve the eigenvalue problem with the\nzeroth order Hamiltonian, H0Ψ0=λΨ0. By following the same approach as in Appendix A1, but with kx=kz= 0,\nwe find straightforwardly the explicit solutions for the surface Fer mi arcs Ψ 0. The corresponding energy parameter is\nfound to be λ=−0.13425 eV. Then, the effective Hamiltonian for the surface states is obtained by integrating over\nthe perpendicular direction y, i.e.,\nH±\nsurf=λ+/integraldisplay∞\n0dyΨ†\n0H1Ψ0=λ+C1k2\nz+C2k2\nx∓vkx1−Q2\n1+Q2+2(γk2\nz−M2k2\nx)Q\n1+Q2\n≈λ∓vsurfkx+γsurfk2\nz. (28)\nwhereQ≈0.5115,vsurf≈1.440 eV˚A andγsurf≈17.38 eV˚A2. Note that the quadratic term in kxvanishes after the\nmodel parameters are used.\nAs is easy to check, the effective Hamiltonian in Eq. (28) reproduces almost perfectly the shape of the Fermi arcs\nin thekxkzplane. However, it does not contain the information about the finite length of the arcs. We could explain\nthis fact in part by pointing out that the corresponding information is encoded in the terms quadratic in momenta\nkxandkz. When such terms are omitted from the zeroth order Hamiltonian H0, the existence of the surface states\nformally appears to be unconstrained. Therefore, the effective H amiltonian in Eq. (28) will be truly useful only when\nsupplemented by its range of validity in the kxkzplane. This, however, seems to diminish its practical value because\nthe corresponding range depends on the energy.\nIV. FERMI ARCS IN REALISTIC MODEL\nIn this section we will consider the complete low-energy theory desc ribed by Hamiltonian (1) with α/negationslash= 0. By\nperforming a unitary transformation in Eq. (1), defined by Uy=1√\n2I2⊗(I2+iσy), we arrive at the following\nequivalent form of the Hamiltonian:\n˜H=/bracketleftbig\nC2(k2\nx−∂2\ny)+C1k2\nz+C0/bracketrightbig\nI2⊗I2−M2(k2\nx−∂2\ny)I2⊗σx\n+\n−vkxv∂y+γ(k2\nz−m)−αkz(kx−∂y)20\n−v∂y+γ(k2\nz−m)vkx 0 αkz(kx−∂y)2\n−αkz(kx+∂y)20 vkxv∂y+γ(k2\nz−m)\n0 αkz(kx+∂y)2−v∂y+γ(k2\nz−m)−vkx\n, (29)\nBy introducing the spinor wave function Ψ = ( ψ1,ψ2,ψ3,ψ4)T, we reduce the eigenvalue problem ( ˜H−E)Ψ = 0 in\nthe semimetal ( y >0) to the following system of equations:\n/bracketleftbig\nC2(k2\nx−∂2\ny)−vkx+C1k2\nz+C0−E/bracketrightbig\nψ1+/bracketleftbig\n−M2(k2\nx−∂2\ny)+v∂y+γk2\nz−γm/bracketrightbig\nψ2−αkz(kx−∂y)2ψ3= 0,(30)\n/bracketleftbig\n−M2(k2\nx−∂2\ny)−v∂y+γk2\nz−γm/bracketrightbig\nψ1+/bracketleftbig\nC2(k2\nx−∂2\ny)+vkx+C1k2\nz+C0−E/bracketrightbig\nψ2+αkz(kx−∂y)2ψ4= 0,(31)\n−αkz(kx+∂y)2ψ1+/bracketleftbig\nC2(k2\nx−∂2\ny)+vkx+C1k2\nz+C0−E/bracketrightbig\nψ3+/bracketleftbig\n−M2(k2\nx−∂2\ny)+v∂y+γk2\nz−γm/bracketrightbig\nψ4= 0,(32)\nαkz(kx+∂y)2ψ2+/bracketleftbig\n−M2(k2\nx−∂2\ny)−v∂y+γk2\nz−γm/bracketrightbig\nψ3+/bracketleftbig\nC2(k2\nx−∂2\ny)−vkx+C1k2\nz+C0−E/bracketrightbig\nψ4= 0.(33)\nOn the vacuum side ( y<0), the corresponding set of equations has the same form, but wit hmreplaced by −˜m. The\ncorresponding full set of equations should be also supplemented by the conditions of continuity of the wave functions\nand their derivatives across the vacuum-semimetal interface at y= 0, see Eqs. (A16) through (A20) in Appendix A2.9\nAs shown in Appendix A2, the spinor structure of the solution on the semimetal side takes the form:\nΨy>0(y) =2/summationdisplay\ni=1\nai\nbi\nci\ndi\ne−piy, (34)\nwhere the explicit expressions for the exponents are given in Eq. (A 23). In the case of surface Fermi arc solutions,\nthe exponents take real values. A nontrivial solution exists when t he following condition is satisfied:\n/parenleftbig\nQ+\n1−Q+\n2/parenrightbig/parenleftbig\nQ−\n1−Q−\n2/parenrightbig\n−/parenleftbig\nT+\n1−T+\n2/parenrightbig/parenleftbig\nT−\n1−T−\n2/parenrightbig\n= 0, (35)\nwhere, by definition, Q±\ni≡Q(pi,±kx) andT±\ni≡T(pi,±kx), and the functions Q(p,kx) andT(p,kx) are defined in\nEqs. (A12) and (A26), respectively.\nBy taking into account that T(p,kx) vanishes at α= 0, one finds that the above condition reduces to its analog\nin Eq. (25) in the two-component model. Indeed, a nontrivial solutio n exists in the model with the two-component\nupper (lower) block Hamiltonian when Q+\n1=Q+\n2(Q−\n1=Q−\n2) is satisfied. We would like to emphasize that the\nclassification of the arc states remains essentially the same also in a g eneral case with α/negationslash= 0. However, because of the\nmixing between the upper and lower block Hamiltonians, the arcs are la beled by the eigenvalues of the Uχoperator,\nsee Appendix B. The eigenstates with χ= +1 (χ=−1) are the generalizations of the arcs from the upper (lower)\nblock Hamiltonian.\nThe numerical results for the surface Fermi arc states are show n in Fig. 3 for α= 1 eV˚A3(left panel) and\nα= 50 eV ˚A3(right panel). At fixed energy, there are two surface Fermi arcs related to two different sectors of\nthe A3Bi (A=Na, K, Rb) compounds with definite eigenvalue of Uχ. One can check that the wave functions that\ndescribe these surface Fermi arcs are related to each other by m eans of the ˜UΠkxtransformation, see Appendix B.\nBy comparing these results with those in the two-component model, see Fig. 2, we find that the quantitative effect\nof a nonzero αon the Fermi arcs is small even when αis moderately large. The only qualitative effect due to αis a\nreconnection of the pair of arcs (from predominantly up and predo minantly down sectors) at negative values of the\nFermi energy. The underlying physics of such an effect is likely to be c onnected with the loss of the chirality as a\ngood quantum number for quasiparticles away from the Dirac/Weyl nodes. Because of the discrete ud-parity, which\nis preserved even at large values of α, there are still two sectors of the theory and there are still small nontrivial arcs\npresent, as we see from the right panel of Fig. 3. It will be interest ing to explore whether the reconnection of the pairs\nof arcs would also appear in the microscopic theory. It may well be an artifact of the low-energy theory used here.\nFIG.3: (Color online)TheFermiarcssolutions intheplaneo ftransversemomentafor α= 1eV˚A3(leftpanel)and α= 50eV˚A3\n(right panel).10\nV. FERMI ARCS AND WEAK BREAKING OF TIME-REVERSAL SYMMETRY\nAs we discussed in detail in Sec. IIB, the low-energy effective Hamilto nian (1) is invariant under the time-reversal\nand inversion symmetries. Moreover, these symmetries play an impo rtant role in defining the physical properties of\nA3Bi semimetals. Thus, it is natural to ask about possible effects on th e structure (and perhaps even the existence) of\nsurface Fermi arcs due to breaking of these symmetries. From th e physics viewpoint, for example, the corresponding\ndiscrete symmetries could be broken explicitly by magnetic doping or a n external magnetic field.\nIn order to study the symmetry breaking effects, we will add to the low-energy Hamiltonian (1) two additional\nterms controlled by parameters m1and ˜µ1:\nHsb=H(k)−/parenleftbigg\n˜µ1I2+σzγm1 0\n0 −˜µ1I2−σzγm1/parenrightbigg\n. (36)\nBy analyzing the Schwinger–Dyson equation for the quasiparticle pr opagator in A 3Bi semimetals in a magnetic field,\nwe found that these terms are indeed perturbatively generated. Alternatively, these terms can be induced by magnetic\ndoping. The value of ˜ µ1could be interpreted as a mismatch between the chemical potentials of quasiparticle states\nin the Weyl sectors of the theory. The value of m1is a mismatch of the parameter mthat determines the chiral shift\nin the two sectors. This means that whenever these symmetry bre aking parameters appear, the Z2Weyl semimetal\nwill get automatically transformed into a true Weyl semimetal with fo ur non-degenerate Weyl nodes.\nBy performing a unitary transformation in Eq. (36), defined by mat rixUy=1√\n2I2⊗(I2+iσy), we arrive at the\nfollowing equivalent Hamiltonian:\n˜Hsb=/bracketleftbig\nC2(k2\nx−∂2\ny)+C1k2\nz+C0/bracketrightbig\nI2⊗I2−M2(k2\nx−∂2\ny)I2⊗σx\n+\n−vkx−˜µ1v∂y+γ/parenleftbig\nk2\nz−m−m1/parenrightbig\n−αkz(kx−∂y)20\n−v∂y+γ/parenleftbig\nk2\nz−m−m1/parenrightbig\nvkx−˜µ1 0 αkz(kx−∂y)2\n−αkz(kx+∂y)20 vkx+ ˜µ1v∂y+γ/parenleftbig\nk2\nz−m+m1/parenrightbig\n0 αkz(kx+∂y)2−v∂y+γ/parenleftbig\nk2\nz−m+m1/parenrightbig\n−vkx+ ˜µ1\n.\n(37)\nIt is straightforward, although tedious to repeat the same analys is as in Sec. IV.\nThe general surface state solution is of the same type, i.e., Ψ y>0(y) = Ψ0e−py, where Ψ 0≡(a,b,c,d)Tis a constant\nspinor. However, the characteristic equation is considerably more complicated,\n/braceleftig/bracketleftbig\n−C2(p2−k2\nx)+C1k2\nz+C0−˜µ1−E/bracketrightbig2−/bracketleftbig\nM2(p2−k2\nx)+γ(k2\nz−m−m1)/bracketrightbig2+v2(p2−k2\nx)−α2k2\nz(p2−k2\nx)2/bracerightig\n×/braceleftig/bracketleftbig\n−C2(p2−k2\nx)+C1k2\nz+C0+ ˜µ1−E/bracketrightbig2−/bracketleftbig\nM2(p2−k2\nx)+γ(k2\nz−m+m1)/bracketrightbig2+v2(p2−k2\nx)−α2k2\nz(p2−k2\nx)2/bracerightig\n+4α2k2\nz(p2−k2\nx)2/parenleftbig\n˜µ2\n1−γ2m2\n1/parenrightbig\n= 0. (38)\nThe important effect of the symmetry breaking terms with nonzero m1and ˜µ1is that the new characteristic equation\nhasfour(instead of two degenerate) pairs of distinct solutions: p=±pi, withi= 1,2,3,4. The generalspinor solution\nin the semimetal takes the following form:\nΨy>0(y) =4/summationdisplay\ni=1\nai\nbi\nci\ndi\ne−piy. (39)\nBy making use of the equation of motion, the components bianddican be expressed in terms of aiandci,\nbi=−C2(p2\ni−k2\nx)+C1k2\nz+C0−˜µ1−E−vkx\n−M2(p2\ni−k2x)−γ(k2z−m−m1)+vpiai−αkz(pi+kx)2\n−M2(p2\ni−k2x)−γ(k2z−m−m1)+vpici,(40)\ndi=−αkz(pi−kx)2\n−M2(p2\ni−k2x)−γ(k2z−m+m1)+vpiai+−C2(p2\ni−k2\nx)+C1k2\nz+C0+ ˜µ1−E−vkx\n−M2(p2\ni−k2x)−γ(k2z−m+m1)+vpici.(41)\nIn order to avoid a possible confusion, let us emphasize that the rem aining two components aiandciare not\nindependent, but fixed unambiguously for each pi. The final solutions for the Fermi arcs are determined after all\nfour independent parameters (e.g., aiwithi= 1,2,3,4) are fixed by satisfying the continuity conditions for the wave\nfunction at the surface of the semimetal. The corresponding solut ions can be obtained by numerical methods.11\nTo slightly simplify the analysis, let us consider a special case of vanish ingαin more detail. In this case, the states\nfrom the two-component upper and lower block Hamiltonians decoup le. Also, the characteristic equation factorizes,\neffectively giving two separate equations, i.e.,\n/bracketleftbig\n−C2(p2−k2\nx)+C1k2\nz+C0−˜µ1−E/bracketrightbig2−/bracketleftbig\nM2(p2−k2\nx)+γ(k2\nz−m−m1)/bracketrightbig2+v2(p2−k2\nx) = 0 (up) ,(42)\n/bracketleftbig\n−C2(p2−k2\nx)+C1k2\nz+C0+ ˜µ1−E/bracketrightbig2−/bracketleftbig\nM2(p2−k2\nx)+γ(k2\nz−m+m1)/bracketrightbig2+v2(p2−k2\nx) = 0 (down) ,(43)\ncf. Eq. (A8). Then, the analysis of the surface Fermi arcs follows very closely the analysis in Sec. IIIB.\n/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(a)m1= 10−4, ˜µ1= 0.05/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(b)m1= 10−4, ˜µ1=−0.05/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(c)m1= 0.005, ˜µ1= 10−4/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(d)m1=−0.005, ˜µ1= 10−4\n/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(e)m1= 0.005, ˜µ1= 0.05/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(f)m1=−0.005, ˜µ1=−0.05/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(g)m1= 0.005, ˜µ1=−0.05/Minus0.4/Minus0.2 0.0 0.2 0.4/Minus0.4/Minus0.20.00.20.4\nakx/Slash1Πckz/Slash1Π\n(h)m1=−0.005, ˜µ1= 0.05\nFIG. 4: The Fermi arcs solutions (thick black lines) in the mo del with the symmetry breaking parameters m1and ˜µ1atE= 0.\nThe shaded regions represent the projections of the bulk Fer mi surfaces onto the kxkzplane. The values of m1and ˜µ1are\ngiven in units of ˚A−2and eV, respectively.\nA number of representative numerical solutions for the Fermi sur face arcs in the model with the symmetry breaking\nparameters m1and ˜µ1are shown in Fig. 4. The results are obtained for the Fermi energy E= 0. In order to shed\nlight on the origin of the individual arcs, in the same figure we also show the projections (shaded regions) of the bulk\nFermi surfaces onto the kxkzplane. Such a representation reveals that some of the Fermi arcs linkdisconnected sheets\nof the bulk Fermi surface [27], while others link different points of the samebulk Fermi surface sheet.\nAs suggested by the physical meaning of the symmetry breaking pa rameters,m1and ˜µ1, the Fermi surface arcs for\nthe up and down Weyl sectors of the theory are not transformed into each other by a mirror symmetry. In addition\nto the expected effects of (i) changing the length of the arcs (prim arily due to nonzero m1) and (ii) shifting the arcs’\nposition in the kxdirection (primarily due to nonzero ˜ µ1), we also see some qualitative changes in the shape and\nbranching of the arcs. By comparing Eqs. (42) and (43) for the tw o sectors of the theory, we find that the whole\nasymmetric sets of the Fermi arcs turn into their mirror reflection s when both parameters m1and ˜µ1change their\nsigns. Examples of two pairs of such mirror configurations are show n in panels (e)–(f) and (g)–(h) in Fig. 4. [Strictly\nspeaking, the other two pairs of configurations, see (a)–(b) and (c)–(d), are not exact mirror reflections of each other\nbecause one of the symmetry breaking parameters does not chan ge the sign. Because of a smallness of the parameter,\nthere is an appearance of approximate mirror configurations.]\nIt is interesting to point out that different topologies of the global ( bulk-plus-arcs) Fermi hypersurfaces, including\nthe bulk sheets and the surface Fermi arcs, are possible. For exa mple, for a range of symmetry breaking parameters,\nrepresentedbypanels(c), (d), (e)and(f) inFig.4, wefindthatt heglobalFermihypersurfacesconsistofpairsofclearly\ndisconnected parts. This is in contrast to the configurations in pan els (a) and (b), where different parts touch at four\npoints, and in contrast to the configurations in panels (g) and (h), where all parts of the global Fermi hypersurfaces12\nare linked by the Fermi arcs. If samples with completely disconnecte d parts of the global Fermi hypersurfaces are\nindeed possible, they will be very interesting to study in experiments .\nAs we see from panels (g) and (h) in Fig. 4, there are also qualitatively new types of the Fermi arcs possible for a\nrange of symmetry breaking parameters. In particular, we find a p air of “short” branches of the Fermi arcs that split\noff from the usual “long” arcs. To the best of our knowledge, the c orresponding short arcs have not been predicted\nbefore. So far, we could not establish a general criterion for the e xistence of the short arcs. In the configurations in\npanels (g) and (h), they play a profound role by linking two disconnec ted sheets of the bulk Fermi surface.\nVI. CONCLUSION\nIn this paper, we studied the surface Fermi arc states by employin g a continuum low-energy effective model. The\nuse of analytical methods and a realistic low-energy model provide a deeper insight into the physical properties and\ncharacteristics of the surface Fermi arcs. In particular, we wer e able to classify the Fermi arcs with respect to the\nud-parity and reconfirm the Z2Weyl structure of A 3Bi semimetals [33]. In this context, it should be noted that the\nexperimental observation of the corresponding Fermi arc state s have been recently reported for Na 3Bi [36]. While in\nagreement with the claimed topological semimetal structure, such an observation does not confirm it unambiguously.\nThat is because the Fermi arc states are also possible in Dirac mater ials where the Z2Weyl structure is absent [5, 31].\nThe unambiguous confirmation of the Z2Weyl structure could, however, be established via the quantum os cillations,\nwhose period should dependent on the thickness of the semimetal in the same way as in true Weyl semimetals [31, 32].\nBy introducing the effects of several possible symmetry breaking t erms, we show that the Z2Weyl structure of A 3Bi\nis destroyed in a very special way: the compounds become true Wey l semimetals. We suggest that this finding can be\ntested in experiment. For example, by taking into account that the mirror-symmetric pairs of surface Fermi arcs in\nclean A 3Bi get distorted upon the introduction of explicit symmetry breakin g (e.g., by magnetic doping), a number\nof specific features (size, shape and number of branches) should be seen in the surface Fermi arcs. The corresponding\nproperties could be studied, for example, by analyzing the quantum oscillations sensitive to the surface states of this\ntype [31]. In the absence of symmetry breaking, there will be a uniqu e period of oscillations dependent in a specific\nway on the thickness of the semimetal slab [32]. On the other hand, t he breaking of symmetry will produce pairs\nof inequivalent arcs of different lengths and the observation of two incommensurate periods of oscillations will be\nexpected. In principle, by making use of the analytical results in this study, the details of the oscillations could be\nused to estimate the magnitude of the symmetry breaking terms.\nAcknowledgments\nThe work of E.V.G. was supported partially by the Ukrainian State Fou ndation for Fundamental Research. The\nwork of V.A.M. was supported by the Natural Sciences and Engineer ing Research Council of Canada. The work of\nI.A.S. was supported by the U.S. National Science Foundation under Grant No. PHY-1404232.\nAppendix A: Derivations of surface Fermi arcs solutions\nIn this Appendix we present the key technical details of deriving the surface Fermi arcs solutions in the 2 ×2 model,\nintroduced in Sec. IIIB, and in the 4 ×4 model, introduced in Sec. IV.\n1. Surface Fermi arcs in 2×2model\nLet us start with the analysis of the surface Fermi arc states in th e 2×2 model, introduced in Sec. IIIB. The\nproblem reduces to solving the eigenvalues problem given by Eqs. (22 ) and (23) at y >0 (semimetal), as well as a\nsimilar set of equations at y <0 (vacuum), but mreplaced by −˜m. The corresponding set of equations should be\nalso supplemented by the boundary conditions at the vacuum-semim etal interface, i.e.,\n˜ψ1(−0) =ψ1(+0), (A1)\n˜ψ2(−0) =ψ2(+0), (A2)\n−C2∂y˜ψ1(−0)+M2∂y˜ψ2(−0) =−C2∂yψ1(+0)+M2∂yψ2(+0), (A3)\n−M2∂y˜ψ1(−0)+C2∂y˜ψ2(−0) =−M2∂yψ1(+0)+C2∂yψ2(+0), (A4)13\nwhere˜ψ1,2(y) correspond to the vacuum region at y<0.\nInside the semimetal ( y>0), the surface state solutions should have the following form:\nΨy>0(y) =/parenleftbigg\na\nb/parenrightbigg\ne−py. (A5)\nBy substituting this ansatz in Eqs. (22) and (23), we arrive at the f ollowing set of linear equations for the spinor\ncomponents aandb:\n/bracketleftbig\nC2(k2\nx−p2)−vkx+C1k2\nz+C0−E/bracketrightbig\na+/bracketleftbig\n−M2(k2\nx−p2)−vp+γk2\nz−γm/bracketrightbig\nb= 0, (A6)/bracketleftbig\n−M2(k2\nx−p2)+vp+γk2\nz−γm/bracketrightbig\na+/bracketleftbig\nC2(k2\nx−p2)+vkx+C1k2\nz+C0−E/bracketrightbig\nb= 0. (A7)\nA nontrivial solution exists when the following characteristic equatio n is satisfied:\n/bracketleftbig\n−C2(p2−k2\nx)+C1k2\nz+C0−E/bracketrightbig2−/bracketleftbig\nM2(p2−k2\nx)+γk2\nz−γm/bracketrightbig2+v2(p2−k2\nx) = 0. (A8)\nThe solutions to this equation are p=±p1andp=±p2, where\np1=/radicaligg\nk2x−X+√\nX2+Y\n2(M2\n2−C2\n2), p 2=/radicaligg\nk2x−X−√\nX2+Y\n2(M2\n2−C2\n2). (A9)\nHere we introduced the following shorthand notations:\nX≡2C2(C1k2\nz+C0−E)+2γM2(k2\nz−m)−v2, (A10)\nY≡4(M2\n2−C2\n2)/bracketleftbig\n(C1k2\nz+C0−E)2−γ2(k2\nz−m)2/bracketrightbig\n. (A11)\nThe spinor components aandbof the corresponding nontrivial solution satisfy the constraint\nb\na=Q(p,kx)≡−C2(p2−k2\nx)+C1k2\nz+C0−E−vkx\n−M2(p2−k2x)−γ(k2z−m)+vp. (A12)\nInside the semimetal ( y >0), the wave function should fall off with increasing y. Thus, we use only the negative\nexponents in the general solution, i.e.,\nΨy>0(y)≃a1/parenleftbigg\n1\nQ1/parenrightbigg\ne−p1y+a2/parenleftbigg\n1\nQ2/parenrightbigg\ne−p2y, (A13)\nwhereQi≡Q(pi,kx) withi= 1,2.\nIn order to find the vacuum solution ( y <0), we replace m→ −˜mand take the limit ˜ m→ ∞. This leads to the\nfollowing general solution on the vacuum side:\nΨy<0(y)≃˜a1√γ˜m/parenleftbigg\n1\n−1/parenrightbigg\ne˜p1y+˜a2√γ˜m/parenleftbigg\n1\n1/parenrightbigg\ne˜p2y, (A14)\nwhere, for convenience, we took the overall constants to be inve rsely proportional to√γ˜m. The exponents in the\nvacuum solution are determined by\n˜p1≃/radicalbigg\nγ˜m\n−M2−C2,˜p2≃/radicalbigg\nγ˜m\n−M2+C2. (A15)\nIt is interesting to note that the conditions of the wave function co ntinuity, see Eqs. (A1) and (A2), are the only\nimportant conditions to be satisfied. Indeed, the nontrivial solutio n of Eq. (A1) in the limit ˜ m→ ∞implies that\na1=−a2/negationslash= 0. This is consistent with Eq. (A2) only when Q1=Q2. Concerning the remaining boundary conditions\nin Eqs. (A3) and (A4), enforcing the continuity of the wave functio n derivative, they do not add any additional\nconstraints. In fact, they are needed only for determining the va cuum spinor components ˜ a1and ˜a2in terms of the\nnontrivial components a1anda2in the semimetal. Such (finite) solutions always exist. However, as is c lear from\nEq. (A14), the vacuum solution have no much physical content bec ause it vanishes in the limit ˜ m→ ∞.\nIn conclusion, the boundary conditions at y= 0 are satisfied and, therefore, a nontrivial solution exists when\nQ1=Q2. The explicit form of the corresponding condition is given in Eq. (25) in the main text.14\n2. Surface Fermi arcs in 4×4model\nThe analysis of the realistic 4 ×4 model introduced in Sec. IV is slightly more involved, but qualitatively similar.\nThe eigenvalues problem in this case is given by Eqs. (30) through (33 ) aty>0 (semimetal), as well as a similar set\nof equations at y<0 (vacuum), but with mreplaced by −˜m. The conditions of continuity of the wave functions and\ntheir derivatives across the vacuum-semimetal surface at y= 0 are given by\n˜ψi(−0) =ψi(+0),fori= 1,2,3,4, (A16)\n−C2∂y˜ψ1(−0)+M2∂y˜ψ2(−0)−αkz∂y˜ψ3(−0) =−C2∂yψ1(+0)+M2∂yψ2(+0)−αkz∂yψ3(+0),(A17)\nM2∂y˜ψ1(−0)−C2∂y˜ψ2(−0)+αkz∂y˜ψ4(−0) =M2∂yψ1(+0)−C2∂yψ2(+0)+αkz∂yψ4(+0),(A18)\n−αkz∂y˜ψ1(−0)−C2∂y˜ψ3(−0)+M2∂y˜ψ4(−0) =−αkz∂yψ1(+0)−C2∂yψ3(+0)+M2∂yψ4(+0),(A19)\nαkz∂y˜ψ2(−0)+M2∂y˜ψ3(−0)−C2∂y˜ψ4(−0) =αkz∂yψ2(+0)+M2∂yψ3(+0)−C2∂yψ4(+0).(A20)\nIn the semimetal ( y>0), we look for a general surface state solution in the form\nΨy>0(y) =\na\nb\nc\nd\ne−py. (A21)\nA nontrivial solution of this type exists when pis a solution to the following characteristic equation:\n/bracketleftbig\n−C2(p2−k2\nx)+C1k2\nz+C0−E/bracketrightbig2−/bracketleftbig\nM2(p2−k2\nx)+γk2\nz−γm/bracketrightbig2+v2(p2−k2\nx)−α2k2\nz(p2−k2\nx)2= 0.(A22)\n(Strictly speaking, the characteristic equation has the square on the left hand side, implying that the degeneracy of\nits solutions should be doubled.) This equation has twopairs of distinct solutions: p=±p1andp=±p2, where\np1,2=/radicaligg\nk2x−X±√\nX2+Y\n2(M2\n2−C2\n2+α2k2z), (A23)\ncf. Eq. (A9). Here, the expression for Xis the same as in the 2 ×2 model (α= 0) in Eq. (A10), but the expression\nforYis slightly different, i.e.,\nY≡4(M2\n2−C2\n2+α2k2\nz)/bracketleftbig\n(C1k2\nz+C0−E)2−γ2(k2\nz−m)2/bracketrightbig\n. (A24)\nTherefore, in the half-space occupied by the semimetal ( y >0), the wave function should have the following general\nform:\nΨy>0(y)≃\na1\nQ+\n1a1−T+\n1c1\nc1\n−T−\n1a1+Q−\n1c1\ne−p1y+\na2\nQ+\n2a2−T+\n2c2\nc2\n−T−\n2a2+Q−\n2c2\ne−p2y, (A25)\nwhere we also introduced the shorthand notation: Q±\ni≡Q(pi,±kx) andT±\ni≡T(pi,±kx). Here the function Q(p,kx)\nis the same as in Eq. (A12) and\nT(p,kx) =αkz(p+kx)2\n−M2(p2−k2x)−γ(k2z−m)+vp. (A26)\nIn the vacuum solution ( y <0), we replace m→ −˜mand take the limit ˜ m→ ∞. In this case, a simple analysis\nleads to the following solution:\nΨy<0(y)≃1√γ˜m\n˜a1\n˜a1C2+αkz˜c1 √\nC2\n2−α2k2z\n˜c1\nαkz˜a1+C2˜c1 √\nC2\n2−α2k2z\ne˜p1y+1√γ˜m\n˜a2\n−˜a2C2+αkz˜c2 √\nC2\n2−α2k2z\n˜c2\n−αkz˜a2+C2˜c2 √\nC2\n2−α2k2z\ne˜p2y, (A27)15\nwhere, for convenience, we introduced an overall constant inver sely proportional to√γ˜m. The explicit form of the\nexponents in this solution is determined by\n˜p1≃/radicaligg\nγ˜m\n−M2+/radicalbig\nC2\n2−α2k2z,˜p2≃/radicaligg\nγ˜m\n−M2−/radicalbig\nC2\n2−α2k2z. (A28)\nNote that the signs in the exponents of the vacuum solution (A27) a re chosen so that the wave function vanishes at\ny→ −∞.\nThe conditions of the continuity of the wave function in Eq. (A16) lea d to the following constraints:\n0 =Q+\n1a1−T+\n1c1+Q+\n2a2−T+\n2c2, (A29)\n0 =T−\n1a1−Q−\n1c1+T−\n2a2−Q−\n2c2, (A30)\ntogether with a2=−a1andc2=−c1. Here, we took into account that the left hand side of Eq. (A16) va nishes in\nthe limit ˜m→ ∞.\nAs in the case of a two-component model, discussed in Appendix A1, t here is no need to satisfy the continuity\nconditions for the wave function derivatives, given by Eqs. (A17) t hrough (A20). The reason is that these conditions\nadd no additional constraints on the spinor solutions in the semimeta l. They are needed only for determining the\ncomponents of the vacuum solution at y <0. Since the latter has no physical content in the limit ˜ m→ ∞, we can\nsafely ignore the conditions in Eqs. (A17) through (A20).\nIn order to have a nontrivial solution to Eqs. (A29) and (A30), the condition in Eq. (35) in the main text should\nbe satisfied.\nAppendix B: Symmetries and surface Fermi arcs bispinors\nIn this Appendix, we discuss the properties of the Fermi arc surfa ce states with respect to the discrete symmetries\nUχ≡UΠkzand˜UΠkx, introduced in Sec. IIB. To startwith, let us note that the genera lFermi arcspinor in Eq. (A25)\ncontains all possible solutions. It is possible to classify these solution s with respect to the discrete symmetry Uχby\nchoosing them as eigenstates of the operator Uχ.\nIn order to construct the first group of solutions, we use the rela tions in Eqs. (A29) and (A30) and rewrite the\nFermi arc spinor in Eq. (A25) in the following form:\nΨ+=a1\n1\nQ+\n1−T+\n1T−\n1−T−\n2\nQ−\n1−Q−\n2\nT−\n1−T−\n2\nQ−\n1−Q−\n2\n−T−\n1+Q−\n1T−\n1−T−\n2\nQ−\n1−Q−\n2\ne−p1y−a1\n1\nQ+\n2−T+\n2T−\n1−T−\n2\nQ−\n1−Q−\n2\nT−\n1−T−\n2\nQ−\n1−Q−\n2\n−T−\n2+Q−\n2T−\n1−T−\n2\nQ−\n1−Q−\n2\ne−p2y. (B1)\nBy noting that pi’s (withi= 1,2), defined in Eq. (A23), contain only quadratic terms in momenta kxandkz, we\nconclude that both of them are invariant under the Π kzand Π kxtransformations. The other quantities, used in\nEq. (B1), transform as follows:\nΠkzQ±\ni=Q±\ni,ΠkzT±\ni=−T±\ni, (B2)\nΠkxQ±\ni=Q∓\ni,ΠkxT±\ni=T∓\ni. (B3)\nIt is straightforward to check that the spinor in Eq. (B1) is an eigen state of the operator Uχwith the eigenvalue\nχ= +1. Indeed, by making use of the definition of the matrix U, we find that\nUΨ+=a1\n1\nQ+\n1−T+\n1T−\n1−T−\n2\nQ−\n1−Q−\n2\n−/bracketleftig\nT−\n1−T−\n2\nQ−\n1−Q−\n2/bracketrightig\n−/bracketleftig\n−T−\n1+Q−\n1T−\n1−T−\n2\nQ−\n1−Q−\n2/bracketrightig\ne−p1y−a1\n1\nQ+\n2−T+\n2T−\n1−T−\n2\nQ−\n1−Q−\n2\n−/bracketleftig\nT−\n1−T−\n2\nQ−\n1−Q−\n2/bracketrightig\n−/bracketleftig\n−T−\n2+Q−\n2T−\n1−T−\n2\nQ−\n1−Q−\n2/bracketrightig\ne−p2y. (B4)\nThen, by taking into account that Uχ≡UΠkz, we see that UχΨ+= Ψ+, as claimed.16\nBy using the relations in Eqs. (A29) and (A30), the Fermi arc spinor in Eq. (A25) can be also rewritten in the\nfollowing alternative form:\nΨ−=c1\nT+\n1−T+\n2\nQ+\n1−Q+\n2\nQ+\n1T+\n1−T+\n2\nQ+\n1−Q+\n2−T+\n1\n1\n−T−\n1T+\n1−T+\n2\nQ+\n1−Q+\n2+Q−\n1\ne−p1y−c1\nT+\n1−T+\n2\nQ+\n1−Q+\n2\nQ+\n2T+\n1−T+\n2\nQ+\n1−Q+\n2−T+\n2\n1\n−T−\n2T+\n1−T+\n2\nQ+\n1−Q+\n2+Q−\n2\ne−p2y. (B5)\nIn this case, as is easy to check, the spinor is an eigenstate of the o peratorUχwith eigenvalue χ=−1. Indeed, by\nmaking use of the definition of the matrix U, we find that\nUΨ−=c1\nT+\n1−T+\n2\nQ+\n1−Q+\n2\nQ+\n1T+\n1−T+\n2\nQ+\n1−Q+\n2−T+\n1\n−1\n−/bracketleftig\n−T−\n1T+\n1−T+\n2\nQ+\n1−Q+\n2+Q−\n1/bracketrightig\ne−p1y−c1\nT+\n1−T+\n2\nQ+\n1−Q+\n2\nQ+\n2T+\n1−T+\n2\nQ+\n1−Q+\n2−T+\n2\n−1\n−/bracketleftig\n−T−\n2T+\n1−T+\n2\nQ+\n1−Q+\n2+Q−\n2/bracketrightig\ne−p2y, (B6)\nwhich implies that UχΨ−=−Ψ−. In other words, the eigenstate Ψ −corresponds to χ=−1, as claimed.\nNow, let us explore the implications of the ˜UΠkxsymmetry in the model at hand. By applying the corresponding\noperator to the eigenstates Ψ ±, we arrive at the following results:\n˜UΠkxΨ+=a1\nQ−\n1−Q−\n2\nT−\n1−T−\n2\n−T+\n1+Q+\n1Q−\n1−Q−\n2\nT−\n1−T−\n2\n1\nQ−\n1−T−\n1Q−\n1−Q−\n2\nT−\n1−T−\n2\ne−p1y−a1\nQ−\n1−Q−\n2\nT−\n1−T−\n2\n−T+\n2+Q+\n2Q−\n1−Q−\n2\nT−\n1−T−\n2\n1\nQ−\n2−T−\n2Q−\n1−Q−\n2\nT−\n1−T−\n2\ne−p2y, (B7)\n˜UΠkxΨ−=c1\n1\n−T+\n1T−\n1−T−\n2\nQ−\n1−Q−\n2+Q+\n1\nT−\n1−T−\n2\nQ−\n1−Q−\n2\nQ−\n1T−\n1−T−\n2\nQ−\n1−Q−\n2−T−\n1\ne−p1y−c1\n1\n−T+\n2T−\n1−T−\n2\nQ−\n1−Q−\n2+Q+\n2\nT−\n1−T−\n2\nQ−\n1−Q−\n2\nQ−\n2T−\n1−T−\n2\nQ−\n1−Q−\n2−T−\n2\ne−p2y. (B8)\nThese results show that Ψ ±are not eigenstates of the operator ˜UΠkx. However, by taking into account the constraint\nin Eq. (35), one can check that the operator ˜UΠkxinterchanges the two types of the states, i.e., Ψ +↔Ψ−.\nIn conclusion, the results of this Appendix confirm the claim in the main text of the paper concerning the symmetry\nproperties of the low-energy theory for A 3Bi (A=Na, K, Rb) semimetals, as well as the classification of their Fer mi arc\nstates. These are in complete agreement with the claim that the cor responding compounds are Z2Weyl semimetals.\n[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zha ng, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov,\nScience306, 666 (2004).\n[2] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, a nd A. M. Rappe, Phys. Rev. Lett. 108, 140405 (2012).\n[3] J. L. Manes, Phys. Rev. B 85, 155118 (2012).\n[4] Z. Wang, Y. Sun, X. Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012).\n[5] Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Phys. Rev. B 88, 125427 (2013).\n[6] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Buchner, and R. J. Cava, Phys. Rev. Lett. 113, 027603 (2014).\n[7] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. L iu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin, A. Bansil ,\nF. Chou, and M. Z. Hasan, Nat. Commun. 5, 3786 (2014).\n[8] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabh akaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain,\nand Y. L. Chen, Science 343, 864 (2014).\n[9] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84, 235126 (2011).\n[10] A. M. Turner and A. Vishwanath, arXiv:1301.0330 [cond- mat.str-el].\n[11] O. Vafek and A. Vishwanath, Ann. Rev. Cond. Mat. Phys. 5, 83 (2014).\n[12] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, P hys. Rev. B 83, 205101 (2011).17\n[13] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).\n[14] G. Y. Cho, arXiv:1110.1939 [cond-mat.str-el].\n[15] H. Weng, C. Fang, Z. Fang, A. Bernevig, and X. Dai, Phys. R ev. X5, 011029 (2015).\n[16] S.-M. Huang, Su-Y. Xu, I. Belopolski, C.-C. Lee, G. Chan g, B. Wang, N. Alidoust, G. Bian, M. Neupane, A. Bansil,\nH. Lin, and M. Z. Hasan, arXiv:1501.00755 [cond-mat.mtrl-s ci].\n[17] C. Zhang, Z. Yuan, S. Xu, Z. Lin, B. Tong, M. Z. Hasan, J. Wa ng, C. Zhang, and S. Jia, arXiv:1502.00251 [cond-mat.mtrl-\nsci]\n[18] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, C. Zha ng, R. Sankar, S.-M. Huang, C.-C. Lee, G. Chang, B. Wang,\nG. Bian, H. Zheng, D. S. Sanchez, F. Chou, H. Lin, S. Jia, and M. Z. Hasan, arXiv:1502.03807 [cond-mat.mtrl-sci].\n[19] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. R ichard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang,\nX. Dai, T. Qian, and H. Ding, arXiv:1502.04684 [cond-mat.mt rl-sci].\n[20] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Li ang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen,\narXiv:1503.01304 [cond-mat.mtrl-sci].\n[21] L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, M. Soljaˇ ci´ c, arXiv:1502.03438 [cond-mat.mtrl-sci].\n[22] E. V. Gorbar, V. A. Miransky, and I. A. Shovkovy, Phys. Re v. B88, 165105 (2013).\n[23] H.-J. Kim, K.-S. Kim, J. F. Wang, M. Sasaki, N. Satoh, A. O hnishi, M. Kitaura, M. Yang, and L. Li, Phys. Rev. Lett.\n111, 246603 (2013).\n[24] E. V. Gorbar, V. A. Miransky and I. A. Shovkovy, Phys. Rev . B89, 085126 (2014).\n[25] D. T. Son and B. Z. Spivak, Phys. Rev. B 88, 104412 (2013).\n[26] H. B. Nielsen and M. Ninomiya, Phys. Lett. B 130, 389 (1983).\n[27] F. D. M. Haldane, arXiv:1401.0529 [cond-mat.str-el].\n[28] V. Aji, Phys. Rev. B 85, 241101 (2012).\n[29] R. Okugawa and S. Murakami, Phys. Rev. B 89, 235315 (2014).\n[30] P. Hosur, Phys. Rev. B 86, 195102 (2012).\n[31] A. C. Potter, I. Kimchi, and A. Vishwanath, Nature Commu n.5, 5161 (2014).\n[32] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and P. O. Suk hachov, Phys. Rev. B 90, 115131 (2014).\n[33] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and P. O. Suk hachov, Phys. Rev. B 91, 121101 (2015).\n[34] W. Zhang, R. Yu, H.J. Zhang, X. Dai, and Z. Fang, New J. Phy s.12, 065013 (2010).\n[35] S. Q. Shen, Topological Insulators (Springer, Heidelberg, 2012).\n[36] S.-Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T.-R. Chang ,\nH.-T. Jeng, C.-Y. Huang, W.-F. Tsai, H. Lin, P. P. Shibayev, F .-C. Chou, R. J. Cava, and M. Z. Hasan, Science 347, 294\n(2015)."    },    {        "title": "1206.6530v1.Continuous_transitions_between_composite_Fermi_liquid_and_Landau_Fermi_liquid__a_route_to_fractionalized_Mott_insulators.pdf",        "content": "arXiv:1206.6530v1  [cond-mat.str-el]  27 Jun 2012Continuous transitions between composite Fermi liquid and Landau Fermi liquid:\na route to fractionalized Mott insulators\nMaissam Barkeshli1and John McGreevy2\n1Department of Physics, Stanford University, Stanford, CA 9 4305\n2Department of Physics, Massachusetts Institute of Technol ogy, Cambridge, MA 02139\nOne of the most successful theories of a non-Fermi liquid met allic state is the composite Fermi\nliquid (CFL) theory of the half-filled Landau level. In this p aper, we study continuous quantum\nphase transitions out of the CFL state and intoa Landau Fermi liquid, in the limit of nodisorder and\nfixed particle number. This transition can be induced by tuni ng the bandwidth of the Landau level\nrelative to the interaction energy, for instance through an externally applied periodic potential.\nWe find a transition to the Landau Fermi liquid through a gaple ss Mott insulator with a Fermi\nsurface of neutral fermionic excitations. In the presence o f spatial symmetries, we also find a direct\ncontinuous transition between the CFL and the Landau Fermi l iquid. The transitions have a number\nof characteristic observable signatures, including the pr esence of two crossover temperature scales,\nresistivity jumps, and vanishing compressibility. When th e composite fermions are paired instead,\nour results imply quantum critical points between various n on-Abelian topological states, including\ntheν= 1/2 Moore-Read Pfaffian (Ising ×U(1) topological order), a version of the Kitaev B\nphase (Ising topological order), and paired electronic sup erconductors. To study such transitions,\nwe use a projective construction of the CFL, which goes beyon d the conventional framework of\nflux attachment to include a broader set of quantum fluctuatio ns. These considerations suggest a\npossible route to fractionalized Mott insulators by starti ng with FQH states and tuning the Landau\nlevel bandwidth.\nI. INTRODUCTION\nDespite decades of work, the breakdown of Landau\nFermi liquid theory in metallic states still poses some\nof the most challenging, unsolved problems in condensed\nmatterphysics. Thisbreakdownoftenoccursinthevicin-\nity of quantum phase transitions in Fermi liquid met-\nals, although there are some situations, such as in the\nhalf-filled Landau level in two-dimensional electron gases\n(2DEGs), where entire non-Fermi liquid metallic phases\nhave been found to exist. Perhaps the most experimen-\ntally successful theory of any non-Fermi liquid metal in\nmore than one dimension is the composite Fermi liquid\n(CFL) theory of the half-filled Landau level, which has\nhadanumberofstrikingtheoreticalpredictionsthathave\nbeen experimentally verified in GaAs 2DEGs.1–5The\nCFL also provides structural insight: as the magnetic\nfield is tuned, the conventional series of fractional quan-\ntum Hall (FQH) plateaus in the lowest Landau level in\nGaAs can be understood as integer quantum Hall (IQH)\nstates of the composite fermions1–6.\nThe crucial reason that the half-filled Landau level\ngives rise to this distinct non-Fermi liquid state of elec-\ntrons is that the Landau level has essentially zero band-\nwidth, thus quenching the kinetic energy and leaving the\ninteraction energy to dominate. As the bandwidth is in-\ncreasedto be on the orderof the interaction strength, the\nsystem should pass through a quantum phase transition.\nIn the limit that the bandwidth is large compared to\nthe interaction strength, the resulting state will be well-\ndescribed by Landau Fermi liquid theory. This raises the\nquestion of whether bandwidth-tuned transitions out of\nquantum Hall states and into more conventional states\ncan be continuous, and if so, what the possible criti-cal theories are. For example, can there be a continu-\nous quantum phase transition out of the CFL state and\ninto a Landau Fermi liquid in a clean system? Such a\ncontinuous phase transition between a non-Fermi liquid\nmetal and a Fermi liquid metal would be quite exotic;\nstarting from the Landau Fermi liquid side, it would de-\nscribe the continuous destruction of the electron Fermi\nsurface and the emergence of a Fermi surface of compos-\nite fermions with singular gauge interactions. Previous\nwork on the fate of incompressible (Abelian) FQH states\nincludes Ref. 7–14. Our analysis below will allow us to\nconsideralsothefateofincompressiblenon-AbelianFQH\nstates, such as the Moore-Read Pfaffian state15, as the\nbandwidth is increased.\nOne possible way to experimentally induce such\nbandwidth-tuned transitions is by imposing a periodic\npotential on a 2DEG subjected to an external magnetic\nfield. When there is 2 πp/qflux per unit cell ofa weak pe-\nriodicpotential, eachLandaulevelsplitsinto psubbands,\nwhere the bandwidths and gaps between the subbands\nare on the order of the strength of the periodic potential.\nFor 2πflux per plaquette, the periodic potential does not\nsplit the bands, but only gives a bandwidth to the Lan-\ndau levels and may be used to reach the regime in which\nthe bandwidth is comparableto, ormuchlargerthan, the\ninteraction strength. There are potentially other physi-\ncal realizations as well. Recent attention has focussed\non bands with non-zero Chern number in lattice systems\nwithout an external magnetic field16–25. Partially filling\nsuch Chern bands can lead to various fractional quantum\nHall (FQH) states. The existence of incompressible FQH\nstates has already been numerically established, and it is\nnatural to expect that the CFL state can also be realized\nin suchsituations. The bandwidth ofthe Chernband can2\nVp\nB ~ VpCFL LFL?(a) (b)\nFQH Conventional\nstateB\nFIG. 1: (a) Schematic illustration of the effect of a periodic\npotential Vpwith 2πflux per plaquette on a half-filled Lan-\ndau level. The Landau level acquires a bandwidth (B) on\nthe order of Vp; in the limit that Vpis much larger than the\ninteraction energy, the system should be well-described by\nLandau Fermi liquid theory. (b) As the bandwidth (B) of the\nflat band is tuned, the system will transition out of the CFL\nand ultimately into the Landau Fermi liquid. More generally ,\nthe system will transition out of a fractionalized QH state a nd\nultimately into a more conventional state.\nbe increased by applying pressure, eventually resulting\nin a Landau Fermi liquid and providing another possible\nphysical realization of the transitions discussed in this\npaper. In the context of the partially filled Chern bands,\nthere is no background magnetic field, so the conven-\ntional framework of flux attachment and flux-smearing\nmean-field theory is inapplicable, raising a further fun-\ndamental conceptual question of how to understand the\neffective field theory of the CFL in such a situation.23\nIn this paper, we address the questions discussed\nabove. We begin by studying a lesser known, alternative\nformulation of the theory of the CFL state using the pro-\njective construction.74This construction has several ad-\nvantagesovertheconventionalfluxattachmentapproach.\nMost importantly it allows us to include a broader range\nof quantum fluctuations, which provides access to nearby\nstates, including ones in which the gauge fluctuations are\ndestroyed to yield more conventional electronic states.\nWe use this construction to develop the CFL theory for\npartially filled Chern bands without an external mag-\nnetic field, where the conventional notion of flux attach-\nment may not be applicable in general. Subsequently, we\nstudy severalpossible states that can be described within\nthe same low energy effective theory. For gapless states,\nthese include the composite Fermi liquid, a gapless Mott\ninsulator (GMI) with a Fermi surface of emergent neu-\ntral fermions, and the Landau Fermi liquid. When the\nfermions are paired instead, the CFL and GMI descend\ninto either paired composite fermion states, such as the\nν= 1/2 Moore-Read (MR) Pfaffian, or gapped topolog-\nical states with an emergent Z2gauge field coupled to\nthe fermions. The Landau Fermi liquid is then replaced\nby a paired electronic superconductor. These consider-\nations then allow us to study zero temperature phase\ntransitions out of the quantum Hall states and into other\nexotic fractionalized states, such as the GMI or its paired\ndescendants, and ultimately into conventional electronic\nstates, such as Landau Fermi liquids and superconduc-\ntors (Fig. 1).We find a transition between the CFL and the Lan-\ndau FL through an intervening gapless Mott insulator\nstate with emergent Fermi surface. In the presence of\nsomespatialsymmetry, suchasinversion,wefind aquan-\ntum critical point directly separating the CFL and the\nLandau FL. When we consider px+ipypairing of the\ncomposite fermions, these considerations lead to a tran-\nsition between the ν= 1/2 MR Pfaffian15and apx+ipy\npaired superconductor of electrons through an interven-\ning non-Abelian topological state that is topologically\nequivalent to the non-Abelian B phase of Kitaev’s honey-\ncombmodel26. Inthe presenceofsomespatialsymmetry,\nour results then imply a quantum critical point directly\nseparatingthe ν= 1/2MR Pfaffian and the conventional\npx+ipyelectronic superconductor; in the presence of a\nstrong magnetic field this may be induced by tuning a\nperiodic potential with 2 πflux per plaquette.\nRecently, building on a slave-rotormean-fieldtheory,27\nT. Senthil has developed a theory of a continuous Mott\ntransitionbetween a U(1) spin liquid Mott insulatorwith\na spinon Fermi surface and a Landau FL by including the\neffects of the U(1) gauge fluctuations28. That theory de-\nscribes how the Fermi surface is continuously destroyed\nas the transition is approached, and it contains a number\nofstriking,experimentallytestablepredictions. Thesein-\nclude the existence of two crossover temperature scales\nand therefore two distinct quantum critical regimes, a\nuniversal resistivity jump at the transition, and diverg-\ning quasiparticle effective masses and Landau parame-\nters as the transition is approached from the Fermi liq-\nuid side. It has been conjectured29,30that such a gapless\nspin liquid state may be realized in a number of different\ncompounds,31–33which would then provide the possibil-\nityofobservingsuchabandwidth-tuned continuousMott\ntransition.\nThe transitions between the CFL, GMI, and Landau\nFermiliquiddisplayphenomenologysimilartothatfound\nintheU(1)spinliquidMotttransitionofRef.28. Wefind\nthat they can be continuous, and there are two crossover\ntemperature scales and resistivity jumps at the transi-\ntions. Remarkably, we find that although the CFL and\nLandau FL are both compressible states, the quantum\ncriticalpointbetweenthemisincompressibleatzerotem-\nperature. On the composite Fermi liquid side of the tran-\nsitions, the second crossover temperature scale does not\nexist in the presence of long-range Coulomb interactions,\nbut can be made to appear by adding a metallic gate to\nscreen the long-range interactions.\nWhile the compounds studied in Refs. 31–33 provide\nexperimentally promising venues for the observation of\nsuch a continuous Mott transition, our work suggests an-\nother experimentally promisingvenue for similar physics.\nIftheCFLtheoryistakenseriouslyasdescribingthehalf-\nfilled Landau level, then our results predict that there\nis a GMI state nearby with a Fermi surface of neutral\nfermions and it may be realized by tuning appropriate\nperiodic potentials with a wavelength on the order of the\ninterparticle spacing; this state is a non-trivial spinless3\nanalog of the U(1) spin liquid Mott insulator with spinon\nFermi surface. We may refer to this as a U(1) orbital liq-\nuid Mott insulator. The proximity to a Landau Fermi\nliquid by tuning the bandwidth would then also allow\nfor another experimentally promising venue to study the\ncontinuous destruction of the Fermi liquid and the ap-\npearance of a fractionalized state with a Fermi surface\ncoupled to an emergent U(1) gauge field.\nThispaperisorganizedasfollows. InSectionII, webe-\nginwithabriefreviewoftheconstructionoftheCFLthe-\nory of the half-filled Landau level, and we review some of\nthe results of the Halperin-Lee-Read theory of the gauge\nfluctuations of this state. In Section III, we develop a\nprojective/partonconstructionforthe CFL state, and we\nshow how it can be applied to situations without an ex-\nternalmagneticfield, andcanbegeneralizedtoanyfilling\nfraction, including odd-denominator fillings. In Section\nIV, we discuss statesproximateto the CFL at half-filling,\nincluding the GMI with emergent Fermi surface, and the\nLandauFermi liquid. In Section V, westudy the continu-\nous transitions separating these states. In Section VI we\ndiscuss the consequences of this theory for various paired\nnon-Abelian states.\nII. BRIEF REVIEW OF THE CFL THEORY\nThe CFL theory of the compressible FQH state at\nν= 1/2mbegins by performing an exact transforma-\ntion by which 2 munits of flux quanta are attached to\neach electron. Similar flux transmutations have been\nused to derive effective field theories for a variety of\nQH states.1,2,34,35In the Lagrangian formulation, in the\nimaginary time formalism, we have\nZ=/integraldisplay\nDf†DfDae−/integraltextβ\n0dτd2rL, (1)\nwithβ= 1/T,Tis the temperature, and the Lagrangian\ndensity is\nL=f†(∂τ−ia0−µ)f−1\n2mbf†(∂−iA−ia)2f\n+1\n2/integraldisplay\nd2r′V(r−r′)f†(r)f†(r′)f(r′)f(r)\n+i\n4π(2m)ǫµνλaµ∂νaλ. (2)\nfis the composite fermion field and Ais the background\nelectromagnetic field. The interaction may be chosen to\nbe of the form V(r)∼1/rη, whereη= 1 corresponds\nto the case of Coulomb interactions. If the fluctuations\nofaare treated exactly, this is an exact transformation\nof the original theory. The value of this rewriting of the\noriginal theory is that it allows for a mean-field approx-\nimation that was not available before. Since the filling\nfraction isν=Ne\nNΦ=1\n2m, there are 2 mflux quanta for\neach electron. Since we have also added 2 munits ofa\nflux to each electron, we can consider a mean-field statewhere∝an}b∇acketle{ta∝an}b∇acket∇i}ht=−A. With this mean-field approximation,\nthe composite fermions fon averagedo not feel anymag-\nnetic field.\nIt is found that the theory above describes a\ncompressible,75metallic state, which describes well the\nphenomenology that is experimentally observed in the\nhalf-filled Landau level.1\nThe single-particle properties of fexhibit various in-\nfrared singularities.3For example, the self-energy Σ f(ω)\noffhas a leading singularity that, for Coulomb interac-\ntions (η= 1), behaves as\nΣf(ω)∼ωln(iω). (3)\nFor shorter-range interactions, where 1 <η≤2, the self-\nenergy gives rise to stronger singularities:\nΣf(ω)∼(iω)2\nη+1 (4)\nHowever,since fisnotgauge-invariant,itssingle-particle\ncorrelation functions cannot be directly measured. In-\nstead, the physical observables are the gauge-invariant\nresponse functions, which are found to be non-singular\nand Fermi liquid-like36.\nThe electron operator, c(r), in this theory is described\nas\nc(r) =ˆM2m(r)f(r), (5)\nwhereˆM(r) is an instanton operator for the gauge field\nathat annihilates 2 πunits of flux. The electron is sim-\nply the composite fermion f, together with 2 munits of\nflux of the agauge field. Using this electron operator,\nthe equal-space electron Green’s function was computed\nin Ref. 37 using a semi-classical approximation for the\ninstanton action, with the result:\nG+(τ)≡ ∝an}b∇acketle{tc(0,τ)c†(0,0)∝an}b∇acket∇i}ht ≈G0(τ)e−SM¯M(τ),(6)\nwhereG0(τ) is an algebraically decaying function of τ\nandSM¯M∝τs, where the exponent sdepends on the\nform of the interactions between the composite fermions.\nIt was found that the spectral function, A+(ω), defined\nas the inverse Laplace transform of G+(τ), behaves like\nA+(ω)∼e−α/ωβ, (7)\nwhereαandβagain depend on the interactions between\nthe composite fermions. This indicates a strong expo-\nnential suppresion of the tunneling density of states at\nlow frequencies (see also Ref. 38 for a different derivation\nwith the same conclusion).\nIn the presence of long-range Coulomb interactions\n(η= 1), the specific heat was calculated to scale as3\nCv∼TlnT. (8)\nWith short-range interactions, the specific heat instead\nscales as\nCv∼T2/3. (9)4\nThe behavior of the specific heat and the exponentially\ndecayingspectralfunction areboth signaturesofstrongly\nnon-Fermi liquid behavior.\nA wave function for the CFL state that is expected to\ncapture its long-wavelength properties is of the form39:\nΨ({ri}) =PLLL/parenleftbig\n(zi−zj)2mDet[eikirj]/parenrightbig\n,(10)\nwherePLLLindicates the projection to the lowest Lan-\ndau level. The factor ( zi−zj)2mcan be thought of\nas attaching 2 munits of flux quanta to the composite\nfermions, which are filling a Fermi sea.\nThe above theory has been quite successful in explain-\ning many long-wavelength phenomena observed experi-\nmentallyat ν= 1/2. Nevertheless,theaboveformulation\nhas three shortcomings that we address in this paper76.\nFirst, it yields a limited framework for understanding\ncontinuous phase transitions out of the state. While this\nformulation is useful for studying transitions of the com-\nposite fermion Fermi surface, such as the pairing insta-\nbility that leads to the Moore-Read Pfaffian state, more\ngeneral continuous transitions, such as into a conven-\ntional Fermi liquid, cannot be understood through the\nabove construction.\nThe second shortcoming is that while this flux attach-\nment procedure and the associated flux-smearing mean-\nfield theory can be defined in models where the Chern\nbands are induced by an external magnetic field, it is in\ngeneralunclearhowto extendthis tolattice modelswith-\nout an external magnetic field (see Ref. 23 for a recent\ndiscussion). In fact, in many cases, the flux attachment\nprocedure for a half-filled Chern band appears to fail en-\ntirely.\nThirdly, the above formulation of the composite parti-\ncletheoriesona compactspaceis problematic40: because\nthe Chern-Simons level is not quantized, the partition\nsum on a surface of genus ≥1 is not invariant under\nlarge gauge transformations41,42.\nIn order to address these shortcomings, in the follow-\ning section we develop a theory of the CFL through a\nprojective construction.\nIII. PROJECTIVE CONSTRUCTION OF THE\nCFL STATE\nHere we will study a derivation of the CFL through\na totally different approach that does not begin from\nnotions of flux attachment and flux smearing. Instead,\nwe use a projective/parton construction, which provides\nseveral crucial advantages: most importantly, it incorpo-\nrates a broader set of quantum fluctuations, which yields\na path towards understanding transitions out of the CFL\nand into, for instance, Fermi liquids. Additionally, this\nformulation can be extended to lattice models without\nan external magnetic field, where the conventional flux\nattachment picture fails; it yields insight into the CFL\nstateanditstopologicalproperties; andfinallyitsuggestsgeneralizations of the CFL state to odd-denominator fill-\ning fractions or to states at the same filling fraction but\nwhich differ in their topological properties. Such pro-\njective constructions are more familiar in the study of\nquantum spin liquids43, although they have been devel-\noped both for non-Abelian FQH states44–47, and more\nconventional Abelian states20,24,25,48,49.\nFor concreteness, let us consider a system of spinless\ninteracting fermions on a square lattice in the presence\nof a background external magnetic field:\nH=/summationdisplay\nij[tijc†\nicjeiAij−(µ+A0\ni)δijni+Vijninj],(11)\nwith\nni=c†\nici. (12)\nLetφ= 2πp/qbe the amount of flux of Aper plaque-\ntte. In the absence of interactions, the above model is\nthen a tight-binding model with qbands; in the limit\nq→ ∞, the lowest bands become flat and equally spaced\nin energy, corresponding to Landau levels. We will sup-\npose that the averagedensity ofelectrons is such that the\nlowest Landau level has a filling ν= 1/2m.\nWebeginbyperforminganexactrewritingoftheabove\nmodel by decomposing ciin terms of different bosonic\nand fermionic variables:\nci=bifi, (13)\nwherebiannihilates a boson carrying the electric charge,\nandfiannihilates a neutral fermion. This introduces a\nU(1) gauge symmetry associated with the local transfor-\nmation\nbi→eiθibi, fi→e−iθifi, (14)\nwhich keeps the electron operator invariant. We see:\nni=nb\ninf\ni, (15)\nwherenb\ni=b†\nibiandnf\ni=f†\nifi. The states at each\nsite are now labelled as |nb\ni,nf\ni∝an}b∇acket∇i}ht. These states form an\nexpanded Hilbert space; the physical states in the ex-\npanded Hilbert space are the gauge-invariant ones: |0,0∝an}b∇acket∇i}ht\nand|1,1∝an}b∇acket∇i}ht, which correspond to the state with zero or one\nelectron(s), respectively. We see that the physical states\nsatisfy the constraint nb=nf, so\nni=nb\ni=nf\ni. (16)\nInserting this into the original Hamiltonian (11) gives:\nH=/summationdisplay\nij[tijb†\nibjeiAijf†\nifj−(µ+A0\ni)δijnb\ni+Vijnb\ninb\nj].\n(17)\nNow we can decouple the quartic terms in Husing a\nself-consistent mean-field approximation:\nb†\nibjeiAijf†\nifj≈1\n2[χijb†\nibjeiAij+ηijf†\nifj+χijηij],(18)5\nwhere\nχij=∝an}b∇acketle{tf†\nifj∝an}b∇acket∇i}ht, ηij=eiAij∝an}b∇acketle{tb†\nibj∝an}b∇acket∇i}ht.(19)\nIn the mean-field ansatz, we will assume that the ground\nstateissuchthat bformsaν= 1/2mLaughlinFQHstate\nwhilefforms a Fermi surface. This implies that ηijand\nχijare both real-valued and translationally invariant. It\nis simple to verify that such a mean-field state can be\nself-consistent, because if ηijis real-valued, then fforms\na Fermi surface with no average magnetic field, and its\nreal-space averageswill be real, which will imply that χij\nis real. Similarly, if χijis real, then bfeels a magnetic\nfield around a plaquette set by Aij, and therefore ηijcan\nbe real.\nThe fluctuations about the mean-field state can be in-\ncluded in the manner familiar from parton constructions\nof other states: the amplitude fluctuations of χandηare\ngapped, so we will only include the phase fluctuations.\nThis leads to the following Hamiltonian:\nH=/summationdisplay\nij[tij(χijb†\nibjeiAij+iaij+ηije−iaijf†\nifj)\n−(µ+A0\ni)nb\ni+a0\ni(nb\ni−nf\ni)+Vijnb\ninb\nj].(20)\na0\niis introduced as a Lagrange multiplier to enforce the\nconstraint that nb\ni=nf\ni. While we derived this effective\ntheory from a mean-field ansatzand included the allowed\nlong-wavelengthfluctuations, itcanalsobederivedsome-\nwhat differently, following the analysis of Ref. 30 for the\nU(1) spin liquid, as a saddle-point of the path integral\nby introducing additional fields to exactly decouple the\nquartic terms. Such mean-field approximations can yield\nstable deconfined fixed points of the resulting gauge the-\nory. Whether they correspond to a global minimum in\nthe energy is a more detailed question of energetics that\nmust be answered through numerical studies or by com-\nparing the phenomenology of these theories with experi-\nmental observations.\nIn aν= 1/2mLaughlin state, the insertion of 2 m\nunits of flux quanta at sufficiently long wavelengths will\ncreate a single unit of charge. Therefore, at energies well\nbelow the gap of the bosonic Laughlin state, the boson\ncan be represented by the instanton operator:\nb=ˆM2m, (21)\nwhereˆMis the instanton operator which annihilates 2 π\nflux ofa. Therefore, at energies well below the gap of the\nbosonic state, the electron operator can be represented\nas\nc=ˆM2mf, (22)\njustasintheconventionalCFLtheory. Inparticular,this\nmeans that at low energies, the boson number density is\nnb=1\n2π1\n2mǫij∂iaj, (23)which means that the boson interaction term in the\nHamiltonian (20) can be rewritten in terms of a.\nWe can describe the ν= 1/2mbosonic Laughlin state\nby using a U(1)2mCS theory associated with a second\nemergent gauge field43. Then, the effective theory be-\ncomes, to lowest order in a continuum approximation,\nL=Lb+Lf+Lint,\nLb=2m\n4πǫµνλ˜aµ∂ν˜aλ+1\n2πǫµνλ(aµ+Aµ)∂ν˜aλ,\nLf=f†(i∂t+a0)f+1\n2m∗f†(∂+ia)2f,\nLint=/integraldisplayd2r′\n(4πm)2V(r−r′)(ǫij∂iaj(r))(ǫab∂aab(r′)).\n(24)\nThe boson current jµ\nbis given in terms of ˜ a:\njµ\nb=1\n2πǫµνλ∂ν˜aλ. (25)\nNote this is consistent with (23), because integrating\nout ˜a0in (24) in the absence of Ayields the constraint\nǫij∂iaj= 2mǫij∂i˜aj.\nRelabelinga→ −(a+A) and subsequently integrating\nout ˜ayields the following Lagrangian density:77\nL=f†(i∂t−a0−A0)f+1\n2mf†(∂−iA−ia)2f\n+1\n21\n4πmǫµνλaµ∂νaλ+···\n+/integraldisplayd2r′\n(4πm)2V(r−r′)(ǫij∂iaj(r))(ǫab∂aab(r′)),\n(26)\nwhich is precisely the conventional CFL Lagrangian (2).\nIt immediately follows that this construction yields a\nstate with all of the same thermodynamic and transport\nproperties ofthe conventionalCFL state. A point ofsub-\ntlety is that this construction yields a compressible state,\ndespite the fact that bforms an incompressible Laughlin\nFQH state. Formally, this is possible because the polar-\nizability of the electrons, Π e, is given by the Ioffe-Larkin\nsum rule50\nΠ−1\ne= Π−1\nb+Π−1\nf, (27)\nwhere Π band Π fare the polarizability tensors of the b\nandfparticles, respectively. If the polarization tensors\nwere diagonal, the inverse compressibilities and inverse\nconductivities of the slave particles would add to give\nthe inverse compressibility or inverse resistivity of the\nelectron system. However, in this case, the polarization\ntensors have off-diagonal components, Π 0i∝ne}ationslash= 0; thus, the\ncompressibility of the electron system can be finite, de-\nspite the incompressibility of the boson sector. Less for-\nmally, the coupling to the gapless gauge flux of aallows\nthe Laughlin state of the bosons to respond continuously\nto the introduction of extra particles.6\nThe single-electron Green’s function is\nGe(r−r′,t−t′) =∝an}b∇acketle{tTc(r,t)c†(r′,t′)∝an}b∇acket∇i}ht\n=∝an}b∇acketle{tTb(r,t)b†(r′,t′)f(r,t)f†(r′,t′)∝an}b∇acket∇i}ht.\n(28)\nIn the slave-particle mean-field approximation, where we\nignore gauge fluctuations,\nGmf\ne(r,t) =Gb(r,t)Gf(r,t). (29)\nSincebforms a gapped FQH state, its bulk correlations\nwill clearly decay exponentially in space and time, and\ntherefore so will Gmf\ne(r,t). The result of Ref. 37 is recov-\nered by considering gauge fluctuations. At low energies,\nwe can represent bby the instanton operator, so that the\nelectron operatoris given by (22). Now the calculationof\nthe electron Green’s function reduces to that of Ref. 37.\nThis mean-field theory of the CFL state can actu-\nally be derived at any filling fraction νby allowing the\nbosons to form more generic incompressible Abelian or\nnon-Abelian FQH states and is therefore not limited to\neven-denominator fillings. Even staying at ν= 1/2, it\nis possible to consider distinct incompressible states at\nν= 1/2 for the bosons instead of the 1 /2 Laughlin state.\nA possible example is the ν= 1/2 bosonic orbifold state\nstudied in Ref. 47. These states would have different\ntopological orders in the boson sector, but would yield\nthe same low energy dynamics described by (2) if the\nbosons are integrated out. The different possible topo-\nlogicalordersfor the bosonic sectorsuggest that the CFL\nshould be viewed as a topological non-Fermi liquid . An\nopen conceptual problem is to develop tools to charac-\nterize the topologicalorderin such topologicalnon-Fermi\nliquid metals.\nA. CFL in the absence of an external magnetic\nfield: partially-filled Chern bands and the\n‘factorization’ of bandstructure\nRecently, therehasbeenwideinterestinstudyingFQH\nstates in lattice models without an external magnetic\nfield, but with partially filled flat bands with non-trivial\ntopology. Since a flat band with Chern number 1 is topo-\nlogically equivalent to a Landau level, it is expected that\nwhen such a band is partially filled, it is possible to re-\ncover conventional FQH states. So far, the focus has\nmostly been on the incompressible FQH states, however\none may expect that the CFL can also appear in such\nsystems. This raises the question of how to construct the\ngroundstate wavefunction and low-energy effective de-\nscription for such a state. It is simple to see that the con-\nventional flux attachment and flux-smearing mean-field\napproximation fails for such a situation.78For example,\nconsider a square lattice with a half-filled flat band with\nChernnumber 1. Since the band is athalf-filling, this im-\nplies that there is one electron for every two sites. In the\nflux attachment picture, we must attach a multiple of 4 πflux to obtain composite fermions. However, attaching\n4πflux to each electron is equivalent to adding 2 πflux\nper plaquette. In the flux-smearing mean-field approxi-\nmation, 2πflux per plaquette is equivalent to zero flux\nper plaquette, so the statistical transmutation appearsto\nbe ineffective.\nIn contrast, the projective construction of the previ-\nous section can straightforwardly be generalized to the\ncase of a partially filled Chern band without an external\nmagnetic field. In Refs. 24,25, a prescription was given\nfor a mean field theory with the parton decomposition\nc=f1f2...fn; the idea is that the hopping elements for\nthefs should be nth roots of those of c. This was justi-\nfied by a strong coupling expansion of the parton lattice\ngauge theory. Here we wish to extend this to the parton\ndecomposition c=fb; the hopping elements for fandb\nmust then be factorsof those ofc, but need not be equal.\nConsider a tight-binding model defined by the follow-\ning hamiltonian\nH=−/summationdisplay\nrr′∈Etc\nrr′c†\nrc′\nr+h.c.+interactions .(30)\nHerec†,cdenote creation and annihilation operators for\nelectrons; Edenotes links in the lattice. Our mean field\nansatz for the “factorization” of this lattice is\nHmf=−/summationdisplay\nrr′∈E/parenleftBig\ntf\nrr′f†\nrfr′+tb\nrr′b†\nrbr′+h.c./parenrightBig\n(31)\nwhere we demand that for each link\ntc\nrr′=tf\nrr′tb\nrr′h−1\nrr′. (32)\nThe scalehrr′is a parameter of the parton lattice gauge\ntheory representing the energy cost for allowing gauge\nfluxalongthelink rr′; theargumentfromstrongcoupling\nexpansion proceeds as in Ref. 24, to which we refer the\nreader for the details. The phases in tcwhich make the\nresulting bands topological must be distributed between\ntfandtb– this choice is a lattice analog of the choice\nin the continuum parton construction of where to assign\nthe electric charge amongst the partons. The analog of\nlettingthebosonscarrythechargeistoputallthephases\nintb.\nFordefiniteness, consideranelectronmodel atquarter-\nfilling on the checkerboard lattice. Some relatively-flat\nChern bands are realized by the following tight-binding\nmodel with next-nearest neighbor interactions51. The\nHamiltonian is H=/summationtext\nk∈BZ(h2N+h3N) with\nh2N=−teiϕ/bracketleftBig\nβ†\nkαk/parenleftBig\nei\n2(kx+ky)+ei\n2(−kx+ky)/parenrightBig\n(33)\nα†\nkβk/parenleftBig\nei\n2(kx−ky)+ei\n2(−kx+ky)/parenrightBig/bracketrightBig\n+h.c.\nh3N=−α†\nkαk/parenleftbig\nt′\n1eikx+t′\n2eiky/parenrightbig\n−β†\nkβk/parenleftbig\nt′\n2eikx+t′\n1eiky/parenrightbig\nwhereαandβare fourier modes of the electron creation\noperators on the two sublattices:\nαk=1√Na/summationdisplay\nxaeikxacxa, βk=1√Nb/summationdisplay\nxbeikxbcxb,(34)7\nThe bands may be further flattened by the addition of\nthird-neighborinteractions52. At quarter-filling,theelec-\ntrons fill the lowest band halfway; this band has Chern\nnumber unity.\nTo construct a lattice analogof the HLR state, we take\nthe ansatz:\ntb=teiφ,(t′\n2)b=−(t′\n1)b, tf=t,(t′\n2)f= +(t′\n1)f.\n(35)\nThis puts the fermion in a state which half-fills a\ntopologically-trivial band, and puts the boson in a state\nwhich half-fills the lowest topologically non-trivial band\nwith Chern number one. Note that unlike the case of\nRef. 24, no enlargement of the unit cell was required so\nfar; if we wanted to explicitly describe the lattice analog\nof theν= 1/2 Laughlin wavefunction of this boson, we\nwould have to use a second layer of parton construction\nwhich in which the unit cell is doubled.\nIV. STATES PROXIMATE TO CFL: GAPLESS\nMOTT INSULATOR AND LANDAU FERMI\nLIQUID\nThe main utility of the projective construction of the\nCFL state in the previous section is that now, within the\nsame theory, we can find saddle points associated with\nother many-body states.\nFor example, consider the ν= 1/2 Landau level prob-\nlem and suppose that we add a periodic potential, Vp,\nwith a multiple of 2 πflux per plaquette. Vpcouples to\nthe boson density (see (20)), and therefore will increase\nthe bandwidth of the Landau level for b. WhenVpis\non the order of the boson gap, the bosons can undergo\na transition into either a Mott insulating state for bor\na superfluid.14Alternatively, in the case of the partially\nfilled Chern band without an external magnetic field, we\ncan consider increasing the pressure to increase the hop-\nping matrix elements. This can also cause transitions\nout of the bosonic Laughlin state and into the MI or the\nsuperfluid.\nIt is also possible to consider fractionalized Mott insu-\nlators forb, such asZ2topologically ordered states. In\nthis paper, we will ignore these more complicated scenar-\nios, as they are expected to be less likely. Instead we will\nfocus on the simplest possible states for the bosons and\nthe correspondingly simplest ways of transitioning out of\nthe CFL and into the LFL.\nA. Gapless Mott Insulator\nWhen the strength of the periodic potential is on the\norderof the interaction strength, we expect the bosons to\ntransition out of the ν= 1/2 Laughlin state. A simple,\ngeneric possibility is that the bosons continuously transi-\ntion into a trivial Mott insulator without any topological\norder. For this to occur, the bosons must be at integerfilling per unit cell; this can happen by either explicit or\nspontaneous translation symmetry breaking.\nFrom the constraint nb=nf, integer filling of the\nbosons implies integer filling of the fermions. This means\nthat the fermions can generally form either a band insu-\nlator or Fermi surfaces with equal area of particles and\nholes. In the mean-field Hamiltonian, the fermion hop-\nping is given by\ntf\nij∝ ∝an}b∇acketle{tb†\nibj∝an}b∇acket∇i}ht. (36)\nDeep in the Mott insulating phase, it is possible that\n∝an}b∇acketle{tnb\ni∝an}b∇acket∇i}ht ≈1 or 0, depending on the lattice site. Similarly,\nit is possible that ∝an}b∇acketle{tb†\nibj∝an}b∇acket∇i}ht ≈0 fori∝ne}ationslash=j. In this limit, the\nfermions cannot form a Fermi surface; they are localized\nto the sites where the bosons are located, and do not\ndisperse.\nCloser to the ν= 1/2 Laughlin and superfluid state,\n∝an}b∇acketle{tnb\ni∝an}b∇acket∇i}htis approximately uniform, and the bosons have a\nlarger correlation length, so tf\nijcan be appreciable. In\nsuch a situation, the mean-field state of the fermions\ncan be such that they form a Fermi surface. Since the\nfermions are at integer filling, the area of hole and parti-\ncle pockets must be equal. This state is closely related to\nthe gapless U(1) spin liquid Mott insulators with spinon\nFermi surface and an emergent U(1) gaugefield.29,30The\ndifference is that the gapless Mott insulator considered\nhere consists of spinless electrons. While there is an in-\nteger number of electrons per unit cell, each unit cell\nconsists of multiple sites; therefore, there is a pseudospin\nindex that plays the role of spin and which can allow the\nsystem to have gapless degrees of freedom, despite the\ncharge being localized and gapped.\nStarting from the CFL, where the fermions form a\nFermi surface, the introduction of a periodic potential\nwill fold the Brillouin zone; if the Fermi surface has\nno nesting, as in the conventional scenarios, there will\ncontinue to be stable particle/hole pockets for the f\nfermions. As the bosons undergo a continuous transi-\ntion to the MI state, the fsector changes gradually, as\nthere is generically no change of spatial symmetry at the\nboson FQH -MI transition. Within the MI state, as the\nbosons are taken deeper into the MI and become fully\nlocalized, the Fermi surface can gradually shrink to zero\nand eventually the ffermions will form a band insulator.\nPhysically, we can understand the appearance of the\nGMI state by analogy with the CFL state. In the CFL\nstate, the electron fractionalizes into a boson b, which\ncarries the electric charge, and the neutral ffermion.\nThe bosonbthen forms an incompressible Laughlin FQH\nstate in order to minimize the strong interaction energy\nand the phase frustration induced by the magnetic field,\nwhilefis free to form a Fermi surface. It is therefore\nreasonable to imagine that instead of bforming a Laugh-\nlin FQH state, it will form a Mott insulator in order to\nminimize the strong repulsive interaction energies. The\nmany-body wave function that describes such a state is8\nthe usual projected wave function43:\nΨe({ri}) =∝an}b∇acketle{t0|/productdisplay\nic(ri)|Φmf∝an}b∇acket∇i}ht\n= Ψb({ri})Ψf({ri}), (37)\nwhere|Φmf∝an}b∇acket∇i}htis the mean-field ground state of bandf,\nwherefis forming a Fermi surface and bis forming a\nMott insulator. This is equivalent to taking the boson\nMI wave function and fermion FS wave function and pro-\njecting the bosons and fermions to the same location.\nThe effective theory for this state takes the form\nL=Lb+Lf+La, (38)\nwhereLbdescribes the gapped bosonic excitations, Lf\ndescribes the fermions in a Fermi sea and coupled to a,\nandLa=1\ng2(∇ ×a)2+···is the action for the U(1)\ngauge field a. When the bosons are in the gapped, trivial\nMI state, they can be integrated out to give, to lowest\norder,\nL=Lf+1\ng2(∇×a)2. (39)\nThegaplessMott insulatoriselectricallyinsulating, ther-\nmally conducting, and incompressible. As a result, its\nexistence cannot be established through DC electrical\ntransportmeasurements. Instead, itcanbe distinguished\nfromthetrivialMottinsulatorbyprobingthermalbehav-\nior, such as specific heat or thermal conductivity. The\nspecific heat and thermal conductivity of such a state\nscale asCv∼T2/3, andK/T∼T2/3, respectively.53–57\nThe possibility of thermal conduction in this spinless\nMott insulator comes from the fact that the thermal con-\nductivity of the electrons is equal to the sum of the ther-\nmal conductivity of the bandfsystems;53this is in con-\ntrast to the electrical transport, where typically it is the\nresistivities that add.\nTheFermisurfacecanbemoredirectlyprobedthrough\nFriedel oscillations, as proposed recently in Ref. 58 for\ntheU(1) spin liquid Mott insulator; in both cases, the\ndensity-density correlation function has algebraic corre-\nlations displaying signatures of a Fermi surface.\nB. Landau Fermi liquid\nWhen the interaction energy is small compared to the\nbandwidth, then it will be preferable for bto condense\nintothebottomofitsbandandformasuperfluid. Insuch\na case, the emergent U(1) gauge symmetry associated\nwithais spontaneously broken, and the resulting state of\nthe electrons is described by Landau Fermi liquid theory.\nIntheLandauFermiliquid, thequasiparticleresidue Z∼\n|∝an}b∇acketle{tb∝an}b∇acket∇i}ht|2.\nSince time-reversal symmetry is broken, with interac-\ntions the boson superfluid state will in general consist of\na normal component with non-vanishing orbitalcurrents.Mott \nInsulator \nSuperfluid1/2 Laughlin \nFQH3D XY\nm+ (m  ,m  )=(0,0)+-|m | -Boson Phase Diagram\nFIG. 2: Proposed phase diagram and renormalization-group\nflows including the Mott insulator, superfluid, and ν= 1/2\nLaughlin FQH state, for fixed average particle number.14We\nhave defined m±≡m1±m2(see eq. 40 with Nf= 2);m−\ncan be viewed as a symmetry-breaking field, so the direct\ntransition between the FQH state and the SF can occur if the\nsymmetry is preserved. The red points on the horizontal and\nvertical axes indicate the three stable phases, while the bl ue\npoints at the origin and the diagonals indicate the unstable\ncritical fixed points.\nTheelectronstatewillthereforebeaLandauFermiliquid\nwith orbital loop currents and a non-zero Hall conduc-\ntance, as time-reversal symmetry is explicitly broken.\nIf the electrons are at integer filling per unit cell of\nthe periodic potential, then the resulting state is either a\nband insulator or a Landau Fermi liquid with equal area\nfor electron and hole pockets. Within our construction,\nif we start from a CFL with non-nested Fermi surface\nand add a periodic potential, the metallic case will be\nthe generic situation.\nIf the electrons are at fractional filling, then the result-\ning state is a Landau Fermi liquid with no constraints on\nelectron and hole pockets.\nV. CONTINUOUS TRANSITIONS\nFrom this framework, we can now understand transi-\ntions out of the CFL state and ultimately into the Fermi\nliquid by considering transitions in the boson sector be-\ntween theν= 1/2 Laughlin FQH state, the Mott insu-\nlator, and the superfluid. In Ref. 14, we found that in\nthe absence of any special symmetries and for fixed par-\nticle number, there are continuous transitions between\nthe bosonic Laughlin FQH state and the Mott insulator,\nandbetweenthebosonMottinsulatorandthesuperfluid.\nHowever, certain spatial symmetries, such as inversion,\ncan stabilize a direct continuous transition between the\nFQH state and the superfluid. These critical points are\ndescribed by massless Dirac fermions coupled to a U(1)9\nCS gauge field:\nLNf,k=Nfk\n4πǫµνλaµ∂νaλ+Nf/summationdisplay\ni=1[¯ψiγµDµψi+mi¯ψiψi].\n(40)\nIt was found that the MI-SF transition is described as\nm1→0+inL1,1/2, the FQH-MI transition is described\nbym1→0+inL1,3/2, and the FQH-SF transition is de-\nscribed bym1=m2≡m→0+inL2,1/2(see Fig. 2)8,14.\nNote that for the above theory to have a well-defined\nlattice regularization, Nfkmust be an integer when Nf\nis even, and half-integer when Nfis odd. The critical\ntheory atm= 0 may be modified in the presence of\nlong-range interactions, depending on the value of k10.\nIt was found that in the large Nflimit, fork > k c1,\nCoulomb interactions are marginally irrelevant, while for\nkc2< k < k c1, Coulomb interactions are relevant and\ncause a flow to a different stable fixed point with dy-\nnamic critical exponent z= 1 and correlation length ex-\nponentν >1. Fork < k c2, Coulomb interactions are\nrelevant and flow to strong coupling. It was found that\nkc1≈0.35 andkc2≈0.28. Thus for the cases of interest\nhere, it is possible that Coulomb interactions will either\nbe marginally irrelevant or flow to the controlled fixed\npoint with z= 1 andν >1.\nWe note that since time-reversal symmetry is strongly\nbroken in the situation under consideration, a possibility\nis that at the FQH - SF critical point, the initial tran-\nsition out of the FQH state is into a vortex state of the\nsuperfluid. If the vortices form a vortex lattice, then\nboth translation and U(1) charge symmetry are broken\nat the transitions. We expect that such a scenario would\nbe multicritical and would not be described by (40); any\ntranslation symmetry breaking should generically occur\naway from the FQH to SF critical point.\nGiven the above critical points between the different\nbosonic states, it is possible that the electron system will\nalso undergo continuous transitions between the CFL,\ngaplessMI,andFLstatesasthebosonicsectorofthethe-\nory undergoes transitions between the ν= 1/2 Laughlin\nstate, the Mott insulator, and the superfluid.\nIn order to analyze whether the resulting transitions\nof the electron system are continuous, we must analyze\nthe coupling of the bosons to the fermions and the gauge\nfield. The effective theory takes the form\nL=Lb+Lf+La+Lbf, (41)\nwhereLbis the action for the boson sector, Lfis the\naction for the ffermions, which fill a Fermi surface, and\nLa=1\ng2(∇×a)2, (42)\nandLbfcontains direct boson-fermion couplings; gis\na phenomenological parameter in the effective theory.\nFirst, we will consider the transitions at the mean-field\nlevel, where we ignore fluctuations of the emergent U(1)gauge field a. In the absence of Lbf, then, the boson\ncritical point will be described by LNf,k, for a suitable\nchoice ofNfandk. Let us consider the possible effects\nofLbf. An operator Ofrom the boson sector can couple\nto the particle-hole continuum of the fFermi surface at\nlow momenta. We can use the arguments of Ref. 59 to\nsee whether such a coupling is relevant at the boson crit-\nical point. Integrating out the ffermions gives rise to a\nperturbation\nv/integraldisplay|ω|\nq|O(q,ω)|2, (43)\nforω≪q. The most relevant operator is expected to be\nO=b†b, whichhasscalingdimension3 −1/νattheboson\ncritical point. For this operator, vis therefore irrelevant\nat the boson critical point if ν >2/3, and relevant if\nν <2/3.\nFrom the large Nfexpansion of LNf,k,8,10\nν−1= 1+512φ(1−2φ)\n3π2(1+φ)31\nNf+O(1/N2\nf),(44)\nwhereφ= (θ/16)2andθ= 2π/k. By comparing\nthe value of ν1,1/2with the known 3D XY value, we\nconclude that the large Nfexpansion is unreliable for\nNf= 1. ForNf= 2, we expect the large Nfexpan-\nsion to be more accurate, and we find that to O(1/N2\nf),\nν2,1/2= 1.4182>2/3. Thus in the absence of gauge fluc-\ntuations and for short-ranged interactions, we see that\nthe direct symmetry-protected CFL to FL transition is\ncontinuous according to the leading order 1 /Nfapproxi-\nmation. Since we don’t have accurate estimates of ν1,3/2,\nwe cannot conclude that the CFL to GMI transition is\nalsocontinuous. The GMI to FLtransition iscontinuous:\nthere, the boson transition is in the 3D XY universality\nclass, for which ν3DXY>2/3. Since both the GMI-FL\nandthe CFL -FL arecontinuous, in what follows, wewill\nconsider the possibility that the CFL to GMI transition\nis also continuous and study the phenomenology of such\na transition, along with that of the CFL - FL transition.\nIn the presence of the long-ranged Coulomb interac-\ntions, there are several possibilities. If the bosonic sec-\ntor flows to the new fixed points found in Ref. 10, then\nν >1, and sovwill be an irrelevant perturbation in all\ncases. Alternatively, if the long-ranged Coulomb inter-\nactions are marginally irrelevant, then scaling functions\nwill receive logarithmic corrections.\nA. Effect of gauge fluctuations\nIn Ref. 28, the effect of gauge fluctuations has been\nstudied in the case of the gapless MI to FL transition,\nwhere the bosons are undergoing a 3D XY transition and\ntime-reversal symmetry is preserved. The phenomenol-\nogy here is similar; the main non-trivial differences arise\nfrom the non-zero Hall conductivity of the boson sector\nat the critical points. We note that the analysis of the10\nGapless Mott \nInsulator \nLandau\nFermi Liquid\nm+ (m  ,m  )=(0,0)+-|m | -Electron Phase Diagram\nComposite \nFermi liquid\nFIG. 3: The red points on the horizontal and vertical axes\nindicate the three stable phases, while the blue points at th e\norigin and the diagonals indicate the unstable critical fixe d\npoints.\ngauge fluctuations relies on results obtained from RPA\ncalculations, whose validity in large Nexpansions is dis-\ncussed in Refs. 55–57,60.\nFirst, consider integrating out all matter fields. The\nresulting action for the gauge field, to quadratic order in\na, is\nSeff[a] =1\n2/integraldisplay\naµ(Πf\nµν+Πb\nµν)aν.(45)\nSince the boson critical point has dynamical scaling ex-\nponentz= 1 and rotation invariance, the general form\nof the boson polarization tensor Πb\nµνatT= 0 is\nΠb\nµν= (δµν−kµkν\nk2)Πb\n+(k)+ǫµνλkλΠb\n−(k),(46)\nwhere Πb\n+and Πb\n−are functions of k≡/radicalbig\nq2+ω2, andkµ\nis a three-component vector ( ω,/vector q), where/vector qis the wave\nvector. At the boson critical point, Πb\n−(k)∼ O(k0) and\nΠ+(k)∼k.8\nIn Coulomb gauge, ∂·a= 0, the gauge field consists\nof (a0,a⊥), wherea⊥is the component transverse to the\nwave vector. The polarization tensors can then be writ-\nten as 2×2 matrices for the time and transverse compo-\nnents:\nΠb=/parenleftbiggq2\nkf1(ξk)−qg(ξk)\nqg(ξk)kf2(ξk)/parenrightbigg\n, (47)\nwhereξis the correlation length of the boson critical\npoint. On the FQH and Mott insulating side of the crit-\nical points,\nfi(x→ ∞)→f0,\nfi(x→0)→x,\ng(x→ ∞)→σc\nxy, (48)\nwheref0is a constant, and σc\nxyis a constant setting the\nHall conductivity at the critical point. In the FQH state,g(x→0)→1\n2; in the Mott insulating state, g(x→0)→\n0. On the superfluid side of the critical points,\nfi(x→ ∞)→f0,\nfi(x→0)→1/x,\ng(x→ ∞)→σc\nxy,\ng(x→0)→σs\nxy, (49)\nwhere we allow for the possibility of a non-zero Hall con-\nductivityσs\nxyin the superfluid phase. The fermion po-\nlarization Π fis, forq≪kFandω≪vFq,\nΠf=/parenleftbiggκfΠf;xy(q,ω)\nΠf;yx(q,ω)k0|ω|\nq+χdq2/parenrightbigg\n,(50)\nwhereκf,χd, andk0are constants. We note that in\ngeneral, since time-reversal symmetry is broken, the f\nfermions may have a Hall conductance σf\nxy, so Π fcan\nalso have off-diagonal components: Π f;xy(q,ω= 0) =\nΠ∗\nf;yx(−q,ω= 0) =−qσf\nxy.\nThe arguments of Ref. 28 imply that the gauge fluctu-\nations also do not modify the boson critical point. This\ncan be understood by observing that the gauge fluctu-\nations, using the RPA propagator from (50), only lead\nto analytic corrections to the boson self-energy at low ω,\nq, and therefore do not modify the critical singularities\ncoming from the boson sector. Alternatively, the ω/q\nterm acts like a Higgs mass for the transverse gauge fluc-\ntuations in the boson sector, since ωandqscale the same\nway at the boson critical point.\nUsing(27), (47), and(50), wecanobtainthe compress-\nibilityκeat zerotemperaturecloseto the transition. The\ninverse electron polarizability satisfies\nΠ−1\ne(q,ω= 0,T= 0) =\nχdq2\n|Πf|+qf2\n|Πb|qσf\nxy\n|Πf|+qg\n|Πb|\n−qσf\nxy\n|Πf|−qg\n|Πb|κf\n|Πf|+qf1\n|Πb|\n,\n(51)\nwhere|Πf| ∝q2and|Πb|=q2(f1f2+g2) are the de-\nterminants of Π fand Π b. We are interested in the case\nwhereqandqξare small, while ξdiverges. On the CFL\nside, in this limit |Πb| ∝q2,fi∼qξ, andg∼const.. On\nthe LFL side, fi∼1/qξ,g∼const., and|Πb| ∼1/ξ2.\nTherefore for small qandqξ, on the CFL and LFL side,\nthe dominant terms are\nΠ−1\ne(q,ω= 0,T= 0)≈/parenleftBiggqf2\n|Πb|α\nq\n−α\nqκf\n|Πf|/parenrightBigg\n,(52)\nfor some constant α, and so\nΠe;00(q,ω= 0,T= 0)∼/parenleftbiggqf2\n|Πb|+α2|Πf|\nq2κf/parenrightbigg−1\n.(53)\nAsξ→ ∞, the first term dominates and we get κe=\nlimq→0|Πb/qf2| ∼1/ξ.11\nTherefore, as the critical point is approached from ei-\nther the CFL side or the LFL side, we find that the com-\npressibility κeat zero temperature close to the transition\nis dominated by the correlation length of the boson sec-\ntor:\nκe∼1/ξ. (54)\nOn the LFL side, 1 /ξ∼ρs, while on the CFL side, 1 /ξ∼\n∆b, where ∆ bis the gap of the boson sector and ρsis the\nsuperfluid density.\nB. Transport and thermodynamics at the critical\npoints\nFrom the Ioffe-Larkin composition rule (27), the in-\nverse polarizability of the electrons is given by the sum\nofthe inversepolarizabilitiesof fandb. The bosonic sec-\ntor has a Hall conductance, Π b,0i∝ne}ationslash= 0 and therefore the\ninversecompressibilitiesandresistivitiesof bandfdonot\nseparately add to yield the electron inverse compressibil-\nity and resistivity. This implies that while the boson re-\nsistivity exhibits a universal jump,61there is also a jump\nin the resistivityofthe electronsystem; in contrastto the\ntime-reversal invariant case, the resistivity jump appears\nto be non-universal because of the off-diagonal terms in\nthe polarization tensors.\nAt finite temperature and zero frequency, by scaling\nwe have\nΠb;µν(q,ω= 0,T) =Tfµν(q/T), (55)\nwherefµνis a scaling function satisfying\n0<lim\nq→0|fµν(q/T)|<∞. (56)\n(56) can be understood by observing that the polariza-\ntion function Π µν(q,ω,T= 0) vanishes as q,ω→0; at\nfinite temperature, the current-current correlation func-\ntions should be more short-ranged in real space, and\ntherefore should continue to vanish as q,ω→0. Further-\nmore, generically |fµν(0)|>0 as there is no symmetry\nforcing it to be zero.\nThe fermion polarization at ω= 0 and small qis tem-\nperature independent and given by (50).\nUsing (27), (56), (55) (50), the inverse electron polar-\nizability is:\nΠ−1\ne(q,ω= 0,T) =1\nTf−1+1\n|Πf|/parenleftbiggχdq2qσf\nxy\n−qσf\nxyκf/parenrightbigg\n,\n(57)\nwhere the fermion polarization determinant |Πf| ∝q2.\nIn the limit q/T→0 andT→0, the dominant terms\nare:\nΠ−1\ne∼\n1\nT(f−1)00qσf\nxy\n|Πf|\n−qσf\nxy\n|Πf|κf\n|Πf|\n. (58)Therefore:\nΠe;00∼/parenleftBigg\n1\nT(f−1)00+q2(σf\nxy)2\nκf|Πf|/parenrightBigg−1\n.(59)\nThe first term dominates at low temperatures, from\nwhich we find that the compressibility is κe(T)≡\nlimq→0limω→0Πe;00(q,ω,T),\nκe∼T+O(T2). (60)\nA remarkable prediction of the above scaling of the com-\npressibility is that the critical point between the compos-\nite Fermi liquid and Landau Fermi liquid is incompress-\nible at zero temperature, even though both the CFL and\nFL are compressible states.\nAt the critical point, the gauge propagator in RPA is\nequivalent to that considered in the Halperin-Lee-Read\ntheory of CFL with long-range interactions. Since the\nspecific heat of the boson sector is Cv∼T2, we therefore\nexpect the specific heat to be dominated by the contri-\nbution of the fermion-gauge system:\nCv∼Tln(1\nT). (61)\nTheexpectationsfortransportareverysimilartothose\nof Ref. 28,62.\nC. Crossover out of criticality\nAn important signature of this class of transitions is\nthat while the boson critical point has a quantum crit-\nical region associated with some crossover temperature\nscaleT∗∼1/ξ, the gauge propagator is not significantly\nmodified until a lower temperature scale, T∗∗∼1\nξ2ck0,\nwherecis the characteristic velocity of excitations at the\nboson critical point (previously we had set c= 1), and\nk0is defined in (50).28Recallξ∼1/∆b, where ∆ bis\nthe boson gap when the bosons are in the gapped FQH\nor MI states, and ξ∼1/ρswhen the bosons are in the\nsuperfluid state. This implies the existence of two fi-\nnite temperature crossoverregimes, each with distinctive\nproperties.\nIn the crossover regime T∗∗< T < T∗, the gauge\npropagator still takes the form that it did in the quan-\ntum critical regime of the boson critical point. On the\ncomposite Fermi liquid side, this means that the the-\nory formally is similar to the composite Fermi liquid\ntheory with long-range Coulomb interactions and we ex-\npect the physics of this theory to dominate this crossover\nregime. In such a situation, the composite fermions form\na“marginal”Fermiliquid. Iftheproblemdoeshavelong-\nrange Coulomb interactions to begin with, there will be\nnosignificantmodification ofthe physicson the CFL side\nas one crosses T∗∗. On the other hand, if the Coulomb\ninteraction is screened by a gate, then T∗∗will mark12\na crossover between shorter-ranged and effectively long-\nranged interactions. The fact that this crossover scale\nT∗∗can effectively be made to disappear and reappear\nwith a screening gate is a non-trivial prediction of this\ntheory.\nThe compressibility below T∗is dominated by the\nbosons and will be a temperature-independent constant\nκe∼1/ξ.\nThe crossover behaviors on the LFL and GMI sides of\nthese transitions are spinless analogs of the cases studied\nin Ref. 28; we will briefly review a few of the key fea-\ntures here; the results are summarized in Figs. 4 and 5\nrespectively.\nOn the GMI side, in the second quantum critical\nregime,T∗∗<T <T∗, the specific heat is still behaving\nas\nCv∼TlnT. (62)\nThe system can be viewed as a marginal Fermi liquid\nof the neutral fexcitations, although the single-particle\nproperties of fare not gauge invariant and cannot be di-\nrectlymeasured. Thethermalconductivityinthisregime\nbehaves as K/T∼1/T.\nAt the lowest temperatures away from the critical\npoint,T < T∗∗, for short-range interactions the system\nhas a specific heat\nCv∼T2/3, (63)\nand the electron single-particle Green’s function decays\nexponentially in frequency, indicating an exponential\nsuppression of the electronic density of states. In this\nregime, the thermal conductivity behaves as K/T∼\n1/T2/3.\nOn the Landau FL side, at T= 0 and1\nξ2ck0< ω <\n1/ξthe electron Green’s function has the marginal Fermi\nliquid form,63with a self-energy\nΣ(ω)∼ωlniω. (64)\nIn the crossover temperature regime T∗∗<T <T∗, the\nspecific heat behaves as\nCv∼TlnT, (65)\nwhile the thermal conductivity is\nK/T∼T. (66)\nThe compressibilitybelow T∗isdominated by the bosons\nand is a temperature-independent constant, κe∼1/ξ.\nAs the transitionisapproachedfrom the LandauFLside,\nthequasiparticleeffectivemassesandLandauparameters\nare expected to diverge in the same manner as described\nin Ref. 28.\nVI. ONSET OF FERMION PAIRING\nAnother possible neighbor of the composite Fermi liq-\nuid state of the half-filled Landau level is a paired su-\nperconducting state of the composite fermions. For ex-\nample, when the composite fermions are paired into aCFL -- FL\nLandau FLComposite FLMarginal FLT\nT*\nT**\nm+Quantum Critical\nNon-Fermi Liquid\nMarginal \nComposite FL\nFIG. 4: Schematic phasediagram showing finitetemperature\ncrossover regimes between the CFL and LFL. For T > T∗, the\nbosons are in their quantum critical regime. For T∗∗< T <\nT∗, on the CFL side the system is a marginal Fermi liquid of\ncomposite fermions; on the LFL side, the system is a marginal\nFermi liquid of electrons.\nCFL -- GMI\nGapless Mott \nInsulatorComposite FLMarginal f\nFermi liquidT\nT*\nT**\nm+Quantum Critical\nNon-Fermi Liquid\nMarginal \nComposite FL\nFIG. 5: Schematic phasediagram showing finitetemperature\ncrossover regimes between the CFL and GMI. For T > T∗,\nthe bosons are in their quantum critical regime. For T∗∗<\nT < T∗, on the CFL side the system is a marginal Fermi\nliquid of composite fermions; on the GMI side, the system is\na marginal Fermi liquid of the emergent f-fermions.\ntopologically non-trivial px+ipysuperconducting state,\ntheresultisadescriptionofthenon-AbelianMoore-Read\nPfaffian state, which is a candidate state for the plateau\natν= 5/2 in GaAs quantum wells15,64. The properties\nof the transition from CFL to the Moore-Read state are\nthe subject of ongoing investigations by others.\nIn the context of this paper, it is possible in princi-\nple that the pairing transition of the composite fermions\noccurs before the transition out of the composite Fermi\nliquid state and into the gapless Mott insulator or the\nLandau Fermi liquid, depending on the interactions be-\ntween the composite fermions. In this case, the Lan-\ndau Fermi liquid is replaced by a paired electronic super-\nconductor, while the gapless Mott insulator is replaced\nby a different topologically ordered state, consisting of\nthe fermion condensate coupled to an emergent Z2gauge13\nKitaev B Phase [Ising] \np+ip\ntopological SC\nm+ (m  ,m  )=(0,0)+-|m | -Electron Phase Diagram\nv=1/2 MR\nPfaffian \n[Ising x U(1)]\nFIG. 6: Schematic phase diagram of the electron system\nwhen the fermions are paired into a topologicaly non-trival\npx+ipystate. The transitions are drivenbythebosonic sector\nundergoing ν= 1/2 Laughlin FQH - MI – SF transitions.\nSince the fermions are gapped, the critical theories are the\nsame as for the boson system, given by eq. (40).\nfield. For example, the Moore-Read Pfaffian has a topo-\nlogical ground state degeneracy of six on the torus, and\nthe gaplesschiraledgetheory consistsofthe Ising ×U(1)\nchiral conformal field theory (CFT) with central charge\nc= 3/2. When thefermionsofthegaplessMottinsulator\nare paired into a topologically non-trivial (weak-paired)\npx+ipystate, we instead obtain a topological phase with\na torus ground state degeneracy of three, and a c= 1/2\nMajoranafermion edgestate; such astatehasthe “Ising”\ntopologicalorder,65andiswell-knownasthe non-Abelian\nstate in the “B phase” of Kitaev’s honeycomb model.26\nWhile the superconducting state cannot exist in the\npresence of a strong magnetic field, these transitions may\nbe induced by starting with the half-filled Landau level\nand turning on a periodic potential with 2 πflux per pla-\nquette. In the limit where the superconducting state is\nobtained, the effect of the magnetic field disappears be-\ncause the electrons are confined to paths along the peri-\nodic potential and therefore do not feel a net magnetic\nfield, as the phase through all closed paths will be a mul-\ntiple of 2π. In these paired states, the U(1) emergent\ngauge symmetry is broken to Z2. Since the gauge fluctu-\nations are gapped, at the critical point they can be inte-\ngrated out to obtain short-range interactions among the\nbosons. At the boson critical points, these short-ranged\ninteractions are irrelevant. Therefore, the critical theo-\nries separating these gapped states are the same critical\ntheories (40) that separate the ν= 1/2 Laughlin state,\nthe Mott insulator, and the superfluid of bosons. Fur-\nthermore, since the gauge fluctuations are gapped, one\nobtains a single crossover temperature scale associated\nwith the quantum critical points. Interestingly, since the\nfermions remain in a gapped state throughout, the elec-\ntron operator is not a scaling operator at these quantum\nphase transitions and has exponentially decaying corre-\nlations at the quantum critical points.\nThe existence of a continuous bandwidth-tuned transi-\ntion between the ν= 1/2 Moore-Read Pfaffian state andthe Kitaev B (Ising) phase, and the symmetry-protected\nquantum critical point between the ν= 1/2 Moore-Read\nPfaffian and the px+ipyelectronic superconductor are\nhighly non-trivial predictions of the theories presented\nhere.\nVII. DISCUSSION\nIn this paper, we developed a theory of transitions out\nof the composite Fermi liquid and ultimately into a Lan-\ndau Fermi liquid. These transitions can be induced by\ntuning the bandwidth of the partially filled band. In the\nLandau level problem, this can be done by using an ex-\nternal periodic potential; if the composite Fermi liquid\nis instead realized in a partially filled Chern band, the\nbandwidth can be tuned with pressure.\nWe found that generically, the transition to the Lan-\ndau Fermi liquid occurs through an intermediate gapless\nMott insulating phase, with a Fermi surface of neutral\nfermions. In the presence of certain spatial symmetries,\nsuch as inversion, we found a direct continuous transi-\ntion between the composite Fermi liquid and the Landau\nFermi liquid, providing a highly non-trivial example of\na quantum critical point between a fractionalized non-\nFermi liquid and a conventional Fermi liquid.\nIn order to establish the above results, we had to de-\npart from the conventional understanding of the compos-\nite Fermiliquid, wherethe electronisviewedasafermion\nattached to flux quanta. Instead, the electron fraction-\nalizes into a boson band a fermion f. Whenbforms a\nν= 1/2 incompressible Laughlin state, at low energies\nbis effectively created by inserting 2 flux quanta, so the\nelectron can be viewed as 2 flux quanta attached to the\nfermions, which leads to the composite Fermi liquid de-\nscription. Another natural possibility is that depending\non the nature of the interactions and the bandwidth, b\ncan form a Mott insulator to minimize the interaction\nenergy. In this case, the state of the electrons is a Mott\ninsulator, which is gapless when fhas a Fermi surface.\nFinally, when the bandwidth is large enough, the bosons\nwill condense into a superfluid, leading to a description\nof the Landau Fermi liquid.\nThe direct CFL – FL transition found here is not sta-\nble to disorder, as it relies on the presence of spatial\nsymmetries that can protect a pair of Dirac cones. Nev-\nertheless, in the presence of weak disorder and small but\nfiniteT∝ne}ationslash= 0, the properties of the direct CFL – FL quan-\ntum critical point can determine the physics. Despite\nthe fact that weak disorder will ultimately render the di-\nrect transition unstable at the lowest temperatures, the\nrelevance of this depends on the relative strength of dis-\norder,W, and temperature T. ForW≪T, this ultimate\ninstability is of no measurable consequence.\nThesituation issomewhatlessclearforthe CFL -GMI\ntransition, because there is currently no reliable estimate\nfor its correlation length exponent ν. While the boson\nFQH - MI transition does not require any spatial sym-14\nmetries, it is expected to be unstable to disorder if ν <1;\nin this case, the theory will flow to a different critical\npoint, with ν≥1, that is stable to disorder. Whether\nthe putative clean critical point or the disordered one is\nthe relevant one for describing the physics depends again\nonW/T.\nInterestingly, the currently understood direct\ntransition14between the Laughlin state and the\nsuperfluid exists only for the ν= 1/2 Laughlin state.79\nThis implies that the direct continuous transition be-\ntween the CFL and the Fermi liquid can currently only\nbe understood at half-filling. For other more general\nCFL states at other filling fractions, the route to the\nLandau Fermi liquid appears to always be separated by\nan intervening gapless Mott insulator, although there\ncan be a multi-critical point directly separating the\nCFL and LFL. This difference in the properties between\nν= 1/2 andν= 1/2mform >1 appears to form\na counterexample to the law of corresponding states7\nthat was suggested for previously understood FQH\ntransitions.\nThe experimentally observable phenomenology of the\ntransitions includes the existence of two crossover tem-\nperature scales and resistivity jumps at the transition,\nand a vanishing compressibility at the critical points,\nsimilar to the U(1) spin liquid Mott transition28. We\nfind that the composite Fermi liquid providesanother ex-perimentally promising venue where the physics of such\nslave-particle gauge theory transitions in the presence of\na Fermi surface, and novel exotic fractionalized phases,\ncan be studied.\nThe theory developed here assumes that the system\ndoes not hit a first-order transition out of the composite\nFermi liquid. The validity of this assumption depends on\nmicroscopic details of the interactions. If there is a first-\norder transition, then both disorder and/or long-range\ninteractions can render the first-order transition to be\ncontinuous.66Such continuous transitions would exhibit\ncompletely different physics from that studied in this pa-\nper and would require a completely different theory.\nAcknowledgements – We thank N. Bonesteel, S. Kivel-\nson, S. Raghu, S. Parameswaran, B. Swingle, Cenke\nXu, M.P.A. Fisher, and especially T. Senthil for helpful\ndiscussions. We also acknowledge the KITP programs\n“Topological Insulators and Superconductors,” and\n“Holographic Duality and Condensed Matter Physics”\nfor hospitality while part of this work was done. This\nwork was supported by a Simons Fellowship (MB) and\nby the U.S. Department of Energy (D.O.E.) under coop-\nerative research agreement DE-FG0205ER41360, and by\nthe Alfred P. Sloan Foundation (JM).\n1O. Heinonen, ed., Composite Fermions (World Scientific,\n1998).\n2J. K. Jain, Composite Fermions (Cambridge University\nPress, 2007).\n3B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47,\n7312 (1993).\n4V. Kalmeyer and S.-C. Zhang, Phys. Rev. B 46, 9889\n(1992).\n5N. Read, Semiconductor Science and Technology 9, 1859\n(1994).\n6J. K. Jain, Phys. Rev. Lett. 63, 199 (1989).\n7S. Kivelson, D.-H. Lee, and S.-C. Zhang, Phys. Rev. B 46,\n2223 (1992).\n8W. Chen, M. P. A. Fisher, and Y.-S. Wu, Phys. Rev. B\n48, 13749 (1993).\n9X.-G.WenandY.-S.Wu, Phys.Rev.Lett. 70, 1501 (1993).\n10J. Ye and S. Sachdev, Phys. Rev. Lett. 80, 5409 (1998).\n11S. Sachdev, Phys. Rev. B 57, 7157 (1998).\n12X.-G. Wen, Phys. Rev. Lett. 84, 3950 (2000).\n13M. Barkeshli and X.-G. Wen, Phys. Rev. Lett. 105, 216804\n(2010).\n14M. Barkeshli and J. McGreevy (2012), arXiv:1201.4393.\n15G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991).\n16T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys.\nRev. Lett. 106, 236804 (2011).\n17E. Tang, J.-W. Mei, and X.-G. Wen, Phys. Rev. Lett. 106,\n236802 (2011).\n18D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nature\nCommunications 2, 389 (2011), 1102.2658.\n19N. Regnault and B. A. Bernevig (2011), 1105.4867.20A. Vaezi (2011), arXiv:1105.0406.\n21Y.-F. Wang, Z.-C. Gu, C.-D. Gong, and D. N. Sheng,\narXiv:1103.1686v1 (2011).\n22X.-L. Qi, arXiv:1105.4298v1 (2011).\n23G. Murthy and R. Shankar (2011), arXiv:1108.5501.\n24J. McGreevy, B. Swingle, and K.-A. Tran, Phys. Rev. B\n85, 125105 (2012).\n25Y.-M. Lu and Y. Ran, Phys. Rev. B 85, 165134 (2012).\n26A. Kitaev, Annals of Physics 321, 2 (2006), ISSN 0003-\n4916, january Special Issue.\n27S.Florens andA.Georges, Phys.Rev.B 70, 035114(2004).\n28T. Senthil, Phys. Rev. B 78, 045109 (2008).\n29O. I. Motrunich, Phys. Rev. B 72, 045105 (2005).\n30S.-S.LeeandP.A.Lee, Phys.Rev.Lett. 95, 036403(2005).\n31Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and\nG. Saito, Phys. Rev. Lett. 91, 107001 (2003).\n32J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M.\nBartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-H.\nChung, et al., Phys. Rev. Lett. 98, 107204 (2007).\n33Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi,\nPhys. Rev. Lett. 99, 137207 (2007).\n34S. Zhang, T. Hansson, and S. Kivelson, Phys. Rev. Lett.\n62, 82 (1989).\n35S. Zhang, Int. J. Mod. Phys. B 6, 25 (1992).\n36Y. B. Kim, A. Furusaki, X.-G. Wen, and P. A. Lee, Phys.\nRev. B50, 17917 (1994).\n37Y. B. Kim and X.-G. Wen, Phys. Rev. B 50, 8078 (1994).\n38S. He, P. M. Platzman, and B. I. Halperin, Phys. Rev.\nLett.71, 777 (1993).\n39E. Rezayi and N. Read, Phys. Rev. Lett. 72, 900 (1994).15\n40A. Lopez and E. Fradkin, Phys. Rev. B 59, 15323 (1999).\n41E. Witten, Commun.Math.Phys. 121, 351 (1989).\n42X. G. Wen and Q. Niu, Phys. Rev. B 41, 9377 (1990).\n43X.-G. Wen, Quantum Field Theory of Many-Body Systems\n(Oxford Univ. Press, Oxford, 2004).\n44X. G. Wen, Phys. Rev. Lett. 66, 802 (1991).\n45X.-G. Wen, Phys. Rev. B 60, 8827 (1999).\n46M. Barkeshli and X.-G. Wen, Phys. Rev. B 81, 155302\n(2010).\n47M. Barkeshli and X.-G. Wen, Phys. Rev. B 84, 115121\n(2011).\n48J. K. Jain, Phys. Rev. B 40, 8079 (1989).\n49X. Wen, Int. J. Mod. Phys. B6, 1711 (1992).\n50L. B. Ioffe and A. I. Larkin, Phys. Rev. B 39, 8988 (1989).\n51K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson, Phys. Rev.\nLett.103, 046811 (2009).\n52K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Phys. Rev.\nLett.106, 236803 (2011), 1012.5864.\n53P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621 (1992).\n54B. L. Altshuler, L. B. Ioffe, and A. J. Millis, Phys. Rev. B\n50, 14048 (1994).\n55S.-S. Lee, Phys. Rev. B 80, 165102 (2009).\n56D. F. Mross, J. McGreevy, H. Liu, and T. Senthil, Phys.\nRev. B82, 045121 (2010).\n57M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127\n(2010).\n58D. F. Mross and T. Senthil, Phys. Rev. B 84, 041102\n(2011).\n59S. Sachdev and T. Morinari, Phys. Rev. B 66, 235117\n(2002).\n60C. Nayak and F. Wilczek, Nuclear Physics B 417, 359\n(1994), ISSN 0550-3213.\n61M. P. A. Fisher, G. Grinstein, and S. M. Girvin, Phys.\nRev. Lett. 64, 587 (1990).\n62W. Witczak-Krempa, P. Ghaemi, T. Senthil, and Y. B.\nKim (2012), arXiv:1206.3309.\n63C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abra-\nhams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996(1989).\n64N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).\n65C. Nayak, S. H. Simon, A. Stern, M. Freedman, and\nS. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).\n66B. Spivak and S. A. Kivelson, Annals of Physics 321, 2071\n(2006), ISSN 0003-4916.\n67J. Alicea, O. I. Motrunich, G. Refael, and M. P. A. Fisher,\nPhys. Rev. Lett. 103, 256403 (2009).\n68N. Read, Semicond. Sci. Technol. 9, 1859 (1994), cond-\nmat/9501090.\n69V. Pasquier and F. Haldane, Nuclear Physics B 516, 719\n(1998), ISSN 0550-3213.\n70G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101\n(2003).\n71D.-H. Lee, Phys. Rev. Lett. 80, 4745 (1998).\n72A. Stern, B. I. Halperin, F. von Oppen, and S. H. Simon,\nPhys. Rev. B 59, 12547 (1999).\n73S. H. Simon, Journal of Physics: Condensed Matter 8,\n10127 (1996).\n74See e.g. Ref. 67 for a recent appearance in the literature.\n75By finite compressibility, it is meant that the zero fre-\nquency density-density correlation function χρρ(q) is finite\nas the wave vector q→0 for short-range interactions. For\nlong-range interactions V(q), finite compressibility means\nthatχρρ(q)∼1/V(q).\n76An orthogonal set of shortcomings of a more microscopic\nnature is addressed in egRef. 68–73.\n77Note that integrating out ˜ athis way is inconsistent on\na genus g≥1 manifold and leads to a gauge anomaly\nassociated with large gauge transformations, as there\nare multiple topological sectors that must be properly\nconsidered.40–42\n78We would like to thank Sri Raghu for useful conversations\nregarding this point\n79For other boson FQH states, the direct transition to the\nsuperfluid appears to be highly multi-critical."    },    {        "title": "0902.0669v1.Low_energy_Landau_levels_of_AB_stacked_zigzag_graphene_ribbons.pdf",        "content": "arXiv:0902.0669v1  [cond-mat.mes-hall]  4 Feb 2009Low-energy Landau levels of AB-stacked zigzag graphene rib bons\nY. C. Huang\nCenter for General Education, Kao Yuan University, 821 Kaoh siung, Taiwan\nC. P. Chang∗and W. S. Su\nCenter for General Education, Tainan University of Technol ogy, 710 Tainan, Taiwan\nM. F. Lin†\nDepartment of Physics, National Cheng Kung University, 701 Tainan,Taiwan\nAbstract\nLow-energy Landau levels of AB-stacked zigzag graphene rib bons in the presence of a uniform\nperpendicular magnetic field ( B) are investigated by the Peierls coupling tight-binding mo del.\nState energies and associated wave functions are dominated by theB-field strength and the kz-\ndependent interribbon interactions. The occupied valence bands are asymmetric to the unoccupied\nconductionbandsabouttheFermilevel. Manydoublydegener ateLandaulevels andsinglet curving\nmagnetobands exist along kxandkzdirections, respectively. Such features are directly refle cted\nin density of states, which exhibits a lot of asymmetric prom inent peaks because of 1D curving\nbands. The kz-dependent interribbon interactions dramatically modify the magnetobands, such as\nthe lift of double degeneracy, the change of state energies, and the production of two groups of\ncurving magnetobands. They also change the characteristic s of the wave functions and cause the\nredistribution of the charge carrier density. The kz-dependent wave functions are further used to\npredict the selection rule of the optical transition.\n∗Electronic address: t00252@mail.tut.edu.tw\n†Electronic address: mflin@mail.ncku.edu.tw\n1I. INTRODUCTION\nCarbon-based materials, such as, graphite, carbon nanotubes,[1 ] two-dimensional (2D)\ngraphene and few-layer graphenes,[2, 3, 4, 5] and graphene ribbo ns, have been widely\nstudied both experimentally and theoretically. Graphene ribbons co uld be synthesized\nby using heat treatment,[6, 7, 8] pulsed-laser deposition technique ,[9] patterning epitaxi-\nally grown graphenes,[10, 11] tailoring exfoliated graphenes[12, 13] by scanning tunneling\nmicroscopy,[14] or chemical vapor deposition.[15] A one-dimensiona l (1D) graphene ribbon\nis obtained by cutting a 2D graphene, planar hexagonal lattices of c arbon atom, along the\nlongitudinal direction. As a result of the special geometric structu re, graphene ribbons have\nmotivated many interesting studies on magnetic properties,[16, 17, 18, 19, 20] optical prop-\nerties, [19, 20, 21] electronic excitations,[22] or electronic proper ties.[23] The objective of\nthis work is to investigate the magnetoelectronic structures of Be rnal graphene ribbons.\n1D zigzag and armchair graphene ribbons, sided with two parallel zigz ag and armchair\nstructures along the longitudinal direction, respectively, are inte nsively studied. Many theo-\nreticalstudiespredictthatazigzagribbonhaspeculiaredgestate s. Suchedgestatesproduce\nflatbands at low energy and give rise to a conspicuous peak in density of states.[24] An arm-\nchair ribbon does not exhibit such states. Its electronic propertie s, such as the band gap,\ndependontheribbonwidth. Furthermore, 2Dmultilayer zigzag(arm chair)grapheneribbons\nare the stack of infinite identical 1D zigzag (armchair) graphene rib bons along the stacking\ndirection [figure 1]. This stacked system exhibits the anisotropic ene rgy dispersions between\nin-ribbon-pane and the stacking direction for the intraribbon coup ling is much stronger\nthan the interribbon interaction. Moreover, the interribbon inter actions also modify the\nin-ribbon-pane electronic properties, such as state energies, en ergy dispersions, band-edge\nstates, and size of band gap. The study results show that the geo metrical structures (the\nstacking types (AA or AB stacking), ribbon width, and ribbon edge s tructure) have a signifi-\ncant effect ontheelectronic andoptical properties ofmultilayer gr aphene ribbons.[25, 26, 27]\nThe optical measurement might serve a method to determine the ge ometrical structures of\ngraphene ribbons.[28]\nWhen a monolayer graphene ribbon is submitted to a perpendicular ma gnetic field B,\nits magnetoelectronic properties, e.g., the magnetic bands and the related wave functions,\nare determined by the competition between the magnetic confineme nt and the quantum\n2confinement.[19, 20, 29] When the spatial extent of Landau wave f unctions is smaller than\nribbon width, i.e., the magnetic confinement predominates over the q uantum confinement,\nthe Landau states exist and the state energy follows E∝/radicalbig\n|n|B (n meaning the subband\nindex).[20] Such a simple characteristic is absent from a bilayer Berna l graphene ribbon. Ac-\ncording to literature,[30, 31, 32, 33] bilayer graphene ribbons exhib it different B-dependent\nLandau levels as a result of the interribbon interaction. In the ener gy region |E| ≤50 meV,\nthe Landau-level energies are linearly dependent on the magnetic fi eld strength. In the\nhigher energy region, they deviate from the B-dependence to the√\nB-dependence with the\nincrease of the field strength. Based on the findings mentioned abo ve, Bernal graphene rib-\nbons (AB-stacked graphene ribbons) are thus expected to have different magnetoelectronic\nproperties due to the different geometrical structure and interr ibbon interactions.\nOn the other hand, experimental measurements confirm that Lan dau levels of mono-\nlayer graphene and epitaxial graphene obey the relation E∝/radicalbig\n|n|B, Landau levels of\nDirac fermions.[33, 34, 35] Interestingly, Landau levels of Dirac fer mions in graphite are\nexperimentally observed.[36] Recently, Far infrared magnetotran smission measurements on\na thin graphite show the√\nB-dependent absorption lines at the Hpoint of graphite.[37] The\nmagneto-transmission of graphite at Kpoint, where interlayer interactions are maximum\nalong theHKHedge of Brillouin zone, exhibits B-dependent Landau levels.[37] Both the\ntheoretical and experimental studies on the electronic and magne to-optical properties of\ngraphitic systems motivate us to study the magnetoelectronic pro perties of Bernal graphene\nribbons. The organization of the present paper is as follows. The an alytic Hamiltonian ma-\ntrix elements of the tight-binding method for magnetoelectronic pr operties of AB-stacked\ngraphene ribbons are first derived in Sec. 2. Then, Sec. 3 investiga tes the effects of mag-\nnetic fields and the interlayer interactions on the band dispersions, Landau plot, and wave\nfunctions, following by the prediction of the selection rule of the opt ical transition. Lastly,\nconclusions are drawn in Sec. 4.\nII. THEORY\nThe AB-stacked zigzag graphene ribbon is chosen for the model st udy. It has hydrogen-\nterminated zigzag edges along the longitudinal (ˆ x) direction [Fig. 1(a)]. The ribbon width\nis defined by the number ( Ny) of zigzag lines along the yaxis.BandAatoms are different\n3sublattices of biparticles hexagonal lattice of a graphene ribbon. T here are 2Nycarbon\natoms in a primitive cell of a monolayer graphene ribbon. To obtain the 2D AB-stacked\nzigzag graphene ribbon, the same 1D zigzag ribbons are piled in the AB AB sequence along\nthezaxis with the stacking distance Iz= 3.35˚A. TheAatoms have the corresponding\natoms on the x−yplane directly above or below them, while the projections of the Batoms\nare located at the center of the hexagonal rings directly above or below them [Fig. 1(b)].\nIn the presence of a perpendicular magnetic field B= (0,0,B), the electronic properties\nof AB-stacked zigzag ribbons are modeled by the Peierls coupling tigh t-binding model. The\nHamiltonian is\nH=/summationdisplay\ni,j(γi,jei2πθi,jc+\nicj+H.c.), (1)\nwhere the subindices i,jdenote the summation over all sites in a primitive unit cell. c+\ni(cj)\nis the creation (annihilation) operation, which generates (destroy s) an electron at i(j) site.\nThe details of the atom-atom interactions[38] γi,j=γ0, γ1,···,γ6are described in Fig.\n1(b).ei2πθi,jis the Peierls phase shift due to the applied magnetic field. θi,jis the line\nintegral of vector potential Afromitojin a unit of the flux quantum Φ 0=ch/e. To keep\nthe translation invariance along the xaxis, within the Landau gauge, the vector potential\nA= (−By,0,0) is deliberatively adopted. In this gauge, AB-stacked zigzag ribbo ns have\ntwo ribbons or 4 Nycarbon atoms in a primitive cell. The first Brillouin zone is a rectangle\ndefined by −π/Ix≤kx≤π/Ix, and−π/2Iz≤kz≤π/2Iz.\nThe tight-binding Bloch function is the linear combination of basis |A1\nm∝an}b∇acket∇i}ht,|B1\nm∝an}b∇acket∇i}ht,|A2\nm∝an}b∇acket∇i}ht,\nand|B2\nm∝an}b∇acket∇i}ht, which are the linear superposition of the 2 pzorbitals located at A1\nm, B1\nm, A2\nm,\nandB2\nmcarbon atoms. The wave function is\n|Φ(kx,y,kz)∝an}b∇acket∇i}ht=Ny/summationdisplay\nm=1aA1m|A1\nm∝an}b∇acket∇i}ht+bB1m|B1\nm∝an}b∇acket∇i}ht+Ny/summationdisplay\nm=1aA2m|A2\nm∝an}b∇acket∇i}ht+bB2m|B2\nm∝an}b∇acket∇i}ht, (2)\nwhere wave function coefficients aA1m,bB1m,aA2mandbB2mare the site amplitudes.\nThe Hamiltonian representation is a 4 Ny×4NyHermitian matrix and could be regarded\nas\nH=\nh1h12\nh21h2\n, (3)\n4where each Hermitian block matrix h1,h2,h12orh21has 2Ny×2Nyelements. The block\nmatrixh1, Hamiltonian representation of the lower ribbon, is a tridiagonal mat rix and its\nmatrix elements h1(i,j) are:\n\n\nh1(2m−1,2m−1) =γ6+β2γ5/2,\nh1(2m,2m) =β2γ2/2,h1(2m−1,2m) =h1(2m,2m−1) = 2γ0cos{√\n3bkx/2−π(m−[N])Φ},\nh1(2m,2m+1) =h1(2m+1,2m) =γ0,(4)\nwherem= 1, 2,···,Nyandβ= 2cos(kzIz). [N] = (Ny+1)/2 is used to locate the origin of\ncoordinate in the center of the ribbon. πΦ, the Peierls phase shift, results from the magnetic\nflux passing half a hexagonal ring. The diagonal term γ6+β2γ5/2 (β2γ2/2), due to the\ninterribbon interactions, can be treated as the equivalent site ene rgy ofA(B) atoms. The\ncenters of the two ribbons in a unit cell do not coincide with each othe r [Fig. 1(a)]. This\nleads to a different tridiagonal matrix h2, which is\n\n\nh2(2m−1,2m−1) =γ6+β2γ5/2,\nh2(2m,2m) =β2γ2/2,\nh2(2m−1,2m) =h2(2m,2m−1) = 2γ0cos{√\n3bkx/2−π(m−[N]+1\n3)Φ},\nh2(2m−1,2m+2) =h2(2m+2,2m−1) =γ0.(5)\nThe termπΦ/3 in Eq. (5) is caused by the difference between the centers of the t wo ribbons\nin a unit cell.\nThe two off-diagonal block matrices satisfy a simple relation h12=h21. As a result of\nthe interribbon interactions, the nonzero matrix elements h12(i,j) of the off-diagonal block\nmatrixh12are as follows:\n\n\nh12(2m−1,2m−1) =βγ1,\nh12(2m,2m) =βγ3,\nh12(2m−1,2m) = 2βγ4cos{√\n3bkx/2−π(m−[N]+1\n3)Φ},\nh12(2m,2m−1) = 2βγ4cos{√\n3bkx/2−π(m−[N])Φ},\nh12(2m,2m+1) =βγ4,\nh12(2m−1,2m+2) =βγ4,\nh12(2m,2m+2) = 2βγ3cos{√\n3bkx/2−π(m−[N]+1\n6)Φ}.(6)\n5The effect of magnetic fields is reflected in the interaction 2 γicos(√\n3bkx/2−π(m−[N])Φ).\nAll the nonzero elements of h12depend on β= 2cos(kzIz). The interribbon interactions\nare equal to zero at kzIZ=π/2, and they are maximum at kzIz= 0. Eigenvalues and\neigenvectors are obtained after the diagonalization of the Hamilton ian matrix. Eigenvalues\nare the energy dispersions Ec,v(kxIx,kzIz), wherec(v) represents the unoccupied (occupied)\nstates. The eigenvector |ac,v\nA1\n1,bc,v\nB1\n1,···,ac,v\nA1m,bc,v\nB1m,···,ac,v\nA2\n1,bc,v\nB2\n1,···,ac,v\nA2m,bc,v\nB2m,···∝an}b∇acket∇i}htis thekz-\ndependent envelope function |Ψc,v(y,kzIz)∝an}b∇acket∇i}htalong theydirection.\nIII. MAGNETO-ELECTRONIC PROPERTIES\nThe band structures of the 2D multilayer graphene ribbons exhibit a highly anisotropic\nstructure. Figs. 2(a)-2(c) show the kx-dependent low-energy bands for the AB-stacked\nzigzag graphene ribbon with width Ny=3000 (639 nm) at B = 0, B = 20 T, and B = 10\nT, respectively. The unit of energy is eV. The Fermi level is set to EF= 0. At B = 0,\nthe low energy bands are similar to those obtained from the LDA calcu lations.[39] These\nbands include parabolic bands and flatbands [Fig. 2(a)]. The occupied valence bands are\nasymmetric to the unoccupied conduction bands about EF= 0 due to the energy-dependent\ninterribbon interactions. The Ny= 3000 AB-stacked zigzag graphene ribbon is a semimetal\nbecause the valence bands lightly touch the conduction bands at EF= 0. The degeneracy\nof the flatbands at E= 0, a special feature of a single zigzag ribbon due to the zigzag-edg e\nboundaries, is lifted by the interribbon interactions. The partial fla tbands near EF= 0\ncorrespond to the edge states mainly localized at the outmost zigza g positions. According\nto Figs. 2(b)-2(d), the magnetic field drastically modifies the energ y bands, such as the\nalteration of the band feature, the shift of the subbands, and th e production of the Landau\nlevels. At B = 20 T, as shown in Fig. 2(b), the original parabolic bands m ight become the\ncomplete Landau levels. The occupied states Evare asymmetric to the unoccupied states\nEcaboutEF= 0. The Landau-level energies do not follow the simple relation E∝/radicalbig\n|n|B.\nThe presence of Binduces a longer and weak splitting energy subband near EF, namely, the\nzero mode. There are more Landau subbands and a shorter range in band structure as the\nmagnitude of the magnetic field decreases [Fig. 2(c)]. The zigzag bou ndaries make narrow\ngraphene ribbons exhibit special electronic and magnetic propertie s.[40, 41]\nInterestingly, the interribbon interactions induce the concave-u pward and -downward\n6magnetobands along ˆkz. The energy dispersions near EFat B = 20 T with Ny= 3000, B =\n10 T withNy= 3000, and B = 20 T with Ny= 6000 (1278 nm) along the kz-axis are shown\nin Figs. 2(e)-2(h), respectively, to illustrate the effect of interrib bon interactions. There\nare oscillating subbands, concave-upward and -downward bands, and nearly dispersionless\nbands nearby EF[Figs. 2(e) and 2(f)]. The occupied states Evare asymmetric to the\nunoccupied states EcaboutE= 0. The main feature of the energy dispersions remains\nunchanged for the variation of magnetic flux. However, the state energies are sensitive to\nthe magnitude of the magnetic field [Fig. 2(g)]. The zero mode near EF, including two\noscillating subbands, has a dispersion of 0.04 eV at the low field limit. The upper subband\nof the zero mode, indexed by n= 0, shows the cosine energy bands E∝cos(kzIz) along\nthekzdirections, while the feature of the lower subband, indexed by n′= 0, is distorted.\nA pseudogap ∆ associated with the zero mode near EFis observed. Thus, the size of the\npseudogap ∆ varies with kz. The upper limit of the pseudogap ∆, occurring at kzIz=π/2,\nis equal toγ6, the chemical difference between Aatoms andBatoms. The concave-upward\nand -downward subbands, are classified into two groups. One grou p, denoted as Group I,\nmoves towards EFwhile the other, Group II, moves away from EF. The first three subbands\nof each group are shown in Fig. 2(f) as examples. The unoccupied st atesEcand occupied\nstatesEvof Group I are denoted by subband indices nandn′, respectively. Those of Group\nII are indexed, respectively, by ¯ nand ¯n′. The subband crossings occur at different kz’s.\nThe band edge states, the local minimum and maximum, of the n, ¯n, and ¯n′subbands are\nlocated atkze= 0.\nThe Landau plot, the Landau-level energies vs. the field strength , at different kz’s can\nreveal the effect of the interribbon interactions. The Landau plot of theNy= 3000 AB-\nstacked zigzag ribbon at kzIz=π/2 is shown in Figs. 3(a) and 3(b). The state energies of\nthe zero mode are independent of the field strength. The energy s pacing between the upper\nand lower subbands, the pseudogap ∆ at kzIz=π/2, is equal to γ6. Except the zero mode,\nLandau-level energies follow the simple relation Ec,v∝√\nB. The chief reason is that the\ntwo ribbons in a primitive unit cell are regarded as two independent rib bons because the\ninterribbon interactions are turned off at kzIz=π/2. The magnetoelectronic properties are\ndominated by the magnetic confinement rather than the quantum e ffect. Accordingly, the\nLandau-level energies, indexed by n, follow the relation E∝/radicalbig\n|n|B.[20] On the other hand,\nthe interribbon interactions between the two ribbons aremaximum a tkzIz= 0. The Landau\n7plots, as shown in Figs. 3(c) and 3(d), exhibit different behaviors. T he state energies of the\nzero mode vary with the field strength. In the energy region |E|<0.02γ0≃50 meV, the\nstate energies of the first and second unoccupied (occupied) Lan dau levels exhibit the linear-\nin-B dependence at B≤50 T. The high energy Landau levels, indexed by n≥3, exhibit\nthe linear-in-B dependence at B <15 T. Then, they evolve from a linear B-dependence to\na square-root B dependence with the increase in the field strength . Finally they exhibit the\nlinear-in-√\nB dependence at B≥20T.\nDensity of state (DOS) of multilayer graphene ribbons is evaluated b yD(ω) =/summationtext\nk,h\n|∇kEh(k;φ)|−1\nEh=ω. As shown in Fig. 4(a), in the absence of a magnetic field, the featur eless\nDOS of the Ny= 3000 graphene ribbon, asymmetric about ω= 0, shows a small bump\natω= 0, which results from the tiny overlap between the occupied bands and unoccupied\nbands [Fig. 2(a)]. The Ny= 3000 graphene ribbon is a semimetal. The magnetic field\nmodifies electronic states [Fig. 2] and the aspect of DOS [Fig. 4(b)]. D OS at B = 20 T\nis asymmetric about EF. It chiefly exhibits three kinds of structures, including the delta-\nfunction-like peak, the compound peak near ω= 0, and the 1D power-law divergences. The\nfirst kind of peaks are related to the Landau levels along ˆkx[Fig. 2(b)] and dispersionless\nbands along ˆkz[the red curves in Fig. 4(b)], the second kind to compound band along ˆkx\nandˆkz, and the third kind to concave-upward and -downward bands along ˆkz[the red curves\nin Fig. 4(b)]. The sharp peaks at ω=−0.3 and−0.45 eV origin in the Landau levels along\nˆkxand dispersionless bands along ˆkz. DOS of the concave-upward and -downward diverges\nin the forms 1 //radicalbig\nω−E(kze) and 1//radicalbig\nE(kze)−ω, respectively, where kzeis the band edge\nof the subband. DOS exhibits more peaks in the decreases of the ma gnitude of magnetic\nfields, as shown in Fig. 4(c). In other words, the peak heights and p ositions are strongly\ndependent on the magnetic flux. Fig. 4(d) shows DOS of the Ny= 6000 graphene ribbon\nat B = 20 T. Remarkably, density of states is independent of the ribb on width [Fig. 4(b)\nand Fig. 4(d)].\nWhentheribbonwidthissufficientlylarge( Ny≥3000),theAB-stackedgrapheneribbons\nat B = 0 are expected to be almost identical to the bulk graphite in DOS except at very low\nfrequency. The zigzag boundary induces partial flat bands near EF= 0 and thus strongly\naffects the low-frequency physical properties. The two kinds of s ystems might differ from\neach other in the transport and magnetic properties. Neverthele ss, the large graphene\nribbons are useful in understanding the bulk graphite, e.g., magnet oelectronic properties[43]\n8and magneto-optical absorption spectra. The main reason is that the calculations of the\nmagnetoelectronic structures are very complicated for the latte r at B<10 T (a very large\nHamiltonian matrix). That is to say, the magnetoelectronic propert ies (state energies and\nassociated wave functions) remain the same when the ribbon width is sufficiently wide. It\nis deduced to be the same with that of a 3D bulk graphite.\nThe effect of interribbon interactions on the envelope functions de serves a closer exam-\nination. Envelope functions of the Ny= 3000 zigzag ribbon subjected to B = 20 T at\ndifferentkz’s are shown in Figs. 5(a)-5(d). Each envelope function Ψ( y,kzIz) is separated\ninto eight components, ψ(A1\no),ψ(B1\no),ψ(A1\ne),ψ(B1\ne),ψ(A2\no),ψ(B2\no),ψ(A2\ne), andψ(B2\ne).A1\no\n(B2\ne) denotesA(B) atoms located at the odd (even) zigzag lines at the lower (upper) r ib-\nbon plane. The subenvelope function ψ(A1\no), for example, is |aA1\n1,aA1\n3,aA1\n5,···∝an}b∇acket∇i}ht. Because\nψ(A1\no) =−ψ(A1\ne),ψ(B1\no) =−ψ(B1\ne),ψ(A2\no) =−ψ(A2\ne), andψ(B2\no) =−ψ(B2\ne), only four\ncomponents ψ(A1\no),ψ(B1\no),ψ(A2\no), andψ(B2\no) are displayed. At kzIz=π/2, the interribbon\ninteractions are closed due to β= 2cos(kzIz) = 0 [Eqs. (4)-(6)]. The two ribbons in a\nprimitive unit cell are decoupled, i.e., there are two isolated graphene ribbons. Except the\nzero mode, the state energies are double degenerate. Their asso ciated envelope functions,\nΨ1(y,kzIz=π/2) and Ψ 2(y,kzIz=π/2), are orthogonal to each other and different in\nthe sign of some components. One of the two envelope functions is s hown in Fig. 5(a).\nIt is exactly described as/bracketleftBig\nφn(A1\no),φn−1(B1\no),φn−1(A2\no),φn(B2\no)/bracketrightBig\n,whereφnis the harmonic\noscillator, product of the Hermite polynomial Hnand the Gaussian function.[20] Each φn\nhasnnodes, where the subenvelope function changes the signs. The no de number nof\nthe subenvelope function ψn(A1\no) is identical to the subband index n. Thus, we use the\nsubband indices n= 1,2,···andn′= 1,2,···to specify the characteristic of envelope\nfunction [Fig. 5(a)]. φn(A1\no) andφn(B2\no) [φn−1(B1\no) andφn−1(A2\ne)] belong to the n[n−1]\nmode. The envelope functions are the same as those of a single grap hene ribbon.[20] Due\nto the orthogonality, Ψ nand Ψ n′are different in the sign of some components. Besides,\nΨn=0=φ0(B2\no)−φ0(B2\ne), the envelop function of the n= 0 subband, is located in the B\nsites of the upper ribbon. φ0(A1\no)−φ0(A1\ne), the envelop function of the n′= 0 subband,\nis mainly controlled by the Aatoms on the lower ribbon plane. The chemical difference\nγ6changes the site energies of atoms Aand, thus, causes the energy split and produces a\npseudogap ∆ = γ6between Ψ n=0and Ψ n′=0atkzIz= 2/π[Fig. 2(e)].\nAway from kzIz=π/2, the switched-on interribbon interactions not only produce two\n9groups of subbands, but also change the aspects of envelop func tions. AtkzIz= 0.499π,\nthe envelope functions related to nand ¯nsubbands concentrate on the lower and upper\nribbon, respectively [Fig. 5(b)]. The subenvelope functions related to thensubband are\nφn(A1\no) andφn−1(B1\no), wheren= 1,2,3,···(n′= 1,2,3,···). As for the ¯ nsubbands,\nthe associated subenvelope functions are φ¯n(A2\no) andφ¯n+1(B2\no), where ¯n= 0,1,2,3···.\nNotably, ¯nis different from nand it begins from ¯ n= 0 (¯n′= 0). For example, the envelop\nfunction of n= 1 [¯n= 0] subband is Ψ n=1=φ1(A1\no)+φ0(B1\no) [Ψ¯n=0=φ0(A2\no)+φ1(B2\no)]. The\neffect caused by the interribbon interactions on the envelope func tions can be interpreted\nby the first order perturbation theory. Around kzIz= 0.5π, the matrix elements of h12\nare proportional to ( π/2−kzIz) because of β= 2cos(kzIz)∼2(π/2−kzIz). Thus, the\nweak interribbon interactions lift the degeneracy at kzIz=π/2 and produces a two-level\nsystem, for example, n= 1 and ¯n= 0 subbands. The related envelope functions are\nΨ(y,kzIz) = [Ψ 1(y,kzIz=π/2)±Ψ2(y,kzIz=π/2)]/√\n2, the bonding or antibonding\nof the envelope functions at kzIz=π/2. As a result, the envelope functions of the n(¯n)\nsubbandschieflydistributeonthelower (upper)ribbonwiththediffe rentspatialsymmetries.\nIn addition, interribbon interactions have no influence on the envelo p functions Ψ n=0and\nΨn′=0.\nThe feature of the wave function at kzIz= 0.4π, as shown in Fig. 5(c), is dissimilar\nto that atkzIz= 0.499π[Fig. 5(b)]. The charge carriers of the former distribute on two\nribbon planes and each subenvelope function is of the same importan ce, whereas those of\nthe latter concentrate only on one ribbon plane. The varying interr ibbon interactions causes\nthe redistribution of charge density. The definition of subband indic esnand ¯n(n′and ¯n′)\nis based on the spatial symmetry of the subenvelope function ψn(A1\no). The subenvelope\nfunctions are proportional to the harmonic oscillator φn, for example, ψn=1(A1\no)∼φ1(A1\no).\nThe envelope functions of the |n| ≥1 subband are/bracketleftbig\nφn(A1\no),φn−1(B1\no),φn(A2\no),φn+1(B2\no)/bracketrightbig\n.\nThe subband index n(n′) is assigned to be n= 1,2,3,···(n′= 1,2,3,···). Notably,\nsubenvelope functions ψ(A1\no) andψ(A2\no) belong to the nmode andψ(B1\no) [φ(B2\no)] to then−1\n[n+1] mode. The spatial symmetry [ n,n−1,n,n+1] is different from that [ n,n−1,n−1,n]\natkz= 0. On the other hand, the envelope functions of the ¯ nsubband is the linear\ncombinationofcomponents φ¯n(A1\no),φ¯n−1(B1\no),φ¯n(A2\no), andφ¯n+1(B2\no). Thespatialsymmetry\n[¯n,¯n−1,¯n,¯n+1] of the ¯ nsubband is the same as that of the nsubband [n,n−1,n,n+1].\nAccording to the spatial symmetry of ψn(A1\no), the subband indices are assigned as ¯ n=\n100,1,2,···. Notably, ¯ nstarts at zero. Due to the interactions between the n=±6 and\n¯n=±1 subbands, the shapes of ¯ n= 0 envelope functions is modified and its component\nψ(B1\no) almost disappears.\nFromkzIz= 0.4πtokzIz= 0, the gradually enhancing interribbon interactions signifi-\ncantly modify the feature or the site amplitude of the nenvelope functions, as shown in Fig.\n5(d). The maximum interribbon interactions, occurring at kzIz= 0, dramatically change\nthe main feature of the envelope function [heavy dots in Fig. 5(d)], w hich now can not\nsimply be described by harmonic oscillator φn. For comparison, those at kzIz= 0.3πare\nshown by light dots in Fig. 5(d), whose main feature of the envelope f unctions is still similar\nto those at kzIz= 0.4π. Thus, the envelope function is approximately linear superposition\nof components φn(A1\no),φn−1(B1\no),φn(A2\no), andφn+1(B2\no). The interribbon interactions dra-\nmatically alter the site amplitude. ψc,v\nn−1(B1\no) andψc,v\nn+1(B2\no) dominate the envelope function.\nThe comparison shows that the increase in the interribbon interact ions not only change the\nmain feature of the subenvelope functions but also enhance the sit e amplitude of Batoms.\nIt should benoted thatthe interribbon interactions do not breakt he spatial symmetry of the\nenvelope function, which is denoted as/bracketleftbig\nψn(A1\no),ψn−1(B1\no),ψn(A2\no),ψn+1(B2\no)/bracketrightbig\n. The spatial\nsymmetry of the envelope function has a critical effect on the optic al selection rule.\nThe dependence of magnetoelectronic properties of the ¯ nsubbands on the wave vector\nkzis deliberated. For comparison purpose, the kz-dependent state energies of the n= 1 and\n¯n= 0 subbands and the associated envelop functions are shown in Fig. 6. Atkz=π/2, two\nsubbands aredegenerate andmagnetoelectronic propertiesmain ly exhibit thecharacteristics\nof an isolated graphene because of the close of the interribbon inte ractions. As kzchanges\nfromπ/2 to zero, the two degenerate subbands are lifted by the kz-dependent interribbon in-\nteractions and form concave-upward and -downward bands along kz[Fig. 6(b)]. Meanwhile,\nthe interribbon interactions have different influences on the shape of wave functions and the\ncharge-carrier distribution [Fig. 6(a)]. The subenvelope functions of ¯n= 0 subband [heavy\ndots in Fig. 6(a)] can be clearly described by the harmonic oscillator φn. By contrast, the\nfeatures of subenvelope functions related to n= 1 subband are seriously modified by the kz-\ndependent interribbon interactions as kzIz→0. Thekz-dependent interribbon interactions\ncause the change of the site amplitudes. The charge carriers of th e ¯n= 0 subbands tend to\nconcentrate on Aatoms, while those of n= 1 subbands are mainly located in Batoms [Fig.\n6(a)].\n11The characteristics of wave functions are applicable to determine t he optical transition\nchannels. The optical absorption spectra is A(ω)∝/summationtext\nµ,ν/integraltext\n1stBZdkx\n2πdkz\n2πD(ω)× |Mµ,ν|2,\nwhereµ=n,¯nandν=n′,¯n′are subband indices. D(ω) is the joint density of states\nandMµ,νis the velocity matrix element.[19, 20, 32] According to the result of R ef. [32], the\nmagnitude of velocity matrix is |Mµ,ν|=|Mµ,ν|intraribbon +|Mµ,ν|inter−ribbon, where|Mµ,ν|is\nproportional to ∝an}b∇acketle{tΨµ(y,kzIz)|Ψν(y,kzIz)∝an}b∇acket∇i}ht, the projection of the envelope function of initial\nstate,|Ψν(y,kzIz)∝an}b∇acket∇i}ht, onto that of final state, |Ψµ(y,kzIz)∝an}b∇acket∇i}ht. The subscript intraribbon (inter-\nribbon) denotes the initial and final states located at the same (different ) ribbons. The\nintraribbon -projection, for example, is\n∝an}b∇acketle{tΨn(y,kzIz)|Ψn′(y,kzIz)∝an}b∇acket∇i}htintraribbon =∝an}b∇acketle{tψn(A1)|ψn′(B1)∝an}b∇acket∇i}ht+∝an}b∇acketle{tψn(B1)|ψn′(A1)∝an}b∇acket∇i}ht+ (7)\n∝an}b∇acketle{tψn(A2)|ψn′(B2)∝an}b∇acket∇i}ht+∝an}b∇acketle{tψn(B2)|ψn′(A2)∝an}b∇acket∇i}ht.(8)\nThe velocity matrix element Mµ,νdepends on the spatial symmetries of the initial\nand final states and, thus, determines the selection rule and the e ffective transition\nchannels.[19, 20, 42] At kzIz=π/2, only the intraribbon optical transition, |Mµ,ν|intraribbon,\nmakes contribution to the velocity matrix element |Mµ,ν|. There is only one Group of sub-\nbands, i.e., µ=nandν=n′. Therefore, the optical selection rule is predicted to be\nδn−n′=±1, wheren= 0,1,2,3···andn′= 0,1,2,3···. The selection rule is the same\nas that of a monolayer graphene ribbon.[20]\nAway from kzIz=π/2, there arefour possible optical transition channels n′→n, ¯n′→¯n,\n¯n′→nandn′→¯n[Fig. 6(b)]. According to the spatial symmetry of envelope function s,\nthe corresponding selection rules of the four transitions are pred icted to be δn−n′=±1,\nδ¯n−¯n′=±1,δn−¯n′=±1, andδ¯n−n′=±1. The change of the state energies caused by\ninterribbon interactions might alter the peak positions of joint dens ity of states, and the\nmodification of the shape and site amplitude of envelope functions mig ht give rise to the\nvivid variation of the magnitude of velocity matrix element Mµ,ν. Thus, the optical spectra\noftheAB-stackedribbonareexpectedtobemorecomplicatedinpe aknumber, peakposition\nand peak height than those of a monolayer or bilayer ribbon. The wor k to calculate the\noptical spectra of the AB-stacked ribbon is on-doing.\n12IV. CONCLUSIONS\nThe Peierls coupling tight-binding method is employed to investigate th e anisotropic\nmagnetoelectronic structures of multilayer graphene ribbons. Th ey are strongly dependent\non theB-field strength and the interribbon interactions. Bcan induce Landau levels along\nkxand change energy spacing. The interribbon interactions significan tly affect state de-\ngeneracy, energy dispersions, Landau plot. They destroy the sy mmetry of magnetoband\nstructures. They also produce concave and convex subbands alo ngkz. The magnetoband\nstructures are strongly anisotropic. There are many doubly dege nerate Landau levels and\nsinglet 1D curving bands along kxandkzdirections. Such features are directly reflected\nin density of states. DOS exhibits a lot of asymmetric prominent peak s, for example, the\ndelta-function-like peak, the 1D power-law divergence, and the co mpound peak. Moreover,\nthekz-dependent interribbon interactions modify the shape of wave fun ctions, alter the\nspatial symmetry, and arouse the change of charge-carrier dist ribution. The findings are\nfurther used to predict the optical selection rule. Most important ly, the large AB-stacked\ngraphene ribbons are expected to be almost identical to the bulk gr aphite in low-energy\nmagnetoelectronic properties. Our work is useful in understandin g the bulk graphite, e.g.,\nmagnetoelectronic excitations and magneto-optical absorption s pectra. The predicted mag-\nnetoelectronic properties could be examined by transport and opt ical measurements.\nAcknowledgments\nThis work was supported by the Taiwan National Science Council (NS C 96-2112-M-165-\n001 MY3; NSC 95-2112-M-006-0002).\n[1] S. Iijima, Nature 35456 (1991).\n[2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zha ng, S. V. Dubonos, I. V.\nGrigorieva, and A. A. Firsov, Science 306, 666 (2004).\n[3] J. S. Bunch, Y. Yaish, M. Brink, K. Bolotin, and P. L. McEue n, Nano. Lett. 5, 287 (2005).\n[4] Y. B. Zhang, J. P. Small, M. E. S. Amori, and P. Kim, Phys. Re v. Lett.94, 176803 (2005).\n13[5] Y. H. Wu, B. J. Yang, B. Y. Zong, H. Sun, Z. X. Shen, and Y. P. F eng, J. Mat. Chem. 14,\n469 (2004).\n[6] M. Murakami, S. Iijima, and S. Yoshimura, J. Appl. Phys. 603856 (1986).\n[7] B. L. V. Prasad, H. Sato, T. Enoki, Y. Hishiyama, Y. Kabura gi, A. M. Rao, P. C. Eklund, K.\nOshida, and M. Endo, Phys. Rev. B 6211209 (2000).\n[8] M. Zhang, D. H. Wu, C. L. Xu, Y. F. Xu and W. K. Wang, Nanostru ct Matt. 10\n[9] M. Yudasaka, Y. Tasaka, M. Tanaka, H. Kamo, Y. Ohki, S. Usa mi, and S. Yoshimura, Appl.\nPhys. Lett. 643237 (1994).\n[10] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayo u, T. Li, J. Hass, A. N.\nMarchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, Scien ce3121191 (2006).\n[11] C. Berger, Z. Song, T. Li, X. Li, A. Y. Ogbazghi, R. Feng, Z . Dai, A. N. Marchenkov, E. H.\nConrad, P. N. First, and W. A. de Heer, J. Phys. Chem. B 10819912 (2004).\n[12] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S.\nV. Dubonos, and A. A. Firsov, Nature 438197 (2005).\n[13] Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature 438201 (2005).\n[14] H. Hiura, Applied Surface Science 222374 (2004).\n[15] J. Campos-Delgado, J. M. Romo-Herrera, X. Jia, D. A. Cul len, H. Muramatsu, Y. A. Kim,\nT. Hayashi, Z. Ren, D. J. Smith, Y. Okuno, T. Ohba,H. Kanoh, K. Kaneko, M. Endo, H.\nTerrones, M. S. Dresselhaus, M. Terrones, Nano Lett., 82773, 2008.\n[16] K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Phy s. Rev. B 598271 (1999).\n[17] B. L. V. Prasad, H. Sato, T. Enoki, Y. Hishiyama, Y. Kabur agi, A. M. Rao, G. U. Sumanasek-\nera, and P. C. Eklund, Phys. Rev. B 64235407 (2001).\n[18] C. Ritter, S. S. Makler, and A. Latge, Phys. Rev. B 77195443 (2008).\n[19] Y. C. Huang, C. P. Chang, and M. F. Lin, Nanotechnology 18495401 (2007).\n[20] Y. C. Huang, C. P. Chang, and M. F. Lin, J. of Appl. Phys. 103073709 (2008).\n[21] M. F. Lin, and F. L. Shyu, J. Phys. Soc. Jpn. 693529 (2000).\n[22] M. F. Lin, M. Y. Chen, and F. L. Shyu, J. Phys. Soc. Jpn. 702513 (2001).\n[23] C. P. Chang, Y. C. Huang, C. L. Lu, J. H. Ho, T. S. Li, and M. F . Lin, Carbon 44508 (2006).\n[24] K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselh aus, Phys. Rev. B 5417954 (1996).\n[25] F. L. Shyu, M. F. Lin, C. P. Chang, R. B. Chen, J. S. Shyu, Y. C. Wang, and C. H. Liao, J.\nPhys. Soc. Jpn. 703348 (2001). (the figures for armchair ribbons in this manusc ript will be\n14revised because of the numerical calculation errors).\n[26] C. W. Chiu, F. L. Shyu, C. P. Chang, R. B. Chen, and M. F. Lin , J. Phys. Soc. Jpn. 72170\n(2003).\n[27] C. P. Chang, C. L. Lu, F. L. Shyu, R. B. Chen, Y. C. Huang, an d M. F.Lin, Physica E 2782\n(2005).\n[28] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D.\nJiang, K. S. Novoselov, S. Roth, and A. K. Geim, Phys. Rev. Let t.97187401 (2006).\n[29] L. Brey and H. A. Fertig, Phys. Rev. B 73, 235411 (2006).\n[30] E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805 (2006).\n[31] E. A. Henriksen, Z. Jiang, L.-C. Tung, M. E. Schwartz, M. Takita, Y. -J. Wang, P. Kim, and\nH. L. Stormer, Phys. Rev. Lett. 100, 087403 (2008).\n[32] Y. C. Huang, C. P. Chang, and M. F. Lin, Phys. Rev. B 78, 115422 (2008).\n[33] P. Plochocka, C. Faugeras, M. Orlita, M. L. Sadowski, G. Martinez, and M. Potemski, Phys.\nRev. Lett. 100, 087401 (2008).\n[34] Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M. E. Sc hwartz, M. Y. Han, P. Kim, and\nH. L. Stormer1, Phys. Rev. Lett. 98, 197403 (2007)\n[35] M. L. Sadowski, G. Martinez, M. Potemski, C. Berger and W . A. de Heer, Phys. Rev. Lett.\n97, 266405 (2006).\n[36] G. Li, E. Y. Andrei, Nature Phys. 3623 (2007).\n[37] M. Orlita, C. Faugeras, G. Martinez, D. K. Maude, M. L. Sa dowski, and M. Potemski, Phys.\nRev. Lett. 100136403, (2008).\n[38] J. C. Charlier, X. Gonze and J. P. Michenaud, Phys. Rev. B 434579 (1991).\n[39] Y. Miyamoto, K. Nakada, and M.Fujita, Phys. Rev. B 599858 (1999).\n[40] H. Lee, Y.-W. Son, N. Park, S. Han, and J. Yu, Phys. Rev. B 72174431 (2005).\n[41] Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97216803 (2006).\n[42] J. M. Pereira, F. M. Peeters, and P. Vasilopoulos, Phys. Rev. B76, 115419 (2007).\n[43] Y. Kopelevich and P. Esquinazi, Advanced Matrials 194559 (2007).\n15Figure Captions\nFIG. 1. (a)Thegeometricstructureofthe Ny=3AB-stackedzigzaggraphenenanoribbons. (b)\nγ0= 2.598eVistheintralayerinteraction. γ1= 0.364eV(γ5= 0.036eV)represents the\ninteraction between two Aatoms from two neighboring ribbons (two next-neighboring\nribbons), and γ3= 0.319eV(γ2=-0.014eV) is for Batoms.γ4= 0.177eVcorresponds\nto the interribbon interaction between Aatoms and Batoms.γ6=−0.026eVis the\nchemical-shift between Aatoms andBatoms. The values of γi’s are the same as those\nof AB-stacked graphite.[38]\nFIG. 2. The kx-dependent low-energy bands for the AB-stacked zigzag graphe ne ribbons with\n(a)Ny=3000 at B = 0, (b) Ny=3000 at B = 20 T, (c) Ny=3000 at B = 10 T,\nand (d)Ny=6000 at B = 20 T. Figs. (e)-(h), same plot as Figs.(a)-(d), but for the\nkz-dependence.\nFIG. 3. (a) Landau plot of the Ny= 3000 ribbon at kzIz=π/2. (b) The Landau level energies\nof theNy= 3000 ribbon at kzIz=π/2 vs.√\nB. (c) Landau plot of the Ny= 3000\nribbon atkz= 0. (d) The relation between Landau level energies of the Ny= 3000\nribbon atkz= 0 and√\nB.\nFIG. 4. The low-energy DOS for the AB-stacked zigzag graphene r ibbons with (a) Ny=3000\nat B = 0, (b) Ny=3000 at B = 20 T, (c) Ny=3000 at B = 10 T, and (d) Ny=6000 at\nB = 20 T.\nFIG. 5. The envelop functions for the AB-stacked zigzag graphen e ribbonsNy=3000 subjected\nto B = 20 T at kzIz=π/2,0.499π,0.4πand 0 are shown, respectively, in (a)-(d).\nFIG. 6. (a) The energy dispersions of the n= 1 and ¯n= 0 subbands. (b) The variation of the\nenvelop functions, related to the n= 1 and ¯n= 0 subbands, with kzIZ.FIG.  1.ӫ˃ ӫ˄ӫˆ\nӫˇIz\nӫˈ1A1,2A1\n3A1B11B11B11B12B12B12B1\n2B22B21A1,2A11A1,2A11A1,2A1\n1A2,2A22B22B2\n1A2,2A2 1A2,2A21A2,2A21A2,2A2\n1A3,2A31A3,2A31A3,2A31A3,2A31B21B21B21B2\n2B32B32B32B32B3\n1B31B31B3\n1A2A\n1BIzӫ˅\n2A3B\n2A\n2A2B\n2B2B( a )\n( b )m = 1\n2\n  Ny = 33\n"    },    {        "title": "1008.0748v1.Anomalous_integer_quantum_Hall_effect_in_AA_stacked_bilayer_graphene.pdf",        "content": "arXiv:1008.0748v1  [cond-mat.mes-hall]  4 Aug 2010Anomalous integer quantum Hall effect in AA-stacked bilayer graphene\nYa-Fen Hsu1and Guang-Yu Guo2,1,∗\n1Department of Physics and Center for Theoretical Sciences,\nNational Taiwan University, Taipei 106, Taiwan\n2Graduate Institute of Applied Physics, National Chengchi U niversity, Taipei 116, Taiwan\n(Dated: October 30, 2018)\nRecent experiments indicate that AA-stacked bilayer graph enes (BLG) could exist. Since the\nenergy bands of the AA-stacked BLG are different from both the monolayer and AB-stacked bilayer\ngraphenes, different integer quantum Hall effect in the AA-st acked graphene is expected. We have\ntherefore calculated the quantized Hall conductivity σxyand also longitudinal conductivity σxxof\nthe AA-stacked BLG within the linear response Kubo formalis m. Interestingly, we find that the AA-\nstacked BLG could exhibit both conventional insulating beh avior (the ¯ ν= 0 plateau) and chirality\nfor|¯µ|< t, where ¯ νis the filling factor (¯ ν=σxyh/e2), ¯µis the chemical potential, and tis the\ninterlayer hopping energy, in striking contrast to the monl ayer graphene (MLG) and AB-stacked\nBLG. We also find that for |¯µ| /negationslash= [(√n2+√n1)/(√n2−√n1)]t, wheren1= 1,2,3,···,n2= 2,3,4,···\nandn2> n1, the Hall conductivity is quantized as σxy=±4e2\nhn, n= 0,1,2,· · ·,if|¯µ|< t\nandσxy=±4e2\nhn, n= 1,2,3,· · ·,if|¯µ|> t. However, if |¯µ|= [(√n1+√n2)/(√n2−√n1)]t,\nthe ¯ν=±4(n1+n2)nplateaus are absent, where n= 1,2,3,· · ·, in comparison with the AB-\nstacked BLG within the two-band approximation. We show that in the low-disorder and high-\nmagnetic-field regime, σxx→0 as long as the Fermi level is not close to a Dirac point, where Γ\ndenotes the Landau level broadening induced by disorder. Fu rthermore, when σxyis plotted as a\nfunction of ¯ µ, a ¯ν= 0 plateau appears across ¯ µ= 0 and it would disappear if the magnetic field\nB=πt2/Nehυ2\nF,N= 1,2,3,· · ·. Finally, the disappearance of the zero-Hall conductivity plateau\nis always accompanied by the occurence of a 8 e2/h-step at ¯ µ=t.\nPACS numbers: 73.43.Cd,71.70.Di,72.80.Vp,73.22.Pr\nI. INTRODUCTION\nGraphene exhibits many peculiar properties[1] and has\ngreatly intrigued physicists in recent years. Charge carri-\nersin the monolayergraphene(MLG) possessalinearen-\nergydispersion (see Fig. 1a)[2] and areofchiralnature[3]\nnear each Dirac point. The quasiparticles in the AB-\nstackedbilayergraphene(BLG) are alsochiral. However,\nunlike MLG, the energy spectra of the AB-stacked BLG\nare parabolic (Fig. 1b). One of the interesting properties\nof graphene is quantum Hall effect (QHE). Indeed, both\ntheoretical[4] and experimental works[5–7] show that in-\nteger quantum Hall effect (IQHE) in MLG is unconven-\ntional. The Hall conductivity in MLG is quantized as\nσxy=±4e2\nh(n+1\n2), wheren= 0,1,2,· · ·. The factor\n4 comes from the fourfold (spin and valley) degeneracy.\nFurthermore, beacuse the states are shared by electron\nand hole at the zeroth Landau level (LL), the shift of 1 /2\noccurs. In contrast, in the AB-stacked BLG, within the\ntwo-bandparabolicapproximation,the Hall conductivity\nwas shown to be σxy=±4e2\nhn, wheren= 1,2,3,·· ·[8].\nThe zeroth and first Landau levels are degenerate and\nhence the first quantum Hall plateau appears at 4 e2/h\ninstead of 2 e2/h. This phenomenon has been observed\nexperimentally[9]. QHE of the AB-stacked BLG was also\nstudied based on a four-band Hamiltoian in Ref. [10].\n∗Electronic address: gyguo@phys.ntu.edu.twAlthough AB stacking is predicted to be energetically\nfavored over AA stacking in ab initio density functional\ntheory(DFT) calculations, the energydifference ofabout\n0.02 eV/cell is small[11, 12]. Moreover, Lauffer et al.\nfound that scanning tunneling microscopy (STM) images\nof BLG resemble that of MLG, and hence they regarded\nit as a consequence of the BLG confiquration being close\nto AA stacking[13]. Moreover, Liu et al.reported that\nin their high-resolution transmission electron microscope\n(HR-TEM) experiments a high proportion of thermally\ntreated samples are AA-stacked BLG[14]. These findings\nindicate the possibilty of fabrication of the AA-stacked\nBLG.SincetheenergybandsoftheAA-stackedBLG(see\nFig. 1c) aredifferent fromboth the AB-stackedBLGand\nmonolayer graphene, the quantum Hall effect in the AA-\nstacked is expected to be quite different from that in the\nlatter two systems.\nWe have therefore carried out a theoretical study of\nIQHE as well as the longitudinal conductivity σxxin the\nAA-stacked bilayer graphene using the Kubo formalism.\nIn this paper, we present a general analytical form of\nthe Hall conductivity [ σxy(¯µ,B)] as a function of both\nchemical potential (¯ µ) and magnetic field ( B) of the AA-\nstacked BLG. Our presentation will be divided into two\nparts: i) the variation of the σxyvs. 1/Bcurve with\nsome fixed ¯ µ’s and ii) the effect of the magnetic field on\ntheσxyvs. ¯µcurve. Our main findings are as follows.\nFirstly, for |¯µ| /ne}ationslash= [(√n2+√n1)/(√n2−√n1)]t, wheretis\nthe interlayer hopping energy, n1andn2are any integers\nlarger than 1 and 2, respectively, and n1< n2, the Hall2\n( )a ( )b ( )c\nt\ntE\nk kE E\nkt\nt\nFIG. 1: (color online) The energy bands of (a) mono-\nlayer graphene, (b) AB-stacked bilayer graphene, and (c)AA -\nstacked bilayer graphene.\nconductivity is quantized as\nσxy=±4e2\nhn, n= 0,1,2,···,if|¯µ|<t\nσxy=±4e2\nhn, n= 1,2,3,···,if|¯µ|>t. (1)\nHowever, if |¯µ|= [(√n2+√n1)/(√n2−√n1)]t, the Hall\nconductivity is given by\nσxy=±4e2\nhn, excluding ±4e2\nh(n1+n2)n, n= 1,2,3,···.\n(2)\nSecondly, in the low-disorder and high-magnetic-field\nregime [Γ →0 and/planckover2pi14ω4\nc≫(¯µ+νt)4],σxx≈\n(8e2/πh)[(¯µ2+t2)Γ2/(¯µ2−t2)2+ 2Γ2//planckover2pi12ω2\nc+ 5(¯µ2+\nt2)Γ2//planckover2pi14ω4\nc)]→0 if the Fermi level is not close to a\nDirac point, where /planckover2pi1ωcis the cyclotron energy. That is\nto say, if the magnetic field is high enough, the applied\nelectric field cannot drive any current for |¯µ|< t, while\nthe current is perpendicular to external electric field for\n|¯µ|> t. Thirdly, we find that the σxy= 0 (the fill-\ning factor ¯ν= 0) plateau across ¯ µ= 0 would disap-\npear when B=πt2/Nehυ2\nF,N= 1,2,3,···, and that a\n8e2/h-step at ¯µ=toccurs while a zero-Hall conductivity\nplateau disappears. Interestingly, unlike the monolayer\nand AB-stacked bilayer graphenes, the AA-stacked bi-\nlayer graphene could display a unusual ¯ ν= 0 plateau\neven though it contains chiral quasiparticles. We argue\nthat the occurrence of the ¯ ν= 0 plateau is due to the\nshift of level anomalies by the interlayer hopping energy\nt.\nII. THEORETICAL MODEL AND\nANALYTICAL CALCULATION\nA. Model Hamiltonian and Landau levels\nWe firstderiveaneffective four-bandHamiltoniannear\neach Dirac point for the AA-stacked bilayer graphene in\ntight-binding aprroximation via k·pexpansion[1]. We\nthen diagonalize this Hamiltonian to obtain the energy\nbands of the AA-stacked BLG. In Fig. 1, we display theenergy bands of the MLG, AB-stacked, and AA-stacked\nBLG together. We find that the energy bands of the\nAA-stacked BLG are just two copies of the MLG band\nstructure shifted up and down by t, respectively. Hence,\nE=±tare the Dirac points for the AA-stacked BLG.\nWhen a magnetic field is applied, we should replace the\nmomentum operator pwithp+eA/c, where the exter-\nnal magnetic field B=∇×Aand -e is the charge of an\nelectron. The magnetic field is applied along the positive\nz-axis (i.e. out of plane) and hence the vector poten-\ntial can be written as A= (−By,0,0) in the Landau\ngauge. Therefore, the effective four-band Hamiltonian in\nthe presence of the magnetic field is given by\nH±=/parenleftbigg\nυF(σxπx±σyπy) −tI\n−tI υ F(σxπx±σyπy)/parenrightbigg\n:/braceleftigg\nπx=−i/planckover2pi1∂x−eBy/c,\nπy=−i/planckover2pi1∂y.(3)\nHere±label the two valleys of the band structure at K\nandK′, respectively. υFdenotes the Fermi velocity. In\nthe Laudau gauge, we can substitute the eigenfunction\nψ=eikxφ(y) of the Hamiltonian into the schr¨ odinger\nequationHψ=Eψ. Hereφ(y) can be written as\n(φ1(y),φ2(y))T, whereφ1andφ2are two-component\ncolumn vectors. Then, we make the transformations:\nσ±=σx±iσy,ξ=y/ℓB−ℓBkandO∓= (ξ±∂ξ)/√\n2 ,\nwhere the magnetic length ℓB=/radicalbig\n/planckover2pi1c/|eB|. Finally, for\ntheKvalley, the schr¨ odinger equation reads\n/parenleftigg−/planckover2pi1υF√\n2ℓB(O−σ++O+σ−) −tI\n−tI−/planckover2pi1υF√\n2ℓB(O−σ++O+σ−)/parenrightigg/parenleftbigg\nφ1\nφ2/parenrightbigg\n=E/parenleftbigg\nφ1\nφ2/parenrightbigg\n. (4)\nSinceO∓satisfythecommutationrelation: [ O−,O+]=\n1,O∓are the annihilation and creation operators of one-\ndimensional (1-D) simple harmonic oscillator (SHO), re-\nspectively. Similarly, σ±are the raising and lowering\noperators of pesudospin angular momentum. Obviously,\nthe eigenstates of O−σ++O+σ−are (|N−1/an}b∇acket∇i}ht,±|N/an}b∇acket∇i}ht)Tif\nN≥1 and (0,|0/an}b∇acket∇i}ht)TifN= 0, where |N/an}b∇acket∇i}htare the eigen-\nstates of the 1-D SHO. All non-zero vectors are eigenvec-\ntors of−tI. Therefore, we can infer that the eigenvalues\nof Eq. (4) are\nEµν\nN=−µ√\nN/planckover2pi1ωc−νt (5)\nwithωc=√\n2υF/ℓBand the eigenstates of Eq. (4) are\n\n\n|0,+,ν/an}b∇acket∇i}ht=1√\n2\n0\n|0/an}b∇acket∇i}ht\n0\nν|0/an}b∇acket∇i}ht\n,ifN= 0,\n|N,µ,ν/an}b∇acket∇i}ht=1\n2\n|N−1/an}b∇acket∇i}ht\nµ|N/an}b∇acket∇i}ht\nν|N−1/an}b∇acket∇i}ht\nµν|N/an}b∇acket∇i}ht\n,ifN≥1,(6)3\nHere the indice µ=±andν=±. Clearly, the LLs of\nthe AA-stacked BLG are just two copies of the LLs of\nthe MLG shifted up and down by t, respectively.\nB. Linear response calculation\nThe conductivity can be calculated using the Kubo\nformula within the linear response theory[15]. The Kubo\nformula for the DC-conductivity is given by\nσij= lim\nΩ→0ImΠR\nij(Ω+i0)\n/planckover2pi1Ω. (7)\nHere the retarded current-current correlation ΠR\nijin the\nMatsubara form reads[10]\nΠR\nij(iνm) =−4e2\n2πℓ2\nBβ/planckover2pi1× (8)\n∞/summationdisplay\nn=−∞∞/summationdisplay\nk,ℓ=0/summationdisplay\nµ,ρ\n=±/summationdisplay\nν,σ\n=±/an}b∇acketleftk,µ,ν|vi|ℓ,ρ,σ/an}b∇acket∇i}ht/an}b∇acketleftℓ,ρ,σ|vj|k,µ,ν/an}b∇acket∇i}ht\n(i˜ωn−˜Eµν\nk)(i˜ωn+iνm−˜Eρσ\nℓ),\nwhere the factor 4 is due to the fourfold (spin and val-\nley) degeneracy, the velocity operator vi= [xiH+]/i/planckover2pi1,\nand˜Eµν\nk=Eµν\nk//planckover2pi1.ωnandνmare Matsubara fre-\nquencies of fermion and boson, respectively. When the\nchemicalpotentialanddisorderscatteringareconsidered,\nMatsubara frequency of fermion has to be corrected as\ni˜ωn=iωn+ ¯µ//planckover2pi1+isgn(ωn)Γ//planckover2pi1. Γ is the Landau level\nbroadening due to the presence of disorder. Substituting\nthe eigenstates of H+into Eq. (8), we obtain\nΠR\nxy(iνm) =−ie2v2\nF\n2πℓ2\nBβ/planckover2pi1(2/summationdisplay\nµ,ν\n=±χµν,+ν\n1,0+/summationdisplay\nℓ≥1/summationdisplay\nµ,ρ\n=±/summationdisplay\nν=±χµν,ρν\nℓ+1,ℓ),\n(9)\nwhereχµν,ρσ\nk,ℓis defined as\nχµν,ρσ\nk,ℓ(iνm) =n=∞/summationdisplay\nn=−∞[1\n(i˜ωn−˜Eµν\nk)(i˜ω+iνm−˜Eρν\nℓ)\n−(iνm→ −iνm)]. (10)\nIn the DC and clean limit, we let Ω+ i0→0 and set\nΓ = 0. Then, after evaluating the Matsubara sums[15],\nwe find that\n1\nβ/planckover2pi1χµν,ρσ\nk,ℓ|iνm=Ω+i0→0≈−2(Ω+i0)[f(Eµν\nk)−f(Eρσ\nℓ)]\n(˜Eµν\nk−˜Eρσ\nℓ)2,\n(11)\nwheref(E) is the Fermi-Dirac distribution. Further-\nmore, we define ˜f(E) =f(E)−1/2. Then, using Eqs.\n(7), (9) and (11) and considering the variation of direc-\ntion of conductivity with the signs of magnet field and\ncarrier charge, we can derive that\nσxy=−4e2\nhsgn(eB)\n/summationdisplay\nν=±˜f(E+ν\n0)+∞/summationdisplay\nℓ≥1/summationdisplay\nµ,ν\n=±˜f(Eµν\nℓ)\n\n(12)At zero temperature, f(E) = 1 andf(E) = 0 for the\noccupied and unoccupied LLs, respectively. The LLs lo-\ncated atE≤¯µare ocuppied and the others are empty.\nThatis tosay, ˜f(E) = 1/2forE≤¯µ, while˜f(E) =−1/2\nforE >¯µ. Therefore, we only need to calculate the num-\nberofLLsbetween −¯µand ¯µtodeterminethemagnitude\nof the Hall conductivity. Moreover, for −|¯µ|< E <|¯µ|,\nthe number of the up-shifted LLs ( ν=−) is equal to\nthat of the down-shifted LLs ( ν= +). Hence, we find\nthat the zero-temperature Hall conductivity is given by\nσxy=−4e2\nhsgn(¯µ)sgn(eB)[θ(|¯µ|+t)θ(|¯µ|−t)\n+∞/summationdisplay\nℓ≥1/summationdisplay\nµ=±θ(|¯µ|−Eµ−\nℓ)θ(|¯µ|+Eµ−\nℓ) ].(13)\nEq. (13) indicates that the Hall conductivity would sim-\nply change its sign as ¯ µ→ −¯µand that the Hall conduc-\ntivity is equal to 4 e2/htimes the number of up-shifted\nLLs between −|¯µ|and|¯µ|. Eq. (13) can also be written\nin the form\nσxy=−4e2\nhsgn(¯µ)sgn(eB)×\n/braceleftbigg\nθ(|¯µ|−t−/radicalig\n2/planckover2pi1v2\nF|eB|/c)/bracketleftbiggc(|¯µ|−t)2\n2/planckover2pi1v2\nF|eB|/bracketrightbigg\n+θ(/radicalig\n2/planckover2pi1v2\nF|eB|/c−t+|¯µ|)/bracketleftbiggc(|¯µ|−t)2\n2/planckover2pi1v2\nF|eB|/bracketrightbigg\n(14)\n+/bracketleftbiggc(|¯µ|+t)2\n2/planckover2pi1v2\nF|eB|/bracketrightbigg\n−/bracketleftbiggc(|¯µ|−t)2\n2/planckover2pi1v2\nF|eB|/bracketrightbigg\n+θ(|¯µ|+t)θ(|¯µ|−t)}.\nHere [x] means the integer part of x. The last term is\nthe contribution of level anomalies ( N= 0)[4].\nWe also calculate the longitudinal conductivity ( σxx)\nvia the Kubo formula. The longitudinal conductivity\nhave to be evaluated under the effect of disorder scat-\ntering (Γ /ne}ationslash= 0). Using Cauchy’s integral theorem as in\nRefs. [16, 17], we can obtain\nσxx=e2\nπhω2\nc/braceleftbigg/integraldisplay∞\n−∞dE/parenleftbigg\n−∂f\n∂E/parenrightbigg\n×\n\n/summationdisplay\nµ,ρ\n=±/summationdisplay\nν=±∞/summationdisplay\nℓ≥1Im˜gµν\nℓ+1(E)Im˜gρν\nℓ(E)+2/summationdisplay\nµ,ν\n=±Im˜gµν\n1(E)Im˜g+ν\n0(E)\n\n\n,\n(15)\nwhere ˜gρ\nℓ= 1/(E//planckover2pi1−˜Eρν\nℓ+iΓ//planckover2pi1). Applying the tech-\nniques of partial-fraction decomposition similar to that\nused in Ref. [18] and after some cumbersome algebra, we\nfinally find that the zero-temperature longitudinal con-\nductivity can be written in terms of the digamma func-4\ntion (ϕ0)[19],\nσxx=4e2\nπh/summationdisplay\nν=±/braceleftbigg\ni/bracketleftbigg\nϕ0/parenleftbigg\n−(¯µ+νt)2−Γ2\n/planckover2pi12ω2c+1−i2Γ(¯µ+νt)\n/planckover2pi12ω2c/parenrightbigg\n−ϕ0/parenleftbigg\n−(¯µ+νt)2−Γ2\n/planckover2pi12ω2c+1+i2Γ(¯µ+νt)\n/planckover2pi12ω2c/parenrightbigg/bracketrightbigg\n×/bracketleftbig\n2(¯µ+νt)3Γ+2(¯µ+νt)Γ3/bracketrightbig\np\n+/bracketleftbig\n/planckover2pi16ω6\nc+/planckover2pi14ω4\nc(¯µ+νt)2−12/planckover2pi12ω2\nc(¯µ+νt)4+8(¯µ+νt)6/bracketrightbig\nΓ2\nq\n+/bracketleftbig\n/planckover2pi14ω4\nc+4/planckover2pi12ω2\nc(¯µ+νt)2+16(¯µ+νt)4/bracketrightbig\nΓ4\nq\n+8(¯µ+νt)2Γ6\nq\n+/bracketleftbig\n/planckover2pi12ω2\nc(¯µ+νt)2+/planckover2pi14ω4\nc/bracketrightbig\nΓ2+/planckover2pi12ω2\ncΓ4\n[(¯µ+/planckover2pi1ωc+νt)2+Γ2][(¯µ−/planckover2pi1ωc+νt)2+Γ2]\n×1\n(¯µ+νt)2+Γ2/bracerightbigg\n. (16)\nHere,\np=/planckover2pi14ω4\nc+16Γ2(¯µ+νt)2,\nq=/braceleftbig\n[(¯µ+νt)2−/planckover2pi12ω2\nc−Γ2]2+4Γ2(¯µ+νt)2/bracerightbig\n×p.\n(17)\nUnlike Ref. 20, we do not adopt low-magnetic field\napproximation here and therefore Eq. (16) is general\nand suitable for all values of the applied magnetic field.\nHowever, when we adopt the low-disorder and high-\nmagnetic-field limit, we first consider Γ as being small\nand keep the terms in the order of Γ2. Then, we let\n(¯µ+νt)/(/planckover2pi14ω4\nc)→0, Eq. (16) could be simplified as\nσxx≈4e2\nπh/bracketleftbigg(¯µ2+t2)Γ2\n(¯µ2−t2)2+2Γ2\n/planckover2pi12ω2c+5¯µ2+t2\n/planckover2pi12ω2cΓ2\n/planckover2pi12ω2c/bracketrightbigg\n.\n(18)\nThe first term is independent of B. The second and third\nterms are proportional to 1 /Band 1/B2, respectively.\nIII. DEPENDENCE OF CONDUCTIVITY ON\nMAGNETIC FIELD AND CHEMICAL\nPOTENTIAL\nSince both the magnetic field and chemical potential\ncould be tuned experimentally, we display the calculated\nconductivity as a function of 1 /Band ¯µin this Sec. Here\nwe use the interlayer hopping energy t= 0.2 eV for the\nAA-stackingand t= 0.4eVfor the AB-stackingasdeter-\nmined by the abinitio DFT calculations within the local\ndensityapproximation(LDA)[21]. Moreover,becausethe\nquantized values of the Hall conductivity is independent\nof the presence of disorder scattering[22], we show only\nthe Hall conductivity in the clean limit (i.e., Γ = 0) and\nanalyze the Hall plateaus qualitatively.-0.03 -0.02 -0.01 0\n1/B (1/T)0204060σxy (e2/h)AA µ = 0.600 eV\nAB µ = 0.516 eV-0.04 -0.02 0 0.02 0.04-20020σxy (e2/ h)µ = 0.10 eV\nµ = 0.15 eV\nµ = 0.25 eV\nµ = 0.30 eV\n(a)\n(b)\nFIG. 2: (color online) (a) The quantized Hall conductivity\nσxyof the AA-stacked bilayer graphene as a function of 1 /B\nfor several values of chemical potential ¯ µ. (b) The quantized\nHall conductivity σxyof both AA-stacked and AB-stacked bi-\nlayer graphenes as a function of 1 /B. The interlayer hopping\nenergytused is 0.2 eV for the AA-stacking and 0.4 eV for the\nAB-stacking, and υF= 1.0×106m/s. The Hall conductivity\nσxyfor the AB-stacking was obtained by using Eq. 15 from\nRef. 10.\nA. Conductivities vs inverse of magnetic field\nIn Fig. 2, the Hall conductivity is plotted as a func-\ntion of 1/Band we should discuss the effect of chem-\nical potential on the σxyvs. 1/Bcurve. From Fig.\n2(a), we see that σxy=±4e2\nhn, withn= 0,1,2,· · ·,\nfor|¯µ|< t, andn= 1,2,3,· · ·, for|¯µ|> t, excluding\n|¯µ|= [(√n2+√n1)/(√n2−√n1)]t, wheren1= 1,2,3,···\nandn2= 2,3,4,···. It is clearfrom either Eq. (13) orEq.\n(14)thatσxy(−¯µ) =−σxy(¯µ), andhencewedidnotshow\nany curves for ¯ µ<0 in Fig. 2(a). Interestingly, in con-\nstrast to the MLG and AB-stackedBLG, the AA-stacked\nBLG displays the pronounced ¯ ν= 0 plateau for |¯µ|<t,\nwhere the filling factor ¯ ν=σxyh/e2. The MLG and AB-\nstacked BLG lack the ¯ ν= 0 plateau because their level\nanomaliesarelocated at E= 0. The level anomalyof the\nMLG is the zeroth Landau level while those of the AB-\nstacked BLG are the zeroth and first Landau levels. The\noccurrence of level anomalies is a remarkable manifesta-\ntion of the unique property of chiral quasiparticles[8, 23].\nIn the AA-stacked BLG, similarly, level anomalies also5\n-0.08 -0.04 0 0.04 0.08-16-80816σij (e2/h)σxxΓ=0.05 eV\nσxyΓ=0\n-0.04 -0.02 0 0.02 0.04\n1/B (1/T)-24-16-8081624σij (e2/h)σxxΓ=0.05 eV\nσxyΓ=0µ<t\nµ=0.1eV\nµ>t\nµ=0.25eV(a)\n(b)\nFIG. 3: (color online) The longitudinal ( σxx) and transverse\n(σxy) conductivities of the AA-stacked bilayer graphene as a\nfuction of 1 /B, (a) for |¯µ|< t, and (b) for |¯µ|> t. The rest\nparameters are the same as in Fig. 2.\nexist in the LL spectrum. However, for the AA-stacked\nBLG, the level anomalies are shifted up and down by\nt, respectively. Thus, these level anomalies are unique in\nthe sensethat they arealwayslocated at the Diracpoints\nregardless of the magnitude of B, in contrast with the\nother Landau levels[4, 8, 10]. It is worth mentioning that\nthe Dirac points often exhibit interesting electronic prop-\nerties, suchaselectron-holepuddleformation[24,25], and\nAndreev reflection type transitions.[26] Untill now, some\ntransport properties at these Dirac points remain to be\nunderstood[27]. The level anomaly is one of the interest-\ning propertiesof the Dirac points and recently the nature\nof its electronic states (being metallic or insulating) in\nthe high magnetic field-low temperature regime is hotly\ndebated[28].\nDisplayed in Fig. 2(b) are the Hall plateaus of the\nAA-stacked BLG for |¯µ|= [(√n2+√n1)/(√n2−√n1)]t.\nWe find that in addition to the ¯ ν= 0 plateau, other dif-\nferences between the AA-stacked and AB-stacked BLGs\nexist. In particular, when |¯µ|= [(√n2+√n1)/(√n2−√n1)]t, some 8e2/h-steps appear at 1 /B/ne}ationslash= 0. In other\nwords, in comparison with the other cases of |¯µ|> t\n(|¯µ| /ne}ationslash= [(√n1+√n2)/(√n2−√n1)]), someplateauswould\nbe missing for |¯µ|= [(√n2+√n1)/(√n2−√n1)]t. Fur-\nthermore, these 8 e2/h-steps appear periodically. Taking\n¯µ= 3t(i.e.n2= 4,n1= 1), for example, between anytwo 8e2/h-steps, the curve passes through three 4 e2/h-\nsteps and four plateaus. Only the Hall plateaus σxy=\n4e2\nhn,n=±1,±2,±3,±4,±6,±7,±8,±9,±11,±12,· ·\n·, appear. In other words, the Hall plateaus n=\n±5,±10,±15,· · ·are absent here. When |¯µ|> t, the\nquantum Hall effect of the AB-stacked BLG must be\nstudied based on a four-band Hamiltonian[10]. Based\non the four-band model[10], for |¯µ|> t, the AB-stacked\nBLG can also exhibit a 8 e2/h-step, as shown in Fig.\n2(b). Although the 8 e2/h-step is not specific to the AA-\nstacked BLG, the periodic appearance of the 8 e2/h-steps\nhas never been seen in the AB-stacked BLG and hence\nis a unique characteristic of the AA-stacked BLG.\nThelongitudinalandtransverseconductivitiesasafuc-\ntion of 1/Bare plotted together in Fig. 3. It is seen\nfrom Fig. 3 that the longitudinal conductivity goes to\na local minima at the position of the Hall plateaus and\nreaches a local maxima as the steps appear except the\nstep at 1/B= 0. The unique ¯ ν= 0 Hall plateau of\nthe AA-stacked bilayer graphene for |¯µ|<tis especially\ninteresting. From Fig. 3(a), we find that σxxfalls to\nzero as the ¯ ν= 0 Hall plateau emerges. As stated previ-\nously, for |¯µ|> t, the AA-stacked BLG lacks the ¯ ν= 0\nHall plateau. The first Hall plateau occurs at ¯ ν= 4 or\n¯ν=−4. This encourages us to investigate the difference\nbetween the longitudinal conductivities at the ¯ ν= 0 Hall\nplateau for |¯µ|< tand at the ¯ ν=±4 Hall plateaus\nfor|¯µ|> t. Fig. 3(b) shows that for |¯µ|> t, theσxx\ngoes to zero as the first Hall plateau occurs. This implies\nthat at the high magnetic field, the external electric field\ncannot drive any current for |¯µ|< twhile the current\nis perpendicular to the external electric field for |¯µ|>t.\nHere ¯µandtare in the order of 0.1 eV while the order\nof magnitude of Γ [ O(Γ)] is 0.01 eV. ¯ µ+νtare about\none order of magnitude higher than Γ (¯ µ+νt∼10Γ).\nTherefore, the condition of low disorder is satisfied. If\n/planckover2pi14ω4\nc≫(¯µ+νt), Eq. (18) can be applied here. This\nneeds/planckover2pi1ωcto be larger than 1.78 (¯ µ+νt) andBis esti-\nmated to be at least a few times larger than 10 T. In the\nlow-disorder and high-magnetic-field regime, we roughly\nestimate from Eq. (18)\nσxx∼4e2\nh/bracketleftbigg2\nπ(¯µ2+t2)Γ2\n(¯µ2−t2)2+O(0.001)/bracketrightbigg\n.(19)\nWhenO(|¯µ|−t)≥1Γ,σxx∼0.1(4e2/h)→0.\nInterestingly, the results shown in Fig. 2 could be\nexplained in terms of Fig. 4. Let us account for Fig.\n2(a) first. It is clear from Fig. 4 that for |¯µ|< t, level\nanomalies are outside the range of −|¯µ| ∼ |¯µ|. Con-\nversely, for |¯µ|>t, level anomalies are inside the range of\n−|¯µ| ∼ |¯µ|. In the high magnetic field regime, for |¯µ|<t,\nno Landau level exists between −|¯µ|and|¯µ|and hence\nthe AA-stackedBLG displaysthe conventionalinsulating\nbehaviour(a ¯ ν= 0 Hallplateau)eventhough it possesses\nchirality. Such behaviour of the AA-stacked BLG is in\nstark contrast to the MLG and AB-stacked BLG. How-\never, for |¯µ|>t, level anomaliesarelocated between −|¯µ|6\nC\nDC\nD\nE F( )a ( )b\n--\nFIG. 4: (color) The Landau level spectrum of the AA-stacked\nbilayer graphene (a) with |¯µ|< tand (b) with |¯µ|> t, re-\nspectively. Here circles mark the positions of Landau lev-\nels. Red, green, and purple circles represent the locations of\nlevel anomalies as well as other up-shifted and down-shifte d\nLandau levels, respectively. Mazarine and blue lines denot e\nthe up-shifted and down-shifted energy bands, respectivel y.\nShaded is the region between −¯µand|¯µ|.\nand|¯µ|even in the high magnetic field regime and hence\na ¯ν= 0 plateau cannot emerge.\nThe Hall plateaus displayed in Fig. 2(b) can be ex-\nplainedasfollows. Aup-shifted LLof E−µ,−\nkandadown-\nshifted LLof Eµ,+\nkarepartnersbecause Eµ,+\nk=−E−µ,−\nk.\nAs the up-shifted LL of E−µ,−\nkgoes through the |¯µ|-level,\nthe down-shifted LL of Eµ,+\nkpasses through the −|¯µ|-\nlevel. They always enter the region between |¯µ|and−|¯µ|\n(i.e. the shaded region) together and hence each con-\ntribute 4e2/hto the Hall conductivity. Therefore, the\nHall conductivity is equal to 4 e2/htimes the number of\neither up-shifted or down-shifted LLs between −|¯µ|and\n|¯µ|. Therefore, in order to form a 8 e2/h-step, either two\nup-shifted or down-shifted LLs must enter or leave the\nshaded region together. Hence we only need to focus on\neither up-shifted or down-shifted Dirac cones and discuss\nthe movement of the LLs located in this cone to explain\nthe origin of 8 e2/h-steps. Let us consider the up-shifted\nDiraccone. Asthemagneticfielddecreasesgradually,the\nup-shifted LLs would go close to its level anomaly, E=t.\nTherefore, we can infer that for |¯µ|< t, the LLs above\nthe|¯µ|-level (called the upper LLs) go far away from the\n|¯µ|-level, while the LLs below the −|¯µ|-level (called the\nlower LLs) move toward the −|¯µ|-level, as shown in Fig.\n4(a). The lower LLs would enter the shaded region one\nby one but the upper LLs can never get into the shaded\nregion. However, for |¯µ|> t, both the upper and lower\nLLs can go close to the shaded region [see Fig. 4(b)], i.e.,\ntwoup-shifted LLs mayenter the shadedregiontogether.\nTherefore, the 8 e2/h-steps can only appear when |¯µ|>t.\nFor brevity, we use indices ( k,µ,ν) to denote the Lan-\ndau level of Eµ,ν\nkbelow. Let us label the two LLs which\nenter the shaded region together as the ( k1,−,−) and\n(k2,+,−) LLs. Then, k1/k2= (|¯µ| −t)2/(|¯µ|+t)2.TABLEI:Thecharacteristics ofintegerquantumHalleffecti n\nmonolayer(ML),AB-stackedbilayer(AB)andAA-stackedbi-\nlayer (AA) graphenes as well as conventional two-dimension al\nsemiconductor structures (2D).\nML AB AA 2D\nplateau steps (e2\nh) 4 4, 8a4, 8b2\n¯ν= 0 plateau no no yescyes\naThe 8e2\nhplateau step only occurs (aperiodically) in the four band\nmodel (Ref. [10]).\nbThe 8e2\nhplateau step appears periodically for |¯µ|> tonly.\ncThe ¯ν= 0 plateau occurs only if |¯µ|< t.\n-0.2 -0.1 0\nµ (eV)-808162432404856σxy (e2/h)B = 10.1 T\nB = 12.0 T\nB = 15.2 T \nFIG. 5: (color online) The quantized Hall conductivity σxy\nof the AA-stacked bilayer graphene as a function of chemical\npotential ¯ µfor several values of magnetic field B. The rest\nparameters are the same as in Fig. 2.\n(n1,n2) satisfies this condition and n1/n2is an irre-\nducible fraction. Then, ( k1,k2) = (pn1,pn2), wherep=\n1,2,3···,isasetofsolutionsof k1/k2= (|¯µ|−t)2/(|¯µ|+t)2.\nBetween the entries of [( p−1)n1,(p−1)n2] and [pn1,pn2]\nLLs, (n1+n2−2) LLs get into the shaded region sequen-\ntially as 1/Bdecreases. Hence, ( n1+n2−2) 4e2/h-steps\noccur between any two 8 e2/h-steps. The Hall conduc-\ntivity is quantized as σxy=±4e2\nhnwith the exception\nof±4e2\nh(n1+n2)n, wheren= 1,2,3,· · ·, as shown in\nFig. 2(b). Unlike the AB-stacked BLG and the other\ncases of |¯µ|> tof the AA-stacked BLG, the Hall con-\nductivity for |¯µ|= [(√n2+√n1)/(√n2−√n1)]tlacks\nthe ¯ν=±4(n1+n2)nplateaus. Furthermore, it is clear\nfrom Fig. 2(b) that when |¯µ|> t, the AB-stacked BLG\ncan also exhibit a 8 e2/h-step but the appearance of the\n8e2/h-steps is not periodical. The main findings here are\nsummarized in Table I.\nB. Hall conductivity vs chemical potential\nFig. 5 is a plot of σxyversus ¯µ, showing how the Hall\nplateaus are influenced by the magnetic field. It is clear\nfrom Fig. 4 that unlike the MLG and AB-stacked BLG,\na ¯ν= 0 plateau centered at ¯ µ= 0 appears for B= 127\nT in the AA-stacked BLG. However, when the condition\nof√\n2N/planckover2pi1υF/ℓB=tis reached by tuning the magnetic\nfield, the (N,+,−) and (N,−,+) LLs would be located\natE= 0. As a result, the ¯ ν= 0 plateaudisappearsand a\n8e2/h-step at ¯µ= 0 forms, like the AB-stacked BLG. In\nother words, the absence of the ¯ ν= 0 plateau needs the\nmagnetic field B=πt2/Nehυ2\nF, whereN= 1,2· ··. In\nFig. 5,B= 10.1 T andB= 15.2 T satisfy this condition\nwithN= 3 andN= 2, respectively. Thus, these curves\nlack the ¯ν= 0 plateau.\nIn addition, we note that a 8 e2/h-step occurs at ¯ ν=t\nforB= 10.1 T andB= 15.2 T. ForB= 12 T, all the\nLLs are nondegenerate and hence all the steps are of the\nheight of 4e2/h. However, if the ( N,−,+) and (o,+,−)\nLLs are degenerate at E=t, a 8e2/h-step appears at\n¯µ=t. This level degeneracy happens as the magnetic\nfieldB= 4πt2/Nehυ2\nF, whereN= 1,2,3,···.B= 10.1T\nandB= 15.2 T fit the condition with N= 3 andN= 2,\nrespectively, and thus a 8 e2/h-step appears at ¯ µ=t. We\nalsofind that the disappearanceof a zero-Hallconductiv-\nity plateau is always accompanied by the occurence of a\n8e2/h-stepat ¯µ=t, becauseif B=πt2/Nehυ2\nF,Bwould\nsatisfyB= 4πt2/N′ehυ2\nFwithN′= 4N. Interestingly,\nhere we find that the structure of the Hall plateaus of the\nAA-stacked BLG would be significantly affected by the\napplied magnetic field, which is quite different from the\nMLG and the AB-stacked BLG.\nIV. SUMMARY\nIn conclusion, we have calculated both the quantized\nHall conductivity and longitudinal conductivity of the\nAA-stacked bilayer graphene within linear response the-\nory by using Kubo formula. We find that the depen-\ndence of the Hall plateau of the AA-stacked BLG on themagnetic field is distinctly different from both the MLG\nand AB-stacked BLG as well as the conventional quan-\ntun Hall materials. In particular, the AA-stacked bilayer\ngraphene could possess the unique ¯ ν= 0 plateau, in con-\ntrast to other graphene materials such as monolayer and\nAB-stacked bilayer graphene. The shift of level anoma-\nlies due to interlayer hopping energy is attributed to be\nthe origin of the ¯ ν= 0 plateau. Nonetheless, the ¯ ν= 0\nplateau across ¯ µ= 0 would disappear if magnetic field\nB=πt2/Nehυ2\nF. Inaddition, wefindthatthedisappear-\nance of a zero-Hall conductivity plateau is alwaysaccom-\npanied by the occurence of a 8 e2/h-step at ¯µ=t. Fur-\nthermore, when |¯µ|= [(√n1+√n2)/(√n2−√n1)]t, the\nAA-stacked BLG lacks the ¯ ν=±4(n1+n2)nplateaus,\nwhich exist in the AB-stacked BLG. We also find that\nwhen Γ→0 and/planckover2pi14ω4\nc≫(¯µ+νt)4,σxx→0 if the Fermi\nlevel is not a few Γs above and below a Dirac point. This\nimplies that at the high magnetic field, the external elec-\ntric field cannot driveany currentfor ¯ µ<twhile the cur-\nrent would be perpendicular to the external electric field\nfor ¯µ>t. We hope that our predicted interesting charac-\nteristics of quantum Hall effect in the AA-stacked BLG,\nwhich are not seen in both the MLG and BLG as well\nas the conventional quantum Hall materials, would stim-\nulate experimental effort on fabricating the AA-stacked\nBLG and also on measurement of its transport property\nin the near future.\nACKNOWLEGEMENTS\nTheauthorsthankHung-YuYeh, Tsung-WeiChenand\nMing-Che Chang for valuable discussions. The authors\nalso acknowledge financial supports from National Sci-\nence Council and NCTS of Taiwan.\n[1]A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and\nA. K. Geim, Rev. Mod. Phys. 81, 109 (2009).\n[2]P. R. Wallace, Phys. Rev. 71, 622 (1947)\n[3]M. I. Katsnelson, K. S. Novoselov, And A. K. Geim, Nat.\nPhys.2, 620 (2006).\n[4]V.P. Gusynin, and S.G. Sharapov, Phys. Rev. Lett. 95,\n146801 (2005).\n[5]K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nM. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A.\nA. Firsov, Nature (London) 438, 197 (2005).\n[6]Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature\n(London) 438, 201 (2005).\n[7]K.S. Novoselov,Z. Jiang, Y. Zhang, S. V. Morozov, H.\nL. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P.\nKim, and A.K. Geim, Science 315, 1379 (2007).\n[8]E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805\n(2006).\n[9]K. S. Novoselov1, E. McCann, S. V. Morozov, V. I.\nFal’ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin\nand A. K. Geim, Nat. Phys. 2, 177 (2006).[10]M. Nakamura, L. Hirasawa, and K.-I. Imura, Phys. Rev.\nB78, 033403 (2008).\n[11]M. Aoki and H. Amawashi, Solid State Commun. 142,\n123(2007).\n[12]P. L. de Andres, R. Ram ´1rez, and J. A. Verg´ es, Phys.\nRev. B77, 045403 (2008).\n[13]P. Lauffer, K. V. Emtsev, R. Graupner, Th. Seyller, and\nL. Ley, Phys. Rev. B 77, 155426 (2006).\n[14]Z. Liu, K. Suenagam, H. Suzuura, P. J. F. Harris, and S.\nIijima, Phys. Rev. Lett. 102, 015501 (2009).\n[15]H. Bruus, and K. Flensberg, Many-Body Quantum The-\nory in Condensed Matter Physics (Oxford, New York,\n2007), Chaps. 6 and 11.\n[16]G. D. Mahan, Many-Particle Physics (Plenum Press,\nNew York, 2000), Chap. 4.\n[17]M. Iwasaki, H. Ohnishi and T. Fukutome, J. Phys. G:\nNucl. Part. Phys. 35, 035003 (2008).\n[18]A. Endo, N. Hatano2, H. Nakamura, and R. Shirasaki,\nJ. Phys. : Condens. Matter 21, 345803 (2009).\n[19]N. N. Lebedev, Special Functions and Their Applications8\n(Dover, New York, 1972), Chap. 1.\n[20]V. P. Gusynin, and S. G. Sharapov, Phy. Rev. B 71,\n125124 (1972).\n[21]H. Y. Yeh (private communication ).\n[22]M. Janβen, O. Veihweger, U. Fastenrath, and J. Hajdu,\nIntroduction to the Theory of the Integer Quantum Hall\nEffect (VCH, Weinheim, 1994).\n[23]D.Yoshioka, The Quantum Hall effect (Spring-Verlag,\nBerlin Heidelberg, 2002), Chap. 7.\n[24]E. H. Hwang, S. Adam, and S. Das Saima, Phys. Rev.Lett.98, 186806 (2007).\n[25]J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H.\nSmet, K. Von Klitzing, and A. Yacoby, Nat. Phys. 4, 144\n(2008).\n[26]Y. F. Hsu, and G. Y. Guo, Phys. Rev. B 81, 045412\n(2010).\n[27]A. K. Geim, Science 324, 1530 (2004).\n[28]L. Zhang, Y. Zhang, M. Khodas, T. Valla, and I. A.\nZaliznyak, arXiv:1003.2738 (2010)."    },    {        "title": "1710.11602v2.Quasiparticles_and_charge_transfer_at_the_two_surfaces_of_the_honeycomb_iridate_Na__2_IrO__3_.pdf",        "content": "Quasiparticles and charge transfer at the two surfaces of the honeycomb iridate Na 2IrO 3\nL. Moreschini,1,\u0003I. Lo Vecchio,2N. P. Breznay,2, 3S. Moser,1S. Ulstrup,1R. Koch,1J. Wirjo,2, 3\nC. Jozwiak,1K. S. Kim,4E. Rotenberg,1A. Bostwick,1J. G. Analytis,2, 3and A. Lanzara2, 3\n1Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA\n2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA\n3Department of Physics, University of California Berkeley, Berkeley, California 94720, USA\n4Department of Physics, Yonsei University, Seoul 03722, Korea\nDirect experimental investigations of the low energy electronic structure of the Na 2IrO3iridate insulator are\nsparse and draw two conflicting pictures. One relies on flat bands and a clear gap, the other involves dispersive\nstates approaching the Fermi level, pointing to surface metallicity. Here, by a combination of angle resolved\nphotoemission, photoemission electron microscopy and x-ray absorption, we show that the correct picture is\nmore complex and involves an anomalous band, arising from charge transfer from Na atoms to Ir-derived states.\nBulk quasiparticles do exist, but in one of the two possible surface terminations the charge transfer is smaller\nand they remain elusive.\nThe honeycomb iridate Na 2IrO3represents an ideal exam-\nple of a 5d5system with complete removal of the orbital de-\ngeneracy by the spin-orbit interaction [1], and for this rea-\nson has been object of considerable attention in recent years.\nMagnetically, it has been proposed to realize the Kitaev model\ndue to the hexagonal symmetry of the abplanes. The ground\nstate was instead shown to have a zigzag antiferromagnetic\n(AF) order [2–4], accounted for by direct 5 d-5doverlap [5],\nnext nearest-neighbor coupling [4, 6–9], or possibly inter-\norbital hopping [10]. Electronically, theoretical calculations\nhave tentatively categorized this compound as a topological\ninsulator [11, 12]. Transport and optical measurements sug-\ngest rather a Mott insulator picture [8, 13–15], and an ongoing\ndebate exists on whether spin-orbit coupling plays a decisive\nrole [5, 16, 17] or a collaborative one [7, 14, 18, 19] in opening\nthe gap.\nThe latter point essentially comes down to whether a rel-\nativistic approach with two effective J e\u000b=1/2 and J e\u000b=3/2\nlevels, as generally accepted in the description of the\nRuddlesden-Popper series iridates [20], holds for this honey-\ncomb lattice, or if spin-orbit coupling only “assists” a band\ngap. While experiments targeting the magnetic order have\nbeen numerous and exhaustive [2, 3, 13, 21, 22], measure-\nments of the electronic structure have not been as successful.\nUnlike in perovskite iridates, which with a more symmetric\ncrystal structure represent an ideal playground for angle re-\nsolved photoemission (ARPES) [23–29], photoemission data\nfor Na 2IrO3are limited to two instances [30, 31], owing to the\ndifficulty of obtaining sufficiently large cleaves.\nIn one case, only remarkably flat bands with <100 meV\nbandwidth were observed, and no quasiparticles. The Fermi\nlevel was pinned at the top of a valence band with a density\nof states (DOS) reminiscent of a pseudogap [30]. In the other\ncase, bands dispersing over more than 1 eV , more compat-\nible with typical 5dbandwidths, were found, and more im-\nportantly a weak intensity in the vicinity of the Fermi level,\nwhich suggested surface metallicity [31]. In this Letter we use\nspatially-resolved ARPES to determine the electronic struc-\nture of Na 2IrO3and its dependence upon the surface termina-\ntion. We find that a clear quasiparticle at the Fermi level canbe measured owing to charge transfer from the Na atoms, but\nonly for Na-terminated surfaces. For O-terminated surfaces,\nthe charge transfer between Na and Ir is reduced, leading to a\nstrong suppression of the quasiparticle and a large gap.\nIn Na 2IrO3the characteristic building blocks of iridates, the\nIrO6octahedra, are edge-sharing and form a layered stack-\ning alternating with pure Na layers. The crystal structure,\nwithC2=mspace group, is less symmetric than in cubic per-\novskites. As shown in Fig. 1(a), it presents a monoclinic c\naxis tilted by\u0018109\u000e[4]. The interlayer hopping terms are\nnegligible, as we will discuss later, and therefore we will re-\nfer throughout this paper to the hexagonal surface Brillouin\nzone (BZ) of the abhoneycomb lattice, with lattice constant\na= 5:427˚A. The surface magnetic unit cell, arising from the\nzigzag AF order, is rectangular and twice as small in kspace,\nbut as we will show the nonmagnetic BZ is the logical choice,\nsince the electron periodicity is clearly hexagonal.\nInspecting the crystal structure of Fig. 1(a), it appears ev-\nident that two cleavage planes are geometrically equivalent\nand therefore equally probable. They are indicated as A and\nB in the figure and throughout the rest of this Letter, and cor-\nrespond to an exposed Na 0:5IrO3and Na 1:5surface, respec-\ntively. In a standard band insulator/semiconductor, one should\nexpect from ARPES to observe the same bulk band structure\n(possibly with a different band bending) and different surface\nelectronic bands depending on the termination [32]. For this\nMott insulator we will draw instead a more intricate picture.\nThe presence of two different surfaces can be easily veri-\nfied by x-ray photoemission spectroscopy (XPS) of the Na 2 p\nlevels, as discussed in the following. Therefore these can be\nmapped in real space with a photoemission electron micro-\nscope (PEEM). The image in Fig. 1(b) has been recorded in\na dark field mode by accepting through the lens system only\nthe photoelectrons in the energy range indicated by the shaded\narea in Fig. 1(c), so that the intensity difference in XPS pro-\nvides the color contrast in the dark field image. We found\nseparate regions of a typical size of 10 to 40 \u0016m, but small\ncracks and spurious areas are almost always present. The core\nlevel data in Fig. 1(c) and the ARPES data shown next were\nobtained in a \u0016ARPES endstation with a spot size of \u001810-arXiv:1710.11602v2  [cond-mat.str-el]  6 Nov 20172\nFIG. 1: (a) crystal structure of Na 2IrO3. The A and B arrows in-\ndicate the two possible cleavage points for the crystal; (b) a PEEM\nimage measured on a sample cleaved in UHV . The energy window of\nthe photoelectrons selected by the aperture is indicated by the shaded\narea in (c); (c) Na 2 pspectra measured on the two surface termina-\ntions indicated in (a). The inset shows, for surface A, two spectra\ntaken at different emission angles; (d) Stacking of the top layers for\nboth A and B. Next to each layer we indicate the valence of the Na\natoms referred to a relative scale (see text) and the label of the corre-\nsponding peak in (c).\n20\u0016m, sufficient to separate the signal from the two domains.\nAt first we discuss the core level intensities in the context of\na structural model, and then we explain the binding energies\nand their correlation to the electronic structure. These two as-\npects give a self-consistent qualitative understanding of all of\nthe valence band data discussed later.\nExperimentally we observe for cleaving plane A two com-\nponents, as shown in Fig. 1(c). These two peaks, A 2and\nA3, are assigned to the outermost Na 0:5IrO3layer of the\noctahedral-terminated surface [Fig. 1(d)], and the next deeper,\npure Na 1:5layer, respectively. The higher intensity of A 3\nis accounted for by the number of Na atoms, which is three\ntimes larger in A 3than A 2, but this intensity ratio is reduced\nfrom 3 to\u00181.5 by the attenuation of the A 3signal due to its\ndepth. This assignment is confirmed by XPS at grazing emis-\nsion, in which A 3is further suppressed [inset of Fig. 1(c)].\nFor the B surface two weak photoemission peaks (B 2, B3) are\nobserved with a very similar branching ratio as A 2:A3, and\nare ascribed to Na atoms in the same respective layers, now\nburied by an additional Na 1:5layer, with corresponding peak\nB1largely dominant. The overall weaker intensities of (B 2,\nB3) compared to (A 2, A3) are explained simply by attenua-\ntion due to this new outer layer.\nThe binding energies of all the peaks can be understood by\nsimple considerations on the initial and final states of photoe-\nmission from ionic insulators [33], where the valence of the\ndifferent Na atoms are indicated in the model of Fig. 1(d). Na\nis least oxidized in the Na 1:5layers and most oxidized in the\nNa0:5IrO3layers, where it donates charge to the IrO 6octahe-dra. For the sake of argument we call these valences “0” and\n“+1”, respectively, although these should be understood on a\nrelative scale.\nFor the core level peaks B 3, A3and B 1(listed in order of\nincreasing binding energy) of the structurally and chemically\nequivalent Na 1:5layers, the variation of position corresponds\nto the progressively weaker screening response to the core\nhole as the surface is approached [33]. The remaining peaks\nB2and A 2, representative of the Na 0:5IrO3layers, are shifted\nto higher binding energy due to charge transfer from Na to the\nIrO6octahedra. The decrease in binding energy of A 2com-\npared to B 2is due to the lower valence of the former ( +\u000ewith\n\u000e<1) compared to the latter (+1) due to the altered environ-\nment of the surface octahedra which causes a smaller charge\ntransfer. We will show that this smaller charge transfer within\nthe Na 0:5IrO3layer at the A surface compared to B (and to\nthe bulk) has profound consequences on the bands observed\nby ARPES on this compound, since it hides the quasiparticles\nof the bulk electronic structure.\nFigure 2(a,b) shows the band structure measured at \u0000with\nthe sample oriented along \u0000Kfor both A and B terminations.\nAll the spectral features, similarly to the layered iridate per-\novskites [23, 26], proved to be hardly dispersive along the kz\ndirection normal to the surface, hence our choice to discuss\nthe data only in terms of the surface BZ. We set the photon\nenergy between 80 and 90 eV in all the measurements to max-\nimize the intensity of the valence states. Likewise, no obvious\nchange was observed as a function of temperature down to\n\u0018100K, where the sample started to show some charging.\nThe valence band presents clearly dispersive hole-like\nstates with maximum at \u00180.7 eV . These were seen in Ref. 31\nbut missed in Ref. 30 with a He lamp as excitation source, due\nto a small cross section. The flat bands seen in Ref. 30 are in-\nstead generally weak at higher photon energy [31]. Here, they\ngain a strong intensity at \u00180.7-0.9 eV , where they overlap with\nthe maximum of the hole-like parabolae. A hint of the differ-\nent orbital character of these two states – the flat bands and the\ndispersive parabolae – comes from the constant energy maps\nin Fig. 2(c,d). The hexagonal contours are 30\u000erotated with\nrespect to each other and in the higher energy one a three-fold\nmodulation is clearly visible.\nThe most apparent difference between Fig. 2(a) and\nFig. 2(b) is that the B surface hosts an additional electronic\nstate with respect to the A surface. A wide range map at \u001860\nmeV below the Fermi level [Fig. 2(e)] proves that this band is\ncentered at \u0000and no other potentially metallic states exist in\nother points of the reciprocal space. Some residual high in-\ntensity spots hint at a (p\n3\u0002p\n3)R30\u000ereconstruction, as seen\nby scanning tunneling microscopy (STM) [34]. Notice also\nthat clearly the fine details in the 0.7-1.2 eV binding energy\nrange are different between A and B, indicative of a different\nhybridization between the dispersive and the flat bands.\nWe propose that the determining factor in the formation of\nthe quasiparticles is the presence (B) or the absence (A) of the\noutermost Na 1:5layer. When present, this layer regenerates\nfor the top Na 0:5IrO3layer the same coordination as in the3\nFIG. 2: (a,b) valence band dispersion along ky, measured at \u0000for the A and B terminations. The dashed curves are guides to the eye for the\nhole-like dispersive states; (c,d) constant energy maps in the vicinity of \u0000at the two energies indicated by the dashed lines in (a); (e) wide\nrange constant energy map measured on surface B at 60 meV binding energy. The periodicity is the one of the nonmagnetic surface BZ with\n\u0000M =\u0019=a'0:58˚A\u00001and\u0000K'0:67˚A\u00001; (f) evolution of the EDC intensity at \u0000with Na deposition on surface A. Two different color\nscales have been used above and below the horizontal dashed line to enhance the weak low-energy state with respect to the other valence states;\n(g) The band structure at \u0000after Na and K deposition is shown on the left and right side of the image, respectively; (h) close up of the surface\nstate in surface B. The markers indicate the peak positions as extracted by the EDC fits and the error bars define a 98% confidence level; (i)\nimage obtained applying the curvature method to (h); (j) The EDC at kyindicated by the dashed line in (h) is compared to the Fermi level\nmeasured on the Cu sample holder. The data in (h-j) were measured at T'200K. Note that in all the ARPES figures ky, withydefined as in\nFig. 1(a), is the horizontal axis\nbulk and the quasiparticles are observed. The Ir-O octahedra\nare embedded between two Na layers, as opposed to A where\nthe absence of the Na overlayer appears to hinder the charge\ntransfer from Na to Ir within the Na 0:5IrO3plane and there-\nfore to hide the bulk low-energy state. A critical test of this\nhypothesis and of our structural model is to deposit Na atoms\non the A termination and see if the electronic structure of B is\nrecovered.\nAlkali atoms are frequently used with ARPES on metals or\nsemiconductors to access more states in the conduction band\nor the value of the band gap, respectively. Here, certainly due\nto a different pinning of the Fermi level on the bare surface,\nthe total energy shift upon Na deposition [see the slight move-\nment of the energy distribution curve (EDC) peak at \u00180.8 eV\nin Fig. 2(f)] is only \u0018100 meV , but the most striking effect is\nthe gradual appearance of the low energy band, as predicted\nabove. Notice that the intensity does not cross the Fermi level\nfrom above, as in a rigid shift of the conduction band with\nthe progressive addition of electrons, but forms instead a new\nelectronic state. This effect is not element-specific either, and\nseems rather linked only to the sp1valence of the Na atoms.\nBy physisorption of K, which is not present in the bulk mate-rial, exactly the same end point can be reached [see Fig. 2(g)].\nThe evolution of the Na core levels upon Na and K deposi-\ntion, as shown in the supplementary information [35], is fully\nconsistent with our discussion of Fig. 1.\nPrevious ARPES studies [30, 31] were conducted with\nlarger probe sizes and therefore averaged signals from both\nA and B terminations. Their interpretation of the low energy\nstates was constrained by a weaker signal to background ratio\nand some key details remained hidden. In Ref. 31 the dis-\npersion was uncertain and was tentatively assigned to a hole-\nlike band. No clear Fermi level crossing could be identified,\nin which case surface metallicity would arise only via ther-\nmally excited carriers. The DOS at the Fermi cut looked like\na pseudogap leading edge, similarly to Ref. 30 which saw no\nin-gap states. This in turn led to a scenario where quasiparti-\ncles would be suppressed by the interference between the two\nsublattices of the honeycomb structure [36].\nFigure 2(h) shows a more detailed measurement of the re-\ngion close to the Fermi level. In this image the dispersion is\nunambiguously electron-like. Notice that even at first sight\nit is incompatible with a quantum well state as observed, for\ninstance, on the surface of Sr 2IrO4upon K deposition (sup-4\nFIG. 3: XAS spectra at the N 7(full line) and K (dashed line) edges\nmeasured on A and B are shown in blue and red, respectively. The\nphoton energy on the horizontal axis has been offset by the binding\nenergy of the final state core hole, namely 62.3 eV for Ir 4 f7=2and\n529.3 eV for O 1 s; the inset shows a qualitative sketch of the DOS\nat the two surfaces after addition of charge to the nominal d5con-\nfiguration, as inferred by ARPES and XAS. Note that the symmetry\nbetween occupied and unoccupied states is clearly an oversimplifica-\ntion.\nplementary information of Ref. 28). The EDC fits super-\nposed to the image plot, as well as the curvature method treat-\nment shown in Fig. 2(i) [37], give a positive effective mass\nof\u00180:9me. It would be tempting to extract from the band\ndispersion a 2D charge density ( \u00181:5\u00021015cm\u00002), but this\nwould imply a Fermi level crossing which in fact does not oc-\ncur.\nIn Fig. 2(j) we plot the EDC at the kvector of the supposed\nFermi level crossing. Since the actual band dispersion cannot\nbe followed reliably outside the range fitted in Fig. 2(h), the\nEDC is integrated over a safe window of 0:08˚A\u00001. Despite\nthe relatively high ( \u0018200K) sample temperature, the slope of\nthe leading edge cannot be explained with the natural broaden-\ning of the Fermi-Dirac function, as shown by the Fermi level\nmeasured on the Cu sample holder at the same temperature.\nWhether the lineshape should be better referred to as a gap\nor a pseudogap [36], it cannot be explained by a surface state\nderived from the Na spelectrons.\nAn elegant way of exploring the character of the low-energy\nstate is by employing the chemical sensitivity of x-ray absorp-\ntion (XAS). Although the strength of the l!l\u00001excitation\nchannel is notoriously low at these energies [38], Ir allows\nfor the “unusual” transition 4f!5dwhich provides a rea-\nsonably good intensity. Therefore we used the N 7edge to\nresonantly probe the portion of the conduction band with Ir\ncharacter. The data for both the A and B surfaces, obtained\nby a standard total electron yield measurement, are shown in\nFig. 3, together with those measured at the O K ( 1s!2p)\nedge. The horizontal scale is shifted by the energy of the final\nstate core hole as measured by XPS.\nAt the threshold the two absorption spectra are very similar,\nexcept for a broader leading edge in the O K case due to trivialintrinsic (shorter core hole lifetime) and extrinsic (experimen-\ntal resolution) factors. However, the Ir N 7XAS presents a\nclear pre-edge feature for the B surface. The strong difference\nin the pre-edge intensity is reproducible over several cleaves\nand is in some sense surprising given that XAS in total elec-\ntron yield is generally considered rather bulk sensitive. The\npresence of an absorption channel below the threshold is the\neffect of a reorganization of the conduction states upon cre-\nation of the 4fcore hole. The interaction with the final state\ncore hole can shift the energy of the empty electronic state\nbelow the Fermi level and bring the XAS leading edge below\nthe nominal threshold. As a consequence, the shape of the\nXAS spectrum may bear little resemblance to the density of\nthe unoccupied 5dstates [39, 40].\nIn a band insulator or semiconductor the addition of charge\ncarriers fills defect states and induces a gradual shift of the\nbands, generally until a band edge is reached. Here, due to\nelectron correlations, the addition of carriers results in a qual-\nitatively different DOS, as sketched in the inset of Fig. 3 [41].\nThe states at the Fermi level in B provide new available ab-\nsorption channels at lower energy than the first unoccupied\nJe\u000b=1/2 level in A and yield the broad pre-edge peak which\nis vanishing in A [42]. In A the bulk DOS is hidden by the\nabsence of the Na layer and ARPES does not detect the low-\nest energy state. This explains the poor agreement between\nthe binding energy of the flat bands seen here and in Ref. 30,\nand the\u00180.4 eV gap measured by optics. Notice that in this\npicture the lowest energy state is populated by electrons trans-\nferred from the Na spto the Ir 5dstates, and therefore “in-\nherits” the Mott gap from the J e\u000b=1/2 electrons. Further sup-\nport for the presence of Ir 5dspectral weight in the vicinity of\nthe Fermi level state comes from resonant photoemission data\n[35].\nThe available first principle calculations, in order to find\nconsistency with the measured optical gap, tend to introduce\na rather high value for U, typically 3-4 eV [19, 30]. Probably\nfor this reason, they also tend to overestimate correlations\nand yield bandwidths of 0.2-0.3 eV [14, 16, 18, 30], at\nvariance with what is observed here both for the dispersive\nband bottom of the lowest energy state, and for the >1 eV\nbandwidth of the hole-like states. However, in both Refs. 14\nand 30 the onset of the optical conductivity is very gradual.\nIt is entirely possible that the actual gap could be slightly\nsmaller than reported and that calculations should be tuned\nto a lower value of U. Perhaps more importantly, our results\nshow the decisive role of the Na electrons in shaping the\nmagnetic and electronic structure of Na 2IrO3. The ground\nstate configuration may therefore have less d5character than\ngenerally assumed, with additional multiplet terms to take\ninto account aside from the d5initial state J e\u000b=1/2 doublet\nand J e\u000b=3/2 quartet.\nThis research used resources of the Advanced Light Source,\nwhich is a DOE Office of Science User Facility under Con-\ntract No. DE-AC02-05CH11231. I.L.V . and A.L. acknowl-\nedge support from the U.S. Department of Energy, Office of5\nScience, Office of Basic Energy Sciences, Materials Sciences\nand Engineering Division, under Contract No. DE-AC02-\n05CH11231 (Quantum materials KC2202). S.M. acknowl-\nedges support by the Swiss National Science Foundation un-\nder Grant No. P300P2-171221. S.U. acknowledges financial\nsupport from the Danish Council for Independent Research,\nNatural Sciences under the Sapere Aude program (Grant No.\nDFF-4090-00125). J.G.A acknowledges support towards the\nsynthesis of materials from the Department of Energy Early\nCareer program, Office of Basic Energy Sciences, Materials\nSciences and Engineering Division, under Contract No. DE-\nAC02-05CH11231. N.P.B. and J.G.A. acknowledge support\nfrom the Gordon and Betty Moore Foundations EPiQS Initia-\ntive through Grant GBMF4374\n\u0003Electronic address: lmoreschini@lbl.gov\n[1] W. Witczak-Krempa, G. Chen, Y . B. Kim, and L. Balents, Ann.\nRev. of Cond. Matter Phys. 5, 57 (2014).\n[2] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y .-J. Kim,\nH. Gretarsson, Y . Singh, P. Gegenwart, and J. P. Hill, Phys. Rev.\nB83, 220403 (2011).\n[3] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. Fernandez-\nBaca, R. Custelcean, T. F. Qi, O. B. Korneta, and G. Cao, Phys.\nRev. B 85, 180403 (2012).\n[4] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I.\nMazin, S. J. Blundell, P. G. Radaelli, Y . Singh, P. Gegenwart,\nK. R. Choi, et al., Phys. Rev. Lett. 108, 127204 (2012).\n[5] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.\n105, 027204 (2010).\n[6] I. Kimchi and Y .-Z. You, Phys. Rev. B 84, 180407 (2011).\n[7] K. Foyevtsova, H. O. Jeschke, I. I. Mazin, D. I. Khomskii, and\nR. Valent ´ı, Phys. Rev. B 88, 035107 (2013).\n[8] Y . Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku,\nS. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, 127203\n(2012).\n[9] M. Kargarian, A. Langari, and G. A. Fiete, Phys. Rev. B 86,\n205124 (2012).\n[10] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.\n110, 097204 (2013).\n[11] A. Shitade, H. Katsura, J. Kune ˇs, X.-L. Qi, S.-C. Zhang, and\nN. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009).\n[12] C. H. Kim, H. S. Kim, H. Jeong, H. Jin, and J. Yu, Phys. Rev.\nLett.108, 106401 (2012).\n[13] Y . Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010).\n[14] C. H. Sohn, H.-S. Kim, T. F. Qi, D. W. Jeong, H. J. Park, H. K.\nYoo, H. H. Kim, J.-Y . Kim, T. D. Kang, D.-Y . Cho, et al., Phys.\nRev. B 88, 085125 (2013).\n[15] Z. Alpichshev, F. Mahmood, G. Cao, and N. Gedik, Phys. Rev.\nLett.114, 017203 (2015).\n[16] B. H. Kim, T. Shirakawa, and S. Yunoki, Phys. Rev. Lett. 117,\n187201 (2016).\n[17] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y . Singh,\nS. Manni, P. Gegenwart, J. Kim, A. H. Said, et al., Phys. Rev.\nLett.110, 076402 (2013).\n[18] I. I. Mazin, H. O. Jeschke, K. Foyevtsova, R. Valent ´ı, and D. I.\nKhomskii, Phys. Rev. Lett. 109, 197201 (2012).\n[19] I. I. Mazin, S. Manni, K. Foyevtsova, H. O. Jeschke, P. Gegen-\nwart, and R. Valent ´ı, Phys. Rev. B 88, 035115 (2013).[20] B. J. Kim, H. Jin, S. J. Moon, J.-Y . Kim, B.-G. Park, C. S. Leem,\nJ. Yu, T. W. Noh, C. Kim, S.-J. Oh, et al., Phys. Rev. Lett. 101,\n076402 (2008).\n[21] H. Gretarsson, J. P. Clancy, Y . Singh, P. Gegenwart, J. P. Hill,\nJ. Kim, M. H. Upton, A. H. Said, D. Casa, T. Gog, et al., Phys.\nRev. B 87, 220407 (2013).\n[22] S. H. Chun, J.-W. Kim, J. Kim, H. Zheng, C. C. Stoumpos,\nC. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y . Singh, Y . Choi,\net al., Nat. Phys. 11, 462 (2015).\n[23] P. D. C. King, T. Takayama, A. Tamai, E. Rozbicki, S. M.\nWalker, M. Shi, L. Patthey, R. G. Moore, D. Lu, K. M. Shen,\net al., Phys. Rev. B 87, 241106 (2013).\n[24] Q. Wang, Y . Cao, J. A. Waugh, S. R. Park, T. F. Qi, O. B.\nKorneta, G. Cao, and D. S. Dessau, Phys. Rev. B 87, 245109\n(2013).\n[25] S. Moser, L. Moreschini, A. Ebrahimi, B. D. Piazza, M. Isobe,\nH. Okabe, J. Akimitsu, V . V . Mazurenko, K. S. Kim, A. Bost-\nwick, et al., New Journal of Physics 16, 013008 (2014).\n[26] L. Moreschini, S. Moser, A. Ebrahimi, B. Dalla Piazza, K. S.\nKim, S. Boseggia, D. F. McMorrow, H. M. Rønnow, J. Chang,\nD. Prabhakaran, et al., Phys. Rev. B 89, 201114 (2014).\n[27] A. de la Torre, E. C. Hunter, A. Subedi, S. McKeown Walker,\nA. Tamai, T. K. Kim, M. Hoesch, R. S. Perry, A. Georges, and\nF. Baumberger, Phys. Rev. Lett. 113, 256402 (2014).\n[28] Y . K. Kim, O. Krupin, J. D. Denlinger, A. Bostwick, E. Roten-\nberg, Q. Zhao, J. F. Mitchell, J. W. Allen, and B. J. Kim, Science\n345, 187 (2014).\n[29] J. He, T. Hogan, T. R. Mion, H. Hafiz, Y . He, J. D. Denlinger,\nS.-K. Mo, C. Dhital, X. Chen, Q. Lin, et al., Nature Materials\n14, 577 (2015).\n[30] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veenstra,\nJ. A. Rosen, Y . Singh, P. Gegenwart, D. Stricker, J. N. Hancock,\net al., Phys. Rev. Lett. 109, 266406 (2012).\n[31] N. Alidoust, C. Liu, S.-Y . Xu, I. Belopolski, T. Qi, M. Zeng,\nD. S. Sanchez, H. Zheng, G. Bian, M. Neupane, et al., Phys.\nRev. B 93, 245132 (2016).\n[32] A. Crepaldi, L. Moreschini, G. Aut `es, C. Tournier-Colletta,\nS. Moser, N. Virk, H. Berger, P. Bugnon, Y . J. Chang, K. Kern,\net al., Phys. Rev. Lett. 109, 096803 (2012).\n[33] E. Rotenberg and M. A. Olmstead, Phys. Rev. B 46, 12884\n(1992).\n[34] F. L ¨upke, S. Manni, S. C. Erwin, I. I. Mazin, P. Gegenwart, and\nM. Wenderoth, Phys. Rev. B 91, 041405 (2015).\n[35] Supplementary information is attached to this paper. It includes\nexperimental details, transport data, resonant photoemission\nand a detailed analysis of the XPS data upon alkali deposition.\n[36] F. Trousselet, M. Berciu, A. M. Ole ´s, and P. Horsch, Phys. Rev.\nLett.111, 037205 (2013).\n[37] P. Zhang, P. Richard, T. Qian, Y .-M. Xu, X. Dai, and H. Ding,\nReview of Scientific Instruments 82, 043712 (2011).\n[38] J. Min ´ar, H. Ebert, G. Ghiringhelli, O. Tjernberg, N. B.\nBrookes, and L. H. Tjeng, Phys. Rev. B 63, 144421 (2001).\n[39] A. Kotani and Y . Toyozawa, Journal of the Physical Society of\nJapan 35, 1073 (1973).\n[40] A. Kotani and Y . Toyozawa, Journal of the Physical Society of\nJapan 35, 1082 (1973).\n[41] The presence of two different DOS on the surface of Na 2IrO3\nwas already noticed by STM in Ref. 34. Here we reveal the\nconnection to the surface termination and show that the Na-rich\nsurface is the one with the narrower gap.\n[42] We believe that the small pre-edge seen there be due to deeper\nNa layers and to the larger probing depth of XAS with respect\nto ARPES."    },    {        "title": "1304.1983v1.Phenomenological_model_of_anomalous_magnon_softening_and_damping_in_half_metallic_manganites.pdf",        "content": "Phenomenological model of anomalou s magnon softening and damping in \nhalf-metallic manganites. \n \nA. Solontsov1,2,* \n \n1N.L. Dukhov Research Institute for Auto matics, Suschevskaya str., 22, 127055 Moscow, \nRussia, \n2State Center for Condensed Matter Physics,  M. Zakharova str., 6-3, 115569 Moscow, Russia \nDated: April 7, 2013 \n \n To describe anomalous zone-boundary soften ing and damping of magnons in manganites we \npresent a phenomenological two-fluid model contai ning ferromagnetic Fermi-liquid and non-Fermi-\nliquid components. The Fermi-liquid component ac counts for softening of zone-boundary magnons \nand for the Landau damping of magnons in the Stone r continuum arising at low frequencies due to \nzero-point effects. Coupling of the Fermi-liquid an d non-Fermi-liquid fluids yi elds conventional long \nwavelength magnons damped due to their coup ling with longitudinal spin fluctuations. \n \nPACS numbers: 75.30.Ds, 75.40.Gb, 75.47.Lx  \n \nMetallic ferromagnetic manganites with co lossal magnetoresistan ce besides numerous \npotential applications  demonstrate fascinating physics in cluding their anomalous magnetic \ndynamics below the Curie temperature . In the high-  manganites like LaCTCT 0.7Sr0.3MnO 3 \n( K) and La 378 CT= 0.7Pb0.3MnO 3 ( 355CT= K) magnon spectrum can  be well explained \nwithin the effective Heisenberg model1, which for manganites fo llow from the canonical \nVonsovsky and Zener description usually  addressed as the double-exchange  (DE) model2. \nIn the low-  manganites PrCT 0.63Sr0.37MnO 3 ( 301 CT= K), La 0.7Ca0.3MnO 3 \n( K), and Nd 238 CT= 0.7Sr0.3MnO 3 ( 198 CT= K) magnons exhibit appr eciate damping in the \nlong wavelength limit and on approaching the Brillouin zone boundary they are softened and \nstrongly damped3, which cannot be attributed to the He isenberg-type interactions. According \nto Ref. 3 at low temperatures magnon damping in these systems abruptly increases near the \nwavevector  (in the reciprocal lattice units, r.l.u.) where magnons merge with \nlongitudinal optical phonons. However, no signs  of magnon-phonon coupling could be seen in \nthe magnon spectra including an energy gap between magnon and phonon frequencies. For \nthe wavevectors  the magnon dispersion curves in  the [001] and [110] directions  \nfollow almost flat optical phonon dispersi ons, and higher-frequency magnons are wiped \nout1,3. Probably, it would be more adequate to speak about locking of magnon modes on the \noptical phonon branches and about magneto-v ibrational nature of the zone-boundary \nmagnons4. ~0 . 3ck\n0.3ck> 2\n Several mechanisms were proposed to a ccount for the anomalies of zone-boundary \nmagnon spectra in the low-  manganites using the effective Heisenberg Hamiltonian or the \nDE approach2 to describe i) four-magnon scattering1, ii) magnon-phonon scattering with \nemission (absorption) of a phonon by magnons5,6, iii) two-magnon scattering by itinerant \nelectrons without spin-flip7, iv) scattering of magnons by orbital fluctuations8. An additional \nmechanism due to v) scattering of magnons by longitudinal spin fluctuations (SF) with \nemission (absorption) of a SF by magnons first discussed in Ref. 9 could also describe \nmagnon anomalies in manganites and will be di scussed elsewhere. Here we only apply the \nlatter mechanism to account for the long wavelength magnon damping. CT\n First, we comment on the scattering mech anisms i)-iii) which result in magnon \ndamping with the model-independent temper ature and wavevector dependencies (model \ndependent parameters only affect the coup ling matrix elements). Magnon damping due to \nfour-magnon scattering processes i) were analyzed in the Heisenberg magnets10,11 and \nferromagnets with itinerant electrons12 more than half a century ago and was shown to be \nnegligibly small  at low temperatures (22 2\n0 ~l n [ / ( ) ]Bm T Tk T ω→→ kk = 0 ()BmkTω<< k= ), \nvanishing in the ground state ( 0T=), where  and k ()mωk are the wavevect or and magnon \nfrequency,  is the temperature. So , four-magnon scattering ca nnot contribute to the \nobserved anomalous magnon damping at low temperatures in manganites. T\n Damping of magnons due to ii) magnon- phonon scattering processes was also \ndiscussed long ago for the Heisenberg magnets13 and itinerant electron systems14,15 and was \nshown to be exponentially small at low temperatures14,15 \n (where 42\n0 ~ / [exp( / ( ) ] 0BD m T Tk Tϑω→→ kk =Dϑ is the Debye temperat ure), vanishing in the \nground state. Three decades later this result wa s independently reexamin ed with respect to \nmanganites using the DE5 and Heisenberg6 models. However, the authors5,6 analyzing this \nmechanism arrive at absolutely different results, e.g., they find finite magnon damping in the \nground state, which is in the strong di sagreement with the previous findings13-15. Besides, it \nwas reported finite magnon soft ening in the ground state6. These results are questionable and \nshould be reexamined16. \n Two-magnon scattering by itinerant electrons wi thout spin-flip iii) was first analyzed \nabout five decades ago and shown to give finite magnon damping in the ground state of \nferromagnets with itinerant electrons17 . Later this result was generalized to account for \nfinite temperatures and complicated Fermi surfaces18 and applied for the s-d-model for \nferromagnets19. Independently similar results were  obtained within the DE-model for \nmanganites7. 6~k 3\n A joint feature of the approaches7,8 describing the effects of scattering of magnons by \nelectrons and orbital fluctu ations is the continuous increase of magnon damping on \napproaching the Brillouin zone boundaries which disagr ees with the observe d abrupt rise in \nmagnon damping near1,3  r.l.u. Finally, all mentioned mechanisms i)-iv) are based \non the assumption of weak magnon damping although they are used to explain strong \ndamping of magnons on approachin g the Brillouin zone boundaries. ~0 . 3c kk≈\n On the other hand, the obvious reason for the abrupt increase of magnon damping \ncould be related to a possible intersection of  the magnon dispersion curv e with the continuum \nof Stoner excitations20 leading to the strong Landau dampi ng. However, Landau damping is \nruled out if to use the canonica l DE description of manganites or band structure calculations \nleading to their half-metallic character21. \n Here it is necessary to comment on the nature of half-metallic behavior of manganites. \nAccording to the mean field treatment of the one band DE exchange model itinerant gе \nelectrons of Mn are 100% polar ized and occupy only majority-spin subband due to strong \nHund's-rule coupling. W ithin this treatment ma nganites have only one spin channel for the \nelectron conductivity at the Fermi surface and s hould be considered as half-metallic. Indeed, \nspectroscopic measurements in La 0.7Sr0.3MnO 3 ( 378 CT= K) using spin-polarized \nphotoemission22 at K and inverse photoemission23 at 40 T= 100 T= K discovered 100% \npolarization which was questioned in Ref. 24.  On the other hand, methods involving tunnel \njunctions give an esse ntially less polarization25,26 ~ 80%. P\n Besides that, there are fundame ntal limitations for the half metallic behavior caused by \nmixing of electronic spins due to the effects of magnons and phonons coupled to the magnetic \nsystem and appearance of minority spin electron s at the Fermi surface at finite temperatures24. \nThis conclusion was generali zed by accounting for spin fluc tuations in the classical \napproximation, which were shown to lead to a significant decrease of the polarization above \n and disappearance of th e half-metallicity above27 . The main finding of \nRefs. 24 and 27 is that half-metallicity in mang anites may be at the be st expected at zero \ntemperature. However, this ignores quantum effects of zero-point excitations which mix \nelectronic spins and decrease spin  polarization in the ground st ate. In manganites zero-point \neffects are mainly caused by longitudinal SF in the continuum of electr on-hole pairs with the \nmajority spin polarization, which yiel ds the following rough estimation for the \npolarization24,27 , where28,29 0.4C T> T T\np Mx m0.7C T>\n0 ~/S PMM22 1 / 2\n.. [(4 ) 3 ]Sz Bμ ≈−−0(4 )B  and M xμ≈−  are \nthe values of spontaneous magnetization per unit cell with and w ithout account of SF,  is \nthe squared amplitude of longitudinal zero-point SF,28 and 2\n..zpm\nx describes manganite doping. To  4\nestimate  we use the recently measured by inelasti c neutron scattering squared amplitude \nof zero-point spin fluctu ations in iron pnictides30 2\n..zpm\n2\n..3.2zp Bm2μ≈  (this value is surprisingly \nsimilar in most of the strongly correlated electron systems), which for half-metallic \nmanganites gives 2\n..~0 . 5 3zp B m2μ. This allows us to estimate the ground state polarization of \nmanganites with  doping, 0.3x= 0.94P≈ . In real samples used in neutron scattering \nmeasurements the polarization may be less due to the effects of surfaces, interfaces, and \nstructural inhomogeneties24 which may lead to the observed value25,26 ~80%. All this breaks \ndown the concept of the half-m etallic character of manganite s even in the ground state and \nallow for the low-frequency Stoner excitations which may play an essential role in their \nmagnetic dynamics and give rise to the strong Landau damping of magnons. \nIn the present Letter we argue that the obs erved effects of an abrupt increase of \nmagnon damping at wavevectors  r.l.u. accompanied by their softening in manganites \nhave a magnetic origin and can be well e xplained in the mean-f ield approximation by \ncoupling of magnons with Stoner excitations aris ing due to the destruction of half-metallic \nground state of manganites. On the other ha nd, the observed sudden increase of magnon \ndamping in manganites may be viewed as a signature of Stoner ex citations and non-half-\nmetallicity of the ground state of manganites. 0.3ck>\nTo analyze the transverse magnetic dynamics and anomalies of the magnon spectra of \nmanganites we use a phenomenological approach  based on the concept of the generalized \nmagnetic susceptibility (, )χωk  which is associated with the transverse fluctuations of the \nmagnetic order parameter with an amplitude (, )mωk , where ω is the frequency of SF. This \napproach is based on th e assumption that all variables othe r than magnetic (e.g., individual \nFermi excitations, charge and lattice fluctuations) are integrated out,31,32 which means that all \nnon-magnetic collective modes including lattice vibrations adiabatic ally follow magnetic \n(spin) fluctuations which can  be viewed then as coupl ed spin-lattice fluctuations31. The \napproach is advantageous for manganites where spin-lattice coupling is believed to be strong1. \nTo calculate the susceptibility (, )χωk  we use a phenomenologi cal two-fluid model \ndecoupling the amplitude 12 (, ) (, ) (, )mm mω ω ω = + kkk  into two transverse components \n1(, )mωk  and 2(, )mωk . The first one describes a ferroma gnetic Fermi-liquid and is giving \nrise to \"bare\" magnons which are propaga ting in the long wavelength region (c kk<) and \nstrongly damped in the Stoner continuum ( ). The second is related to a non-Fermi-\nliquid component of the magnetization. Both co mponents are associated with the “partial” c kk> 5\ndynamical susceptibilities 1,2(, )χωk  of two fluids and descri be their linear response \n1,2 1,2 (, ) (, )(, )m B ωχω= kk k ω to the transverse magnetic field (, )Bωk . \n Allowing for the coupling of two fluids one  arrives at the following explicit form for \nthe generalized dynamical susceptibility of the total system in  the mean-field approximation \n(cf. Ref. 33) \n   12 1 2\n2\n12( ,) ( ,)2 ( ,) ( ,)(, )1( , ) ( , )χωχ ω λ χω χ ωχωλχ ωχ ω+ +=−kk k kkkk.  (1) \nHere λ accounts for coupling of fluids which we assume to be ferromagnetic and neglect its \nspatial and time dispersions, .0 constλ=> . The susceptibility (1) founds the basis for the \npresented here phenomenologica l two-fluid model for transv erse magnetic dynamics of \nmanganites. Similar models were used for the de scription of the neutr on scattering spectra in \nheavy fermion systems33,34 and for the interpretation of the density functional calculations in \niron pnictides35. \n It should be noted that Eq.(1) formally gives the dynamical susceptibility in the DE \nmodel if to consider 1(, )χωk  and 2(, )χωk  as \"partial\" susceptibilities of itinerant and \nlocalized moments subsystems and λ as a coupling constant de scribing interaction between \nthem36. Unlike the DE model the susceptibility (1) is not related to any microscopic \nHamiltonian and founds the basis for a phenome nological approach being an alternative to \nmicroscopic descriptions, which is justified belo w by the analysis of the anomalies of the \nmagnon spectra in manganites. \nHere we use a minimal phenomenological mode l in order to explain anomalies of the \nmagnon spectrum of manganites and take the partial Fermi-liq uid susceptibility 1(, )χωk  in \nthe following form, \n   0\n11\n0()(, ) ()() (, ) iωχωχωωω=−− Γkkkkk,   (2) \nwhere 0()ωk is the frequency of \"bare\" optical ma gnons not vanishing in the long wavelength \nlimit 00( ( 0) 0)ω ω==≠k . We emphasize that th ese \"bare\" magnons describing the poles of \nthe \"partial\" magnetic susceptibility (2) are not the normal magnon modes of the total system \nand due to strong spin-lattice c oupling in manganites they must be better viewed as magneto-\nvibrational modes.14 We emphasize that these \"bare\" ma gnons are defined by the pole of the \nHere \n   0()(, ) [ () ]()S\nflωωωθ ω ωωΓ= −kkkk    (3)  6\nis a conventional Fermi-liquid term account ing for the Landau damping in the Stoner \ncontinuum, ( )flωk is the characteristic frequency of transverse SF, the function [(S) ] θωω− k \nis unity inside the Stoner continuum (when ()Sωω> k) and zero otherwise, and ()Sωk is the \nlower boundary of the Stoner continuum. As we shall see later, 0()ωk appears to be close to \nthe frequencies of longitudina l optical phonons, confir ming strong spin-latti ce coupling in the \nlow-  manganites.  CT\n For the non-Fermi-liquid component of the magnetic susceptibility we use a static \napproximation, 2(, ) ()2 χωχ≈ k k. Here we do not consider th e magnetic relaxation aside the \nStoner continuum which we account for later. \nThe static partial susceptibilities 1,2()χk we present in the form \n     ,    (4) \nwhere the terms  account for their spatial dispersion which in the long wavelength \nlimit we assume to be isotropic and quadr atically dependent on the wavevector, 11\n1,2 1,2 1,2 () [ 1 () ] a χχ−−=+k k\nk1,2()ak\n     ,      ( 5 )  2\n1,2 1,2 () ( )aξ=k\nwhere 1,2ξ are correlation lengths. \n Using Eq. (2) and assuming that ( )flωω<< k we present the generalized dynamical \nsusceptibility (1) near the magnon disper sion in the following explicit form \n  1()(, ) ()()() ()m\nmziωχωχωω τ−=−−kkk kkk,    (6) \nwhich  has a pole at the magnon frequency \n    ,    (7) 2\n01 2 () () [ 1 () () ]mωω λ χ χ=−kk k k\ndescribing  is a true normal mode of our two-fluid system. Here \n    12 1 2\n2\n12() () 2 () ()()1( ) ( )χχλ χ χχλχ χ++=−kk kkkkk    (8) \nis the static transverse susceptibility, \n    2\n2\n1\n12 1 2[1 2 ( )]() ()() () 2 () ()zλχχχχ λ χ χ+=++kkkkk k k ,  (9) \nis the relative weight of the magnon mode, and  describes damping of \nmagnons in the Stoner continuum. 1() (, () )m τω−=Γkk k\n The factor  plays the role of an enhancement factor for the \nsusceptibility (8), and vanishes in the long wavelength limit, 2\n12 [ 1 () () ]λχ χ− kk 7\n     2\n12 1λχχ 0 −=.      ( 1 0 )  \nThis equality could plays a role of the ma gnetic equation of state if one would know the \ndependencies of 1,2χ on the magnetic order parameter. \n The important consequence of Eq.(10) is the gapless character of the magnon \nspectrum (7) which with account of Eq.(4) takes the form \n    121 2\n0\n12() () () ()() ()[1 ( )][1 ( )]maaa a\naaωω++=++kkkkkkkk,   (11) \nand vanishes in the long wavelength limit \n    2\n1,2 1,2 () ( ) 1aξ≈<< kk ,     ( 1 2 )  \nwhere the magnon dispersion is isotropic and quadratic \n       ( 1 3 )  22\n01 2 ( ) ( )[( ) ( ) ]m D ωω ξξ≈+kk kk2≈k\n2) and 22\n01(0 ) ( Dω ξξ ==+ k  is the magnon stiffness. Eq.(13)  agrees with the measured \nisotropic long wave-vecto r spectrum of manganites1. As it follows from Eq. (9) the weight \nfactor ()zk in this limit equals unity ( ). (0 )z→≈k 1\n It should be noted that within our two-fluid model the high-frequency mode \n0()ωω= k of the Fermi-liquid fluid is transferre d into the acoustical magnon (11) due the \nexchange coupling of the fluids. In a sense, this  is similar to the transfer of the plasma ion \nmode in metals into the ac oustical longitudinal phonon due to  the electron Coulomb screening \nof ions. \n The magnon spectrum and damping given by Eqs. (6) and (11) may account for \nanomalous magnon softening and damping on a pproaching the Brillouin zone boundaries \nobserved in manganites1,3. To explain the anomalies of the magnon spect rum in manganites \nwithin our phenomenological model one should take into account anisot ropic dependencies of \n0()ωk, ( )flωk, and  on the wavevector, which must be inferred from the measured \ndata.  1,2()ak\nDue to the lack of the experimental data for the magnon spectra in the whole Brillouin \nzone we shall limit ourselves to the qualitativ e analysis of magnon anomalies near the \nBrillouin zone boundaries along several direc tions. Namely, we analyze magnon spectrum \nalong the [001] and [110] directions in the low-  manganites LaCT 0.7Ca0.3MnO 3, \nPr0.63Sr0.37MnO 3, and Nd 0.7Sr0.3MnO 3. \n This allows us to make further approximations minimizing the number of parameters \nthat should be taken from experiments. Namely , we neglect the spatial dispersion of the  8\nfrequency ( ) (0)flf l fl ωω≈=k ω1  and the partial susceptibility 1()χχ≈k  (or set 1() 0a=k  and \n10ξ=), and use an approximate equality  throughout the Brillouin zone. 2\n22() ( )aξ≈kk\n Then the magnon frequency (11) is given by the following explicit formula \n     2\n2\n0 2\n2()() ()1( )mξωωξ=+kkkk.     ( 1 4 )  \nIt has a quadratic dispersion in the long wavelength limit  with the magnon \nstiffness 2\n2()ξ<<k 1\n2\n02 Dωξ= , softens at 1\n2~skξ−= k  and saturates 0 () ()mωω≈k k\n1 at short wavelengths \n.  2\n2()ξ>>k\n Magnon damping in Eq.(6)  \n    2\n1 0()() [ () () ]mS\nflωτθ ωω−=−kkk ωk    ( 1 5 )  \nexhibits a discontinuous jump when th e magnon dispersion curve crosses the Stoner \ncontinuum boundary at the wavevector c kk= defined by \n      () ()mSωω=k k,     ( 1 6 )  \nIn microscopic descriptions magnon dispersi on possesses logarithmic softening when crosses \nthe Stoner continuum boundary36, so, one should expect that the wavevector  is close to the \nvalue ck\nsk at which the magnon dispersion demonstrate softening. \n In the low-  manganites PrCT 0.63Sr0.37MnO 3, La 0.7Ca0.3MnO 3, and Nd 0.7Sr0.3MnO 3 the \nmagnon dispersions measured by inelastic neutron scattering1 are quadratic up to the \nwavevector  r.l.u. At higher wavevectors ma gnon energies exhibit softening and \nsaturation, which is strongly anisotropic being about 20 an d 40 meV along [001] and [110], \nand are close to the energies of longitudinal optical phonons in th ese directions. It should be \nnoted that in Nd~0 . 2sk\n0.7Sr0.3MnO 3 zone boundary magnons in the [110]  direction are overdamped \nto be observed experimentally3. To account for the anisotropy of magnon softening and \nsaturation in the low-  manganites we are to allow for the wavevector dependence of the \n“bare” magnon frequency of the ferromagnetic fluid CT\n   ,     ( 1 7 )  2\n02\n2\n01 2\n2\n2, 2, ( ) 1 ,\n() ,  ( ) 1 ,  [ 0 0 1 ]\n( ) 1,  [110]  ωξ\nωω ξ\nωξ⎧ <<\n⎪≈> > ⎨\n⎪>>⎩k\nkk\nk 9\nwhere we take the saturation energies in the [001] and [110] directions 122ω≈=  and \n240ω≈=  meV from Refs. 1 and 3, which ap pear to be close to the energies  and \n meV of the optical  longitudinal phonons3.  127Ω≈=\n249Ω≈=\n It should be emphasized that the frequencies 1,2ω are the magnon (or magneto-\nvibrational) frequencies defi ned by the pole of the transverse magnetic susceptibility (6). \nTheir proximity to the optical phonons frequencies 1,2Ω probably results from the spin-lattice \ncouplings incorporated into magnetic dynami cs of our model. This proximity in our \nphenomenological approach is not necessary fo r the description of magnon softening and \nbroadening unlike the microscopic approach5,6 assuming an intersection of magnon and \nphonon dispersions. \n First, we analyze the long wave length magnon spectrum in the low-  manganites \nPrCT\n0.63Sr0.37MnO 3, La 0.7Ca0.3MnO 3, and Nd 0.7Sr0.3MnO 3 basing on Eq.(11).  To minimize the \nnumber of the parameters of our model we assume the energy 0ω= in Eq.(17) to be equal to \nits minimal zone-boundary value 01 22 ωω=≈ ==  meV. Using the measured magnon stiffness \n meVA2 one estimates the correlation length 2\n02 165 Dωξ=≈==22.74ξ≈  A and the \nwavevector  r.l.u. which agrees with the measu red vector where starts softening \nof the magnon spectra1,3 and is close to the value 1\n20.2skξ−=≈\n0.3ck≈  marking the abrupt increase of \nmagnon damping in [001] and [1 10] directions. From the anal ysis of magnon damping in the \n[001] direction we can estimate the SF frequency flω in the Stoner continuum. Using the \nequality 12\n1/fl τωω−=  following from Eq.(15) and the measured abrupt increase of magnon \ndamping from ~4 to 12 meV at 0.3ck≈  we get the estimate 60flω≈=  meV which looks \nreasonable for SF in itinerant electron magnets36 and satisfies the assumption ( )fl mωω>> k \nwe used in Eq.(6). \n Up to now we analyzed magnon damping and softening on the basis of the mean-field \nmagnetic susceptibility (6) which is the linear approximation to non-linear magnetic \ndynamics32  accounting for mode-mode couplings and gi ve rise to the mentioned above i)-v) \nscattering mechanisms.  \n Finally, we comment on the long wa velength magnon damping in the low-  \nmanganites which is also anomalou sly high and at the wavevector  r.l.u. is about 4 \nmeV. The most effective mechanism of magnons in magnets with itinerant electrons (besides \ndamping in the Stoner continuum) is cau sed by the scattering processes with emission CT\n0.2sk≈ 10\n(absorption) of longitudinal SF by magnons which was first proposed  in Ref. 37. In the Born \napproximation it gives the following explicit expression for damping of low-temperature \n( ()BmkTω<< k) long wavelength magnons9 \n    2\n1 () 4(, ) ~5m\nSFTωτμω−=kk4k     ( 1 8 )  \nwhere μ is the magnetic moment of the unit cell in the units of th e Bohr magneton and SFω \nis the characteristic frequency of longitudinal SF (different from the analogous frequency flω \nof transverse SF in the Stone r continuum). Comparing magnon da mping (18) with the inverse \nlifetime due to, e.g., magnon-electron scattering18 one finds that the latter contains an \nadditional small factor  making magnon-electron scatteri ng mechanism less effective at \nlong wavelengths. 2~k\n Eq.(18) is in reasonable agreement w ith the wavevector dependence of long \nwavelength damping in the low-  manganites1,3. Using the values3  meV and CT14τ−≈=\n14mω≈=  meV for magnons at the wavevector 0.3ck≈  r.l.u. in the [001] direction and the \nmagnetic moment 3.7μ≈ , we find the energy of longitudinal SF 26SFω≈=  meV. The \ndifference between energies 1\n1~flωχ−=  and 1~SF lωχ−=  of the transverse and longitudinal SF \nproportional to the appropriate in verse magnetic susceptibilities37 should be due to the \nexchange enhancement of the longitudinal susceptibility lχ compared to the \"partial\" \ntransverse one 1χ, 1 /~ / 2 .fl SF l 3 ωωχ χ ≈ , which looks reasonable. \n To conclude, our main finding is that zone-boundary magnon anomalies in the low-  \nmanganites cannot be understood without using a concept of the Ston er excitations which \nshould arise even in the ground st ate of half-metallic ferromagnets  due to mixing of electronic \nspins by zero-point longitudinal spin fluctuatio ns. We also argue that softening of magnons \nnear the Brillouin zone boundari es is closely related to thei r magneto-vibrational character \nand in our minimal description can be well desc ribed within a two-fluid model containing a \nferromagnetic Fermi liquid and non-Fermi-liquid components. Aside the Stoner continuum \nmagnon damping can be described by various  mode-mode scattering processes among which \nprocesses with emission (absorption) of long itudinal SF by magnons are the most important. CT\n The work was supported by the State Atomic Energy Corporation of Russia \n“ROSATOM”. \n* E-mail address : asolontsov@mail.ru . \n \n  11\nLiterature \n1J. Zhang, F.Ye, P. Dai et al., J. Phys.: Condens. Matter 19, 315204 (2007). \n2S.V. Vonsovsky, J. Phys. USSR 10, 468 (1946); C. Zener, Phys. Rev. Lett.  82, 403 (1951). \n3P. Dai et al., Phys. Rev. B 61, 9553 (2000). \n4P. Reutler et.al., J. Magn. Magn. Mater. 242-245, 689 (2002). \n5N. Furukawa, J. Phys. Soc. Jap. 68, 2522 (1999). \n6L.M. Woods, Phys. Rev. B 65, 014409 (2001) \n7D.I. Golosov, Phys. Rev. Lett. 84, 3974 (2000). \n8S. Krivenko et al., J. Magn. Magn. Mater. 272-276, 458 (2004). \n9A.Z. Solontsov and A.N.Vasil'ev, Phys. Lett. A  177, 362 (1993). \n10V.N. Kashcheev and M.A. Krivoglaz, Sov. Phys. - Solid State 3, 117 (1961). \n11V.G. Vaks, A.N. Larkin, and C. A. Pikin, Sov. Phys.-JETP 26 (1968) 647. \n12J. Morkowski, Acta Phisica Polonica A 43, 809 (1973). \n13A.I. Akhiezer J. Phys. USSR, 10, 217 (1946); A.I. Akhiezer, V. G. Bar'akhtar, and S.V. \nPeletminskii, Spin waves  (North Holland Publ. Co., Amsterdam, 1968). \n14J. Morkowski, Phys. Lett. A 21, 146 (1966). \n15H. Yamada, J. Phys. Soc. Jap. 41, 753 (1976). \n15N. Furukawa (Ref. 5) and L.M. Woods (Ref. 6) in the calculations of magnon damping set \nthe occupation numbers of magnons and phonons in the integrands of th eir expressions (2) \nand (26) for the self-energy of magnons, r espectively, to zero, which was shown by Y. \nMorkowski (Ref. 14) and H. Yamada (Ref. 15)  to be inadequate and leads to a strong \ndisagreement between their resu lts. The same inappropriate a pproximation was made by L.M. \nWoods (Ref. 6) in his calcu lation of magnon softening.  \n17E.D. Thompson, J. Appl. Phys. 31, 1133 (1965). \n18V.P. Silin, A.Z. Solontsov, Fiz. Met. Metalloved.  52, 231 (1981). \n19V. Yu. Irkhin and M.I. Katsnelson,  J. Phys. C: Solid State Phys. 18, 4173 (1985). \n20X. Wang, Phys. Rev. B 57, 7427 (1998). \n21P.K. de Boer, R.A. de Groot, Comput. Mater. Sci. 10, 240 (1998). \n22J.H. Park et al., Nature 392, 794 (1998). \n23R. Bertacco et al., J.  Magn. Magn. Mater. 242-245, 710 (2002). \n24P.A. Dowben and R. Skomski, J. Appl. Phys. 95, 7453 (2004). \n25R.J. Soulen, Jr et al., Science  282, 85 (1998). \n26S.K. Upadhyay et.al., Phys.Rev. Lett. 81, 3247 (1998). \n27H. Itoh, T. Ohsawa, and J. Inoue, Phys. Rev. Lett. 84, 2501 (2000).  12\n28A.Z. Solontsov and D. Wagner, J. Phys.: Condens. Matter 6, 7395 (1994); Phys. Rev. B 51, \n12410 (1995). \n29A.J. Millis, P.B. Littlewood, and B.I. Shraiman, Phys. Rev. Lett. 74, 5144 (1995). \n30M. Liu et al.,. arXiv: 1202.2827. \n31V.P. Silin and A.Z. Solontsov, Sov. Phys. - JETP 62, 829 (1985); Sov. Phys. - JETP 65, \n1013 (1987). \n32A. Solontsov, Int. J. Mod. Phys. B 19, 3621 (2005). \n33N.R. Bernhoeft and G.G. Lonzaric h, J. Phys.: Condens. Matter, 7, 7325 (1995). \n34N. Bernhoeft et al., In: Itinerant Electron Magnetism: Fluctuation Effects , edited by D. \nWagner, W. Brauneck, and A. Solontsov (Dor drecht, Kluwer Academ ic Publishers, 1998). \n35A. Smerald, N. Shannon, and H.H. Wills, EPL, 92, 47005 (2010). \n36V.P. Silin and A.Z. Solontsov, Sov. Phys.- JETP 42, 1089 (1975). \n37T. Moriya, Spin  Fluctuations in Itin erant Electron Magnetism  (Berlin, Springer, 1985).  "    },    {        "title": "2208.12456v2.Quasiparticle_characteristics_of_the_weakly_ferromagnetic_Hund_s_metal_MnSi.pdf",        "content": "Quasiparticle characteristics of the weakly ferromagnetic Hund’s metal MnSi\nYuan Fang,1,∗Huali Zhang,1,∗Ding Wang,1,∗Guowei Yang,1Yi Wu,1Peng Li,1\nZhiguang Xiao,1Tianyun Lin,1Hao Zheng,1Xiao-Long Li,2Huan-Hua Wang,3\nFanny Rodolakis,4Yu Song,1Yilin Wang,5Chao Cao,1and Yang Liu1, 6, 7,†\n1Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China\n2Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute,\nChinese Academy of Sciences, Shanghai 201204, China\n3Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China\n4Advanced Photon Source, Argonne National Laboratory,\n9700 South Cass Avenue, Argonne, Illinois 60439, USA\n5Hefei National Laboratory for Physical Sciences at Microscale,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n6Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China\n7Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China\n(Dated: October 17, 2022)\nHund’s metals are multi-orbital systems with 3 dor 4delectrons exhibiting both itinerant character\nand local moments, and they feature Kondo-like screenings of local orbital and spin moments, with\nsuppressed coherence temperatures driven by Hund’s coupling JH. They often exhibit magnetic\norder at low temperature, but how the interaction between the Kondo-like screening and long-range\nmagnetic order is manifested in the quasiparticle spectrum remains an open question. Here we\npresent spectroscopic signature of such interaction in a Hund’s metal candidate MnSi exhibiting\nweak ferromagnetism. Our photoemission measurements reveal renormalized quasiparticle bands\nnear the Fermi level with strong momentum dependence: the ferromagnetism manifests through\npossibly exchange-split bands (Q1) below TC, while the spin/orbital screenings lead to gradual\ndevelopment of quasiparticles (Q2) upon cooling. Our results demonstrate how the characteristic\nspin/orbital coherence in a Hund’s metal could coexist and compete with the magnetic order to form\na weak itinerant ferromagnet, via quasiparticle bands that are well separated in momentum space\nand exhibit distinct temperature dependence. Our results imply that the competition between the\nspin/orbital screening and the magnetic order in a Hund’s metal bears interesting similarities to the\nKondo lattice systems.\nUnderstanding weak itinerant ferromangets represents\nan outstanding question in condensed matter physics\n[1, 2]. In these materials, the ordered moments in the\nferromagnetic (FM) phase are much smaller than the lo-\ncal moments inferred from the Curie-Weiss law above\nthe Curie temperature TC, which indicates that mod-\nels based on the local-moment physics (e.g. the Heisen-\nberg model) do not hold. On the other hand, a simple\nitinerant band picture is also inadequate, as it does not\ncapture effects due to spin fluctuations and electronic\ncorrelations. As an archetypal weak ferromagnet, MnSi\nplays an important role in the understanding of weak\nferromagnetism and validating the famous spin fluctua-\ntion theory by Moriya [3]. Nevertheless, the importance\nof electron correlation and the underlying mechanism of\ncoexisting local moments and itinerant electrons remain\nopen questions.\nIn multi-orbital (3 dor 4d) weak ferromagnets, strong\nelectron correlations could arise from the inter-orbital\nHund’s coupling JH, leading to the so-called Hund’s\nmetal (HM) [4–8]. The HM physics is thought to play\nan important role in many correlated electron systems,\nincluding iron pnictides, ruthenates, transition metal\nchalcogenides, etc [4–15]. A recent study from inelastic\nneutron scattering (INS) and dynamic mean-field theory(DMFT) suggested that MnSi is a HM exhibiting strong\norbital and spin fluctuations [16], which can be crucial for\nits non-Fermi-liquid behaviors over a large phase space\n[17–20]. It was further proposed that the orbital coher-\nence scale might be smaller than the spin one in MnSi,\nimplying possibly an unconventional type of HM as a\nresult of strong electron correlations [16]. A key charac-\nteristic of the HM is the quasiparticle (QP) bands that\nemerge gradually at low temperature, due to the low co-\nherence temperature, but these QP bands remain elusive\nin MnSi. In addition, how the QP coherence as a result\nof the HM physics interacts with the weak FM is an-\nother interesting question. While a previous study from\nangle-resolved photoemission spectroscopy (ARPES) on\nbulk MnSi(001) suggested weak electron correlation [21],\nanother ARPES study on thick MnSi(111) films empha-\nsized the importance of Fermi surface (FS) nesting in\ndriving the strong magnetic fluctuations [22]. Neverthe-\nless, neither the FM exchange splitting below TCnor the\ncharacteristic QP bands as a result of the HM physics\nhave been identified so far.\nHere, combining thin film growth by molecular beam\nepitaxy (MBE) and in-situelectronic structure measure-\nments by ARPES, we demonstrate the QP characteristics\nof the weakly ferromagnetic HM MnSi. Our MnSi filmsarXiv:2208.12456v2  [cond-mat.str-el]  14 Oct 20222\nFIG. 1. Characterization of MnSi films. (a) RHEED patterns\nof the Si(111) 7x7 surface and a typical MnSi film. (b,c) X-\nray diffraction scans of a thick MnSi film near the MnSi Bragg\npeak centered at (H,K,L)=(2.069,0,3.582), where H,K,L are\ndefined in units of Si substrate [23]. (d) XAS spectrum near\nthe MnLedge taken at 29 K. (e) Resistance vs temperature\nfor a MnSi film, showing a kink at TC∼45 K.\nwere grown on Si(111) substrates and transferred under\nultrahigh vacuum to a Helium-lamp ARPES system for\nelectronic structure measurements. Details can be found\nin the Supplemental Material [23] (see also references\n[24–26] therein). Figure 1(a) shows the reflection high\nenergy electron diffraction (RHEED) patterns from the\nstarting Si(111)-7x7 surface and a as-grown thick MnSi\nfilm, confirming epitaxial growth of high-quality MnSi\nfilms. The epitaxial growth is further confirmed by ex-\nsitu X-ray scattering measurements (Fig. 1(b,c)), show-\ning that the in-plane and out-of-plane lattice constants\nare∼3.243 ˚A and ∼7.872 ˚A, respectively [23]. This in-\ndicates that the epitaxial MnSi film undergoes a tensile\nstrain of ∼1% compared to bulk MnSi, although it does\nnot fully match with the underlying Si substrate, similar\nto previous studies [22, 27]. The strain gives rise to an en-\nhancedTC∼45 K (compared to TC∼30 K in bulk MnSi),\nas evidenced by the characteristic kink in the resistiv-\nity vs temperature data shown in Fig. 1(e). Due to the\nnon-centrosymmetric crystal structure, there is a weak\nDzyaloshinskii-Moriya (DM) interaction which leads to\na very small rotation ( ∼5◦) of the in-plane Mn moments\nbetween the adjacent (111) planes [28]. However, the DM\ninteraction is too weak to generate any obvious effect on\nthe observed QP bands (see Fig. S1 in [23]), and there-fore is ignored in this paper. Measurements from X-ray\nabsorption spectroscopy (XAS) near the Mn Ledge indi-\ncate that the ground state is mixed-valent with both 3 d5\nand 3d6configurations, similar to bulk MnSi [29]. The\npresence of the 3 d6configuration is unusual for Mn and\nis likely caused by hybridization with Si orbitals. Note\nthat the 3d6configuration is one electron more than half\nfilling, implying an important role of Hund’s physics [16].\nFigure 2(a) shows the energy-momentum dispersion\ntaken with He I (21.2 eV) and He II (40.8 eV) photons,\nrevealing dispersive QP bands near the Fermi level ( EF).\nFor comparison, Fig. 2(b) shows the calculated band\nstructure from density functional theory (DFT) overlaid\non top of the ARPES data, with kz= 0.4 ˚A−1and 1.196\n˚A−1based on an estimated inner potential of ∼12.8 eV\n(see Fig. S2 in [23] for more details). The Mn moment\nwas set to the experimental value of 0.4 µB/Mn in DFT\ncalculations [30], which yields a FM exchange splitting of\n∼0.2 eV (see Fig. S3 in [23]). The overall valence bands\nfrom the 21.2 eV data can be partially explained by the\nDFT calculations, particularly the broad parabolic band\nnear -1.2 eV (labelled B1) centered at ¯Γand the hole-like\nbands near EF(labelled B2) centered at ky∼1.12 ˚A−1,\ni.e., ¯Γin the second surface Brillouin zone (BZ). For the\n40.8 eV data, different set of bands (labelled B3,B4 in\nFig. 2(a)) can be observed. The obvious spectral differ-\nence between the first and second surface BZs confirms\nthat the observed spectral features are bulk states, as\nthe bulk BZ is twice as large as the surface BZ (see Fig.\n2(b)). Despite these similarities between ARPES results\nand DFT calculations, the ARPES spectra also show\nwell-defined flat bands right near EFextending over a\nlarge momentum region ( ±0.6˚A−1), marked by black ar-\nrows in Fig. 2(c,d), which cannot be easily accounted for\nby DFT calculations. We mention that DMFT calcula-\ntions indeed suggest enhanced effective mass (or flatness)\nof quasiparticle bands near EFcompared to DFT [16],\nalthough they cannot reproduce quantitatively the pro-\nnounced flat bands observed experimentally (note that\nDMFT calculations in [16] were performed at room tem-\nperature). In addition, the experimental constant energy\nmaps (Fig. 2(e)) show obvious deviations from the DFT\ncalculations (see Fig. S4 in [23]). Such a discrepancy im-\nplies appreciable band renormalization near EF, suggest-\ning that correlation effects are likely important in MnSi.\nThe flat QP bands near EFare consistent with the en-\nhanced effective mass from de Haas-van Alphen [31, 32]\nand specific heat measurements [33, 34]. Note that the\nQP bands are well-defined near EFand become much\nbroader away from EF, in agreement with the strong QP\nscattering as a result of correlation effect.\nTo understand the electron correlation from the HM\nphysics and the origin of FM, the temperature evo-\nlution of the QP bands is crucial. Fig. 3(a) shows\nthe momentum-dependent energy distribution curves\n(EDCs) along the ¯Γ-¯Mdirection above and below TC,3\nFIG. 2. ARPES spectra of MnSi films taken at ∼6 K. (a) ARPES data of a MnSi film taken with 21.2 eV and 40.8 eV photons,\nmeasured along ¯Γ-¯M. (b) Results from DFT calculations overlaid on top of ARPES data. The kzvalues are based on an\nestimated inner potential of ∼12.8 eV. (c,d) Zoomed-in view of the QP bands near EF. Black arrows highlight flat QP bands\nnearEF. (e) Constant energy maps at various energies using 21.2 eV photons. The green hexagon at the bottom marks the\nsurface BZ.\nwhere clear spectral changes across TCwith strong mo-\nmentum dependence can be observed. In particular, near\nky∼1.118 ˚A−1(Fig. 3(b)), there is a broad asymmetric\npeak (Q1) at ∼-0.15 eV in the paramagnetic (PM) phase\n(see the 80 K data), which evolves into two peaks at ∼-0.3\nand∼-0.15 eV in the FM phase, and at the same time,\nthe spectral weight at EFis reduced (see the 6 K data).\nAccording to DFT calculations in the PM phase (see Fig.\nS3(a) in [23]), the broad asymmetric Q1 peak could con-\ntain contributions from two close-by bands, i.e., Q1-1 at\n∼-0.15 eV and Q1-2 very close to EF. Upon entering the\nFM phase, the FM exchange splittings of Q1-1 and Q1-2\ncould lead to emergence of a new satellite band at ∼-0.3\neV (from the majority band of Q1-1), a more pronounced\ncentral peak at ∼-0.15 eV (due to overlapping contri-\nbutions from split Q1-1 and Q1-2 peaks) and reduced\ndensity of states (DOS) at EF(as the minority band of\nQ1-2 is pushed above EF). See Fig. S6 in [23] for a\nschematic illustration of our proposed scenario. Detailed\ntemperature-dependent scans (Fig. 3(b,c)) further show\nthat this temperature-driven transition takes place near\nTC∼45 K, consistent with its FM origin. Although our\ninterpretation of the Q1 temperature evolution in terms\nof FM exchange splitting is the most plausible one based\non current data, other possibilities cannot be completely\nruled out. In contrast to a momentum-independent FM\nexchange splitting of ∼0.2 eV obtained from DFT calcu-\nlations [23], the experimental FM splitting seems to be\nhighly anisotropic in momentum space. For instance, the\nQP band near ky∼0.6˚A−1shows almost no sign of FM\nsplitting, while the EDCs at ky∼0.4 and 0.2 ˚A−1show\nthe opposite trend compared to that at ky∼1.118 ˚A−1(Fig. 3(a)), i.e., the spectral weight is now transferred\nfrom deeper energies to EFupon cooling. This leads to\ndevelopment of sharp quasiparticle peak near EFat low\ntemperature, e.g., the Q2 peak at ky∼0.4˚A−1.\nAlthough it is tempting to attribute the observed mo-\nmentum anisotropy to momentum-dependent FM ex-\nchange splitting, as observed in some FM systems [35–\n37], the gradual enhancement of the Q2 peak upon cool-\ning over a wide temperature range implies that its spec-\ntral change is not caused by the FM order (Fig. 3(d)). In-\nstead, such behavior is consistent with the QPs expected\nin a HM, i.e., QPs with low coherence temperature and\nenergy scale. The suppressed coherence temperature is\ndue to Hund’s coupling, i.e., large Hund’s coupling will\nhelp formation of large local moments with low coher-\nence temperature [8]. Interestingly, the growth of the\nQ2 peak appears to slow down below ∼30 K, as shown\nin Fig. 3(d). To recover the full spectral function near\nEF, we divide the temperature-dependent EDCs by the\nresolution-convoluted Fermi-Dirac function (for details,\nsee Fig. S5 in [23]). The integrated peak intensity after\nsuch analysis is summarized in Fig. 3(e), which indicates\nthat the peak intensity grows monotonically above TC,\nand shows sign of slowing down well below TC[23].\nIt is interesting to note that the Q2 QP development\nat low temperature is limited to a small energy region\nnearEF(Fig. 3(d)), and its intensity over a wide tem-\nperature range can be roughly described by the −log(T)\ndependence (Fig. S5 in [23]). These characteristics bear\nintriguing similarity to the renowned Kondo resonances\nin KL systems [38–41]. There, the spin-flip scattering of\nconduction electrons with the local moments leads to spin4\nFIG. 3. Temperature evolution of QP bands. (a) Momentum-dependent EDCs well below (blue curves, ∼6 K) and above TC\n(red curves, ∼80 K) along ¯Γ-¯Mdirection. The cyan dashed box and green dashed circle highlight the Q1 (b,c) and Q2 (d,e)\nbands, respectively. (b,d) Temperature evolution of the EDCs. (c,e) The integrated intensity of Q1-2 (c) and Q2 (e) from (b,d)\nas a function of temperature. The integration region is [-0.145 eV, -0.01 eV] for Q1-2 and [-0.125 eV, 0.005 eV] for Q2.\nscreening and a gradual buildup of the Kondo resonance\nnearEF. In the HM, the screening could occur in both\nthe spin and orbital channels [16, 42], with separated en-\nergy scales, which can give rise to QP bands with strong\ntemperature dependence, as exemplified in ruthenates\n[12–14, 43]. The low coherence energy also results in a\nbroad regime with bad metal or non-Fermi-liquid (NFL)\nbehaviors [44]. Indeed, MnSi is well-known for its ex-\ntended NFL region over a large pressure/temperature re-\ngion [17–20].\nThe possible slowdown of the Q2 QP development be-\nlowTC(see Fig. S5(i) in [23]) suggests that the QP co-\nherence as a result of orbital/spin screening might be in-\nterrupted by the FM order. This implies interesting anal-\nogy to the f-electron KL systems: there, the magnetic\norder (via the Ruderman-Kittel-Kasuya-Yosida (RKKY)\nexchange interaction) competes with the heavy QP for-\nmation (via the Kondo screening), resulting in the well-\nknown Doniach phase diagram that lies at the heart of\nthe heavy fermion physics [45]. Often, magnetic KL sys-\ntems exhibit some degree of Kondo screening, and re-\ncent ARPES studies on CeSb and USb 2indicate that\nthe competition between the RKKY and Kondo inter-\nactions is manifested by the momentum-separated QP\nbands [46, 47]. The itineracy of the felectrons can beestimated by the magnitude of the screened moment, i.e.,\nthe ratio between the local moment above TCand the or-\ndered moment below TC. In bulk MnSi, this ratio is ∼5\n[28, 48]: such moderately large value is probably a direct\nconsequence of the competition between the spin/orbital\nscreening from the HM physics and the long-range FM\norder. Our results therefore suggest that the FM order\nand the spin/orbital screening can coexist and compete\nin a multi-orbital 3 dHM, via momentum-separated QP\nbands with distinct temperature dependence, resulting\nin a weak itinerant ferromagnet. Such insight might help\nexplain the dichotomy of itinerant and local moments\nobserved in recent INS studies [16, 49].\nTo conclude, we present systematic temperature-\ndependent ARPES results for a HM candidate MnSi with\nweak FM. We observed well-defined QP bands with pro-\nnounced temperature dependence: the Q1 band shows\na possible FM exchange splitting below TC, originating\nfrom its weak itinerant FM; by contrast, the Q2 peak\ngrows monotonically upon cooling above TC, and be-\ncomes fully developed inside the FM phase, as a result of\ncharacteristic orbital/spin screening in a HM. Our results\ntherefore provide spectroscopic insight for understand-\ning the coexistence and competition between the weak\nFM (forming ordered moments) and the HM physics (fa-5\nvoring screened moments at low temperature): it can\nbe achieved through QP bands that are well separated\nin momentum space and exhibit distinct temperature\ndependence. Such competition in a ferromagnetic HM\nbears striking similarity with the classical KL systems.\nOur study motivates future works to understand the un-\nderlying mechanism of the momentum-dependent elec-\ntron correlation, as well as the inherent connection be-\ntween the Kondo physics and HM physics. Interestingly,\na very recent DMFT study shows that the QP self-energy\ncan indeed exhibit strong momentum anisotropy in a\nmulti-orbital Hund’s ferromagnet [50].\nACKNOWLEDGMENTS\nThis work is supported by the Key R&D Program of\nZhejiang Province, China (2021C01002), the State Key\nproject of Zhejiang Province (No. LZ22A040007), the\nNational Science Foundation of China (No. 12174331,\n11674280), the National Key R&D Program of China\n(Grant No. 2017YFA0303100, 2016YFA0300203) and the\nFundamental Research Funds for the Central Universi-\nties (2021FZZX001-03). The surface X-ray diffraction\nexperiments were performed at Beamline 1W1A in Bei-\njing Synchrotron Radiation Facility (BSRF) and Beam-\nline BL02U2 in Shanghai Synchrotron Radiation Facility\n(SSRF). The XAS measurements were performed at 29-\nID IEX beamline at Advanced Photon Source (APS).\nAPS is supported by the US Department of Energy\n(DOE) Office of Science under Contract No. DE-AC02-\n06CH11357; additional support by the National Science\nFoundation under Grant No. DMR-0703406 is also ac-\nknowledged. We thank Dr. Chenchao Xu, Prof. Xin Lu,\nProf. Huiqiu Yuan, Prof. Zhi-Cheng Zhong, Prof. Yi-\nfeng Yang for experimental help and useful discussions.\n∗These authors contributed equally to this paper\n†yangliuphys@zju.edu.cn\n[1] J. M. Santiago, C.-L. Huang, and E. Morosan, Journal\nof Physics: Condensed Matter 29, 373002 (2017).\n[2] M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirk-\npatrick, Rev. Mod. Phys. 88, 025006 (2016).\n[3] T. Moriya, Spin Fluctuations in Itinerant Electron Mag-\nnetism (Springer, Berlin, Heidelberg, 1985).\n[4] Z. P. Yin, K. Haule, and G. Kotliar, Nature Materials\n10, 932 (2011).\n[5] Z. P. Yin, K. Haule, and G. Kotliar, Nature Physics 7,\n294 (2011).\n[6] L. de’Medici, J. Mravlje, and A. Georges, Phys. Rev.\nLett.107, 256401 (2011).\n[7] J. Mravlje, M. Aichhorn, T. Miyake, K. Haule,\nG. Kotliar, and A. Georges, Phys. Rev. Lett. 106, 096401\n(2011).[8] A. Georges, L. d. Medici, and J. Mravlje, Annual Review\nof Condensed Matter Physics 4, 137 (2013).\n[9] M. Yi, D. H. Lu, R. Yu, S. C. Riggs, J.-H. Chu, B. Lv,\nZ. K. Liu, M. Lu, Y.-T. Cui, M. Hashimato, S.-K. Mo,\nZ. Hussain, C. W. Chu, I. R. Fisher, Q. Si, and Z.-X.\nShen, Phys. Rev. Lett. 110, 067003 (2013).\n[10] H. Miao, Z. P. Yin, S. F. Wu, J. M. Li, J. Ma, B.-Q.\nLv, X. P. Wang, T. Qian, P. Richard, L.-Y. Xing, X.-\nC. Wang, C. Q. Jin, K. Haule, G. Kotliar, and H. Ding,\nPhys. Rev. B 94, 201109(R) (2016).\n[11] A. Damascelli, D. H. Lu, K. M. Shen, N. P. Armitage,\nF. Ronning, D. L. Feng, C. Kim, Z.-X. Shen, T. Kimura,\nY. Tokura, Z. Q. Mao, and Y. Maeno, Phys. Rev. Lett.\n85, 5194 (2000).\n[12] S.-C. Wang, H.-B. Yang, A. K. P. Sekharan, H. Ding,\nJ. R. Engelbrecht, X. Dai, Z. Wang, A. Kaminski,\nT. Valla, T. Kidd, A. V. Fedorov, and P. D. Johnson,\nPhys. Rev. Lett. 92, 137002 (2004).\n[13] T. Kondo, M. Ochi, M. Nakayama, H. Taniguchi,\nS. Akebi, K. Kuroda, M. Arita, S. Sakai, H. Namatame,\nM. Taniguchi, Y. Maeno, R. Arita, and S. Shin, Phys.\nRev. Lett. 117, 247001 (2016).\n[14] A. Tamai, M. Zingl, E. Rozbicki, E. Cappelli, S. Ricc` o,\nA. D. la Torre, S. M. Walker, F. Y. Bruno, P. D. C. King,\nW. Meevasana, M. Shi, M. Radovi´ o, N. C. Plumb, A. S.\nGibbs, A. P. Mackennzie, C. Berthod, H. U. R. Strand,\nM. Kim, A. Georges, and F. Baumberger, Phys. Rev. X\n9, 021048 (2019).\n[15] B. G. Jang, G. Han, I. Park, D. Kim, Y. Y. Koh, Y. Kim,\nW. Kyung, H.-D. Kim, C.-M. Cheng, K.-D. Tsuei, K. D.\nLee, N. Hur, J. H. Shim, C. Kim, and G. Kotliar, Nature\nCommunications 12, 1208 (2021).\n[16] X. Chen, I. Krivenko, M. B. Stone, A. I. Kolesnikov,\nT. Wolf, D. Reznik, K. S. Bedell, F. Lechermann, and\nS. D. Wilson, Nature communications 11, 3076 (2020).\n[17] C. Pfleiderer, S. R. Julian, and G. G. Lonzarich, Nature\n414, 427 (2001).\n[18] N. Doiron-Leyraud, I. R. Walker, L. Taillefer, M. J.\nSteiner, S. R. Julian, and G. G. Lonzarich, Nature 425,\n595 (2003).\n[19] C. Pfleiderer, D. Reznik, L. Pintschovius,\nH. v. L¨ ohneysen, M. Garst, and A. Rosch, Nature\n427, 227 (2004).\n[20] R. Ritz, M. Halder, M. Wagner, C. Franz, A. Bauer, and\nC. Pfleiderer, Nature 497, 231 (2013).\n[21] K. Kura, K. Takano, Y. Takeichi, A. Harasawa,\nT. Okuda, I. Matsuda, and A. Kakizaki, Journal of the\nPhysical Society of Japan 77, 024709 (2008).\n[22] A. Nicolaou, M. Gatti, E. Magnano, P. Le F` evre,\nF. Bondino, F. Bertran, A. Tejeda, M. Sauvage-Simkin,\nA. Vlad, Y. Garreau, A. Coati, N. Gu´ erin, F. Parmigiani,\nand A. Taleb-Ibrahimi, Phys. Rev. B 92, 081110(R)\n(2015).\n[23] See online supplementary material at xxx, which includes\nRefs.[24-26] and details on thin film growth, ARPES\nmeasurements and DFT calculations.\n[24] B. Geisler, P. Kratzer, T. Suzuki, T. Lutz, G. Costantini,\nand K. Kern, Phys. Rev. B 86, 115428 (2012).\n[25] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\n[26] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n[27] E. Karhu, S. Kahwaji, T. L. Monchesky, C. Parsons,\nM. D. Robertson, and C. Maunders, Phys. Rev. B 82,\n184417 (2010).6\n[28] G. Shirane, R. Cowley, C. Majkrzak, J. B. Sokoloff,\nB. Pagonis, C. H. Perry, and Y. Ishikawa, Phys. Rev.\nB28, 6251 (1983).\n[29] F. Carbone, M. Zangrando, A. Brinkman, A. Nicolaou,\nF. Bondino, E. Magnano, A. A. Nugroho, F. Parmigiani,\nT. Jarlborg, and D. van der Marel, Phys. Rev. B 73,\n085114 (2006).\n[30] T. Jeong and W. E. Pickett, Phys. Rev. B 70, 075114\n(2004).\n[31] L. Tailefer, G. G. Lonzarich, and P. Strange, J. Magn.\nMagn. Mater. 54, 957 (1986).\n[32] M. A. Wilde, M. Dodenh¨ oft, A. Niedermayr, A. Bauer,\nM. M. Hirschmann, K. Alpin, A. P. Schnyder, and\nC. Pfleiderer, Nature 594, 374 (2021).\n[33] A. Bauer, A. Neubauer, C. Franz, W. M¨ onzer, M. Garst,\nand C. Pfleiderer, Phys. Rev. B 82, 064404 (2010).\n[34] A. Mishra, M. Krishnan, M. Gangrade, D. Singh,\nR. Venkatesh, and V. Ganesan, J. Phys.: Conf. Ser. 755,\n012037 (2016).\n[35] J. S´ anchez-Barriga, J. Braun, J. Min´ ar, I. D. Marco,\nA. Varykhalov, O. Rader, V. Boni, V. Bellini, F. Manghi,\nH. Ebert, M. I. Katsnelson, A. I. Lichtenstein, O. Eriks-\nson, W. Eberhardt, H. A. D¨ urr, and J. Fink, Phys. Rev.\nB85, 205109 (2012).\n[36] H. Miyazaki, T. Ito, H. J. Im, S. Yagi, M. Kato, K. Soda,\nand S. Kimura, Phys. Rev. Lett. 102, 227203 (2009).\n[37] M. Kim and B. I. Min, Phys. Rev. B 91, 205116 (2015).\n[38] J. D. Denlinger, G.-H. Gweon, J. W. Allen, C. G. Ol-\nson, M. B. Maple, J. Sarrao, P. Armstrong, Z. Fisk, and\nH. Yamagami, Journal of Electron Spectroscopy and Re-\nlated Phenomena 117, 347 (2001).\n[39] J. W. Allen, Journal of the Physical Society of Japan 74,\n34 (2005).[40] S.-i. Fujimori, Journal of Physics: Condensed Matter 28,\n153002 (2016).\n[41] S. Kirchner, S. Paschen, Q. Chen, S. Wirth, D. Feng,\nJ. D. Thompson, and Q. Si, Rev. Mod. Phys. 92, 011002\n(2020).\n[42] X. Deng, K. M. Stadler, K. Haule, A. Weichselbaum,\nJ. von Delft, and G. Kotliar, Nature communications 10,\n2721 (2019).\n[43] Y. Liu, H. P. Nair, J. P. Ruf, D. G. Schlom, and K. M.\nShen, Phys. Rev. B 98, 041110(R) (2018).\n[44] E. Walter, K. M. Stadler, S. S. B. Lee, Y. Wang,\nG. Kotliar, A. Weichselbaum, and J. von Delft, Phys.\nRev. X 10, 031052 (2020).\n[45] S. Doniach, Physica B 91, 231 (1977).\n[46] S. Jang, R. Kealhofer, C. John, S. Doyle, J.-S. Hong,\nJ. H. Shim, Q. Si, O. Erten, J. D. Denlinger, and J. G.\nAnalytis, Sci. Adv. 5, eaat7158 (2019).\n[47] Q. Y. Chen, X. B. Luo, D. H. Xie, M. L. Li, X. Y. Ji,\nR. Zhou, Y. B. Huang, W. Zhang, W. Feng, Y. Zhang,\nL. Huang, Q. Q. Hao, Q. Liu, X. G. Zhu, Y. Liu,\nP. Zhang, X. C. Lai, Q. Si, and S. Y. Tan, Phys. Rev.\nLett.123, 106402 (2019).\n[48] H. Yasuoka, V. Jaccarino, R. C. Sherwood, and J. H.\nWernick, Journal of the Physical Society of Japan 44,\n842 (1978).\n[49] Z. Jin, Y. Li, Z. Hu, B. Hu, Y. Liu, K. Iida, K. Ka-\nmazawa, M. B. Stone, A. I. Kolesnikov, D. L. Abernathy,\nX. Zhang, H. Chen, Y. Wang, C. Fang, B. Wu, I. A.\nZaliznyak, J. M. Tranquada, and Y. Li, arXiv preprint\narXiv:2206.13699 (2022).\n[50] Y. Nomura, S. Sakai, and R. Arita, Phys. Rev. Lett. 128,\n206401 (2022).Supplementary Information for:\nQuasiparticle characteristics of the weakly ferromagnetic Hund’s metal MnSi\nYuan Fang,1,∗Huali Zhang,1,∗Ding Wang,1,∗Guowei Yang,1Yi Wu,1Peng Li,1\nZhiguang Xiao,1Tianyun Lin,1Hao Zheng,1Xiao-Long Li,2Huan-Hua Wang,3\nFanny Rodolakis,4Yu Song,1Yilin Wang,5Chao Cao,1and Yang Liu1, 6, 7,†\n1Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China\n2Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute,\nChinese Academy of Sciences, Shanghai 201204, China\n3Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China\n4Advanced Photon Source, Argonne National Laboratory,\n9700 South Cass Avenue, Argonne, Illinois 60439, USA\n5Hefei National Laboratory for Physical Sciences at Microscale,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n6Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou, China\n7Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China\n(Dated: October 17, 2022)\nI. MBE THIN FILM GROWTH AND IN-SITU ARPES MEASUREMENT\nOur growth method of epitaxial MnSi films was adapted from previous studies [22,24,27], with small adjustments\nto yield best quality films for ARPES measurements. MnSi films were grown on Si(111) substrates by MBE under\nultrahigh vacuum with a pressure better than 2 ×10−10mbar. The Si(111) substrates were cut into a typical size of\n5x10 mm2. The substrates were heated in ultrahigh vacuum to 900 −1000◦C to yield the 7x7 reconstructed surface,\nas verified by RHEED. After that, ∼4.2˚A of Mn was deposited on the 7x7 surface at 70◦C or 200◦C and annealed\nto∼260◦C to form a good buffer layer for subsequent MnSi film growth. We found that a lower growth temperature\n(70◦C) of the buffer layer yields slightly better film quality. Mn was then deposited on the buffer layer at 260◦C,\nallowing Si from the underlying substrate to diffuse to the top layers to form epitaxial MnSi fimls. Codeposition of\nMn and Si (from a separate Si effusion cell) was also performed and the ARPES results were found to be similar,\ndespite slightly weaker quasiparticle bands for very thin films. The structural quality of the MnSi films were monitored\nin-situby RHEED. The deposition of Mn was done with an effusion cell set to 790◦C to yield a rate of approximately\n0.7 ML per min, as determined by a quartz crystal monitor. All the data presented in this paper were taken from\nfilms with thicknesses ranging from a few nm to a few tens of nm.\nAll the ARPES data were taken in a Helium-lamp ARPES system at Zhejiang University, which is connected under\nultrahigh vacuum to the MBE system. ARPES measurements were performed immediately after the film growth,\nby transferring the sample from the MBE growth chamber to the ARPES chamber. A five-axis manipulator cooled\nby a closed-cycle helium refrigerator was employed for sample alignment and for the temperature control during the\nARPES measurements. The photon source was a VUV-5k Helium lamp coupled to a grating monochromator for\nselecting either the He-I (21.2 eV) or He-II (40.8 eV) lines. The base pressure of the ARPES system was ∼7.6×10−11\nmbar, which increased to ∼2.5×10−10mbar during the Helium lamp operation. Most of the ARPES data were\ntaken with He-I photons, unless mentioned otherwise. The typical energy and momentum resolution is ∼12 meV and\n∼0.01 ˚A−1, respectively.\nII. DFT CALCULATIONS\nElectronic structure calculations were performed using DFT and a plane-wave basis projected augmented wave\nmethod, as implemented in the Vienna ab initio simulation package (VASP) [25,26]. The Perdew, Burke and Ernzerhof\n(PBE) generalized gradient approximation for exchange correlation potential was used for the DFT calculation. Spin-\norbital coupling (SOC) was taken into account in all DFT calculations. An energy cutoff of 450 eV and a 12 ×12×12\ngamma-centered k-mesh were employed to converge the calculation to 1 meV/atom. The optimized ordered moment\nfor the FM phase of MnSi is 1 µB/Mn from the DFT calculation, which is substantially larger than the experimental\nvalue of 0.4 µB/Mn. For better comparison with experiments, the magnetic moment is constrained to the experimental\nvalue, which yields a FM exchange splitting of ∼0.2 eV.arXiv:2208.12456v2  [cond-mat.str-el]  14 Oct 20222\nIII. X-RAY DIFFRACTION, TRANSPORT AND X-RAY ABSORPTION MEASUREMENTS\nThe lattice constant of the epitaxial MnSi films were determined by synchrotron X-ray diffraction. The measure-\nments were performed at room temperature for a thick film, in both Beamline 1W1A in Beijing Synchrotron Radiation\nFacility (BSRF) and Beamline BL02U2 in Shanghai Synchrotron Radiation Facility (SSRF). We used the standard\nsurface coordinate of the substrate Si to present our diffraction data (Fig. 1(b,c)): a=b= 3.84˚A,c= 9.406˚A,\nα=β=90◦,γ=120◦. Therefore, H,K directions lie within the (111) plane, while the L direction points to the out-\nof-normal [111] direction. From the measured positions of MnSi Bragg peaks, the lattice constants of MnSi film is\ndetermined to be a=b= 3.243˚A,c= 7.872˚A,α=β=90◦,γ=120◦. This indicates a tensile strain of ∼1% compared\nwith the bulk MnSi.\nThe resistance measurements were conducted via the standard four-probe method using a quantum design physical\nproperty measurement system (PPMS).\nThe XAS measurements near the Mn Ledge were performed using the total electron yield mode at ∼29 K, at sector\n29 in Advanced Photon Source (APS). A 100 nm MnSi film was employed for the XAS measurements.\nFIG. S1. Comparison of ARPES spectra of a 10 monolayer (ML) MnSi film (a) and a 27 ML MnSi film (b). Here one ML\n(=2.624 ˚A) refers to the basic repeating structure along the [111] direction, consisting of two Mn and two Si layers with different\natomic densities. Note that the helical order has a long periodicity of ∼20 nm along the [111] direction, due to very weak DM\ninteraction. Therefore, the film thickness studied here is much smaller than the periodicity of the helix, i.e., the film is not\nthick enough to accommodate one complete helix. The fact that the ARPES spectra show no obvious change with thickness\nindicates that the DM interaction plays a minor role on the observed electronic structure.\nFIG. S2. (a) Direct comparison of the ARPES data and DFT calculations by overlaying the DFT calculations (cyan curves)\non top of the experimental data (same as Fig. 2(b)). (b) The corresponding waterfall plot of the EDCs from the ARPES data.3\nFIG. S3. (a,b) Comparison of calculated band structure in a wide energy range for kz= 0.4 ˚A−1(corresponding to the kz\ncut using He I photons) for the PM (a) and FM (b) phase. A momentum-independent FM exchange splitting of ∼0.2 eV can\nbe readily identified from the comparison. Note that the calculated FM exchange splitting is roughly uniform in momentum\nspace, in sharp contrast to the large momentum anisotropy seen in experiments. (c-d) The ARPES spectra taken with He I\nphotons in the PM (c) and FM (d) phases. (e-f) The corresponding DFT calculations in the PM (e) and FM (f) phases for\ndirect comparison with the experimental data in (c,d). The spectral change across TCis rather small and strongly dependent\non the momentum (see EDCs in Fig. 3). The flat QP bands near EFin the momentum range of ±0.6˚A−1cannot be explained\nby the DFT calculation.\nFIG. S4. (a) Calculated constant energy contours for MnSi from DFT, for comparison with the experimental results (b) (same\nas Fig. 2(e)). The green hexagons in (b) indicate surface BZs. At -0.05 eV, the DFT calculation predicts elliptical pockets\ncentered at ¯M(see black arrows), which qualitatively agrees with the ARPES data, although the detailed shapes are quite\ndifferent. At -0.21 eV, the DFT calculation shows large pockets centered at ¯Γin the second surface Brillouin zone (see green\narrows), which is quite different from the experimental data. At -0.42 eV, the DFT calculation predicts a pocket at ¯Γwith\na triangular shape, which agrees reasonably with the ARPES data. The large difference between DFT and ARPES could be\nattributed to strong electron correlation and complex band structure of MnSi.4\nFIG. S5. Detailed analysis of the temperature-dependent ARPES data shown in Fig. 3. (a) Temperature-dependent EDCs\ncorresponding to Q1 (same as Fig. 3(b)). Three spectral features (Q1-1, Q1-2, S) are labelled. (b,c) Integrated spectral\nintensity for Q1-2 and S marked in (a) as a function of temperature. The integration range for Q1-2 and S is [-0.145 eV,\n-0.01 eV] (b) and [-0.445 eV, -0.205 eV] (c), respectively. (d) The reference spectra for obtaining the resolution-convoluted\nFermi-Dirac function (RC-FDD). (e,f) The temperature-dependent EDCs corresponding to the Q2 peak (e) and those divided\nby the resolution-convoluted Fermi-Dirac distribution (RC-FDD) (f). (g,h) Peak height (g) and integrated peak intensity (h)\nof Q2 as a function of temperature. Here the RC-FDD divided spectra (f) were used. The integration was done from -0.125\neV to 0.005 eV. (i) Same as (h), but plotted with the log( T) scale.5\nFIG. S6. (a) Calculated bands from DFT for the PM and FM phases (same as Fig. S3). The vertical dashed line indicates the\nmomentum point corresponding to the experimental EDCs shown in Fig. 3(b). (b) A schematic showing how the FM exchange\nsplitting of the Q1 peaks could lead to the observed spectral change in Fig. 3(b). Q1-1 and Q1-2 are two peaks that are already\npresent in the PM phase (see the vertical dashed line in (a)). FM exchange splittings of Q1-1 and Q1-2 lead to emergence of a\nnew satellite band (labelled S in Fig. S5(a)) at ∼-0.3 eV (from Q1-1), a more pronounced central peak at ∼-0.15 eV (due to\noverlapping contributions from split Q1-1 and Q1-2 peaks) and reduced density of states (DOS) at EF(as the minority band\nof Q1-2 is pushed above EF).6\n∗These authors contributed equally to this paper\n†yangliuphys@zju.edu.cn"    },    {        "title": "0912.3434v1.Droplet_like_Fermi_surfaces_in_the_anti_ferromagnetic_phase_of_EuFe__2_As__2___an_Fe_pnictide_superconductor_parent_compound.pdf",        "content": "APS/123-QED\nDroplet-like Fermi surfaces in the anti-ferromagnetic phase of EuFe 2As2, an\nFe-pnictide superconductor parent compound\nS. de Jong,1,\u0003E. van Heumen,1S. Thirupathaiah,2R. Huisman,1F. Massee,1J. B.\nGoedkoop,1R. Ovsyannikov,3J. Fink,3, 4H. A. D urr,3A. Gloskovskii,5H. S. Jeevan,6P.\nGegenwart,6A. Erb,7L. Patthey,8M. Shi,8R. Follath,2A. Varykhalov,2and M. S. Golden1\n1Van der Waals-Zeeman Institute, University of Amsterdam, NL-1018XE Amsterdam, The Netherlands\n2Helmholtz Zentrum Berlin, Albert-Einstein-Strasse 15, 12489 Berlin, Germany\n3Helmholtz Zentrum Berlin GmbH, Albert-Einstein-Strasse 15, 12489 Berlin, Germany\n4Leibniz Institute of Solid and Materials Research Dresden, P.O. Box 270116, D-01171, Dresden, Germany\n5Institut f ur Anorganische Chemie und Analytische Chemie,\nJohannes Gutenberg-Universit at, 55099 Mainz, Germany\n6I. Physikalisches Institut, Georg-August-Universit at, G ottingen 37077, Germany\n7Walther-Meissner-Institut, Walther-Mei\u0019ner Strasse 8, 85748 Garching, Germany\n8Paul Scherrer Institut, Swiss Light Source, 5232 Villigen, Switzerland\n(Dated: November 9, 2018)\nUsing angle resolved photoemission it is shown that the low lying electronic states of the iron\npnictide parent compound EuFe 2As2are strongly modi\fed in the magnetically ordered, low tem-\nperature, orthorhombic state compared to the tetragonal, paramagnetic case above the spin density\nwave transition temperature. Back-folded bands, re\rected in the orthorhombic/ anti-ferromagnetic\nBrillouin zone boundary hybridize strongly with the non-folded states, leading to the opening of en-\nergy gaps. As a direct consequence, the large Fermi surfaces of the tetragonal phase fragment, the\nlow temperature Fermi surface being comprised of small droplets, built up of electron and hole-like\nsections. These high resolution ARPES data are therefore in keeping with quantum oscillation and\noptical data from other undoped pnictide parent compounds.\nPACS numbers: 74.25.Jb, 74.70.b, 79.60.-i\nThe electronic structure and properties of the newly\ndiscovered iron-pnictide superconductors [1] are a fo-\ncus of much research. A central theme is the under-\nstanding of the origins of the magnetic ordering and\nits possible interplay with superconductivity. The char-\nacteristics of the low-temperature, orthorhombic, anti-\nferromagnetic (AFM) phase have been experimentally\ndetermined, for example for the undoped 122-parent\ncompounds, MFe2As2(withM= Ca, Sr, Ba, Eu. . . ).\nA common proposition is that the magnetism is of an\nitinerant, spin density wave (SDW) type, connected to\nnesting of the warped cylindrical Fermi surfaces (FS's)\ncentered at the (0 ;0) (\u0000) and ( \u0019;\u0019) points (X) of the\ntetragonal Brillouin zone (BZ) [2]. Such an SDW would\nhave dramatic consequences for the band structure and\nFermi surface, leading to reconstruction, the opening\nof gaps and major Fermi surface depletion. Recently,\nquantum oscillation (QO) experiments have presented\nevidence for Fermi surfaces comprising small pockets -\ndue to the e\u000bects of the SDW order - in SrFe 2As2[3]\nand BaFe 2As2[4]. Both studies \fnd the existence of\nthree distinct orbits, with FS areas of only 0.3%, 0.6%\nand 1.5% of the tetragonal BZ (compared to a total FS\narea larger than 20% in the tetragonal phase). In ad-\ndition, the opening of gaps as well as a dramatic re-\nduction of the free charge carrier density upon entering\nthe orthorhombic AFM state has been concluded from\noptical conductivity measurements [5, 6]. Given that\na FS-nesting-driven SDW is rooted in k-space, it is ofgreat importance whether the SDW `\fngerprints' of re-\nconstruction, gapping and FS depletion really take place\nin theE(k)-space hosting the electronic states of these\nmaterials. Angle-resolved photoemission (ARPES) is a\npowerful probe of such issues, and consequently there\nhave been numerous studies of the parent compounds of\nthe pnictide superconductors using ARPES. Early stud-\nies were unable to detect signi\fcant changes below the\nmagnetic ordering temperature [7, 8]. More recently,\nARPES studies have shown the existence of small, ad-\nditional FS pockets (either hole or electron-like) around\nthe (0;0) or (\u0019;\u0019) point [9, 10, 11, 12], and a detailed\ntemperature-dependent study is reported in [13]. Despite\nthis progress, it could be argued that there are no stud-\nies that cover the states around \u0000 and Xon an equal\nfooting, avoiding matrix-element-induced extinction of\nimportant states, and there are, in any case, no stud-\nies reported of the europium parent compound. On top\nof this, the issues of the dimensionality of the electronic\nstates [14, 15, 16, 17] and the nature of the cleavage sur-\nface of theMFe2As2systems [18, 19] are both matters of\nquite some discussion.\nIn this paper we present the \frst photoemission study\non the pnictide parent compound Eu122, showing data\nof a clarity that makes use of second di\u000berentials unnec-\nessary. The local iron moment, \u00191\u0016B[20], for Eu122 is\nsimilar to the other M122 compounds, yet in Eu122 the\nEu moments also order at low temperatures, having a lo-\ncal moment of nearly 7 \u0016B. Our low temperature ARPESarXiv:0912.3434v1  [cond-mat.supr-con]  17 Dec 20092\nFIG. 1: Temperature dependent Fermi surface (FS) topology\nof Eu122. ( a) Measured FS (E F\u000610 meV, h\u0017= 84 eV) of\nEu122 atT= 10 K. The light polarization \u000fis indicated.\nThe inset shows a schematic high temperature FS in a two-\ndimensional projection covering a quarter of the tetragonal\nBrillouin zone. ( b) FS maps (h \u0017= 75 eV) around the ( \u0019;\u0019)-\npoint ink-space (purple box in the inset of [a]) at di\u000berent\ntemperatures. The polarization vector ( \u000f) is parallel to the\nanalyzer slit. The top-most 20 K FS map is taken on a fresh\ncleavage surface, the bottom one after cycling the sample tem-\nperature up to 300 K [28].\ndata however do closely resemble those reported for other\nparent compounds [10, 13], indicating that the ordering\nof the Eu moments is of little in\ruence on the ordering\nof the iron moments.\nExploiting matrix elements and polarisation-\ndependent selection rules, we uncover hybridization\ne\u000bects between back-folded hole and electron-like bands\nboth at the ( \u0019;\u0019) and (0;0) regions of k-space and\nprovide clear evidence of the resulting energy gaps.\nAs a result, the large FS's of the paramagnetic phase\nbreak up into very small hole and electron pockets we\ndub `droplets' on entering the magnetically ordered\nstate. The exact nature of the AFM phase (localized vs.\nitinerant) is still subject of debate [21, 22]. We note that\nour ARPES measurements cannot a priori distinguish\nbetween localized versus itinerant magnetic order. Even\nmore so, the orthorhombic structural transition should\nalso give rise to an electronic reconstruction. Since\nthe orthorhombic and magnetic transitions coincide\nin the parent compounds, it is virtually impossible to\ndisentangle the e\u000bect of the combined electronic recon-\nstructions. Regardless of its origin, the back-folding,\nhybridization and gapping we observe in our ARPES\ndata are consistent with the QO and optical conductivity\nresults on related compounds.\nTemperature dependent ARPES was performed at the\nUE112-PGM beamline at BESSYII, coupled to a Sci-\nenta SES100 analyzer with a total energy resolution of27 meV (at T= 20 K, h\u0017= 75 eV) and a resolution\nof better than 0 :02\u0019=aalong the slit. The analyzer slit\nwas perpendicular to the scattering plane of the incoming\nlight and the outgoing photoelectrons and the polariza-\ntion vector of the light was perpendicular to this plane.\nAdditional, detailed FS mapping experiments were per-\nformed at the SIS-HRES beamline of the Swiss Light\nSource, using a Scienta R4000 analyzer with a total en-\nergy resolution of 16 meV (at T= 10 K, h\u0017= 84 eV)\nand a momentum resolution better than 0 :05\u0019=aalong\nthe slit. FS maps were collected with an angular step\nof 0.5\u000e, scanning in the direction perpendicular to the\nanalyzer slit (denoted k0\nkforthwith), which is compara-\nble to the resolution along the analyzer slit. The small\nbeam spot-size (\u0019100\u0016m) assured a minimal in\ruence of\nsurface roughness on the ARPES experiments and aided\nin the achievement of very high e\u000bective contrast in k-\nspace. During the SLS experiments the analyzer slit and\nthe polarization vector of the synchrotron radiation were\nparallel (p-polarized) to the scattering plane formed by\nthe incoming light and the outgoing photoelectrons.\nSingle crystals of Eu122 were grown using the Bidge-\nman and Sn-\rux method at Goettingen University [23].\nTheir magnetic transition temperature was determined\nto be about 190 K, with an additional AFM ordering\nof the Eu moments around 18 K [24]. Crystal surfaces\nwere cleaved at T < 50 K in a vacuum better than\n1\u000210\u000010mbar. The quality of the cleavage surfaces was\nchecked using low energy electron di\u000braction (LEED).\nAll measured cleaves yielded very sharp di\u000braction pat-\nterns, displaying, apart from the expected 1 \u00021 spots,\nprominent 2\u00021, 1\u00022, and less clearp\n2\u0002p\n2 recon-\nstruction spots. These surface reconstructions were not\nrobust against thermal cycling up to room temperature,\nin keeping with their origins in ordering/disordering pro-\ncesses within the partial Eu layer that comprises the ter-\nmination of the cleaved crystal [18, 19].\nWe start with a discussion of the `as cleaved', T= 10 K\ndata from EuFe 2As2shown in Fig. 1a. The inset shows\na cartoon of the k-space region under scrutiny, with a\nsimpli\fed representation of the band theory predictions\nfor the tetragonal phase: three hole pockets around (0 ;0)\n(blue circles) and two electron pockets at ( \u0019;\u0019) (red el-\nlipses) [25]. The main part of Fig. 1a is a wide k-range,\nlow temperature FS map which obviously exhibits quite\na di\u000berent FS topology than the tetragonal expectation.\nThe most eye-catching di\u000berence is the fact that both\nthe large hole-like FS's at (0 ;0) and the electron-like FS's\nat (\u0019;\u0019) and equivalent points have broken up into a\nnumber of clearly disconnected `droplets' [26]. We stress\nthat our data clearly resolves the droplet structure of the\nFS's at both high symmetry points: there are no large\nFS's left. Droplet, or petal-like FS's have been reported\nin ARPES studies of di\u000bering k-space locations of the\nSr122 (Ref. [10]), Ba122 (Ref. [12, 13]) and Ca122 (Ref.\n[29]) parent compounds, as well as from the doped super-3\nconductor (Ba,K)Fe 2As2(Ref. [9]). We will zoom in on\nthe \u0000 andX k-space regions for Eu122 later in the pa-\nper. Before doing this, we deal with the issue of surface\nreconstructions.\nSeeing as the simplest SDW scenario involves shifted\ncopies of bands appearing at new k-space locations com-\npared to the high- Ttetragonal phase, it is important\nto test the impact of (surface) structural reconstructions\non the data, particularly as such reconstructions are a\ncommon feature of scanning tunneling microscopy data\nfrom theMFe2As2materials [18, 30]. Given the fact that\nthe extra LEED spots, arising from the ordering of the\nM-cations in the surface layer, disappear irreversibly on\nwarming the (cold cleaved) sample up to room tempera-\nture, re-cooling then o\u000bers us the test we require.\nIn Fig. 1b, we show that the FS's around ( \u0019;\u0019) remain\nunchanged upon raising the temperature from 10 K to\n20 K, at which point the Eu moments have become dis-\nordered [31], illustrating that the magnetic ordering of\nthe Eu sub-lattice does not have a big e\u000bect on the Fe-\nderived electronic states at the Fermi level as seen with\nARPES. The situation is very di\u000berent at T= 200 K:\nupon exiting the AFM phase of the Fe-planes, the droplet\nFS structure (indicated with an arrow and appearing as\na dumb-bell in the measurement geometry used in Fig.\n1b due to intensity from a below- EFband at the X point\nitself) is no longer observed. At 200 K, the LEED pat-\ntern shows only 1x1 tetragonal spots. Upon re-cooling\nto 20K (bottom panel of Fig. 1b), the extra, 2 \u00021 andp\n2\u0002p\n2 LEED spots do not re-appear, yet the droplet FS\nat (\u0019;\u0019) does, proving that this remarkable FS topology\nre\rects the bulk, magnetic state of the pnictide parent\ncompound, and is de\fnitely not a result of `di\u000braction\nreplicas' due to surface reconstructions.\nFIG. 2: Low temperature Fermi surface topology of Eu122\n(T= 10 K, h\u0017= 84 and 100 eV). The ( E;k)-maps show\nraw data taken from the cuts in k-space indicated with pink\nlines in the small FS map. For an explanation for colored the\narrows, see text.\nFIG. 3: Detailed FS topology around the (0 ;0)-points of\nEu122 (T= 10 K, h\u0017= 84 (Z) and 100 eV (\u0000). For each\nof (a)-(d), the left panels show schematic (top) and measured\n(bottom) (E;k)-images for the green k-space cuts shown in\nthe small FS maps (top right panel in each case). In each\nband structure scheme, electron-pocket bands are labeled `e'\nand indicated in red, whereas hole-pocket bands get an `h',\nand are shown in blue. Barely distinguishable features are\nindicated with dotted lines. Red shading indicates areas in\nwhich the symmetry selection rules can lead to suppression of\nsome of the orbital states. The measured FS map in panel (c)\nhas been replaced with a sketch of the low TFermi surface.\nThe FS predicted for the tetragonal phase is shown in gray.\nNow we are sure that the observed low- TEu122 FS is\nsensitive to the bulk magnetism, we return to the band\nstructure in the orthorhombic phase in more detail. In\nFig. 2 several E(k)-cuts through the \u0000 and Z points,\nparallel to \u0000 Xare shown, taken with h \u0017= 100 and\n84 eV. The data taken with these two photon energies\nlook, in general, very similar - with FS crossings at very\nsimilarkF's, albeit with (at times strongly) varying spec-\ntral weight. As these two photon energies correspond to\nkz= 0 (100 eV) and \u0019/c (84 eV), this illustrates that\nthekz-dispersion for this compound is modest [27]. A\nsecond point is that the destruction of the large FS's to\ngive droplets is clearly present in Eu122 at both kz= 0\nandkz=\u0019=c, unlike what has been reported for Ba and\nCa122 [12].\nWe note that the surface quality and favorable exper-\nimental conditions mean that a large number of band\nfeatures can be distinguished in the E(k) images shown\nin Fig. 2. We deal \frstly with a group of features that\nclearly possess reversed character with respect to the\nhigh-T, tetragonal situation, being hole-like around X4\nand electron-like around \u0000(Z). These states are marked\nin Fig. 2 using blue and red arrows, for the hole-like and\nelectron-like features, respectively, and originate from the\ne\u000bects of magnetic order in the Fe plane.\nFurther inspection of Fig. 2, however, shows that there\nare also ARPES features (marked with green arrows)\nwhich seem to match those expected in the paramagnetic,\ntetragonal state, such as the electron-like feature clearly\nvisible around ( \u0019;\u0019) for h\u0017= 100eV. This apparent di-\nchotomy - that both reconstructed and unreconstructed\nbands coexist - has been reported in ARPES investiga-\ntions of other parent compounds,[10, 12] and will be re-\nturned to further on. Dealing with these kind of issues\none is confronted with the intrinsic complexity of the low\nlying electronic structure of these multiband materials.\nEven adopting the simplest picture of only back-folding\nthe tetragonal band structure yields 8 or 10 bands (start-\ning from 4 or 5 FS's at high temperature), and if the\ncrystal contains small-scale magnetic twin domains we\naverage over in the ARPES experiment, then this num-\nber doubles. It is therefore imperative to zoom-in and\ntake a closer look at the di\u000berent high symmetry points\nink-space.\nTo start o\u000b, we zoom-in on the \u0000-point in Fig. 3.\nFourk-space cuts for the two di\u000berent photon energies\nare presented, partnered with a band structure scheme\nas a guide to the E(k) images. Panel (a) clearly shows\nthe existence of shallow electron-like bands, appearing in\nthe FS map as small, triangular shaped pockets. Cuts\nthrough the center of this feature along the kx-direction\nusing h\u0017= 84 and 100 eV are shown in panels (c) and\n(d), and reveal that, in fact, it consists of two concen-\ntric electron pockets. The cuts shown in Fig. 3(b) illus-\ntrate a d-wave like gapping operative around the (0 ;0)\npoint, with no low energy pockets such as those seen at\nkx\u00180:2\u0019=ain panels (c) and (d). The concentric elec-\ntron pockets have Fermi surface areas of about 1% and\n0.3% of the tetragonal BZ. Such small FS areas match\nthose observed in the QO studies of Sr122 and Ba122,\nalthough we note that the details of the band structure\nof the di\u000berent pnictide parent compounds will depend\nsomewhat on the identity of the M-cation.\nThe important point here is the fact that we are able\nto detect the disintegration of the large \u0000-centered FS's\ninto very small droplets. As can be seen in the E(k)\nimages, the two di\u000berent photon energies either more\nclearly resolve the left (84 eV) or the right side (100 eV)\nof the small pockets, and this matrix element di\u000berence\nis, in fact, a signal of di\u000berent orbital character for ei-\nther side of the pocket. A natural explanation for this\nwould be that these multiple, orbitally-split pockets con-\nsist of back-folded and hybridized hole-like and electron-\nlike bands which started life as the \u0000 and X-point band\nstructure features in the tetragonal phase, respectively.\nAs theE(k)-sketches indicate, the data still contains\nremnants resembling the tetragonal bands. However,\nFIG. 4: Detailed X-point Fermi surface topology of Eu122\n(T= 10 K, h\u0017= 84 eV). ( a) FS map covering the ( \u0019;\u0019)-\n(\u0000\u0019;\u0019)-(\u0019;\u0019) region of k-space (left) with a 'guide' sketch\n(right), including the high temperature Fermi surfaces in gray.\n(b) (E;k)-maps taken from the cuts in k-space indicated in\n(a) partnered by sketches in which electron-like bands are\nindicated in red and hole-like in blue. ( c) X-point FS map,\nh\u0017= 100 eV, including a sketch showing the location of k-\ncuts 5 and 6. As in Fig. 3, the red shading marks the e\u000bect\nof the symmetry selection rules for certain orbital states.\nthese interact with the back-folded bands such that the\nformer do not cross the Fermi energy, and thus do not\ngive rise to large FS's around \u0000. In Fig. 3(c) one can\nalso see a further case of interaction between a folded\nand non-folded band in that the blue hole-like band -\nwhich is well below the Fermi level - is not symmetric\naroundkx= 0 but is further pushed down by the pres-\nence of the lower binding energy back-folded bands on\nthe right. These data, then, do not support a picture of\nselective modi\fcation of bands from the tetragonal band\nstructure,[10] in which only some bands interact with the\nback-folded structures, a point we will return to in the\nfollowing discussion of the X-point zoom-in.\nWe now turn to the states around the X or ( \u0019;\u0019) point,\nwhich - in general - have received less attention in ARPES\npapers on the parent compounds than have those around\n\u0000. In Fig. 4(a) and (b), data recorded with 84eV pho-\ntons are presented. Cut 1 in panel (b) clearly shows\ntwo pairs of hole-like features crossing EF, giving rise to\nthe two pairs of blue features shown along cut 1 in the\nmain FS sketch [panel (a)]. This is in contrast to cut 2,5\nwhich reveals that the hole-like bands do not approach\nEF: their spectral weight is suppressed up to a binding\nenergy of about 70 meV, which, as also noted in Ref.\n[10], corresponds rather well to the gap observed in op-\ntical conductivity measurements (with 2\u0001 \u001990 meV for\nEu122 [6]). Thus, cut 2 of Fig. 4 attests to the opening\nof a gap, leading to the break-up of the large elliptical\nX-point FS's into signi\fcantly smaller structures.\nThe data of cut 3 take us a step further: here the cut\njust grazes the end of the longer hole-like arc, and if we\ntake this result at face value the droplet-like FS would, in\nfact, have to be `open'. Thus, for the kzvalue close to \u0019/c\n(i.e., that relevant for h \u0017= 84 eV) if we were conduct-\ning a quantum oscillation experiment, there would be no\nclosed FS orbits measurable in the kx;yplane near ( \u0019;\u0019).\nTheE(k)-image along k-space cut 4 clearly shows that\nthe hole-like bands bend back into an electron-like feature\naround X that has its band bottom at a binding energy\nBE\u0019100 meV. This delivers the clue that the crescent-\nshaped, gapped, FS segments around the X-point in the\nAFM orthorhombic phase arise from hybridization of the\nX-centered electron pockets with the back-folded hole-\nlike bands from the \u0000 point.\nWe now switch in Fig. 4(c) to data recorded with\nh\u0017= 100 eV (for which kz\u00190). Comparing the FS\nmaps of panels (a) and (c) and the data of cuts 1 and\n5, we note that for 100 eV photons a clear electron-like\nfeature is observed, located at the opening of the hole-like\narc seen also for 100 eV photons in cut 6. These electron-\nlike sections can be seen to act as a `gate', closing o\u000b the\narc into a continuous, droplet-like FS.\nThus, these data from the X-point can support two\nscenarios: i) \frstly, if the lack of hole-like bands in cut 1\nis taken to be an initial state e\u000bect (i.e., is not simply due\nto a low matrix element), then we see a kz-dependence\nin which for kz= 0 (h\u0017= 100 eV) there are small FS\ndroplets around the ( \u0019;\u0019) points, whereas for kz=\u0019/c\n(h\u0017= 84 eV) the hole-like arcs remain open and thus no\nclosed FS exists near X for this kzvalue. ii) the alterna-\ntive is to posture that there are electron-like `gates' clos-\ning the hole-like arcs for all kzvalues (i.e., the droplets\nare, in fact thin, roughly triangular prisms with their long\naxis alongkz), but that they carry no spectral weight for\nh\u0017= 84 eV.\nWe believe scenario (ii) to be more likely, but irrespec-\ntive of thekzdependence, the large FS pockets present in\nthe paramagnetic phase are broken up into much smaller\nstructures on entering the AFM phase, either with or\nwithout continuity along kz.\nThis \frst ARPES study of the EuFe 2As2parent com-\npound of the pnictide high Tcsuperconductors has\nyielded data of high quality, with excellent `contrast' in\nk-space. This enables a detailed discussion of all Fermi\nsurfaces near both the (0 ;0) and (\u0019;\u0019) points on an equal\nfooting. Thus, bringing together the conclusions fromboth the \u0000 and X-point data, we can state the following\nby way of summary:\n1. On cooling well below the ordering temperature of\nthe iron planes, the large FS's (reversibly) break-up\ninto small droplet-like structures, supporting only a\nfraction of the original, high temperature FS areas.\n2. The fact that both electron-like features are found\nat (0;0) and hole-like at ( \u0019;\u0019) suggests that back-\nfolding of the high-T band structure takes place via\nre\rection in the new Brillouin zone boundaries.\n3. The back-folding picture is strongly supported by\nclear evidence for hybridization between the back-\nfolded and non-folded bands, resulting in shifting of\nbands and the opening of d-wave-like energy gaps\nof order 50-70meV.\n4. Comparison of data probing di\u000berent locations\nalong thekzaxis suggest that either the small FS\nstructures near the ( \u0019;\u0019) point might indeed be\ndroplets {without continuity along kz{ or they take\nthe form of prisms stretching along kz.\n5. Finally, the ordering of the Eu moments below\nca. 20K does not have a noticeable impact on the\nARPES data.\nTaking points 2)\u00004) together, we can close by stating\nthat these ARPES data are in keeping with the conclu-\nsions from recent quantum oscillation and optical data on\notherMFe2As2parent compounds [3, 4, 5, 6], that of-\nfer support for a large suppression of the number of near\nEFelectronic states in the low temperature magnetically\nordered orthorhombic phase.\nWe acknowledge the IFW Dresden for the provision\nof their SES100 spectrometer via the Helmholtz Zen-\ntrum Berlin, and Huib Luigjes for expert technical sup-\nport. This work is part of the research programme of\nthe `Stichting voor Fundamenteel Onderzoek der Ma-\nterie' (FOM). We also acknowledge funding from the EU\n(via I3 contract RII3-CT-2004-506008).\n\u0003Electronic address: sdejong@science.uva.nl\n[1] Yoichi Kamihara, Takumi Watanabe, Masahiro Hirano,\nand Hideo Hosono, J.Am.Chem.Soc. 130, 3296 (2008),\nRen Zhi-An, Lu Wei, Yang Jie, Yi Wei, Shen Xiao-Li,\nZheng-Cai, Che Guang-Can, Dong Xiao-Li, Sun Li-Ling,\nZhou Fang and Zhao Zhong-Xian, Chinese Phys. Lett.\n25, 2215 (2008).\n[2] J. Dong, H. J. Zhang, G. Xu, Z. Li, G. Li, W. Z. Hu, D.\nWu, G. F. Chen, X. Dai, J. L. Luo, Z. Fang and N. L.\nWang, Euro. Phys. Lett. 83, 27006 (2008).\n[3] Suchitra E. Sebastian, J. Gillet, N. Harrison, P. H. C.\nLau, D. J. Sing, C. H. Mielke and G. G. Lonzarich, J.\nPhys.: Condens. Matter 20, 422203 (2008).6\n[4] J. G. Analytis, R. D. McDonald, J.-H. Chu, S. C. Riggs,\nA. F. Bangura, C. Kucharczyk, M. Johannes, I. R. Fisher,\nPhys. Rev. B 80, 064507 (2009).\n[5] W. Z. Hu, J. Dong, G. Li, Z. Li, P. Zheng, G. F. Chen,\nJ. L. Luo and N. L. Wang, Phys. Rev. Lett. 101, 257005\n(2008).\n[6] D. Wu, N. Barisic, N. Drichko, S. Kaiser, A. Faridan, M.\nDressel, S. Jiang, Z. Ren, L. J. Li, G. H. Cao, Z. A. Xu,\nH. S. Jeevan and P. Gegenwart, Phys. Rev. B 79, 155103\n(2009).\n[7] C. Liu, G. D. Samolyuk, Y. Lee, N. Ni, T. Kondo, A.\nF. Santander-Syro, S. L. Bud'ko, J. L. McChesney, E.\nRotenberg, T. Valla, A. V. Fedorov, P. C. Can\feld, B.\nN. Harmon, A. Kaminski, Phys. Rev. Lett. 101, 177005\n(2008).\n[8] J. Fink, S. Thirupathaiah, R. Ovsyannikov, H. A. D urr,1\nR. Follath, Y. Huang, S. de Jong, M. S. Golden, Yu-\nZhong Zhang, H. O. Jeschke, R. Valent, C. Felser, S.\nDastjani Farahani, M. Rotter, and D. Johrendt, Phys.\nRev. B 79, 155118 (2009).\n[9] V. B. Zabolotnyy, D. S. Inosov, D. V. Evtushinsky,\nA.Koitzsch, A. A. Kordyuk, G. L. Sun, J. T. Park, D.\nHaug, V. Hinkov, A. V. Boris, C. T. Lin, M. Knupfer, A.\nN. Yaresko, B. B uchner, A. Varykhalov, R. Follath and\nS. V. Borisenko, Nature London 457, 569 (2008).\n[10] D. Hsieh, Y. Xia, L. Wray, D. Qian, K. K. Gomes, A.\nYazdani, C. F. Chen, J. L. Luo, N. L. Wang and M. Z.\nHasan, arXiv:0812.2289v1 [cond-mat.supr-con] (2008).\n[11] P. Richard, K. Nakayama, T. Sato, M. Neupane, Y.-M.\nXu, J. H. Bowen, G. F. Chen, J. L. Luo, N. L. Wang,\nH. Ding, T. Takahashi, arXiv:0909.0574 [cond-mat.supr-\ncon] (2009)\n[12] Takeshi Kondo, R. M. Fernandes, R. Khasanov, Chang\nLiu, A. D. Palczewski, Ni Ni, M. Shi, A. Bostwick, E.\nRotenberg, J. Schmalian, S. L. Bud'ko, P. C. Can\feld\nand A. Kaminski, arXiv:0905.0271v1 [cond-mat.supr-\ncon] (2009).\n[13] M. Yi, D. H. Lu, G. Analytis, J.-H. Chu, S.-K. Mo, R.-H.\nHe, M. Hashimoto, R. G. Moore, I. I. Mazin, D. J. Singh,\nZ. Hussain, I. R. Fisher and Z.-X. Shen, Phys. Rev. B 80,\n174510 (2009).\n[14] Chang Liu, Takeshi Kondo, Ni Ni, A. D. Palczewski,\nA. Bostwick, G. D. Samolyuk, R. Khasanov, M. Shi, E.\nRotenberg, S. L. Budko, P. C. Can\feld and A. Kaminski,\nPhys. Rev. Lett 102,167004 (2009).\n[15] P. Vilmercati, A. Fedorov, I. Vobornik, Manju U., G.\nPanaccione, A. Goldoni, A. S. Sefat, M. A. McGuire,\nB. C. Sales, R. Jin, D. Mandrus, D. J. Singh and N.\nMannella, Phys. Rev. B 79, 220503(R) (2009)\n[16] V. Brouet, M. Marsi, B. Mansart, A. Nicolaou, A. Taleb-\nIbrahimi, P. Le Fvre, F. Bertran, F. Rullier-Albenque, A.\nForget, and D. Colson, Phys. Rev. B 80, 165115 (2009).\n[17] S. Thirupathaiah, S. de Jong, R. Ovsyannikov, H.A.\nD urr, A. Varykhalov, R. Follath, Y. Huang, R. Huisman,\nM.S. Golden, Yu-Zhong Zhang, H.O. Jeschke, R. Valenti,\nA. Erb, A. Gloskovskii and J. Fink, arXiv:0910.0147v1[cond-mat.supr-con] (2009).\n[18] F. Massee, S. de Jong, Y. Huang, J. Kaas, E. van\nHeumen, J. B. Goedkoop and M. S. Golden, Phys. Rev.\nB80, 140507(R) (2009).\n[19] Miao Gao, Fengjie Ma, Zhong-Yi Lu, and Tao Xiang,\narXiv:0909.5136v1 [cond-mat.mtrl-sci] (2009).\n[20] Y. Xiao and Y. Su and M. Meven and R. Mittal and C. M.\nN. Kumar and T. Chatterji and S. Price and J. Persson\nand N. Kumar and S. K. Dhar and A. Thamizhavel and\nTh. Brueckel, Phys. Rev. B 80, 174424 (2009).\n[21] Myung Joon Han, Quan Yin, Warren E. Pickett, and\nSergey Y. Savrasov, Phys. Rev. Lett. 102, 107003 (2009).\n[22] S. J. Moon, J. H. Shin, D. Parker, W. S. Choi, I. I. Mazin,\nY. S. Lee, J. Y. Kim, N. H. Sung, B. K. Cho, S. H. Khim,\nJ. S. Kim, K. H. Kim, and T. W. Noh, arXiv:0909.3352v1\n[cond-mat.supr-con] (2009).\n[23] H.S. Jeevan and P. Gegenwart, Proc. ICM 2009, J. Phys.:\nConf. Ser., in press.\n[24] H. S. Jeevan, Z. Hossain, Deepa Kasinathan, H. Rosner,\nC. Geibel and P. Gegenwart, Phys. Rev. B 78, 052502\n(2008).\n[25] Fengji Ma, Zhong-Yi Lu and Tao Xiang,\narXiv:0806.3526v2 [cond-mat.mtrl-sci] (2008).\n[26] At the ( \u0019;\u0019) point in Fig. 1a, only two of the four droplet\nFS's are visible, due to the strong polarisation depen-\ndence of the photoemission matrix elements, linked to\nthe presence of a mirror plane in the Fe-plane and the p-\npolarised synchrotron radiation used. For k0\nkvalues well\naway from zero, this selection rule no longer operates and\nall four droplet FS's are seen, such as at ( \u0000\u0019;\u0019).\n[27] The assignment of h \u0017= 84 eV as kz=\u0019/c and h\u0017=\n100 eV askz= 0 is strictly speaking only valid at the\n\u0000 point: the kzvectors at the X-point are about 15% of\nthe reciprocal c-axis lattice vector smaller.\n[28] Between the two data sets recorded at T= 20 K in\npanel (c) of Fig. 1, several warm up- cool down cycles\nwere performed, with slightly di\u000berent spectral weight\ndistribution and kFloci for the droplet-like FS seen be-\ntween the two spectra. The mechanism leading to these\nsmall changes is not clear, but the important, quantita-\ntive point to note is the dis- and re-appearance of the\nsmall FS droplets above and below the iron magnetic or-\ndering temperature.\n[29] C. Liu, T. Kondo, N. Ni, A. D. Palczewski, A. Bostwick,\nG. D. Samolyuk, R. Khasanov, M. Shi, E. Rotenberg, S.\nL. Bud'ko, P. C. Can\feld and A. Kaminski, Phys. Rev.\nLett.102, 167004 (2009).\n[30] Yi Yin, M. Zech, T.L. Williams and J.E. Ho\u000bman, Phys-\nica C 469, 535 (2009).\n[31] We carried out further checks whether the ordering of the\nEu moments had any in\ruence on the electronic structure\nof Eu122 as probed with ARPES, but no di\u000berence was\nfound for several di\u000berent k-cuts taken at 10K and well\nabove 20K."    },    {        "title": "0912.3698v1.A_Search_for_Candidate_TeV_Emitters_in_the_High_latitude_Fermi_Unassociated_Source.pdf",        "content": "2009 Fermi Symposium, Washington, D.C., Nov. 2-5 1\nA Search for Candidate TeV Emitters in the High-latitude Fermi Unassociated\nSources\nP. Fortin and D. Horan\nLaboratoire Leprince Ringuet / \u0013Ecole Polytechnique / CNRS / IN2P3, Palaiseau, France\nE. Ferrara\nNASA Goddard Space Flight Center, Greenbelt, MD, USA\non behalf of the Fermi-LAT Collaboration\nWe report the results of an analysis to identify candidates for very-high-energy (VHE; E >100\nGeV) emission from the high-latitude ( jbj>10) unassociated sources in the year-1 catalog under\ndevelopment by the Fermi Large Area Telescope (LAT) team. These are sources with no known\ncounterparts at other wavelengths. Since VHE instruments are pointed instruments with small\n\felds of view and low duty cycles, their observing programs need to be planned carefully to identify\nthe most promising targets for observation. The scienti\fc potential of combined Fermi and VHE\nobservations has already been demonstrated with a number of joint VHE- Fermi papers. The goal\nof this work is to select the most promising unassociated Fermi sources for joint observations with\nFermi and the VHE instruments.\nI. INTRODUCTION\nThe goal of this work is to \fnd candidate very-high-\nenergy (VHE) emitters from the sources in the Fermi\nyear-1 catalogue. Many of the sources that have al-\nready been detected at VHE are in the catalog so it is\nalready clear that Fermi can provide further targets\nof interest for VHE observatories.\nII. METHOLDOLOGY\nA. Preliminary Scan\nTo begin, a preliminary scan of the catalog was per-\nformed to select unassociated sources above galactic\nlatitudes of 10\u000ethat had potential to be VHE emit-\nters based on their \rux, spectra and redshift. Unas-\nsociated sources that met the following criteria were\nselected:\n\u000fjGalactic Latitude j>10\u000e\n\u000fFlux (E>100 MeV)>2\u000210\u00009cm\u00002s\u00001\n\u000fPhoton Index <2:0\n\u000fNumber of predicted photons >20\nThis resulted in a list of 80 candidates correspond-\ning to\u001811% of the total number of unassociated\nsources in the year-1 catalog. A skymap of these\nsources in galactic coordinates is shown in Figure 1.\nB. Detailed Analysis\nThese 80 sources were then re-analyzed to search\nfor curvature in their spectra; the sources in the cat-\nFIG. 1: Spatial distribution of the 80 candidates selected\nfor this study in galactic coordinates. The solid red circles\ncorrespond to the top 10 candidates shown in Table I. The\ngreen-shaded region corresponds to the \u000610\u000eexclusion re-\ngion centered on the galactic plane.\nalog were all \ft with power-law spectra, which gives,\nto \frst order, a good indication of their properties\nacross the Fermi energy band but does not always\ndescribe accurately their spectra above 1 GeV. We in-\ncluded only those data at energies above 1 GeV from\nthese sources to see if the resulting \rux agrees with the\npower-law \ft in the catalog. For each source, events\nwere selected from a region of 10\u000eradius centered on\nthe catalog coordinates for that source. These data\nwere analyzed using the standard Fermi analysis soft-\nware ( ScienceTools v9r15p3; IRF P6 V3DIFFUSE )\navailable from the HEASARC. All of the Fermi\nsources in the \feld of view were modeled and the\nbackground emission was modeled using a galactic dif-\nfuse emission model and an isotropic component [1].\nEvents were analyzed using an unbinned maximum\nlikelihood method [2], [3].\nThe sources were modeled using a power law cover-\ning two overlapping energy ranges: 100 MeV - 300 GeV\nand 1 GeV - 300 GeV. In addition, a log parabola cov-\nering the full energy range (100 MeV - 300 GeV) was\nused to search for curvature. The di\u000berential \rux,\neConf C091122arXiv:0912.3698v1  [astro-ph.HE]  18 Dec 20092 2009 Fermi Symposium, Washington, D.C., Nov. 2-5\nFIG. 2: SED of good TeV candidate showing no evidence\nfor curvature in the Fermi energy band.\nFIG. 3: SED of bad TeV candidate showing clear evidence\nfor curvature.\nF(E) as a function of energy, E, for the power-law\nand the log-parabola spectral functions are shown in\nequations 1 and 2, respectively.\nF(E) =dN\ndE=F0\u0012E\nE0\u0013\u0000\u0000\n(1)\nF(E) =dN\ndE=F0\u0012E\nEb\u0013\u0000(\u000b+\fln\u0000\nE\nEb\u0001\n)\n(2)\nF0is the di\u000berential \rux at the decorrelation energy,\nE0; \u0000 is the photon index; Ebis the break energy; \u000b\nand\fare the parameters of the log-parabola \ft. The\nlikelihood values of the power-law and log-parabola\n\fts over the full energy range were compared and a\n\u001f2statistical test was used to calculate the probability\nof curvature in the photon spectrum. The covariance\nmatrices were used to calculate the 1-sigma con\fdence\nintervals of the spectral energy distribution (SED) for\neach spectral model. The energy range of the SED\nstarts at 100 MeV and terminates at the energy of the\nmost energetic photon associated with each source.\nFigures 2 and 3 show SED examples for sources with\nand without evidence for curvature.C. Predicting the TeV Flux\nFor sources where there was no evidence for cur-\nvature, the \rux above 100 MeV and the spectral in-\ndex obtained from the power-law \ft were used to es-\ntimate the \rux in the VHE band (200 GeV - 1 TeV).\nTheFermi spectrum was extrapolated to higher ener-\ngies assuming no break in the spectrum and the \rux\nwas absorbed for the extragalactic background light\n(EBL) with the model of Franceschini et al. [4]. A\nredshift value of z= 0:2 was assumed for all sources.\nFor those sources where a log-parabola was a bet-\nter \ft, the \rux and spectral index obtained using the\npower-law \ft above 1 GeV were used for the extrap-\nolation to the VHE band. Given that curvature was\ndetected in the Fermi energy band, it is likely that the\npredicted TeV \rux is overly optimistic. But, given the\nlack of knowledge about the redshift of these sources,\nwe consider the brightest ones as good \\\fller\" tar-\ngets in under-populated RA bands. Table I lists the\n10 most promising unassociated Fermi sources for the\nVHE band using our methodology.\nD. Acknowledgments\nThe Fermi LAT Collaboration acknowledges the\ngenerous support of a number of agencies and insti-\ntutes that have supported the Fermi LAT Collabo-\nration. These include the National Aeronautics and\nSpace Administration and the Department of Energy\nin the United States, the Commissariat \u0012 a l'Energie\nAtomique and the Centre National de la Recherche\nScienti\fque / Institut National de Physique Nucl\u0013 eaire\net de Physique des Particules in France, the Agenzia\nSpaziale Italiana and the Istituto Nazionale di Fisica\nNucleare in Italy, the Ministry of Education, Cul-\nture, Sports, Science and Technology (MEXT), High\nEnergy Accelerator Research Organization (KEK)\nand Japan Aerospace Exploration Agency (JAXA)\nin Japan, and the K. A. Wallenberg Foundation, the\nSwedish Research Council and the Swedish National\nSpace Board in Sweden.\nAdditional support for science analysis during the\noperations phase from the following agencies is also\ngratefully acknowledged: the Istituto Nazionale di As-\ntro\fsica in Italy and the K. A. Wallenberg Foundation\nin Sweden.\nThis research has made use of the NASA/IPAC\nExtragalactic Database (NED) which is operated by\nthe Jet Propulsion Laboratory, California Institute of\nTechnology, under contract with the National Aero-\nnautics and Space Administration. This research has\nmade use of the SIMBAD database, operated at CDS,\nStrasbourg, France.\nThe authors wish to thank JACoW for their guid-\nance in preparing this template. Work supported by\neConf C0911222009 Fermi Symposium, Washington, D.C., Nov. 2-5 3\nSource RA Dec Fermi Flux Fermi Index Emax TeV Flux\nName (J2000) (J2000) [E > 100 MeV] [GeV] 10\u000012cm\u00002s\u00001%Crab\n[E-09]\nTeV 01 00 22 22.1 -18 48 42.9 4:7\u00061:4 1:65\u00060:11 96 10.0 4\nTeV 02 03 38 59.5 +13 12 39.7 4:1\u00062:2 1:64\u00060:18 133 8.91 4\nTeV 03 21 18 18.4 -32 38 27.5 3:3\u00061:7 1:61\u00060:18 35 8.64 3\nTeV 04 05 05 52.4 +61 22 32.7 4:1\u00062:4 1:65\u00060:18 131 8.39 3\nTeV 05 13 07 40.4 -43 00 19.9 13\u00064:1 1:84\u00060:11 31 7.77 3\nTeV 06 23 29 13.4 +37 55 29.9 6:5\u00062:5 1:74\u00060:13 71 7.51 3\nTeV 07 04 39 53.6 -18 58 02.3 3:3\u00061:5 1:65\u00060:17 49 6.65 3\nTeV 08 21 46 40.8 -13 45 30.3 7:9\u00063:1 1:79\u00060:15 52 6.60 3\nTeV 09 20 14 26.0 -00 45 53.2 2:5\u00062:2 1:65\u00060:28 114 5.39 2\nTeV 10 04 27 21.6 +20 26 06.0 2:2\u00062:2 1:64\u00060:29 71 4.89 2\nTABLE I: Predicted \ruxes in the VHE energy band (200 GeV - 1 TeV) for a the top 10 candidates for VHE emission\nfrom the Fermi unassociated sources in the year-1 catalog.\nDepartment of Energy contract DE-AC03-76SF00515.\n[1] http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html\n[2] Cash, W. 1979, ApJ, 228, 939\n[3] Mattox, J.R., et al. 1996, ApJ, 461, 396[4] Franceschini, A., Rodighiero, G., & Vaccari, M. 2008,\nA&A, 487, 837\neConf C091122"    },    {        "title": "1805.04851v1.Evolution_of_individual_quantum_Hall_edge_states_in_the_presence_of_disorder.pdf",        "content": "Evolution of individual quantum Hall edge states in the presence of disorder\nKai-Tong Wang,1Fuming Xu,2Yanxia Xing,3,\u0003and Hong-Kang Zhao1,y\n1School of Physics, Beijing Institute of Technology, Beijing 100081, China\n2Shenzhen Key Laboratory of Advanced Thin Films and Applications,\nCollege of Physics and Energy, Shenzhen University, Shenzhen 518060, China\n3Beijing Key Laboratory of Nanophotonics and Ultra\fne Optoelectronic Systems,\nSchool of Physics, Beijing Institute of Technology, Beijing 100081, China\nEmploying the Bloch eigenmode matching approach, we numerically study the evolution of indi-\nvidual quantum Hall edge states with respect to disorder. As shown by the two-parameter renor-\nmalization group \row of the Hall and Thouless conductances, quantum Hall edge states with high\nChern number nare completely di\u000berent from that of n= 1 case. Two categories of individual\nedge modes are evaluated in a quantum Hall system with high Chern number. Edge states from\nthe lowest Landau level have similar eigenfunctions which are well localized at the system edge and\nindependent of the Fermi energy. On the other hand, at \fxed Fermi energy, the edge state from\nhigher Landau levels has larger expansion, which leads to less stable quantum Hall states at high\nFermi energies. By presenting the local current density distribution, the in\ruence of disorder on\neigenmode-resolved edge states is vividly demonstrated.\nPACS numbers: 73.23.-b, 73.43.Nq, 72.10.Bg, 73.50.-h\nI. INTRODUCTION\nInteger quantum Hall e\u000bect(IQHE) has attracted in-\ntensive research attention since proposed in 1980.1Dif-\nferent from the mono-gapped system such as quantum\nanomalous Hall(QAH) and quantum spin Hall(QSH) sys-\ntems, the band structure of IQHE system is multi-\ngapped, which is characterized by high Chern num-\nbern. As a result, the quantum phase transition\nof IQHE is of particular interest, such as the metal-\ninsulator transition(MIT).2,3To study the transition be-\nhavior of QH states, the two-parameter scaling theory\nwas proposed4,5and numerical investigations were per-\nformed on a QH system to verify the two-parameter\n\row.6,7Besides, global phase diagram8of IQHE pre-\ndicted that there was no direct transition from general\nQH states to insulator under the in\ruence of perturba-\ntions or disorders for high occupation case. But later\nexperiments9{11and theories12{14indicated that direct\nMIT ofn>1 QH state was achievable.\nAll the prominent features of MIT in QH systems are\ndue to the appearance of multiple edge states at the de-\nvice's boundaries15,16. These edge states originate from\ndi\u000berent Landau levels induced by external magnetic\n\feld, and the number of edge states equals the Chern\nnumbern. Unlike the QAH chiral edge states and QSH\nhelical edge states17,18, QH edge states are robust against\nany type of disorder, static or spin-dependent. During\nthe metal-insulator transition of QH states, edge states\nfrom di\u000berent Landau levels respond distinctly to disor-\nder, since those gapped Landau levels are closely linked\nthrough edge states. Therefore, the in\ruence of disor-\nder on edge states from di\u000berent Landau levels remains\na fascinating issue, and the underlying mechanism trig-\ngers our research interest. For simplicity, the QH states\ndiscussed here and after all refer to IQHE.\nIn this work, we \frst investigate the two-parameterrenormalization group(RG) \row of QH states in\n2-dimensional lattice system based on a supercell\nsystem19{22. The renormalization group \row for the\nn= 2 Landau level is completely di\u000berent from that of\nn= 1, suggesting the fascinating MIT feature of QH sys-\ntems with high Chern number. To study the phase tran-\nsition of QH systems, we calculate the Hall and Thouless\nconductances7,23{25within the lattice gauge7. Numerical\nresults con\frm that the n > 1 QH state directly trans-\nforms into insulator but the critical transition disorder\nstrength is smaller compared with the case of n= 1.\nFurthermore, by adopting non-equilibrium Green's func-\ntion(NEGF) formalism26and Bloch eigenmode match-\ning approach27{29, we calculate the eigenmode-resolved\ntransmission coe\u000ecient of individual QH edge states with\nrespect to disorder. The results show that, probability\ndistractions of edge states from the lowest Landau level\nare always localized at the lattice edge despite of the\nvariation of Fermi energy, which keeps these edge states\nrobust and stable against disorder. On the other hand,\nthe edge state originated from higher Landau levels has\nlarger expansion across the system and they are more\nsensitive to disorder. The quality of QH states at the\nsame high Chern number with di\u000berent Fermi energy is\nalso discussed. By showing the local current density dis-\ntributions, the evolution of individual edge states with\nrespect to disorder is intuitively displayed.\nThe rest of the paper is organized as follows. In Sec.II,\nthe theoretical formalisms are introduced and we derive\nthe transmission coe\u000ecients and the local current den-\nsity for the propagating modes. Sec.III is the numerical\nresults with discussions about our work. Finally, a brief\nsummary is presented in Sec.IV.arXiv:1805.04851v1  [cond-mat.mes-hall]  13 May 20182\nII. MODEL AND FORMALISM\nIn this section, the system Hamiltonian and related\nnumerical formalisms are introduced.\nA. Hall and Thouless Conductances\nFor a two-dimensional square lattice with external\nmagnetic \feld, the Hamiltonian is expressed as30\nH=X\ni\u000fidy\nidi\u0000tX\n<i;j>ei\u001eijdy\nidj+ H.c. (1)\nwheredy\ni/diis the creation/annihilation operator for an\nelectron on site i, and<i;j>denotes nearest neigh-\nboring lattice sites. The random on-site potential \u000fiis\nuniformly distributed in the interval [-W/2,W/2] where\nW is the disorder strength, known as the Anderson-type\ndisorder.tis nearest-neighbor coupling strength, which\nis set as the unit of energy and disorder in the calcula-\ntion. In the presence of a perpendicular magnetic \feld,\nan extra phase is induced in the adjacent coupling, which\nis de\fned as \u001eij=e\n\u0016hRj\niA\u0001dlwithAthe magnetic vector\npotential. In the numerical implementation, two di\u000berent\ngauges are adopted. For the eigenmode-resolved trans-\nmission through a ribbon system, we choose the Coulomb\ngauge and the vector potential is simply A= [\u0000By;0;0]\nin Cartesian coordinate. To calculate the Hall and Thou-\nless conductances, we use the lattice gauge with special\ntreatment on the lattice edge7, which allows to reduce\nthe system size at a \fxed magnetic \feld. In the calcula-\ntion, the extra phase through unit lattice measures the\nmagnetic \feld strength. Spin degree of freedom is not\nconsidered in this work.\nTo calculate the transverse Hall conductance gH, it is\nstraightforward to adopt the relation between gHand the\ntotal Chern number\ngH=e2\nh\u0006ncn (2)\nwherecnis the Chern number of the nth band and the\nsummation is taken over all the bands below the Fermi\nenergy. The Chern number de\fned in kspace can be\nwritten as23,31{33\ncn=1\n2\u0019iZ\nBZd2kr\u0002A0\nwithA0=< nkj@j nk>the Berry connection34{36and\n nkthe normalized wave function of the nth Bloch band.\nOn the other hand, the longitudinal conductance can\nbe measured by the band sensitivity subject to changing\nin boundary conditions25,37. For a disordered system,\nThouless and Edwards25,37proposed a relation between\nthe average longitudinal conductance and the band cur-\nvature at the Fermi energy, which is called the Thoulessconductance. At Fermi energy Ef, the Thouless conduc-\ntance is expressed as7,23\ngT(Ef) =e2\nh<j\u0019\n\u0001(Ef)@2Ef\n@k2xj> (3)\nwhere@2Ef\n@k2xdenotes the band curvature at Efand \u0001(Ef)\nis the mean level spacing. The angle bracket stands for\nensemble averaging over the disorder and j:::jmeans tak-\ning the absolute value. Since both gHandgTare in unit\nofe2\nh, the unit is omitted in the numerical results shown\nbelow.\nB. Local Current Density and Eigenmode-resolved\nTransmission Coe\u000ecients\nFor the lattice Hamiltonian Hshown in Eq.(1), based\non the Green's function formalism38{40and the de\fni-\ntion of current density41,42, the di\u000berential local current\ndensity vector at zero temperature reads\ndJx=y=d(eV) =1\n2e[^\u001a^v+ ^v^\u001a] (4)\nwith\n^\u001a=1\n2\u0019Gr\u0000sGa\n^v=\u0000i\n\u0016h[r;H]\nwhere ^\u001ais the nonequilibrium density matrix and ^ vde-\nnotes the velocity matrix in the scattering region. \u0000s=\ni(\u0006r\ns\u0000\u0006a\ns) is the linewidth function and \u0006r\nsdenotes the\nretarded self energy of source lead s, which can be gen-\nerally calculated using the transfer matrix method43,44.\nTo distinguish di\u000berent propagating modes, we employ\nthe mode matching technique27,28,45,46. For a system di-\nvided into slices with onsite Hamiltonian H0and hopping\nHamiltonian H1, the Bloch equation is written as\n(K0+K1eikn+K\u00001e\u0000ikn) n= 0 (5)\nwhereKa=Ha\u0000ESa(a= 0;\u00061),Sis the overlap matrix\nand ndenotes the eigenvector of the n\u0000theigenmode.\nSolving this equation, we can separate right-going and\nleft-going modes. Selecting the propagating modes and\nsubstituting them into Eq.(4), the linewidth function \u0000 s\nbecomes\n\u0000s=X\nsnQsn1\nvsnQy\nsn(6)\nwith\nQsn=Gr\u00001\n00 sn\nvsn=ia\n\u0016h[ yK1 eika\u0000e\u0000ika yK\u00001 ]3\n1.01.21.41.61.82.00.00.10.20.30.40.5gTg\nH(b)0\n.00.20.40.60.81.00.00.10.20.30.40.5gTg\nH(a)\nFIG. 1. (Color online)The renormalization group \rows ex-\ntracted from \fnite size scaling, with system size ranging from\nwid=12 to 27 and disorder strength W2[1,1.5] for magnetic\n\feld\u001e= 2\u0019=9. The arrows indicate the increasing in system\nsize. The \rows in panels (a) and (b) corresponds to Landau\nlevelsn= 1 andn= 2, respectively. The numerical results\nare averaged over 10 ;000 random con\fgurations.\nSimilarly, the eigenmode-resolved transmission coe\u000e-\ncient can be written as\nT\u000bm;\fn=jt\u000bm;\fnj2(7)\nwheret\u000bm;\fnis the transmission matrix element. Pa-\nrameters\u000b;\f andm;n label the leads and modes, re-\nspectively. The transmission matrix element t\u000bm;\fnis\ncalculated from27,28\nt\u000bm;\fn=p\njv\u000bmj~ y\n\u000bmGr\n\u000b\fQ\fn1p\njv\fnj(8)\nHere Gr\n00is the Green's function of an in\fnite ribbon.\nThe eigenvector  msatis\fes the orthonormalization con-\ndition ~ y\nn m=\u000en;m.v\u000bmis the group velocity of in-\njecting or outgoing electrons of the m\u0000theigenmode in\nlead\u000b. Detailed numerical procedures can be found in\nrelevant references cited above.\nIII. NUMERICAL RESULTS AND DISCUSSION\nTo investigate the critical behavior in the topolog-\nical phase transition, we calculate the two-parameter\nrenormalization group(RG) \row of Hall conductance gH\nand Thouless conductance gTdriven by the system size.\nIn the presence of disorder, discrete Landau levels are\nbroadened into Landau bands. A critical energy appears\nin the center of the Landau band, supporting an extended\nstate and separating adjacent QH phases7. Adopting the\nlattice gauge, the extra phase per unit lattice associated\nwith the magnetic \feld is set as \u001e= 2\u0019=9. The cor-\nresponding RG \rows are shown in Fig.1. In Fig.1(a),\nwe plot the RG \row related to the \frst Landau level,\nwhere the \row links the zeroth and the \frst QH states\nwithgH= 0 andgH= 1, respectively. The arrows show\nthe behaviors of gTandgHas the system size increases.\nClearly,gTalways decreases with the increasing of sys-\ntem size. On contrary, the Hall conductance decays to\n0.00.20.40.60.81.00\n2 4 6 8 1 00.00.51.01.52.0 gH wid=12 \nwid=18 \nwid=27(a)0\n.00.51.01.52.0g\nTgHW\n wid=12 \nwid=18 \nwid=27(b)0\n.00.51.01.52.0g\nTFIG. 2. (Color online) Hall conductance gHand Thouless\nconductance gTversus the disorder strength Wfor di\u000berent\nsystem sizes. Panels (a) and (b) correspond to Fermi energies\nEf=\u00003 andEf=\u00002, which locate in the n= 1 andn= 2\nQH plateaus, respectively. The red dashed lines indicate the\ncritical transition disorder strengthes W1= 4:7 andW2= 4:2.\nThe magnetic \feld is \fxed at \u001e= 2\u0019=9.\ngH= 0 at the left and grows to gH= 1 at the right. As\na result, a transition point corresponding to the critical\nenergy emerges in the middle of the \row map, which is\ntypical for mono-gapped topological insulators. The nu-\nmerical results in Fig.1(a) are perfectly consistent with\nprevious theoretical predictions4,5. On the other hand,\nthe RG \row corresponding to the second Landau level\n(n= 2) shown in Fig.1(b) is severely unsymmetric be-\ntweengH= 1 andgH= 2. The critical transition point\nseriously deviates from the \row center but is close to\nthegH= 1 QH state. The deviation and di\u000berence in\nFig.1(b) indicate that the RG \row from the \frst Landau\nlevel is more robust against the disorder. Furthermore,\nthese numerical results show that the quality of di\u000ber-\nent Landau levels are not the same, which will have an\nimpact on the evolution of QH states at metal-insulator\ntransition(MIT) in the presence of disorder.\nTo explore the transition behaviors of QH states from\nconductor to insulator, we investigate the evolution of\nHall and Thouless conductances with disorders at \fxed\nmagnetic \feld \u001e= 2\u0019=9. The Fermi energy is \frst set as\nEf=\u00003, which locates at the center of the n= 1 QH\nplateau. The ensemble averaged Hall and Thouless con-\nductances versus disorder strength Ware shown in Fig.2.\nAt weak disorder, the Hall conductance is well quantized\nasgH= 1 and the Thouless conductance is zero. With\nthe increasing of disorder strength, gHis swiftly reduced4\n02040y\n(a) (c)\n0 20 40 6002040\nxy\n(b)\n0 20 40 60\nx(d)\nFIG. 3. (Color online) Local di\u000berential current density dis-\ntributions for both n= 1 andn= 2 QH states with \fxed mag-\nnetic \feld\u001e= 2\u0019=9. The arrow at the lattice site denotes the\ndirection of the density vector and its length shows the den-\nsity magnitude. The color of arrow changes from blue(light)\nto red(dark) means the increasing of magnitude. Other pa-\nrameters in each panel: panel (a), Ef=\u00003 andW= 0; panel\n(b),Ef=\u00003 andW= 2; panel (c), Ef=\u00002 andW= 0;\npanel (d),Ef=\u00002 andW= 2. The width of the system is\nwid= 54 lattices and xis the transport direction. The results\nare averaged over 10 ;000 con\fgurations.\nand eventually goes to zero, and gTquickly arises to max-\nimum and then gradually decreases. For di\u000berent systems\nsizes, allgHcurves cross at a critical disorder point as\nindicated by the red dashed line in Fig.2(a). Meanwhile,\nmaximal Thouless conductances appear at the same dis-\norder, which are independent of system sizes. Except\nthis critical point, the averaged gTdepends on the sys-\ntem size and performs the behavior of localized states\nat large disorder47,48. The critical disorder strength at\nEf=\u00003 isW1= 4:7, which characterizes MIT of QH\nstates. In Fig.2(b), we plot the dependence of gHand\ngTon the disorder strength WatEf=\u00002, which labels\nthen= 2 QH states. Similar behaviors of gHandgTare\nobserved, and we also \fnd a critical disorder strength\nW2= 4:2 as shown by the red dashed line in Fig.2(b).\nThe results prove that the n= 2 QH states can directly\ntransform to insulator without intermediate QH state.\nBesides, the critical transition disorder strength will de-\ncrease with the increase of Chern number, which means\nhigh QH states are more vulnerable to disorder. Together\nwith the renormalized group \row shown in Fig.1, the\nproperties of high QH states are more attractive since\nthere are multi edge states originating from multi-gapped\nLandau levels. In the following, in order to demonstrate\nthe evolution of individual edge states in the presence of\ndisorder, we will intuitively visualize the edge states and\nexhibit the eigenmode-resolved transmission.\nIn real space, QH states display as a series of edge\nstates in the system with \fnite size. The destruction of\nthese edge states by disorder causes the metal-insulator\n-3- 2- 10 1 2 3 -4-3-2-10b\n1b 2b3b4b5b 6a 6a\n5a\n4a\n3a\n2Ek\nxa1FIG. 4. (Color online) The band structure of a lattice system\nin clean limit with magnetic \feld \u001e= 2\u0019=20 and width wid=\n40. The blue points(a1-a6) corresponds to the same Landau\nlevel in di\u000berent gaps, and the red points(b1-b3) belongs to\ndi\u000berent Landau levels at the same Fermi energy Ef=\u00000:95.\ntransition. To intuitively visualize the edge states, we cal-\nculate the local current density for the n= 1 andn= 2\nQH states and show the numerical results in Fig.3. The\nlattice system under investigation has width wid= 54\nand magnetic \feld \u001e= 2\u0019=9. In Fig.3(a) and (c), we\nplot the local current density of QH states for energy\nEf=\u00003(n= 1) andEf=\u00002(n= 2) with disorder\nstrengthW= 0. Clearly, the n= 1 andn= 2 QH states\nare perfectly localized at the upper edge of the lattice and\nno current density exists in the bulk, which is the signa-\nture of edge states. An obvious di\u000berence between the\ndensity patterns of two QH states is that the n= 2 state\nhas a larger expansion in y-direction, which is double-\ndegenerate with gH= 2. When introducing disorder in\nthe lattice, the local current density of both edge states\nare shown in Fig.3(b) and (d) with the same disorder\nstrengthW= 2. The results are obtained through av-\neraging over 10 ;000 random con\fgurations. It's obvious\nthat both edge states are a\u000bected by the disorder. The\nedge-moving electrons injecting from the clean lead are\nscattered into the bulk of the central scattering region.\nThe thick black lines indicate the border between clean\nleads and the disordered region. Comparing Fig.3(b) and\n(d), we \fnd that the n= 1 QH state still has a major\nedge portion with small density distributing in the whole\nscattering region. On the other hand, the edge state of\nn= 2 is almost completely destroyed under the same dis-\norder. We can con\frm from these numerical results that,\nthe edge state of n= 1 QH state is more robust than the\nn= 2 QH state. Hence a larger critical MIT disorder is\nexpected for lower QH state. The n= 2 QH state has\ndouble-degenerate edge states, which belongs to the 1 st\nand 2ndLandau levels, respectively.\nTo further understand the properties of edge states as-\nsociated with di\u000berent Landau levels, we separate degen-5\n0.00.10.20.30.40.50\n1 02 03 04 00.00.10.20.30.40.5Probability Density a1 \na2 \na3 \na4 \na5 \na6(a)Probability Densityy\n grid b1 \nb2 \nb3 \nb4 \nb5 \nb6(b)\nFIG. 5. (Color online) The probability density distributions\nof edge states corresponding to the crossed points in Fig.4.\nPanel (a): density distributions of edge states from the lowest\nLandau level, and the Fermi energies lie in the center of each\ngap. Panel (b): density distributions of edge states from dif-\nferent Landau levels with the same fermi energy Ef=\u00000:95.\nThe lattice width is wid= 40.\nerate edge states and investigate them individually for\nthe high occupation case with Chern number n >1. In\nFig.4, the band structure of a QH system is presented.\nThe lattice under investigation has width wid= 40 with\nmagnetic \feld \u001e= 2\u0019=20. Due to the magnetic \feld,\nthe original parabolic spectrum becomes highly degener-\nated and transforms to discrete Landau levels as shown\nin the \fgure. The edge states cross the gaps and link\nadjacent Landau levels. When scanning Fermi energy\nover the energy band, the edge state emerges one by one.\nThese edge states fall into two distinct categories. At\na high Fermi energy such as Ef=\u00000:95, there are six\nindividual edge states, which are labeled as red points\nfrom b1 to b6. Among these edge states, b1 comes from\nthe \frst Landau level, with b6 from the sixth. Since\nall the Landau levels are gapped and linked, these edge\nstates may have di\u000berent characteristics. On the other\nhand, the blue points (from a1 to a6) are chosen from\nthe same Landau level(the lowest) but locate in the cen-\nter of di\u000berent gaps with various injecting energies. The\nedge qualities of these two categories will be intensively\nstudied in the following. Notice that these energy points\nare non-degenerate.\nIn Fig.5, we present the lattice probability densities of\ntwo series of edge states, the blue points a1-a6 and the\nred points b1-b6 in Fig.4. The probability densities of\nthe blue points, i.e., edge states from the same Landau\nlevel with di\u000berent energies are plotted in Fig.5(a). The\n010203040y\n(b)\n0 10 20 30 40 50 60010203040\nxy\n(c)0.5 1 1.5 2 2.50.40.60.81\nWT\n  \na1\na2\na3\na4\na5\na6\n(a)FIG. 6. (Color online) Panel (a) plots the eigenmode-\nresolved transmission coe\u000ecients versus disorder for the \frst\nLandau level, with Fermi energies locating at the center of dif-\nferent gaps, as shown by blue points (a1-a6) in Fig.4; Panels\n(b) and (c) are the local di\u000berential current density distribu-\ntions corresponding to edge states in the 1 stand 6thgaps with\ndisorder strength W= 1:5. The magnetic \feld is \u001e= 2\u0019=20\nand the lattice has width wid= 40, which are same as in\nFig.4.\nresult shows that in the absence of disorder, edge states\nfrom the lowest Landau level for di\u000berent Fermi ener-\ngies present the same behavior and have almost identical\ndistributions. The eigenfunctions are strictly restrained\non the boundary, acting as edge modes. On contrary,\nin Fig.5(b), the probability densities of edge states from\ndi\u000berent Landau levels with the same energy Ef=\u00000:95\nhave distinct distributions. Clearly, density distribution\nof the eigenmode from the lowest Landau level is perfectly\nlocalized at the lattice edge as indicated by the black solid\nline. Starting from b2, node appears in the eigenfunction\nof edge states and local minimums grow in the probability\ndistributions in Fig.5(b). With the increasing of Chern\nnumbern, the probability distribution gradually spread\ninto the lattice center, which implies that the quality of\ncorresponding edge state decays. For instance, the den-\nsity distribution corresponding to b6 edge state expands\nto one fourth of the lattice in y-direction. These numeri-\ncal facts reveal the generic di\u000berence between edge states\nfrom the two categories. In the following, we will show\nthe properties of these edge states in the presence of dis-\norder, accompanies with local di\u000berential current density\ndistributions de\fned in Eq.(4).\nWith the mode matching method, we are able to study6\nthe properties of individual edge states against disorder.\nSpeci\fcally, the eigenmode-resolved transmission coe\u000e-\ncientTof each propagating mode as well as its current\ndensity distribution in the lattice can be numerically cal-\nculated. We \frst study the evolution of individual edge\nmodes from the \frst Landau level with di\u000berent ener-\ngies. We focus on the lowest 6 gaps as shown by blue\npoints in Fig.4. Here, the injecting energy is set at the\ncenter of each gap. The eigenmode-resolved transmission\ncoe\u000ecients for di\u000berent energies versus disorder strength\nWare shown in Fig.6(a). As presented in Fig.5(a), the\neigenfunctions of edge states from the same Landau level\nare almost identical. However, from Fig.6(a), We can\nsee that these edge states respond di\u000berently to large\ndisorder. The eigenmode-resolved T satis\fes the order\nTa1> Ta2> Ta3> Ta4> Ta5> Ta6in the whole dis-\norder range. We also plot the local di\u000berential current\ndensity distributions with di\u000berent energies in Fig.6(b)\nand (c). It is obvious that at the same disorder W= 1:5,\nthe edge state with lowest energy survives well, and the\n6thedge mode has been destructed in the central scat-\ntering region. Since the energy gap between adjacent\nLandau levels becomes more and more narrow at high\noccupation, the edge state from the same Landau level\nwith higher energy is more vulnerable to disorder.\nFinally, we \fx the Fermi energy and investigate\nedge modes from the same QH state. The n= 6 QH\nstates with two Fermi energies are evaluated and the\neigenmode-resolved transmission coe\u000ecient Tversus dis-\norder strength Ware presented in Fig.7(a). The data are\nobtained through averaging over 10 ;000 con\fgurations.\nHere we have chosen two Fermi energies, Ef1=\u00001:07\n(gray lines) and Ef2=\u00000:95 (color lines), which are\nat the the center and top edge of the 6 thenergy gap,\nrespectively. For weak disorder, all transmission coe\u000e-\ncients are well quantized as T= 1. With the increasing\nof disorder, the eigenmode-resolved Tstarts to decrease\ngradually. A remarkable contrast is that, the resolved\ntransmissions at Ef1is always larger that that at Ef2\nfor the same disorder, which suggests that with the same\nhigh Chern number the QH state at the gap center is\nmore robust than that close to the top edge of the same\ngap. Another intuitive fact is, for both Ef1andEf2, the\norderTb1> Tb2> Tb3> Tb4> Tb5> Tb6at the same\nWis observed in the entire disorder window. This fact\nshows that, for the same Fermi energy, the resolved edge\nstate of a lower Landau level is more stable than that of\na higher Landau level. This result is consistent with the\nconclusion in Fig.5(b), since the edge state from a higher\nLandau level has larger expansion and is more easily\na\u000bected by disorder. It is noticeable that, the resolved\nTfor the QH state at the top gap edge(color lines) is\nmore easily to be distinguished from each than those for\nthe QH state at gap center(gray lines), suggesting that\nMIT of this sensitive QH state is mainly caused by the\nedge states from high Landau levels. In Fig.7(b) and\n(c), we show the local current density distributions of\nedge states b1 and b5 for Ef2=\u00000:95 of then= 6 QH\n010203040y\n(c)\n0 10 20 30 40 50 60010203040\nxy\n(c)0.5 1 1.5 2 2.50.40.60.81\nWT\n  \nb1\nb2\nb3\nb4\nb5\nb6\n(b)(a)FIG. 7. (Color online) Panel (a) shows the eigenmode-\nresolved transmission coe\u000ecients as a function of the disorder\nstrengthWfor \fxed Fermi energies Ef1=\u00001:07 (grey lines)\nandEf2=\u00000:95 (color lines), respectively. Tb1> Tb2>\nTb3>Tb4>Tb5>Tb6is observed in the whole disorder win-\ndow for both Fermi energies. Panels (c) and (d) are the local\ndi\u000berential current density distributions of eigenmodes corre-\nsponding to the 1 stand 5thLandau levels of the n= 6 QH\nstate withEf=\u00000:95 and disorder strength W= 0:7. Other\nparameters are the same as in Fig.6.\nstate. In the clean lead ( x2[0;10]), the 1stedge state\nis sharply localized at the lattice edge and the 5 thone\nhas a larger expansion in y-direction. In the scattering\nregion with the same disorder W= 0:7, the 1stedge\nmode is well preserved while the 5 thedge mode is de-\nstroyed and spread in the whole region. These evidences\nclearly show for the QH state with high Chern num-\nber the edge state from higher Landau level is less stable.\nIV. CONCLUSION\nIn summary, we have studied the evolution of edge\nstates in disordered 2-dimensional quantum Hall sys-\ntems. The two-parameter renormalization group \row\nshows that the QH state with high Chern number n>1\nis completely di\u000berent from that of n= 1. By adopting\nnon-equilibrium Green's function formalism and Bloch\neigenmode matching approach, evolution of eigenmode-\nresolved edge states with respect to disorder is studied\nin detail. We have veri\fed that for the lowest Landau\nlevel, edge states injecting from di\u000berent band gaps have7\nsimilar probability distributions. For a quantum Hall\nstate at \fxed Fermi energy with multiple edge states,\nthe edge state from a lower Landau level is more local-\nized at the system edge, and always more robust against\ndisorder. On contrary, edge modes from higher Landau\nlevels exhibit larger expansion across the system, and\nmore sensitive to disorder. Furthermore, the quality of\nQH states with the same high Chern number heavily de-\npend on the Fermi energy. When Efis at the top gap\nedge, the eigenmode-resolved transmission is more easilyto be distinguished from each other than the gap center\ncase. By presenting the local current density distribu-\ntions, the evolution of eigenmode-resolved edge states in\nthe presence of disorder is intuitively visualized.\nACKNOWLEDGMENTS\nThis work is \fnancially supported by the National Nat-\nural Science Foundation of China (Grants No. 11674024\nand No. 11504240).\n\u0003xingyanxia@bit.edu.cn\nyzhaohonk@bit.edu.cn\n1K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.\n45, 494 (1980).\n2A. Bao, Y.-H. Chen, H.-F. Lin, H.-D. Liu, and X.-Z. Zhang,\nFront. Phys. 10, 106401 (2015).\n3F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355\n(2008).\n4A. M. M. Pruisken, Phys. Rev. Lett. 61, 1297 (1988).\n5D. E. Khmel'nitskii, Pis'ma Zh. Eksp. Teor. Fiz. 38, 454\n(1983)[JETP Lett. 38, 552 (1983)].\n6J. Song and E. Prodan, Eur. Phys. Lett. 105, 37001 (2014).\n7M. A. Werner, A. Brataas, F. von Oppen, and G. Zar\u0013 and,\nPhys. Rev. B 91, 125418 (2015).\n8S. Kivelson, D.-H. Lee, and S.-C. Zhang, Phys. Rev. B 46,\n2223 (1992).\n9S.-H. Song, D. Shahar, D. C. Tsui, Y. H. Xie, and D. Mon-\nroe, Phys. Rev. Lett. 78, 2200 (1997).\n10T. Wang, K. P. Clark, G. F. Spencer, A. M. Mack, and\nW. P. Kirk, Phys. Rev. Lett. 72, 709 (1994).\n11S. V. Kravchenko, W. Mason, J. E. Furneaux, and V. M.\nPudalov, Phys. Rev. Lett. 75, 910 (1995).\n12D. N. Sheng and Z. Y. Weng, Phys. Rev. Lett. 80, 580\n(1998).\n13D. N. Sheng and Z. Y. Weng, Phys. Rev. B 62, 15363\n(2000).\n14X. C. Xie, D. Z. Liu, B. Sundaram, and Q. Niu, Phys. Rev.\nB54, 4966 (1996).\n15B. I. Halperin, Phys. Rev. B 25, 2185 (1982).\n16M. B uttiker, Phys. Rev. B 38, 9375 (1988).\n17Y. Ren, Z. Qiao, and Q. Niu, Rep. Prog. Phys. 79, 066501\n(2016).\n18Y. Ren, J. Zeng, K. Wang, F. Xu, and Z. Qiao, Phys. Rev.\nB96, 155445 (2017).\n19Q. Niu, D. J. Thouless, and Y.-S. Wu, Phys. Rev. B 31,\n3372 (1985).\n20Y. Hatsugai, K. Ishibashi, and Y. Morita, Phys. Rev. Lett.\n83, 2246 (1999).\n21A. M. Essin and J. E. Moore, Phys. Rev. B 76, 165307\n(2007).\n22Y.-Y. Zhang, R.-L. Chu, F.-C. Zhang, and S.-Q. Shen,\nPhys. Rev. B 85, 035107 (2012).\n23Z.-G. Song, Y.-Y. Zhang, J.-T. Song, and S.-S. Li, Sci.\nRep.6, 19018 (2016).\n24K. Nomura, S. Ryu, M. Koshino, C. Mudry, and A. Fu-rusaki, Phys. Rev. Lett. 100, 246806 (2008).\n25J. T. Edwards and D. J. Thouless, J. Phys. C 5, 807\n(1972).\n26J.-S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, Front.\nPhys 9, 673 (2014).\n27P. A. Khomyakov, G. Brocks, V. Karpan, M. Zwierzycki,\nand P. J. Kelly, Phys. Rev. B 72, 035450 (2005).\n28T. Ando, Phys. Rev. B 44, 8017 (1991).\n29L. Zhang, Y. Xing, and J. Wang, Phys. Rev. B 86, 155438\n(2012).\n30Y. Xing, Q.-F. Sun, and J. Wang, Phys. Rev. B 77, 115346\n(2008).\n31T. Fukui and Y. Hatsugai, J. Phys. Soc. Jpn. 76, 053702\n(2007).\n32T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn.\n74, 1674 (2005).\n33T. Fukui and Y. Hatsugai, Phys. Rev. B 75, 121403 (2007).\n34D. J. Thouless, M. Kohmoto, M. P. Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 49, 405 (1982).\n35M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).\n36B. Simon, Phys. Rev. Lett. 51, 2167 (1983).\n37D. Braun, E. Hofstetter, A. MacKinnon, and G. Montam-\nbaux, Phys. Rev. B 55, 7557 (1997).\n38A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B\n50, 5528 (1994).\n39Y. Xing, J. Wang, and Q.-F. Sun, Phys. Rev. B 81, 165425\n(2010).\n40J. Li and S.-Q. Shen, Phys. Rev. B 76, 153302 (2007).\n41H. Jiang, L. Wang, Q.-F. Sun, and X. C. Xie, Phys. Rev.\nB80, 165316 (2009).\n42Y. Xing, L. Zhang, and J. Wang, Phys. Rev. B 84, 035110\n(2011).\n43D. H. Lee and J. D. Joannopoulos, Phys. Rev. B 23, 4997\n(1981).\n44D. H. Lee and J. D. Joannopoulos, Phys. Rev. B 23, 4988\n(1981).\n45S. Sanvito, C. J. Lambert, J. H. Je\u000berson, and A. M.\nBratkovsky, Phys. Rev. B 59, 11936 (1999).\n46I. Rungger and S. Sanvito, Phys. Rev. B 78, 035407 (2008).\n47T. Ando, J. Phys. Soc. Jpn. 52, 1740 (1983).\n48T. Ando, J. Phys. Soc. Jpn. 53, 3101 (1984).\n49C. W. J. Beenakker and H. van Houten, Solid State Phys.\n44, 1 (1991)."    },    {        "title": "0901.2691v1.A_high_resolution__hard_X_ray_photoemission_investigation_of_BaFe__2_As__2___moderate_influence_of_the_surface_and_evidence_for_a_low_degree_of_Fe_3d___As_4p_hybridization_of_the_near_E__F__electronic_states.pdf",        "content": "APS/123-QED\nA high resolution, hard X-ray photoemission investigation of BaFe 2As2: moderate\nin\ruence of the surface and evidence for a low degree of Fe 3d - As 4p hybridization\nof the near-E Felectronic states\nS. de Jong,1,\u0003Y. Huang,1R. Huisman,1F. Massee,1S. Thirupathaiah,2\nM. Gorgoi,2R. Follath,2J. B. Goedkoop,1and M. S. Golden1\n1Van der Waals-Zeeman Institute, University of Amsterdam, NL-1018XE Amsterdam, The Netherlands\n2Helmholz Zentrum Berlin, Albert-Einstein-Strasse 15, 12489 Berlin, Germany\n(Dated: June 4, 2021)\nPhotoemission data taken with hard X-ray radiation on cleaved single crystals of the barium\nparent compound of the MFe2As2pnictide high temperature superconductor family are presented.\nMaking use of the increased bulk-sensitivity upon hard X-ray excitation, and comparing the results\nto data taken at conventional VUV photoemission excitation energies, it is shown that the BaFe 2As2\ncleavage surface provides an electrostatic environment that is slightly di\u000berent to the bulk, most\nlikely in the form of a modi\fed Madelung potential. However, as the data argue against a di\u000berent\nsurface doping level, and the surface-related features in the spectra are by no means as dominating\nas seen in systems such as YBa 2Cu3Ox, we can conclude that the itinerant, near-E Felectronic states\nare almost una\u000bected by the existence of the cleavage surface. Furthermore, exploiting the strong\nchanges in photoionisation cross section between the Fe and As states across the wide photon energy\nrange employed, it is shown that the degree of energetic overlap between the iron 3d and arsenic\n4p valence bands is particularly small at the Fermi level, which can only mean a very low degree of\nhybridization between the Fe 3d and As 4p states near and at E F. Consequently, the itinerancy of\nthe charge carriers in this group of materials involves mainly the Fe 3d - Fe 3d overlap integrals with\nat best a minor role for the Fe 3d - As 4p hopping parameters, and that the states which support\nsuperconductivity upon doping are essentially of Fe 3d character.\nPACS numbers: 74.25.Jb, 74.70.b, 79.60.-i\nIntroduction\nThe recent discovery of an entirely new class of high TC\nsuperconductors based on (quasi) two-dimensional FeAs-\nlayers, rather than on CuO-layers,1has caused signi\fcant\nexcitement in the \feld of condensed matter physics. The\ndiscovery of a new family of superconductors with high\ntransition temperatures, large critical \felds and more\nisotropic properties than the cuprates also gives a window\nof opportunity in terms of future applications. Moreover,\nmany hope that these iron pnictides can help us to gain\nmore insight into the mechanisms that lead to uncon-\nventional superconductivity in general, or may even help\nunravel a now twenty year old mystery: what makes the\nhighTCcuprates superconduct? From the beginning of\nthe cuprate era, surface sensitive probes such as (angle\nresolved) photoemission (AR)PES, and scanning tunnel-\ning spectroscopy (STS), have played an important role\nin determining the electronic structure of the high TC\nsuperconductors.2,3Crucial pieces of the high TCpuzzle\nhave been supplied by these experimental techniques, for\ninstance insight in the superconducting order parameter\nand coupling to identi\fable bosonic modes.2In this light,\nit will be no surprise that also the iron pnictides are al-\nready being studied intensively using the aforementioned\ntechniques.5,6,7,8,9,10,11\nOne important point to keep in mind, however, is that\nthe surface electronic structure of a material can di\u000ber\nfrom the bulk electronic structure, in which case detailed\nknowledge regarding the origin and nature of these dif-ferences is required in order to fully exploit the strong\npoints of techniques such as ARPES and STS to investi-\ngate bulk superconductivity.\nThe pnictides are quasi two-dimensional materials, like\nthe cuprates, where one can assign a formal charge to\neach layer of the crystal structure. As the crystal symme-\ntry at the surface is broken, one can be left with a polar\nsurface and the possibility of having a diverging electri-\ncal \feld at the cleavage plane. To avoid this `polariza-\ntion catastrophe', the surface of a material can be recon-\nstructed, both structurally and/or electronically. An in-\nteresting example of the latter is thought to occur at the\ninterface of perovskite heterostructures, such as LaAlO 3\n/ SrTiO 3. Such heterointerfaces are conducting,12al-\nthough both oxides have a band gap of several electron\nvolts. The diverging electrical \feld is quenched here by\nmeans of a partial charge transfer, giving the interface\nlayer half the charge and opposite sign with respect to\nthe charge of the layer below.\nIn order to compare the surface and bulk properties of\na material one would want to be able to tune the probing\ndepth of the experiment. In a photoemission experiment\nthe escape depth of the photoelectrons is sensitive to their\nkinetic energy, i.e. the excitation energy of the photons.\nConventional (AR)PES experiments are performed with\nphoton energies typically between 20 and 100 eV [in the\n(V)UV range], coinciding with a minimum in the elec-\ntron escape depth and thus probing only the \frst few\n\u0017Angstroms below the surface. By choosing higher exci-\ntation energies in the hard X-ray regime ( h\u0017's of severalarXiv:0901.2691v1  [cond-mat.supr-con]  19 Jan 20092\nFIG. 1: Hard X-ray photoemission data from Ba122. All data taken at room temperature. ( a) Overview spectrum taken with\nh\u0017= 3000 eV. The inset shows a scanning tunneling microscopy topograph (150 \u0002150\u0017A2) from room temperature cleaved\nBa122, displaying a clear surface reconstruction that lacks long range order, taken from Ref. [15], ( b) Zoom of the near valence\nband region displaying the Ba5p (green shaded) and As4s core levels (red shaded) and an approximate inelastic background\nin blue. Representative core level spectra ( h\u0017= 2010 eV) for all three elements are shown in ( c). Displayed are spectra taken\nin a normal emission geometry (black) and 35\u000eo\u000b normal (red), the latter decreasing the bulk sensitivity, as the mean free\npath length of the electron is determined by the photon energy and the escape depth thus by the emission angle (illustrated\nschematically in the inset of the right-most panel). The Fe 2p 3=2peak is \ftted with a single Doniac-Sunjic line shape, displayed\nin green. The inset to the Fe 2p 3=2core level spectrum shows a broadened and background corrected version of this spectrum\nincluding the Fe 2p 1=2line (blue) together with Fe2p core lines from metallic Fe(111) taken from Ref. [20] in green. Note the\nexcellent resemblance. For the Ba 3d 3=2line shown in the right-most panel of (c), the di\u000berence between normal and o\u000b-normal\nemission is highlighted in green. (color online)\nkeV), the probing depth is increased to nanometers or\neven tens of nanometers.\nIn this paper, we present results from a photoemission\nstudy of the undoped parent compound of one of the main\npnictide high-T Csuperconductor families, BaFe 2As2(ab-\nbreviated forthwith as Ba122), presenting core level and\nk-integrated valence band spectra for this composition.\nBa122 has been reported to become superconducting by\nboth hole doping (substituting K for Ba) and electron\ndoping (substituting Co for Fe), with a maximum TCof\n40 K and 22 K, respectively.13,14\nReports in the literature have pointed out that the\ncleavage plane of single crystalline Ba122 is most likely\nto be the Ba block-layer.9,15As this compound con-\nsists of FeAs layers separated by a single Ba layer, thismeans that (in order to obtain a symmetric cleave and\nmaintain charge neutrality) the top barium layer on the\nsurface termination after the cleave should contain half\nthe number of atoms compared to the bulk Ba layers.\nThe surface of Ba122 is thus expected to di\u000ber from the\nbulk, structurally. This has been shown to be the case\nby several groups doing scanning tunneling microscopy,\nwhere a clear reconstruction of the tetragonal unit cell is\nvisible.9,11,15The nature of the reconstruction has been\nreported to be 2x1 [11], or with larger period,15but upon\ncleavage at room temperatures it lacks su\u000ecient long\nrange order to be observed using low energy electron\ndi\u000braction.15An example of such a disordered, recon-\nstructed Ba122 surface is shown in the inset of Fig. 1a.\nThe question is, of course: what will be the e\u000bect of these3\nFIG. 2: Comparison of x-ray and VUV photoemission valence\nband data ( a) Valence band spectra taken with h\u0017= 2010 eV,\nT=room temperature and h\u0017= 125 eV, T=20K [22]. The\ninset shows a broadened version of the h\u0017= 125 eV spectrum,\nrepresenting the Fe 3d pDOS of the valence band. This curve\nhas been subtracted from the one in the left panel to obtain\nthe As 4p pDOS (shown as the red curve in the inset), ( b)\nShallow core levels of Ba122 recorded with h\u0017= 125 eV,\nT=20K. The inset shows a zoom of the Ba 5p and As core\nlevels, with their Lorentzian peak \fts (green shaded and red\nshaded areas, respectively). (color online)\nspeci\fcally surface-related phenomena on the electronic\nstructure of the surface and near surface region?\nBy comparing photoemission measurements carried\nout using hard X-ray radiation and data taken at con-\nventional excitation energies, we \fnd that the former give\ninformation mainly about the bulk electronic structure of\nBa122, while the latter data show signs of an additional\nsurface electronic structure component, representing a\nminor alteration of the electronic environment at the ter-\nmination surface compared to the bulk. We can therefore\nconclude that the surface of this parent compound of the\npnictide 122 high T Csuperconductors is electronically\nreconstructed with respect to the bulk electronic struc-\nture, but that the deviation from the bulk situation is\nmodest. The implications of this altered surface struc-\nture are expected to be small for the near-E Felectronic\nstates.TABLE I: Binding energies (BE) in eV of the measured core\nlevels of Ba122, h\u0017= 3000 eV, T= room temperature. The\nFermi level was determined from the Fermi cut-o\u000b of piece of\ngold foil. The accuracy of the binding energy determination\nis\u0006100 meV. The Fe 3s line coincides with the Ba 4d lines\nand could thus not be resolved. The Fe 3p spin-orbit splitting\ncould also not be resolved.\nCore level BE (eV ) Core level BE (eV)\nAs 2p 1=2 1358.1 Ba 4p 3=2 178.1\nAs 2p 3=2 1322.4 As 3p 1=2 145.0\nBa 3s 1292.8 As 3p 3=2 140.1\nBa 3p 1=2 1136.0 Fe 3s - - -\nBa 3p 3=2 1062.1 Ba 4d 3=2 92.0\nFe 2s 844.9 Ba 4d 5=2 89.5\nBa 3d 3=2 795.1 Fe 3p 1=2;3=2 52.7\nBa 3d 5=2 779.7 As 3d 3=2 41.4\nFe 2p 1=2 719.6 As 3d 5=2 40.7\nFe 2p 3=2 706.5 Ba 5s 29.6\nBa 4s 252.9 Ba 5p 1=2 16.3\nAs 3s 204.2 Ba 5p 3=2 14.4\nBa 4p 1=2 192.2 As 4s 11.6\nHaving clari\fed this point, we are able to use data\nrecorded with widely di\u000bering photon energies to show\nthat the majority of the As 4p and Fe 3d states that\nmake up the valence band (VB) of Ba122, possess rela-\ntively little overlap in energy. This means that the degree\nof hybridization between iron 3d and arsenic 4p orbitals\nis quite small for this compound compared to the Cu-O\nhybridization in the high T Ccuprates. Importantly, the\nnear-E Fstates are almost exclusively Fe 3d, thus the hop-\nping of the itinerant charge carriers in this parent com-\npound of the pnictide high temperature superconductors\ninvolves mainly the Fe 3d- Fe 3d overlap integrals.\nExperimental\nExperiments with photon energies h\u0017= 2010 and\n3000 eV were performed at the KMC-1 beamline at\nthe Helmholtz-Zentrum Berlin, coupled to the Scienta\nR4000 analyzer of the HiKE endstation.16Experiments\nwere carried out at room temperature in a grazing inci-\ndence geometry with a total energy resolution of 300 and\n450 meV for h\u0017= 2010 and 3000 eV, respectively, as de-\ntermined from the width of the Fermi edge of a piece of\ngold foil. Single crystals of Ba122 larger than 1 \u00021 mm2\nwere grown from Sn \rux and cleaved in a vacuum bet-\nter than 1 \u000210\u00009mbar, resulting in shiny, \rat cleavage\nsurfaces. The level of tin impurities in the crystals was\nestimated from core-level intensities to be in the order of\n7 atomic % and these single crystals show a magnetic and\nstructural transition at a reduced temperature of 60 K,\ninstead of the familiarly observed 140 K for Sn impurity-\nfree samples.14\nExperiments with h\u0017= 125 and 140 eV were carried\nout at the UE112{PGMa beamline at the Helmholtz-4\nZentrum Berlin using an SES100 electron analyzer and\nwere performed both at room temperature and at low\ntemperature (25 K), with an experimental energy resolu-\ntion of 30 meV. These experiments were performed on the\nsame batch of single crystals as the HiKE experiments,\nand on sample surfaces cleaved both at room and at low\ntemperature in a vacuum better than 2 \u000210\u000010mbar.\nThe results obtained did not depend on the temperature\nat which the crystal was cleaved. Spectra were taken in\ntransmission ( k-integrating) mode of the electron energy\nanalyzers.\nHard X-ray data\nIn Fig. 1a an overview spectrum of Ba122 taken with\nh\u0017= 3 keV is shown, displaying many core level lines.\nThe small O 1s, C 1s and Ag 3d signals come from sur-\nface contamination of the sample holder and the silver\nloaded epoxy that was used to attach the Ba122 crystal\nto the sample holder17. The binding energies of identi\f-\nable core levels are listed in Table I. Figure 1b shows a\nzoom of the near- EFregion with clearly distinguishable,\nspin orbit split Ba 5p lines at a binding energy (BE) of\n15 eV, the As 4s line at 12 eV and the valence band\n(VB) between BE \u00198 eV andEF. We note that the As\n4s line at 12eV has been confused in the literature with a\ncharge transfer satellite coming from the Fe 3d states of\nthe valence band.18This spectral feature however is un-\nlikely to originate from a satellite, as its spectral weight\nath\u0017= 3 keV is comparable to the entire valence band.\nAt lower excitation energies its relative weight becomes\nsigni\fcantly smaller, see for instance the inset of Fig.\n2b taken with h\u0017= 125 eV. The spectral weight of the\n12 eV feature traces the tabulated photoionisation As 4s\ncross-section values19for these two photon energies, thus\nsupporting an assignment to the As 4s shallow core level.\nFigure 1c shows spectra of representative core levels\nfrom the three elements in Ba122: As 2p, Fe 2p and Ba\n3d, taken with a photon energy of 2010 eV. Comparing\nthe line shapes of the three core levels, it is immediately\nclear that while the As 2p and Ba 3d peaks are quite\nsymmetric, the Fe 2p line is not. The Fe 2p peak can be\n\ftted with a single, Gaussian-broadened Doniac-Sunjic\nline shape with asymmetry parameter \u000b= 0:44, a value\nthat is identical to reported values for elemental Fe.20If\nthe asymmetry in the Ba122 Fe core lines was caused by\nthe presence of high BE charge transfer satellites, one\ncould expect the signature of a shoulder in the spectrum,\nbut instead the high BE side of the core line is com-\npletely smooth. The inset in Fig. 1c shows the measured\nFe 2p spectrum together with a Fe (111) 2p line from\nthe literature.20The measured spectrum has been broad-\nened with a Gaussian with full width of half maximum\nFWHM= 700 meV to compensate for the higher resolu-\ntion of our experiment and additionally a smooth Shirley-\nlike background has been subtracted from the hard X-ray\nexcited data presented here so as to enable a reasonablecomparison of the two spectra. One can see that the two\nspectra coincide perfectly.21The fact that the Ba122 Fe\ncore level lines are so identical to those of elementary\nFe means that the asymmetry of the core lines is best\nthough of as caused by the signi\fcant partial density of\nstates (pDOS) of Fe at the Fermi level. This enables a\ncontinuum of possible excitations during the creation of\nthe core hole, leading to a smooth `loss-tail' at the high\nBE side of the core level line. In turn, the fact that the\nAs 2p core line in Fig. 1c is so symmetric (a \ft with a\nDoniac-Sunjic line form yields an \u000bof only 0.06), means\nthat the As pDOS at the Fermi level is almost negligible\nin comparison with the Fe3d contribution. The core level\nlines were recorded both in normal emission geometry, as\nwell as with an emission angle of 35\u000eo\u000b-normal. The lat-\nter geometry increases the surface sensitivity of the ex-\nperiment, see the inset in 1c, but yields almost identical\nresults as normal emission for the Fe 2p and As 2p core\nlines. The Ba 3d line, however, is broader when recorded\nin o\u000b-normal emission, as can be seen in the right-most\npanel of Fig. 1c. Although the di\u000berence is quite small,\nit is evident that there is a small surface contribution for\nthe Ba core lines, causing a (modest) asymmetry in the\nBa 4d line shape.\nValence band data\nNow focusing on the near EFelectronic states, a zoom\nof the valence band region taken with h\u0017= 2010 eV is\ndepicted in Fig. 2a. One can see that the total band-\nwidth of the valence band is about 6 \u00008 eV, displaying\na sharp peak close to EF(the maximum being centered\nat about 350 meV BE) and a broad hump between 3\nto 7 eV. From our core level data it is already evident\nthat the DOS closest to EFconsists primarily of Fe3d\nstates, but at this particular photon energy ( h\u0017= 2010\neV), the photoionization cross-sections (PIXs),19for Fe\n3d and As 4p are roughly equal, meaning that we can-\nnot - a priori - distinguish between Fe and As pDOS.\nAt photon energies around 100 eV however, the PIXs fa-\nvor Fe 3d with respect to the As 4p states by a factor 70,\nmaking spectra measured with these photon energies rep-\nresentative for the Fe 3d pDOS. In Fig. 2a one can also\nsee a valence band spectrum recorded with h\u0017= 125 eV\n[23], with a large peak at EFand hardly any sign of a\nhump at higher binding energies, showing that the Fe\n3d states are indeed located close to or at EFand that\nthe As 4p states are mainly responsible for the spectral\nweight between 3 and 7 eV in Fig. 2a. The inset in Fig.\n2a shows the di\u000berence between the measured spectra in\npanel (a), whereby the h\u0017= 125 eV spectrum has been\nbroadened with a Gaussian of 350 meV to account for\nthe di\u000berence in resolution and temperature between the\ntwo measurements. This subtraction spectrum gives an\nestimate for the As 4p density of states. One minor re-\nmark is that the relative spectral weight of the As 4p\nto the Fe 3d states would then be roughly 2:1 for the5\nFIG. 3: Surface versus bulk electronic states of Ba122 from\ncomparison of x-ray and VUV photoemission data ( a) Ba\n4d (left) and As 3d (right) core levels recorded with h\u0017=\n140 eV (black). The surface (S), and bulk (B), contributions\nhave been determined by a \ft with 4 Gaussian broadened\nLorentzians and are shown with blue and green shaded ar-\neas, respectively, ( b) For comparison, the same core levels\nrecorded with h\u0017= 3000 eV (red) are shown together with\nthe data from panel (a) (thin black) and Gaussian broadened\nversions of the latter (blue) to compensate for the di\u000berence\nin experimental resolution, ( c) Ba 4d core levels of cleaved\nYBa 2Cu3Ox(x= 6:75) taken with di\u000berent experimental ge-\nometries and photon energies in the hard X-ray regime. Data\nshown courtesy of K. Maiti and co-workers. Note the large\nspectral weight of the surface contribution here and the large\nenergy di\u000berence between bulk (B) and surface (S) states,\nindicated in the \fgure with green and blue vertical lines, re-\nspectively. (color online)hard X-ray data, judging from the area under the curves\nin the inset of Fig. 2, which is a factor two too high\nif compared to tabulated photoionization cross-sections.\nThese cross-sections however are listed for atomic val-\nues and it is not certain they are strictly applicable to\nthe valence band of a material such as Ba122. The im-\nportant, qualitatively robust point we make here is that\nthere is little overlap in energy between the main max-\nima of the Fe 3d (350 meV) and the As 4p states (4 eV).\nIn the literature, several studies, both theoretical24and\nexperimental,25have suggested that the Fe 3d and As 4p\nstates are strongly hybridized, thereby trying to rational-\nize the small measured Fe magnetic moment compared\nto predictions from LDA calculations. Others, however,\nhave argued that the Fe-As overlap is small.26Although,\nfrom our data it is hard to make a quantitative statement\nabout the hybridization between Fe and As, it is clear\nthat electronic states between E Fand the \frst 1.5 eV,\nthat determine the physical properties such as magnetism\nand superconductivity, are (almost) exclusively of Fe 3d\ncharacter. This would not mean the As 4p states are\nof no importance: our data would not exclude an im-\nportant role for the As atoms as a source of screening\nof the iron on-site Coulomb repulsion.26We stress here\nthat our use of the Ba122 compound enables us to draw\nthese conclusions regarding the Fe-As covalence in a re-\nliable manner. Many photoemission studies in the liter-\nature have been carried out on the structurally related\n1111-pnictides25,27,28that, besides arsenic, also contain\noxygen. The O 2p states of these compounds overlap\nin energy with the As 4p states26,29and give a strong\ncontribution to the valence band spectral weight, both\nat low as well as at high photon energies, thus making\na trustworthy disentanglement of the As and Fe partial\ndensity of states practically impossible.\nShallow core level data\nNow zooming in on the form of the As 3d and Ba 4d\nlevels, it is evident that for both core levels the main lines\nhave shoulders, see panel (a) of Fig. 3, taken with h\u0017=\n140 eV. The weight of these features varies slightly from\ncleavage surface to cleavage surface. Note that whereas\nthe shoulder of the As 3d line is always located at the low\nbinding energy side of the main line, the Ba4d shoulders\nare reproducibly situated at the high binding energy side\nof the main line. The size of the binding energy shift\nfor both elements is similar: +( \u0000) 400 meV for Ba (As).\nThe appearance of these extra core level features is most\nnaturally explained in terms of the existence of a surface\nand a bulk contribution. The relative weight of these\ncontributions has been determined by \ftting the spin-\norbit split doublets with, in total, 4 Gaussian broadened\nLorentzians, for both the As 3d and the Ba 4d core levels.\nWe mention in passing that the spin-orbit splitting of the\nAs 3d core level is 670 meV, which is very close to the\nvalue found for metallic arsenic,30and thus we are of the6\nopinion that this value itself cannot be taken as a signal\nof strong Fe - As hybridization, as was done recently in\na photoemission study of a member of the 1111- pnictide\nfamily.25\nFrom Fig. 3a one can see that with respect to that of\nAs, the surface contribution for Ba is very large and sig-\nni\fcantly broadened when compared to the bulk states,\nwhich would support a picture in which the Ba atoms in-\ndeed make up the termination surface after UHV cleavage\nof Ba122 single crystals. The broad energy distribution\nof the Ba surface states might partly be due to adsorp-\ntion of residual gas from the vacuum onto the surface,\nor, more signi\fcantly, to the absence of long range struc-\ntural order in the topmost Ba layer (as shown in the inset\nin Fig. 1a).11,15. The fact that the surface contribution\nfor arsenic is much more narrow than for Ba can also be\nrationalized within the same picture, as the `sub-surface'\nAs layer is far less perturbed by the cleavage, and pos-\nsesses a markedly reduced variation in electronic environ-\nments compared to the top-most Ba layer. Furthermore,\nthe small spectral weight of the surface features in the\nAs spectra compared to the Ba case can be taken as ev-\nidence that the e\u000bect of the presence of the surface is\nrapidly screened away for the atomic layers further be-\nlow the Ba termination layer.\nIn Fig. 3b the same core level spectra as in panel (a)\nare plotted, overlain with the same core levels measured\nwithh\u0017= 3 keV. It is evident that the (much more sur-\nface sensitive) h\u0017= 140eVdata for Ba 4d (left panel) are\nmuch broader at the base, although they were recorded\nwith much higher resolution than the hard X-ray data.\nBroadening of the VUV-excited Ba 4d data, to correct\nfor the di\u000berence in experimental resolution, gives an\nasymmetric line shape with much more spectral weight\nat the high binding energy side than the h\u0017= 3 keV\ndata, emphasizing the fact that the hard X-ray data is\nindeed probing the electronic structure representative for\nthe bulk of Ba122. The As 3d peaks, as measured with\nh\u0017= 140eV, are signi\fcantly narrower compared to the\nX-ray data and broadening yields a spectrum that is, de-\nspite the non-negligible surface component, almost iden-\ntical to the 3 keV data. This also explains why the surface\ncontribution for As could not be resolved from compari-\nson between the 3 keV and 2010 eV X-ray data, while it\nwas evident for the Ba core levels. Also the Fe 3p core\nlevel (not shown) shows an altered peak form when mea-\nsured withh\u0017= 140eVcompared to the hard X-ray data,\nalthough the VUV excited signal is very weak, and thus\nnot amenable to further analysis. In addition, the small\nspin-orbit splitting and the Doniac-Sunjic peak form for\nthis line disqualify a disentanglement of the surface and\nbulk contributions for the Fe 3p signal.\nAn important point to unravel is, of course, the exact\norigin of the di\u000berence between the surface and the bulk\nelectronic environments of Ba122. In general, the shift\nof a core level \u0001 Ecan be explained by a number of\nterms given by the following formula:\u0001E= \u0001\u0016+K\u0001Q+ \u0001VM\u0000\u0001ER,\nwhere \u0001\u0016is the change in the chemical potential,\n\u0001Qthe change in the number of valence electrons of\nthe atom under consideration, \u0001 VMthe change in the\nMadelung potential, and \u0001 ERis the change in the\nextra-atomic relaxation energy due to polarizability of\nthe atoms and the conduction electrons surrounding the\ncreated core hole.31In the case of Ba122 it is a non-\ntrivial exercise to determine exactly the role played by\neach of these terms at the surface. One may expect, at\nleast, that the polarization and the Madelung potential\nare di\u000berent at the surface, although it is di\u000ecult to\ndisentangle these two contributions. A good starting\npoint would therefore be to compare the measured data\nfor Ba122 to a known case from the literature, in which\n(electronic) surface renormalization is known to play an\nimportant role.\nLet us \frst consider the possibility of a (grossly) dif-\nferent charge carrier concentration at the surface, i.e. an\naltered surface doping level. From photoemission inves-\ntigations it is known that certain cuprates which lack a\nnatural cleavage plane, such as YBa 2Cu3Ox(x\u00197:0)\n- YBCO for short - show such a surface doping e\u000bect.32\nFor YBCO this is caused by the fact that the CuO chains\nare ruptured upon cleavage, remaining behind - in the\nform of debris - on the termination surface. These chain\nfragments donate extra holes to the underlying copper-\noxygen plane bilayer. The analogy with Ba122, where\nalso only half of a bulk crystal layer remains on the sur-\nface after cleavage, is obvious. This type of electronic\nsurface reconstruction for YBCO, however, has a drastic\ne\u000bect on the measured core level spectra, as can be seen\nin Fig. 3c. Depicted are Ba 4d core levels taken, in or-\nder of decreasing bulk sensitivity, with h\u0017\u00196 keV and\nh\u0017\u00192 keV in normal emission geometry and h\u0017\u00192 keV\nwith 35\u000eo\u000b-normal emission. The di\u000berence between\nspectra recorded with the latter two sets of experimen-\ntal conditions is still very signi\fcant, showing that one\nhas a sizable surface contribution even with an excita-\ntion energy of 2 keV. This is not the case for Ba122, Fig.\n1c, where the di\u000berence between normal and o\u000b-normal\nemission is nigh on indistinguishable. Also, the shift of\nthe surface states of YBCO is quite large: about 1 eV\nwith respect to the bulk states, more than twice that\nseen in Ba122. Moreover, the increased surface doping\nof YBCO leads to a shift of surface states toward higher\nbinding energy both for the cations and anions (O2\u0000),\nso this mechanism can most likely be disquali\fed as the\ncause of the electronic surface reconstruction of Ba122.\nThe picture for Ba122 shown in Fig. 3a is, in fact, very\nreminiscent of the situation seen in X-ray photoemission\nfrom the GaAs (110) surface, where the core levels are\nknown to contain a surface contribution.33This surface\ncomponent has a shift in binding energy (with respect\nto the main line) that is opposite for the negatively and\npositively charged As and Ga ions: toward lower and\nhigher binding energy, respectively. This shift has been7\nexplained by a change in the Madelung potential at the\nsurface34with a core level shift that happens to be close\nto what we \fnd here for Ba122. Therefore, it is quite\ncredible that the altered surface electronic structure of\nBa122 is caused by a change in the Madelung potential,\nwith the surface doping level itself being very close or\nequal to the bulk value for the Ba122 material. The\nquestion then arises as to what e\u000bect this has on the\nvalence band states, and in particular those close to E F,\nwhich are intimately involved in the superconductivity in\nthe doped Ba122 and related materials.\nWhile one may expect the altered surface Madelung\npotential to have some e\u000bect, one should bear in mind\nthat the near-E Fstates in these systems are band-\nlike and itinerant in nature, whereas the concept of a\nMadelung energy is more generally applicable to local-\nized, ionic electronic levels. Secondly, in the case of GaAs\n(which possesses a band gap in the order of 1 eV), calcula-\ntions have indicated the presence of occupied surface elec-\ntronic states attributable to the altered Madelung poten-\ntial in that system, lying at about 500 meV below the va-\nlence band edge. Yet, no sizable, structured contribution\nfrom these states has been identi\fed in photoemission\nexperiments.35Taking all these considerations together,\nwe conclude that the near- EFelectronic structure at the\nsurface of Ba122 is very close to that of the bulk. Fur-\nthermore, the fact that the surface doping level of Ba122\nlooks to be the same as the bulk (despite the presence\nof a reconstructed and potentially polar Ba termination\nlayer) provides con\fdence that STM and ARPES should\nbe representative probes to investigate the bulk proper-\nties of Ba122, such as spin ordering transition temper-\natures and superconductivity, both in terms of critical\ntemperature and superconducting gap sizes as a function\nof temperature and doping.\nConclusions\nWe have presented high-resolution photoemission data\ntaken with hard X-ray and VUV photon beams on single\ncrystals of the undoped parent compound of the electron\nand hole doped pnictide high temperature superconduc-\ntor, BaFe 2As2. From the line shape of the core levels\nit could be deduced that the near- EFelectronic statesare primarily of itinerant Fe 3d character. By comparing\nthe hard X-ray excited data with results obtained using\nconventional VUV excitation energies, we were able to\ndisentangle the approximate Fe and As partial densities\nof states in the valence band and show that the contri-\nbution of the As 4p states near the Fermi level is very\nsmall indeed. Seeing as this energy region is where the\nFe 3d partial density of states is maximal, this strongly\nsuggests that the degree of hybridization between the Fe\n3d and As 4p states is minimal at and near E Ffor these\nstates, a fact that could be of considerable signi\fcance in\nrelation to the issue of the small magnetic moment found\nat the Fe sites in these compounds.\nThe exploitation of two widely di\u000bering regimes of\nphotoelectron kinetic energies has also enabled us to\nexamine whether the electronic structure of the cleaved\nBa122 surface is the same as or close to that of the bulk.\nBy comparison with the well studied case of GaAs, we\n\fnd that the termination surface - which comprises the\nBa layer of the quasi-2D crystal structure - is likely to\npossess a modi\fed Madelung potential, compared to the\nbulk. However, the core level data do provide evidence\nagainst the existence of a surface region with di\u000bering\ndoping level. The departure of the surface contribution\nto the electronic structure in Ba122 from the bulk\nsituation is very modest in the iron pnictide. Therefore,\nthe distorting e\u000bects of the real cleavage surface on the\ninvestigation of bulk-representative, near- EFelectronic\nproperties of Ba122 and related compounds with surface\nsensitive probes such as angle resolved photoemission\nand scanning tunneling spectroscopy are found to be\nminor in nature.\nWe thank J. Fink and K. Maiti for making the\nYBCO data available to us and for valuable discussion.\nWe would also like to acknowledge the IFW Dresden\nfor the provision of their SES100 spectrometer via the\nHelmholz Zentrum Berlin, and Huib Luigjes for expert\ntechnical support. This work is part of the research\nprogramme of the `Stichting voor Fundamenteel Onder-\nzoek der Materie (FOM)', which is \fnancially supported\nby the `Nederlandse Organisatie voor Wetenschappelijk\nOnderzoek (NWO)'. We also acknowledge funding\nfrom the EU (via I3 contract RII3-CT-2004-506008 at\nBESSY).\n\u0003Electronic address: sdejong@science.uva.nl\n1Yoichi Kamihara, Takumi Watanabe, Masahiro Hirano,\nand Hideo Hosono, J.Am.Chem.Soc. 130, 3296 (2008),\nRen Zhi-An, Lu Wei, Yang Jie, Yi Wei, Shen Xiao-Li,\nZheng-Cai, Che Guang-Can, Dong Xiao-Li, Sun Li-Ling,\nZhou Fang and Zhao Zhong-Xian, Chinese Phys. Lett. 25,\n2215 (2008).\n2A. Damascelli, Z. Hussain and Z.-X. Shen, Rev. Mod. Phys.\n75, 473 (2003).\n3\u001f. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod andC. Renner, Rev. Mod. Phys. 79, 353 (2007).\n4Ding, H., A. F. Bellman, J. C. Campuzano, M. Randeria,\nM. R. Norman, T. Yokoya, T. Takahashi, H. Katayama-\nYoshida, T. Mochiku, K. Kadowaki, G. Jennings, and G.\nP. Brivio, Phys. Rev. Lett. 76, 1533 (1996).\n5D. H. Lu, M. Yi, S.-K. Mo, A. S. Erickson, J. Analytis,\nJ.-H. Chu, D. J. Singh, Z. Hussain, T. H. Geballe, I. R.\nFisher and Z.-X. Shen, Nature London 455, 81 (2008).\n6C. Liu, G. D. Samolyuk, Y. Lee, N. Ni, T. Kondo, A.\nF. Santander-Syro, S. L. Bud'ko, J. L. McChesney, E.8\nRotenberg, T. Valla, A. V. Fedorov, P. C. Can\feld, B. N.\nHarmon and A. Kaminski, Phys. Rev. Lett. 101, 177005\n(2008).\n7H. Ding, P. Richard, K. Nakayama, K. Sugawara, T.\nArakane, Y. Sekiba, A. Takayama, S. Souma, T. Sato, T.\nTakahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L.\nLuo, and N. L. Wang, Europ. Phys. Lett 83, (2008).\n8D. V. Evtushinsky, D. S. Inosov, V. B. Zabolotnyy,\nA.Koitzsch, M. Knupfer, B. Buchner, G. L. Sun, V.\nHinkov, A. V. Boris, C. T. Lin, B. Keimer, A. Varykhalov,\nA. A. Kordyuk and S. V. Borisenko, arXiv:0809.4455v1\n[cond-mat.supr-con] (2008)\n9M. C. Boyer, Kamalesh Chatterjee, W. D. Wise, G.\nF. Chen, J. L. Luo, N. L. Wang and E. W. Hudson,\narXiv:0806.4400v2 [cond-mat.supr-con] (2008).\n10Yi Yin, M. Zech, T. L. Williams, X. F. Wang, G. Wu, X. H.\nChen, J. E. Ho\u000bman, arXiv:0810.1048v1 [cond-mat.supr-\ncon] (2008).\n11D. Hsieh, Y. Xia, L. Wray, D. Qian, K. Gomes, A. Yaz-\ndani, G.F. Chen, J.L. Luo, N.L. Wang and M.Z. Hasan,\narXiv:0812.2289v1 [cond-mat.supr-con] (2008).\n12A. Ohtomo and H.Y. Hwang, Nature London 427, 423\n(2004).\n13H. Chen, Y. Ren, Y. Qiu, Wei Bao, R. H. Liu, G. Wu, T.\nWu, Y. L. Xie, X. F. Wang, Q. Huang and X. H. Chen,\narXiv:0807.3950v1 [cond-mat.supr-con] (2008).\n14Jiun-Haw Chu, James G. Analytis, Chris Kucharczyk\nand Ian R. Fisher, arXiv:0811.2463v1 [cond-mat.supr-con]\n(2008).\n15F. Massee, Y. Huang, R. Huisman, S. de Jong, M. S.\nGolden, and J. B. Goedkoop, arXiv:0812.4536v1 [cond-\nmat.supr-con] (2008).\n16F. Scheafers, M. Mertin and M. Gorgoi, Rev. Sci. Instrum.\n78, 123102 (2007).\n17Due to the extreme grazing incidence geometry necessary\nto better match the inelastic mean-free path length of the\nphotoelectrons and the penetration depth of the hard X-\nrays, it is unavoidable to also pick up signals from the\nsample mounting.\n18H. Ding, K. Nakayama, P. Richard, S. Souma, T. Sato,\nT. Takahashi, M. Neupane, Y.-M. Xu, Z.-H. Pan, A.V.\nFederov, Z. Wang, X. Dai, Z. Fang, G.F. Chen, J.L. Luo\nand N.L. Wang, arXiv:0812.0534v1 [cond-mat.supr-con]\n(2008).\n19J. J. Yeh and I. Lindau, Atomic Data and Nuclear Data\nTables 32, l (1985). These listed values, used throughout\nthis document, are values for the atomic PIXs and are not\nnecessarily directly applicable to more complicated com-\npounds, but can be well used as an estimate to the real\nvalues.\n20A. Fanelsa, R. Schellenberg, F. U. Hillebrecht, E. Kisker,\nA. P. Kaduwela, C. S. Fadley and M. A. Van Hove, Phys.\nRev. B 54, 17962 (1996).\n21The literature spectrum of Fe (111) shown in Fig. 1c has a\nsmall satellite around a binding energy of 712 eV, which is\ndue to the fact that this data was taken with a Mg k\u000blab\nsource. This feature is thus not a property of the Fe (111)Fe 2p core level line shape.\n22Although the hard X-ray and VUV valence band spectra\nin Fig. 2a are taken with di\u000berent temperatures, it was\nchecked and con\frmed that data taken at room tempera-\nture with VUV radiation did not di\u000ber signi\fcantly from\nthe low temperature data.\n23We note that, due to the small angular acceptance of the\nSES100 analyzer, the k-space integration of the h\u0017=\n125 eV spectra is over only part of the 2D Brillouin\nzone. The depicted spectra are integrated over 40% of the\nBrilllouin zone area around \u0000. Using a polar rotation of\nthe sample it was checked and con\frmed that integration\naround di\u000berent parts of the Brillouin zone, for instance\n(\u0019;0), yielded very similar spectra.\n24J. Wu, Ph. Phillips and A. H. Castro Neto, Phys. Rev.\nLett. 101, 126401 (2008).\n25D.R. Garcia, C. Jozwiak, C.G. Hwang, A. Fedorov, S.M.\nHanrahan, S.D. Wilson, C.R. Rotundu, B.K. Freelon,\nR.J. Birgeneau, E. Bourret-Courchesne, and A. Lanzara,\narXiv:0810.3034v1 [cond-mat.supr-con] (2008).\n26G.A. Sawatzky, I.S. El\fmov, J. van den Brink and J. Zaa-\nnen, arXiv:0808.1390v1 [cond-mat.supr-con] (2008).\n27W. Malaeb, T. Yoshida, T. Kataoka, A. Fujimori, M. Kub-\nota, K. Ono, H. Usui, K. Kuroki, R. Arita, H. Aoki,\nY. Kamihara, M. Hirano, H. Hosono, arXiv:0806.3860v1\n[cond-mat.supr-con] (2008).\n28A. Koitzsch, D. Inosov, J. Fink, M. Knupfer, H.Eschrig, S.\nV. Borisenko, G. Behr, A. K ohler, J. Werner, B. B uchner,\nR. Follath and H. A. D urr, Phys. Rev. B 78,180506(R).\n29E. Z. Kurmaev, R. G. Wilks, A. Moewes, N. A. Skorikov,\nYu. A. Izyumov, L. D. Finkelstein, R. H. Li and X. H.\nChen, arXiv:0805.0668v3 [cond-mat.supr-con] (2008).\n30M. K. Bahl and R. L. Watson, Surface Science 54, 540\n(1976).\n31S. H ufner, Photoelectron Spectroscopy (Springer-Verlag,\nBerlin), 1995.\n32M. C. Schabel, C.-H. Park, A. Matsuura, Z.-X. Shen, D.\nA. Bonn, Ruixing Liang, and W. N. Hardy, Phys. Rev. B\n57, 6090 (1998), V. B. Zabolotnyy, S. V. Borisenko, A. A.\nKordyuk, J. Geck, D. S. Inosov, A. Koitzsch, J. Fink, M.\nKnupfer, B. Bchner, S.-L. Drechsler, H. Berger, A. Erb, M.\nLambacher, L. Patthey, V. Hinkov and B. Keimer, Phys.\nRev. B 76, 064519 (2007), M. A. Hossian, J. D. F. Mot-\ntershead, D. Fournier, A. Botswick, J. L. McChesney, E.\nRotenberg, R. Liang, W. N. Hardy, G. A. Sawatzky, I. S.\nEl\fmov, D. A. Bonn and A. Damascelli, Nature Phys. 4,\n527 (2008).\n33D. E. Eastman, T.-C. Chiang, P. Heimann and F. J.\nHimpsel, Phys. Rev. Lett. 45, 656 (1980).\n34A. B. McLean, Surface Science Lett. 220, 671 (1989), J.W.\nDavenport, R.E. Watson, M.L. Perlman and T.K. Sham,\nSolid State Commun. 40, 999 (1981), W. M onch, Solid\nState Commun. 58, 215 (1986).\n35P. E. Gregory and W. E. Spicer, Phys. Rev. B 13, 725\n(1976), D. J. Chadi, Phys. Rev. B 18, 1800 (1978)."    },    {        "title": "2307.09299v2.Analog_of_cosmological_particle_production_in_Dirac_materials.pdf",        "content": "Analog of cosmological particle production in Dirac materials\nMireia Tolosa-Simeón,1Michael M. Scherer,1and Stefan Floerchinger2\n1Institut für Theoretische Physik III, Ruhr-Universität Bochum, D-44801 Bochum, Germany\n2Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany\nTwo-dimensional van der Waals materials have recently been established experimentally as a\nhighly-tunable condensed matter platform, facilitating the controlled manipulation of band structures\nand interactions. In several of these materials, Dirac cones are present in the low-energy regime near\nthe Fermi level. Thus, fermionic excitations emerging in these materials close to the Dirac cones have\na linear dispersion relation near the Fermi surface as massless relativistic Dirac fermions. Here, we\nstudy low-energy fermionic excitations of such Dirac materials in the presence of a mass gap that may\nbe generated by symmetry breaking. Introducing a dynamical Fermi velocity and/or time-dependent\nmass gap for the Dirac quasiparticles, we exhibit the emergence of an analog of cosmological fermion\npair production in terms of observables such as the expected occupation number or two-point\ncorrelation functions. We find that it is necessary and sufficient for quasiparticle production that\nonly the ratio between the mass gap and the Fermi velocity is time-dependent. In this way, we\nestablish that highly-tunable Dirac materials can serve as analog models for cosmological spacetime\ngeometries, in particular, for Friedmann-Lemaître-Robertson-Walker expanding cosmologies. We\nbriefly discuss possibilities for experimental realization.\nI. INTRODUCTION\nCondensed matter systems can be used in various sce-\nnarios to emulate and study phenomena from completely\ndifferent fields of physics such as elementary particle\nphysics or gravity [ 1,2]. Such analog condensed matter\nmodels provide a novel perspective to approach questions\nthat are not directly accessible in the original systems\nas they can potentially be realized experimentally in a\nwell-controlled setup. An important example for suc-\ncessful condensed matter realizations of analog gravity\nscenarios are Bose-Einstein condensates realizing expand-\ning spacetime geometries by exploiting the highly tunable\nenvironment provided by cold-atom setups [ 3–12]. Similar\nstudies for the case of fermions are still missing.\nRecently, two-dimensional moiré materials, such as\ntwisted bilayer graphene (TBG), have been established\nas another highly tunable condensed matter platform al-\nlowing us to manipulate electronic band structures and\ninteraction effects in a controlled manner [ 13–18]. A key\nfeature of moiré materials is that electron bands near the\nFermi level can be tuned to become very narrow. This\nleads to an enhancement of interaction effects and tenta-\ntively supports the formation of correlated states, which\nhave been confirmed experimentally, see, e.g., Refs. [ 15–\n20]. Some moiré materials, including TBG but also Γ-\nvalley transition metal dichalcogenide bilayers [ 21], even\nbelong to the class of Dirac materials [ 22,23], i.e. they\nare characterized by the presence of fermionic low-energy\nexcitations, describedbyaquasirelativisticDiracequation\nwhere the velocity of light is replaced by the Fermi veloc-\nityvF. Dirac cones are typically protected by symmetries,\ne.g., time-reversal and spatial inversion symmetry in the\ncase of TBG [ 24–26]. Hence, upon tuning the bandwidth\nby a symmetry-compatible moiré potential, the Dirac\ncones continue to be present, however, with a modified\nFermi velocity, which can become very small, e.g., close\nto the magic angle in TBG [25, 27].Here, we take these developments as a motivation to\ntheoretically explore an analog gravity scenario for two-\ndimensional moiré Dirac materials in which the Fermi\nvelocity can be tuned dynamically over several orders of\nmagnitude. In a geometric formulation, we show how\nto obtain an effectively time-dependent metric for the\nDirac fermions. In addition, we consider the presence\nof time-dependent Dirac masses that may originate from\nsymmetry breaking in Dirac materials and lead to a finite\nband gap in the energy dispersion [ 26,28–32]. These in-\ngredients are relevant traits of highly tunable moiré Dirac\nmaterials, which allow us to construct an analog model\nfor the phenomenon of cosmological fermion production\nin expanding universes, arising due to a time-dependent\nmetric and conformal symmetry breaking [33–43].\nAccording to the modern understanding of cosmology,\nfermionic matter was created mainly during an epoch\ncalled reheating, following the early fast expansion re-\nferred to as inflation [ 44,45]. All fermionic matter that\nmight have existed before inflation would be diluted dur-\ning this rapid expansion so much that it is believed that\nthe matter existing today must have been created af-\nterwards. The precise mechanism for fermionic matter\nproduction is at present not known, and it contains a\nnumber of mysteries. In particular, one would like to\nunderstand why more matter than anti-matter exists to-\nday, how precisely leptons and baryons have been created,\nand whether any physics beyond the standard model of\nelementary particle physics played a role for this.\nUnfortunately, it is difficult to clarify these questions\nfrom astrophysical observations alone. Most of the infor-\nmation about the reheating phase has been lost during\nthe evolution of the universe and a direct probe of the\nrelevant high energy physics in laboratories (e.g., with\nparticle colliders) also has limitations. This motivates us\nto explore here the possibilities to do quantum simula-\ntions, or analog experiments, with accessible condensed\nmatter systems.arXiv:2307.09299v2  [cond-mat.mes-hall]  21 May 20242\nProducing relativistic fermions (usually as particle-anti-\nparticle pairs) is possible by different mechanisms. One\npossibility would be inelastic collisions or reactions, as\none could also imagine for classical particles in a kinetic\ndescription. But there are also other possibilities. For ex-\nample, a coupling to an oscillating scalar field would make\nthe effective Hamiltonian for the fermions time-dependent\nsuch that excitations get produced when the time de-\npendence is strong enough to make the time evolution\nappreciably non-adiabatic (in the quantum-mechanical\nsense) [37–40]. Time dependencies in the Hamiltonian can\nalso appear for different reasons and another interesting\npossibility besides a time-dependent mass term is the time\ndependence of spacetime metric due to the cosmological\nexpansion itself [46, 47].\nOn first sight one would think that particle production\nis most effective for massless relativistic fermions because\nthere is then no energy gap to overcome. There is a\nsubtlety here, however. Non-interacting, massless rela-\ntivistic fermions are an example for a conformal quantum\nfield theory, and such theories cannot be taken out-of-\nequilibrium by only an isotropic expansion of space as in\nan expanding universe. Only when conformal symmetry is\nbroken, e.g., by a mass term (or energy gap in condensed\nmatter notation) does the time-dependent cosmological\nmetric by itself induce particle production [48, 49].\nIn the condensed matter analogy, the effect of a time-\ndependent spacetime metric can be modeled by a time-\ndependent Fermi velocity vF. However, by the above\narguments, this only induces particle production together\nwith a non-vanishing energy band gap ∆. In fact, we find\nthat it is the time dependence of the ratio ∆/vFthat\nmatters. This implies that the effect we want to study\ncould be induced either by a constant energy gap ∆with\ntime-dependent Fermi velocity vF, or a time-dependent\ngap∆with constant Fermi velocity, or a combination\nthereof. We further exhibit that at non-zero temperature,\nwhen some modes have a thermal occupation, particle\nproduction is suppressed by Pauli blocking. We discuss\ndifferent options for realizing time-dependent ∆/vFin\nour conclusion for several material platforms.\nInteresting observables to confirm and further investi-\ngate fermionic quasiparticle production in Dirac materials\nare various two-point correlation functions. We therefore\nexplore their time- and wavenumber-dependence in detail\nand, finally, we briefly discuss possibilities towards an\nexperimental realization of the proposed quantum simula-\ntion.\nThe remainder of this work is organized as follows.\nIn Sec. II, we derive the spacetime metric correspond-\ning to the low-energy fermionic excitations of a general\nDirac material with dynamical Fermi velocity and we\nmakeaone-to-onemappingwithanexpandingFriedmann-\nLemaître-Robertson-Walker (FLRW) metric. In Sec. III,\nwe discuss the phenomenon of quasiparticle production\nthrough the study of observables such as the occupation\nnumber or two-point correlation functions. In Sec. IV, we\nsummarize and discuss further directions.Notation. We work in SI units. For convenience, we\ndrop the operator hats. The indices from the beginning\nof the Greek alphabet, α, β, correspond to the constant\nspacetime running from 0to2, while the indices from the\nend, µ, ν, refer to the expanding spacetime coordinates\nt,x. Also, vectors are denoted by bold symbols and\nEinstein summation convention is used.\nII. FERMIONIC FIELD IN A SPATIALLY FLAT\nDIRAC MATERIAL\nThe aim of this chapter is to establish the space-\ntime metric formulation of a dynamically tunable, two-\ndimensional, but spatially flat Dirac material, e.g., twisted\nbilayer graphene, and to find a correspondence with\nthe spatially flat Friedmann-Lemaître-Robertson-Walker\n(FLRW) metric [ 50–55]. The FLRW metric provides a\nmathematical framework for modeling the geometry and\ndynamics of the universe on cosmological scales. It as-\nsumes an expanding (or contracting), homogeneous and\nisotropic universe, meaning that, on large scales, the uni-\nverse appears the same in all directions and at all points.\nThisisafundamentalassumptionincosmology, supported\nby observational evidence, such as the isotropic distribu-\ntion of galaxies and the cosmic microwave background\nradiation.\nA. Dirac material metric and graphene\nrepresentation\nGenerally, the dynamics of Dirac fermions in a two-\ndimensional Dirac material can be described by the action\nS[Ψ,¯Ψ] =R\ndtd2xLwith Lagrangian density\nL[Ψ,¯Ψ] =−¯Ψ\u0002\nℏγ0∂t+ℏvF(t)γ·∂+ ∆ j(t)Γj\u0003\nΨ,(1)\nwhere Ψis a four-component Dirac spinor, describing\nlow-energy fermionic excitations near the Fermi surface,\nand ¯Ψ=iΨ†γ0is the Dirac adjoint. The Fermi veloc-\nityvFwill be considered as being time-dependent in the\nfollowing and the spatial part of the Lagrangian contains\nthe4×4matrices γ= (γ1, γ2), which are typically con-\nstructed from an underlying lattice-hopping model. It is\nconvenient to choose the basis of the Dirac field as\nΨ =\nψ+A\nψ+B\nψ−A\nψ−B\n, (2)\nwhere the superscripts AandBmay denote different sub-\nlattices of the considered Dirac material and +and−may\nlabel the two points KandK′in momentum space where\nDirac cones appear as, e.g., in the case of spinless fermions\nhopping on a honeycomb lattice [ 56–60]. We note that\nmoiré Dirac materials can also have larger spinors or have\ncomponents of a different origin, e.g., from layer, spin,3\nor orbital degrees of freedom [ 21,26,61]. Here, we con-\nsider the four-component case as a minimal paradigmatic\nmodel and do not further restrict the microscopic origin\nof the indices A, Band±.\nThe gamma matrices generate a Clifford algebra in\nMinkowski spacetime,\b\nγα, γβ\t\n= 2ηαβ, where ηαβis\nthe Minkowski metric with signature (−,+,+). The\ntime-like gamma matrix can be chosen as γ0=i I2⊗σ3\nand the space-like gamma matrices as γ1=σ3⊗σ2\nandγ2=−I2⊗σ1, with I2being the 2×2identity ma-\ntrix and σthe standard Pauli matrices [ 58,59]. The\nphysical results do not depend on the explicit choice\nof the representation. The two remaining anticommut-\ning gamma matrices can be taken as γ3=σ1⊗σ2and\nγ5=iγ0γ1γ2γ3=σ2⊗σ2. Further, we also introduce\nγ35=γ3γ5=iσ3⊗I2. This specific choice of gamma ma-\ntrices is often referred to as the graphene representation.\nIn addition, we consider energy band gap openings at\nthe Dirac points of the Dirac material. To this end, a\nlinear superposition of mass terms in the Lagrangian den-\nsity(1)is included, where ∆jrepresents the amplitude of\nthe mass gap and can be time-dependent, and the tensor\nΓjdepends on the shape of the gap. Such masses can\nbe caused by symmetry breaking, e.g., when a sample\nof TBG is suspended on a substrate of hexagonal boron\nnitride (h-BN), a band gap is opened due to breaking of\nC2symmetry [ 62–64]. Another possible source can be the\nformation of strongly-correlated states and spontaneous\nsymmetry breaking [ 26,28–32,58–60]. Here, we exclu-\nsively examine masses that preserve Lorentz symmetry.\nThis approach is motivated by our aim to establish a\nconnection with cosmological scenarios, where Lorentz in-\nvariance is typically realized at the level of the Lagrangian.\nTherefore, in a (2+1)-dimensional spacetime the possible\nmasses are proportional to Γj∈ {I4, iγ3, iγ5, iγ35}and\ncombinations of them. However, Lorentz breaking gap\ntermscanbeinterestingaswellfromthecondensedmatter\nperspective and could be studied further in the future.\nFor illustration, we briefly discuss the meaning of the\ndifferent mass tensors for the paradigmatic case of Dirac\nexcitations originating from spinless fermions hopping\non a honeycomb lattice [ 59]: A mass term proportional\nto the identity matrix corresponds to the breaking of\nspatial inversion symmetry. Mass terms proportional\ntoγ3andγ5also break spatial inversion and imply a\nlinear mixing between the two Lorentz group irreducible\nrepresentations, cf. Appendix A for more details. Their\ncombination (γ3cosα+γ5sinα)can be understood as\na Kekulé modulation of the nearest neighbor hopping.\nFinally, a mass term proportional to γ35corresponds to a\nHaldane mass, preserving handedness but breaking time-\nreversal symmetry [ 57,59]. As a result of time-reversal\nsymmetry breaking, a quantum Hall effect is generated\nwithout need of an external magnetic field when this kind\nof mass term is considered. In the extended context of\nmoiré Dirac materials, the meaning of the different mass\ngap contributions may be changed as compared to the\ncase of spinless fermions on a honeycomb lattice, due tothe different roles of the spinor components.\nThe general case allowed by Lorentz symmetry corre-\nsponds to a linear superposition of the different gap terms\nin Eq.(1). This band gap contribution in the Lagrangian\nfor the Dirac material is related to a fermionic mass term\nin the cosmological Lagrangian density in an FLRW ex-\npanding universe. This contribution is essential for the\nphenomenon of particle production.\nB. Dirac fermions in curved spacetime\nFor the following discussion it is helpful to recall briefly\nhow Dirac fermion fields are described in general space-\ntimes, possibly curved. We concentrate on non-interacting\nfermions but allow for a general mass gap.\nThe Lagrangian density is of the form [48, 49, 65, 66]\nLD[Ψ,¯Ψ] =−√g¯Ψ\u0002\nℏγαeµ\nα(∂µ+ Ωµ) +mjΓj\u0003\nΨ.(3)\nDifferent geometric fields enter here besides the actual\nDirac spinor field Ψ(x). One is eµ\nα(x), the inverse of the\nvielbein eα\nµ(x), which can be seen as a differential one-\nform, eα\nµ(x)dxµ, labeled by the index α. The vielbein\nestablished a connection between a local orthonormal\nframe (with metric ηαβ) and the coordinate frame where\nthe metric is\ngµν(x) =eα\nµ(x)eβ\nν(x)ηαβ. (4)\nThe orthonormal frame is where the Clifford algebra with\n{γα, γβ}= 2ηαβis situated. We also use the abbreviation\nfor the determinant of the metric g=−det(gµν(x)).\nBesides the vielbein field we are also using the spin con-\nnection Ωµ(x) =ωµαβ(x)\u0002\nγα, γβ\u0003\n/8, in the appropriate\nrepresentation for Dirac fermions. One can see the spin\nconnection coefficient ωµαβ(x)as a gauge field for local\nLorentz transformations. It can be expressed through\nωµαβ(x) =−ηαγ\u0002\n∂µeγ\nν(x)−Γρ\nµν(x)eγ\nρ(x)\u0003\neν\nβ(x),\n(5)\nin terms of the vielbein field and the connection in the\nmetric frame Γρ\nµν(x). The latter is the Levi-Civita con-\nnection (see Appendix B) when torsion and non-metricity\nare absent.\nThe action in (3)is invariant under general coordinate\nchanges (diffeomorphisms) and local Lorentz transforma-\ntions. In addition to this, one can introduce another\nspacetime related transformation that will be very use-\nful in the following: local Weyl scaling transformations.\nHere, the Dirac field transform with their scaling dimen-\nsion∆Ψ= (d−1)/2 = 1ind= 1+2spacetime dimensions\nas\nΨ(x)→e−∆Ψζ(x)Ψ(x),\n¯Ψ(x)→e−∆Ψζ(x)¯Ψ(x).(6)\nAt the same time the vielbein gets transformed like\neα\nµ(x)→eζ(x)eα\nµ(x)and the metric in the coordinate4\nframe accordingly like gµν(x)→˜gµν(x) =e2ζ(x)gµν(x).\nFor the spin connection one has [49]\nωµαβ→ωµαβ+h\neαµeν\nβ−eβµeν\nαi\n∂νζ .(7)\nUsing γα[γα, γβ] = 2( d−1)γβone can see that the La-\ngrangianin (3)isindeedinvariantunderlocalWeylscaling\ntransformations in the massless case, mjΓj= 0. If this\ngap term is non-vanishing, a Weyl transformation changes\neffectively\nmjΓj→eζ(x)mjΓj. (8)\nIn other words, a previously constant mass parameter\nbecomes after the Weyl scaling in general space- and time-\ndependent. On the other side, one can use a Weyl scaling\nto go from a reference frame where the gap parameter\ndepends in a factorizable way on time or space to another\nreference system where it is constant. Such a transforma-\ntion changes the spacetime metric gµν(x)and the spin\nconnection Ωµ(x). It is in this sense that a non-trivial\nspacetime metric and a time- or space-dependent gap\nparameter are two sides of the same coin.\nC. From Dirac material to an expanding universe\nLow-energy fermionic excitations in Dirac materials in\nthe vicinity of Dirac cones have the characteristic linear\ndispersion relation of quasirelativistic fermions. These\nexcitations propagate within the spacetime geometry set\nby a line element which is governed by a potentially time-\ndependent Fermi velocity.\nWe can now compare the Lagrangian density in (1)with\ntime-dependent Fermi velocity vF(t)and gap parameter\n∆j(t)to the Lagrangian density for a Dirac field in curved\nspacetime in Eq. (3). We find an agreement of the two\ntheories for the choice\n(eµ\nα) =diag(1/vF(t),1,1). (9)\nOne finds therefore in this frame the effective metric of\na Dirac material to be (gµν) =diag\u0000\n−v2\nF(t),1,1\u0001\n, with\ndeterminant√g=vF(t). A small calculation (see Ap-\npendix C) reveals that the spin connection Ωµindeed\nvanishes in this case. One also has to identify\nmj= ∆ j(t)/vF(t). (10)\nLet us initially assume that mjis constant. Despite first\nappearance, this corresponds to a flat spacetime, as it\nbecomes clear when a new time variable ηis chosen such\nthatdη=vF(t)dt, and the invariant spacetime element is\nsimplyds2=−dη2+dx2. Accordingly, in the case that\nthe ratio on the right hand side of (10)is independent\nof time, we do not expect curved spacetime effects like\nparticle production.\nHowever, in general, the ratio on the right hand side of\nEq.(10)isnotindependent of time. In this case, one cando a time-dependent Weyl scaling as specified in Eq. (6).\nWith the choice\neζ(t)=∆j(0)vF(t)\n∆j(t)vF(0), (11)\nafter the transformation, we find then a constant mass\nterm\neζ(t)∆j(t)Γj\nvF(t)=mjΓj. (12)\nNotethatthisassumesthat ∆j(t)isnon-vanishingonlyfor\none index jor, more generally, that the time-dependence\ncan be factored out. This Weyl scaling has changed the\nspin connection, the vielbein fields and the metric,\nds2= ˜gµν(x)dxµdxν\n=e2ζ(η)\u0002\n−dη2+dx2\u0003\n,(13)\nand now they correspond to the case of curved spacetimes,\nsee Appendix C for further details.\nThus, considering a time-evolving Fermi velocity vF(η)\nand gap ∆(η), one can introduce the time-dependent scale\nfactor a(η), establishing a connection with cosmology,\na(η)≡eζ(η), (14)\nand reshape the former Dirac material line element into\nan FLRW line element in conformal time η\nds2=a2(η)\u0002\n−dη2+dx2\u0003\n, (15)\nwith a(η)dη=dt. In a cosmological context, the scale\nfactor a(η)governs the expansion of an FLRW universe\nand, hence, increasing (decreasing) the ratio between the\nFermi velocity and the gap, vF(η)/∆(η), corresponds to\na Dirac material analog of an expanding (contracting)\nuniverse.\nIII. PARTICLE PRODUCTION\nWe now address the phenomenon of fermionic pair pro-\nduction, which arises when a band gap in terms of a Dirac\nmass is considered and a time-dependent perturbation\nmanifests through the Fermi velocity and/or the band\ngap.\nA. Dirac equation and mode functions\nThe generally covariant equation of motion for a Dirac\nfield Ψin a spacetime background given by the time-\ndependent metric (9) reads\n0 =\u0014\nℏγ0∂η+ℏγ·∂+∆j(η)\nvF(η)Γj\u0015\nΨ,(16)\nwhich is the well-known massive Dirac equation.5\nThe Dirac field Ψobeys the equal-time canonical anti-\ncommutation relations. Within the graphene representa-\ntion, the Dirac field satisfies\nn\nΨa(t,x),Ψ†\nb(t,x′)o\n=δ(2)(x−x′)δab,(17)\nwhere a, brefer to the spinor indices [ 49], cf. Eq. (B3)in\nAppendix B for more details.\nIn flat space, it is convenient to expand the components\nof the Dirac field, cf. Eq. (2), in Fourier modes\nψξλ(η,x) =ZΛd2q\n(2π)2ψξλ\nq(η)eiq·x,(18)\nwhere the subscripts ξ={+,−}andλ={A, B}de-\nnote the respective spinor indices. For our theory, we\nexclusively consider the quasirelativistic linear part of\nthe energy bands, i.e. we do not take into account any\nbending of the bands that appears at some energy further\naway from the Fermi level. Therefore, the UV cutoff Λ\nshould be adapted to the extent of the Dirac cone, e.g.\nfor moiré Dirac materials Λ≃1/aMwith aMbeing the\nmoiré lattice constant. For convenience, we suppress Λin\nall expressions unless explicitly needed.\nThe Grassmann fields ψξλ\nqmay be expanded as [48]\nψξλ\nq(η) =uξλ\nq(η)cξ\nq+vξλ\n−q(η)dξ†\n−q,(19)\nwhere we introduce annihilation (creation) operators for\nfermionic cξ(†)\nqandantifermionic dξ(†)\nqDiracquasiparticles.\nThey satisfy the fermionic anticommutation relations\nn\ncξ\nq, cξ′†\nq′o\n=n\ndξ\nq, dξ′†\nq′o\n= (2π)2δ(2)(q−q′)δξξ′,(20)\nwith all the other anticommutators equal to zero so that\nEq. (17) is satisfied.\nIn addition, we introduced the mode functions for\nfermions, uξλ\nq(t), and antifermions, vξλ\nq(t), which are solu-\ntions of the Dirac equation (16)and fully contain the time\ndependence of the Dirac field. They are constrained by\nthe canonical anticommutation relations of the Dirac field\nand by charge conjugation symmetry, as we will discuss\nin the next section.\nB. Symmetry transformations and mode functions\nIn Dirac materials, including the case of spinless\nfermions on a honeycomb lattice and TBG, Dirac cones\nare typically protected by time-reversal and spatial in-\nversion symmetry and they are robust as long as these\ntwo fundamental discrete symmetries are obeyed. For\nexample, for the case of spinless fermions on a honeycomb\nlattice, the time-reversal operator interchanges the Dirac\npoints leaving the sublattices AandBinvariant [24, 56]\nTψξλ=ψξ′λ, (21)with ξ̸=ξ′andT=σ1⊗I2=iγ1γ5; while the spatial\ninversion goes one step further and, apart from reversing\nthe Dirac points, it also switches the sublattices\nPψξλ=ψξ′λ′, (22)\nwhere ξ̸=ξ′andλ̸=λ′, and the spatial inversion matrix\nis written as P=σ1⊗σ1=γ0γ3. For the case of other\nDirac materials, these operators have to be adjusted,\naccordingly. Both symmetries can be broken by certain\nmass terms leading to the opening of a band gap.\nA charge conjugation operator Cis a complex linear uni-\ntary operator exchanging fermions with antifermions, sat-\nisfying C−1γµC=−(γµ)TandC−1=CT=C†=−C[67].\nConsequently, it depends on the choice of the represen-\ntation for the gamma matrices. Within the graphene\nrepresentation, the charge conjugation operator can be\nchosen as C=iγ0γ2C=iI2⊗σ2C. Here, Ctransforms\nannihilation (creation) operators as\nCcξ(†)\nq=dξ(†)\n−q, Cdξ(†)\nq=cξ(†)\n−q. (23)\nThe charge conjugation operator acts on the field as\nCΨ(η,x) = Ψ∗(η,x), (24)\nwhere the ∗superscript on a multi-component quantum\nfield denotes hermitian conjugation of each component of\nthe field, but without transposing the components.\nCharge conjugation invariance implies the following\nrelations between the fermionic and antifermionic mode\nfunctions\nvξA\nq=uξB∗\n−qand vξB\nq=−uξA∗\n−q,(25)\nreducing the number of independent mode functions per\nfield by half. Additionally, the mode functions are re-\nstricted by the equal-time canonical anticommutation\nrelations of the Dirac field, Eq. (17), and by the fermionic\nanticommutation relations satisfied by the annihilation\nand creation operators, Eq. (20). This yields\nuξλ\nquξλ′∗\nq+vξλ\nqvξλ′∗\nq=δλλ′. (26)\nBy using Eq. (25), Eq. (26) becomes\nX\nλ\f\fuξλ\nq\f\f2=\f\fuξA\nq\f\f2+\f\fuξB\nq\f\f2= 1,(27)\nreducing the number of independent mode functions per\nDirac field to two.\nC. Mode equations\nIn a non-static situation, where the Fermi velocity\nand/or the gap are not constant, the Dirac field cannot\nbe expanded in plane waves with respect to time. Instead,\nwe have to solve the corresponding Dirac equation (16)6\nto find the time dependence of the mode functions. Im-\nportantly, we observe that Eq. (16)depends only on the\nratio ∆j(η)/vF(η)and only if this ratio exhibits a time\ndependence, non-trivial effects like pair production can\nappear. In particular, if the gap vanishes, i.e. ∆j(η) = 0,\na time-dependent Fermi velocity vFwould not influence\nEq.(16). This can be interpreted as a consequence of\nconformal symmetry of massless Dirac fermions.\nFor mass terms anticommuting with the massless differ-\nential Dirac operator, which includes γαwith α= 0,1,2,\nwe require (Γj)2=−I4. Correspondingly, for commuting\nmass terms, we need (Γj)2=I4. Then, applying another\nDirac differential operator to the Dirac equation (16),\nwith the sign of the mass term changed only for com-\nmuting mass terms, leads to the second order differential\nequation\n0 =\"\n∂2\nη−∂2+∆2\nj(η)\nℏ2v2\nF(η)−1\nℏ∂η\u0012∆j(η)\nvF(η)\u0013\nγ0Γj#\nΨ.(28)\nGap terms proportional to block diagonal matrices, e.g.,\nthe identity or γ35, do not mix the Dirac points. Conse-\nquently, the mode functions of the different valleys can\nbe analyzed separately. Otherwise, the four components\nof the field have to be considered simultaneously.\nTo solve the Dirac equation (16)we take into account\nthe mode expansion of the Dirac field, cf. Eq. (19), which\nleads to a coupled system of differential equations for\nthe mode functions. For example, for the band gap pro-\nportional to the identity matrix, we obtain for a specific\nchoice of valley ξ=±1,\n0 =\u0012\ni∂η+ ∆/(ℏvF) ξqe−ξiφ\n−ξqeξiφ−i∂η+ ∆/(ℏvF)\u0013\u0012uξA\nq\nuξB\nq\u0013\n.(29)\nHere, the momentum vector has been rewritten in polar\ncoordinates qx+ξiqy=qeξiφ. Equivalently, Eq. (28)\nwith a gap proportional to the identity matrix yields the\nuncoupled second-order differential equations\n0 =\u0014\n∂2\nη+ω2\nq(η)(+)\n−i\nℏ∂η\u0012∆(η)\nvF(η)\u0013\u0015\nuξA(B)\nq (η),(30)\nwhere ω2\nq(η) =q2+ ∆2(η)/(ℏvF(η))2and the sign is fixed\nthrough the diagonal values of γ0entering the last term\nin Eq. (28).\nWe note that Eq. (30)corresponds to a differential har-\nmonic oscillator equation with complex, time-dependent\nfrequency. The mode functions uξλ\nqcan be determined by\nsolving Eq. (29) for a given set of initial conditions.\nD. Hamiltonian and time segments\nThe Hamiltonian of the system is given by\nH=H0+H∆=Z\nd2x¯Ψ\u0000\nℏvFγ·∂+ ∆ jΓj\u0001\nΨ,(31)and, for a time-dependent vFand/or ∆, it depends ex-\nplicitly on time. Accordingly, the energy in excitations\nneed not to be conserved. The kinetic part of the Dirac\nHamiltonian in Eq. (31) in the basis of Eq. (19) reads\nH0=−ℏvFZd2q\n(2π)2X\nξξq\n×n\n2Re\u0002\ne−ξiφuξA∗\nquξB\nq\u0003\u0000\ncξ†\nqcξ\nq−dξ\nqdξ†\nq\u0001\n+h\ne−ξiφ\u0000\nuξB\nq\u00012−eξiφ\u0000\nuξA\nq\u00012i\ndξ\n−qcξ\nq\n+h\neξiφ\u0000\nuξB∗\nq\u00012−e−ξiφ\u0000\nuξA∗\nq\u00012i\ncξ†\nqdξ†\n−qo\n.(32)\nIn the following, we exclusively study a specific mass term\nfor Eq.(31), i.e. the one proportional to the identity,\nΓj=I4,\nH∆= ∆Z\nd2x¯Ψ Ψ. (33)\nOther forms of the mass gap are investigated in Ap-\npendix D. In the basis of Eq. (19)the massive part of\nEq. (31) reads\nH∆= ∆Zd2q\n(2π)2X\nξ\u0002\u0000\n|uξB\nq|2−|uξA\nq|2\u0001\u0000\ncξ†\nqcξ\nq−dξ\nqdξ†\nq\u0001\n−2uξA\nquξB\nqdξ\n−qcξ\nq−2uξA∗\nquξB∗\nqcξ†\nqdξ†\n−q\u0003\n.\nThe full Dirac Hamiltonian then reads [36–38]\nH=Zd2q\n2πX\nξ\u0002\nEξ\nq(η)\u0000\ncξ†\nqcξ\nq−dξ\nqdξ†\nq\u0001\n+Fξ\nq(η)dξ\n−qcξ\nq+Fξ∗\nq(η)cξ†\nqdξ†\n−q\u0003\n,(34)\nwhere\nEξ\nq=−2ξ��vFqRe\u0002\ne−ξiφuξA∗\nquξB\nq\u0003\n+ ∆\u0010\n1−2\f\fuξA\nq\f\f2\u0011\n,\nFξ\nq=ξℏvFqh\neξiφ\u0000\nuξA\nq\u00012−e−ξiφ\u0000\nuξB\nq\u00012i\n−2∆uξA\nquξB\nq,(35)\nwith\u0000\nEξ\nq\u00012+|Fξ\nq|2=ℏ2v2\nFω2\nq.\nWe choose the Hamiltonian to be initially diagonal in\nterms of the set of operators cξ(†)\nqanddξ(†)\nqin a time seg-\nment where the Fermi velocity, vF(η≤ηi)≡v0, and the\nband gap, ∆(η≤ηi)≡∆0, are constant. We refer to this\nstationary time segment as region I, cf. Fig. 1. The initial\nHamiltonian (34)becomes diagonal by taking a conve-\nnientinitialconfigurationofmodefunctionscorresponding\nto a no-particle state, leading to Eξ\nq(η≤ηi) =ℏωI\nqand\nFξ\nq(η≤ηi) = 0. Note that the last term of Eq. (28)\nvanishes in stationary regions and it becomes a massive\nKlein-Gordon equation. Consequently, expanding the\nDirac field in Fourier modes, the mode functions in this7\nηi ηfηΔ0/v0\nΔf/vfΔ/vF\nFIG. 1.Time dependence of the ratio ∆(η)/vF(η)for the\nthree different temporal regions. In regions I (η≤ηi)and III\n(η≥ηf)the mass gap and the Fermi velocity are held constant,\nwhile in region II they become time-dependent. If their time\ndependence in region II is cancelled out, then no particles will\nbe produced. The time dependencies shown in the plot are\ngiven by Eqs. (39) and (38).\nregion are solutions of a harmonic oscillator differential\nequation and can be taken as standard Minkowski modes.\nAccordingly, a possible initial configuration of the mode\nfunctions, diagonalizing the initial Hamiltonian for a band\ngap proportional to the identity matrix, is given by\nuξA,I\nq(η≤ηi) =e−ξiφ/2\n√\n2s\n1−∆0\nℏv0ωIqe−iωI\nqη,\nuξB,I\nq(η≤ηi) =−ξeξiφ/2\n√\n2s\n1 +∆0\nℏv0ωIqe−iωI\nqη,(36)\nwith positive frequency ωI\nq≡ωq(ηi) =p\nq2+ ∆2\n0/(ℏv0)2.\nWith this, the behavior is indeed compatible with the one\nfor Lorentz transformations, see Appendix E. In a similar\nway one can find for each band gap a set of initial mode\nfunctions corresponding to the no-particle state and with\nan initial Hamiltonian in the standard diagonal form. The\n(creation) annihilation fermionic cξ(†)\nqand antifermionic\ndξ(†)\nqoperators associated to the mode functions uξλ\nqde-\nfine an initial \" c-\" and \" d-vacuum\" state |Ω⟩for such\nexcitations,\ncξ\nq|Ω⟩=dξ\nq|Ω⟩= 0. (37)\nHere, the vacuum state |Ω⟩is a free state of excitations,\nwhich describes the ground state for a Fermi system with\na filling up to a certain level. More generally, one can\nconsider a different initial state with excitations, such as\none characterized by a fixed temperature T.\nNow, we assume that at time ηia dynamical time seg-\nment begins. There, the Fermi velocity vF(η)and/or the\nband gap ∆(η), but in particular their ratio ∆(η)/vF(η),\nbecome time-dependent up to a time ηf. We refer to this\ntime segment as region II in the following and the static\nregimefollowingregionIIisregionIII,cf. Fig.1. Inthisre-\ngion, the mode functions are obtained by solving Eq. (16)\ntaking into consideration their initial conditions (36). Thesolution will not be the simple plane waves of Minkowski\nspace, but a more involved function of time.\nThe functional forms of the mass gap ∆(η)and the\nFermi velocity vF(η)for region II are chosen for simplicity\nas linear in the conformal time η,\n∆(η) = ∆ 0−D(η−ηi), (38)\nand\nvF(η) =v0+V(η−ηi), (39)\nwhere VandDare two real, positive, constant factors.\nTheir ratio is shown in Fig. 1. From the condensed matter\npoint of view, tuning the ratio of the band gap and Fermi\nvelocity in a real Dirac material may be achieved by differ-\nent strategies, depending on the actual material system.\nFor example, in a correlated moiré Dirac material, increas-\ning the Fermi velocity, tentatively suppresses the effect of\nthe interactions. Consequently, a decrease of the band gap\ncan be expected. In other Dirac materials, it may even\nbe possible to use an external field to modify the band\ngap. However, note that the choice of time dependence\nis flexible and can be tailored to suit both experimental\nfeasibility and scientific interest. While our work presents\na specific choice of such time dependencies, it can be ad-\njusted to align with the constraints of a real experimental\nsetup. Here, we choose their time-dependencies to be\nlinear in conformal time for illustrative purposes, allowing\nus to present results on the characteristic signatures of\nfermion production in the Figs. 2,3 and 4. From a cos-\nmological point of view, different time dependencies of\nthe ratio can be chosen to study different cosmological\nscenarios, as a(t)∝vF/∆. For instance, vF/∆∝eHt\ncorresponds to a de Sitter universe, vF/∆∝ |t−t0|γ\nwith γ= 2/3to a radiation dominated universe and with\nγ= 1to a matter dominated universe. Other interesting\ncosmologies can be studied by choosing a proper time\ndependence of this ratio.\nAfter the time-dependent perturbation has ceased, i.e.\nfor times η > η f, the Hamiltonian of the system is not\ndiagonal in the basis cξ(†)\nqanddξ(†)\nq. Instead, a new set\nof creation and annihilation operators ˜cξ(†)\nqand˜dξ(†)\nqcan\nbe introduced such that\nH=ℏvFZd2q\n(2π)2ωqX\nξ\u0010\n˜cξ†\nq˜cξ\nq−˜dξ\nq˜dξ†\nq\u0011\n.(40)\nIn this stationary region, we assume the Fermi velocity,\nvF(η≥ηf)≡vf, and the band gap, ∆(η≥ηf)≡∆f, to be\nconstant, again. A set of solutions for the mode functions\ncan be found in terms of Minkowski modes with positive\nfrequencies as for region I by solving Eq. (28). For a gap\nterm proportional to the identity matrix one can choose\n˜uξA,III\nq (η≥ηf) =e−ξiφ/2\n√\n2s\n1−∆f\nℏvfωIIIqe−iωIII\nqη,\n˜uξB,III\nq (η≥ηf) =−ξeξiφ/2\n√\n2s\n1 +∆f\nℏvfωIIIqe−iωIII\nqη,(41)8\nwhere now ωIII\nq≡ωq(ηf) =p\nq2+ ∆2\nf/(ℏvf)2.\nE. Bogoliubov transformation\nThe new set of annihilation (creation) operators ˜cξ(†)\nq\nand ˜dξ(†)\nqassociated to the mode functions ˜uξλ\nqthat are\nMinkowski modes in region III define a new vacuum state\n|˜Ω⟩for such excitations,\n˜cξ\nq|˜Ω⟩=˜dξ\nq|˜Ω⟩= 0. (42)\nThe old mode functions uξλ\nqcan be expressed in terms of\nthe new set ˜uξλ\nqthrough the Bogoliubov transformation\n\u0012uξA\nq\nuξB∗\nq\u0013\n=X\nζ\u0012αξζ\nq−\u0000\nβξζ\nq\u0001∗\nβξζ\nq\u0000\nαξζ\nq\u0001∗\u0013\u0012˜uζA\nq\n˜uζB∗\nq\u0013\n(43)\nwhich, in region III, corresponds to a linear superposition\nof positive and negative frequency-mode solutions, with\nαζξ\nqandβζξ\nqbeing complex-valued and time-independent\nBogoliubov coefficients. Here, we have introduced a new\nsuperscript ζfor the new set of operators, which is not\nequivalent to ξwhen the mass term mixes the Dirac\npoints. Otherwise, when the mass term is block diagonal\nas the one studied here, there are only four non-vanishing\nBogoliubov coefficients matrix elements for ζ=ξ.\nTherefore, the Bogoliubov transformation reduces to\nuξA\nq=αξ\nq˜uξA\nq−βξ∗\nq˜uξB∗\nq,\nuξB\nq=αξ\nq˜uξB\nq+βξ∗\nq˜uξA∗\nq,(44)\nwhere the Bogoliubov coefficients are simplified by drop-\nping one of the superscripts αξξ\nq≡αξ\nqandβξξ\nq≡βξ\nq.\nFurthermore, one can introduce the general Bogoliubov\ntransformation that connects the two sets of creation and\nannihilation operators\n\u0012˜cζ\nq\n˜dζ†\n−q\u0013\n=X\nξ\u0012αζξ\nq βζξ\nq\n−\u0000\nβζξ\nq\u0001∗\u0000\nαζξ\nq\u0001∗\u0013\u0012cξ\nq\ndξ†\n−q\u0013\n(45)\nwhich for a non-mixing mass term reduces to\n˜cξ\nq=αξ\nqcξ\nq+βξ\nqdξ†\n−q,\n˜dξ†\n−q=−βξ∗\nqcξ\nq+αξ∗\nqdξ†\n−q.(46)\nThe new set of operators also satisfies the canonical an-\nticommutation relations in Eq. (20)from which the fol-\nlowing identities for the Bogoliubov coefficients matrix\nelements can be derived\nX\nξ\u0010\nαζξ\nqαζ′ξ∗\nq+βζξ\nqβζ′ξ∗\nq\u0011\n≡δζζ′(47)\nand\nX\nξ\u0010\nαζξ\nqβζ′ξ\nq−βζξ\nqαζ′ξ\nq\u0011\n≡0. (48)Eq.(47)guarantees that the space density cannot exceed\nunity, as expected from the Pauli exclusion principle [ 43].\nFor a mass that does not mix the Dirac points the former\nidentity reduces to |αξ\nq|2+|βξ\nq|2= 1[35, 38].\nThe Bogoliubov coefficients are obtained from diago-\nnalizing the Hamiltonian after the dynamical process by\nperforming a Bogoliubov transformation (46) such that\nβξ\nq\nαξ\nq=Fξ∗\nq\nℏωq+Eξ\nqand|βξ\nq|2=ℏωq−Eξ\nq\n2ℏωq.(49)\nF. Number of excitations and correlation functions\nIn this section, we study observables that give signa-\ntures of quasiparticle production. These observables can\nbe investigated starting from an initial vacuum state or\nfrom a state in which there are already fermionic excita-\ntions, e.g., a thermal state.\nAn initial state at tiwith non-vanishing occupation\nnumber leads to an initial fermionic and antifermionic\nquasiparticle distribution given by\nNξ\nin,c(q) =⟨cξ†\nqcξ\nq⟩and Nξ\nin,d(q) =⟨dξ†\nqdξ\nq⟩,(50)\nrespectively, and consequently, to Pauli blocking. There-\nfore, quasiparticle creation will be suppressed, when start-\ning from an initial state that is different to the vacuum of\nexcitations, as fermions cannot be created in states that\nare already occupied. For instance, one can consider an\ninitial thermal state, in which the amount of fermions\nand antifermions is balanced, i.e. their initial occupation\nnumber will be the same, Nξ\nin≡Nξ\nin,c=Nξ\nin,d, following\nthe Fermi-Dirac statistics\nNξ\nin(T, q) =1\neℏωIq/(kBT)+ 1, (51)\nwith temperature Tand Boltzmann constant kB.\n1. Expected number of quasiparticles\nTo study the occupation number of Dirac fermion-\nantifermion pairs created per mode q, we evaluate the\nnumber operator of the new set of (creation) annihilation\noperators, i.e. ˜cξ(†)\nq,˜dξ(†)\nq, at the initial vacuum state |Ω⟩\nNξ\nq=⟨Ω|˜cξ†\nq˜cξ\nq|Ω⟩=|βξ\nq|2(2π)2δ(2)(0),(52)\nwhere the divergent factor δ(2)(0)is a consequence of\nconsidering an infinite spatial volume, which arises from\nthe anticommutation relation (20)evaluated at equal\nmomentum. The divergent factor can be substituted by\nconsidering a finite spatial volume V. From now on, we\nomit the factors of volume assuming that the number\nof particles are always referred to a unit of volume for\nsimplicity of notation [9, 68].9\n0.0 0.2 0.4 0.6 0.8 1.00.000.050.100.150.200.250.300.35\nq[1/aM]βq2(1-2Nin)0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.5\nq[1/aM]NqT=0T=0.01T=0.02T=0.03T=0.05T=0.1T=0.5T=1\nFIG. 2.Spectrum of produced particles including ther-\nmal Pauli blocking, cf. the last term of the expected number\nof quasiparticles in Eq. (54). The inset shows the expected\nnumber of quasiparticles, Nq, of fermionic excitations in one\nDirac point as a function of momentum qin units of 1/aM.\nThis has been calculated for a band gap term proportional to\nthe identity matrix starting with an initial no-particle state\nwith βin\nq= 0. The same plot is obtained for the other non-\nsymmetric Dirac point. The dynamical process (region II) has\na duration of tf−ti=aM/(5v0). The Fermi velocity vFis\nchanged here from v0tovf= 102v0and the band gap from\n∆0= 50 ℏv0/aMto∆f=ℏv0/aMwith the time dependence\ngiven in Eqs. (39)and(38). The colors correspond to dif-\nferent thermal initial states with temperature Tin units of\nℏv0/(kBaM). The same plot is obtained for a gap proportional\ntoγ35starting with its corresponding initial no-particle state.\nAs a consequence of charge conjugation symmetry, the\nexpected occupation number of fermions and antifermions\nis the same. Integrating over all momentum modes one\ngets the quasiparticle number density\nn=Zd2q\n(2π)2X\nξNξ\nq. (53)\nTaking an initial excited state instead of the vacuum or\nground state, the expected number of quasiparticles is\nNξ\nq=⟨˜cξ†\nq˜cξ\nq⟩=Nξ\nin+|βξ\nq|2\u0010\n1−2Nξ\nin\u0011\n,(54)\nwhere the last term corresponds to the Pauli blocking,\nshown in Fig. 2 as well as the expected number of quasi-\nparticles for one Dirac point in the inset. One can observe\nthat the expected number of quasiparticles saturates at\n1/2for all momentum modes for higher temperatures,\nas expected. The higher the temperature, the higher\nthe initial number of excitations and, consequently, fewer\nquasiparticles are produced due to Pauli blocking and the\nmain contribution to the expected number of quasiparti-\ncles comes from the initial number of excitations.2. Two-point correlation functions\nTwo-point correlation functions are also a good indi-\ncator of particle production. Let us start studying the\nequal-time two-point correlation functions in momentum\nspace ⟨¯Ψq(t)γαΨq′(t)⟩forα= 0,1,2. The zero compo-\nnent of this set of two-point correlation functions corre-\nsponds to the electronic one-particle density matrix ρ,\nwhich evaluated at t≥tfis given by\nρ(t;q) = 2 , (55)\nwith ρ(t;q)(2π)2δ(2)(q−q′)=⟨Ψ†\nq(t)Ψq′(t)⟩For the\nequal-time two-point correlation function with α= 1\nevaluated at t≥tf\n˜G¯Ψγ1Ψ(t,q) =−2X\nξξ\u0010\n1−2Nξ\nin\u0011\n×n\nRe\u0002\n˜uξA∗\nq˜uξB\nq\u0003\u0000\n2|βξ\nq|2−1\u0001\n+Reh\u0010\u0000\n˜uξB\nq\u00012−\u0000\n˜uξA\nq\u00012\u0011\nαξ\nqβξ\nqio\n,(56)\nwith ˜G¯Ψγ1Ψ(t,q)(2π)2δ(2)(q−q′) =i⟨¯Ψq(t)γ1Ψq′(t)⟩and\nfor the component α= 2\n˜G¯Ψγ2Ψ(t,q) =−2X\nξ\u0010\n1−2Nξ\nin\u0011\n×n\nIm\u0002\n˜uξA∗\nq˜uξB\nq\u0003\u0000\n2|βξ\nq|2−1\u0001\n+Imh\u0010\u0000\n˜uξB\nq\u00012+\u0000\n˜uξA\nq\u00012\u0011\nαξ\nqβξ\nqio\n,(57)\nwith ˜G¯Ψγ2Ψ(t,q)(2π)2δ(2)(q−q′) = i⟨¯Ψq(t)γ2Ψq′(t)⟩.\nThese two-point correlation functions in real space are\nrelated to the expectation value of the electric (vector)\ncurrent evaluated at equal position, Jα\nV=−Ψ†γ0γαΨ[49,\n69], which vanish for α= 1,2as a consequence of rota-\ntional symmetry. Another two-point correlation function\nwe consider is\n˜G¯Ψγ35Ψ(t,q) =−X\nξξ\u0010\n1−2Nξ\nin\u0011\n×n\u0000\n2|˜uξB\nq|2−1\u0001\u0000\n2|βξ\nq|2−1\u0001\n−4Re\u0002\n˜uξA\nq˜uξB\nqαξ\nqβξ\nq\u0003o\n,(58)\nwith ˜G¯Ψγ35Ψ(t,q)(2π)2δ(2)(q−q′) =i⟨¯Ψq(t)γ35Ψq′(t)⟩.\nWe also study statistical two-point correlation functions\nat equal-time, e.g.,\n˜GS\n¯ΨΨ(t,q) =X\nξ\u0010\n1−2Nξ\nin\u0011\n×n\u0000\n2|˜uξB\nq|2−1\u0001\u0000\n2|βξ\nq|2−1\u0001\n−4Re\u0002\n˜uξA\nq˜uξB\nqαξ\nqβξ\nq\u0003o\n,(59)10\n0.0 0.2 0.4 0.6 0.8 1.0-2-10123\nq[1/aM]G\nΨ_\nγ35Ψ×1015[(aM)2]\n03691215-0.6-0.4-0.20.00.20.40.6\nL[aM]GΨ_\nγ35Ψ×1014T=0T=0.01T=0.02T=0.03T=0.05T=0.1T=0.5T=1\n0.00.20.40.60.81.0-2-10123\nq[1/aM]GΨ_γ35Ψ×1015[(aM)2]03691215-0.6-0.4-0.20.00.20.40.6L[aM]GΨ_γ35Ψ×1014\n0.0 0.2 0.4 0.6 0.8 1.0-50510\nq[1/aM]G\nΨ_\nΨ×1015[(aM)2]03691215-2-1012\nL[aM]GΨ_\nΨ×1014T=0T=0.01T=0.02T=0.03T=0.05T=0.1T=0.5T=1\n0.00.20.40.60.81.0-50510\nq[1/aM]GΨ_Ψ×1015[(aM)2]03691215-2-1012L[aM]GΨ_Ψ×1014\nFIG. 3.Equal-time two-point correlation functions of fermionic excitations. From top to bottom: Equal-time two\npoint correlation function of fermionic excitations in momentum space ˜GS\n¯Ψγ35Ψand statistical equal-time two-point correlation\nfunction of fermionic excitations (also known as static structure factor) in momentum space ˜GS\n¯ΨΨin units of (aM)2as a function\nof momentum qin units of 1/aM. This has been calculated for a band gap term proportional to the identity matrix. The time\ndynamical process in region II has a duration of tf−ti=aM/(5v0). The Fermi velocity vFis changed here from v0tovf= 102v0\nand the band gap from ∆0= 50 ℏv0/aMto∆f=ℏv0/aMwith a time dependence given in Eqs. (39)and(38). In the inset the\ndimensionless equal-time two-point correlation functions in position space as a function of the distance Lare shown obtained\nafter a regularization with a Gaussian window function of width w=aM. In the left panels, the colors correspond to different\ninitial thermal states with temperature Tin units of ℏv0/(kBaM). The two-point correlation functions are evaluated at final\ntime tf. In the right panels, the initial state is taken to be the vacuum at T=0K. The colors correspond to different holding\ntimes after the expansion has ceased with ∆t=aM/(50v0).\nwith ˜GS\n¯ΨΨ(t,q)(2π)2δ(2)(q−q′) =1\n2P\nb⟨\u0002¯Ψb\nq(t),Ψb\nq′(t)\u0003\n⟩,\nwhere the sum over bindicates a sum over the compo-\nnents of the Dirac field. For example, for the case of\nspinless fermions on a honeycomb lattice, the latter two-\npoint correlation function provides information about the\nstaggered density contributions on the sublattices.\nThe two-point correlation functions related to super-\nconductivity vanish for all points\n⟨¯Ψq(t)Ψ∗\nq′(t)⟩= 0 = ⟨¯Ψ∗\nq(t)Ψq′(t)⟩,(60)\nand so do the ones corresponding to anomalous densities\n⟨Ψ†\nq(t)Ψ∗\nq′(t)⟩= 0 = ⟨ΨT\nq(t)Ψq′(t)⟩.(61)Correlation functions in momentum space are regular,\nbut an ultraviolet regularization is needed to represent\nthem in position space (otherwise they would be distri-\nbutions). To this end, we convolute the Dirac field in\nposition space with a window function [68]\nΨW(t,x)≡Z\nd2x′W(x−x′)Ψ(t,x′).(62)\nThe window function is chosen as a normalized Gaussian\nW(x) =1\n2πw2exp\u0012\n−|x|2\n2w\u0013\n, (63)\nwith wbeing the standard deviation or width which we\nchoose to be w=aM. In Fourier space, the window11\n0.0 0.2 0.4 0.6 0.8 1.0-10-5051015\nq[1/aM]G\nΨ_\nγ35Ψ×1015[(aM)2]03691215-2-101\nL[aM]GΨ_\nγ35Ψ×1014T=0T=0.01T=0.02T=0.03T=0.05T=0.1T=0.5T=1\n0.00.20.40.60.81.0-10-5051015\nq[1/aM]GΨ_γ35Ψ×1015[(aM)2]03691215-2-101L[aM]GΨ_γ35Ψ×1014\nFIG. 4.Equal-time two-point correlation function of fermionic excitations in momentum space ˜G¯Ψγ35Ψin units of\n(aM)2as a function of the momentum qin units of 1/aM. This has been calculated for a band gap term proportional to γ35.\nThe time dynamical process (region II) has a duration of tf−ti=aM/(5v0). The Fermi velocity vFis changed here from v0to\nvf= 102v0and the band gap from ∆0= 50 ℏv0/aMto∆f=ℏv0/aMwith a time dependence given in Eqs. (39)and(38). In\nthe insets the dimensionless equal-time two-point correlation functions in position space as a function of the distance Lshown\nobtained after a regularization with a Gaussian window function of width w=aM. In the left panel, the colors correspond to\ndifferent initial thermal states with temperature Tin units of ℏv0/(kBaM). The two-point correlation function is evaluated at\nfinal time tf. In the right panel, the initial state is taken to be the vacuum at T=0K. The colors correspond to different holding\ntimes after the expansion has ceased with ∆t=aM/(50v0).\nfunction acts as an ultraviolet regulator and it is also of\nGaussian form, ˜W(q) =e−q2w2/2, such as the regularized\nexpression for the two-point correlation functions becomes\nG(t, L) =Zd2q\n(2π)2e−iqLcosφ˜G(t,q)˜W∗(q)˜W(q),(64)\nwithL=|x−x′|beingthecomovingdistancebetweenthe\ntwo spatial positions xandx′. For w→0one formally\nrecovers the full form of G(t, L)as a distribution.\nIn Fig. 3, the equal-time two-point correlation function\n˜G¯Ψγ35Ψand the statistical equal-time two-point correla-\ntion function ˜GS\n¯ΨΨgiven in Eqs. (58)and(59)respectively,\nare shown for a band gap proportional to the identity after\nthe time dynamical process. On the left panel, the initial\nstate is taken as a thermal state for different temperatures\nand one can observe again the effect of Pauli blocking.\nOn the right panel, the time evolution of the statistical\nequal-time two-point correlation function at T= 0K is\nshown after the time dynamical process has ceased. Here,\none can observe that the characteristic features of the\ntwo-point correlation function evolve with twice the speed\nof the final Fermi velocity vf.\nIn Fig. 4, the two-point correlation functions ˜G¯Ψγ35Ψis\nshown in momentum space and in real space for different\ninitial thermal states (left panel) and for different holding\ntimes (right panel) after the time-dependent process has\nceasedforabandgapproportionalto γ35giveninEq. (58).\nThe characteristic features of this two-point correlation\nfunction evolve also with twice the final Fermi velocity.The evaluation of the statistical two-point correlation\nfunction, cf. Eq. (59), for an initial no-particle state\nvanishes for all distances for this gap shape.\nThe equal-time two-point correlation functions in mo-\nmentum space ˜G¯Ψγ1Ψand˜G¯Ψγ2Ψgiven in Eqs. (56),(57)\ndepend on the direction of momentum in contrast to\n˜G¯Ψγ35Ψand ˜GS\n¯ΨΨ, which depend only on the momentum\namplitude. Therefore, they have not been plotted in order\nto not choose a specific direction of the momentum.\nIV. CONCLUSIONS AND OUTLOOK\nIn summary, we discussed how a situation analogous to\ncosmological particle production in an expanding space-\ntime could be realized in highly-tunable Dirac materials as\nthey may be realized in moiré heterostructures. The key\ningredient is a controllable time-dependent ratio of an en-\nergy gap and the Fermi velocity. The energy gap, or mass\nterm in relativistic nomenclature, is important because\nmassless relativistic fermions are invariant under a scaling\nsymmetry that would allow to absorb a time-dependent\nchange in the metric scale factor or Fermi velocity. In\nthe presence of a finite gap this changes and a time de-\npendence of the ratio ∆/vFleads to the production of\nquasiparticle excitations.\nWe have investigated a scenario with unbroken charge\nconjugation symmetry and, here, the number of particle\nand antiparticle excitations are precisely equal for every\nmomentum mode. At non-zero temperature, when some12\nmodes have a thermal occupation, particle production is\nsuppressed by Pauli blocking.\nInteresting observables to confirm and further investi-\ngate fermionic quasiparticle production are various two-\npoint correlation functions as we exhibited in this contri-\nbution, which depend on the moiré Dirac material chosen\nas the analog platform. Their dependence on time as well\nas wavenumber contains characteristic information that\ncould be compared between theory and a possible experi-\nment. It is of particular interest to reproduce these results\nexperimentally to test the validity of highly tunable Dirac\nmaterials as quantum simulators.\nIn the present paper we have not discussed the possibil-\nities for experimental realizations of our proposal in much\ndetail. However, we believe that moiré materials are par-\nticularlyinterestinginthisregard. Anotoriousexampleof\nmoiré Dirac materials is twisted bilayer graphene, whose\nlow energy fermionic excitations at charge neutrality have\na Fermi velocity vF(θ)which is a function of the twist an-\ngleθoriginating from the interlayer coupling. Generically,\nthe interlayer coupling in this van-der-Waals structure\ncan be considered weak. However, it turns out that near\nthe magic angle θM∼1.1◦, the Fermi velocity vF(θ)is\nstrongly modified and approaches zero [ 14,15,27]. The\nmagic angle itself is determined by the interlayer coupling,\nwhich can be increased by applying pressure δpto the\nsystem [ 19,70], i.e. θM=θM(δp). Hence, even at a\nfixed angle near θM, the Fermi velocity of the system\ncan be strongly modified by an external perturbation,\nsince it is very sensitive to any change in the system. In\nother words, around the magic angle, the Fermi velocity\nof TBG’s Dirac excitations can be tuned over a wide\nrange of values, potentially covering more than an order\nof magnitude. This may be experimentally achieved in\na TBG system with a twist close to a magic angle θM\nby different methods, e.g., (1) by varying the hydrostatic\npressure δp(t)with respect to time; (2) by applying ul-\ntrafast (non-interacting) light pulses, which produce the\nsame mechanical effect as hydrostatic pressure; and (3) by\nperforming continuous rotation, which has been achieved\nat room temperature [ 71], but work is currently underway\nto achieve this at low temperatures. We think that the\nsecond one can be performed in the right time frame to\nbring the system out of equilibrium. Another interesting\nmaterial system could be the case of Bernal-stacked bi-\nlayer graphene, which hosts Dirac cones due to trigonal\nwarping [ 72]. Experimentally, the Dirac cones can be\ngapped out in a controlled way [ 73], which could possibly\nrealize a scenario where the Fermi velocity is fixed and\nthe gap is time-dependent. An alternative to choosing\nbilayer graphene to perform the experiment could be to\ndesign a honeycomb optical lattice to trap cold atoms.\nTime dependencies could be introduced into the system by\nmodifying the potential depths in the sublattices, leading\nto a modification of the band gap. Therefore, in this case,\nall the time dependence would be contained in the mass\nterm, but as demonstrated in the paper, we only need\na time-dependent ratio between the mass and the Fermivelocity, so that would be enough to produce particles.\nFor future work, many extensions of the present setup\nare conceivable. One can add electromagnetic fields to\nstudy electromagnetic response and correlation functions\nin more detail. This would allow us to study the analog\nof Schwinger effect, for example. Further, it would be\ninteresting to study also spatial curvature, which may be\nrealized by making the Fermi velocity space dependent,\nor through inhomogeneous lattice configurations. Also,\nit would be highly interesting to investigate whether an\nanalog to the production of the baryon-antibaryon asym-\nmetry in the early universe could be realized in Dirac\nmaterials. In analogy to the famous Sakharov conditions\nthis would likely need an explicit breaking of charge con-\njugation symmetry as well as time-reversal. Finally, it\nwould be nice to study dynamically evolving spacetime\ngeometries in Dirac materials and a possible interplay\nbetween the geometric and electronic degrees of freedom.\nACKNOWLEDGMENTS\nThe authors would like to thank María A. H. Vozmedi-\nano, AlbertoCortijoandIlyaEreminforusefuldiscussions.\nMMS and MTS acknowledge funding from the Deutsche\nForschungsgemeinschaft (DFG, German Research Foun-\ndation) within Project-ID 277146847, SFB 1238 (project\nC02). MMS is supported by the DFG Heisenberg pro-\ngramme (Project-ID 452976698).\nAppendix A: Graphene representation and\nreducibility\nThe set of time- and space-like gamma matrices ( γα\nwith α= 0,1,2) in the graphene representation are block\ndiagonal, but the two blocks differ in γ1. This shows\nthat it is a reducible representation from the point of\nview of the Lorentz group and it is composed out of an\nirreducible representation with the two-component spinor\u0000\nψ+A, ψ+B\u0001Tand a second irreducible representation\nthat differs by a parity transform x1→ −x1. It is possible\nto perform a similarity transform in the Clifford algebra to\nmake the reducibility manifest. Define the transformation\nmatrix\nˆγ±=\u0012\nI20\n0±σ2\u0013\nwith (ˆγ±)2=I4,(A1)\nand introduce the transformed representation of the Clif-\nford algebra\n˜γµ= ˆγ−γµˆγ+, (A2)\nwhich is concretely ˜γ0=I2⊗iσ3,˜γ1=I2⊗iσ2and\n˜γ2=I2⊗(−σ1). The transformed spinors can be taken\nas\n˜Ψ = ˆγ+Ψ,\n¯˜Ψ = ¯Ψˆγ−=iΨ†γ0ˆγ−=i(ˆγ+Ψ)†γ0.(A3)13\nThe Dirac-like action is invariant under the transforma-\ntion. It becomes now clear that the transformed spinor\n˜Ψ =\nψ+A\nψ+B\n−iψ−B\n+iψ−A\n(A4)\ncan be seen as two copies, or families, of Lorentz group\nirreducible two-component spinors. It becomes sensible\nto organize the spinors as\nΨξ=\u0012˜ψξA\n˜ψξB\u0013\n, (A5)\nwhere ξis the additional family index.\nAppendix B: Dirac field, Dirac matter metric and\nenergy-momentum tensor\nThe Dirac field Ψcan be quantized by introducing the\nequal-time anticommutation relations [49]\n{Ψa(t,x), πb(t,x′)}=iℏδ(2)(x−x′)δab,(B1)\nwhere a, brefer to the spinor indices and the conjugate\nmomentum density π(t,x)is given by\nπ(t,x) =∂L[Ψ,¯Ψ]\n∂(∂tΨ)=−√gℏ¯Ψ(t,x)γ0et\n0(t,x).(B2)\nTherefore, the anticommutator (B1) can be written as\n\b\nΨa(t,x),¯Ψb(t,x′)\t\n=−i√gδ(2)(x−x′)\u0002\nγ0et\n0(t,x)\u0003−1\nab.\n(B3)\nThe energy-momentum tensor for a Dirac field is ob-\ntained by the variation of the action with respect to the\nspacetime metric [48]\nTµν=2√gδS\nδgµν. (B4)\nFor an action corresponding to the one studied here, cf.\nEq. (9), it reads\nTµν=−¯Ψℏγαeµ\nα(∂ν+ Ων) Ψ\n+gµν¯Ψ\u0000\nℏγαeσ\nα∂σ+ ∆ jΓj\u0001\nΨ,(B5)\nwith the diagonal components being (no sum implied)\nTα\nα=¯Ψ\u0000\nℏvFγα∂α+ ∆ jΓj\u0001\nΨ. (B6)\nIn particular, the time-time component of the energy-\nmomentum tensor leads to the Dirac Hamiltonian density,\nwhich in our case is given by\nTη\nη≡ H =¯Ψ\u0000\nℏvFγ·∂+ ∆ jΓj\u0001\nΨ.(B7)Appendix C: Vielbein, metric and connections\nWe investigate here a vielbein field eα\nµ(x)with the non-\nvanishing temporal component e0\n0(t) =eζ(t)vF(t), spatial\ncomponent e1\n1(t) =e2\n2(t) =eζ(t), and all non-diagonal\ncomponents vanishing. Here vF(t)is the Fermi velocity,\nandeζ(t)is a scaling factor factor that we keep unspecified\nfor the time being. The corresponding spacetime metric\ngµν(x) =eα\nµ(x)eβ\nν(x)ηαβhas the non-vanishing compo-\nnents g00(t) =−e2ζ(t)v2\nF(t)andg11(t) =g22(t) =e2ζ(t).\nFor this metric with eζ(t)= 1, i.e. Eq. (9), one can\ndetermine the non-vanishing Christoffel symbol\nΓt\ntt=˙vF(t)\nvF(t), (C1)\nwith dot notation indicating a partial derivative respect to\nreal time. All spin connection components vanish. Thus,\nthis metric corresponds to a Minkowski spacetime with\ntime-dependent Fermi velocity leading to a zero Ricci\ncurvature.\nFor this metric with eζ(t)=∆j(0)vF(t)\n∆j(t)vF(0), i.e. Eq. (13),\none can obtain the non-vanishing Christoffel symbols\nΓt\nxx=1\nv2\nF(t) \n˙vF(t)\nvF(t)−˙∆j(t)\n∆j(t)!\n= Γt\nyy,\nΓx\ntx=˙vF(t)\nvF(t)−˙∆j(t)\n∆j(t)= Γx\nxt= Γy\nyt= Γy\nyt,\nΓt\ntt= 2˙vF(t)\nvF(t)−˙∆j(t)\n∆j(t)(C2)\nTherefore, the non-vanishing spin connection coefficients\nare\nωx01=1\nvF(t) ˙∆j(t)\n∆j(t)−˙vF(t)\nvF(t)!\n=ωy02(C3)\nand their respective antisymmetric components. Thus,the\nnon-zero components of the spin connection are given by\nΩx=1\n2vF(t) ˙∆j(t)\n∆j(t)−˙vF(t)\nvF(t)!\nγ0γ1,\nΩy=1\n2vF(t) ˙∆j(t)\n∆j(t)−˙vF(t)\nvF(t)!\nγ0γ2.(C4)\nThe Ricci curvature, R=gµνRµν, is given by\nR= 2v2\nF(0)∆2\nj(t)\nv2\nF(t)∆2\nj(0)\"\n−3˙v2\nF(t)\nv2\nF(t)+˙∆2\nj(t)\n∆2\nj(t)\n−2¨∆j(t)\n∆j(t)+ 2¨vF(t)\nvF(t)#\n,(C5)\nwhich indicates a curvature deviation from an Euclidean\nspacetimewhenthereisabandgapandatimedependence\non the band gap and/or the Fermi velocity.14\nAppendix D: General band gap\n1. Mode equations and Hamiltonians\nLet us introduce the corresponding mode equations and\nHamiltonians for different tensor shapes of the gap.\na. Haldane mass\nMass terms proportional to γ35are block diagonal and,\nconsequently, do not mix the Dirac points, such that one\ncan study the mode equations for the different valleys\nindependently, as we did previously with a mass term\nproportional to the identity matrix.\nTaking the mass gap tensor as Γj=iγ35, the coupled\nsystem of first order differential equations (16)for a Dirac\nfield expanded in the basis (19) becomes\n0 =\u0012i∂η−ξ∆\nℏvFξqe−ξiφ\n−ξqeξiφ−i∂η−ξ∆\nℏvF\u0013\u0012uξA\nq\nuξB\nq\u0013\n.(D1)\nEquivalently, the second order differential equation (28)\nfor this kind of mass term yields\n0 =\u0014\n∂2\nη+ω2\nq(η)+\n(−)ξi\nℏ∂η\u0012∆(η)\nvF(η)\u0013\u0015\nuξA(B)\nq (η),(D2)\nwhich again corresponds to a harmonic oscillator differen-\ntial equation with complex and time-dependent frequency.\nFor this kind of band gap, the massive part of the\nHamiltonian, which corresponds to the Haldane mass in\nthe case of spinless fermions on a honeycomb lattice, is\ngiven by\nHHald. =i∆Z\nd2x¯Ψγ35Ψ\n=−∆Zd2q\n(2π)2X\nξξ\n×\u0002\u0000\n|uξB\nq|2− |uξA\nq|2\u0001\u0000\ncξ†\nqcξ\nq−dξ\nqdξ†\nq\u0001\n−2uξA\nquξB\nqdξ\n−qcξ\nq−2uξA∗\nquξB∗\nqcξ†\nqdξ†\n−q\u0003\n,(D3)\nwhere the Dirac field has been expanded again in the\nbasis of Eq. (19). Thus, the Hamiltonian of the system\nreads\nH=Zd2q\n(2π)2X\nξ\u0002\nEξ\nq(η)\u0000\ncξ†\nqcξ\nq−dξ\nqdξ†\nq\u0001\n+Fξ\nq(η)dξ\n−qcξ\nq+Fξ∗\nq(η)cξ†\nqdξ†\n−q\u0003\n,(D4)\nwhere now the functions Eξ\nqandFξ\nqare given by\nEξ\nq=−2ξℏvFqRe\u0002\ne−ξiφuξA∗\nquξB\nq\u0003\n−ξ∆\u0010\n1−2\f\fuξA\nq\f\f2\u0011\n,\nFξ\nq=ξℏvFqh\neξiφ\u0000\nuξA\nq\u00012−e−ξiφ\u0000\nuξB\nq\u00012i\n+ 2ξ∆uξA\nquξB\nq.(D5)The Bogoliubov coefficients are obtained by diagonaliz-\ning the Hamiltonian of the system through the Bogoli-\nubov transformation (46)after the dynamical process has\nceased. As this kind of mass gap is block diagonal, the\nBogoliubov coefficients are also given by Eqs. (49)for the\ncorresponding functions (D5).\nA possible initial configuration of the mode functions\nleading to the no-particle initial state, i.e. an initial\ndiagonal Hamiltonian, for this kind of gap term is given\nby\nuξA,I\nq(η≤ηi) =e−ξiφ/2\n√\n2s\n1 +ξ∆0\nℏv0ωIqe−iωI\nqη,\nuξB,I\nq(η≤ηi) =−ξeξiφ/2\n√\n2s\n1−ξ∆0\nℏv0ωIqe−iωI\nqη.(D6)\nIn region III, the set of mode functions associated to the\ncreation and annihilation operators that diagonalize the\nHamiltonian is given by\n˜uξA,III\nq (η≥ηf) =e−ξiφ/2\n√\n2s\n1 +ξ∆f\nℏvfωIIIqe−iωIII\nqη,\n˜uξB,III\nq (η≥ηf) =−ξeξiφ/2\n√\n2s\n1−ξ∆f\nℏvfωIIIqe−iωIII\nqη.(D7)\nb. Kekulé modulation of the nearest neighbor hopping\nA mass term i∆(γ3cosθ+γ5sinθ), with real amplitude\n∆and phase θ, mixes the Dirac points and the sublattices\nin the case of spinless fermions on a honeycomb lattice.\nConsequently, the mode functions for the different Dirac\npoints cannot be studied independently as in the previous\nSecs. IIIC and D1a.\nFor a band gap proportional to i(γ3cosθ+γ5sinθ)and\ntaking into consideration the expansion of the Dirac field\nin Fourier modes (19), the mode equation is\n0 =\ni∂ηqe−iφ0¯∆∗\nℏvF\n−qeiφ−i∂η−¯∆∗\nℏvF0\n0¯∆\nℏvFi∂η−qeiφ\n−¯∆\nℏvF0qe−iφ−i∂η\n\nu+A\nq\nu+B\nq\nu−A\nq\nu−B\nq\n,(D8)\nwith ¯∆ = ∆ eiθand, equivalently, Eq. (28)\n0 =\u0002\n∂2\nη+ω2\nq(η)\u0003\nu+λ\nq(η) +i\nℏ∂η\u0012¯∆∗(η)\nvF(η)\u0013\nu−λ′\nq(η),\n0 =\u0002\n∂2\nη+ω2\nq(η)\u0003\nu−λ\nq(η) +i\nℏ∂η\u0012¯∆(η)\nvF(η)\u0013\nu+λ′\nq(η),\n(D9)\nwhere the superindex λ̸=λ′.\nThe corresponding massive part of the Hamiltonian is\ngiven\nHKekulé =iZ\nd2xΨ†\u0012\n0¯∆∗σ1\n¯∆σ10\u0013\nΨ.(D10)15\nTherefore, the Hamiltonian of the system\nH=Zd2q\n(2π)2(X\nξh\nE(0)ξ\nq\u0000\ncξ†\nqcξ\nq−dξ\nqdξ†\nq\u0001\n+F(0)ξ\nqdξ\n−qcξ\nq+F(0)ξ∗\nqcξ†\nqdξ†\n−qi\n+h\nJq\u0010\nc−†\nqc+\nq−d+\n−qd−†\n−q\u0011\n+Gq\u0010\nd−\n−qc+\nq+d+\n−qc−\nq\u0011\n+h.c.i)\n,(D11)\nwith\nE(0)ξ\nq=−2ξℏvFqRe\u0002\ne−ξiφuξA∗\nquξB\nq\u0003\n,\nF(0)ξ\nq=ξℏvFqh\neξiφ\u0000\nuξA\nq\u00012−e−ξiφ\u0000\nuξB\nq\u00012i\n,\nJq=i¯∆\u0000\nu−A∗\nqu+B\nq+u−B∗\nqu+A\nq\u0001\n,\nGq=i¯∆\u0000\nu+B\nqu−B\nq−u+A\nqu−A\nq\u0001\n,(D12)\nis diagonalized by using the general Bogoliubov transfor-\nmations given in Eq. (45).\nAppendix E: Dirac spinor and Lorentz\ntransformations\nThe generators of the Lorentz group are identified as\nSµν=i\n4[γµ, γν]. (E1)\nIn a (2+1) dimensional spacetime this can be decomposed\ninto a single spatial-spatial generator corresponding to a\nunique rotation\nS12=i\n2γ1γ2, (E2)and into two spatial-temporal generators corresponding\nto boosts\nS01=i\n2γ0γ1and S02=i\n2γ0γ2.(E3)\nIn the graphene representation S12=−σ3⊗σ3/2is di-\nagonal, while S01=iσ3⊗σ1/2andS02=iσ0⊗σ2/2.\nA Dirac spinor transforms under Lorentz transforma-\ntions as\nΨ(x)→Ψ′(x′) =S[Λ]Ψ(Λ−1x) (E4)\nand\n¯Ψ(x)→¯Ψ′(x′) =¯Ψ(Λ−1x)S−1[Λ],(E5)\nwith S[Λ]being the exponentiation of the Lorentz gener-\nators\nS[Λ] = exp\u0012i\n2ΩµνSµν\u0013\n, (E6)\nwhere Ωµνare six antisymmetric numbers which define\nthe angle of rotation and the rapidity for the boosts.\nAppendix F: Parameter values\nThe plots shown in Sec. IIIF are computed for the\nfollowing experimental parameters. The moiré lattice\nconstant is given by aM=a0/[2 sin( θ/2)], where a0\nis taken as the graphene lattice constant a0≃2.46Å\nand for the reported narrow bands the twist angle is\ntaken to be θ≃1.1◦, which leads to the moiré lat-\ntice constant aM≃a0/θ≃15 nm[74]. The Fermi ve-\nlocity is changed from v0=c/105tovf=c/103, where\ncdenotes the speed of light, while the mass gap from\n∆0= 50 ℏv0/aM≃16.9 meVto∆f=ℏv0/aM≃0.3 meV.\nThe time dynamical process has a duration of\ntf−ti=aM/(5v0) = 10−12sand the holding time steps\n∆t=aM/(50v0) = 10−13s. The initial temperatures T\nare taken from 0 Ktoℏv0/(kBaM)≃1.5 K. The window\nfunction for the regularization of the correlation functions\nin position space has Gaussian form with a width w=aM.\n[1]G. E. Volovik, The Universe in a Helium Droplet (Oxford\nUniversity Press, 2009).\n[2]C. Barceló, S. Liberati, and M. Visser, Analogue gravity,\nLiving Rev. Relativ. 14, 3 (2011).\n[3]M. Novello, M. Visser, and G. E. Volovik, eds., Artifi-\ncial Black Holes (World Scientific Publishing, Singapore,\n2002).\n[4]C. Barceló, S. Liberati, and M. Visser, Towards the obser-\nvation of Hawking radiation in Bose–Einstein condensates,\nInt. J. Mod. Phys. A 18, 3735 (2003).\n[5]C. Barceló, S. Liberati, and M. Visser, Analogue models\nfor FRW cosmologies, Int. J. Mod. Phys. D 12, 1641\n(2003).[6]U. R. Fischer, Quasiparticle universes in Bose–Einstein\ncondensates, Mod. Phys. Lett. A 19, 1789 (2004).\n[7]U. R. Fischer and R. Schützhold, Quantum simulation of\ncosmic inflation in two-component Bose-Einstein conden-\nsates, Phys. Rev. A 70, 063615 (2004).\n[8]M.Uhlmann, Y.Xu,andR.Schützhold,Aspectsofcosmic\ninflation in expanding Bose-Einstein condensates, New J.\nPhys.7, 248 (2005).\n[9]E. A. Calzetta and B. L. Hu, Early Universe quantum\nprocesses in BEC collapse experiments, Int. J. Theor.\nPhys.44, 1691 (2005).\n[10]S. Weinfurtner, P. Jain, M. Wisser, and C. W. Gardiner,\nCosmological particle production in emergent rainbow\nspacetimes, Class. Quantum Gravity 26, 065012 (2009).16\n[11]M. Tolosa-Simeón, A. Parra-López, N. Sánchez-Kuntz,\nT. Haas, C. Viermann, M. Sparn, N. Liebster, M. Hans,\nE. Kath, H. Strobel, M. K. Oberthaler, and S. Flo-\nerchinger, Curved and expanding spacetime geometries\nin Bose-Einstein condensates, Phys. Rev. A 106, 033313\n(2022).\n[12]C. Viermann, M. Sparn, N. Liebster, M. Hans, E. Kath,\nÁ. Parra-López, M. Tolosa-Simeón, N. Sánchez-Kuntz,\nT. Haas, H. Strobel, S. Floerchinger, and M. K.\nOberthaler, Quantum field simulator for dynamics in\ncurved spacetime, Nature 611, 260 (2022).\n[13]A. H. MacDonald and R. Bistritzer, Graphene moiré\nmystery solved?, Nature 474, 453 (2011).\n[14]J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H.\nCastro Neto, Continuum model of the twisted graphene\nbilayer, Phys. Rev. B 86, 155449 (2012).\n[15]Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,\nJ. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,\nT. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-\nHerrero, Correlated insulator behaviour at half-filling\nin magic-angle graphene superlattices, Nature 556, 80\n(2018).\n[16]Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,\nE. Kaxiras, and P. Jarillo-Herrero, Unconventional su-\nperconductivity in magic-angle graphene superlattices,\nNature556, 43 (2018).\n[17]D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J.\nMillis, J. Hone, C. R. Dean, D. Basov, A. N. Pasupathy,\nand A. Rubio, Moiré heterostructures as a condensed-\nmatter quantum simulator, Nature Physics 17, 155 (2021).\n[18]Q. Zhang, T. Senaha, R. Zhang, C. Wu, L. Lyu, L. W.\nCao, J. Tresback, A. Dai, K. Watanabe, T. Taniguchi,\nand M. T. Allen, Dynamic twisting and imaging of moiré\ncrystals, arXiv preprint arXiv:2307.06997 (2023).\n[19]M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan-\nabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean,\nTuning superconductivity in twisted bilayer graphene,\nScience363, 1059 (2019).\n[20]Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora,\nR. Polski, Y. Zhang, H. Ren, J. Alicea, G. Refael, F. von\nOppen, K. Watanabe, T. Taniguchi, and S. Nadj-Perge,\nElectronic correlations in twisted bilayer graphene near\nthe magic angle, Nature Physics 15, 1174 (2019).\n[21]M. Angeli and A. H. MacDonald, Γvalley transition metal\ndichalcogenide moiré bands, Proceedings of the National\nAcademy of Sciences 118, e2021826118 (2021).\n[22]T. Wehling, A. Black-Schaffer, and A. Balatsky, Dirac\nmaterials, Advances in Physics 63, 1 (2014).\n[23]O. Vafek and A. Vishwanath, Dirac Fermions in Solids:\nFrom High- TcCuprates and Graphene to Topological\nInsulators and Weyl Semimetals, Annual Review of Con-\ndensed Matter Physics 5, 83 (2014).\n[24]J. L. Mañes, F. Guinea, and M. A. H. Vozmediano, Exis-\ntence and topological stability of Fermi points in multi-\nlayered graphene, Phys. Rev. B 75, 155424 (2007).\n[25]M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi,\nK. Kuroki, and L. Fu, Maximally Localized Wannier Or-\nbitals and the Extended Hubbard Model for Twisted\nBilayer Graphene, Phys. Rev. X 8, 031087 (2018).\n[26]H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Origin\nof Mott Insulating Behavior and Superconductivity in\nTwisted Bilayer Graphene, Phys. Rev. X 8, 031089 (2018).\n[27]R. Bistritzer and A. H. MacDonald, Moiré bands in\ntwisted double-layer graphene, Proceedings of the Na-tional Academy of Sciences 108, 12233 (2011).\n[28]J. S. Hofmann, E. Khalaf, A. Vishwanath, E. Berg, and\nJ. Y. Lee, Fermionic Monte Carlo Study of a Realistic\nModel of Twisted Bilayer Graphene, Phys. Rev. X 12,\n011061 (2022).\n[29]X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir,\nI. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang,\net al., Superconductors, orbital magnets and correlated\nstates in magic-angle bilayer graphene, Nature 574, 653\n(2019).\n[30]P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe,\nT. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and\nD. K. Efetov, Untying the insulating and superconducting\norders in magic-angle graphene, Nature 583, 375 (2020).\n[31]Y. Da Liao, Z. Y. Meng, and X. Y. Xu, Valence bond\norders at charge neutrality in a possible two-orbital ex-\ntended hubbard model for twisted bilayer graphene, Phys.\nRev. Lett. 123, 157601 (2019).\n[32]N. Parthenios and L. Classen, Twisted bilayer graphene\nat charge neutrality: competing orders of SU(4) Dirac\nfermions, arXiv preprint arXiv:2305.06949 (2023).\n[33]L. Parker, Quantized fields and particle creation in ex-\npanding universes. ii, Phys. Rev. D 3, 346 (1971).\n[34]A. Dolgov and K. Freese, Calculation of particle pro-\nduction by Nambu-Goldstone bosons with application to\ninflation reheating and baryogenesis, Physical Review D\n51, 2693 (1995).\n[35]D. H. Lyth, D. Roberts, and M. Smith, Cosmological\nconsequencesofparticlecreationduringinflation,Physical\nReview D 57, 7120 (1998).\n[36]G. F. Giudice, A. Riotto, I. Tkachev, and M. Peloso,\nProduction of massive fermions at preheating and leptoge-\nnesis, Journal of High Energy Physics 1999, 014 (1999).\n[37]D. J. H. Chung, E. W. Kolb, A. Riotto, and I. I. Tkachev,\nProbing planckian physics: Resonant production of parti-\ncles during inflation and features in the primordial power\nspectrum, Phys. Rev. D 62, 043508 (2000).\n[38]M. Peloso and L. Sorbo, Preheating of massive fermions\nafter inflation: analytical results, Journal of High Energy\nPhysics2000, 016 (2000).\n[39]P. B. Greene and L. Kofman, Theory of fermionic pre-\nheating, Phys. Rev. D 62, 123516 (2000).\n[40]T. Asaka and H. Nagao, Nonperturbative corrections to\nparticle production from coherent oscillation, Progress of\nTheoretical Physics 124, 293 (2010).\n[41]P. Adshead and E. I. Sfakianakis, Fermion production\nduring and after axion inflation, Journal of Cosmology\nand Astroparticle Physics 2015(11), 021.\n[42]A. Scardua, L. F. Guimarães, N. Pinto-Neto, and G. S.\nVicente, Fermion production in bouncing cosmologies,\nPhys. Rev. D 98, 083505 (2018).\n[43]Y. Ema, K. Nakayama, and Y. Tang, Production of purely\ngravitational dark matter: the case of fermion and vector\nboson, Journal of High Energy Physics 2019, 60 (2019).\n[44]E. W. Kolb and M. S. Turner, The Early Universe , Vol. 69\n(1990).\n[45]S.Weinberg, Cosmology (OxfordUniversityPress,Oxford,\n2008).\n[46]V. Kuzmin and I. Tkachev, Matter creation via vacuum\nfluctuations in the early universe and observed ultra-\nhigh energy cosmic ray events, Physical Review D 59,\n10.1103/physrevd.59.123006 (1999).\n[47]D. Chung, L. Everett, H. Yoo, and P. Zhou, Gravitational\nfermion production in inflationary cosmology, Physics17\nLetters B 712(2011).\n[48]N. D. Birrell and P. C. W. Davies, Quantum Fields in\nCurved Space , Cambridge Monographs on Mathematical\nPhysics (Cambridge University Press, Cambridge, 1982).\n[49]L. Parker and D. Toms, Quantum Field Theory in Curved\nSpacetime: Quantized Fields and Gravity , Cambridge\nMonographs on Mathematical Physics (Cambridge Uni-\nversity Press, 2009).\n[50]A. Friedman, Über die Krümmung des Raumes, Z. Phys.\n10, 377 (1922).\n[51]A. Friedman, Über die Möglichkeit einer Welt mit kon-\nstanter negativer Krümmung des Raumes, Z. Phys. 21,\n326 (1924).\n[52]A. G. Lemaître, A Homogeneous Universe of Constant\nMass and Increasing Radius accounting for the Radial\nVelocity of Extra-galactic Nebulæ, Mon. Not. R. Astron.\nSoc.91, 483 (1931).\n[53]H. P. Robertson, Kinematics and World-Structure, Astro-\nphys. J.82, 284 (1935).\n[54]H. P. Robertson, Kinematics and World-Structure II,\nAstrophys. J. 83, 187 (1936).\n[55]H. P. Robertson, Kinematics and World-Structure III,\nAstrophys. J. 83, 257 (1936).\n[56]M. I. Katsnelson, Graphene: Carbon in Two Dimensions\n(Cambridge University Press, 2012).\n[57]F. D. M. Haldane, Model for a quantum hall effect without\nlandau levels: Condensed-matter realization of the \"parity\nanomaly\", Phys. Rev. Lett. 61, 2015 (1988).\n[58]I. F. Herbut, Interactions and Phase Transitions on\nGraphene’s Honeycomb Lattice, Phys. Rev. Lett. 97,\n146401 (2006).\n[59]I. F. Herbut, V. Juričić, and B. Roy, Theory of interacting\nelectrons on the honeycomb lattice, Phys. Rev. B 79,\n085116 (2009).\n[60]R. Boyack, H. Yerzhakov, and J. Maciejko, Quantum\nphase transitions in dirac fermion systems, The European\nPhysical Journal Special Topics 230, 979 (2021).\n[61]M. Christos, S. Sachdev, and M. S. Scheurer, Supercon-\nductivity, correlated insulators, and wess–zumino–witten\nterms in twisted bilayer graphene, Proceedings of the\nNational Academy of Sciences 117, 29543 (2020).[62]T. Cea, P. A. Pantaleón, and F. Guinea, Band structure\nof twisted bilayer graphene on hexagonal boron nitride,\nPhys. Rev. B 102, 155136 (2020).\n[63]M. Long, P. A. Pantaleón, Z. Zhan, F. Guinea, J. Á.\nSilva-Guillén, and S. Yuan, An atomistic approach for\nthe structural and electronic properties of twisted bilayer\ngraphene-boron nitride heterostructures, npj Computa-\ntional Materials 8, 73 (2022).\n[64]M. Long, Z. Zhan, P. A. Pantaleón, J. A. Silva-Guillén,\nF. Guinea, and S. Yuan, Electronic properties of twisted\nbilayer graphene suspended and encapsulated with hexag-\nonal boron nitride, Phys. Rev. B 107, 115140 (2023).\n[65]L. Parker, Quantized fields and particle creation in ex-\npanding universes. i, Phys. Rev. 183, 1057 (1969).\n[66]M. Vozmediano, M. Katsnelson, and F. Guinea, Gauge\nfields in graphene, Physics Reports 496, 109–148 (2010).\n[67]J. D. Bjorken and S. D. Drell, Relativistic quantum me-\nchanics, International series in pure and applied physics\n(McGraw-Hill, New York, NY, 1964).\n[68]V. Mukhanov and S. Winitzki, Introduction to Quantum\nEffects in Gravity (Cambridge University Press, Cam-\nbridge, 2007).\n[69]L. Parker, One-electron atom as a probe of spacetime\ncurvature, Phys. Rev. D 22, 1922 (1980).\n[70]S. Carr, S. Fang, P. Jarillo-Herrero, and E. Kaxiras, Pres-\nsure dependence of the magic twist angle in graphene\nsuperlattices, Phys. Rev. B 98, 085144 (2018).\n[71]A.Inbar, J.Birkbeck, J.Xiao, T.Taniguchi, K.Watanabe,\nB. Yan, Y. Oreg, A. Stern, E. Berg, and S. Ilani, The\nquantum twisting microscope, Nature 614, 682 (2023).\n[72]E. McCann and M. Koshino, The electronic properties\nof bilayer graphene, Reports on Progress in physics 76,\n056503 (2013).\n[73]A. M. Seiler, F. R. Geisenhof, F. Winterer, K. Watanabe,\nT. Taniguchi, T. Xu, F. Zhang, and R. T. Weitz, Quantum\ncascade of correlated phases in trigonally warped bilayer\ngraphene, Nature 608, 298 (2022).\n[74]T. Cea, P. A. Pantaleón, N. R. Walet, and F. Guinea, Elec-\ntrostatic interactions in twisted bilayer graphene, Nano\nMaterials Science 4, 27 (2022), special issue on Graphene\nand 2D Alternative Materials."    },    {        "title": "1903.05968v1.Observation_of_topological_transition_in_high_T___c___superconductor_FeTe___1_x__Se___x___SrTiO___3___001__monolayers.pdf",        "content": "1 \n \nObservation of topological transition  in high -Tc \nsuperconductor FeTe 1-xSex/SrTiO 3(001)  monolayers  \nX.-L. Peng1,2, Y. Li1,2, X.-X. Wu3, H.-B. Deng1,2, X. Shi1,2, W.-H. Fan1,2, M. Li4, Y.-B. \nHuang4, T. Qian1,5,6, P. Richard7, J.-P. Hu1,2,5,6, S.-H. Pan1,2,5,6, H.-Q. Mao1†, Y.-J. Sun1,5,6† &    \nH. Ding1,2,5,6† \n1Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, \nChinese Academy of Sciences, Beijing 100190, China  \n2School of Physics, University of Chinese Academy of Sciences, Beijing 100190, China  \n3Institut f ür Theoretische Physik und Astrophysik, Julius -Maximilians -Universit ät Würzburg, \nWürzburg,  Germany.  \n4Shanghai Synchrotron Radiation Facility, Shanghai Institute of Applied Physics, Chinese \nAcademy of Sciences, Shanghai, China.  \n5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China  \n6CAS Center for Excellence in Topological Quantum Computation, University of Chinese \nAcademy of Sciences, Beijing 100190, China  \n7Institut quantique , Université de Sherbrooke, 2500 boulevard de l’Université, Sherbrooke, \nQuébec J1K 2R1, Canada  \n \nABSTRACT  \n        Superconductors  with topological surface  or edge states have been intensively explored  \nfor the prospect of realizing Majorana  bound states, which obey  non-Abelian statistics and \nare crucial for topological quantum computation.  The t raditional  route s for making \ntopological insulator/superconductor and semiconductor/superconductor heterostructures  \nsuffer  fabricati on difficult ies and can only work at low temperature. Here, we use angle -\nresolved photoe mission  spectroscopy  to directly observe  the evolution  of a topological  \ntransition of band  structure  nearby the Fermi level  in two-dimensional  high-Tc \nsuperconductor FeTe 1-xSex/SrTiO 3(001)  monolayers , fully consistent with our theoretical \ncalculations . Furthermore , evidence of edge states  is revealed  by scanning tunneling 2 \n \nspectroscopy  with assistance of theoretical calculations . Our study provides  a simple and \ntunable  platform for realizing and manipulating Majorana states at high temperature.   \n \nI.    INTRODUCTION  \n        Topological superconductors, which can host  Majorana bound states on their surface or \nedge, have drawn enormous attention in condensed matter physics  [1, 2] . This is because  the \nnon-Abelian  braiding properties of Majorana  states  are predicted to play an essential role in \ntopological quantum computation . Intrinsic p-wave superconductors [3, 4]  and \nheterostructures of s-wave superconductors and topological insulators /Rashba -type \nsemiconductors  [5, 6]  are two typical routes for realiz ing topological superconductors  (or \nmore precisely, superconducting topological states) . However, the former  suffers from the \nscarcity  of candidate mate rials and their sensitivity to disorder , whereas the latter suffers \nfrom structural complexities  and low working t emperature s, hindering  manipulation of \nMajorana states in applica tions. Recently, bulk superconductors with topological surface \nstates [7-16] provide a new route , where ordinary  superconductivity and nontrivial topology \ncan be  integrated in a single material , such as FeTe 0.55Se0.45 [9-12], Bi2Pd [13] and other iron -\nbased superconductors [14-16]. Among them, the most interesting systems are the high -Tc \nsuperconductors, which can significantly raise  the working temperature.  \n        Monolayer FeSe grown on SrTiO 3(001) (STO) hold s the highest Tc (above 65 K  [17-24]) \namong all iron -based superconductors and can potentially  combines  superconductivity and \ntopological properties in to a single material [25-27]. One way of integration is to adjust \nlattice parameters , for example by substituting selenium by  tellurium , which maintains high \nTc for monolayer FeTe 1-xSex films  within a wide range of substitution , as revealed by \nprevious scanning tunneling microscopy / spectroscopy  (STM /STS ) [28] and angle -resolved \nphotoemission spectroscopy (ARPES) results [8]. Moreover,  a theoretical work predict ed that \nthe monolayer FeTe 1-xSex system can realize topological nontrivial states through band 3 \n \ninversion at the  point [29] and some supporting evidences were found later in ARPES  \nmeasurements  [8]. However , it is far from settled that this monolayer system can serve as a \nsimple and tunable  platform for realizing topological superconductivity at high temperature.   \n        In this paper, we perform systematic ARPES measurements with photon e nergy and \npolarization dependence  on F eTe1-xSex/STO monolayer s with different Se content ( x) which \nare grown by in situ molecular beam epitaxy (MBE) . In addition to the observation of two \nbands  intersecting , the spectral weight  and effective mass of bands change dramatically  \naround x = 0.21, further supporting  a transition  of topological band inversion nearby the \nFermi level ( EF). Furthermore,  evidence of edge states of monolayer  FeTe/STO  around 50  \nmeV below EF is observed by STM/STS measurements . Their topological origin c an be \ninterpreted by our LDA  calculations and band structure results .  \nII.    EXPERIMENTAL AND COMPUTATIONAL METHODS  \n        The crystal structure of monolayer FeTe 1-xSex/STO is presented in Fig. 1 (a). Monolayer \nFeTe 1-xSex for in situ  ARPES measurements are grown on 0.7wt% Nb -doped STO  substrates \nafter degassing for 10 hours at 600 °C and then annealing for 1.5  hours at 950 °C in an \nultrahigh vacuum MBE chamber. Substrates are kept at 310 °C for monolayer FeSe and \n275 °C for monolayer FeTe 1-xSex during the film growth. Fe (99.98%), Se (99.999%) and Te \n(99.99%) are co -evaporated from Knudsen cells. The flux ratio of Fe to Se/Te is 1:10 which \nare measured by a quart crystal balance. The growth rate is determined by Fe flux and \nequaled to 0.7 UC/min. The Se content ( x) is controlled by the ratio of Te to Se flux speed \nduring growth. After growth, the monolayer FeSe films are annealed at 370 °C and \nmonolayer FeTe 1-xSex at 260 °C for 20 hours. During the growth process, the sample quality \nis monitored using reflection high -energy electron diffraction (RHEED), as shown in Fig. \n1(b). Then the samples are transferred in situ  to the ARPES chamber for measur ements . The \nexact S e content x of the samples is determined by the in situ  X-ray photoelectron \nspectroscopy (XPS) method. The results are shown in Fig. 1 (c), where the value of [ x/(1- x)] 4 \n \nof six different samples is proportional to the ratio of the characteristic peak area associated \nto Se and Te. ARPES measurements are recorded at the BL -09U “Dreamline” beamline of \nShanghai Synchrotron Radiation Facility (SSRF), using a VG DA30 electron an alyzer under \nultrahigh vacuum better than 5 × 10-11 torr. The energy resolution is set to ~12 meV for the \nband structure and ~16 meV for Fermi surface mapping, while the angular resolution is set to \n0.2°. Spectra are recorded at 30 K except for special cla rification. Growth p rocedures of  \nsamples  for in situ  STM/STS measurement s are similar to that introduced above. STM \nmeasurements are carried out in ultra -high-vacuum condition  at 4.3  K with a home -made \nSTM -MBE combined system.  W Tips are carefully calibrated on a Ag island before \nmeasurements. The tunneling conductance spectra are obtained using standard lock -in \namplifier techniques with a root -mean -square oscillation voltage of 3  meV and a lock -in \nfrequency of 791.1 Hz.  \n        Density f unction al theory (DFT)  calculations employ the projector augmented wave \n(PAW) method encoded in the Vienna ab initio  simulation package (VASP) [30-32]. The \nprojector augmented -wave method is used to describe the wavefunctions near the core and \nthe generalized gradient approximatio n within the Perdew -Burke -Ernzerhof (PBE) \nparameterization is employed as the electron exchange -correlation  functional [33]. For band \nstructure calculations, the cutoff energy of 500 eV is taken for expanding the wave functions \ninto plane -wave basis. In the calculation s, the Brillouin zone is sampled in the k space with in \nMonkhorst -Pack scheme [34]. The spin -orbit coupling is taken into account by the second \nvariation method. For 2D free standing FeSe monolayer, the numbers of these k points are \n11× 11. In the edge states calculations, a 24 Å  vacuum layer  is adopted and there are 59 Fe \natoms a nd 59 Se atoms for [100] edge and 60 Fe atoms and 62 Se atoms for [110] edge.  \nIII.    RESULTS  \n        The unit cell  of monolayer FeTe 1-xSex/STO is tetragonal and contains two iron atoms.  \nThe in-plane lattice constant a is fixed at 3.905  Å by the STO substrate. The main effect of  5 \n \nthe isovalent substitution of Te by Se is increasing  the height of  Se/Te surface H, which is \nlabelled  in Fig. 1(a). In addition, this substitution can enhance the spin-orbit coupling  (SOC ) \nsplitting for d-orbital bands  due to the intrinsic p-d hybridization.  The Fermi surface of \nmonolayer FeTe 1-xSex/STO ( x = 0.19) is shown in Fig. 1(d). There are electron pocket s at the \nM point , which have been observed in all supercondu cting monolayer FeTe 1-xSex/STO \nsamples.  They are  believed to play an important role in  promoting high-Tc superconductivity \nin this system  [8]. Our local -density approximation ( LDA ) calculations reveal that a  band \ninversion can be induced  around the  point of freestanding monolayer  FeSe  with varying \nanion height . Note that f reestanding monolayer films lack electron doping  from substrates  \ncompared with samples grown on STO.  Figure 1(e) depicts the band structure  with a low \nanion height  before band inversion . Around the  point, there are  three hole -like bands (, , \n) and one electron -like band (). With increasing H, the hybridization between Fe dxy and Se \npz weakens and the  band starts to sink . As the  band  and the , ,  bands have  opposite \ninversion -symmetry  eigenvalue s, a band inversion between them will drive the system into a \ntopologically nontrivial phase.  As shown in Fig. 1(f) for H = 1.54 Å, the orbital character \nchange of the  band bottom from pz/dxy to dyz/dxz clearly demonstrates this band inversion. \nSimultaneously, the  band top changes from dxz to pz/dxy and this induces a parity change \nfrom even to odd with respect to the xz plane, which is a direct evidence for the band \ninversion  that can be verified by  ARPES experiments .  \n        In ARPES measurement s, orbital and parity characters of bands  can be determined by \nchanging the photo n energy and the polarization  according to the matrix element effect [35, \n36]. We demonstrate the observation  of two hole -like band s (, ) and one electron -like band \n() around the  point of monolayer  FeTe 1-xSex (x = 0.21)  in Fig. 2  using photons with \ndifferent energy and polariz ations . Using our experimental facility, as shown in Fig. 2(a), odd \n(even ) orbital with respect to the M x mirror plane can be observed  with p- (s-) polarized \nphotons . The dyz orbital is odd while dxz is even. Moreover, pz orbital can be observed b y p-\npolarized photons , which  also contain the z component of photon polarization ( vector \npotential Az) in our facility geometry . The results of the matrix element effect analysis are 6 \n \nsummarized in Table 1. Furthermore, t o distinguish p and d orbital s, we can take advantage \nof the photoemission cross section difference of Fe 3 d and Te 5 p (Se 4p) in the energy range \nfrom 15 eV to 40  eV [37], as shown in Fig. 2(b).  \n        Figures 2(c)-2(d) display  ARPES results  and the correspond ing second derivative \nspectrum  measured with p-polarized, 22 eV  photo ns. The  and  band s can be observed \nclearly. These two bands form a Dirac -cone -like band structure , indicating that the system is \nclose to the topological transition  critical point . The energy of the Dirac point (DP)  can be  \nestimat ed in the momentum d istribution  curves (MDC s) plot , as shown in Fig. 2(i). When the \nphoton energy increase s to 32 eV , as shown in Figs. 2(e)-2(f), the spectral weight  of the  \nand  bands is obviously  suppressed , confirming  the pz orbital character  in these  band s. The \nresidual weak spectral weight  of the  band is attributed to  its dxy orbital character according \nto our LDA  analysis  [Fig. S4 of  Supplementa l Material  [38] ]. On the other hand,  the  band \nbecomes more apparent , suggesting its dyz orbital character with odd parity. In Figs. 2(g)-2(h), \nwhen we change the polarization geometry  from p to s, only the  band  can be detected , \nconfirming its dxz orbital character with even parity . In the measurements, t he third  band \nwith dominant  dxy orbital component can hardly  be observed because the spectral  weight  of \ndxy is the smallest  due to the strong est correlat ion among all the orbitals in iron -based \nsuperconductors [39]. By fitting the experimental  bands to E=𝐶0+𝐶1|𝑘|+𝐶2𝑘2 , we \ndetermine  that the band top of the  and  band s are located at 21.7  0.5 meV and 25.7  1.2 \nmeV below  EF, respectively,  and that the band bottom of  band is at 20.3  1.2 meV below \nEF. We summari ze the results of the band structure and orbital  analysis of the x = 0.21 sample \nin Fig. 2(j).  \n        If there is a topological transition in monolayer FeTe 1-xSex, as predicted in the \naforementioned LDA calculations,  the orbital and parity characters of the ,  and  band s \nshould exhibit a dramatic change through it. Moreover, a s shown above,  the sample with Se \ncontent  x = 0.21  is already close to the critical point . To visualize the band inversion process , 7 \n \nwe systematic ally study  the band structure  evolution  of monolayer FeTe 1-xSex as a function of \nthe Se content  x.  \n        Figure s 3(a)-3(c) show the band structures of six different Se content samples detected \nby 22 eV p-, 32 eV p- and 32 eV s-polarized photons. With one particular photon  energy , we \ncan observe dramatic changes of bands in samples with different content. In Fig. 3(a ), the \nspectra weight of the  band increases dramatically with decreasing Se content  indicating the \nincreases of pz orbital character in the  band. Moreover, as indicated by a red arrow in Fig. \n3(a), the  band top cannot be observed before the band inversion ( x > 0.21). As the orbital \ncharacter of the  band top changes to pz/dxy during the band inversion  process , the  band \ntop becomes signif icantly enhanced  after band inversion ( x < 0.21) . The enhancement of  \nband can also be found in Fig. 3(b), which shows the evolution of d-orbital bands with odd \nparities. The  band is expected to be clear and the  band to be completely suppressed \nbefore the band inversion due to its even parity nature. However, the missing spectral weight \nindicated by a red arrow becomes apparent after the band inversion. Compared with Fig. 3(a), \nthe re -emerging band is attributed to the  band, which clearly suggests that the parity of the \n band top changes from even to odd. In contrast to results of p-polarized photons, the \nevolution of the  band exhibits the opposite behavior when measurements are performed \nwith 32 eV s-polariz ed photons and only orbitals with even parities can be detected , as shown \nin Fig. 3(c). The spectral weight of the  band top is clear  indicated by a green arrow before \nthe band inversion and is compar ative  suppressed after the band inversion , which is als o \nconsistent with the above analysis.  \n        We can also trace the evolution of  the  band . In contrast to the  band, no obvious \nchange  occurs  for the  band, except for a band shift to a lower binding energy . To see it \nmore clearly, w e show the MDCs plot of the  and  band s in Figs. 3(d)-3(e), respectively.  \nThe peak position of the  band continuously evolve s toward the  point , while that of the  \nband is almost unchanged  with the Se content decreasing . Then, we calculate the effective \nmass of the ,  and  band s. The results are  shown in Fig. 3(f). The effective mass of the 8 \n \nthree bands exhibits rather different trends with the variation of the Se content. When the Se \ncontent  decreases , the value of the  band decreases dramatically  while that of the  band \nincreas es rapidly . However , the effective mass of the   band  remains almost unchanged . \nThese changes of effective masses can be interpreted in terms of the orbital character change \ndue to band inversion . Because  the d orbital s exhibit stronger correlation effect than the p \norbitals , the  band becomes more correlated due to the increas ing d orbital  components , \nwhile the  band behaves opposite ly due to the increasing p orbital component , and the  \nband shows nearly no change , in agreement with i ts relative d and p components  remaining \nthe same in the process of band inversion.  \n        According to the bulk -boundary correspondence, topological edge states (TES s) will \nappear at the edge of nontrivial monolayer  FeTe 1-xSex/STO films  [1]. We carry out LDA \ncalculations for Fe-Te [100] and Fe-Fe [110] edges  of monolayer FeTe , as shown in Figs. 4(a) \nand 4(d) . The obtained band structures and local density -of-states (LDOS)  for bulk and edge \natoms are displayed in Figs. 4(b)-4(c) for the [100] edge  and Figs. 4(e)-4(f) for the [110] edge.  \nIn the calculations, the nontrivial gap is located about 0.2 eV above the Fermi level and \ngapless Dirac -cone -like edge state s should appear in the region.  Actually, this is the case for \nthe (100) surface, as shown in Fig. 4(b). The edge state s extend  over nearly the whole \nBrillouin zone  and merge into the bulk states near the M point . Due to the 1D nature of the \nDirac band dispersion, the edge LDOS is almost a constant near the nontrivial gap. With \ndecreasing energy, the edge LDOS increases rapidly and develops a peak at 0.2 eV below the \ngap, which is attributed to the flat dispersion of edge states.  A similar pe ak can also be found \nin the LDOS of the [110] edge , although a constant LDOS nearly disappears due to the strong \noverlap between the edge and bulk states, as shown in Figs. 4(e)-4(f). \n        To demonstrate the edge states experimentally , in situ STM/STS measurements are \ncarried out after growth of  monolayer FeTe/STO . Topographic images of monolayer FeTe \ncontaining Fe -Te [100] and Fe -Fe [110] edges are presented in Figs. 4(g)-4(h), respectively . \nThe inset s of Figs. 4(g) -4(h) show the atomically -resolved topography  indicating the high-9 \n \nquality of our sample. We take STS measurements along the lines indicated by the yellow  \narrows in Figs. 4(g) and 4( h) from the edges of the Fe-Te and Fe -Fe directions to the inside \nfilm, as shown  in Figs. 4(i) and 4( j). The common  feature is that  noticeable  enhancement s \naround  50 meV  below EF (30 meV below the  band bottom) appear for the edges  compared \nwith the bulk, which decay to the bulk value within several nanomete rs. According to the \ncalculated results in Fig. 4(b), we find that the edge states connect ing the  band at the  \npoint and the electron  band at the M point  have a section of flat dispersion around the band \nbottom of the  band, which probably  correspond s to the enhanced DOS. Although  the exact \nvalue of the calculated binding energy is not reliable  due to strong correlations in FeTe , the \nband bottom of the  band is reported experimentally to reside around 50  meV below EF [8], \nwhich is consistent with the energy observed in STM/STS measurements . Therefore , the \nincreased DOS around this energy  level  is expected to be induced by topologic al edge  states \naccording to our LDA calculations . Furthermore, for the spectra of the Fe-Te edge, a plateau \nof DOS  appears around 0  meV to  20 meV below E F in Fig. 4 (i), which probably correspond s \nto the Dirac point of the edge state s. The energy value  of the Dirac point  matches well with \nthe band structure  results shown in Fig. 2(c).  \nIV.    CONCLUSION   \n       Our ARPES and LDA calculation  results  provide direct evidence of topological \ntransition  in high-Tc FeTe 1-xSex/STO  monolayers .  The consistency of STM/STS experiments  \nand ARPES/LDA results further support s it. The topological transition  can be  tuned  by Te \ndoping. Moreover, the position  of topological bands can be tuned by  abundant methods, \nincluding post anneal treatment [19], alkali metal doping [40, 41] , changing various  \nsubstrates [42-44] and electric field regulation [45-47]. To d irectly observe the topological \nedge states by ARPES, we suggest th at FeTe 1-xSex monolayers should be grown on STO \nsubstrates with densely parallel edges, which can be obtained by chamfer ing STO at  specific \nangle.  10 \n \nFeTe 1-xSex/STO monolayers  are the two -dimensional counterpart of FeTe 0.5Se0.5 single \ncrystal  but with much higher Tc. Majorana bound states have  been  observed  on the surface of \nthe latter [11,12]. The superconductivity on topological edge states of FeTe 1-xSex/STO \nmonolayers  can be introduced by the bulk proximity effect.  If a magnetic defect  is deposited \nat the superconducting edges  to suppress  the superconductivity , Majorana bound states are \nexpected to appear at the  domain wall between the superconducting and magnetic insulating \nregions.  Due to the high superconducting transition temperatu re in FeTe 1-xSex/STO  \nmonolayers [8, 28] , our discovery of its topological nontrivial properties provide s us a simple \nand tunable  platform  for realizing  topological superconduct ivity and manipulat ing Majorana \nbound states at high temperature.   \nACKNOWLEDG MENTS  \n        We thank P. Zhang , Z.-Q. Han and L.-Y. Kong  for useful discussions. This work is \nsupported by grants from the Ministry of Science and Technology of China \n(2016YFA0401000, 2015CB921000 , 2016YFA0300600) and the National Natural Science \nFoundation of China ( 11574371, 11622435, U1832202 , 11888101 ). Y.B.H. acknowledges \nsupport by the CAS Pioneer ”Hundred Talents Program” (type C). P.R. acknowledges the \nfunding from the Canada First Research Excellence Fund.  \n        X.L.P ., Y.L ., H.B.D ., H.Q.M . and Y.J.S . synthesized the samples. X.L.P . performed the \nARP ES measurements and analy zed the data with help from X.S ., W.H.F ., M.L . and Y.B.H. .  \nY.L., H.B.D . and H.Q.M . performed the STM/STS measurements and analy zed the data. \nX.X.W . and J.P.H . performed the LDA calculations and provided theoretical input. X.L.P ., \nY.L., X.X.W ., P.R., J.P.H., H.Q.M ., Y.J.S . and H.D . wrote the manuscript. All authors \ndiscussed the manuscript. H.D. and Y.J.S. supervised the project.  \n        X.L.P ., Y.L. and X.X.W . contributed equally to this work.  \n       †Correspondence  author s: mhq@iphy.ac.cn; yjsun@iphy.ac.cn; dingh@iphy.ac.cn  11 \n \n[1]   X.L. Qi, S.C. Zhang, Topological insulators and superconductors , Rev . Mod . Phys . 83,  \n1057  (2011).  \n[2]   S.R. Elliott, M. Franz, Colloquium: Majorana fermions in nuclear, particle, and solid -\nstate physics , Rev . Mod . Phys .  87,  137  (2015).  \n[3]   C. Kallin, Chiral p -wave order in Sr 2RuO 4, Rep . Prog. Phys .  75,  042501  (2012).  \n[4]   A.P. Mackenzie, Y. Maeno, The superconductivity of Sr 2RuO 4 and the physics of spin -\ntriplet pairing , Rev . Mod . Phys .  75,  657  (2003).  \n[5]   L. Fu, C.L. Kane, Superconducting proximity effect and Majorana fermions at the \nsurface of a topological insulator , Phys . Rev. Lett.  100,  096407  (2008).  \n[6]   H.H. Sun, K.W. Zhang, L.H. Hu, C. Li, G.Y. Wang, H.Y. Ma, Z.A. Xu, C.L. Gao, D.D. \nGuan, Y.Y. Li, et al., Majorana Zero Mode Detected with Spin Selective Andreev \nReflection in the Vortex  of a Topological Superconductor , Phys . Rev. Lett.  116,  \n257003  (2016).  \n[7]   N. Hao, J. Hu, Topological quantum states of matter in iron -based superconductors: \nFrom concepts to material realization ,  arxiv :1811.03802  (2018) . \n[8]   X. Shi, Z.Q. Han, P. R ichard, X.X. Wu, X.L. Peng, T. Qian, S.C. Wang, J.  P. Hu, Y.J. \nSun, H. Ding, FeTe 1-xSex monolayer films: towards the realization of high -temperature \nconnate topological superconductivity , Sci. Bull.  62,  503  (2017).  \n[9]   Z.J. Wang, P. Zhang, G. Xu, L.K. Zeng, H. Miao, X.Y. Xu, T. Qian, H.M. Weng, P. \nRichard, A.V. Fedorov, et al., Topologi cal nature of the FeSe 0.5Te0.5 superconductor , \nPhys . Rev. B  92,  115119  (2015).  \n[10]  G. Xu, B. Lian, P.Z. Tang, X.L. Qi, S.C. Zhang, Topological Superconductivity on the \nSurface of Fe -Based Superconductors ,  Phys . Rev. Lett. 117,  047001  (2016).  \n[11] P . Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z.J. Wang, J.S. Wen, \nG.D. Gu, H. Ding, et al., Observation of topological superconductivity on the surface of \nan iron -based superconductor , Science   360,  182  (2018).  \n[12] D.F. Wang, L.Y. Kong,  P. Fan, H. Chen, S.Y. Zhu, W.Y. Liu, L. Cao, Y.J. Sun, S.X. Du, \nJ. Schneeloch, et al., Evidence for Majorana bound states in an iron -based \nsuperconductor , Science   362,  333  (2018).  \n[13] Y.F. Lv, W.L. Wang, Y.M. Zhang, H. Ding, W. Li, L.L. Wang, K. He, C.L. Song, X.C. \nMa, Q.K. Xue, Experimental signature of topological superconductivity and Majorana \nzero modes on -Bi2Pd thin films , Sci. Bull.  62,  852-856  (2017).  12 \n \n[14] X. Wu, S. Qin , Y. Liang, C. Le, H. Fan, J. Hu, CaFeAs 2: A staggered intercalation of \nquantum spin Hall and high -temperature superconductivity , Phys . Rev. B  91, 081111(R)    \n(2015).  \n[15] P. Zhang, Z.J. Wang, X.X. Wu, K. Yaji, Y. Ishida, Y. Kohama, G.Y. Dai, Y. Sun, C. \nBareille, K. Kuroda, et al., Multiple topological states in iron -based superconductors , \nNat. Phys . 15,  41  (2019).  \n[16] Q. Liu, C. Chen, T. Zhang, R. Peng, Y.J. Yan, C.H.P. Wen, X. Lou, Y.L. Huang, J.P. \nTian, X.L. Dong, et al., Robust and Clean Majorana Ze ro Mode in the Vortex Core of \nHigh -Temperature Superconductor (Li 0.84Fe0.16)OHFeSe , Phys Rev X   8,  041056  \n(2018).  \n[17] Q.Y. Wang, Z. Li, W.H. Zhang, Z.C. Zhang, J.S. Zhang, W. Li, H. Ding, Y.B. Ou, P. \nDeng, K. Chang, et al., Interface -Induced High -Temperature Superconductivity in \nSingle Unit -Cell FeSe Films on SrTiO 3, Chin. Phys. Lett.   29,  037402  (2012).  \n[18] D.F. Liu, W.H. Zhang, D.X. Mou, J.F. He, Y.B. Ou, Q.Y. Wang, Z. Li, L.L. Wang, L. \nZhao, S.L. He, et al., Electronic origin of high -temperature superconductivity in single -\nlayer FeSe superconductor , Nat Commun   3,  931  (2012).  \n[19] S.L. He, J.F. He, W.H. Zhang,  L. Zhao, D.F. Liu, X. Liu, D.X. Mou, Y.B. Ou, Q.Y. \nWang, Z. Li, et al., Phase diagram and electronic indication of high -temperature \nsuperconductivity at 65 K in single -layer FeSe films , Nat Mater   12,  605-610  (2013).  \n[20] S.Y. Tan, Y. Zhang, M. Xia, Z.R . Ye, F. Chen, X. Xie, R. Peng, D.F. Xu, Q. Fan, H.C. \nXu, et al., Interface -induced superconductivity and strain -dependent spin density waves \nin FeSe/SrTiO 3 thin films , Nat Mater   12,  634-640  (2013).  \n[21] J.J. Lee, F.T. Schmitt, R.G. Moore, S. Johnston, Y.T. Cui, W. Li, M. Yi, Z.K. Liu, M. \nHashimoto, Y. Zhang, et al., Interfacial mode coupling as the origin of the enhancement \nof T c in FeSe films on SrTiO 3, Nature   515,  245  (2014).  \n[22] J.F. Ge, Z.L. Liu, C.H. Liu, C.L. Gao, D. Qian, Q.K. Xue, Y. Liu, J. F. Jia, \nSuperconductivity above 100 K in single -layer FeSe films on doped SrTiO 3, Nat Mater, \n14,  285-289  (2015).  \n[23] Z.C. Zhang, Y.H. Wang, Q. Song, C. Liu, R. Peng, K.A. Moler, D.L. Feng, Y.Y. Wang, \nOnset of the Meissner effect at 65 K in FeSe thin fil m grown on Nb -doped SrTiO 3 \nsubstrate , Sci. Bull.  60,  1301 -1304  (2015).  \n[24] X. Shi, Z.Q. Han, X.L. Peng, P. Richard, T. Qian, X.X. Wu, M.W. Qiu, S.C. Wang, J.P. \nHu, Y.J. Sun, et al., Enhanced superconductivity accompanying a Lifshitz transition in \nelectron -doped FeSe monolayer , Nat . Commun .  8,  14988  (2017).  13 \n \n[25] N.N. Hao, J.P. Hu, Topological Phases in the Single -Layer FeSe , Phys . Rev. X  4,  \n031053  (2014).  \n[26] N.N. Hao, S.Q. Shen, Topological superconducting states in monolayer FeSe/SrTiO 3, \nPhys . Rev. B  92,  165104  (2015).  \n[27] Z.F. Wang, H.M. Zhang, D.F. Liu, C. Liu, C.J. Tang, C.L. Song, Y. Zhong, J.P. Peng, \nF.S. Li, C.N. Nie, et al., Topological edge states in a high -temperature superconductor \nFeSe/SrTiO 3(001) film , Nat . Mater .  15,  968-973  (2016).  \n[28] F.S. Li, H. Ding, C.J. Tang, J.P. Peng, Q.H. Zhang, W.H. Zhang , G.Y. Zhou, D. Zhang, \nC.L. Song, K. He, et al., Interface -enhanced high -temperature superconductivity in \nsingle -unit-cell FeTe 1-xSex films on SrTiO 3, Phys. Rev. B   91,  220503 (R)  (2015).  \n[29] X.X. Wu, S.S. Qin, Y. Liang, H. Fan, J.P. Hu, Topological characters in Fe(Te 1-xSex) \nthin films , Phys . Rev. B  93,  115129  (2016).  \n[30] G. Kresse, J. Hafner, Abinitio Molecular -Dynamics for Liquid -Metals ,  Phys . Rev. B  47,  \n558-561  (1993).  \n[31] G. Kresse, J. Furthmuller, Efficiency of ab -initio total energy calculations for metals and \nsemiconductors using a plane -wave basis set , Comp . Mater . Sci.  6,  15-50  (1996).  \n[32] G. Kresse, J. Furthmuller, Efficient iterative schem es for ab initio total -energy \ncalculations using a plane -wave basis set , Phys . Rev. B, 54  11169  (1996).  \n[33] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple , \nPhys . Rev. Lett.  77,  3865  (1996).  \n[34] H.J. Monkhorst, J. D. Pack, Special Points for Brillouin -Zone Integrations , Phys . Rev. B  \n13,  5188 -5192  (1976).  \n[35] X.P. Wang, P. Richard, Y.B. Huang, H. Miao, L. Cevey, N. Xu, Y.J. Sun, T. Qian, Y.M. \nXu, M. Shi, et al., Orbital characters determined from Fermi surface intensity patterns \nusing angle -resolved photoemission spectroscopy , Phys . Rev. B  85,  214518  (2 012).  \n[36] P. Richard, T. Qian, H. Ding, ARPES measurements of the superconducting gap of Fe -\nbased superconductors and their implications to the pairing mechanism , J. Phys . \nCondens . Matter  27,  293203  (2015).  \n[37] J.J. Yeh, I. Lindau, Atomic Subshell Pho toionization Cross -Sections and Asymmetry \nParameters : 1 ⩽ Z ⩽ 103, At. Data Nucl . Data  Tables   32,  1  (1985).  \n[38] See Supplemental Materials at http://link.aps.org/  for more details., DOI.  \n[39] M. Yi, Y. Zhang, Z. -X. Shen, D. Lu, Role of the orbital deg ree of freedom in iron -based \nsuperconductors , npj Quantum Materials   2,  57  (2017).  14 \n \n[40] Y. Miyata, K. Nakayama, K. Sugawara, T. Sato, T. Takahashi, High -temperature \nsuperconductivity in potassium -coated multilayer FeSe thin films , Nat . Mater .  14,  775  \n(2015).  \n[41] G.N. Phan, K. Nakayama, S. Kanayama, M. Kuno, K. Sugawara, T. Sato, T. Takahashi, \nHigh -Temperature Superconductivity and Lattice Relaxation in Lithium -Deposited FeSe \non SrTiO 3, J. Phys . Soc. Jpn.  86,  033706  (2017).  \n[42] R. Peng, H.C.  Xu, S.Y. Tan, H.Y. Cao, M. Xia, X.P. Shen, Z.C. Huang, C.H.P. Wen, Q. \nSong, T. Zhang, et al., Tuning the band structure and superconductivity in single -layer \nFeSe by interface engineering , Nat . Commu m.  5,  5044  (2014).  \n[43] P. Zhang, X.L. Peng, T. Qian,  P. Richard, X. Shi, J.Z. Ma, B.B. Fu, Y.L. Guo, Z.Q. Han, \nS.C. Wang, et al., Observation of high -Tc superconductivity in rectangular \nFeSe/SrTiO 3(110) monolayers , Phys . Rev. B 94,  104510  (2016).  \n[44] S.N. Rebec, T. Jia, C. Zhang, M. Hashimoto, D.H. Lu, R .G. Moore, Z.X. Shen, \nCoexistence of Replica Bands and Superconductivity in FeSe Monolayer Films , Phys . \nRev. Lett.  118,  067002  (2017).  \n[45] B. Lei, J.H. Cui, Z.J. Xiang, C. Shang, N.Z. Wang, G.J. Ye, X.G. Luo, T. Wu, Z. Sun, \nX.H. Chen, Evolution of High -Temperature Superconductivity from a Low -Tc Phase \nTuned by Carrier Concentration in FeSe Thin Flakes , Phys . Rev. Lett. 116,  077002  \n(2016).  \n[46] J. Shiogai, Y. Ito, T. Mitsuhashi, T. Nojima, A. Tsukazaki, Electric -field-induced \nsuperconductivity in e lectrochemically etched ultrathin FeSe films on SrTiO 3 and MgO , \nNat. Phys . 12,  42  (2016).  \n[47] K. Hanzawa, H. Sato, H. Hiramatsu, T. Kamiya, H. Hosono, Electric field -induced \nsuperconducting transition of insulating FeSe thin film at 35 K , Proc. Natl. Ac ad. Sci. \nU.S.A. 113,  3986  (2016).  \n \n 15 \n \n \nFIG. 1.    (a) Schematic crystal structure of monolayer  FeTe 1-xSex/STO. H is defined as the \nTe/Se height relative to the Fe plane. (b) Typical RHEED pattern of STO substrate (upper \npanel) after treatment in vacuum and FeTe 1-xSex/STO monolayer (lower panel) after \nannealing. (c) In situ  XPS results of six samples with different Se content x detected by 100 \neV photons. (d) Definition of the Brillouin Zone of the two Fe unit cell. The Fermi surface of \nmonolayer  FeTe 0.81Se0.19/STO is measured  at T = 15 K. The intensity ha s been integrated in \nthe 10 meV  energy range.  White dashed line represents the cut position of ARPES spectrum \nshown as followed. (e)-(f) LDA  calculation results  of band structure  for freestanding \nmonolayer  FeSe /STO with Se height H = 1.50 Å and H = 1.54 Å, respectively.  Three hole -\nlike bands ( , , ) and one electron -like band  are predicted around Γ. The main orbital of \nthe  band is d xy according to our calculation  [Fig. S4 of  Supplementa l Material  [38] ] .  \n \n16 \n \n \nFIG. 2.     Electronic structure of monolayer  FeTe 0.79Se0.21/STO  around the Brillouin Zone \ncenter (Γ). (a) Schematic e xperimental geometry of ARPES measurements. (b) Comparison \nof cross section of Fe 3 d and Se 4 p (Te 5 p) [37]. (c)-(d) Electronic structure around the Γ \npoint detected by 22 eV p-polarized photon s and the corresponding second derivative \nspectrum. Red dots in (d) represent peak position of MDC fitted by Lorentz function . Blue \ndashed curves are  fitting results of the MDC peak position , using the function E=𝐶0+\n𝐶1|𝑘|+𝐶2𝑘2. (e)-(h) Same as (c)-(d) but measured with  (e)-(f) 32 eV p-polarized photon s \nand (g)-(h) 32 eV s-polarized photon s, respectively. (i) MDC plot of the dashed rectangular \nregion in (c). Green dots show MDC peak position fitted by Lorentz function. Black dashed \nlines are schematic of the Dirac -cone -like dispersion after tracing the position of MDC peaks. \nThe MDC s at energ ies of the Fermi level ( EF) and the Dirac point  (DP) are highlighted by \nblack and blue, respectively. (j) Summary  of the band structure  and orbital analysis  around . \nThe t hree solid curves are results of fitting of MDC peaks  of the ,  and  band s. Black \ndashed curve s are the schematic results of band hybridization due to spin -orbit coupling \n(SOC) .  \n \n \n \n \n \n \n \n17 \n \n \nFIG. 3.     Band inversion in monolayer FeTe 1-xSex/STO . (a)-(c) Electronic structure evolution \nwith Se content, detected by (a) 22 eV p-polariz ed, (b) 32 eV p-polariz ed and (c) 32 eV s-\npolariz ed photons , respectively. The Se content s x of six samples (from left to right) are 1, \n0.47, 0.31, 0.21, 0.12 and 0.07 , which are determined by in situ XPS method. The red arrows \nin (a) and (b) show the relatively weak intensity around  Γ before the topological transition.  \nThe green arrows in (c) indicate the appearance of band top.  (d)-(e) MDC plot of six samples \nat energy position s pointed by  #1 in (a) and #2 in (b), respectively . Position of #1 and #2 is \n80 meV below the  and  band top s. Four -Lorentzian -function fitting results for the x = 1 \nsample are presented at the bottom of (d) and (e). Black dots on each MDC are fitting results \nof the peak position s. Lorentzian function  curves  associated with the  and  bands are \nshaded in (d) and (e), respectively.  The black arrow  in (e) indicate s the appearance of  the  \nband after the topological transition. (f) Effective mass evolution with Se content . The value s \nof the ,  and  bands  are shown by red, green and blue , respectively .  \n \n18 \n \nFIG. 4.     Topological edge states in monolayer  FeTe /STO . (a) Model of atomic chain along \nthe Fe-Te direction . (b) One-dimensional band structure for  the atomic chain in (a). The \ngreen arrow represents the position of the Dirac point (DP) and the sizes of the blue circles \nrepresent the contributions of the left edge . (c) Calculated local density -of-states (LDOS) of \nbulk and edge for model (a). (d)-(f) Same as (a)-(c) but for the atomic chain  along  the Fe-Fe \ndirection.  (g)-(h) Experimental STM Topography  of the Fe -Te edge  (V = 500 mV, I = 50 pA)  \nand the Fe -Fe edge  (V = 1 V, I = 50 pA)  of FeTe/STO monolayer . Notice that the roughness \nof the film is mainly due to STO substrate instead of Te/Se height difference [Fig. S5 of  \nSupplementa l Material  [38] ] . The inset shows the a tomically resolved topogr aphy with the \nedge along Fe -Te (V  = 100 mV, I  = 0.5nA) and Fe -Fe direction (V  = 100 mV, I  = 1 nA), \nrespectively.  (i)-(j) dI/dV spectra (Junction setpoint V  = 100mV, I  = 0.2 nA) from edge to \nbulk detected along the yellow arrow shown in (g)-(h), respectively. The black arrows show \nthe density -of-states enhancement  near the edge . The plat eau labelled DP  in (i) indicates the \nexistence of Dirac point.  \n \n \n19 \n \n \n \n \nTAB. 1.    ARPES matrix element effect of orbital s relevant to the band inversion process.  ”✓” \nindicates that the orbital can be observed by the corresponding polarized photons, while ” ×” \nmeans that it can not be observed. “k≠0” indicates that the orbital can be observed away from \nthe  point.  \n \n"    },    {        "title": "2312.02054v1.From_High_to_Low__Simulating_Nondeterminism_and_State_with_State.pdf",        "content": "arXiv:2312.02054v1  [cs.PL]  4 Dec 2023Preprint submitted to JFP ,120pages, 2023. 1\nFrom High to Low: Simulating Nondeterminism\nand State with State\nWENHAO TANG\nThe University of Edinburgh\n(e-mail: wenhao.tang@ed.ac.uk)\nTOM SCHRIJVERS\nKU Leuven Department of Computer Science\n(e-mail: tom.schrijvers@kuleuven.be)\nAbstract\nSome effects are considered to be higher-level than others. H igh-level effects provide expressive and\nsuccinct abstraction of programming concepts, while low-l evel effects allow more fine-grained control\nover program execution and resources. Yet, often it is desir able to write programs using the convenient\nabstraction offered by high-level effects, and meanwhile sti ll benefit from the optimisations enabled\nby low-level effects. One solution is to translate high-leve l effects to low-level ones.\nThis paper studies how algebraic effects and handlers allow u s to simulate high-level effects in\nterms of low-level effects. In particular, we focus on the int eraction between state and nondeterminism\nknown as the local state, as provided by Prolog. We map this hi gh-level semantics in successive steps\nonto a low-level composite state effect, similar to that mana ged by Prolog’s Warren Abstract Machine.\nWe first give a translation from the high-level local-state s emantics to the low-level global-state\nsemantics, by explicitly restoring state updates on backtr acking. Next, we eliminate nondeterminsm\naltogether in favor of a lower-level state containing a choi cepoint stack. Then we avoid copying the\nstate by restricting ourselves to incremental, reversible state updates. We show how these updates can\nbe stored on a trail stack with another state effect. We prove t he correctness of all our steps using\nprogram calculation where the fusion laws of effect handlers play a central role.\n1 Introduction\nThe trade-off between “high-level” and “low-level” styles o f programming is almost as\nold as the field of computer sciences itself. In a high-level s tyle of programming, we lean\non abstractions to make our programs easier to read and write , and less error-prone. We\npay for this comfort by giving up precise control over the und erlying machinery; we forego\noptimisation opportunities or have to trust a (usually opaq ue) compiler to perform low-level\noptimisations for us. For performance-sensitiveapplicat ions, compiler optimisations are not\nreliable enough; instead we often resort to lower-level pro gramming techniques ourselves.\nAlthough they allow a fine-grained control over program exec ution and the implementation\nof optimisation techniques, they tend to be harder to write a nd not compose very well. This\nis an important trade-off to take into account when choosing a n appropriate programming\nlanguage for implementing an application.2\nMaybe surprisingly, as they are rarely described in this way , there is a similar pattern for\nside-effects within programming languages: some effects can be described as “lower-level”\nthan others. We say that an effect is lower-level than another effect when the lower-level\neffect can simulate the higher-level effect. In other words, i t is possible to write a program\nusing lower-level effects that has identical semantics to th e same program with higher-level\neffects. Yet, due to the lack of abstraction of low-level effec ts, writing a faithful simulation\nrequires careful discipline and is quite error-prone.\nThis article investigates how we can construct programs tha t are most naturally expressed\nwith a high-level effect, but where we still want access to the optimisation opportunities of a\nlower-level effect. In particular, inspired by Prolog and Co nstraint Programming systems, we\ninvestigate programs that rely on the high-level interacti on between the nondeterminismand\nstate effects which we call local state . Following low-level implementation techniques for\nthese systems, like the Warren Abstract Machine (WAM) (Warr en,1983; A¨ıt-Kaci, 1991),\nwe show how these can be simulated in terms of the low-level global state interaction of\nstate and nondeterminism, and finally by state alone. This al lows us to incorporate typical\noptimisations like exploiting mutable state for efficient ba cktracking based on trailing as\nopposed to copying or recomputing the state from scratch (Sc hulte, 1999).\nOur approach is based on algebraic effects and handlers (Plot kin and Power, 2003;\nPlotkin and Pretnar, 2009,2013) to cleanly separate the syntax and semantics of effects.\nFor programs written with high-level effects and interprete d by their handlers, we can define\na general translation handler to transform these high-leve l effects to low-level effects, and\nthen interpret the translated programs with the handlers of low-level effects.\nOf particular interest is the way we reason about the correct ness of our approach. There\nhas been much debate in the literature on different equationa l reasoning approaches for\neffectful computations. Hutton and Fulger ( 2008) break the abstraction boundaries and\nuse the actual implementation in their equational reasonin g approach. Gibbons and Hinze\n(2011) promote an alternative, law-based approach to preserve ab straction boundaries and\ncombine axiomatic with equational reasoning. In an earlier version of this work (Pauwels\net al., 2019), we have followed the latter, law-based approach for reaso ning about the\ncorrectness of simulating local state with global state. Ho wever, we have found that approach\nto be unsatisfactory because it incorporates elements that are usually found in the syntactic\napproachfor reasoning about programminglanguages (Wrigh t and Felleisen, 1994), leading\nto more boilerplate and complication in the proofs: notions of contextual equivalence\nand explicit manipulation of program contexts. Hence, for t hat reason we return to the\nimplementation-based reasoning approach, which we believ e works well with algebraic\neffects and handlers. Indeed, we prove all of our simulations correct using equational\nreasoning techniques, exploiting in particular the fusion property of handlers (Wu and\nSchrijvers, 2015; Gibbons, 2000).\nAfter introducing the reader to the appropriate background material and motivating the\nproblem ( Section 2 andSection 3 ), this paper makes the following contributions:\n•We distinguish between local-state and global-state seman tics, and simulate the\nformer in terms of the latter ( Section 4 ).\n•We simulate nondeterminism using a state that consists of a c hoicepoint stack\n(Section 5 ).3\n•We combine the previous two simulations and merge the two sta tes into a single state\neffect ( Section 6 ).\n•By only allowing incremental, reversible updates to the sta te we can avoid holding\non to multiple copies of the state ( Section 7 ).\n•By storing the incremental updates in a trail stack state, we can restore them in batch\nwhen backtracking ( Section 8 ).\n•We prove all simulations correct using equational reasonin g techniques and the fusion\nlaw for handlers in particular ( Appendix 2 ,Appendix 3 ,Appendix 4 ,Appendix 5 ,\nAppendix 6 andAppendix 7 ).\nFinally, we discuss related work ( Section 9 ) and conclude ( Section 10 ).Table 1 gives an\noverview of the simulations of high-level effects with low-l evel effects we implemented\nand proved in the paper. Throughout the paper, we use Haskell as a means to illustrate our\nfindings with code.\nTranslations Descriptions Correctness\nlocal2global local state to global state ( §4.3) Theorem 1\nnondet2state nondeterminism to state ( §5.2) Theorem 3\nstates2state multiple states to a single state ( §6.1) Theorem 4\nlocal2global Mlocal state to global state with reversible updates ( §7.2)Theorem 6\nlocal2trail local state to global state with trail stacks ( §8.1) Theorem 7\nTable 1: Overview of translations from high-level effects to low-level effects in the paper.\n2 Background and Motivation\nThis section summarises the main prerequisites for equatio nal reasoning with effects and\nmotivates our translations from high-level effects to low-l evel effects. We discuss the two\ncentral effects of this paper: state and nondeterminism.\n2.1 Functors and Monads\nFunctors In Haskell, a functor f::∗→∗ instantiates the functor type class, which has a\nsingle functor mapping operation.\nclass Functor f where\nfmap ::(a→b)→f a→f b\nFurthermore, a functor should satisfy the following two fun ctor laws:\nidentity : fmap id=id, (2.1)\ncomposition :fmap(f◦g)=fmap f◦fmap g . (2.2)\nWe sometimes use the operator /a\\}bracketle{t$/a\\}bracketri}htas an alias for fmap.4\n(/a\\}bracketle{t$/a\\}bracketri}ht)::Functor f⇒(a��b)→f a→f b\n(/a\\}bracketle{t$/a\\}bracketri}ht)=fmap\nMonads Monadic side-effects (Moggi, 1991), the main focus of this paper, are those that\ncan dynamically determine what happens next. A monad m::∗→∗ is a functor instantiates\nthe monad type class, which has two operations return ( /u1D702) and bind ( > >=).\nclass Functor m⇒Monad m where\n/u1D702 ::a→m a\n(> >=)::m a→(a→m b)→m b\nFurthermore, a monad should satisfy the following three mon ad laws:\nreturn-bind :/u1D702x> >=f=f x, (2.3)\nbind-return : m> >=/u1D702=m, (2.4)\nassociativity :(m> >=f)> >=g=m> >=(/u1D706x→f x> >=g). (2.5)\nHaskell supports doblocks as syntactic sugar for monadic computations. For exa mple,\ndox←m;f xis translated to m> >=f. Two convenient derived operators are > >and/a\\}bracketle{t∗/a\\}bracketri}ht./one.sup\n(> >)::Monad m⇒m a→m b→m b\nm1> >m2=m1> >=/u1D706→m2\n(/a\\}bracketle{t∗/a\\}bracketri}ht)::Monad m⇒m(a→b)→m a→m b\nmf/a\\}bracketle{t∗/a\\}bracketri}htmx=mf> >=/u1D706f→mx> >=/u1D706x→/u1D702(f x)\n2.2 Nondeterminism and State\nFollowing both the approaches of Hutton and Fulger ( 2008) and of Gibbons and Hinze\n(2011), we introduce effects as subclasses of the Monad type class.\nNondeterminism The first monadic effect we introduce is nondeterminism. We de fine a\nsubclass MNondet ofMonad to capture the nondeterministic interfaces as follows:\nclass Monad m⇒MNondet m where\n∅::m a\n(/talloblong)::m a→m a→m a\nHere,∅denotes failures and (/talloblong)denotes nondeterministic choices. Instances of the\nMNondet interface should satisfy the following four laws: /two.sup\n1We deviate from the type class hierarchy of Functor ,Applicative andMonad that can be found in Haskell’s\nstandard library because its additional complexity is not n eeded in this article.\n2One might expect additional laws such as idempotence or comm utativity. As argued by Kiselyov ( 2015), these\nlaws differ depending on how the monad is used and how it should interact with other effects. We choose to\npresent a minimal setting for nondeterminism here.5\nidentity : ∅/talloblongm=m=m/talloblong∅, (2.6)\nassociativity :(m/talloblongn)/talloblongk=m/talloblong(n/talloblongk), (2.7)\nright-distributivity :(m1/talloblongm2)> >=f=(m1> >=f)/talloblong(m2> >=f), (2.8)\nleft-identity : ∅> >=f=∅. (2.9)\nThe first two laws state that (/talloblong)and∅should form a monoid, i.e., (/talloblong)should be associative\nwith∅as its neutral element. The last two laws show that (> >=)is right-distributive over\n(/talloblong)and that∅cancels bind on the left.\nThe approach of Gibbons and Hinze ( 2011) is to reason about effectful programs using\nan axiomatic characterisation given by these laws. It does n ot rely on the specific imple-\nmentation of any particular instance of MNondet . In contrast, Hutton and Fulger ( 2008)\nreason directly in terms of a particular instance. In the cas e ofMNondet , the quintessential\ninstance is the list monad, which extends the conventional Monad instance for lists.\ninstance MNondet[]where\n∅=[]\n(/talloblong)=(++)instance Monad[]where\n/u1D702x=[x]\nxs> >=f=concatMap f xs\nState The signature for the state effect has two operations: a getoperation that reads and\nreturns the state, and a putoperation that modifies the state, overwriting it with the gi ven\nvalue, and returns nothing. Again, we define a subclass MState ofMonad to capture its\ninterfaces.\nclass Monad m⇒MState s m|m→swhere\nget::m s\nput::s→m()\nThese operations are regulated by the following four laws:\nput-put : put s> >put s′=put s′, (2.10)\nput-get : put s> >get=put s> >/u1D702s, (2.11)\nget-put : get> >=put=/u1D702(), (2.12)\nget-get :get> >=(/u1D706s→get> >=k s)=get> >=(/u1D706s→k s s). (2.13)\nThe standard instance of MState is the state monad State s .\nnewtype State s a=\nState{runState::s→(a,s)}\ninstance MState s(State s)where\nget=State(/u1D706s→(s,s))\nput s=State(/u1D706→((),s))instance Monad(State s)where\n/u1D702x=State(/u1D706s→(x,s))\nm> >=f=State(/u1D706s→let(x,s′)=runStatem s\ninrunState(f x)s′)6\n2.3 The N-queens Puzzle\nThe n-queens problem used here is an adapted and simplified ve rsion from that of Gibbons\nand Hinze ( 2011). The aim of the puzzle is to place /u1D45Bqueens on a /u1D45B×/u1D45Bchess board such that\nno two queens can attack each other. Given /u1D45B, we number the rows and columns by [1..n].\nSince all queens should be placed on distinct rows and distin ct columns, a potential solution\ncan be represented by a permutation xsof the list[1..n], such that xs!!i=jdenotes that\nthe queen on the ith column is placed on the jth row. Using this representation, queens\ncannot be put on the same row or column.\nA Naive Algorithm We have the following naive nondeterministic algorithm for n-queens.\nqueens naive::MNondet m⇒Int→m[Int]\nqueens naiven=choose(permutations[1..n])> >=filtr valid\nThe program queens naive4 ::[[Int]]gives as result[[2,4,1,3],[3,1,4,2]]. The pro-\ngram uses a generate-and-test strategy: it generates all pe rmutations of queens as candiate\nsolutions, and then tests which ones are valid.\nThe function permutations ::[a]→[[ a]]from Data.Listcomputes all the permutations\nof its input. The function choose implemented as follows nondeterministically picks an\nelement from a list.\nchoose ::MNondet m⇒[a]→m a\nchoose=foldr((/talloblong)◦/u1D702)∅\nThe function filtr p x returns xifp xholds, and fails otherwise.\nfiltr::MNondet m⇒(a→Bool)→a→m a\nfiltr p x=ifp xthen/u1D702xelse∅\nThe pure function valid ::[Int]→Bool determines whether the input is a valid solution.\nvalid ::[Int]→Bool\nvalid[]=True\nvalid(q:qs)=valid qs∧safe q 1qs\nA solution is valid when each queen is safewith respect to the subsequent queens:\nsafe::Int→Int→[Int]→Bool\nsafe[]=True\nsafe q n(q1:qs)=and[q/nequivalenceq1,q/nequivalenceq1+n,q/nequivalenceq1−n,safe q(n+1)qs]\nThe call safe q n qs checks whether the current queen qis on a different ascending and\ndescending diagonal than the other queens qs, where nis the number of columns that qis\napart from the first queen q1inqs.\nAlthough this generate-and-test approach works and is quit e intuitive, it is not very\nefficient. For example, all solutions of the form (1 : 2 : qs)are invalid because the first two\nqueens are on the same diagonal. However, the algorithm stil l needs to generate and test all\n(/u1D45B−2)! candidate solutions of this form.7\nA Backtracking Algorithm We can fuse the two phases of the naive algorithm to obtain\na more efficient algorithm, where both generating candidates and checking for validity\nhappens in a single pass. The idea is to move to a state-based b acktracking implementation\nthat allows early pruning of branches that are invalid. In pa rticular, when placing the new\nqueen in the next column, we make sure that it is only placed in positions that are valid\nwith respect to the previously placed queens.\nWe use a state(Int,[Int])to contain the current column and the previously placed\nqueens. The backtracking algorithm of n-queens is implemen ted as follows.\nqueens ::(MState(Int,[Int])m,MNondet m)⇒Int→m[Int]\nqueens n=loopwhere\nloop=do(c,sol)←get\nifc/greaterorequalslantnthen/u1D702sol\nelse do r←choose[1..n]\nguard(safe r 1sol)\ns←get\nput(s⊕r)\nloop\nThe function guard fails when the input is false.\nguard ::MNondet m⇒Bool→m()\nguard True =/u1D702()\nguard False =∅\nThe function s⊕rupdates the state with a new queen placed on row rin the next column.\n(⊕)::(Int,[Int])→ Int→(Int,[Int])\n(⊕)(c,sol)r=(c+1,r:sol)\nThe above monadic version of queens essentially assume that each searching branch has\nits own state; we do not need to explicitly restore the state w hen backtracking. Though it\nis a convenient high-level programming assumption for prog rammers, it causes obstacles\nto low-level implementations and optimisations. In the fol lowing sections, we investigate\nhow low-level implementation and optimisation techniques , such as those found in Prolog’s\nWarren Abstract Machine and Constraint Programming system s, can be incorporated and\nproved correct.\n3 Algebraic Effects and Handlers\nThis section introduces algebraic effects and handlers, the approach we use to define\nsyntax, semantics, and simulations for effects. Comparing t o giving concrete monad imple-\nmentations for effects, algebraic effects and handlers allow us to easily provide different\ninterpretations for the same effects due to the clear separat ion of syntax and semantics. As\na result, we can smoothly specify translations from high-le vel effects to low-level effects as\nhandlers of these high-level effects, and then compose them w ith the handlers of low-level\neffects to interpret high-level programs. Algebraic effects and handlers also provide us with8\na modular way to combine our translations with other effects, and a useful tool, the fusion\nproperty, to prove the correctness of translations.\n3.1 Free Monads and Their Folds\nWe implement algebraic effects and handlers as free monads an d their folds.\nFree Monads Free monads are gaining popularity for their use in algebrai c effects (Plotkin\nand Power, 2002,2003) and handlers (Plotkin and Pretnar, 2009,2013), which elegantly\nseparate syntax and semantics of effectful operations. A fre e monad, the syntax of an\neffectful program, can be captured generically in Haskell.\ndataFree f a=Var a|Op(f(Free f a))\nThis data type is a form of abstract syntax tree (AST) consist ing of leaves ( Var a ) and\ninternal nodes ( Op(f(Free f a))), whose branching structure is determined by the functor\nf. This functor is also known as the signature of operations.\nA Fold Recursion Scheme Free monads come equipped with a fold recursion scheme.\nfold::Functor f⇒(a→b)→( f b→b)→Free f a→b\nfold gen alg(Var x)=gen x\nfold gen alg(Op op)=alg(fmap(fold gen alg)op)\nThis fold interprets an AST structure of type Free f a into some semantic domain b. It does\nso compositionally using a generator gen::a→bfor the leaves and an algebra alg::f b→b\nfor the internal nodes; together these are also known as a handler .\nThe monad instance of Freeis straightforwardly implemented with fold.\ninstance Functor f⇒Monad(Free f)where\n/u1D702=Var\nm> >=f=fold f Op m\nUnder certain conditions folds can be fused with functions t hat are composed with\nthem (Wu and Schrijvers, 2015; Gibbons, 2000). This gives rise to the following laws:\nfusion-pre :fold(gen◦h)alg=fold gen alg◦fmap h , (3.1)\nfusion-post :h◦fold gen alg =fold(h◦gen)alg′withh◦alg=alg′◦fmap h , (3.2)\nfusion-post’ :h◦fold gen alg =fold(h◦gen)alg′(3.3)\nwithh◦alg◦fmap f=alg′◦fmap h◦fmap f andf=fold gen alg .\nThese three fusion laws turn out to be essential in the furthe r proofs of this paper.\nNondeterminism Instead of using a concrete monad like List, we use the free monad\nFree Nondet Fover the signature Nondet Ffollowing algebraic effects.\ndataNondet Fa=Fail|Or a a\nThis signatures gives rise to a trivial MNondet instance:9\ninstance MNondet(Free Nondet F)where\n∅=Op Fail\n(/talloblong)p q=Op(Or p q)\nWith this representation the right-distributivity law and the left-identity law follow\ntrivially from the definition of (> >=)for the free monad.\nIn contrast, the identity andassociativity laws are not satisfied on the nose. Indeed,\nOp(Or Fail p)is for instance a different abstract syntax tree than p. Yet, these syntactic\ndifferences do not matter as long as their interpretation is t he same. This is where the\nhandlers come in; the meaning they assign to effectful progra ms should respect the laws.\nWe have the following hNDhandler which interprets the free monad in terms of lists.\nhND::Free Nondet Fa→[a]\nhND=fold gen NDalgND\nwhere\ngenNDx=[x]\nalgNDFail=[]\nalgND(Or p q)=p++q\nWith this handler, the identity andassociativity laws are satisfied up to handling as follows:\nhND(∅/talloblongm)=hNDm=hND(m/talloblong∅)\nhND((m/talloblongn)/talloblongo)=hND(m/talloblong(n/talloblongo))\nIn fact, two stronger contextual equalities hold:\nhND((∅/talloblongm)> >=k)=hND(m> >=k)=hND((m/talloblong∅)> >=k)\nhND(((m/talloblongn)/talloblongo)> >=k)=hND((m/talloblong(n/talloblongo))> >=k)\nThese equations state that the intepretations of the left- a nd right-hand sides are indistin-\nguishable even when put in a larger program context > >=k. They follow from the definitions\nofhNDand(> >=), as well as the associativity and identity proprerties of (++).\nWe obtain the two non-contextual equations as a corollary by choosing k=/u1D702.\nState Again, instead of using the concrete State monad in Section 2.2 , we model states via\nthe free monad Free(State Fs)over the state signature.\ndataState Fs a=Get(s→a)|Put s a\nThis state signature gives the following MState s instance:\ninstance MState s(Free(State Fs))where\nget=Op(Get/u1D702)\nput s=Op(Put s(/u1D702()))\nThe following handler h′\nStatemaps this free monad to the State s monad.\nh′\nState::Free(State Fs)a→State s a\nh′\nState=fold gen′\nSalg′\nS\nwhere\ngen′\nSx =State $/u1D706s→(x,s)10\nalg′\nS(Get k)=State $/u1D706s→runState(k s)s\nalg′\nS(Put s′k)=State $/u1D706s→runStatek s′\nIt is easy to verify that the four state laws hold contextuall y up to interpretation with h′\nState.\n3.2 Modularly Combining Effects\nCombining multiple effects is relatively easy in the axiomat ic approach based on type\nclasses. By imposing multiple constraints on the monad m, e.g.,(MState s m ,MNondet m),\nwe can express that mshould support both state and nondeterminism and respect th eir\nassociated laws. In practice, this is often insufficient: we u sually require additional laws\nthat govern the interactions between the combined effects. W e discuss possible interaction\nlaws between state and nondeterminism in details in Section 4.\nThe Coproduct Operator for Combining Effects To combine the syntax of effects given\nby free monads, we need to define a coproduct operator : +: for signatures.\ndata(f:+:g)a=Inl(f a)|Inr(g a)\nNote that given two functors fandg, it is obvious that f:+:gis again a functor. This\ncoproduct operator allows a modular definition of the signat ures of combined effects. For\ninstance, we can encode programs with both state and nondete rminism as effects using\nthe data type Free(State F:+:Nondet F)a. The coproduct also has a neutral element NilF,\nrepresenting the empty effect set.\ndataNilFa-- no constructors\nWe define the following two instances, which allow us to compo se state effects with any\nother effect functor f, and nondeterminism effects with any other effect functors fandg,\nrespectively. As a result, it is easy to see that Free(State Fs:+:Nondet F:+:f)supports both\nstate and nondeterminism for any functor f.\ninstance(Functor f)⇒MState s(Free(State Fs:+:f))where\nget=Op$Inl$Get/u1D702\nput x=Op$Inl$Put x(/u1D702())\ninstance(Functor f ,Functor g)⇒MNondet(Free(g:+:Nondet F:+:f))where\n∅=Op$Inr$Inl Fail\nx/talloblongy=Op$Inr$Inl(Or x y)\nModularly Combining Effect Handlers In order to interpret composite signatures, we use\nthe forwarding approach of Schrijvers et al. ( 2019). This way the handlers can be modularly\ncomposed: they only need to know about the part of the syntax t heir effect is handling, and\nforward the rest of the syntax to other handlers.\nA mediator(#)is used to separate the algebra algfor the handled effects and the\nforwarding algebra fwdfor the unhandled effects.11\n(#)::(f a→b)→( g a→b)→( f:+:g)a→b\n(alg#fwd)(Inl op)=alg op\n(alg#fwd)(Inr op)=fwd op\nThe handlers for state and nondeterminism we have given earl ier require a bit of adjustment\nto be used in the composite setting since they only consider t he signature of their own\neffects. We need to interpret the free monads into composite d omains, StateT(Free f)aand\nFree f[a], respectively. Here StateT is the state transformer from the Monad Transformer\nLibrary (Jones, 1995).\nnewtype StateT s m a =StateT{runStateT ::s→m(a,s)}\nThe new handlers, into these composite domains, are defined a s follows:\nhState::Functor f⇒Free(State Fs:+:f)a→StateT s(Free f)a\nhState=fold gen S(algS#fwdS)\nwhere\ngenSx =StateT $/u1D706s→/u1D702(x,s)\nalgS(Get k)=StateT $/u1D706s→runStateT(k s)s\nalgS(Put s′k)=StateT $/u1D706s→runStateT k s′\nfwdSop =StateT $/u1D706s→Op$fmap(/u1D706k→runStateT k s)op\nhND+f::Functor f⇒Free(Nondet F:+:f)a→Free f[a]\nhND+f=fold gen ND+f(algND+f#fwdND+f)\nwhere\ngenND+f =Var◦/u1D702\nalgND+fFail=Var[]\nalgND+f(Or p q)=liftM2(++)p q\nfwdND+fop=Op op\nAlso, the empty signature NilFhas a trivial associated handler.\nhNil::Free Nil Fa→a\nhNil(Var x)=x\n4 Modelling Local State with Global State\nThis section studies two flavors of effect interaction betwee n state and nondeterminism:\nlocal-state and global-state semantics. Local state is a hi gher-level effect than globale state.\nIn a program with local state, each nondeterministic branch has its own local copy of the\nstate. This is a convenient programming abstraction provid ed by many systems that solve\nsearch problems, e.g., Prolog. In contrast, global state li nearly threads a single state through\nthe nondeterministic branches. This can be interesting for performance reasons: we can\nlimit memory use by avoiding multiple copies, and perform in -place updates to reduce\nallocation and garbage collection, and to improve locality .12\nIn this section, we first formally characterise local-state and global-state semantics,\nand then define a translation from the former to the latter whi ch uses the mechanism of\nnondeterminism to store previous states and insert backtra cking branches.\n4.1 Local-State Semantics\nWhen a branch of a nondeterministic computation runs into a d ead end and the continuation\nis picked up at the most recent branching point, any alterati ons made to the state by the\nterminated branch are invisible to the continuation. We ref er to this semantics as local-state\nsemantics . Gibbons and Hinze ( 2011) also call it backtrackable state.\nThe Local-State Laws The following two laws characterise the local-state semant ics for a\nmonad with state and nondeterminism:\nput-right-identity : put s> >∅=∅, (4.1)\nput-left-distributivity :put s> >(m1/talloblongm2)=(put s> >m1)/talloblong(put s> >m2). (4.2)\nThe equation ( 4.1) expresses that ∅is the right identity of put; it annihilates state updates.\nThe other law expresses that putdistributes from the left in (/talloblong).\nThese two laws only focus on the interaction between putand nondeterminism. The\nfollowing laws for getcan be derived from other laws. The proof can be foundin Appendix 1 .\nget-right-identity : get> >∅=∅, (4.3)\nget-left-distributivity :get> >=(/u1D706x→k1x/talloblongk2x)=(get> >=k1)/talloblong(get> >=k2). (4.4)\nIf we take these four equations together with the left-ident ity and right-distributivity laws\nof nondeterminism, we can say that that nondeterminism and s tate “commute”; if a getor\nputprecedes a ∅or/talloblong, we can exchange their order (and vice versa).\nImplementation Implementation-wise, the laws imply that each nondetermin istic branch\nhas its own copy of the state. For instance, Equation (4.2) gives us\nput42(put21/talloblongget)=(put42> >put21)/talloblong(put42> >get)\nThe state we getin the second branch is still 42, despite the put21 in the first branch.\nOne implementation satisfying the laws is\ntypeLocal s m a =s→m(a,s)\nwhere mis a nondeterministic monad, the simplest structure of whic h is a list. This imple-\nmentation is exactly that of StateT s m a in the Monad Transformer Library (Jones, 1995)\nwhich we have introduced in Section 3.2 .\nWith effect handling (Kiselyov and Ishii, 2015; Wu et al., 2014), we get the local state\nsemantics when we run the state handler before the nondeterm inism handler:\nhLocal::Functor f⇒Free(State Fs:+:Nondet F:+:f)a→(s→Free f[a])\nhLocal=fmap(fmap(fmap fst)◦hND+f)◦runStateT◦hState\nIn the case where the remaining signature is empty ( f=NilF), we get:13\nfmap h Nil◦hLocal::Free(State Fs:+:Nondet F:+:NilF)a→(s→[a])\nHere, the result type (s→[a])differs from s→[( a,s)]in that it produces only a list of\nresults ([a]) and not pairs of results and their final state ( [(a,s)]). The latter is needed for\nLocal s m to have the structure of a monad, in particular to support the modular composition\nof computations with (> >=). Such is not needed for the carriers of handlers, because the\ncomposition of computations is taken care of by the (> >=)operator of the free monad.\n4.2 Global-State Semantics\nAlternatively, one can choose a semantics where state reign s over nondeterminism. In this\ncase of non-backtrackable state, alterations to the state p ersist over backtracks. Because\nonly a single state is shared over all branches of nondetermi nistic computation, we call this\nstate the global-state semantics .\nThe Global-State Law The global-state semantics sets apart non-backtrackable s tate from\nbacktrackable state. In addition to the general laws for non determinism (( 2.6) – (2.9)) and\nstate (( 2.10) – (2.13)), we provide a global-state law to govern the interaction between\nnondeterminism and state.\nput-or :(put s> >m)/talloblongn=put s> >(m/talloblongn). (4.5)\nThis law allows lifting a putoperation from the left branch of a nondeterministic choice .\nFor instance, if m=∅in the left-hand side of the equation, then under local-stat e semantics\n(laws ( 2.6) and ( 4.1)) the left-hand side becomes equal to n, whereas under global-state\nsemantics (laws ( 2.6) and ( 4.5)) the equation simplifies to put s> >n.\nImplementation Figuringout a correct implementationfor the global-state monad is tricky.\nOne might believe that Global s m a =s→(m a,s)is a natural implementation of such a\nmonad. However, the usual naive implementation of (> >=)for it does not satisfy right-\ndistributivity ( 2.8) and is therefore not even a monad. The type ListT(State s)from the\nMonad Transformer Library (Jones, 1995) expands to essentially the same implementation\nwith monad minstantiated by the list monad. This implementation has the same flaws.\nMore careful implementations of ListT (often referred to as “ ListT done right”) satisfying\nright-distributivity ( 2.8) and other monad laws have been proposed by Volkov ( 2014); Gale\n(2007). The following implementation is essentially that of Gale .\nnewtype Global s a =Gl{runGl ::s→(Maybe(a,Global s a),s)}\nTheMaybe in this type indicates that a computation may fail to produce a result. However,\nsince the sis outside of the Maybe , a modified state is returned even if the computation\nfails. This Global s a type is an instance of the MState andMNondet typeclasses.\ninstance MNondet(Global s)where\n∅=Gl(/u1D706s→(Nothing,s))\np/talloblongq=Gl(/u1D706s→caserunGl p s of\n(Nothing,t)→runGl q t\n(Just(x,r),t)→( Just(x,r/talloblongq),t))instance MState s(Global s)where\nget=Gl(/u1D706s→(Just(s,∅),s))\nput s=Gl(/u1D706→(Just((),∅),s))14\nFailure, of course, returns an empty continuation and an unm odified state. Branching\nfirst exhausts the left branch before switching to the right b ranch.\nEffect handlers (Kiselyov and Ishii, 2015; Wu et al., 2014) also provide implementations\nthat match our intuition of non-backtrackable computation s. The global-state semantics can\nbe implemented by simply switching the order of the two effect handlers compared to the\nlocal state handler hLocal.\nhGlobal ::(Functor f)⇒Free(State Fs:+:Nondet F:+:f)a→(s→Free f[a])\nhGlobal=fmap(fmap fst)◦runStateT◦hState◦hND+f◦(⇔)\nThis also runs a single state througha nondeterministiccom putation.The(⇔)isomorphism\nswaps the order of two functors in the co-product signature o f the free monad in order to\nlethLocalandhGlobal have the same type signature.\n(⇔)::(Functor f 1,Functor f 2,Functor f)⇒Free(f1:+:f2:+:f)a→Free(f2:+:f1:+:f)a\n(⇔)( Var x) =Var x\n(⇔)( Op(Inl k))=(Op◦Inr◦Inl)(fmap(⇔)k)\n(⇔)( Op(Inr(Inl k)))=(Op◦Inl) ( fmap(⇔)k)\n(⇔)( Op(Inr(Inr k)))=(Op◦Inr◦Inr)(fmap(⇔)k)\nIn the case where the remaining signature is empty ( f=NilF), we get:\nfmap h Nil◦hGlobal ::Free(State Fs:+:Nondet F:+:NilF)a→(s→[a])\nThe carrier type is again simpler than Global s a because it does not have to support the\n(> >=)operator.\n4.3 Simulating Local State with Global State\nBoth local state and global state have their own laws and sema ntics. Also, both\ninterpretations of nondeterminism with state have their ow n advantages and disadvantages.\nLocal-state semantics imply that each nondeterministic br anch has its own state. This\nmay be expensive if the state is represented by data structur es, e.g. arrays, that are costly\nto duplicate. For example, when each new state is only slight ly different from the previous,\nwe have a wasteful duplication of information.\nGlobal-state semantics, however, threads a single state th rough the entire computation\nwithout making any implicit copies. Consequently, it is eas ier to control resource usage and\napply optimisation strategies in this setting. However, do ing this to a program that has a\nbacktracking structure, and would be more naturally expres sed in a local-state style, comes\nat a great loss of clarity. Furthermore, it is significantly m ore challenging for programmers\nto reason about global-state semantics than local-state se mantics.\nTo resolve this dilemma, we can write our programsin a local- state style and then translate\nthem to the global-state style to enable low-level optimisa tions. In this subsection, we show\none systematic program translation that alters a program wr itten for local-state semantics\nto a program that, when interpreted under global-state sema ntics, behaves exactly the same\nas the original program interpreted under local-state sema ntics. This translation makes\nexplicit copying of the whole state and relies on using the no ndeterminism mechanism to\ninsert state-restoring branches. We will show other transl ations from local-state semantics15\nto global-state semantics which avoid the copying and do not rely on nondeterminism in\nSection 7 andSection 8 .\nState-Restoring Put Central to the implementation of backtracking in the global state\nsetting is the backtracking variant of put. Going forward, such a state-restoring putRmodifies\nthe state as usual, but, when backtracked over, it restores t he old state.\nWe implement putRusing both state and nondeterminism as follows:\nputR::(MState s m ,MNondet m)⇒s→m()\nputRs=get> >=/u1D706s′→put s/talloblongside(put s′)\nHere the sidebranch is executed for its side-effect only; it fails before y ielding a result.\nside::MNondet m⇒m a→m b\nside m=m> >∅\nIntuitively, the second branch generated by putRcan be understood as a backtracking or\nstate-restoring branch. The putRsoperation changes the state to sin the first branch put s,\nand then restores the it to the original state s′in the second branch after we finish all\ncomputations in the first branch. Then, the second branch imm ediately fails so that we can\nkeep going to other branches with the original state s′. For example, assuming an arbitrary\ncomputation comp is placed after a state-restoring put, we have the following calculation.\nputRs> >comp\n={- definition of putR-}\n(get> >=/u1D706s′→put s/talloblongside(put s′))> >comp\n={- right-distributivity ( 2.8) -}\n(get> >=/u1D706s′→(put s> >comp)/talloblong(side(put s′)> >comp))\n={- left identity ( 2.9) -}\n(get> >=/u1D706s′→(put s> >comp)/talloblongside(put s′))\nThis program saves the current state s′, computes comp using state s, and then restores the\nsaved state s′. Figure 1shows how the state-passing works and the flow of execution fo r a\ncomputation after a state-restoring put.\nget→/u1D706s′\n(/talloblong)\nput s put s′\ncomp ∅s′\ns s′s\nFig. 1: State-restoring put-operation.16\nAnother example of putRis shown in Table 2, where three programs are run with\ninitial state s0. Note the difference between the final state and the program re sult for the\nstate-restoring put.\nProgram result Final state\n/u1D702x> >get s0 s0\nput s> >/u1D702x> >get s s\nputRs> >/u1D702x> >get s s 0\nTable 2: Comparing putandputR.\nTranslation with State-Restoring Put The idea is that putR, when run with a global state,\nsatisfies laws ( 2.10) to (4.2) — the state laws and the local-state laws. Then, one can take\na program written for local-state semantics, replace all oc currences of putbyputR, and run\nthe program with a global state. However, to satisfy all of th ese laws, care should be taken\nto replace alloccurrences of put. Particularly, placing a program in a larger context, where\nputhas not been replaced, can change the meaning of its subprogr ams. An example of\nsuch a problematic context is (> >put t), where the get-putlaw (2.12) breaks and programs\nget> >putRand/u1D702()can be differentiated:\n(get> >putR)> >put t\n={- definition of putR-}\n(get> >=/u1D706s→get> >=/u1D706s0→put s/talloblongside(put s 0))> >put t\n={- get-get ( 2.13) -}\n(get> >=/u1D706s→put s/talloblongside(put s))> >put t\n={- right-distributivity ( 2.8) -}\n(get> >=/u1D706s→(put s> >put t)/talloblong(side(put s))> >put t)\n={- left-identity ( 2.9) -}\n(get> >=/u1D706s→(put s> >put t)/talloblongside(put s))\n={- put-put ( 2.10) -}\n(get> >=/u1D706s→put t/talloblongside(put s))/u1D702()> >put t\n=put t\nThose two programs do not behave in the same way when s/nequivalencet. Hence, only provided\nthatalloccurences of putin a program are replaced by putRcan we simulate local-state\nsemantics. The replacement itself as well as the correctnes s statement that incorporates\nthis requirement can be easily epressed with effect handlers . As we will explain below we\nneed to articulate this global replacement in the correctne ss proof. This requires using the\nfusion-post’ rule rather than the more widely used fusion-post rule.\nWe realize the global replacement of putwith a putRwith the effect handler local2global :\nlocal2global ::Functor f\n⇒Free(State Fs:+:Nondet F:+:f)a\n→Free(State Fs:+:Nondet F:+:f)a\nlocal2global =fold Var alg17\nwhere\nalg(Inl(Put t k))=putRt> >k\nalg p=Op p\nThe following theorem shows that the translation local2global preserves the meaning\nwhen switching from local-state to global-state semantics :\nTheorem 1. hGlobal◦local2global =hLocal\nProof Both the left-hand side and the right-hand side of the equati on consist of function\ncompositions involving one or more folds. We apply fold fusi on separately on both sides to\ncontract each into a single fold:\nhGlobal◦local2global =fold genLHS(algS\nLHS#algND\nRHS#fwdLHS)\nhLocal=fold genRHS(algS\nRHS#algND\nRHS#fwdRHS)\nWe approach this calculationally. That is to say, we do not fir st postulate definitions of\nthe unknowns above ( algS\nLHSand so on) and then verify whether the fusion conditions are\nsatisfied. Instead, we discover the definitions of the unknow ns. We start from the known\nside of each fusion condition and perform case analysis on th e possible shapes of input.\nBy simplifying the resulting case-specific expression, and pushing the handler applications\ninwards, we end up at a point where we can read off the definition of the unknown that\nmakes the fusion condition hold for that case.\nFinally, we show that both folds are equal by showing that the ir correspondingparameters\nare equal:\ngenLHS=genRHS\nalgS\nLHS=algS\nRHS\nalgND\nLHS=algND\nRHS\nfwdLHS=fwdRHS\nA noteworthy observation is that, for fusing the left-hand s ide of the equation, we do not\nuse the standard fusion rule fusion-post (3.2):\nhGlobal◦fold Var alg =fold(hGlobal◦Var)alg′\n⇐ hGlobal◦alg=alg′◦fmap h Global\nwhere local2global =fold Var alg . The problem is that we will not find an appropriate alg′\nsuch that alg′(fmap h Global t)restores the state for any tof type(State Fs:+:Nondet F:+:\nf)(Free(State Fs:+:Nondet F:+:f)a).\nFortunately, we do not need such an alg′. As we have already pointed out, we can\nassume that the subterms of thave already been transformed by local2global , and thus all\noccurrences of Putappear in the putRconstellation.\nWe can incorporate this assumption by using the alternative fusion rule fusion-post’ (3.3):\nhGlobal◦fold Var alg =fold(hGlobal◦Var)alg′\n⇐ hGlobal◦alg◦fmap local2global =alg′◦fmap h Global◦fmap local2global18\nThe additional fmap local2global in the condition captures the property that all the subterms\nhave been transformed by local2global .\nIn order to not clutter the proofs, we abstract everywhere ov er this additional\nfmap local2global application, except for the key lemma which expresses that t he syn-\ntactic transformation local2global makes sure that, despite any temporary changes, the\ncomputation trestores the state back to its initial value.\nWe elaborate each of these steps in Appendix 2 . /squaresolid\nN-queens with Global-State Semantics Recall the backtracking algorithm queens for the\nn-queens example in Section 2.3 . It is initially designed to run in the local-state semantic s\nbecause every branch maintains its own copy of the state and h as no influence on other\nbranches. We can handle it with hLocalas follows.\nqueens Local::Int→[[ Int]]\nqueens Local=hNil◦flip h Local(0,[])◦ queens\nWith the simulation local2global , we can also translate queens to an equivalent program in\nglobal-state semantics and handle it with hGlobal.\nqueens Global ::Int→[[ Int]]\nqueens Global=hNil◦flip h Global(0,[])◦ local2global◦queens\n5 Modelling Nondeterminism with State\nIn the previous section, we have translated the local-state semantics, a high-level combi-\nnation of the state and nondeterminism effects, to the global -state semantics, a low-level\ncombination of the state and nondeterminism effects. In this section, we further translate\nthe resulting nondeterminism component, which is itself a r elatively high-level effect, to\na lower-level implementation with the state effect. Our tran slation coincides with the fact\nthat, while nondeterminism is typically modelled using the Listmonad, many efficient\nnondeterministic systems, such as Prolog, use a low-level s tate-based implementation to\nimplement the nondeterminism mechanism.\n5.1 Simulating Nondeterminism with State\nThe main idea of simulating nondeterminism with state is to e xplictly manage\n1. a list of the results found so far, and\n2. a list of yet to be explored branches, which we call a stack.\nThis stack corresponds to the choicepoint stack in Prolog. W hen entering one branch,\nwe can push other branches to the stack. When leaving the bran ch, we collect its result and\npop a new branch from the stack to continue.\nWe define a new type S aconsisting of the results andstack.19\ntypeComp s a =Free(State Fs)a\ndataS a=S{results ::[a],stack ::[Comp(S a)()]}\nThe branches in the stack are represented by computations in the form of free monads over\ntheState Fsignature. We do not allow branches to use other effects here t o show the idea\nmore clearly. In Section 5.2 we will consider the more general case where branches can\nhave any effects abstracted by a functor f.\npopS::Comp(S a)()\npopS=do\nS xs stack←get\ncasestack of\n[] →/u1D702()\np:ps→do\nput(S xs ps);p\n(a) Popping from the stack.push S::Comp(S a)()\n→Comp(S a)()\n→Comp(S a)()\npush Sq p=do\nS xs stack←get\nput(S xs(q:stack))\np\n(b) Pushing to the stack.append S::a\n→Comp(S a)()\n→Comp(S a)()\nappend Sx p=do\nS xs stack←get\nput(S(xs++[x])stack)\np\n(c) Appending a result.\nFig. 2: Auxiliary functions popS,push Sandappend S.\nFor brevity, instead of defining a new stack effect capturing t he stack operations like pop\nand push, we implement stack operations with the state effect . We define three auxiliary\nfunctions in Figure 2 to interact with the stack in S a:\n•The function popSremoves and executes the top element of the stack.\n•The function push Spushes a branch into the stack.\n•The function append Sadds a result to the current results.\nNow, everything is in place to define a simulation function nondet2state Sthat interprets\nnondeterministic programs represented by the free monad Free Nondet Faas state-wrapped\nprograms represented by the free monad Free(State F(S a))().\nnondet2state S::Free Nondet Fa→Free(State F(S a))()\nnondet2state S=fold gen alg\nwhere\ngen x =append Sx pop S\nalg Fail =popS\nalg(Or p q)=push Sq p\nThe generator of this handler records a new result and then po ps the next branch from the\nstack and proceeds with it. Likewise, for failure the handle r simply pops and proceeds with\nthe next branch. For nondeterministic choices, the handler pushes the second branch on the\nstack and proceeds with the first branch.\nTo extract the final result from the Swrapper, we define the extract Sfunction.20\nextract S::State(S a)()→[ a]\nextract Sx=results◦snd$runStatex(S[][])\nFinally, we define the function runNDwhich wraps everything up to handle a non-\ndeterministic computation to a list of results. The state ha ndler h′\nStateis defined in\nSection 3.1 .\nrunND::Free Nondet Fa→[a]\nrunND=extract S◦h′\nState◦nondet2state S\nWe have the following theorem showing the correctness of the simulation via the equivalence\nof the runNDfunction and the nondeterminism handler hNDdefined in Section 3.1 .\nTheorem 2. runND=hND\nThe proof can be found in Appendix 3.1 . The main idea is again to use fold fusion.\nConsider the expanded form\n(extract S◦h′\nState)◦nondet2state S=hND\nBoth nondet2state SandhNDare written as folds. We use the law fusion-post’ (3.3) to fuse\nthe left-hand side into a single fold. Since the right-hand s ide is already a fold, to prove the\nequation we just need to check the components of the fold hNDsatisfy the conditions of\nthe fold fusion, i.e., the following two equations: For the l atter we only need to prove the\nfollowing two equations:\n(extract S◦h′\nState)◦gen=genND\n(extract S◦h′\nState)◦alg◦fmap nondet2state S\n=algND◦fmap(extract S◦h′\nState)◦fmap nondet2state S\nwhere genandalgare from the definition of nondet2state S, and genNDandalgNDare from\nthe definition of hND.\n5.2 Combining the Simulation with Other Effects\nThenondet2state Sfunction only considers nondeterminism as the only effect. I n this section,\nwe generalise it to work in combination with other effects. On e immediate benefit is that\nwe can use it in together with our previous simulation local2global inSection 4.3 .\nFirstly, we need to augment the signature in the computation type for branches with an\nadditional functor ffor other effects. The computation type is essentially chang ed from\nFree(State Fs)atoFree(State Fs:+:f)a. We define the state type SS f a as follows:\ntypeComp SSs f a=Free(State Fs:+:f)a\ndataSS f a=SS{results SS::[a],stack SS::[Comp SS(SS f a)f()]}\nWe also modify the three auxiliary functions in Figure 2 topopSS,push SSandappend SS\ninFigure 3 . They are almost the same as the previous versions apart from being adapted to\nuse the new state-wrapper type SS f a .21\npopSS::Functor f\n⇒Comp SS(SS f a)f()\npopSS=do\nSS xs stack←get\ncasestack of\n[] →/u1D702()\np:ps→do\nput(SS xs ps);p\n(a) Popping from the stack.push SS::Functor f\n⇒Comp SS(SS f a)f()\n→Comp SS(SS f a)f()\n→Comp SS(SS f a)f()\npush SSq p=do\nSS xs stack←get\nput(SS xs(q:stack))\np\n(b) Pushing to the stack.append SS::Functor f\n⇒a\n→Comp SS(SS f a)f()\n→Comp SS(SS f a)f()\nappend SSx p=do\nSS xs stack←get\nput(SS(xs++[x])stack)\np\n(c) Appending a result.\nFig. 3: Auxiliary functions popSS,push SSandappend SS.\nThe simulation function nondet2state is also very similar to nondet2state Sexcept for\nrequiring a forwarding algebra fwdto deal with the additional effects in f.\nnondet2state ::Functor f⇒Free(Nondet F:+:f)a→Free(State F(SS f a):+:f)()\nnondet2state =fold gen(alg#fwd)\nwhere\ngen x =append SSx pop SS\nalg Fail =popSS\nalg(Or p q)=push SSq p\nfwd y =Op(Inr y)\nThe function runND+fputs everything together: it translates the nondeterminis m effect into\nthe state effect and forwards other effects using nondet2state , then handles the state effect\nusing hState, and finally extracts the results from the final state using extract SS.\nrunND+f::Functor f⇒Free(Nondet F:+:f)a→Free f[a]\nrunND+f=extract SS◦hState◦nondet2state\nextract SS::Functor f⇒StateT(SS f a)(Free f)()→ Free f[a]\nextract SSx=results SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT x(SS[][])\nWe have the following theorem showing that the simulation runND+fis equivalent to the\nmodular nondeterminism handler hND+finSection 3.2 .\nTheorem 3. runND+f=hND+f\nThe proof proceeds essentially in the same way as in the non-m odular setting. The main\ndifference, due to the modularity, is an additional proof cas e for the forwarding algebra.\n(extract SS◦hState)◦fwd◦fmap nondet2state S\n=fwdND+f◦fmap(extract SS◦hState)◦fmap nondet2state S\nThe full proof can be found in Appendix 3.2 .22\n6 All in One\nThis section combines the results of the previous two sectio ns to ultimately simulate the\ncombination of nondeterminism and state with a single state effect.\n6.1 Modelling Two States with One State\nWhen we combine the two simulation steps from the two previou s sections, we end up\nwith a computation that features two state effects. The first s tate effect is the one present\noriginally, and the second state effect keeps track of the res ults and the stack of remaining\nbranches to simulate the nondeterminism.\nFor a computation of type Free(State Fs1:+:State Fs2:+:f)athat features two state\neffects, we can go to a slightly more primitive representatio nFree(State F(s1,s2):+:f)a\nfeaturing only a single state effect that is a pair of the previ ous two states.\nThe handler states2state implements the simulation by projecting different getandput\noperations to different components of the pair of states.\nstates2state ::Functor f\n⇒Free(State Fs1:+:State Fs2:+:f)a\n→Free(State F(s1,s2):+:f)a\nstates2state =fold Var(alg1#alg2#fwd)\nwhere\nalg1(Get k)=get> >=/u1D706(s1,)→k s1\nalg1(Put s′\n1k)=get> >=/u1D706(,s2)→put(s′\n1,s2)> >k\nalg2(Get k)=get> >=/u1D706(,s2)→k s2\nalg2(Put s′\n2k)=get> >=/u1D706(s1,)→put(s1,s′\n2)> >k\nfwd op =Op(Inr op)\nWe have the following theorem showing the correctness of states2state :\nTheorem 4. hStates=nest◦hState◦states2state\nOn the left-hand side, we write hStatesfor the composition of two consecutive state handlers:\nhStates::Functor f⇒Free(State Fs1:+:State Fs2:+:f)a→StateT s 1(StateT s 2(Free f))a\nhStatesx=StateT(hState◦runStateT(hStatex))\nh′\nStates::Functor f⇒Free(State Fs1:+:State Fs2:+:f)a→StateT(s1,s2)(Free f)a\nh′\nStatest=StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2\nOn the right-hand side, we use the isomorphism nestto mediate between the two different\ncarrier types. The definition of nestand its inverse flatten are defined as follows:\nnest ::Functor f⇒StateT(s1,s2)(Free f)a→StateT s 1(StateT s 2(Free f))a\nnest t=StateT $/u1D706s1→StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}htrunStateT t(s1,s2)\nflatten ::Functor f⇒StateT s 1(StateT s 2(Free f))a→StateT(s1,s2)(Free f)a\nflatten t=StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(runStateT t s1)s2\nwhere the isomorphism /u1D6FC−1and its inverse /u1D6FCrearrange a nested tuple23\n/u1D6FC::((a,x),y)→( a,(x,y))\n/u1D6FC((a,x),y)=(a,(x,y))/u1D6FC−1::(a,(x,y))→(( a,x),y)\n/u1D6FC−1(a,(x,y))=((a,x),y)\nThe proof of Theorem 4 can be found in Appendix 4 . Instead of proving it directly, we show\nthe correctness of the isomorphism of nestandflatten , and prove the following equation:\nflatten◦hStates=hState◦states2state\nThe following commuting diagram simmuarises the simulatio n.\nFree(State Fs1:+:State Fs2:+:f)a StateT s 1(StateT s 2(Free f))a\nFree(State F(s1,s2):+:f)a StateT (s1,s2)(Free f)aflatten nesthStates\nhStatestates2state\n6.2 Putting Everything Together\nWe have defined three translations for encoding high-level e ffects as low-level effects.\n•The function local2global simulates the high-level local-state semantics with globa l-\nstate semantics for the nondeterminism and state effects ( Section 4 ).\n•The function nondet2state simulates the high-level nondeterminism effect with the\nstate effect ( Section 5 ).\n•The function states2state simulates multiple state effects with a single state effect\n(Section 6.1 ).\nCombining them, we can encode the local-state semantics for nondeterminism and state\nwith just one state effect. The ultimate simulation function simulate is defined as follows:\nsimulate ::Functor f\n⇒Free(State Fs:+:Nondet F:+:f)a\n→s→Free f[a]\nsimulate=extract◦hState◦states2state◦nondet2state◦(⇔)◦ local2global\nSimilar to the extract SSfunction in Section 5.2 , we use the extract function to get the final\nresults from the final state.\nextract ::Functor f\n⇒StateT(SS(State Fs:+:f)a,s)(Free f)()\n→s→Free f[a]\nextract x s =results SS◦fst◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT x(SS[][],s)\nFigure 4 illustrates each step of this simulation.\nIn the simulate function, we first use our three simulations local2global ,nondet2state\nandstates2state to interpret the local-state semantics for state and nondet erminism in terms24\nFree(State Fs:+:Nondet F:+:f)a\nFree(State Fs:+:Nondet F:+:f)a\nFree(Nondet F:+:State Fs:+:f)a\nFree(State F(SS(State Fs:+:f)a):+:State Fs:+:f)()\nFree(State F(SS(State Fs:+:f)a,s):+:f)()\nStateT(SS(State Fs:+:f)a,s)(Free f)()\ns→Free f[a]local2global\n(⇔)\nstates2state\nhState\nextractnondet2state\nFig. 4: An overview of the simulate function.\nof only one state effect. Then, we use the handler hStateto interpret the state effect into a\nstate monad transformer. Finally, we use the function extract to get the final results.\nWe have the following theorem showing that the simulate function exactly behaves the\nsame as the local-state semantics given by hLocal.\nTheorem 5. simulate=hLocal\nThe proof can be found in Appendix 5 .\nWe provide a more compact and direct definition of simulate by fusing all the consecutive\nsteps into a single handler:\ntypeComp s f a =(CP f a s,s)→Free f[a]\ndataCP f a s=CP{results ::[a],cpStack ::[Comp s f a]}\nsimulate F::Functor f\n⇒Free(State Fs:+:Nondet F:+:f)a\n→s\n→Free f[a]\nsimulate Fx s=fold gen(alg1#alg2#fwd)x(CP[][],s)\nwhere\ngen x(CP xs stack ,s)=continue(xs++[x])stack s\nalg1(Get k) (CP xs stack ,s)=k s(CP xs stack ,s)\nalg1(Put t k) (CP xs stack ,s)=k(CP xs(backtracking s :stack),t)\nalg2Fail(CP xs stack ,s)=continue xs stack s\nalg2(Or p q) (CP xs stack ,s)=p(CP xs(q:stack),s)25\nfwd op(CP xs stack ,s)=Op(fmap($(CP xs stack ,s))op)\nbacktracking s(CP xs stack ,)=continue xs stack s\ncontinue xs stack s =casestack of\n[] → /u1D702xs\n(p:ps)→p(CP xs ps,s)\nThe common carrier of the above algebras alg1#alg2#fwdisComp s f a . This is a com-\nputation that takes the current results, choicepoint stack and application state, and returns\nthe list of all results. The first two inputs are bundled in the CPtype.\nN-queens with Only One State With simulate , we can implement the backtracking\nalgorithm of the n-queens problem in Section 2.3 with only one state effect as follows.\nqueens Sim::Int→[[ Int]]\nqueens Sim=hNil◦flip simulate(0,[])◦ queens\n7 Modelling Local State with Undo\nInSection 4.3 , we give a translation local2global which simulates local state with global\nstate by replacing putwith its state-restoring version putR. The putRoperation makes the\nimplicit copying of the local-state semantics explicit in t he global-state semantics. However,\nthis copying still exists and can be rather costly if the stat e is big (e.g., a long array), and\nespecially wasteful if the modifications made to that state a re small (e.g., a single entry\nin the array). Fortunately, low-level features like the glo bal-state semantics give us more\npossibility to apply more fine-grainedoptimisation strate gies. As a result, instead of copying\nthe whole state to implement the backtracking behaviour in t he global-state semantics, we\ncan just keep track of the modifications made to the state, and undo them when necessary.\nIn this section, we formalise this intuition with a translat ion from the local-state semantics\nto the global-state semantics for incremental and reversib le state updates.\n7.1 State Update and Restoration\nWe first need to characterise a specific subset of state effects where all state update operations\ncan be undone. We call them modification-based state effects. For example, the queens\nprogram in Section 2.3 uses the operation s⊕rto update the state. We can undo it using\nthe following⊖roperation which is essentially the left inverse of ⊕r.\n(⊖)::(Int,[Int])→ Int→(Int,[Int])\n(⊖)(c,sol)r=(c−1,tail sol)\nThese two operators satisfy the equation (⊖r)◦(⊕ r)=idfor any r::Int.\nIn general, we define a typeclass Undo s r with two operations (⊕)and(⊖)to characterise\nrestorable state updates. Here, sis the type of states and ris the type of deltas. We can\nimplement the previous state update and restoration operat ions of n-queens as an instance\nUndo(Int,[Int])Intof the typeclass.26\nclass Undo s r where\n(⊕)::s→r→s\n(⊖)::s→r→s\ninstance Undo(Int,[Int])Intwhere\n(⊕)(c,sol)r=(c+1,r:sol)\n(⊖)(c,sol)r=(c−1,tail sol)\nInstances of Undo should satisfy the following law which says ⊖xis a left inverse of⊕x:\nplus-minus :(⊖x)◦(⊕ x)=id. (7.1)\nModification-based state effects restrict the general putoperation of MState to modifica-\ntion oprations. We define a new typeclass MModify s r m which inherits from Monad m and\nUndo s r to capture the interfaces of state updates and restoration. It has three operations:\namget operation that reads and returns the state (similar to the getoperation of MState ),\naupdate r operation that updates the state with the delta r, and a restore r operations that\nrestores the update introduced by the delta r. Note that only the mget andupdate operations\nare expected to be used by programmers; restore operations are automatically generated by\nthe translation to the global-state semantics.\nclass(Monad m ,Undo s r)⇒MModify s r m|m→s,m→rwhere\nmget ::m s\nupdate ::r→m()\nrestore ::r→m()\nThe three operations satisfy the following laws:\nmget-mget : mget> >=(/u1D706s→mget> >=k s)=mget> >=(/u1D706s→k s s), (7.2)\nupdate-mget : mget> >=/u1D706s→update r> >/u1D702(s⊕r)=update r> >mget , (7.3)\nrestore-mget : mget> >=/u1D706s→restore r> >/u1D702(s⊖r)=restore r> >mget , (7.4)\nupdate-restore : update r> >restore r=/u1D702(). (7.5)\nThe first law for mget corresponds to that for get. The second and third law respecively\ncapture the impact of update andrestore onmget. Finally, the fourth law expresses that\nrestore undoes the effect of update .\nAs what we did for the nondeterminism and state effects in Section 3.1 , we define a\nnew signature Modify Frepresenting the syntax of modification-based state effects , and\nimplement the free monad Free(Modify Fs r:+:f)as an instance of MModify s r .\ndataModify Fs r a=MGet(s→a)|MUpdate r a|MRestore r a\ninstance(Functor f ,Undo s r)⇒MModify s r(Free(Modify Fs r:+:f))where\nmget=Op(Inl(MGet/u1D702))\nupdate r=Op(Inl(MUpdate r(/u1D702())))\nrestore r=Op(Inl(MRestore r(/u1D702())))\nThe following handler hModify maps this free monad to the StateT monad transformer using\nthe operations(⊕)and(⊖)provided by Undo s r .27\nhModify ::(Functor f ,Undo s r)⇒Free(Modify Fs r:+:f)a→StateT s(Free f)a\nhModify=fold gen(alg#fwd)\nwhere\ngen x =StateT $/u1D706s→/u1D702(x,s)\nalg(MGet k)=StateT $/u1D706s→runStateT(k s)s\nalg(MUpdate r k)=StateT $/u1D706s→runStateT k(s⊕r)\nalg(MRestore r k)=StateT $/u1D706s→runStateT k(s⊖r)\nfwd y =StateT $/u1D706s→Op(fmap(/u1D706k→runStateT k s)y)\nIt is easy to check that the four laws hold contextually up to i nterpretation with hModify .\nNote that here we still use the StateT monad transformer and immutable states for the\nclarity of presentation and simplicity of proofs. The (⊕)and(⊖)operations also take\nimmutable arguments. To be more efficient, we can use mutable s tates to implement in-\nplace updates or use the technique of functional but in-plac e update (Lorenzen et al., 2023).\nWe leave them as future work.\nSimilar to Section 4.1 andSection 4.2 , the local-state and global-state semantics of\nModify FandNondet Fare given by the following functions hLocalM andhGlobalM , respectively.\nhLocalM ::(Functor f ,Undo s r)\n⇒Free(Modify Fs r:+:Nondet F:+:f)a→(s→Free f[a])\nhLocalM=fmap(fmap(fmap fst)◦hND+f)◦runStateT◦hModify\nhGlobalM ::(Functor f ,Undo s r)\n⇒Free(Modify Fs r:+:Nondet F:+:f)a→(s→Free f[a])\nhGlobalM=fmap(fmap fst)◦runStateT◦hModify◦hND+f◦(⇔)\n7.2 Simulating Local State with Global State and Undo\nWe can implement the translation from local-state semantic s to global-state semantics for\nthe modification-based state effects in a similar style to the translation local2global in\nSection 4.3 . The translation local2global Mstill uses the mechanism of nondeterminism to\nrestore previous state updates for backtracking. In Section 8 we will show a lower-level\nsimulation of local-state semantics without relying on non determinism.\nlocal2global M::(Functor f ,Undo s r)\n⇒Free(Modify Fs r:+:Nondet F:+:f)a\n→Free(Modify Fs r:+:Nondet F:+:f)a\nlocal2global M=fold Var alg\nwhere\nalg(Inl(MUpdate r k))=(update r /talloblongside(restore r))> >k\nalg p=Op p\nCompared to local2global , the main difference is that we do not need to copy and store the\nwhole state. Instead, we store the delta rand undo the state update using restore r in the\nsecond branch. The following theorem shows the correctness oflocal2global M.\nTheorem 6. Given Functor f and Undo s r, the equation28\nhGlobalM◦local2global M=hLocalM\nholds for all programs p ::Free(Modify Fs r:+:Nondet F:+:f)a that do not use the\noperation Op(Inl MRestore).\nThe proof of this theorem can be found in Appendix 6 .\nN-queens with State Update and Restoration We can rewrite the queens program with\nmodification-based state and nondeterminism. Compared to t hequeens inSection 2.3 , we\nonly need to change the getandput(s⊕r)with the update operation update r .\nqueens M::(MModify(Int,[Int])Int m,MNondet m)⇒Int→m[Int]\nqueens Mn=loopwhere\nloop=do(c,sol)←mget\nifc/greaterorequalslantnthen/u1D702sol\nelse do r←choose[1..n]\nguard(safe r 1sol)\nupdate r\nloop\nWe can interpret it using either hLocalM orhGlobalM composed with local2global M.\nqueens LocalM ::Int→[[ Int]]\nqueens LocalM=hNil◦flip h LocalM(0,[])◦ queens M\nqueens GlobalM ::Int→[[ Int]]\nqueens GlobalM=hNil◦flip h GlobalM(0,[])◦ local2global M◦queens M\n8 Modelling Local State with Trail Stack\nIn order to trigger the restoration of the previous state, th e simulations local2global in\nSection 4.3 andlocal2global MinSection 7 introduce a call to orand to failat every\nmodification of the state. The Warren Abstract Machine (WAM) (A¨ıt-Kaci, 1991) does this\nin a more efficient and lower-level way: it uses a trail stack to batch consecutive restorative\nsteps. In this section, we first make use of the idea of trail st acks to implement a lower-level\ntranslation from local-state semantics to global-state se mantics for the modifcation-based\nversion of state effects in Section 7 which does not require extra calls to nondeterminism\noperations. Then, we combine this simulation with other sim ulations to obtain another\nultimate simulation function which uses two stacks, a choic epoint stack and a trail stack,\nsimultaneously.\n8.1 Simulating Local State with Global State and Trail Stack\nLet us consider modification-based version of simulation local2global M. We can use a trail\nstack to contain elements of type Either r(), where ris the type of deltas to the states.\nEach Left x entry represents an update to the state with the delta x, and each Right()is a\nmarker. When we enter a left branch, we push a marker on the tra il stack. For every update29\nwe perform in that branch, we push the corresponding delta on the trail stack, on top of\nthe marker. When we backtrack to the right branch, we unwind t he trail stack down to the\nmarker and restore all deltas along the way. This process is k nown as “untrailing”.\nWe can easily model the Stack data type with Haskell lists.\nnewtype Stack s=Stack[s]\nWe thread the stack through the computation using the state e ffect and define primitive pop\nand push operations as follows.\npopStack ::MState(Stack s)m\n⇒m(Maybe s)\npopStack =do\nStack xs←get\ncasexsof\n[] → /u1D702Nothing\n(x:xs′)→doput(Stack xs′);/u1D702(Just x)pushStack ::MState(Stack s)m\n⇒s→m()\npushStack x =do\nStack xs←get\nput(Stack(x:xs))\nIn order to correctly use state operations to interact with t he trail stack in the translation,\nwe also need to define a new instance of MState .\ninstance(Functor f ,Functor g ,Functor h)\n⇒MState s(Free(f:+:g:+:State Fs:+:h))where\nget=Op◦Inr◦Inr◦Inl$Get/u1D702\nput x=Op◦Inr◦Inr◦Inl$Put x(/u1D702())\nWith these in place, the following translation function local2trail simulates the local-state\nsemantics with global-state semantics by means of the trail stack.\nlocal2trail ::(Functor f ,Undo s r)\n⇒Free(Modify Fs r:+:Nondet F:+:f)a\n→Free(Modify Fs r:+:Nondet F:+:State F(Stack(Either r())):+:f)a\nlocal2trail =fold Var(alg1#alg2#fwd)\nwhere\nalg1(MUpdate r k)=pushStack(Left r)> >update r> >k\nalg1p =Op◦Inl$p\nalg2(Or p q)=(pushStack(Right())> >p)/talloblong(untrail> >q)\nalg2p =Op◦Inr◦Inl$p\nfwd p =Op◦Inr◦Inr◦Inr$p\nuntrail=dotop←popStack\ncasetopof\nNothing→/u1D702()\nJust(Right())→/u1D702()\nJust(Left r) → restore r> >untrail\nAs already informally explained above, this translation fu nction introduces code to push a\nmarker in the left branch of a choice, and untrail in the right branch. Whenever an update\nhappens, it is also recorded on the trail stack. All other ope rations remain as is.30\nNow, we can combine the simulation local2trail with the global-state semantics provided\nbyhGlobalM , and handle the trail stack at the end.\nhGlobalT ::(Functor f ,Undo s r)\n⇒Free(Modify Fs r:+:Nondet F:+:f)a→s→Free f[a]\nhGlobalT=fmap(fmap fst◦flip run StateT(Stack[])◦ hState)◦hGlobalM◦local2trail\nThe following theorem establishes the correctness of hGlobalT with respect to the local-state\nsemantics given by hLocaldefined in Section 4.1 .\nTheorem 7. Given Functor f and Undo s r, the equation\nhGlobalT=hLocalM\nholds for all programs p ::Free(Modify Fs r:+:Nondet F:+:f)a that do not use the\noperation Op(Inl MRestore).\nThe proof can be found in Appendix 7; it uses the same fold fusion strategy as in the\nproofs of other theorems.\n8.2 Putting Everything Together, Again\nWe can further combine the simulation local2trail with the simulation nondet2state of\nnondeterminism in Section 5 and the simulation state2state of multiple states in Section 6.1 .\nThe result simulation encodes the local-state semantics wi th one modification-based state\nand two stacks, a choicepoint stack generated by nondet2state and a trail stack generated\nbylocal2trail . This has a close relationship to the WAM of Prolog. The modif cation-based\nstate models the state of the program. The choicepoint stack stores the remaining branches\nto implement the nondeterministic searching. The trail sta ck stores the privous state updates\nto implement the backtracking.\nThe combined simulation function simulate Tis defined as follows:\nsimulate T::(Functor f ,Undo s r)\n⇒Free(Modify Fs r:+:Nondet F:+:f)a\n→s→Free f[a]\nsimulate Tx s=extractT◦hState\n◦fmap fst◦flip run StateT s◦hModify\n◦(⇔)◦ states2state◦(/anticlockwise)\n◦(⇔)◦ nondet2state◦(⇔)\n◦local2trail $x\nIt uses the auxiliary function extractT to get the final results, and (/anticlockwise)to reorder the signa-\ntures. Note that the initial state used by extractT is(SS[][],Stack[]), which essentially\ncontains an empty results list, an empty choicepoint stack, and an empty trail stack.\nextractT x =results SS◦fst◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT x(SS[][],Stack[])\n(/anticlockwise)::(Functor f 1,Functor f 2,Functor f3 ,Functor f4)\n⇒Free(f1:+:f2:+:f3:+:f4)a→Free(f2:+:f3:+:f1:+:f4)a31\n(/anticlockwise)(Var x) =Var x\n(/anticlockwise)(Op(Inl k)) =(Op◦Inr◦Inr◦Inl)(fmap(/anticlockwise)k)\n(/anticlockwise)(Op(Inr(Inl k)))=(Op◦Inl) ( fmap(/anticlockwise)k)\n(/anticlockwise)(Op(Inr(Inr(Inl k))))=(Op◦Inr◦Inl) ( fmap(/anticlockwise)k)\n(/anticlockwise)(Op(Inr(Inr(Inr k))))=(Op◦Inr◦Inr◦Inr)(fmap(/anticlockwise)k)\nFigure 5 illustrates each step of this simulation. The state type St s r f a is defined as\nSS(Modify Fs r:+:State F(Stack(Either r())):+:f)a.\nFree(Modify Fs r:+:Nondet F:+:f)a\nFree(Modify Fs r:+:Nondet F:+:State F(Stack(Either r())):+:f)a\nFree(Modify Fs r:+:State F(St s r f a):+:State F(Stack(Either r())):+:f)()\nFree(Modify Fs r:+:State F(St s r f a,Stack(Either r())):+:f)()\nFree(State F(St s r f a,Stack(Either r())):+:f)()\nFree f[a]local2trail\n(⇔)◦ nondet2state◦(⇔)\nfmap fst◦flip run StateT s◦hModify(⇔)◦ states2state◦(/anticlockwise)\nextractT◦hState\nFig. 5: An overview of the simulate Tfunction.\nIn the simulate Tfunction, we first use our three simulations local2trail ,nondet2state\nandstates2state (together with some reordering of signatures) to interpret the local-state\nsemantics for state and nondeterminism in terms of a modifica tion-based state and a general\nstate containing two stacks. Then, we use the handler hModify to interpret the modification-\nbased state effect, and use the handler hStateto interpret the two stacks. Finally, we use the\nfunction extractT to get the final results.\nAs in Section 6.2 , we can also fuse simulate Tinto a single handler.\ntypeComp f a s r =(WAM f a s r ,s)→Free f[a]\ndataWAM f a s r =WAM{results ::[a]\n,cpStack ::[Comp f a s r]\n,trStack ::[Either r()]}\nsimulate TF::(Functor f ,Undo s r)\n⇒Free(Modify Fs r:+:Nondet F:+:f)a\n→s\n→Free f[a]\nsimulate TFx s=fold gen(alg1#alg2#fwd)x(WAM[][][],s)\nwhere32\ngen x (WAM xs cp tr ,s)=continue(xs++[x])cp tr s\nalg1(MGet k) ( WAM xs cp tr ,s)=k s(WAM xs cp tr ,s)\nalg1(MUpdate r k)(WAM xs cp tr ,s)=k(WAM xs cp(Left r :tr),s⊕r)\nalg1(MRestore r k)(WAM xs cp tr ,s)=k(WAM xs cp tr ,s⊖r)\nalg2Fail(WAM xs cp tr ,s)=continue xs cp tr s\nalg2(Or p q) ( WAM xs cp tr ,s)=p(WAM xs(untrail q :cp)(Right():tr),s)\nfwd op (WAM xs cp tr ,s)=Op(fmap($(WAM xs cp tr ,s))op)\nuntrail q (WAM xs cp tr ,s)=casetrof\n[]→ q(WAM xs cp tr ,s)\nRight():tr′→q(WAM xs cp tr′,s)\nLeft r :tr′→untrail q(WAM xs cp tr′,s⊖r)\ncontinue xs cp tr s = casecpof\n[] → /u1D702xs\np:cp′→p(WAM xs cp′tr,s)\nHere, the carrier type of the algebras is Comp f a s r . It differs from that of simulate Fin that\nit also takes a trail stack as an input.\nN-queens with Two Stacks With simulate T, we can implement the backtracking algorithm\nof the n-queens problem with one modification-based state an d two stacks.\nqueensSimT ::Int→[[ Int]]\nqueensSimT =hNil◦flip simulate T(0,[])◦ queens M\n9 Related Work\nThere are various related works.\n9.1 Prolog\nProlog is a prominent example of a system that exposes nondet erminism with local state to\nthe user, but is itself implemented in terms of a single, glob al state.\nWarren Abstract Machine The folklore idea of undoing modifications upon backtrackin g\nis a key feature of many Prolog implementations, in particul ar those based on the Warren\nAbstract Machine (WAM) Warren ( 1983); A¨ıt-Kaci ( 1991). The WAM’s global state is\nthe program heap and Prolog programs modify this heap during unification only in a very\nspecific manner: following the union-find algorithm, they ov erwrite cells that contain self-\nreferences with pointers to other cells. Undoing these modi fications only requires knowledge\nof the modified cell’s address, which can be written back in th at cell during backtracking.\nThe WAM has a special stack, called the trail stack, for stori ng these addresses, and the\nprocess of restoring those cells is called untrailing .\nWAM Derivation and Correctness Several authors have studied the derivation of the\nWAM from a specification of Prolog, and its correctness.33\nB¨orger and Rosenzweig ( 1995) start from an operational semantics of Prolog based on\nderivation trees and refine this in successive steps to the WA M. Their approach was later\nmechanized in Isabelle/HOL by Pusch ( 1996). Pirog and Gibbons ( 2011) sketch how the\nWAM can be derived from a Prolog interpreter following the fu nctional correspondence\nbetween evaluator and abstract machine Ager et al. ( 2005).\nNeither of these approaches is based on an abstraction of effe cts that separates them from\nother aspects of Prolog.\nThe 4-Port Box Model While trailing happens under the hood, there is a folklore Pr olog\nprogramming pattern for observing and intervening at differ ent point in the control flow\nof a procedure call, known as the 4-port box model . In this model, upon the first entrance\nof a Prolog procedure it is called ; it may yield a result and exits; when the subsequent\nprocedure fails and backtracks, it is asked to redo its computation, possibly yielding the\nnext result; finally it may fail. Given a Prolog procedure pimplemented in Haskell, the\nfollowing program prints debugging messages when each of th e four ports are used:\n(putStr\"call\" /talloblongside(putStr\"fail\"))> >\np> >=/u1D706x→\n(putStr\"exit\" /talloblongside(putStr\"redo\"))> >\n/u1D702x\nThis technique was applied in the monadic setting by Hinze ( 1996), and it has been our\ninspiration for expressing the state restoration with glob al state.\nFunctional Models of Prolog Various authors have modelled (aspects of) Prolog in func-\ntional programming languages, often using monads to captur e nondeterminism and state\neffects. Notably, Spivey and Seres ( 1999) develop an embedding of Prolog in Haskell.\nMost attention has gone towards modelling the nondetermins m or search aspect of Prolog,\nwith various monads and monad transformers being proposed ( Hinze, 2000; Kiselyov et al.,\n2005). Notably, Schrijvers et al. ( 2014) shows how Prolog’s search can be exposed with a\nfree monad and manipulated using handlers.\nNone of these works consider mapping high-level to low-leve l representations of the\neffects.\n9.2 Reasoning About Side Effects\nThere are many works on reasoning and modelling side effects. Here, we cover those that\nhave most directly inspired this paper.\nAxiomatic Reasoning Gibbons and Hinze Gibbons and Hinze ( 2011) proposed to reason\naxiomatically about programs with effects and provided an ax iomatic characterization of\nlocal state semantics. Our earlier work in Pauwels et al. ( 2019) was directly inspired by\ntheir work: we introduced an axiomatic characterization of global state and used axiomatic\nreasoning to prove handling local state with global state co rrect. We also provided models\nthat satisfy the axioms, whereas their paper mistakenly cla ims that one model satisfies the\nlocal state axioms and that another model is monadic. This pa per is an extension of Pauwels34\net al. ( 2019), but notably, we depart from the axiomatic reasoning appro ach; instead we use\nproof techniques based on algebraic effects and handlers.\nAlgebraic Effects Our formulation of implementing local state with global sta te is directly\ninspired by the effect handlers approach of Plotkin and Pretn ar (2009). By making the\nfree monad explicit our proofs benefit directly from the indu ction principle that Bauer and\nPretnar established for effect handler programs . While Lawv ere theories were originally\nPlotkin’s inspiration for studying algebraic effects, the e ffect handlers community has for a\nlong time paid little attention to them. Yet, Luk ˇsiˇc and Pretnar ( 2020) have investigated a\nframework for encoding axioms or effect theories in the type s ystem: the type of an effectful\nfunction declares the operators used in the function, as wel l as the equalities that handlers\nfor these operators should comply with. The type of a handler indicates which operators\nit handles and which equations it complies with. This allows expressing at the a handles a\nhigher-level effect in terms of a lower-level one.\nWu and Schrijvers ( 2015) first presented fusion as a technique for optimizing compo-\nsitions of effect handlers. They use a specific form of fusion k nown as fold–build fusion\nor short-cut fusion (Gill et al., 1993). To enable this kind of fusion they transform the\nhandler algebras to use the codensity monad as their carrier . Their approach is not directly\nusable because it does not fuse non-handler functions, and w e derive simpler algebras (not\nobfuscated by the condisity monad) than those they do.\nMore recently Yang and Wu ( 2021) have used the fusion approach of Wu and Schrijvers\n(2015) (but with the continuation monad rather than the condensit y monad) for reasoning;\nthey remark that, although handlers are composable, the sem antics of these composed\nhandlers are not always obvious and that determining the cor rect order of composition to\narrive at a desired semantics is nontrivial. They propose a t echnique based on modular\nhandlers (Schrijvers et al., 2019) which considers conditions under which the fusion of\nthese modular handlers respect not only the laws of each of th e handler’s algebraic theories,\nbut also additional interaction laws. Using this technique they provide succinct proofs of the\ncorrectness of local state handlers, constructed from a fus ion of state and nondeterminism\nhandlers.\nEarlier Versions This paper refines and much expands on two earlier works of the last\nauthor.\nPauwels et al. ( 2019) has the same goal as Section 4 : it uses the state-restoring version\nof put to simulate local state with global state. It differs fr om this work in that it relies on\nan axiomatic (i.e., law-based), as opposed to handler-base d, semantics for local and global\nstate. This means that handler fusion cannot be used as a reas oning technique. Moreover,\nit uses a rather heavy-handed syntactic approach to context ual equivalence, and it assumes\nthat no other effects are invoked.\nAnother precursor is the work of Seynaeve et al. ( 2020), which establishes similar results\nas those in Section 5.1 . However, instead of generic definitions for the free monad a nd its\nfold, they use a specialized free monad for nondeterminism a nd explicitly recursive handler\nfunctions. As a consequence, their proofs use structural in duction rather than fold fusion.\nFurthermore, they did not consider other effects either.35\n10 Conclusion and Future Work\nWe studied the simulations of higher-level effects with lowe r-level effects for state and\nnondeterminism. We started with the translation from the lo cal-state semantics of state and\nnondeterminism to the global-state semantics. Then, we fur ther showed how to translate\nnondeterminism to state (a choicepoint stack), and transla te multiple state effects into one\nstate effect. Combining these results, we can simulate the lo cal-state semantics, a high-level\nprogramming abstraction, with only one low-level state effe ct. We also demonstrated that\nwe can simulate the local-state semantics using a trail stac k in a similar style to the Warren\nAbstract Machine of Prolog. We define the effects and their tra nslations with algebraic\neffects and effect handlers respectively. These are implemen ted as free monads and folds\nin Haskell. The correctness of all these translations has be en proved using the technique of\nprogram calculation, especially using the fusion properti es.\nIn future work, we would like to explore the potential optimi sations enabled by mutable\nstates. Mutable states fit the global-state semantics natur ally. With mutable states, we\ncan implement more efficient state update and restoration ope rations for the simulation\nlocal2global M(Section 7 ), as well as more efficient implementations of the choicepoin t\nstacks and trail stacks used by the simulations nondet2state (Section 5.2 ) and local2trail\n(Section 8 ), respectively. We would also like to consider the low-leve l simulations of\nother control-flow constructs used in logical programming l anguages such as Prolog’s cut\noperator for trimming the search space. Since operators lik ecutare usually implemented\nas scoped or higher-order effects (Pir ´og et al., 2018; Wu et al., 2014; Yang et al., 2022;\nvan den Berg and Schrijvers, 2023), we would have to adapt our approach accordingly.\nConflicts of Interest\nNone.\nReferences\nAger, M. S., Danvy, O. & Midtgaard, J. (2005) A functional cor respondence between monadic\nevaluators and abstract machines for languages with comput ational effects. Theoretical Computer\nScience .342(1), 149–172. Applied Semantics: Selected Topics.\nA¨ıt-Kaci, H. (1991) Warren’s Abstract Machine: A Tutorial Reconstruction . MIT Press.\nB¨orger, E. & Rosenzweig, D. (1995) The WAM - definition and comp iler correctness. In Logic\nProgramming: Formal Methods and Practical Applications, S tudies in Computer Science and\nArtificial Intelligence , Beierle, C. & Pl¨ umer, L. (eds). Elsevier Science B.V./Nor th-Holland. pp.\n20–90.\nGale, Y. (2007) ListT done right alternative. https://wiki.haskell.org/ListT_done_right_\nalternative .\nGibbons, J. (2000) Calculating functional programs. Algeb raic and Coalgebraic Methods in the\nMathematics of Program Construction, International Summe r School and Workshop, Oxford, UK,\nApril 10-14, 2000, Revised Lectures. Springer. pp. 149–202 .\nGibbons, J. & Hinze, R. (2011) Just do it: Simple monadic equa tional reasoning. SIGPLAN Not.\n46(9), 2–14.\nGill, A., Launchbury, J. & Peyton Jones, S. L. (1993) A short c ut to deforestation. Proceedings of the\nConference on Functional Programming Languages and Comput er Architecture. New York, NY,36\nUSA. Association for Computing Machinery. p. 223–232.\nHinze, R. (1996) Monadic-style backtracking. Technical Re port IAI-TR-96-9. Institut f¨ ur Informatik\nIII, Universit ¨at Bonn.\nHinze, R. (2000) Deriving backtracking monad transformers . Proceedings of the Fifth ACM\nSIGPLAN International Conference on Functional Programmi ng (ICFP ’00), Montreal, Canada,\nSeptember 18-21, 2000. ACM. pp. 186–197.\nHutton, G. & Fulger, D. (2008) Reasoning about effects: Seeing the wood through the trees (e xtended\nversion) .\nJones, M. P. (1995) Functional programming with overloadin g and higher-order polymorphism.\nKiselyov, O. (2015) Laws of monadplus. http://okmij.org/ftp/Computation/monads.html#\nmonadplus .\nKiselyov, O. & Ishii, H. (2015) Freer monads, more extensibl e effects. SIGPLAN Not. 50(12), 94–105.\nKiselyov, O., Shan, C., Friedman, D. P. & Sabry, A. (2005) Bac ktracking, interleaving, and terminating\nmonad transformers: (functional pearl). Proceedings of th e 10th ACM SIGPLAN International\nConference on Functional Programming, ICFP 2005, Tallinn, Estonia, September 26-28, 2005.\nACM. pp. 192–203.\nLorenzen, A., Leijen, D. & Swierstra, W. (2023) Fp ²: Fully in-place functional programming. Proc.\nACM Program. Lang. 7(ICFP).\nLukˇsiˇc, Z. & Pretnar, M. (2020) Local algebraic effect theories. J. Funct. Program. 30, e13.\nMoggi, E. (1991) Notions of computation and monads. Information and Computation .93(1), 55–92.\nPauwels, K., Schrijvers, T. & Mu, S.-C. (2019) Handling loca l state with global state. International\nConference on Mathematics of Program Construction. Spring er. pp. 18–44.\nPirog, M. & Gibbons, J. (2011) A functional derivation of the warren abstract machine. Unpublished.\nPir´og, M., Schrijvers, T., Wu, N. & Jaskelioff, M. (2018) Syntax a nd semantics for operations with\nscopes. Proceedings of the 33rd Annual ACM/IEEE Symposium o n Logic in Computer Science,\nLICS 2018, Oxford, UK, July 09-12, 2018. pp. 809–818.\nPlotkin, G. & Pretnar, M. (2009) Handlers of algebraic effect s. Programming Languages and Systems.\nBerlin, Heidelberg. Springer Berlin Heidelberg. pp. 80–94 .\nPlotkin, G. D. & Power, J. (2002) Notions of computation dete rmine monads. Proceedings of the 5th\nInternational Conference on Foundations of Software Scien ce and Computation Structures. Berlin,\nHeidelberg. Springer-Verlag. p. 342–356.\nPlotkin, G. D. & Power, J. (2003) Algebraic operations and ge neric effects. Appl. Categorical Struct.\n11(1), 69–94.\nPlotkin, G. D. & Pretnar, M. (2013) Handling Algebraic Effect s.Logical Methods in Computer\nScience .Volume 9, Issue 4 .\nPusch, C. (1996) Verification of compiler correctness for th e wam. Proceedings of the 9th International\nConference on Theorem Proving in Higher Order Logics. Berli n, Heidelberg. Springer-Verlag. p.\n347–361.\nReynolds, J. C. (1983) Types, abstraction and parametric po lymorphism. Information Processing 83,\nProceedings of the IFIP 9th World Computer Congress, Paris, France, September 19-23, 1983.\nNorth-Holland/IFIP. pp. 513–523.\nSchrijvers, T., Pir ´og, M., Wu, N. & Jaskelioff, M. (2019) Monad transformers and m odular alge-\nbraic effects: what binds them together. Proceedings of the 1 2th ACM SIGPLAN International\nSymposium on Haskell, Haskell, ICFP 2019, Berlin, Germany, August 18-23, 2019. pp. 98–113.\nSchrijvers, T., Wu, N., Desouter, B. & Demoen, B. (2014) Heur istics entwined with handlers com-\nbined: From functional specification to logic programming i mplementation. Proceedings of the\n16th International Symposium on Principles and Practice of Declarative Programming, Kent,\nCanterbury, United Kingdom, September 8-10, 2014. ACM. pp. 259–270.\nSchulte, C. (1999) Comparing trailing and copying for const raint programming. Proceedings of the\nSixteenth International Conference on Logic Programming. Las Cruces, NM, USA. The MIT Press.\npp. 275–289.\nSeynaeve, W., Pauwels, K. & Schrijvers, T. (2020) State will do. International Symposium on Trends\nin Functional Programming. Springer. pp. 204–225.37\nSpivey, J. M. & Seres, S. (1999) Embedding Prolog in Haskell.\nvan den Berg, B. & Schrijvers, T. (2023) A framework for highe r-order effects & handlers. CoRR .\nabs/2302.01415 .\nVolkov, N. (2014) list-t: ListT done right. https://github.com/nikita-volkov/list-t .\nWadler, P. (1989) Theorems for free! Proceedings of the four th international conference on Functional\nprogramming languages and computer architecture, FPCA 198 9, London, UK, September 11-13,\n1989. ACM. pp. 347–359.\nWarren, D. H. D. (1983) An abstract prolog instruction set. T echnical Report 309. AI Center, SRI\nInternational. 333 Ravenswood Ave., Menlo Park, CA 94025.\nWright, A. K. & Felleisen, M. (1994) A syntactic approach to t ype soundness. Inf. Comput. 115(1),\n38–94.\nWu, N. & Schrijvers, T. (2015) Fusion for free: Efficient algeb raic effect handlers. MPC 2015.\nWu, N., Schrijvers, T. & Hinze, R. (2014) Effect handlers in sc ope. Proceedings of the 2014 ACM\nSIGPLAN symposium on Haskell, Gothenburg, Sweden, Septemb er 4-5, 2014. pp. 1–12.\nYang, Z., Paviotti, M., Wu, N., van den Berg, B. & Schrijvers, T. (2022) Structured handling of scoped\neffects. Programming Languages and Systems - 31st European S ymposium on Programming, ESOP\n2022, Held as Part of the European Joint Conferences on Theor y and Practice of Software, ETAPS\n2022, Munich, Germany, April 2-7, 2022, Proceedings. Sprin ger. pp. 462–491.\nYang, Z. & Wu, N. (2021) Reasoning about effect interaction by fusion. Proc. ACM Program. Lang.\n5(ICFP), 1–29.38\n1 Proofs for Get Laws in Local-State Semantics\nIn this section we prove two equations about the interaction of nondeterminism and state in\nthe local-state semantics.\nEquation (4.3) :get> >∅=∅\nProof\nget> >∅\n={- definition of(> >)-}\nget> >=(/u1D706s→∅)\n={- Law ( 4.1): put-right-identity - }\nget> >=(/u1D706s→put s> >∅)\n={- Law ( 2.5): associativity of(> >)-}\n(get> >=put)> >∅\n={- Law ( 2.12): get-put - }\n/u1D702()> >∅\n={- Law ( 2.3): return-bind and definition of (> >)-}\n∅\n/squaresolid\nEquation (4.4) :get> >=(/u1D706x→k1x/talloblongk2x)=(get> >=k1)/talloblong(get> >=k2)\nProof\nget> >=(/u1D706x→k1x/talloblongk2x)\n={- Law ( 2.3): return-bind and definition of (> >)-}\n/u1D702()> >(get> >=(/u1D706x→k1x/talloblongk2x))\n={- Law ( 2.5): associativity of(> >=)-}\n(/u1D702()> >get)> >=(/u1D706x→k1x/talloblongk2x)\n={- Law ( 2.12): get-put - }\n((get> >=put)> >get)> >=(/u1D706x→k1x/talloblongk2x)\n={- Law ( 2.5): associativity of(> >)-}\n(get> >=(/u1D706s→put s> >get))> >=(/u1D706x→k1x/talloblongk2x)\n={- Law ( 2.5): associativity of(> >=)-}\nget> >=(/u1D706s→(/u1D706s→put s> >get)s> >=(/u1D706x→k1x/talloblongk2x))\n={- function application - }\nget> >=(/u1D706s→(put s> >get)> >=(/u1D706x→k1x/talloblongk2x))\n={- Law ( 2.11): put-get - }\nget> >=(/u1D706s→(put s> >/u1D702s)> >=(/u1D706x→k1x/talloblongk2x))\n={- Law ( 2.5): associativity of(> >)-}\nget> >=(/u1D706s→put s> >(/u1D702s> >=(/u1D706x→k1x/talloblongk2x)))\n={- Law ( 2.3): return-bind and function application - }\nget> >=(/u1D706s→put s> >(k1s/talloblongk2s))\n={- Law ( 4.2): put-left-distributivity - }\nget> >=(/u1D706s→(put s> >k1s)/talloblong(put s> >k2s))\n={- Law ( 2.3): return-bind (twice) - }39\nget> >=(/u1D706s→(put s> >(/u1D702s> >=k1))/talloblong(put s> >(/u1D702s> >=k2)))\n={- Law ( 2.5): associativity of(> >)-}\nget> >=(/u1D706s→((put s> >/u1D702s)> >=k1)/talloblong((put s> >/u1D702s)> >=k2))\n={- Law ( 2.11): put-get - }\nget> >=(/u1D706s→((put s> >get)> >=k1)/talloblong((put s> >get)> >=k2))\n={- Law ( 2.5): associativity of(> >)-}\nget> >=(/u1D706s→(put s> >(get> >=k1))/talloblong(put s> >(get> >=k2)))\n={- Law ( 4.2): put-left-distributivity - }\nget> >=(/u1D706s→put s> >((get> >=k1)/talloblong(get> >=k2)))\n={- Law ( 2.5): associativity of(> >=)-}\n(get> >=put)> >((get> >=k1)/talloblong(get> >=k2))\n={- Law ( 2.12): get-put - }\n/u1D702()> >((get> >=k1)/talloblong(get> >=k2))\n={- Law ( 2.3): return-bind and definition of (> >)-}\n(get> >=k1)/talloblong(get> >=k2)\n/squaresolid\n2 Proofs for Modelling Local State with Global State\nThis section proves the following theorem in Section 4.3 .\nTheorem 1. hGlobal◦local2global =hLocal\nPreliminary It is easy to see that runStateT◦hStatecan be fused into a single fold defined as\nfollows:\nhState1::Functor f⇒Free(State Fs:+:f)a→(s→Free f(a,s))\nhState1=fold gen S(algS#fwdS)\nwhere\ngenSx s =Var(x,s)\nalgS(Get k)s=k s s\nalgS(Put s k)=k s\nfwdSy s =Op(fmap($s)y)\nFor brevity, we use hState1to replace runStateT◦hStatein the following proofs.\n2.1 Main Proof Structure\nThe main theorem we prove in this section is:\nTheorem 8. hGlobal◦local2global =hLocal\nProof Both the left-hand side and the right-hand side of the equati on consist of function\ncompositions involving one or more folds. We apply fold fusi on separately on both sides to40\ncontract each into a single fold:\nhGlobal◦local2global =fold genLHS(algS\nLHS#algND\nRHS#fwdLHS)\nhLocal=fold genRHS(algS\nRHS#algND\nRHS#fwdRHS)\nWe approach this calculationally. That is to say, we do not fir st postulate definitions of\nthe unknowns above ( algS\nLHSand so on) and then verify whether the fusion conditions are\nsatisfied. Instead, we discover the definitions of the unknow ns. We start from the known\nside of each fusion condition and perform case analysis on th e possible shapes of input.\nBy simplifying the resulting case-specific expression, and pushing the handler applications\ninwards, we end up at a point where we can read off the definition of the unknown that\nmakes the fusion condition hold for that case.\nFinally, we show that both folds are equal by showing that the ir correspondingparameters\nare equal:\ngenLHS=genRHS\nalgS\nLHS=algS\nRHS\nalgND\nLHS=algND\nRHS\nfwdLHS=fwdRHS\nA noteworthy observation is that, for fusing the left-hand s ide of the equation, we do not\nuse the standard fusion rule:\nhGlobal◦fold Var alg =fold(hGlobal◦Var)alg′\n⇐ hGlobal◦alg=alg′◦fmap h Global\nwhere local2global =fold Var alg . The problem is that we will not find an appropriate alg′\nsuch that alg′(fmap h Global t)restores the state for any tof type(State Fs:+:Nondet F:+:\nf)(Free(State Fs:+:NonDetF :+:f)a).\nFortunately, we do not need such an alg′. As we have already pointed out, we can\nassume that the subterms of thave already been transformed by local2global , and thus all\noccurrences of Putappear in the putRconstellation.\nWe can incorporate this assumption by using the alternative e fusion rule:\nhGlobal◦fold Var alg =fold(hGlobal◦Var)alg′\n⇐ hGlobal◦alg◦fmap local2global =alg′◦fmap h Global◦fmap local2global\nThe additional fmap local2global in the condition captures the property that all the subterms\nhave been transformed by local2global .\nIn order to not clutter the proofs, we abstract everywhere ov er this additional\nfmap local2global application, except in the one place where we need it. That is the appeal\nto the key lemma:\nhState1(hND+f((⇔)( local2global t)))s\n=\ndo(x,)←hState1(hND+f((⇔)( local2global t)))s;/u1D702(x,s)\nThis expresses that the syntactic transformation local2global makes sure that, despite any\ntemporary changes, the computation trestores the state back to its initial value.41\nWe elaborate each of these steps below. /squaresolid\n2.2 Fusing the Right-Hand Side\nWe calculate as follows:\nhLocal\n={- definition - }\nhL◦hState1\nwith\nhL::(Functor f)⇒( s→Free(Nondet F:+:f)(a,s))→ s→Free f[a]\nhL=fmap(fmap(fmap fst)◦hND+f)\n={- definition of hState1-}\nhL◦fold gen S(algS#fwdS)\n={- fold fusion-post (Equation 3.2) -}\nfold genRHS(algS\nRHS#algND\nRHS#fwdRHS)\nThis last step is valid provided that the fusion conditions a re satisfied:\nhL◦genS=genRHS\nhL◦(algS#fwdS)=(algS\nRHS#algND\nRHS#fwdRHS)◦fmap h L\nWe calculate for the first fusion condition:\nhL(genSx)\n={- definition of genS-}\nhL(/u1D706s→Var(x,s))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→Var(x,s))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(Var(x,s)))\n={- definition of hND+f-}\n/u1D706s→fmap(fmap fst)(Var[(x,s)])\n={- definition of fmap (twice) -}\n/u1D706s→Var[x]\n={- define genRHSx=/u1D706s→Var[x]-}\n=genRHSx\nWe conclude that the first fusion condition is satisfied by:\ngenRHS::Functor f⇒a→(s→Free f[a])\ngenRHSx=/u1D706s→Var[x]\nThe second fusion condition decomposes into two separate co nditions:\nhL◦algS=algS\nRHS◦fmap h L\nhL◦fwdS=(algND\nRHS#fwdRHS)◦fmap h L\nWe calculate for the first subcondition:\ncaset=Get k42\nhL(algS(Get k))\n={- definition of algS-}\nhL(/u1D706s→k s s)\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→k s s)\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(k s s))\n={- beta-expansion (twice) - }\n=/u1D706s→(/u1D706s1s2→fmap(fmap fst)(hND+f(k s2s1)))s s\n={- definition of fmap (twice) -}\n=/u1D706s→(fmap(fmap(fmap(fmap fst)◦hND+f))(/u1D706s1s2→k s2s1))s s\n={- eta-expansion of k-}\n=/u1D706s→(fmap(fmap(fmap(fmap fst)◦hND+f))k)s s\n={- define algS\nRHS(Get k)=/u1D706s→k s s-}\n=algS\nRHS(Get(fmap(fmap(fmap(fmap fst)◦hND+f))k))\n={- definition of fmap -}\n=algS\nRHS(fmap(fmap(fmap(fmap fst)◦hND+f))(Get k))\n={- definition of hL-}\n=algS\nRHS(fmap h L(Get k))\ncaset=Put s k\nhL(algS(Put s k))\n={- definition of algS-}\nhL(/u1D706→k s)\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706→k s)\n={- definition of fmap -}\n/u1D706→fmap(fmap fst)(hND+f(k s))\n={- beta-expansion - }\n=/u1D706→(/u1D706s1→fmap(fmap fst)(hND+f(k s1)))s\n={- definition of fmap -}\n=/u1D706→(fmap(fmap(fmap fst)◦hND+f)(/u1D706s1→k s1))s\n={- eta-expansion of k-}\n=/u1D706→(fmap(fmap(fmap fst)◦hND+f)k)s\n={- define algS\nRHS(Pus s k)=\\→k s-}\n=algS\nRHS(Put s(fmap(fmap(fmap fst)◦hND+f)k))\n={- definition of fmap -}\n=algS\nRHS(fmap(fmap(fmap fst)◦hND+f))(Put s k))\n={- definition of hL-}\n=algS\nRHS(fmap h L(Put s k))\nWe conclude that the first subcondition is met by taking:\nalgS\nRHS::Functor f⇒State Fs(s→Free f[a])→( s→Free f[a])\nalgS\nRHS(Get k)=/u1D706s→k s s\nalgS\nRHS(Put s k)=/u1D706→k s43\nThe second subcondition can be split up in two further subcon ditions:\nhL◦fwdS◦Inl=algND\nRHS◦fmap h L\nhL◦fwdS◦Inr=fwdRHS◦fmap h L\nFor the first of these, we calculate:\nhL(fwdS(Inl op))\n={- definition of fwdS-}\nhL(/u1D706s→Op(fmap($s)(Inl op)))\n={- definition of fmap -}\nhL(/u1D706s→Op(Inl(fmap($s)op)))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→Op(Inl(fmap($s)op)))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(Op(Inl(fmap($s)op))))\n={- definition of hND+f-}\n/u1D706s→fmap(fmap fst)(algND+f(fmap h ND+f(fmap($s)op)))\nWe split on op:\ncaseop=Fail\n/u1D706s→fmap(fmap fst)(algND+f(fmap h ND+f(fmap($s)Fail)))\n={- defintion of fmap (twice) -}\n/u1D706s→fmap(fmap fst)(algND+fFail)\n={- definition of algND+f-}\n/u1D706s→fmap(fmap fst)(Var[])\n={- definition of fmap (twice) -}\n/u1D706s→Var[]\n={- define algND\nRHSFail=/u1D706s→Var[]-}\nalgND\nRHSFail\n={-definition fo fmap -}\nalgND\nRHS(fmap h Lfail)\ncaseop=Or p q\n/u1D706s→fmap(fmap fst)(algND+f(fmap h ND+f(fmap($s)(Or p q))))\n={- defintion of fmap (twice) -}\n/u1D706s→fmap(fmap fst)(algND+f(Or(hND+f(p s))(hND+f(q s))))\n={- definition of algND+f-}\n/u1D706s→fmap(fmap fst)(liftM2(++)( hND+f(p s))(hND+f(q s)))\n={- Lemma 3-}\n/u1D706s→liftM2(++)( fmap(fmap fst)(hND+f(p s)))(fmap(fmap fst)(hND+f(q s)))\n={- define algND\nRHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)-}\nalgND\nRHS(Or(fmap(fmap fst)◦hND+f◦p)(fmap(fmap fst)◦hND+f◦q))\n={- defintion of fmap (twice) -}\nalgND\nRHS(fmap(fmap(fmap(fmap fst)◦hND+f))(Or p q))44\n={- defintion of hL-}\nalgND\nRHS(fmap h L(Or p q))\nFrom this we conclude that the definition of algND\nRHSshould be:\nalgND\nRHS::Functor f⇒Nondet F(s→Free f[a])→( s→Free f[a])\nalgND\nRHSFail=/u1D706s→Var[]\nalgND\nRHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)\nFor the last subcondition, we calculate:\nhL(fwdS(Inr op))\n={- definition of fwdS-}\nhL(/u1D706s→Op(fmap($s)(Inr op)))\n={- definition of fmap -}\nhL(/u1D706s→Op(Inr(fmap($s)op)))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→Op(Inr(fmap($s)op)))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(Op(Inr(fmap($s)op))))\n={- definition of hND+f-}\n/u1D706s→fmap(fmap fst)(fwdND+f(fmap h ND+f(fmap($s)op)))\n={- definition of fwdND+f-}\n/u1D706s→fmap(fmap fst)(Op(fmap h ND+f(fmap($s)op)))\n={- definition of fmap -}\n/u1D706s→Op(fmap(fmap(fmap fst))(fmap h ND+f(fmap($s)op)))\n={- definition of hL-}|\n/u1D706s→Op(hL(fmap($s)op))\n={-Lemma 2 -}\n/u1D706s→Op(fmap($s)(fmap h Lop))\n={- define fwdRHSop=/u1D706s��Op(fmap($s)op)-}\nfwdRHS(fmap h Lop)\nFrom this we conclude that the definition of fwdRHSshould be:\nfwdRHS::Functor f⇒f(s→Free f[a])→( s→Free f[a])\nfwdRHSop=/u1D706s→Op(fmap($s)op)\n2.3 Fusing the Left-Hand Side\nWe proceed in the same fashion with fusing left-hand side, di scovering the definitions that\nwe need to satisfy the fusion condition.\nWe calculate as follows:\nhGlobal◦local2global\n={- definition of local2global -}\nhGlobal◦fold Var alg\nwhere45\nalg(Inl(Put t k))=putRt> >k\nalg p=Op p\n={- fold fusion-post (Equation 3.2) -}\nfold genLHS(algS\nLHS#algND\nLHS#fwdLHS)\nThis last step is valid provided that the fusion conditions a re satisfied:\nhGlobal◦Var=genLHS\nhGlobal◦alg=(algS\nLHS#algND\nLHS#fwdLHS)◦fmap h Global\nWe calculate for the first fusion condition:\nhGlobal(Var x)\n={- definition of hGlobal -}\nfmap(fmap fst)(hState1(hND+f((⇔)( Var x))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hState1(hND+f(Var x)))\n={- definition of hND+f-}\nfmap(fmap fst)(hState1(Var[x]))\n={- definition of hState1-}\nfmap(fmap fst)(/u1D706s→Var([x],s))\n={- definition of fmap (twice) -}\n/u1D706s→Var[x]\n={- define genLHSx=/u1D706s→Var[x]-}\ngenLHSx\nWe conclude that the first fusion condition is satisfied by:\ngenLHS::Functor f⇒a→(s→Free f[a])\ngenLHSx=/u1D706s→Var[x]\nWe can split the second fusion condition in three subconditi ons:\nhGlobal◦alg◦Inl=algS\nLHS◦fmap h Global\nhGlobal◦alg◦Inr◦Inl=algND\nLHS◦fmap h Global\nhGlobal◦alg◦Inr◦Inr=fwdLHS◦fmap h Global\nLet’s consider the first subconditions. It has two cases:\ncaseop=Get k\nhGlobal(alg(Inl(Get k)))\n={- definition of alg-}\nhGlobal(Op(Inl(Get k)))\n={- definition of hGlobal -}\nfmap(fmap fst)(hState1(hND+f((⇔)( Op(Inl(Get k))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hState1(hND+f(Op(Inr(Inl(fmap(⇔)( Get k)))))))\n={- definition of fmap -}\nfmap(fmap fst)(hState1(hND+f(Op(Inr(Inl(Get((⇔)◦ k)))))))46\n={- definition of hND+f-}\nfmap(fmap fst)(hState1(Op(fmap h ND+f(Inl(Get((⇔)◦ k))))))\n={- definition of fmap -}\nfmap(fmap fst)(hState1(Op(Inl(Get(hND+f◦(⇔)◦ k)))))\n={- definition of hState1-}\nfmap(fmap fst)(/u1D706s→(hState1◦hND+f◦(⇔)◦ k)s s)\n={- definition of fmap -}\n(/u1D706s→fmap fst((hState1◦hND+f◦(⇔)◦ k)s s))\n={- definition of fmap -}\n(/u1D706s→((fmap(fmap fst)◦hState1◦hND+f◦(⇔)◦ k)s s))\n={- define algS\nLHS(Get k)=/u1D706s→k s s-}\nalgS\nLHS(Get(hGlobal◦k))\n={- definition of fmap -}\nalgS\nLHS(fmap h Global(Get k))\ncaseop=Put s k\nhGlobal(alg(Inl(Put s k)))\n={- definition of alg-}\nhGlobal(putRs> >k)\n={- definition of putR-}\nhGlobal((get> >=/u1D706t→put s/talloblongside(put t))> >k)\n={- definitions of side,get,put,(/talloblong),(> >=)-}\nhGlobal(Op(Inl(Get(/u1D706t→Op(Inr(Inl(Or(Op(Inl(Put s k)))\n(Op(Inl(Put t(Op(Inr(Inl Fail)))))))))))))\n={- definition of hGlobal -}\nfmap(fmap fst)(hState1(hND+f((⇔)\n(Op(Inl(Get(/u1D706t→Op(Inr(Inl(Or(Op(Inl(Put s k)))\n(Op(Inl(Put t(Op Inr((Inl Fail))))))))))))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hState1(hND+f(\n(Op(Inr(Inl(Get(/u1D706t→Op(Inl(Or(Op(Inr(Inl(Put s((⇔) k)))))\n(Op(Inr(Inl(Put t(Op(Inl Fail))))))))))))))))\n={- definition of hND+f-}\nfmap(fmap fst)(hState1(\n(Op(Inl(Get(/u1D706t→liftM2(++)( Op(Inl(Put s(hND+f((⇔) k)))))\n(Op(Inl(Put t(Var[]))))))))))\n={- definition of hState1-}\nfmap(fmap fst)\n(/u1D706t→hState1(liftM2(++)( Op(Inl(Put s(hND+f((⇔) k)))))\n(Op(Inl(Put t(Var[]))))) t)\n={- definition of liftM2 -}\nfmap(fmap fst)\n(/u1D706t→hState1(dox←Op(Inl(Put s(hND+f((⇔) k))))\ny←Op(Inl(Put t(Var[])))\nVar(x++y)47\n)t)\n={-Lemma 4 -}\nfmap(fmap fst)\n(/u1D706t→do(x,t1)←hState1(Op(Inl(Put s(hND+f((⇔) k))))) t\n(y,t2)←hState1(Op(Inl(Put t(Var[])))) t1\nhState1(Var(x++y))t2\n)\n={- definition of hState1-}\nfmap(fmap fst)\n(/u1D706t→do(x,)←hState1(hND+f((⇔) k))s\n(y,t2)←Var([],t)\nVar(x++y,t2)\n)\n={- monad law - }\nfmap(fmap fst)\n(/u1D706t→do(x,)←hState1(hND+f((⇔) k))s\nVar(x++[],t)\n)\n={- right unit of(++)-}\nfmap(fmap fst)\n(/u1D706t→do(x,)←hState1(hND+f((⇔) k))s\nVar(x,t)\n)\n={- definition of fmap fst -}\nfmap(fmap fst)\n(/u1D706t→dox←fmap fst(hState1(hND+f((⇔) k))s)\nVar(x,t)\n)\n={- definition of fmap -}\nfmap(fmap fst)\n(/u1D706t→dox←(fmap(fmap fst)(hState1(hND+f((⇔) k))))s\nVar(x,t)\n)\n={- definition of fmap(fmap fst)-}\n\\→dox←(fmap(fmap fst)(hState1(hND+f((⇔) k))))s\nVar x\n={- monad law - }\n\\→(fmap(fmap fst)(hState1(hND+f((⇔) k))))s\n={- definition of hGlobal -}\n\\→(hGlobal k)s\n={- define algS\nLHS(Put s k)=\\→k s-}\nalgS\nLHS(Put s(hGlobal k))\n={- definition of fmap -}\nalgS\nLHS(fmap h Global(Put s))48\nWe conclude that this fusion subcondition holds provided th at:\nalgS\nLHS::Functor f⇒State Fs(s→Free f[a])→( s→Free f[a])\nalgS\nLHS(Get k)=/u1D706s→k s s\nalgS\nLHS(Put s k)=\\→k s\nLet’s consider the second subcondition. It has also two case s:\ncaseop=Fail\nhGlobal(alg(Inr(Inl Fail)))\n={- definition of alg-}\nhGlobal(Op(Inr(Inl Fail)))\n={- definition of hGlobal -}\nfmap(fmap fst)(hState1(hND+f((⇔)( Op(Inr(Inl Fail))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hState1(hND+f(Op(Inl(fmap(⇔)Fail)))))\n={- definition of fmap -}\nfmap(fmap fst)(hState1(hND+f(Op(Inl Fail))))\n={- definition of hND+f-}\nfmap(fmap fst)(hState1(Var[]))\n={- definition of hState1-}\nfmap(fmap fst)(/u1D706s→Var([],s))\n={- definition of fmap twice and fst-}\n/u1D706s→Var[]\n={- define algND\nRHSFail=/u1D706s→Var[]-}\nalgND\nRHSFail\n={- definition of fmap -}\nalgND\nRHS(fmap h Global Fail)\ncaseop=Or p q\nhGlobal(alg(Inr(Inl(Or p q))))\n={- definition of alg-}\nhGlobal(Op(Inr(Inl(Or p q))))\n={- definition of hGlobal -}\nfmap(fmap fst)(hState1(hND+f((⇔)( Op(Inr(Inl(Or p q)))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hState1(hND+f(Op(Inl(fmap(⇔)( Or p q))))))\n={- definition of fmap -}\nfmap(fmap fst)(hState1(hND+f(Op(Inl(Or((⇔) p)((⇔) q))))))\n={- definition of hND+f-}\nfmap(fmap fst)(hState1(liftM2(++)( hND+f((⇔) p))(hND+f((⇔) q))))\n={- definition of liftM2 -}\nfmap(fmap fst)(hState1(dox←hND+f((⇔) p)\ny←hND+f((⇔) q)\n/u1D702(x++y)))\n={-Lemma 4 -}49\nfmap(fmap fst)(/u1D706s0→(do(x,s1)←hState1(hND+f((⇔) p))s0\n(y,s2)←hState1(hND+f((⇔) q))s1\nhState1(/u1D702(x++y))s2))\n={- definition of hState1-}\nfmap(fmap fst)(/u1D706s0→(do(x,s1)←hState1(hND+f((⇔) p))s0\n(y,s2)←hState1(hND+f((⇔) q))s1\nVar(x++y,s2)))\n={- Lemma 1(pandqare in the codomain of local2global ) -}\nfmap(fmap fst)(/u1D706s0→(do(x,s1)←do{(x,)←hState1(hND+f((⇔) p))s0;/u1D702(x,s0)}\n(y,s2)←do{(y,)←hState1(hND+f((⇔) q))s1;/u1D702(x,s1)}\nVar(x++y,s2)))\n={- monad laws - }\nfmap(fmap fst)(/u1D706s0→(do(x,)←hState1(hND+f((⇔) p))s0\n(y,)←hState1(hND+f((⇔) q))s0\nVar(x++y,s0)))\n={- definition of fmap (twice) and fst-}\n/u1D706s0→(do(x,)←hState1(hND+f((⇔) p))s0\n(y,)←hState1(hND+f((⇔) q))s0\nVar(x++y))\n={- definition of fmap,fstand monad laws - }\n/u1D706s0→(dox←fmap fst(hState1(hND+f((⇔) p))s0)\ny←fmap fst(hState1(hND+f((⇔) q))s0)\nVar(x++y))\n={- definition of fmap -}\n/u1D706s0→(dox←fmap(fmap fst)(hState1(hND+f((⇔) p)))s0\ny←fmap(fmap fst)(hState1(hND+f((⇔) q)))s0\nVar(x++y))\n={- definition of hGlobal -}\n/u1D706s0→(dox←hGlobal p s0\ny←hGlobal q s0\nVar(x++y))\n={- definition of liftM2 -}\n/u1D706s0→liftM2(++)( hGlobal p s0)(hGlobal q s0)\n={- define algND\nLHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)-}\nalgND\nLHS(Or(hGlobal p)(hGlobal q))\n={- definition of fmap -}\nalgND\nLHS(fmap hGobal(Or p q))\nWe conclude that this fusion subcondition holds provided th at:\nalgND\nLHS::Functor f⇒Nondet F(s→Free f[a])→( s→Free f[a])\nalgND\nLHSFail=/u1D706s→Var[]\nalgND\nLHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)\nFinally, the last subcondition:50\nhGlobal(alg(Inr(Inr op)))\n={- definition of alg-}\nhGlobal(Op(Inr(Inr op)))\n={- definition of hGlobal -}\nfmap(fmap fst)(hState1(hND+f((⇔)( Op(Inr(Inr op))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hState1(hND+f(Op(Inr(Inr(fmap(⇔)op))))))\n={- definition of hND+f-}\nfmap(fmap fst)(hState1(Op(fmap h ND+f(Inr(fmap(⇔)op)))))\n={- definition of fmap -}\nfmap(fmap fst)(hState1(Op(Inr(fmap h ND+f(fmap(⇔)op)))))\n={-fmap fusion -}\nfmap(fmap fst)(hState1(Op(Inr(fmap(hND+f◦(⇔)) op))))\n={- definition of hState1-}\nfmap(fmap fst)(/u1D706s→Op(fmap($s)(fmap h State1(fmap(hND+f◦(⇔)) op))))\n={-fmap fusion -}\nfmap(fmap fst)(/u1D706s→Op(fmap($s)(fmap(hState1◦hND+f◦(⇔)) op)))\n={- definition of fmap -}\n/u1D706s→fmap fst(Op(fmap($s)(fmap(hState1◦hND+f◦(⇔)) op)))\n={- definition of fmap -}\n/u1D706s→Op(fmap(fmap fst)(fmap($s)(fmap(hState1◦hND+f◦(⇔)) op)))\n={-fmap fusion -}\n/u1D706s→Op(fmap(fmap fst◦($s))(fmap(hState1◦hND+f◦(⇔)) op)))\n={-Lemma 2 -}\n/u1D706s→Op(fmap(($s)◦fmap(fmap fst))(fmap(hState1◦hND+f◦(⇔)) op)))\n={-fmap fission -}\n/u1D706s→Op((fmap($s)◦fmap(fmap(fmap fst)))(fmap(hState1◦hND+f◦(⇔)) op))\n={-fmap fusion -}\n/u1D706s→Op(fmap($s)(fmap(fmap(fmap fst)◦hState1◦hND+f◦(⇔)) op))\n={- definition of hGlobal -}\n/u1D706s→Op(fmap($s)(fmap h Global op))\n={- define fwdLHSop=/u1D706s→Op(fmap($s)op-}\nfwdLHS(fmap h Global op)\nWe conclude that this fusion subcondition holds provided th at:\nfwdLHS::Functor f⇒f(s→Free f[a])→( s→Free f[a])\nfwdLHSop=/u1D706s→Op(fmap($s)op)51\n2.4 Equating the Fused Sides\nWe observe that the following equations hold trivially.\ngenLHS=genRHS\nalgS\nLHS=algS\nRHS\nalgND\nLHS=algND\nRHS\nfwdLHS=fwdRHS\nTherefore, the main theorem holds.\n2.5 Key Lemma: State Restoration\nThe key lemma is the following, which guarantees that local2global restores the initial state\nafter a computation.\nLemma 1 (State is Restored) .\nhState1(hND+f((⇔)( local2global t)))s\n=\ndo(x,)←hState1(hND+f((⇔)( local2global t)))s;/u1D702(x,s)\nProof The proof proceeds by structural induction on t.\ncaset=Var y\nhState1(hND+f((⇔)( local2global(Var y))))s\n={- definition of local2global -}\nhState1(hND+f((⇔)( Var y)))s\n={- definition of(⇔)-}\nhState1(hND+f(Var y))s\n={- definition of hND+f-}\nhState1(Var[y])s\n={- definition of hState1-}\nVar([y],s)\n={- monad law - }\ndo(x,)←Var([y],s);Var(x,s)\n={- definition of local2global ,hND+f,(⇔),hState1and/u1D702-}\ndo(x,)←hState1(hND+f((⇔)( local2global(Var y))))s;/u1D702(x,s)\ncaset=Op(Inl(Get k))\nhState1(hND+f((⇔)( local2global(Op(Inl(Get k)))))) s\n={- definition of local2global -}\nhState1(hND+f((⇔)( Op(Inl(Get(local2global◦k)))))) s\n={- definition of(⇔)-}\nhState1(hND+f(Op(Inr(Inl(Get((⇔)◦ local2global◦k)))))) s52\n={- definition of hND+f-}\nhState1(Op(Inl(Get(hND+f◦(⇔)◦ local2global◦k))))s\n={- definition of hState1-}\n(hState1◦hND+f◦(⇔)◦ local2global◦k)s s\n={- definition of(◦)-}\n(hState1(hND+f((⇔)( local2global(k s))))) s\n={- induction hypothesis - }\ndo(x,)←hState1((⇔)( hND+f(local2global(k s))))s;/u1D702(x,s)\n={- definition of local2global ,(⇔),hND+f,hState1-}\ndo(x,)←hState1(hND+f(local2global(Op(Inl(Get k))))) s;/u1D702(x,s)\ncaset=Op(Inr(Inl Fail))\nhState1(hND+f((⇔)( local2global(Op(Inr(Inl Fail)))))) s\n={- definition of local2global -}\nhState1(hND+f((⇔)( Op(Inr(Inl Fail))))) s\n={- definition of(⇔)-}\nhState1(hND+f(Op(Inl Fail)))s\n={- definition of hND+f-}\nhState1(Var[])s\n={- definition of hState1-}\nVar([],s)\n={- monad law - }\ndo(x,)←Var([],s);Var(x,s)\n={- definition of local2global ,(⇔),hND+f,hState1-}\ndo(x,)←hState1(hND+f((⇔)( local2global(Op(Inr(Inl Fail)))))) s;/u1D702(x,s)\ncaset=Op(Inl(Put t k))\nhState1(hND+f((⇔)( local2global(Op(Inl(Put t k)))))) s\n={- definition of local2global -}\nhState1(hND+f((⇔)( putRt> >local2global k)))s\n={- definition of putR-}\nhState1(hND+f((⇔)(( get> >=/u1D706t′→put t/talloblongside(put t′))> >local2global k)))s\n={- definition of(/talloblong),get,put,sideand(> >=)-}\nhState1(hND+f((⇔)( Op(Inl(Get(/u1D706t′→\nOp(Inr(Inl(Or(Op(Inl(Put t(local2global k))))\n(Op(Inl(Put t′(Op(Inr(Inl Fail))))))))))))))) s\n={- definition of(⇔)-}\nhState1(hND+f(Op(Inr(Inl(Get(/u1D706t′→\nOp(Inl(Or(Op(Inr(Inl(Put t((⇔)( local2global k))))))\n(Op(Inr(Inl(Put t′(Op(Inl Fail)))))))))))))) s\n={- definition of hND+f-}\nhState1(Op(Inl(Get(/u1D706t′→\nliftM2(++)( Op(Inl(Put t(hND+f((⇔)( local2global k))))))\n(Op(Inl(Put t′(Var[])))))))) s\n={- definition of hState1-}53\nhState1(liftM2(++)( Op(Inl(Put t(hND+f((⇔)( local2global k))))))( Op(Inl(Put s(Var[]))))) s\n={- definition of liftM2 -}\nhState1(dox←Op(Inl(Put t(hND+f((⇔)( local2global k)))))\ny←Op(Inl(Put s(Var[])))\nVar(x++y)\n)s\n={- Lemma 4-}\ndo(x,s1)←hState1(Op(Inl(Put t(hND+f((⇔)( local2global k)))))) s\n(y,s2)←hState1(Op(Inl(Put s(Var[])))) s1\nVar(x++y,s2)\n={- definition of hState1-}\ndo(x,s1)←hState1(hND+f((⇔)( local2global k)))t\n(y,s2)←Var([],s)\nVar(x++y,s2)\n={- monad laws - }\ndo(x,)←hState1(hND+f((⇔)( local2global k)))t\nVar(x++[],s)\n={- right unit of(++)-}\ndo(x,)←hState1(hND+f((⇔)( local2global k)))t\nVar(x,s)\n={- monad laws - }\ndo(x,)←do{(x,)←hState1(hND+f((⇔)( local2global k)))t;/u1D702(x,s)}\nVar(x,s)\n={- deriviation in reverse - }\ndo(x,)←hState1(hND+f((⇔)( local2global(Op(Inl(Put t k)))))) s\nVar(x,s)\ncaset=Op(Inr(Inl(Or p q)))\nhState1(hND+f((⇔)( local2global(Op(Inr(Inl(Or p q))))))) s\n={- definition of local2global -}\nhState1(hND+f((⇔)( Op(Inr(Inl(Or(local2global p)(local2global q))))))) s\n={- definition of(⇔)-}\nhState1(hND+f(Op(Inl(Or((⇔)( local2global p))((⇔)( local2global q)))))) s\n={- definition of hND+f-}\nhState1(liftM2(++)( hND+f((⇔)( local2global p)))(hND+f((⇔)( local2global q))))s\n={- definition of liftM2 -}\nhState1(dox←hND+f((⇔)( local2global p))\ny←hND+f((⇔)( local2global q))\nVar(x++y)\n)s\n={- Lemma 4-}\ndo(x,s1)←hState1(hND+f((⇔)( local2global p)))s\n(y,s2)←hState1(hND+f((⇔)( local2global q)))s1\nhState1(Var(x++y))s2\n={- induction hypothesis - }54\ndo(x,s1)←do{(x,)←hState1(hND+f((⇔)( local2global p)))s;/u1D702(x,s)}\n(y,s2)←do{(y,)←hState1(hND+f((⇔)( local2global q)))s1;/u1D702(y,s1)}\nhState1(Var(x++y))s2\n={- monad laws - }\ndo(x,)←hState1(hND+f((⇔)( local2global p)))s\n(y,)←hState1(hND+f((⇔)( local2global q)))s1\nhState1(Var(x++y))s\n={- definition of hState1-}\ndo(x,)←hState1(hND+f((⇔)( local2global p)))s\n(y,)←hState1(hND+f((⇔)( local2global q)))s\n/u1D702(x++y,s)\n={- monad laws - }\ndo(x,)←(\ndo(x,)←hState1(hND+f((⇔)( local2global p)))s\n(y,)←hState1(hND+f((⇔)( local2global q)))s1\n/u1D702(x++y,s)\n)\n/u1D702(x,s)\n={- derivation in reverse (similar to before) - }\ndo(x,)←hState1(hND+f((⇔)( local2global(Op(Inr(Inl(Or p q))))))) s\n/u1D702(x,s)\ncaset=Op(Inr(Inr y))\nhState1(hND+f((⇔)( local2global(Op(Inr(Inr y)))))) s\n={- definition of local2global -}\nhState1(hND+f((⇔)( Op(Inr(Inr(fmap local2global y )))))) s\n={- definition of(⇔);fmap fusion -}\nhState1(hND+f(Op(Inr(Inr(fmap((⇔)◦ local2global)y))))) s\n={- definition of hND+f;fmap fusion -}\nhState1(Op(Inr(fmap(hND+f◦(⇔)◦ local2global)y)))s\n={- definition of hState1;fmap fusion -}\nOp(fmap(($s)◦hState1◦hND+f◦(⇔)◦ local2global)y)\n={- induction hypothesis - }\nOp(fmap((> >=/u1D706(x,)→/u1D702(x,s))◦($s)◦hState1◦hND+f◦(⇔)◦ local2global)y)\n={-fmap fission; definition of (> >=)-}\ndo(x,)←Op(fmap(($s)◦hState1◦hND+f◦(⇔)◦ local2global)y)\n/u1D702(x,s)\n={- deriviation in reverse (similar to before) - }\ndo(x,)←hState1(hND+f((⇔)( local2global(Op(Inr(Inr y)))))) s\n/u1D702(x,s)\n/squaresolid55\n2.6 Auxiliary Lemmas\nThe derivations above make use of two auxliary lemmas. We pro ve them here.\nLemma 2 (Naturality of($s)).($x)◦fmap f=f◦($x)\nProof\n(($x)◦fmap f)m\n={- function application - }\n(fmap f m)x\n={- eta-expansion - }\n(fmap f(/u1D706y◦m y))x\n={- definition of fmap -}\n(/u1D706y◦f(m y))x\n={- function application - }\nf(m x)\n={- definition of◦and $ -}\n(f◦($x))m\n/squaresolid\nLemma 3.\nfmap(fmap fst)(liftM2(++)p q)=liftM2(++)( fmap(fmap fst)p)(fmap(fmap fst)q)\nProof\nfmap(fmap fst)(liftM2(++)p q)\n={- definition of liftM2 -}\nfmap(fmap fst)(do{x←p;y←q;/u1D702(x++y)})\n={- derived property for monad: fmap f(m> >=k)=m> >=fmap f◦k-}\ndo{x←p;y←q;fmap(fmap fst)(/u1D702(x++y))}\n={- definition of fmap -}\ndo{x←p;y←q;/u1D702(fmap fst(x++y))}\n={- naturality of(++)-}\ndo{x←p;y←q;/u1D702((fmap fst x)++(fmap fst y))}\n={- monad left unit law (twice) - }\ndo{x←p;x′←/u1D702(fmap fst x);y←q;y′←/u1D702(fmap fst y)/u1D702(x′++y′)}\n={- definition of fmap -}\ndo{x←fmap(fmap fst)p;y←fmap(fmap fst)q;/u1D702(x++y)}\n={- definition of liftM2 -}\nliftM2(++)( fmap(fmap fst)p)(fmap(fmap fst)q)\n/squaresolid56\nLemma 4 (Distributivity of hState1).\nhState1(p> >=k)s=hState1p s> >=/u1D706(x,s′)→hState1(k x)s′\nProof The proof proceeds by induction on p.\ncasep=Var x\nhState1(Var x> >=k)s\n={- monad law - }\nhState1(k x)s\n={- monad law - }\n/u1D702(x,s)> >=/u1D706(x,s′)→hState1(k x)s′\n={- definition of hState1-}\nhState1(Var x)s> >=/u1D706(x,s′)→hState1(k x)s′\ncasep=Op(Inl(Get p))\nhState1(Op(Inl(Get p))> >=k)s\n={- definition of(> >=)for free monad - }\nhState1(Op(fmap(> >=k)(Inl(Get p))))s\n={- definition of fmap for coproduct(:+:)-}\nhState1(Op(Inl(fmap(> >=k)(Get p))))s\n={- definition of fmap forGet-}\nhState1(Op(Inl(Get(/u1D706x→p s> >=k))))s\n={- definition of hState1-}\nhState1(p s> >=k)s\n={- induction hypothesis - }\nhState1(p s)s> >=/u1D706(x,s′)→hState1(k x)s′\n={- definition of hState1-}\nhState1(Op(Inl(Get p)))s> >=/u1D706(x,s′)→hState1(k x)s′\ncasep=Op(Inl(Put t p))\nhState1(Op(Inl(Put t p))> >=k)s\n={- definition of(> >=)for free monad - }\nhState1(Op(fmap(> >=k)(Inl(Put t p))))s\n={- definition of fmap for coproduct(:+:)-}\nhState1(Op(Inl(fmap(> >=k)(Put t p))))s\n={- definition of fmap forPut-}\nhState1(Op(Inl(Put t(p> >=k))))s\n={- definition of hState1-}\nhState1(p> >=k)t\n={- induction hypothesis - }\nhState1p t> >=/u1D706(x,s′)→hState1(k x)s′\n={- definition of hState1-}\nhState1(Op(Inl(Put t p)))s> >=/u1D706(x,s′)→hState1(k x)s′57\ncasep=Op(Inr y)\nhState1(Op(Inr y)> >=k)s\n={- definition of(> >=)for free monad - }\nhState1(Op(fmap(> >=k)(Inr y)))s\n={- definition of fmap for coproduct(:+:)-}\nhState1(Op(Inr(fmap(> >=k)y)))s\n={- definition of hState1-}\nOp(fmap(/u1D706x→hState1x s)(fmap(> >=k)y))\n={-fmap fusion -}\nOp(fmap((/u1D706x→hState1(x> >=k)s))y)\n={- induction hypothesis - }\nOp(fmap(/u1D706x→hState1x s> >=/u1D706(x′,s′)→hState1(k x′)s′)y)\n={-fmap fission -}\nOp(fmap(/u1D706x→x> >=/u1D706(x′,s′)→hState1(k x′)s′)(fmap(/u1D706x→hState1x s)y))\n={- definition of(> >=)-}\nOp((fmap(/u1D706x→hState1x s)y))> >=/u1D706(x′,s′)→hState1(k x′)s′\n={- definition of hState1-}\nOp(Inr y)s> >=/u1D706(x′,s′)→hState1(k x′)s′\n/squaresolid\nimport Data.Bitraversable (Bitraversable)\n3 Proofs for Modelling Nondeterminism with State\nIn this secton, we prove the theorems in Section 5 .\n3.1 Only Nondeterminism\nThis section proves the following theorem in Section 5.1 .\nTheorem 2. runND=hND\nProof We start with expanding the definition of runND:\nextract S◦h′\nState◦nondet2state S=hND\nBoth nondet2state SandhNDare written as folds. We use the fold fusion law fusion-\npost’ (3.3) to fuse the left-hand side. Since the right-hand side is alr eady a fold, to prove\nthe equation we just need to check the components of the fold hNDsatisfy the conditions of\nthe fold fusion, i.e., the following two equations:\n(extract S◦h′\nState)◦gen=genND\n(extract S◦h′\nState)◦alg◦fmap nondet2state S\n=algND◦fmap(extract S◦h′\nState)◦fmap nondet2state S58\nFor brevity, we omit the last common part fmap nondet2state Sof the second equa-\ntion in the following proof. Instead, we assume that the inpu t is in the codomain of\nfmap nondet2state S.\nFor the first equation, we calculate as follows:\nextract S(h′\nState(gen x))\n={- definition of gen-}\nextract S(h′\nState(append Sx pop S))\n={- definition of extract S-}\nresults◦snd$runState(h′\nState(append Sx pop S))(S[][])\n={-Lemma 6 -}\nresults◦snd$runState(h′\nStatepopS)(S([]++[ x])[])\n={- definition of(++)-}\nresults◦snd$runState(h′\nStatepopS)(S[x][])\n={-Lemma 7 -}\nresults◦snd$((),S[x][])\n={- definition of snd-}\nresults(S[x][])\n={- definition of results -}\n[x]\n={- definition of genND-}\ngenNDx\nFor the second equation, we proceed with a case analysis on th e input.\ncaseFail\nextract S(h′\nState(alg Fail))\n={- definition of alg-}\nextract S(h′\nStatepopS)\n={- definition of extract S-}\nresults◦snd$runState(h′\nStatepopS)(S[][])\n={-Lemma 7 -}\nresults◦snd$((),S[][])\n={- evaluation of results ,snd-}\n[]\n={- definition of algND-}\nalgNDFail\n={- definition of fmap -}\n(algND◦fmap(extract S◦h′\nState))Fail\ncaseOr p q\nextract S(h′\nState(alg(Or p q)))\n={- definition of alg-}\nextract S(h′\nState(push Sq p))\n={- definition of extract -}\nresults◦snd$runState(h′\nState(push Sq p))(S[][])59\n={-Lemma 9 -}\nresults◦snd$runState(h′\nStatep)(S[][q])\n={-Lemma 5 -}\nresults◦snd$runState(h′\nStatepopS)(S([]++ extract S(h′\nStatep))[q])\n={- definition of(++)-}\nresults◦snd$runState(h′\nStatepopS)(S(extract S(h′\nStatep))[q])\n={-Lemma 8 -}\nresults◦snd$runState(h′\nStateq)(S(extract S(h′\nStatep))[])\n={-Lemma 5 -}\nresults◦snd$runState(h′\nStatepopS)(S(extract S(h′\nStatep)++extract S(h′\nStateq))[])\n={-Lemma 7 -}\nresults◦snd$((),S(extract S(h′\nStatep)++extract S(h′\nStateq))[])\n={- evaluation of results,snd-}\nextract S(h′\nStatep)++extract S(h′\nStateq)\n={- definition of algND-}\nalgND(Or((extract S◦h′\nState)p)((extract S◦h′\nState)q))\n={- definition of fmap -}\n(algND◦fmap(extract S◦h′\nState))(Or p q)\n/squaresolid\nIn the above proof we have used several lemmas. Now we prove th em.\nLemma 5 (pop-extract) .\nrunState(h′\nStatep)(S xs stack)=runState(h′\nStatepopS)(S(xs++extract S(h′\nStatep))stack)\nholds for all p in the codomain of the function nondet2state S.\nProof We prove this lemma by structural induction on p::Free(State F(S a))(). For\neach inductive case of p, we not only assume this lemma holds for its sub-terms (this i s the\nstandard induction hypothesis), but also assume Theorem 2 holds for pand its sub-terms.\nThis is sound because in the proof of Theorem 2 , for(extract S◦h′\nState◦nondet2state S)p=\nhNDp, we only apply Lemma 5 to the sub-terms of p, which is already included in the\ninduction hypothesis so there is no circular argument.\nSince we assume Theorem 2 holds for pand its sub-terms, we can use several useful\nproperties proved in the sub-cases of the proof of Theorem 2 . We list them here for easy\nreference:\n•extract-gen: extract S◦h′\nState◦gen=/u1D702\n•extract-alg1: extract S(h′\nState(alg Fail))=[]\n•extract-alg2: extract S(h′\nState(alg(Or p q)))=extract S(h′\nStatep)++extract S(h′\nStateq)\nWe proceed by structural induction on p. Note that for all pin the codomain of\nnondet2state S, it is either generated by the genor the algofnondet2state S. Thus, we\nonly need to prove the following two equations where p=gen x orp=alg x andxis in the\ncodomain of fmap nondet2state S.60\n1.runState(h′\nState(gen x))(S xs stack)=runState(h′\nStatepopS)(S(xs+\n+extract S(h′\nState(gen x)))stack)\n2.runState(h′\nState(alg x))(S xs stack)=runState(h′\nStatepopS)(S(xs+\n+extract S(h′\nState(alg x)))stack)\nFor the case p=gen x, we calculate as follows:\nrunState(h′\nState(gen x))(S xs stack)\n={- definition of gen-}\nrunState(h′\nState(append Sx pop S))(S xs stack)\n={-Lemma 6 -}\nrunState(h′\nStatepopS)(S(xs++[x])stack)\n={- definition of /u1D702-}\nrunState(h′\nStatepopS)(S(xs++/u1D702x)stack)\n={- extract-gen - }\nrunState(h′\nStatepopS)(S(xs++extract S(h′\nState(gen x)))stack)\nFor the case p=alg x, we proceed with a case analysis on x.\ncaseFail\nrunState(h′\nState(alg Fail))(S xs stack)\n={- definition of alg-}\nrunState(h′\nState(popS))(S xs stack)\n={- definition of[]-}\nrunState(h′\nStatepopS)(S(xs++[]) stack)\n={- extract-alg1 - }\nrunState(h′\nStatepopS)(S(xs++extract S(h′\nState(alg Fail)))stack)\ncaseOr p 1p2\nrunState(h′\nState(alg(Or p 1p2)))(S xs stack)\n={- definition of alg-}\nrunState(h′\nState(push Sp2p1))(S xs stack)\n={-Lemma 9 -}\nrunState(h′\nStatep1)(S xs(p2:stack))\n={- induction hypothesis - }\nrunState(h′\nStatepopS)(S(xs++extract S(h′\nStatep1))(p2:stack))\n={-Lemma 8 -}\nrunState(h′\nStatep2)(S(xs++extract S(h′\nStatep1))stack)\n={- induction hypothesis - }\nrunState(h′\nStatepopS)(S(xs++extract S(h′\nStatep1)++extract S(h′\nStatep2))stack)\n={- extract-alg2 - }\nrunState(h′\nStatepopS)(S(xs++h′\nState(alg(Or p 1p2)))stack)\n/squaresolid\nThe following four lemmas characterise the behaviours of st ack operations.\nLemma 6 (evaluation-append) .61\nrunState(h′\nState(append Sx p))(S xs stack)=runState(h′\nStatep)(S(xs++[x])stack)\nProof\nrunState(h′\nState(append Sx p))(S xs stack)\n={- definition of append S-}\nrunState(h��\nState(get> >=/u1D706(S xs stack)→put(S(xs++[x])stack)> >p))(S xs stack)\n={- definition of get-}\nrunState(h′\nState(Op(Get/u1D702)> >=/u1D706(S xs stack)→put(S(xs++[x])stack)> >p))(S xs stack)\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunState(h′\nState(Op(Get(/u1D706(S xs stack)→put(S(xs++[x])stack)> >p))))( S xs stack)\n={- definition of h′\nState-}\nrunState(State(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→put(S(xs++[x])stack)> >p)s))s))\n(S xs stack)\n={- definition of runState-}\n(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→put(S(xs++[x])stack)> >p)s))s)(S xs stack)\n={- function application - }\nrunState(h′\nState((/u1D706(S xs stack)→put(S(xs++[x])stack)> >p)(S xs stack)))(S xs stack)\n={- function application - }\nrunState(h′\nState(put(S(xs++[x])stack)> >p))(S xs stack)\n={- definition of put-}\nrunState(h′\nState(Op(Put(S(xs++[x])stack)(/u1D702()))> >p))(S xs stack)\n={- definition of(> >)for free monad and Law 2.3: return-bind - }\nrunState(h′\nState(Op(Put(S(xs++[x])stack)p)))(S xs stack)\n={- definition of h′\nState-}\nrunState(State(/u1D706s→runState(h′\nStatep)(S(xs++[x])stack)))(S xs stack)\n={- definition of runState-}\n(/u1D706s→runState(h′\nStatep)(S(xs++[x])stack))(S xs stack)\n={- function application - }\nrunState(h′\nStatep)(S(xs++[x])stack)\n/squaresolid\nLemma 7 (evaluation-pop1) .\nrunState(h′\nStatepopS)(S xs[])=((),S xs[])\nProof\nrunState(h′\nStatepopS)(S xs[])\n={- definition of popS-}\nrunState(h′\nState(get> >=/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op))(S xs[])\n={- definition of get-}62\nrunState(h′\nState(Op(Get/u1D702)> >=/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op))(S xs[])\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunState(h′\nState(Op(Get(/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op))))( S xs[])\n={- definition of h′\nState-}\nrunState(State(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op)s))s))(S xs[])\n={- definition of runState-}\n(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op)s))s)(S xs[])\n={- function application - }\nrunState(h′\nState((/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op)(S xs[])))( S xs[])\n={- function application, case-analysis - }\nrunState(h′\nState(/u1D702()))( S xs[])\n={- definition of h′\nState-}\nrunState(State(/u1D706s→((),s)))(S xs[])\n={- definition of runState, function application - }\n((),S xs[])\n/squaresolid\nLemma 8 (evaluation-pop2) .\nrunState(h′\nStatepopS)(S xs(q:stack))=runState(h′\nStateq)(S xs stack)\nProof\nrunState(h′\nStatepopS)(S xs(q:stack))\n={- definition of popS-}\nrunState(h′\nState(get> >=/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op))(S xs(q:stack))\n={- definition of get-}\nrunState(h′\nState(Op(Get/u1D702)> >=/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op))(S xs(q:stack))\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunState(h′\nState(Op(Get(/u1D706(S xs stack)→\ncasestack of[] → /u1D702()63\nop:ps→doput(S xs ps);op))))( S xs(q:stack))\n={- definition of h′\nState-}\nrunState(State(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op)s))s))(S xs(q:stack))\n={- definition of runState-}\n(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op)s))s)(S xs(q:stack))\n={- function application - }\nrunState(h′\nState((/u1D706(S xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(S xs ps);op)(S xs(q:stack))))( S xs(q:stack))\n={- function application, case-analysis - }\nrunState(h′\nState(put(S xs stack)> >q))(S xs(q:stack))\n={- definition of put-}\nrunState(h′\nState(Op(Put(S xs stack)(/u1D702()))> >q))(S xs(q:stack))\n={- definition of(> >)for free monad and Law 2.3: return-bind - }\nrunState(h′\nState(Op(Put(S xs stack)q)))(S xs(q:stack))\n={- definition of h′\nState-}\nrunState(State(/u1D706s→runState(h′\nStateq)(S xs stack)))(S xs(q:stack))\n={- definition of runState-}\n(/u1D706s→runState(h′\nStateq)(S xs stack))(S xs(q:stack))\n={- function application - }\nrunState(h′\nStateq)(S xs stack)\n/squaresolid\nLemma 9 (evaluation-push) .\nrunState(h′\nState(push Sq p))(S xs stack)=runState(h′\nStatep)(S xs(q:stack))\nProof\nrunState(h′\nState(push Sq p))(S xs stack)\n={- definition of push S-}\nrunState(h′\nState(get> >=/u1D706(S xs stack)→put(S xs(q:stack))> >p))(S xs stack)\n={- definition of get-}\nrunState(h′\nState(Op(Get/u1D702)> >=/u1D706(S xs stack)→put(S xs(q:stack))> >p))(S xs stack)\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunState(h′\nState(Op(Get(/u1D706(S xs stack)→put(S xs(q:stack))> >p))))( S xs stack)\n={- definition of h′\nState-}\nrunState(State(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→put(S xs(q:stack))> >p)s))s))\n(S xs stack)64\n={- definition of runState-}\n(/u1D706s→runState(h′\nState((/u1D706(S xs stack)→put(S xs(q:stack))> >p)s))s)(S xs stack)\n={- function application - }\nrunState(h′\nState((/u1D706(S xs stack)→put(S xs(q:stack))> >p)(S xs stack)))(S xs stack)\n={- function application - }\nrunState(h′\nState(put(S xs(q:stack))> >p))(S xs stack)\n={- definition of put-}\nrunState(h′\nState(Op(Put(S xs(q:stack))(/u1D702()))> >p))(S xs stack)\n={- definition of(> >)for free monad and Law 2.3: return-bind - }\nrunState(h′\nState(Op(Put(S xs(q:stack))p)))(S xs stack)\n={- definition of h′\nState-}\nrunState(State(/u1D706s→runState(h′\nStatep)(S xs(q:stack))))( S xs stack)\n={- definition of runState-}\n(/u1D706s→runState(h′\nStatep)(S xs(q:stack)))(S xs stack)\n={- function application - }\nrunState(h′\nStatep)(S xs(q:stack))\n/squaresolid\n3.2 Combining with Other Effects\nThis section proves the following theorem in Section 5.2 .\nTheorem 3. runND+f=hND+f\nProof The proof is very similar to that of Theorem 2 inAppendix 3.1 .\nWe start with expanding the definition of runND+f:\nextract SS◦hState◦nondet2state =hND\nWe use the fold fusion law fusion-post’ (3.3) to fuse the left-hand side. Since the right-hand\nside is already a fold, to prove the equation we just need to ch eck the components of the\nfoldhNDsatisfy the conditions of the fold fusion. The conditions ca n be splitted into the\nfollowing three equations:\n(extract SS◦hState)◦gen=genND+f\n(extract SS◦hState)◦alg◦fmap nondet2state S\n=algND+f◦fmap(extract SS◦hState)◦fmap nondet2state S\n(extract SS◦hState)◦fwd◦fmap nondet2state S\n=fwdND+f◦fmap(extract SS◦hState)◦fmap nondet2state S\nFor brevity, we omit the last common part fmap nondet2state Sof the second equa-\ntion in the following proof. Instead, we assume that the inpu t is in the codomain of\nfmap nondet2state S.\nFor the first equation, we calculate as follows:65\nextract SS(hState(gen x))\n={- definition of gen-}\nextract SS(hState(append SSx pop SS))\n={- definition of extract SS-}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(append SSx pop SS))(SS[][])\n={-Lemma 11 -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hStatepopSS)(SS([]++[ x])[])\n={- definition of(++)-}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hStatepopSS)(SS[x][])\n={-Lemma 12 -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht/u1D702((),SS[x][])\n={- evaluation of snd,results SS-}\n/u1D702[x]\n={- definition of /u1D702for free monad - }\nVar[x]\n={- definition of /u1D702[]-}\n(Var◦/u1D702)x\n={- definition of genND+f-}\ngenND+fx\nFor the second equation, we proceed with a case analysis on th e input.\ncaseFail\nextract SS(hState(alg Fail))\n={- definition of alg-}\nextract SS(hStatepopSS)\n={- definition of extract SS-}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hStatepopSS)(SS[][])\n={-Lemma 12 -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht/u1D702((),SS[][])\n={- evaluation of snd,results SS-}\n/u1D702[]\n={- definition of /u1D702for free monad - }\nVar[]\n={- definition of algND+f-}\nalgND+fFail\n={- definition of fmap -}\n(algND+f◦fmap(extract SS◦hState))Fail\ncaseInl(Or p q)\nextract SS(hState(alg(Or p q)))\n={- definition of alg-}\nextract SS(hState(push SSq p))\n={- definition of extract SS-}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(push SSq p))(SS[][])\n={-Lemma 14 -}66\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hStatep)(SS[][q])\n={-Lemma 10 -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht\ndo{p′←extract SS(hStatep);runStateT(hStatepopSS)(SS([]++ p′)[q])}\n={- definition of(++)-}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htdo{p′←extract SS(hStatep);runStateT(hStatepopSS)(SS p′[q])}\n={-Lemma 13 -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htdo{p′←extract SS(hStatep);runStateT(hStateq)(SS p′[])}\n={-Lemma 10 -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htdo{p′←extract SS(hStatep);\ndo{q′←extract SS(hStateq);runStateT(hStatepopSS)(SS(p′++q′)[])}}\n={- Law ( 2.5) fordo-notation - }\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htdo{p′←extract SS(hStatep);q′←extract SS(hStateq);\nrunStateT(hStatepopSS)(SS(p′++q′)[])}\n={-Lemma 12 -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht\ndo{p′←extract SS(hStatep);q′←extract SS(hStateq);/u1D702((),SS(p′++q′)[])}\n={- evaluation of snd,results SS-}\ndo{p′←extract SS(hStatep);q′←extract SS(hStateq);/u1D702(p′++q′)}\n={- definition of liftM2 -}\nliftM2(++)(( extract SS◦hState)p)((extract SS◦hState)q)\n={- definition of algND+f-}\nalgND+f(Or((extract SS◦hState)p)((extract SS◦hState)q))\n={- definition of fmap -}\n(algND+f◦fmap(extract SS◦hState))(Or p q)\nFor the last equation, we calculate as follows:\nextract SS(hState(fwd y))\n={- definition of fwd-}\nextract SS(hState(Op(Inr y)))\n={- definition of hState-}\nextract SS(StateT $/u1D706s→Op$fmap(/u1D706k→runStateT k s)(fmap h Statey))\n={- definition of extract SS-}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(StateT $/u1D706s→Op$fmap(/u1D706k→runStateT k s)(fmap h Statey))(SS[][])\n={- definition of runStateT -}\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht(/u1D706s→Op$fmap(/u1D706k→runStateT k s)(fmap h Statey))(SS[][])\n={- function application - }\nresults SS◦snd/a\\}bracketle{t$/a\\}bracketri}htOp$fmap(/u1D706k→runStateT k(SS[][]))( fmap h Statey))\n={- definition of/a\\}bracketle{t$/a\\}bracketri}ht-}\nOp(fmap(/u1D706k→results SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT k(SS[][]))( fmap h Statey))\n={- definition of fwdND+f-}\nfwdND+f(fmap(/u1D706k→results SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT k(SS[][]))( fmap h Statey))\n={- definition of extract SS-}\nfwdND+f(fmap extract SS(fmap h Statey))67\n={- Law ( 2.2) -}\nfwdND+f(fmap(extract SS◦hState)y)\n/squaresolid\nIn the above proof we have used several lemmas. Now we prove th em.\nLemma 10 (pop-extract of SS).\nrunStateT(hStatep)(SS xs stack)\n=do{p′←extract SS(hStatep);runStateT(hStatepopSS)(SS(xs++p′)stack)}\nholds for all p in the codomain of the function nondet2state.\nProof The proof structure is similar to that of Lemma 5 . We prove this lemma by structural\ninduction on p::Free(State F(SS f a):+:f)(). For each inductive case of p, we not only\nassume this lemma holds for its sub-terms (this is the standa rd induction hypothesis), but\nalso assume Theorem 3 holds for pand its sub-terms. This is sound because in the proof\nofTheorem 3 , for(extract SS◦hState◦nondet2state)p=hND+fp, we only apply Lemma 10\nto the sub-terms of p, which is already included in the induction hypothesis so th ere is no\ncircular argument.\nSince we assume Theorem 3 holds for pand its sub-terms, we can use several useful\nproperties proved in the sub-cases of the proof of Theorem 3 . We list them here for easy\nreference:\n•extract-gen-ext: extract SS◦hState◦gen=Var◦/u1D702\n•extract-alg1-ext: extract SS(hState(alg Fail))=Var[]\n•extract-alg2-ext: extract SS(hState(alg(Or p q)))=\nliftM2(++)( extract SS(hStatep))(extract SS(hStateq))\n•extract-fwd extract SS(hState(fwd y))=fwdND+f(fmap(extract SS◦hState)y)\nWe proceed by structural induction on p. Note that for all pin the codomain of\nnondet2state , it is either generated by the gen,alg, orfwdofnondet2state . Thus, we only\nneed to prove the following three equations where xis in the codomain of fmapnondet2state S\nandp=gen x,p=alg x, and p=fwd x , respectively.\n1. runStateT(hState(gen x))(SS xs stack)\n=do{p′←extract SS(hState(gen x));runStateT(hStatepopSS)(SS(xs++p′)stack)}\n2. runStateT(hState(alg x))(SS xs stack)\n=do{p′←extract SS(hState(alg x));runStateT(hStatepopSS)(SS(xs++p′)stack)}\n3. runStateT(hState(fwd x))(SS xs stack)\n=do{p′←extract SS(hState(fwd x));runStateT(hStatepopSS)(SS(xs++p′)stack)}\nFor the case p=gen x, we calculate as follows:68\nrunStateT(hState(gen x))(SS xs stack)\n={- definition of gen-}\nrunStateT(hState(append SSx pop SS))(SS xs stack)\n={-Lemma 11 -}\nrunStateT(hStatepopSS)(SS(xs++[x])stack)\n={- definition of /u1D702[]-}\nrunStateT(hStatepopSS)(SS(xs++/u1D702x)stack)\n={- definition of Varand reformulation - }\ndo{p′←Var(/u1D702x);runStateT(hStatepopSS)(SS(xs++p′)stack)}\n={- extract-gen-ext - }\ndo{p′←extract SS(hState(gen x));runStateT(hStatepopSS)(SS(xs++p′)stack)}\nFor the case p=alg x, we proceed with a case analysis on x.\ncaseFail\nrunStateT(hState(alg Fail))(SS xs stack)\n={- definition of alg-}\nrunStateT(hStatepopSS)(SS xs stack)\n={- definition of[]-}\nrunStateT(hStatepopSS)(SS(xs++[]) stack)\n={- definition of Var-}\ndo{p′←Var[];runStateT(hStatepopSS)(SS(xs++p′)stack)}\n={- extract-alg1-ext - }\ndo{p′←extract SS(hState(alg Fail));runStateT(hStatepopSS)(SS(xs++p′)stack)}\ncaseOr p 1p2\nrunStateT(hState(alg(Or p 1p2)))(SS xs stack)\n={- definition of alg-}\nrunStateT(hState(push SSp2p1))(SS xs stack)\n={-Lemma 14 -}\nrunStateT(hStatep1)(SS xs(p2:stack))\n={- induction hypothesis: pop-extract of p1-}\ndo{p′\n1←extract SS(hStatep1);runStateT(hStatepopSS)(SS(xs++p′\n1)(p2:stack))}\n={-Lemma 13 -}\ndo{p′\n1←extract SS(hStatep1);runStateT(hStatep2)(SS(xs++p′\n1)stack)}\n={- induction hypothesis: pop-extract of p2-}\ndo{p′\n2←extract SS(hStatep2);do{p′\n1←extract SS(hStatep1);\nrunStateT(hStatepopSS)(SS(xs++p′\n1++p′\n2)stack)}}\n={- Law ( 2.5) with do-notation - }\ndo{p′\n2←extract SS(hStatep2);p′\n1←extract SS(hStatep1);\nrunStateT(hStatepopSS)(SS(xs++p′\n1++p′\n2)stack)}\n={- definition of liftM2 -}\ndo{p′←liftM2(++)( extract SS(hStatep2))(extract SS(hStatep1));\nrunStateT(hStatepopSS)(SS(xs++p′)stack)}\n={- extract-alg2-ext - }\ndo{p′←extract SS(hState(alg(Or p 1p2)));runStateT(hStatepopSS)(SS(xs++p′)stack)}69\nFor the case p=fwd x , we proceed with a case analysis on x.\nrunStateT(hState(fwd x))(SS xs stack)\n={- definition of fwd-}\nrunStateT(hState(Op(Inr x)))(SS xs stack)\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→Op(fmap(/u1D706y→runStateT(hStatey s))x)))(SS xs stack)\n={- definition of runStateT -}\nOp(fmap(/u1D706y→runStateT(hStatey(SS xs stack)))x)\n={- induction hypothesis - }\nOp(fmap(/u1D706y→do{p′←extract SS(hStatey);runStateT(hStatepopSS)(SS(xs++p′)stack)})x)\n={- definition of > >=-}\ndo{p′←Op(fmap(/u1D706y→extract SS(hStatey))x);runStateT(hStatepopSS)(SS(xs++p′)stack)}\n={- definition of fwdND+f-}\ndo{p′←fwdND+f(fmap(/u1D706y→extract SS(hStatey))x);runStateT(hStatepopSS)(SS(xs++p′)stack)}\n={- extract-fwd - }\n=do{p′←extract SS(hState(fwd x));runStateT(hStatepopSS)(SS(xs++p′)stack)}\n/squaresolid\nThe following four lemmas characterise the behaviours of st ack operations.\nLemma 11 (evaluation-append-ext) .\nrunStateT(hState(append SSx p))(SS xs stack)=runStateT(hStatep)(SS(xs++[x])stack)\nProof\nrunStateT(hState(append SSx p))(SS xs stack)\n={- definition of append SS-}\nrunStateT(hState(get> >=/u1D706(SS xs stack)→put(SS(xs++[x])stack)> >p))(SS xs stack)\n={- definition of get-}\nrunStateT(hState(Op(Inl(Get/u1D702))> >=/u1D706(SS xs stack)→put(SS(xs++[x])stack)> >p))\n(SS xs stack)\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunStateT(hState(Op(Inl(Get(/u1D706(SS xs stack)→put(SS(xs++[x])stack)> >p)))))( SS xs stack)\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→put(SS(xs++[x])stack)> >p)s))s))\n(SS xs stack)\n={- definition of runStateT -}\n(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→put(SS(xs++[x])stack)> >p)s))s)(SS xs stack)\n={- function application - }\nrunStateT(hState((/u1D706(SS xs stack)→put(SS(xs++[x])stack)> >p)(SS xs stack)))(SS xs stack)\n={- function application - }\nrunStateT(hState(put(SS(xs++[x])stack)> >p))(SS xs stack)\n={- definition of put-}70\nrunStateT(hState(Op(Inl(Put(SS(xs++[x])stack)(/u1D702())))> >p))(SS xs stack)\n={- definition of(> >)for free monad and Law 2.3: return-bind - }\nrunStateT(hState(Op(Inl(Put(SS(xs++[x])stack)p))))( SS xs stack)\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→runStateT(hStatep)(SS(xs++[x])stack)))(SS xs stack)\n={- definition of runStateT -}\n(/u1D706s→runStateT(hStatep)(SS(xs++[x])stack))(SS xs stack)\n={- function application - }\nrunStateT(hStatep)(SS(xs++[x])stack)\n/squaresolid\nLemma 12 (evaluation-pop1-ext) .\nrunStateT(hStatepopSS)(SS xs[])=/u1D702((),SS xs[])\nProof\nrunStateT(hStatepopSS)(SS xs[])\n={- definition of popSS-}\nrunStateT(hState(get> >=/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op))(SS xs[])\n={- definition of get-}\nrunStateT(hState(Op(Inl(Get/u1D702))> >=/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op))(SS xs[])\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunStateT(hState(Op(Inl(Get(/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)))))( SS xs[])\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)s))s))(SS xs[])\n={- definition of runStateT -}\n(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)s))s)(SS xs[])\n={- function application - }\nrunStateT(hState((/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)(SS xs[])))( SS xs[])\n={- function application, case-analysis - }\nrunStateT(hState(/u1D702()))( SS xs[])\n={- definition of hState-}71\nrunStateT(StateT(/u1D706s→/u1D702((),s)))(SS xs[])\n={- definition of runStateT -}\n(/u1D706s→/u1D702((),s))(SS xs[])\n={- function application - }\n((),SS xs[])\n/squaresolid\nLemma 13 (evaluation-pop2-ext) .\nrunStateT(hStatepopSS)(SS xs(q:stack))=runStateT(hStateq)(SS xs stack)\nProof\nrunStateT(hStatepopSS)(SS xs(q:stack))\n={- definition of popSS-}\nrunStateT(hState(get> >=/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op))(SS xs(q:stack))\n={- definition of get-}\nrunStateT(hState(Op(Inl(Get/u1D702))> >=/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op))(SS xs(q:stack))\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunStateT(hState(Op(Inl(Get(/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)))))( SS xs(q:stack))\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)s))s))(SS xs(q:stack))\n={- definition of runStateT -}\n(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)s))s)(SS xs(q:stack))\n={- function application - }\nrunStateT(hState((/u1D706(SS xs stack)→\ncasestack of[] → /u1D702()\nop:ps→doput(SS xs ps);op)(SS xs(q:stack))))( SS xs(q:stack))\n={- function application, case-analysis - }\nrunStateT(hState(put(SS xs stack)> >q))(SS xs(q:stack))\n={- definition of put-}\nrunStateT(hState(Op(Inl(Put(SS xs stack)(/u1D702())))> >q))(SS xs(q:stack))\n={- definition of(> >)for free monad and Law 2.3: return-bind - }\nrunStateT(hState(Op(Inl(Put(SS xs stack)q))))( SS xs(q:stack))72\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→runStateT(hStateq)(SS xs stack)))(SS xs(q:stack))\n={- definition of runStateT -}\n(/u1D706s→runStateT(hStateq)(SS xs stack))(SS xs(q:stack))\n={- function application - }\nrunStateT(hStateq)(SS xs stack)\n/squaresolid\nLemma 14 (evaluation-push-ext) .\nrunStateT(hState(push SSq p))(SS xs stack)=runStateT(hStatep)(SS xs(q:stack))\nProof\nrunStateT(hState(push SSq p))(SS xs stack)\n={- definition of push SS-}\nrunStateT(hState(get> >=/u1D706(SS xs stack)→put(SS xs(q:stack))> >p))(SS xs stack)\n={- definition of get-}\nrunStateT(hState(Op(Inl(Get/u1D702))> >=/u1D706(SS xs stack)→put(SS xs(q:stack))> >p))(SS xs stack)\n={- definition of(> >=)for free monad and Law 2.3: return-bind - }\nrunStateT(hState(Op(Inl(Get(/u1D706(SS xs stack)→put(SS xs(q:stack))> >p)))))( SS xs stack)\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→put(SS xs(q:stack))> >p)s))s))\n(SS xs stack)\n={- definition of runStateT -}\n(/u1D706s→runStateT(hState((/u1D706(SS xs stack)→put(SS xs(q:stack))> >p)s))s)(SS xs stack)\n={- function application - }\nrunStateT(hState((/u1D706(SS xs stack)→put(SS xs(q:stack))> >p)(SS xs stack)))(SS xs stack)\n={- function application - }\nrunStateT(hState(put(SS xs(q:stack))> >p))(SS xs stack)\n={- definition of put-}\nrunStateT(hState(Op(Inl(Put(SS xs(q:stack))(/u1D702())))> >p))(SS xs stack)\n={- definition of(> >)for free monad and Law 2.3: return-bind - }\nrunStateT(hState(Op(Inl(Put(SS xs(q:stack))p))))( SS xs stack)\n={- definition of hState-}\nrunStateT(StateT(/u1D706s→runStateT(hStatep)(SS xs(q:stack))))( SS xs stack)\n={- definition of runStateT -}\n(/u1D706s→runStateT(hStatep)(SS xs(q:stack)))(SS xs stack)\n={- function application - }\nrunStateT(hStatep)(SS xs(q:stack))\n/squaresolid73\n4 Proofs for Modelling Two States with One State\nIn this section, we prove the following theorem in Section 6.1 .\nTheorem 4. hStates=nest◦hState◦states2state\nProof Instead of proving it directly, we show the correctness of th e isomorphism of nest\nandflatten , and prove the following equation:\nflatten◦hStates=hState◦states2state\nIt is obvious that /u1D6FCand/u1D6FC−1form an isomorphism, i.e., /u1D6FC◦/u1D6FC−1=idand/u1D6FC−1◦/u1D6FC=\nid. We show that nest andflatten form an isomorphism by calculation. For all t::\nStateT s 1(StateT s 2(Free f))a, we show that(nest◦flatten)t=t.\n(nest◦flatten)t\n={- definition of flatten -}\nnest$StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(runStateT t s1)s2\n={- definition of nest-}\nStateT $/u1D706s1→StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(runStateT t s1)s2)(s1,s2)\n={- definition of runStateT -}\nStateT $/u1D706s1→StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}ht\n(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(runStateT t s1)s2)(s1,s2)\n={- function application - }\nStateT $/u1D706s1→StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}ht(/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(runStateT t s1)s2)\n={-Equation (2.2) -}\nStateT $/u1D706s1→StateT $/u1D706s2→(fmap(/u1D6FC−1◦/u1D6FC)(runStateT(runStateT t s1)s2))\n={-/u1D6FC−1◦/u1D6FC=id-}\nStateT $/u1D706s1→StateT $/u1D706s2→(fmap id(runStateT(runStateT t s1)s2))\n={-Equation (2.1) -}\nStateT $/u1D706s1→StateT $/u1D706s2→(runStateT(runStateT t s1)s2)\n={-/u1D702-reduction and reformulation - }\nStateT $/u1D706s1→(StateT◦runStateT)(runStateT t s1)\n={-StateT◦runStateT=id-}\nStateT $/u1D706s1→runStateT t s1\n={-/u1D702-reduction - }\nStateT $runStateT t\n={-StateT◦runStateT=id-}\nt\nFor all t::StateT(s1,s2)(Free f)a, we show that(flatten◦nest)t=t.\n(flatten◦nest)t=t\n={- definition of nest-}\nflatten $StateT $/u1D706s1→StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}htrunStateT t(s1,s2)\n={- definition of flatten -}74\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(runStateT(StateT $/u1D706s1→StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}htrunStateT t(s1,s2))s1)s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT((/u1D706s1→StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}htrunStateT t(s1,s2))s1)s2\n={- function application - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(StateT $/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}htrunStateT t(s1,s2))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht(/u1D706s2→/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}htrunStateT t(s1,s2))s2\n={- function application - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht(/u1D6FC−1/a\\}bracketle{t$/a\\}bracketri}htrunStateT t(s1,s2))\n={- definition of/a\\}bracketle{t$/a\\}bracketri}ht-}\nStateT $/u1D706(s1,s2)→fmap/u1D6FC(fmap/u1D6FC−1(runStateT t(s1,s2)))\n={-Equation (2.2) -}\nStateT $/u1D706(s1,s2)→fmap(/u1D6FC◦/u1D6FC−1)(runStateT t(s1,s2))\n={-/u1D6FC◦/u1D6FC−1=id-}\nStateT $/u1D706(s1,s2)→fmap id(runStateT t(s1,s2))\n={-Equation (2.1) -}\nStateT $/u1D706(s1,s2)→runStateT t(s1,s2)\n={-/u1D702-reduction - }\nStateT $runStateT t\n={-StateT◦runStateT=id-}\nt\nThen, we first calculate the LHS flatten◦hStatesinto one function h′\nStateswhich is defined as\nh′\nStates::Functor f⇒Free(State Fs1:+:State Fs2:+:f)a→StateT(s1,s2)(Free f)a\nh′\nStatest=StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2\nFor all t::Free(State Fs1:+:State Fs2:+:f)a, we show the equation (flatten◦hStates)t=\nh′\nStatestby the following calculation.\n(flatten◦hStates)t\n={- definition of hStates-}\n(flatten◦(/u1D706t→StateT(hState◦runStateT(hStatet))))t\n={- function application - }\nflatten(StateT(hState◦runStateT(hStatet)))\n={- definition of flatten -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(runStateT(StateT(hState◦runStateT(hStatet)))s1)s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT((hState◦runStateT(hStatet))s1)s2\n={- definition of h′\nStates-}\nh′\nStatest75\nNow we only need to show that for any input t::Free(State Fs1:+:State Fs2:+:f)a, the\nequation h′\nStatest=(hState◦states2state)tholds. Note that both sides use folds. We can\nproceed with either fold fusion, as what we have done in the pr oofs of other theorems, or\na direct structural induction on the input t. Although using fold fusion makes the proof\nsimpler than using structural induction, we opt for the latt er here to show that the our\nmethods of defining effects and translations based on algebra ic effects and handlers also\nwork well with structural induction.\ncaset=Var x\n(hState◦states2state)(Var x)\n={- definition of states2state -}\nhState(Var x)\n={- definition of hState-}\nStateT $/u1D706s→/u1D702(x,s)\n={- lets=(s1,s2)-}\nStateT $/u1D706(s1,s2)→Var(x,(s1,s2))\n={- definition of /u1D6FC-}\nStateT $/u1D706(s1,s2)→Var(/u1D6FC((x,s1),s2))\n={- definition of fmap -}\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$Var((x,s1),s2)\n={- definition of /u1D702-}\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$/u1D702((x,s1),s2)\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$(/u1D706s→/u1D702((x,s1),s))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$runStateT(StateT $/u1D706s→/u1D702((x,s1),s))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$runStateT(hState(/u1D702(x,s1)))s2\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$runStateT(hState((/u1D706s→/u1D702(x,s))s1))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$runStateT(hState(runStateT(StateT $/u1D706s→/u1D702(x,s))s1))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→fmap/u1D6FC$runStateT(hState(runStateT(hState(Var x))s1))s2\n={- definition of/a\\}bracketle{t$/a\\}bracketri}ht-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hState(Var x))s1))s2\n={- definition of h′\nStates-}\nh′\nStates(Var x)\ncaset=Op(Inl(Get k))\nInduction hypothesis: h′\nStates(k s)=(hState◦states2state)(k s)for any s.\n(hState◦states2state)(Op(Inl(Get k)))\n={- definition of states2state -}\nhState$get> >=/u1D706(s1,)→states2state(k s1)\n={- definition of get-}76\nhState$Op(Inl(Get/u1D702))> >=/u1D706(s1,)→states2state(k s1)\n={- definition of(> >=)-}\nhState(Op(Inl(Get(/u1D706(s1,)→states2state(k s1)))))\n={- definition of hState-}\nStateT $/u1D706s→runStateT((/u1D706(s1,)→hState(states2state(k s1)))s)s\n={- lets=(s1,s2)-}\nStateT $/u1D706(s1,s2)→runStateT((/u1D706(s1,)→hState(states2state(k s1)))(s1,s2))(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→runStateT(hState(states2state(k s1)))(s1,s2)\n={- induction hypothesis - }\nStateT $/u1D706(s1,s2)→runStateT(h′\nStates(k s1))(s1,s2)\n={- definition of h′\nStates-}\nStateT $/u1D706(s1,s2)→runStateT(StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(k s1))s1))s2)(s1,s2)\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(k s1))s1))s2)(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hState(k s1))s1))s2\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState((/u1D706s→runStateT(hState(k s))s)s1))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(StateT $/u1D706s→runStateT(hState(k s))s)s1))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hState(Op(Inl(Get k))))s1))s2\n={- definition of h′\nStates-}\nh′\nStates(Op(Inl(Get k)))\ncaset=Op(Inl(Put s k))\nInduction hypothesis: h′\nStatesk=(hState◦states2state)k.\n(hState◦states2state)(Op(Inl(Put s k)))\n={- definition of states2state -}\nhState$get> >=/u1D706(,s2)→put(s,s2)> >(states2state k)\n={- definition of getandput-}\nhState$Op(Inl(Get/u1D702))> >=/u1D706(,s2)→Op(Inl(Put(s,s2)(/u1D702())))> >(states2state k)\n={- definition of(> >=)and(> >)-}\nhState$Op(Inl(Get(/u1D706(,s2)→Op(Inl(Put(s,s2)(states2state k))))))\n={- definition of hState-}\nalgS(Get(/u1D706(,s2)→hState(Op(Inl(Put(s,s2)(states2state k))))))\n={- definition of algS-}\nStateT $/u1D706s′→runStateT\n((/u1D706(,s2)→hState(Op(Inl(Put(s,s2)(states2state k))))) s′)s′\n={- lets′=(s1,s2)-}77\nStateT $/u1D706(s1,s2)→runStateT\n((/u1D706(,s2)→hState(Op(Inl(Put(s,s2)(states2state k)))))( s1,s2))(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→runStateT\n(hState(Op(Inl(Put(s,s2)(states2state k)))))( s1,s2)\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→runStateT\n(StateT $/u1D706→runStateT(hState(states2state k))(s,s2))(s1,s2)\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→(/u1D706→runStateT(hState(states2state k))(s,s2))(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→runStateT(hState(states2state k))(s,s2)\n={- induction hypothesis - }\nStateT $/u1D706(s1,s2)→runStateT(h′\nStatesk)(s,s2)\n={- definition of h′\nStates-}\nStateT $/u1D706(s1,s2)→runStateT(StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatek)s1))s2)(s,s2)\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatek)s1))s2)(s,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatek)s))s2\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState((/u1D706s′→runStateT(hStatek)s)s1))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(StateT $/u1D706s′→runStateT(hStatek)s)s1))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(Op(Inl(Put s k))))s1))s2\n={- definition of h′\nStates-}\nh′\nStates(Op(Inl(Put s k)))\ncaset=Op(Inr(Inl(Get k)))\nInduction hypothesis: h′\nStates(k s)=(hState◦states2state)(k s)for any s.\n(hState◦states2state)(Op(Inr(Inl(Get k))))\n={- definition of states2state -}\nhState$get> >=/u1D706(,s2)→states2state(k s2)\n={- definition of get-}\nhState$Op(Inl(Get/u1D702))> >=/u1D706(,s2)→states2state(k s2)\n={- definition of(> >=)for free monad - }\nhState(Op(Inl(Get(/u1D706(,s2)→states2state(k s2)))))\n={- definition of hState-}78\nStateT $/u1D706s→runStateT((/u1D706(,s2)→hState(states2state(k s2)))s)s\n={- lets=(s1,s2)-}\nStateT $/u1D706(s1,s2)→runStateT((/u1D706(,s2)→hState(states2state(k s2)))(s1,s2))(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→runStateT(hState(states2state(k s2)))(s1,s2)\n={- induction hypothesis - }\nStateT $/u1D706(s1,s2)→runStateT(h′\nStates(k s2))(s1,s2)\n={- definition of h′\nStates-}\nStateT $/u1D706(s1,s2)→runStateT(StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(k s2))s1))s2)(s1,s2)\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(k s2))s1))s2)(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(k s2))s1))s2\n={- reformulation - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\n(runStateT((hState◦(/u1D706k→runStateT k s1)◦hState◦k)s2)s2)\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\n(/u1D706s→runStateT((hState◦(/u1D706k→runStateT k s1)◦hState◦k)s)s)s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(StateT $/u1D706s→runStateT((hState◦(/u1D706k→runStateT k s1)◦hState◦k)s)s)s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(Op(Inl(Get((/u1D706k→runStateT k s1)◦hState◦k))))) s2\n={- definition of fmap -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(Op$fmap(/u1D706k→runStateT k s1)(Inl(Get(hState◦k))))) s2\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState((/u1D706s→Op$fmap(/u1D706k→runStateT k s)(Inl(Get(hState◦k))))s1))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState\n(runStateT(StateT $/u1D706s→Op$fmap(/u1D706k→runStateT k s)(Inl(Get(hState◦k))))s1))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(Op(Inr(Inl(Get k))))) s1))s2\n={- definition of h′\nStates-}\nh′\nStates(Op(Inr(Inl(Get k))))\ncaset=Op(Inr(Inl(Put s k)))\nInduction hypothesis: h′\nStatesk=(hState◦states2state)k.79\n(hState◦states2state)(Op(Inr(Inl(Put s k))))\n={- definition of states2state -}\nhState$get> >=/u1D706(s1,)→put(s1,s)> >(states2state k)\n={- definition of getandput-}\nhState$Op(Inl(Get/u1D702))> >=/u1D706(s1,)→Op(Inl(Put(s1,s)(/u1D702())))> >(states2state k)\n={- definition of(> >=)and(> >)-}\nhState$Op(Inl(Get(/u1D706(s1,)→Op(Inl(Put(s1,s)(states2state k))))))\n={- definition of hState-}\nStateT $/u1D706s′→runStateT\n((/u1D706(s1,)→hState(Op(Inl(Put(s1,s)(states2state k))))) s′)s′\n={- lets′=(s1,s2)-}\nStateT $/u1D706(s1,s2)→runStateT\n((/u1D706(s1,)→hState(Op(Inl(Put(s1,s)(states2state k)))))( s1,s2))(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→runStateT\n(hState(Op(Inl(Put(s1,s)(states2state k)))))( s1,s2)\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→runStateT(StateT $\n/u1D706s′→runStateT(hState(states2state k))(s1,s))(s1,s2)\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→(/u1D706s′→runStateT(hState(states2state k))(s1,s))(s1,s2)\n={- function application - }\nStateT $/u1D706(s1,s2)→runStateT(hState(states2state k))(s1,s)\n={- induction hypothesis - }\nStateT $/u1D706(s1,s2)→runStateT(h′\nStatesk)(s1,s)\n={- definition of h′\nStates-}\nStateT $/u1D706(s1,s2)→runStateT(StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatek)s1))s2)(s1,s)\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatek)s1))s2)(s1,s)\n={- function application - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatek)s1))s\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\n(/u1D706s′→runStateT(hState(runStateT(hStatek)s1))s)s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(StateT $/u1D706s′→runStateT(hState(runStateT(hStatek)s1))s)s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(Op(Inl(Put s(runStateT(hStatek)s1))))) s2\n={- reformulation - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht80\nrunStateT(hState(Op(Inl(Put s((/u1D706k→runStateT k s1)(hStatek)))))) s2\n={- definition of fmap -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(Op$fmap(/u1D706k→runStateT k s1)(Inl(Put s(hStatek))))) s2\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT\n(hState((/u1D706s′→Op$fmap(/u1D706k→runStateT k s′)(Inl(Put s(hStatek))))s1))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(StateT $\n/u1D706s′→Op$fmap(/u1D706k→runStateT k s′)(Inl(Put s(hStatek))))s1))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(Op(Inr(Inl(Put s k))))) s1))s2\n={- definition of h′\nStates-}\nh′\nStates(Op(Inr(Inl(Put s k))))\ncaset=Op(Inr(Inr y))\nInduction hypothesis: h′\nStatesy=(hState◦states2state)y.\n(hState◦states2state)(Op(Inr(Inr y)))\n={- definition of states2state -}\nhState$Op(Inr(fmap states2state y))\n={- definition of hState-}\nfwdS(fmap(hState◦states2state)y)\n={- induction hypothesis - }\nfwdS(fmap h′\nStatesy)\n={- definition of h′\nStates-}\nfwdS(fmap(/u1D706t→StateT\n$/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2)y)\n={- definition of fwdS-}\nStateT $/u1D706s→Op$fmap(/u1D706k→runStateT k s)(fmap(/u1D706t→StateT\n$/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2)y)\n={-Equation (2.2) -}\nStateT $/u1D706s→Op(fmap((/u1D706k→runStateT k s)◦(/u1D706t→StateT\n$/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2))y)\n={- reformulation - }\nStateT $/u1D706s→Op(fmap(/u1D706t→runStateT(StateT\n$/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2)s)y)\n={- definition of runStateT -}\nStateT $/u1D706s→Op(fmap(/u1D706t→\n(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2)s)y)\n={- lets=(s1,s2)-}\nStateT $/u1D706(s1,s2)→Op(fmap(/u1D706t→\n(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2)(s1,s2))y)\n={- function application - }\nStateT $/u1D706(s1,s2)→Op(fmap(/u1D706t→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht81\nrunStateT(hState(runStateT(hStatet)s1))s2)y)\n={- reformulation - }\nStateT $/u1D706(s1,s2)→Op(fmap(/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\n◦(/u1D706k→runStateT k s2)◦hState◦(/u1D706k→runStateT k s1)◦hState)y)\n={- definition of/a\\}bracketle{t$/a\\}bracketri}ht-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nOp(fmap((/u1D706k→runStateT k s2)◦hState◦(/u1D706k→runStateT k s1)◦hState)y)\n={-Equation (2.2) -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nOp$fmap(/u1D706k→runStateT k s2)(fmap(hState◦(/u1D706k→runStateT k s1)◦hState)y)\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht(/u1D706s→Op$\nfmap(/u1D706k→runStateT k s)(fmap(hState◦(/u1D706k→runStateT k s1)◦hState)y))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(StateT $/u1D706s→Op$\nfmap(/u1D706k→runStateT k s)(fmap(hState◦(/u1D706k→runStateT k s1)◦hState)y))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT\n(hState(Op(Inr(fmap((/u1D706k→runStateT k s1)◦hState)y))))s2\n={-Equation (2.2) -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT\n(hState(Op(Inr(fmap(/u1D706k→runStateT k s1)(fmap h Statey))))) s2\n={- definition of fmap -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT\n(hState(Op$fmap(/u1D706k→runStateT k s1)(Inr(fmap h Statey))))s2\n={-/u1D6FD-expansion - }\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState\n((/u1D706s→Op$fmap(/u1D706k→runStateT k s)(Inr(fmap h Statey)))s1))s2\n={- definition of runStateT -}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(StateT $\n/u1D706s→Op$fmap(/u1D706k→runStateT k s)(Inr(fmap h Statey)))s1))s2\n={- definition of hState-}\nStateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hState(Op(Inr(Inr y))))s1))s2\n={- definition of h′\nStates-}\nh′\nStates(Op(Inr(Inr y)))\n/squaresolid\n5 Proofs for the All in One Simulation\nIn this section, we prove the correctness of the final simulat ion in Section 6.2 .\nTheorem 5. simulate=hLocal82\nProof We calculate as follows, using all our three previous theore msTheorem 1 ,\nTheorem 3 ,Theorem 4 , and an auxiliary lemma Lemma 15 .\nsimulate\n={- definition of simulate -}\nextract◦hState◦states2state◦nondet2state◦(⇔)◦ local2global\n={-Theorem 4 -}\nextract◦flatten◦hStates◦nondet2state◦(⇔)◦ local2global\n={-Lemma 15 -}\nfmap(fmap fst)◦runStateT◦hState◦extract SS◦hState◦nondet2state◦(⇔)◦ local2global\n={- definition of runND+f-}\nfmap(fmap fst)◦runStateT◦hState◦runND+f◦(⇔)◦ local2global\n={-Theorem 3 -}\nfmap(fmap fst)◦runStateT◦hState◦hND+f◦(⇔)◦ local2global\n={- definition of hGlobal -}\nhGlobal◦local2global\n={-Theorem 1 -}\nhLocal\n/squaresolid\nLemma 15. extract◦flatten◦hStates=fmap(fmap fst)◦runStateT◦hState◦extract SS◦hState\nProof As shown in Appendix 4 , we can combine flatten◦hStatesinto one function h′\nStates\ndefined as follows:\nh′\nStates::Functor f⇒Free(State Fs1:+:State Fs2:+:f)a→StateT(s1,s2)(Free f)a\nh′\nStatest=StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)s1))s2\nThen we show that for any input t::Free(State F(SS(State Fs:+:f)a):+:(State Fs:+:\nf))(), we have(extract◦h′\nStates)t=(fmap(fmap fst)◦runStateT◦hState◦extract SS◦hState)t\nvia the following calculation.\n(extract◦h′\nStates)t\n={- function application - }\nextract(h′\nStatest)\n={- definition of h′\nStates-}\nextract(StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatet)s1))s2)\n={- definition of extract -}\n/u1D706s→results SS◦fst◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(StateT $/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatet)s1))s2)(SS[][],s)\n={- definition of runStateT -}\n/u1D706s→results SS◦fst◦snd/a\\}bracketle{t$/a\\}bracketri}ht(/u1D706(s1,s2)→/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatet)s1))s2)(SS[][],s)\n={- function application - }\n/u1D706s→results SS◦fst◦snd/a\\}bracketle{t$/a\\}bracketri}ht(/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht83\nrunStateT(hState(runStateT(hStatet)(SS[][]))) s)\n={-Equation (2.2) -}\n/u1D706s→results SS◦fst◦snd◦/u1D6FC/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hState(runStateT(hStatet)(SS[][]))) s\n={-fst◦snd◦/u1D6FC=snd◦fst-}\n/u1D706s→results SS◦snd◦fst/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hState(runStateT(hStatet)(SS[][]))) s\n={-Equation (2.2) and definition of/a\\}bracketle{t$/a\\}bracketri}ht-}\n/u1D706s→fmap(results SS◦snd)◦fmap fst $\nrunStateT(hState(runStateT(hStatet)(SS[][]))) s\n={- definition of flipand reformulation - }\n/u1D706s→fmap(results SS◦snd)◦fmap fst $\nflip(runStateT◦hState)s(runStateT(hStatet)(SS[][]))\n={- reformulation - }\n/u1D706s→fmap(results SS◦snd)◦(fmap fst◦flip(runStateT◦hState)s)$\nrunStateT(hStatet)(SS[][])\n={- parametricity of free monads - }\n/u1D706s→(fmap fst◦flip(runStateT◦hState)s)◦fmap(results SS◦snd)$\nrunStateT(hStatet)(SS[][])\n={- definition of/a\\}bracketle{t$/a\\}bracketri}ht-}\n/u1D706s→(fmap fst◦flip(runStateT◦hState)s)$results SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hStatet)(SS[][])\n={- function application - }\n/u1D706s→fmap fst(runStateT(hState(results SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hStatet)(SS[][]))) s)\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(runStateT(hState(results SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hStatet)(SS[][])))) s\n={- reformulation - }\n/u1D706s→(fmap(fmap fst)◦runStateT◦hState$results SS◦snd/a\\}bracketle{t$/a\\}bracketri}ht\nrunStateT(hStatet)(SS[][])) s\n={-/u1D702-reduction - }\nfmap(fmap fst)◦runStateT◦hState$results SS◦snd/a\\}bracketle{t$/a\\}bracketri}htrunStateT(hStatet)(SS[][])\n={- definition of extract SS-}\nfmap(fmap fst)◦runStateT◦hState$extract SS(hStatet)\n={- -}\n(fmap(fmap fst)◦runStateT◦hState◦extract SS◦hState)t\nNote that in the above calculation, we use the parametricity (Reynolds, 1983; Wadler, 1989)\nof free monads which is formally stated as follows:\nfmap f◦g=g◦fmap f\nfor any g::∀a.Free F a→Free G a with two functors FandG.\n/squaresolid84\n6 Proofs for Modelling Local State with Undo\nIn this section, we prove the following theorem in Section 7 .\nTheorem 6. Given Functor f and Undo s r, the equation\nhGlobalM◦local2global M=hLocalM\nholds for all programs p ::Free(Modify Fs r:+:Nondet F:+:f)a that do not use the\noperation Op(Inl MRestore).\nThe proof structure is very similar to that in Appendix 2 . We start with the following\npreliminary fusion.\nPreliminary It is easy to see that runStateT◦hModify can be fused into a single fold defined\nas follows:\nhModify1 ::(Functor f ,Undo s r)⇒Free(Modify Fs r:+:f)a→(s→Free f(a,s))\nhModify1=fold gen S(algS#fwdS)\nwhere\ngenSx s =Var(x,s)\nalgS(MGet k) s=k s s\nalgS(MUpdate r k)s=k(s⊕r)\nalgS(MRestore r k)s=k(s⊖r)\nfwdSy s =Op(fmap($s)y)\nFor brevity, we use hModify1 to replace runStateT◦hModify in the following proofs.\n6.1 Main Proof Structure\nThe main proof structure of Theorem 6 is as follows.\nProof Both the left-hand side and the right-hand side of the equati on consist of function\ncompositions involving one or more folds. We apply fold fusi on separately on both sides to\ncontract each into a single fold:\nhGlobalM◦local2global M=fold genLHS(algS\nLHS#algND\nRHS#fwdLHS)\nhLocalM=fold genRHS(algS\nRHS#algND\nRHS#fwdRHS)\nFinally, we show that both folds are equal by showing that the ir corresponding parameters\nare equal:\ngenLHS=genRHS\nalgS\nLHS=algS\nRHS\nalgND\nLHS=algND\nRHS\nfwdLHS=fwdRHS\nWe elaborate each of these steps below. /squaresolid85\n6.2 Fusing the Right-Hand Side\nWe calculate as follows:\nhLocalM\n={- definition - }\nhL◦hModify1\nwith\nhL::(Functor f ,Functor g)\n⇒g(Free(Nondet F:+:f)(a,s))→ g(Free f[a])\nhL=fmap(fmap(fmap fst)◦hND+f)\n={- definition of hModify1 -}\nhL◦fold gen S(algS#fwdS)\n={- fold fusion-post (Equation 3.2) -}\nfold genRHS(algS\nRHS#algND\nRHS#fwdRHS)\nThis last step is valid provided that the fusion conditions a re satisfied:\nhL◦genS=genRHS (1)\nhL◦(algS#fwdS)=(algS\nRHS#algND\nRHS#fwdRHS)◦fmap h L(2)\nFor the first fusion condition (1), we define genRHSas follows\ngenRHS::Functor f⇒a→(s→Free f[a])\ngenRHSx=/u1D706s→Var[x]\nWe show that (1)is satisfied by the following calculation.\nhL(genSx)\n={- definition of genS-}\nhL(/u1D706s→Var(x,s))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→Var(x,s))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(Var(x,s)))\n={- definition of hND+f-}\n/u1D706s→fmap(fmap fst)(Var[(x,s)])\n={- definition of fmap (twice) -}\n/u1D706s→Var[x]\n={- definition of genRHS-}\n=genRHSx\nBy a straightforward case analysis on the two cases InlandInr, the second fusion condition\n(2)decomposes into two separate conditions:\nhL◦algS=algS\nRHS◦fmap h L(3)\nhL◦fwdS◦Inl=algND\nRHS◦fmap h L(4)\nhL◦fwdS◦Inr=fwdRHS◦fmap h L(5)\nFor the subcondition (3), we define algS\nRHSas follows.86\nalgS\nRHS::Undo s r⇒Modify Fs r(s→p)→( s→p)\nalgS\nRHS(MGet k)=/u1D706s→k s s\nalgS\nRHS(MUpdate r k)=/u1D706s→k(s⊕r)\nalgS\nRHS(MRestore r k)=/u1D706s→k(s⊕r)\nWe prove its correctness by a case analysis on the shape of inp utt::State Fs(s→\nFree(Nondet F:+:f)(a,s)).\ncaset=Get k\nhL(algS(Get k))\n={- definition of algS-}\nhL(/u1D706s→k s s)\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→k s s)\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(k s s))\n={- beta-expansion (twice) - }\n=/u1D706s→(/u1D706s1s2→fmap(fmap fst)(hND+f(k s2s1)))s s\n={- definition of fmap (twice) -}\n=/u1D706s→(fmap(fmap(fmap(fmap fst)◦hND+f))(/u1D706s1s2→k s2s1))s s\n={- eta-expansion of k-}\n=/u1D706s→(fmap(fmap(fmap(fmap fst)◦hND+f))k)s s\n={- definition of algRHS -}\n=algS\nRHS(Get(fmap(fmap(fmap(fmap fst)◦hND+f))k))\n={- definition of fmap -}\n=algS\nRHS(fmap(fmap(fmap(fmap fst)◦hND+f))(Get k))\n={- definition of hL-}\n=algS\nRHS(fmap h L(Get k))\ncaset=MUpdate r k\nhL(algS(MUpdate r k))\n={- definition of algS-}\nhL(/u1D706s→k(s⊕r))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→k(s⊕r))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(k(s⊕r)))\n={- beta-expansion - }\n=/u1D706s→(/u1D706s1→fmap(fmap fst)(hND+f(k s1)))(s⊕r)\n={- definition of fmap -}\n=/u1D706s→(fmap(fmap(fmap fst)◦hND+f)(/u1D706s1→k s1))(s⊕r)\n={- eta-expansion of k-}\n=/u1D706s→(fmap(fmap(fmap fst)◦hND+f)k)(s⊕r)\n={- definition of algS\nRHS-}\n=algS\nRHS(MUpdate r(fmap(fmap(fmap fst)◦hND+f)k))87\n={- definition of fmap -}\n=algS\nRHS(fmap(fmap(fmap fst)◦hND+f))(MUpdate r k))\n={- definition of hL-}\n=algS\nRHS(fmap h L(MUpdate r k))\ncaset=MRestore r k\nhL(algS(MRestore r k))\n={- definition of algS-}\nhL(/u1D706s→k(s⊖r))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→k(s⊖r))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(k(s⊖r)))\n={- beta-expansion - }\n=/u1D706s→(/u1D706s1→fmap(fmap fst)(hND+f(k s1)))(s⊖r)\n={- definition of fmap -}\n=/u1D706s→(fmap(fmap(fmap fst)◦hND+f)(/u1D706s1→k s1))(s⊖r)\n={- eta-expansion of k-}\n=/u1D706s→(fmap(fmap(fmap fst)◦hND+f)k)(s⊖r)\n={- definition of algS\nRHS-}\n=algS\nRHS(MRestore r(fmap(fmap(fmap fst)◦hND+f)k))\n={- definition of fmap -}\n=algS\nRHS(fmap(fmap(fmap fst)◦hND+f))(MRestore r k))\n={- definition of hL-}\n=algS\nRHS(fmap h L(MRestore r k))\nFor the subcondition (4), we define algND\nRHSas follows.\nalgND\nRHS::Functor f⇒Nondet F(s→Free f[a])→( s→Free f[a])\nalgND\nRHSFail=/u1D706s→Var[]\nalgND\nRHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)\nTo show its correctness, given op::Nondet F(s→Free(Nondet F:+:f)(a,s))with\nFunctor f , we calculate:\nhL(fwdS(Inl op))\n={- definition of fwdS-}\nhL(/u1D706s→Op(fmap($s)(Inl op)))\n={- definition of fmap -}\nhL(/u1D706s→Op(Inl(fmap($s)op)))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→Op(Inl(fmap($s)op)))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(Op(Inl(fmap($s)op))))\n={- definition of hND+f-}\n/u1D706s→fmap(fmap fst)(algND+f(fmap h ND+f(fmap($s)op)))\nWe proceed by a case analysis on op:88\ncaseop=Fail\n/u1D706s→fmap(fmap fst)(algND+f(fmap h ND+f(fmap($s)Fail)))\n={- defintion of fmap (twice) -}\n/u1D706s→fmap(fmap fst)(algND+fFail)\n={- definition of algND+f-}\n/u1D706s→fmap(fmap fst)(Var[])\n={- definition of fmap (twice) -}\n/u1D706s→Var[]\n={- definition of algND\nRHS-}\nalgND\nRHSFail\n={-definition of fmap -}\nalgND\nRHS(fmap h Lfail)\ncaseop=Or p q\n/u1D706s→fmap(fmap fst)(algND+f(fmap h ND+f(fmap($s)(Or p q))))\n={- defintion of fmap (twice) -}\n/u1D706s→fmap(fmap fst)(algND+f(Or(hND+f(p s))(hND+f(q s))))\n={- definition of algND+f-}\n/u1D706s→fmap(fmap fst)(liftM2(++)( hND+f(p s))(hND+f(q s)))\n={- Lemma 3-}\n/u1D706s→liftM2(++)( fmap(fmap fst)(hND+f(p s)))(fmap(fmap fst)(hND+f(q s)))\n={- definition of algND\nRHS-}\nalgND\nRHS(Or(fmap(fmap fst)◦hND+f◦p)(fmap(fmap fst)◦hND+f◦q))\n={- defintion of fmap (twice) -}\nalgND\nRHS(fmap(fmap(fmap(fmap fst)◦hND+f))(Or p q))\n={- defintion of hL-}\nalgND\nRHS(fmap h L(Or p q))\nFor the last subcondition (5), we define fwdRHSas follows.\nfwdRHS::Functor f⇒f(s→Free f[a])→( s→Free f[a])\nfwdRHSop=/u1D706s→Op(fmap($s)op)\nTo show its correctness, given input op::f(s→Free(Nondet F:+:f)(a,s)), we calculate:\nhL(fwdS(Inr op))\n={- definition of fwdS-}\nhL(/u1D706s→Op(fmap($s)(Inr op)))\n={- definition of fmap -}\nhL(/u1D706s→Op(Inr(fmap($s)op)))\n={- definition of hL-}\nfmap(fmap(fmap fst)◦hND+f)(/u1D706s→Op(Inr(fmap($s)op)))\n={- definition of fmap -}\n/u1D706s→fmap(fmap fst)(hND+f(Op(Inr(fmap($s)op))))\n={- definition of hND+f-}\n/u1D706s→fmap(fmap fst)(fwdND+f(fmap h ND+f(fmap($s)op)))89\n={- definition of fwdND+f-}\n/u1D706s→fmap(fmap fst)(Op(fmap h ND+f(fmap($s)op)))\n={- definition of fmap -}\n/u1D706s→Op(fmap(fmap(fmap fst))(fmap h ND+f(fmap($s)op)))\n={-fmap fusion -}\n/u1D706s→Op(fmap(fmap(fmap fst)◦hND+f)(fmap($s)op))\n={- definition of hL-}|\n/u1D706s→Op(hL(fmap($s)op))\n={-Lemma 2 -}\n/u1D706s→Op(fmap(($s)◦hL)op)\n={-fmap fusion -}\n/u1D706s→Op(fmap($s)(fmap h Lop))\n={- definition of fwdRHS-}\nfwdRHS(fmap h Lop)\n6.3 Fusing the Left-Hand Side\nWe calculate as follows:\nhGlobalM◦local2global M\n={- definition of local2global M-}\nhGlobalM◦fold Var alg\nwhere\nalg(Inl(MUpdate r k))=(update r /talloblongside(restore r))> >k\nalg p =Op p\n={- fold fusion-post’ (Equation 3.3) -}\nfold genLHS(algS\nLHS#algND\nLHS#fwdLHS)\nThis last step is valid provided that the fusion conditions a re satisfied:\nhGlobalM◦Var=genLHS(1)\nhGlobalM◦alg◦fmap local2global M\n=(algS\nLHS#algND\nLHS#fwdLHS)◦fmap h GlobalM◦fmap local2global M(2)\nThe first subcondition (1)is met by\ngenLHS::Functor f⇒a→(s→Free f[a])\ngenLHSx=/u1D706s→Var[x]\nas established in the following calculation:\nhGlobalM(Var x)\n={- definition of hGlobalM -}\nfmap(fmap fst)(hModify1(hND+f((⇔)( Var x))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hModify1(hND+f(Var x)))\n={- definition of hND+f-}90\nfmap(fmap fst)(hModify1(Var[x]))\n={- definition of hModify1 -}\nfmap(fmap fst)(/u1D706s→Var([x],s))\n={- definition of fmap (twice) -}\n/u1D706s→Var[x]\n={- definition of genLHS-}\ngenLHSx\nWe can split the second fusion condition (2)in three subconditions:\nhGlobalM◦alg◦Inl◦fmap local2global M=algS\nLHS◦fmap h GlobalM◦fmap local2global M(3)\nhGlobalM◦alg◦Inr◦Inl◦fmap local2global M=algND\nLHS◦fmap h GlobalM◦fmap local2global M(4)\nhGlobalM◦alg◦Inr◦Inr◦fmap local2global M=fwdLHS◦fmap h GlobalM◦fmap local2global M(5)\nFor brevity, we omit the last common part fmap local2global Mof these equations\nin the following proofs. Instead, we assume that the input is in the codomain of\nfmap local2global M. Also, we use the condition in Theorem 6 that the input program\ndoes not use the restore operation.\nFor the first subcondition (3), we can define algS\nLHSas follows.\nalgS\nLHS::(Functor f ,Undo s r)⇒Modify Fs r(s→Free f[a])→( s→Free f[a])\nalgS\nLHS(MGet k)=/u1D706s→k s s\nalgS\nLHS(MUpdate r k)=/u1D706s→k(s⊕r)\nalgS\nLHS(MRestore r k)=/u1D706s→k(s⊖r)\nWe prove it by a case analysis on the shape of input op::Modify Fs r(Free(Modify Fs r:+:\nNondet F:+:f)a). Note that we only need to consider the case that opis of form MGet k or\nMUpdate r k where restore is also not used in the continuation k.\ncaseop=MGet k\nhGlobalM(alg(Inl(MGet k)))\n={- definition of alg-}\nhGlobalM(Op(Inl(MGet k)))\n={- definition of hGlobalM -}\nfmap(fmap fst)(hModify1(hND+f((⇔)( Op(Inl(MGet k))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hModify1(hND+f(Op(Inr(Inl(fmap(⇔)( MGet k)))))))\n={- definition of fmap -}\nfmap(fmap fst)(hModify1(hND+f(Op(Inr(Inl(MGet((⇔)◦ k)))))))\n={- definition of hND+f-}\nfmap(fmap fst)(hModify1(Op(fmap h ND+f(Inl(MGet((⇔)◦ k))))))\n={- definition of fmap -}\nfmap(fmap fst)(hModify1(Op(Inl(MGet(hND+f◦(⇔)◦ k)))))\n={- definition of hModify1 -}\nfmap(fmap fst)(/u1D706s→(hModify1◦hND+f◦(⇔)◦ k)s s)\n={- definition of fmap -}\n/u1D706s→fmap fst((hModify1◦hND+f◦(⇔)◦ k)s s)\n={- definition of fmap -}91\n/u1D706s→((fmap(fmap fst)◦hModify1◦hND+f◦(⇔)◦ k)s s)\n={- definition of hGlobalM -}\n/u1D706s→(hGlobalM◦k)s s\n={- definition of algS\nLHS-}\nalgS\nLHS(MGet(hGlobalM◦k))\n={- definition of fmap -}\nalgS\nLHS(fmap h GlobalM(MGet k))\ncaseop=MUpdate r k From opis in the codomain of fmap local2global Mwe obtain kis in\nthe codomain of local2global M.\nhGlobalM(alg(Inl(MUpdate r k)))\n={- definition of alg-}\nhGlobalM((update r /talloblongside(restore r))> >k)\n={- definitions of side,update ,restore ,(/talloblong), and(> >)-}\nhGlobalM(Op(Inr(Inl(Or(Op(Inl(MUpdate r k)))\n(Op(Inl(MRestore r(Op(Inr(Inl Fail))))))))))\n={- definition of hGlobalM -}\nfmap(fmap fst)(hModify1(hND+f((⇔)\n(Op(Inr(Inl(Or(Op(Inl(MUpdate r k)))\n(Op(Inl(MRestore r(Op Inr((Inl Fail)))))))))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hModify1(hND+f(\n(Op(Inl(Or(Op(Inr(Inl(MUpdate r((⇔) k)))))\n(Op(Inr(Inl(MRestore r(Op(Inl Fail))))))))))))\n={- definition of hND+f-}\nfmap(fmap fst)(hModify1(\n(liftM2(++) ( Op(Inl(MUpdate r(hND+f((⇔) k)))))\n(Op(Inl(MRestore r(Var[])))))))\n={- definition of liftM2 -}\nfmap(fmap fst)(hModify1(\ndox←Op(Inl(MUpdate r(hND+f((⇔) k))))\ny←Op(Inl(MRestore r(Var[])))\nVar(x++y)\n))\n={-Lemma 17 -}\nfmap(fmap fst)(/u1D706t→\ndo(x,t1)←hModify1(Op(Inl(MUpdate r(hND+f((⇔) k))))) t\n(y,t2)←hModify1(Op(Inl(MRestore r(Var[])))) t1\nhModify1(Var(x++y))t2\n)\n={- definition of hModify1 -}\nfmap(fmap fst)\n(/u1D706t→do(x,) ← hModify1(hND+f((⇔) k))(t⊕r)\n(y,t2) ← Var([],t⊖r)\nVar(x++y,t2)92\n)\n={- monad law - }\nfmap(fmap fst)\n(/u1D706t→do(x,t1)←hModify1(hND+f((⇔) k))(t⊕r)\nVar(x++[],t1⊖r)\n)\n={- right unit of(++)-}\nfmap(fmap fst)\n(/u1D706t→do(x,t1)←hModify1(hND+f((⇔) k))(t⊕r)\nVar(x,t1⊖r)\n)\n={-Lemma 16 -}\nfmap(fmap fst)\n(/u1D706t→do(x,t⊕r)←hModify1(hND+f((⇔) k))(t⊕r)\nVar(x,(t⊕r)⊖r)\n)\n={-Equation (7.1) -}\nfmap(fmap fst)\n(/u1D706t→do(x,)←hModify1(hND+f((⇔) k))(t⊕r)\nVar(x,t)\n)\n={- definition of fmap fst -}\nfmap(fmap fst)\n(/u1D706t→dox←fmap fst(hModify1(hND+f((⇔) k))(t⊕r))\nVar(x,t)\n)\n={- definition of fmap -}\nfmap(fmap fst)\n(/u1D706t→dox←(fmap(fmap fst)(hModify1(hND+f((⇔) k))))( t⊕r)\nVar(x,t)\n)\n={- definition of fmap(fmap fst)-}\n/u1D706t→dox←(fmap(fmap fst)(hModify1(hND+f((⇔) k))))( t⊕r)\nVar x\n={- monad law - }\n/u1D706t→(fmap(fmap fst)(hModify1(hND+f((⇔) k))))( t⊕r)\n={- definition of hGlobalM -}\n/u1D706t→(hGlobalM k)(t⊕r)\n={- definition of algS\nLHS-}\nalgS\nLHS(MUpdate r(hGlobalM k))\n={- definition of fmap -}\nalgS\nLHS(fmap h GlobalM(MUpdate r k))\nFor the second subcondition (4), we can define algND\nLHSas follows.93\nalgND\nLHS::Functor f⇒Nondet F(s→Free f[a])→( s→Free f[a])\nalgND\nLHSFail=/u1D706s→Var[]\nalgND\nLHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)\nWe prove it by a case analysis on the shape of input op::Nondet F(Free(Modify Fs r:+:\nNondet F:+:f)a).\ncaseop=Fail\nhGlobalM(alg(Inr(Inl Fail)))\n={- definition of alg-}\nhGlobalM(Op(Inr(Inl Fail)))\n={- definition of hGlobalM -}\nfmap(fmap fst)(hModify1(hND+f((⇔)( Op(Inr(Inl Fail))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hModify1(hND+f(Op(Inl(fmap(⇔)Fail)))))\n={- definition of fmap -}\nfmap(fmap fst)(hModify1(hND+f(Op(Inl Fail))))\n={- definition of hND+f-}\nfmap(fmap fst)(hModify1(Var[]))\n={- definition of hModify1 -}\nfmap(fmap fst)(/u1D706s→Var([],s))\n={- definition of fmap twice and fst-}\n/u1D706s→Var[]\n={- definition of algND\nRHS-}\nalgND\nRHSFail\n={- definition of fmap -}\nalgND\nRHS(fmap h GlobalM Fail)\ncaseop=Or p q From opis in the codomain of fmap local2global Mwe obtain pandqare\nin the codomain of local2global M.\nhGlobalM(alg(Inr(Inl(Or p q))))\n={- definition of alg-}\nhGlobalM(Op(Inr(Inl(Or p q))))\n={- definition of hGlobalM -}\nfmap(fmap fst)(hModify1(hND+f((⇔)( Op(Inr(Inl(Or p q)))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hModify1(hND+f(Op(Inl(fmap(⇔)( Or p q))))))\n={- definition of fmap -}\nfmap(fmap fst)(hModify1(hND+f(Op(Inl(Or((⇔) p)((⇔) q))))))\n={- definition of hND+f-}\nfmap(fmap fst)(hModify1(liftM2(++)( hND+f((⇔) p))(hND+f((⇔) q))))\n={- definition of liftM2 -}\nfmap(fmap fst)(hModify1(dox←hND+f((⇔) p)\ny←hND+f((⇔) q)\n/u1D702(x++y)))\n={-Lemma 17 -}94\nfmap(fmap fst)(/u1D706s0→(do(x,s1)←hModify1(hND+f((⇔) p))s0\n(y,s2)←hModify1(hND+f((⇔) q))s1\nhModify1(/u1D702(x++y))s2))\n={- definition of hModify1 -}\nfmap(fmap fst)(/u1D706s0→(do(x,s1)←hModify1(hND+f((⇔) p))s0\n(y,s2)←hModify1(hND+f((⇔) q))s1\nVar(x++y,s2)))\n={-Lemma 16 -}\nfmap(fmap fst)(/u1D706s0→(do(x,s1)←do{(x,)←hModify1(hND+f((⇔) p))s0;/u1D702(x,s0)}\n(y,s2)←do{(y,)←hModify1(hND+f((⇔) q))s1;/u1D702(x,s1)}\nVar(x++y,s2)))\n={- monad laws - }\nfmap(fmap fst)(/u1D706s0→(do(x,)←hModify1(hND+f((⇔) p))s0\n(y,)←hModify1(hND+f((⇔) q))s0\nVar(x++y,s0)))\n={- definition of fmap (twice) and fst-}\n/u1D706s0→(do(x,)←hModify1(hND+f((⇔) p))s0\n(y,)←hModify1(hND+f((⇔) q))s0\nVar(x++y))\n={- definition of fmap,fstand monad laws - }\n/u1D706s0→(dox←fmap fst(hModify1(hND+f((⇔) p))s0)\ny←fmap fst(hModify1(hND+f((⇔) q))s0)\nVar(x++y))\n={- definition of fmap -}\n/u1D706s0→(dox←fmap(fmap fst)(hModify1(hND+f((⇔) p)))s0\ny←fmap(fmap fst)(hModify1(hND+f((⇔) q)))s0\nVar(x++y))\n={- definition of hGlobalM -}\n/u1D706s0→(dox←hGlobalM p s0\ny←hGlobalM q s0\nVar(x++y))\n={- definition of liftM2 -}\n/u1D706s0→liftM2(++)( hGlobalM p s0)(hGlobalM q s0)\n={- definition of algND\nLHS-}\nalgND\nLHS(Or(hGlobalM p)(hGlobalM q))\n={- definition of fmap -}\nalgND\nLHS(fmap hGobal(Or p q))\nFor the last subcondition (5), we can define fwdLHSas follows.\nfwdLHS::Functor f⇒f(s→Free f[a])→( s→Free f[a])\nfwdLHSop=/u1D706s→Op(fmap($s)op)\nWe prove it by the following calculation for input op::f(Free(Modify Fs r:+:Nondet F:+:\nf)a).95\nhGlobalM(alg(Inr(Inr op)))\n={- definition of alg-}\nhGlobalM(Op(Inr(Inr op)))\n={- definition of hGlobalM -}\nfmap(fmap fst)(hModify1(hND+f((⇔)( Op(Inr(Inr op))))))\n={- definition of(⇔)-}\nfmap(fmap fst)(hModify1(hND+f(Op(Inr(Inr(fmap(⇔)op))))))\n={- definition of hND+f-}\nfmap(fmap fst)(hModify1(Op(fmap h ND+f(Inr(fmap(⇔)op)))))\n={- definition of fmap -}\nfmap(fmap fst)(hModify1(Op(Inr(fmap h ND+f(fmap(⇔)op)))))\n={-fmap fusion -}\nfmap(fmap fst)(hModify1(Op(Inr(fmap(hND+f◦(⇔)) op))))\n={- definition of hModify1 -}\nfmap(fmap fst)(/u1D706s→Op(fmap($s)(fmap h Modify1(fmap(hND+f◦(⇔)) op))))\n={-fmap fusion -}\nfmap(fmap fst)(/u1D706s→Op(fmap($s)(fmap(hModify1◦hND+f◦(⇔)) op)))\n={- definition of fmap -}\n/u1D706s→fmap fst(Op(fmap($s)(fmap(hModify1◦hND+f◦(⇔)) op)))\n={- definition of fmap -}\n/u1D706s→Op(fmap(fmap fst)(fmap($s)(fmap(hModify1◦hND+f◦(⇔)) op)))\n={-fmap fusion -}\n/u1D706s→Op(fmap(fmap fst◦($s))(fmap(hModify1◦hND+f◦(⇔)) op)))\n={-Lemma 2 -}\n/u1D706s→Op(fmap(($s)◦fmap(fmap fst))(fmap(hModify1◦hND+f◦(⇔)) op)))\n={-fmap fission -}\n/u1D706s→Op((fmap($s)◦fmap(fmap(fmap fst)))(fmap(hModify1◦hND+f◦(⇔)) op))\n={-fmap fusion -}\n/u1D706s→Op(fmap($s)(fmap(fmap(fmap fst)◦hModify1◦hND+f◦(⇔)) op))\n={- definition of hGlobalM -}\n/u1D706s→Op(fmap($s)(fmap h GlobalM op))\n={- definition of fwdLHS-}\nfwdLHS(fmap h GlobalM op)\n6.4 Equating the Fused Sides\nWe observe that the following equations hold trivially.\ngenLHS=genRHS\nalgS\nLHS=algS\nRHS\nalgND\nLHS=algND\nRHS\nfwdLHS=fwdRHS\nTherefore, the main theorem ( Theorem 6 ) holds.96\n6.5 Key Lemma: State Restoration\nSimilar to Appendix 2 , we have a key lemma saying that local2global Mrestores the initial\nstate after a computation.\nLemma 16 (State is Restored) .\nFor any program p ::Free(Modify Fs r:+:Nondet F:+:f)a that do not use the operation\nOP(Inl MRestore), we have\nhModify1(hND+f((⇔)( local2global Mp)))s\n=do(x,)←hModify1(hND+f((⇔)( local2global Mp)))s;/u1D702(x,s)\nProof The proof follows the same structure of Lemma 1 . We proceed by induction on t.\ncaset=Var y\nhModify1(hND+f((⇔)( local2global M(Var y))))s\n={- definition of local2global M-}\nhModify1(hND+f((⇔)( Var y)))s\n={- definition of(⇔)-}\nhModify1(hND+f(Var y))s\n={- definition of hND+f-}\nhModify1(Var[y])s\n={- definition of hModify1 -}\nVar([y],s)\n={- monad law - }\ndo(x,)←Var([y],s);Var(x,s)\n={- definition of local2global M,hND+f,(⇔),hModify1 and/u1D702-}\ndo(x,)←hModify1(hND+f((⇔)( local2global M(Var y))))s;/u1D702(x,s)\ncaset=Op(Inl(MGet k))\nhModify1(hND+f((⇔)( local2global M(Op(Inl(MGet k)))))) s\n={- definition of local2global M-}\nhModify1(hND+f((⇔)( Op(Inl(MGet(local2global M◦k)))))) s\n={- definition of(⇔)-}\nhModify1(hND+f(Op(Inr(Inl(MGet((⇔)◦ local2global M◦k)))))) s\n={- definition of hND+f-}\nhModify1(Op(Inl(MGet(hND+f◦(⇔)◦ local2global M◦k))))s\n={- definition of hModify1 -}\n(hModify1◦hND+f◦(⇔)◦ local2global M◦k)s s\n={- definition of(◦)-}\n(hModify1(hND+f((⇔)( local2global M(k s))))) s\n={- induction hypothesis - }\ndo(x,)←hModify1((⇔)( hND+f(local2global M(k s))))s;/u1D702(x,s)\n={- definition of local2global M,(⇔),hND+f,hModify1 -}\ndo(x,)←hModify1(hND+f(local2global M(Op(Inl(MGet k))))) s;/u1D702(x,s)\ncaset=Op(Inr(Inl Fail))97\nhModify1(hND+f((⇔)( local2global M(Op(Inr(Inl Fail)))))) s\n={- definition of local2global M-}\nhModify1(hND+f((⇔)( Op(Inr(Inl Fail))))) s\n={- definition of(⇔)-}\nhModify1(hND+f(Op(Inl Fail)))s\n={- definition of hND+f-}\nhModify1(Var[])s\n={- definition of hModify1 -}\nVar([],s)\n={- monad law - }\ndo(x,)←Var([],s);Var(x,s)\n={- definition of local2global M,(⇔),hND+f,hModify1 -}\ndo(x,)←hModify1(hND+f((⇔)( local2global M(Op(Inr(Inl Fail)))))) s;/u1D702(x,s)\ncaset=Op(Inl(MUpdate r k))\nhModify1(hND+f((⇔)( local2global M(Op(Inl(Put t k)))))) s\n={- definition of local2global M-}\nhModify1(hND+f((⇔)(( update r /talloblongside(restore r))> >local2global Mk)))s\n={- definition of(/talloblong),update ,restore ,sideand(> >=)-}\nhModify1(hND+f((⇔)(\nOp(Inr(Inl(Or(Op(Inl(MUpdate r(local2global Mk))))\n(Op(Inl(MRestore r(Op(Inr(Inl Fail)))))))))))) s\n={- definition of(⇔)-}\nhModify1(hND+f(\nOp(Inl(Or(Op(Inr(Inl(MUpdate r((⇔)( local2global Mk))))))\n(Op(Inr(Inl(MRestore r(Op(Inl Fail)))))))))) s\n={- definition of hND+f-}\nhModify1(\nliftM2(++) ( Op(Inl(MUpdate r(hND+f((⇔)( local2global Mk))))))\n(Op(Inl(MRestore r(Var[]))))) s\n={- definition of liftM2 -}\nhModify1(do x←Op(Inl(MUpdate r(hND+f((⇔)( local2global Mk)))))\ny←Op(Inl(MRestore r(Var[])))\nVar(x++y)\n)s\n={- Lemma 17-}\ndo(x,s1)←hModify1(Op(Inl(MUpdate r(hND+f((⇔)( local2global Mk)))))) s\n(y,s2)←hModify1(Op(Inl(MRestore r(Var[])))) s1\nVar(x++y,s2)\n={- definition of hModify1 -}\ndo(x,s1)←hModify1(hND+f((⇔)( local2global Mk)))(s⊕r)\n(y,s2)←Var([],s1⊖r)\nVar(x++y,s2)\n={- monad laws - }\ndo(x,s1)←hModify1(hND+f((⇔)( local2global Mk)))(s⊕r)98\nVar(x++[],s1⊖r)\n={- right unit of(++)-}\ndo(x,s1)←hModify1(hND+f((⇔)( local2global Mk)))(s⊕r)\nVar(x,s1⊖r)\n={- induction hypothesis - }\ndo(x,s1)←do{(x,)←hModify1(hND+f((⇔)( local2global Mk)))(s⊕r);/u1D702(x,s⊕r)}\nVar(x,s1⊖r)\n={- monad laws - }\ndo(x,)←hModify1(hND+f((⇔)( local2global Mk)))(s⊕r)\nVar(x,(s⊕r)⊖r)\n={-Equation (7.1) -}\ndo(x,)←hModify1(hND+f((⇔)( local2global Mk)))(s⊕r)\nVar(x,s)\n={- monad laws - }\ndo(x,)←do{(x,)←hModify1(hND+f((⇔)( local2global Mk)))(s⊕r);/u1D702(x,s)}\nVar(x,s)\n={- deriviation in reverse - }\ndo(x,)←hModify1(hND+f((⇔)( local2global M(Op(Inl(MUpdate r k)))))) s\nVar(x,s)\ncaset=Op(Inr(Inl(Or p q)))\nhModify1(hND+f((⇔)( local2global M(Op(Inr(Inl(Or p q))))))) s\n={- definition of local2global M-}\nhModify1(hND+f((⇔)( Op(Inr(Inl(Or(local2global Mp)(local2global Mq))))))) s\n={- definition of(⇔)-}\nhModify1(hND+f(Op(Inl(Or((⇔)( local2global Mp))((⇔)( local2global Mq)))))) s\n={- definition of hND+f-}\nhModify1(liftM2(++)( hND+f((⇔)( local2global Mp)))(hND+f((⇔)( local2global Mq))))s\n={- definition of liftM2 -}\nhModify1(dox←hND+f((⇔)( local2global Mp))\ny←hND+f((⇔)( local2global Mq))\nVar(x++y)\n)s\n={- Lemma 17-}\ndo(x,s1)←hModify1(hND+f((⇔)( local2global Mp)))s\n(y,s2)←hModify1(hND+f((⇔)( local2global Mq)))s1\nhModify1(Var(x++y))s2\n={- induction hypothesis - }\ndo(x,s1)←do{(x,)←hModify1(hND+f((⇔)( local2global Mp)))s;/u1D702(x,s)}\n(y,s2)←do{(y,)←hModify1(hND+f((⇔)( local2global Mq)))s1;/u1D702(y,s1)}\nhModify1(Var(x++y))s2\n={- monad laws - }\ndo(x,)←hModify1(hND+f((⇔)( local2global Mp)))s\n(y,)←hModify1(hND+f((⇔)( local2global Mq)))s1\nhModify1(Var(x++y))s99\n={- definition of hModify1 -}\ndo(x,)←hModify1(hND+f((⇔)( local2global Mp)))s\n(y,)←hModify1(hND+f((⇔)( local2global Mq)))s\n/u1D702(x++y,s)\n={- monad laws - }\ndo(x,)←(\ndo(x,)←hModify1(hND+f((⇔)( local2global Mp)))s\n(y,)←hModify1(hND+f((⇔)( local2global Mq)))s1\n/u1D702(x++y,s)\n)\n/u1D702(x,s)\n={- derivation in reverse (similar to before) - }\ndo(x,)←hModify1(hND+f((⇔)( local2global M(Op(Inr(Inl(Or p q))))))) s\n/u1D702(x,s)\ncaset=Op(Inr(Inr y))\nhModify1(hND+f((⇔)( local2global M(Op(Inr(Inr y)))))) s\n={- definition of local2global M-}\nhModify1(hND+f((⇔)( Op(Inr(Inr(fmap local2global My)))))) s\n={- definition of(⇔);fmap fusion -}\nhModify1(hND+f(Op(Inr(Inr(fmap((⇔)◦ local2global M)y))))) s\n={- definition of hND+f;fmap fusion -}\nhModify1(Op(Inr(fmap(hND+f◦(⇔)◦ local2global M)y)))s\n={- definition of hModify1 ;fmap fusion -}\nOp(fmap(($s)◦hModify1◦hND+f◦(⇔)◦ local2global M)y)\n={- induction hypothesis - }\nOp(fmap((> >=/u1D706(x,)→/u1D702(x,s))◦($s)◦hModify1◦hND+f◦(⇔)◦ local2global M)y)\n={-fmap fission; definition of (> >=)-}\ndo(x,)←Op(fmap(($s)◦hModify1◦hND+f◦(⇔)◦ local2global M)y)\n/u1D702(x,s)\n={- deriviation in reverse (similar to before) - }\ndo(x,)←hModify1(hND+f((⇔)( local2global M(Op(Inr(Inr y)))))) s\n/u1D702(x,s)\n/squaresolid\n6.6 Auxiliary Lemmas\nThe derivations above made use of several auxliary lemmas. W e prove them here.\nLemma 17 (Distributivity of hModify1 ).\nhModify1(p> >=k)s=hModify1 p s> >=/u1D706(x,s′)→hModify1(k x)s′\nProof The proof follows the same structure of Lemma 4 . We proceed by induction on p.100\ncasep=Var x\nhModify1(Var x> >=k)s\n={- monad law - }\nhModify1(k x)s\n={- monad law - }\n/u1D702(x,s)> >=/u1D706(x,s′)→hModify1(k x)s′\n={- definition of hModify1 -}\nhModify1(Var x)s> >=/u1D706(x,s′)→hModify1(k x)s′\ncasep=Op(Inl(MGet p))\nhModify1(Op(Inl(MGet p))> >=k)s\n={- definition of(> >=)for free monad - }\nhModify1(Op(fmap(> >=k)(Inl(MGet p))))s\n={- definition of fmap for coproduct(:+:)-}\nhModify1(Op(Inl(fmap(> >=k)(MGet p))))s\n={- definition of fmap forMGet -}\nhModify1(Op(Inl(MGet(/u1D706x→p s> >=k))))s\n={- definition of hModify1 -}\nhModify1(p s> >=k)s\n={- induction hypothesis - }\nhModify1(p s)s> >=/u1D706(x,s′)→hModify1(k x)s′\n={- definition of hModify1 -}\nhModify1(Op(Inl(MGet p)))s> >=/u1D706(x,s′)→hModify1(k x)s′\ncasep=Op(Inl(MUpdate r p))\nhModify1(Op(Inl(MUpdate r p))> >=k)s\n={- definition of(> >=)for free monad - }\nhModify1(Op(fmap(> >=k)(Inl(MUpdate r p))))s\n={- definition of fmap for coproduct(:+:)-}\nhModify1(Op(Inl(fmap(> >=k)(MUpdate r p))))s\n={- definition of fmap forMUpdate -}\nhModify1(Op(Inl(MUpdate r(p> >=k))))s\n={- definition of hModify1 -}\nhModify1(p> >=k)(s⊕r)\n={- induction hypothesis - }\nhModify1 p(s⊕r)> >=/u1D706(x,s′)→hModify1(k x)s′\n={- definition of hModify1 -}\nhModify1(Op(Inl(MUpdate r p)))s> >=/u1D706(x,s′)→hModify1(k x)s′\ncasep=Op(Inl(MRestore r p))\nhModify1(Op(Inl(MRestore r p))> >=k)s\n={- definition of(> >=)for free monad - }\nhModify1(Op(fmap(> >=k)(Inl(MRestore r p))))s\n={- definition of fmap for coproduct(:+:)-}101\nhModify1(Op(Inl(fmap(> >=k)(MRestore r p))))s\n={- definition of fmap forMRestore -}\nhModify1(Op(Inl(MRestore r(p> >=k))))s\n={- definition of hModify1 -}\nhModify1(p> >=k)(s⊖r)\n={- induction hypothesis - }\nhModify1 p(s⊖r)> >=/u1D706(x,s′)→hModify1(k x)s′\n={- definition of hModify1 -}\nhModify1(Op(Inl(MRestore r p)))s> >=/u1D706(x,s′)→hModify1(k x)s′\ncasep=Op(Inr y)\nhModify1(Op(Inr y)> >=k)s\n={- definition of(> >=)for free monad - }\nhModify1(Op(fmap(> >=k)(Inr y)))s\n={- definition of fmap for coproduct(:+:)-}\nhModify1(Op(Inr(fmap(> >=k)y)))s\n={- definition of hModify1 -}\nOp(fmap(/u1D706x→hModify1 x s)(fmap(> >=k)y))\n={-fmap fusion -}\nOp(fmap((/u1D706x→hModify1(x> >=k)s))y)\n={- induction hypothesis - }\nOp(fmap(/u1D706x→hModify1 x s> >=/u1D706(x′,s′)→hModify1(k x′)s′)y)\n={-fmap fission -}\nOp(fmap(/u1D706x→x> >=/u1D706(x′,s′)→hModify1(k x′)s′)(fmap(/u1D706x→hModify1 x s)y))\n={- definition of(> >=)-}\nOp((fmap(/u1D706x→hModify1 x s)y))> >=/u1D706(x′,s′)→hModify1(k x′)s′\n={- definition of hModify1 -}\nOp(Inr y)s> >=/u1D706(x′,s′)→hModify1(k x′)s′\n/squaresolid\n7 Proofs for Modelling Local State with Trail Stack\nIn this section, we prove the following theorem in Section 8 .\nTheorem 7. Given Functor f and Undo s r, the equation\nhGlobalT=hLocalM\nholds for all programs p ::Free(Modify Fs r:+:Nondet F:+:f)a that do not use the\noperation Op(Inl MRestore).\nThe proof follows a similar structure to those in Appendix 2 andAppendix 6 .\nAs in Appendix 6 , we fuse runStateT◦hModify intohModify1 and use it instead in the\nfollowing proofs.102\n7.1 Main Proof Structure\nThe main proof structure of Theorem 7 is as follows.\nProof The left-hand side is expanded to\nhGlobalT=fmap(fmap fst◦flip run StateT(Stack[])◦ hState)◦hGlobalM◦local2trail\nBoth the left-hand side and the right-hand side of the equati on consist of function composi-\ntions involving one or more folds. We apply fold fusion separ ately on both sides to contract\neach into a single fold:\nhGlobalT=fold genLHS(algS\nLHS#algND\nRHS#fwdLHS)\nhLocalM=fold genRHS(algS\nRHS#algND\nRHS#fwdRHS)\nFinally, we show that both folds are equal by showing that the ir corresponding parameters\nare equal:\ngenLHS=genRHS\nalgS\nLHS=algS\nRHS\nalgND\nLHS=algND\nRHS\nfwdLHS=fwdRHS\nWe elaborate each of these steps below. /squaresolid\n7.2 Fusing the Right-Hand Side\nWe have already fused hLocalM inAppendix 6.2 . We just show the result here for easy\nreference.\nhLocalM=fold genRHS(algS\nRHS#algND\nRHS#fwdRHS)\nwhere\ngenRHS::Functor f⇒a→(s→Free f[a])\ngenRHSx=/u1D706s→Var[x]\nalgS\nRHS::Undo s r⇒State Fs(s→p)→( s→p)\nalgS\nRHS(MGet k)=/u1D706s→k s s\nalgS\nRHS(MUpdate r k)=/u1D706s→k(s⊕r)\nalgS\nRHS(MRestore r k)=/u1D706s→k(s⊕r)\nalgND\nRHS::Functor f⇒Nondet F(s→Free f[a])→( s→Free f[a])\nalgND\nRHSFail=/u1D706s→Var[]\nalgND\nRHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)\nfwdRHS::Functor f⇒f(s→Free f[a])→( s→Free f[a])\nfwdRHSop=/u1D706s→Op(fmap($s)op)\n7.3 Fusing the Left-Hand Side\nAs in Appendix 2 , we fuse runStateT◦hStateintohState1. For brevity, we define\nrunStack=fmap fst◦flip h State1(Stack[])103\nThe left-hand side is simplified to\nfmap runStack◦hGlobalM◦local2trail\nWe calculate as follows:\nfmap runStack◦hGlobalM◦local2trail\n={- definition of local2trail -}\nfmap runStack◦hGlobalM◦fold Var(alg1#alg2#fwd)\nwhere\nalg1(MUpdate r k)=pushStack(Left r)> >update r> >k\nalg1p =Op◦Inl$p\nalg2(Or p q)=(pushStack(Right())> >p)/talloblong(untrail> >q)\nalg2p =Op◦Inr◦Inl$p\nfwd p =Op◦Inr◦Inr◦Inr$p\nuntrail=dotop←popStack ;\ncasetopof\nNothing→/u1D702()\nJust(Right())→/u1D702()\nJust(Left r)→dorestore r ;untrail\n={- fold fusion-post’ (Equation 3.3) -}\nfold genLHS(algS\nLHS#algND\nLHS#fwdLHS)\nThis last step is valid provided that the fusion conditions a re satisfied:\nfmap runStack◦hGlobalM◦Var=genLHS(1)\nfmap runStack◦hGlobalM◦(alg1#alg2#fwd)◦fmap local2trail\n=(algS\nLHS#algND\nLHS#fwdLHS)◦fmap(fmap runStack◦hGlobalM)◦fmap local2trail(2)\nThe first subcondition (1)is met by\ngenLHS::Functor f⇒a→(s→Free f[a])\ngenLHSx=/u1D706s→Var[x]\nas established in the following calculation:\nfmap runStack $hGlobalM(Var x)\n={- definition of hGlobalM -}\nfmap runStack $fmap(fmap fst)(hModify1(hND+f((⇔)( Var x))))\n={- definition of(⇔)-}\nfmap runStack $fmap(fmap fst)(hModify1(hND+f(Var x)))\n={- definition of hND+f-}\nfmap runStack $fmap(fmap fst)(hModify1(Var[x]))\n={- definition of hModify1 -}\nfmap runStack $fmap(fmap fst)(/u1D706s→Var([x],s))\n={- definition of fmap (twice) -}\nfmap runStack $/u1D706s→Var[x]\n={- definition of fmap -}\n/u1D706s→runStack $Var[x]104\n={- definition of runStack -}\n/u1D706s→fmap fst◦flip h State1(Stack[])$Var[x]\n={- definition of hState1-}\n/u1D706s→fmap fst $(/u1D706s→Var([x],s))(Stack[])\n={- function application - }\n/u1D706s→fmap fst(Var([x],Stack[]))\n={- definition of fmap -}\n/u1D706s→Var[x]\n={- definition of genLHS-}\ngenLHSx\nWe can split the second fusion condition (2)in three subconditions:\nfmap runStack◦hGlobalM◦alg1◦fmap local2trail\n=algS\nLHS◦fmap(fmap runStack◦hGlobalM)◦fmap local2trail(3)\nfmap runStack◦hGlobalM◦hGlobalM◦alg2◦fmap local2trail\n=algND\nLHS◦fmap(fmap runStack◦hGlobalM)◦fmap local2trail(4)\nfmap runStack◦hGlobalM◦hGlobalM◦fwd◦fmap local2trail\n=fwdLHS◦fmap(fmap runStack◦hGlobalM)◦fmap local2trail(5)\nFor brevity, we omit the last common part fmaplocal2global Mof these equations.Instead,\nwe assume that the input is in the codomain of fmap local2global M.\nFor the first subcondition (3), we define algS\nLHSas follows.\nalgS\nLHS::(Functor f ,Undo s r)⇒Modify Fs r(s→Free f[a])→( s→Free f[a])\nalgS\nLHS(MGet k)=/u1D706s→k s s\nalgS\nLHS(MUpdate r k)=/u1D706s→k(s⊕r)\nalgS\nLHS(MRestore r k)=/u1D706s→k(s⊖r)\nWe prove it by a case analysis on the shape of input op::Modify Fs r(Free(Modify Fs r:+:\nNondet F:+:f)a). We use the condition in Theorem 6 that the input program does not use\ntherestore operation. We only need to consider the case that opis of form MGet k or\nMUpdate r k where restore is also not used in the continuation k.\ncaseop=MGet k In the corresponding case of Appendix 6.3 , we have calculated that\nhGlobalM(Op(Inl(MGet k)))=/u1D706s→(hGlobalM◦k)s s(★).\nfmap runStack $hGlobalM(alg1(MGet k))\n={- definition of alg1-}\nfmap runStack $hGlobalM(Op(Inl(MGet k)))\n={- Equation (★)-}\nfmap runStack $/u1D706s→(hGlobalM◦k)s s\n={- definition of fmap -}\n/u1D706s→runStack $(hGlobalM◦k)s s\n={- definition of fmap -}\n/u1D706s→(fmap runStack◦hGlobalM◦k)s s\n={- definition of algS\nLHS-}\nalgS\nLHS(MGet(fmap runStack◦hGlobalM◦k))105\n={- definition of fmap -}\nalgS\nLHS(fmap(fmap runStack◦hGlobalM)(MGet k))\ncaseop=MUpdate r k From opis in the codomain of fmap local2global Mwe obtain kis in\nthe codomain of local2global M.\nfmap runStack◦hGlobalM $alg1(MUpdate r k)\n={- definition of alg1-}\nfmap runStack◦hGlobalM $pushStack(Left r)> >update r> >k\n={- definition of pushStack -}\nfmap runStack◦hGlobalM $do\nStack xs←get\nput(Stack(Left r :xs))\nupdate r> >k\n={- definition of get,put, and update -}\nfmap runStack◦hGlobalM $\nOp◦Inr◦Inr◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inr◦Inr◦Inl$Put(Stack(Left r :xs))(\nOp◦Inl$MUpdate r k))\n={- definition of hGlobalM -}\nfmap runStack◦fmap(fmap fst)◦hModify1◦hND+f◦(⇔) $\nOp◦Inr◦Inr◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inr◦Inr◦Inl$Put(Stack(Left r :xs))(\nOp◦Inl$MUpdate r k))\n={- definition of(⇔)-}\nfmap runStack◦fmap(fmap fst)◦hModify1◦hND+f$\nOp◦Inr◦Inr◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inr◦Inr◦Inl$Put(Stack(Left r :xs))(\nOp◦Inr◦Inl$MUpdate r((⇔) k)))\n={- definition of hND+f-}\nfmap runStack◦fmap(fmap fst)◦hModify1 $\nOp◦Inr◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inr◦Inl$Put(Stack(Left r :xs))(\nOp◦Inl$MUpdate r(hND+f◦(⇔) $k)))\n={- definition of hModify1 -}\nfmap runStack◦fmap(fmap fst)$/u1D706s→\nOp◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inl$Put(Stack(Left r :xs))(\n(hModify1◦hND+f◦(⇔) $k)(s⊕r)))\n={- definition of fmap(fmap fst)-}\nfmap runStack $/u1D706s→\nOp◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inl$Put(Stack(Left r :xs))(\n(fmap(fmap fst)◦hModify1◦hND+f◦(⇔) $k)(s⊕r)))\n={- definition of fmap -}\n/u1D706s→runStack $106\nOp◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inl$Put(Stack(Left r :xs))(\n(fmap(fmap fst)◦hModify1◦hND+f◦(⇔) $k)(s⊕r)))\n={- definition of hGlobalM -}\n/u1D706s→runStack $\nOp◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inl$Put(Stack(Left r :xs))(\n(hGlobalM k)(s⊕r)))\n={- definition of runStack -}\n/u1D706s→fmap fst◦flip h State1(Stack[])$\nOp◦Inl$Get(/u1D706(Stack xs)→\nOp◦Inl$Put(Stack(Left r :xs))(\n(hGlobalM k)(s⊕r)))\n={- definition of hState1-}\n/u1D706s→fmap fst $(/u1D706t→\n(/u1D706(Stack xs)→/u1D706→\n((fmap h State1◦hGlobalM $k)(s⊕r))(Stack(Left r :xs))\n)t t\n)(Stack[])\n={- function application - }\n/u1D706s→fmap fst $\n(/u1D706(Stack xs)→/u1D706→\n((fmap h State1◦hGlobalM $k)(s⊕r))(Stack(Left r :xs))\n)(Stack[])(Stack[])\n={- function application - }\n/u1D706s→fmap fst $\n(/u1D706→\n((fmap h State1◦hGlobalM $k)(s⊕r))(Stack(Left r :[]))\n)(Stack[])\n={- function application - }\n/u1D706s→fmap fst $\n((fmap h State1◦hGlobalM $k)(s⊕r))(Stack(Left r :[]))\n={- function application - }\n/u1D706s→fmap fst $\n((fmap h State1◦hGlobalM $k)(s⊕r))(Stack(Left r :[]))\n={- definition of flipand reformulation - }\n/u1D706s→(fmap(fmap fst◦flip h State1(Stack[Left r]))◦hGlobalM $k)(s⊕r)\n={-Lemma 18 and definition of fmap andfst-}\n/u1D706s→(fmap(fmap fst◦flip h State1(Stack[]))◦ hGlobalM $k)(s⊕r)\n={- definition of runStack -}\n/u1D706s→(fmap runStack◦hGlobalM $k)(s⊕r)\n={- definition of algS\nLHS-}\nalgS\nLHS(MUpdate r(fmap runStack◦hGlobalM $k))\n={- definition of fmap -}\nalgS\nLHS(fmap(fmap runStack◦hGlobalM)(MUpdate r k))107\nFor the second subcondition (4), we can define algND\nLHSas follows.\nalgND\nLHS::Functor f⇒Nondet F(s→Free f[a])→( s→Free f[a])\nalgND\nLHSFail=/u1D706s→Var[]\nalgND\nLHS(Or p q)=/u1D706s→liftM2(++)( p s)(q s)\nWe prove it by a case analysis on the shape of input op::Nondet F(Free(Modify Fs r:+:\nNondet F:+:f)a).\ncaseop=Fail In the corresponding case of Appendix 6.3 , we have calculated that\nhGlobalM(Op(Inr(Inl Fail)))=/u1D706s→Var[](★).\nfmap runStack $hGlobalM(alg2(Fail))\n={- definition of alg2-}\nfmap runStack $hGlobalM(Op(Inr(Inl Fail)))\n={- Equation (★)-}\nfmap runStack $/u1D706s→Var[]\n={- definition of fmap -}\n/u1D706s→runStack $Var[]\n={- definition of runStack -}\n/u1D706s→fmap fst◦flip h State1(Stack[])$Var[]\n={- definition of hState1-}\n/u1D706s→fmap fst $Var([],Stack[])\n={- definition of fmap -}\n/u1D706s→Var[]\n={- definition of algND\nRHS-}\nalgND\nRHSFail\n={- definition of fmap -}\nalgND\nRHS(fmap(fmap runStack◦hGlobalM)Fail)\ncaseop=Or p q From opis in the codomain of fmap local2global Mwe obtain pandqare\nin the codomain of local2global M.\nfmap runStack◦hGlobalM $alg2(Or p q)\n={- definition of alg2-}\nfmap runStack◦hGlobalM $(pushStack(Right())> >p)/talloblong(untrail> >q)\n={- definition of(/talloblong)-}\nfmap runStack◦hGlobalM $Op◦Inr◦Inl$Or\n(pushStack(Right())> >p)(untrail> >q)\n={- definition of hGlobalM -}\nfmap runStack◦fmap(fmap fst)◦hModify1◦hND+f◦(⇔) $Op◦Inr◦Inl$Or\n(pushStack(Right())> >p)(untrail> >q)\n={- definition of(⇔)-}\nfmap runStack◦fmap(fmap fst)◦hModify1◦hND+f$Op◦Inl$Or\n(pushStack(Right())> >(⇔)p)(untrail> >(⇔)q)\n={- definition of hND+fandliftM2 -}\nfmap runStack◦fmap(fmap fst)◦hModify1 $do\nx←hND+f((⇔)( pushStack(Right())))> >hND+f((⇔) p)108\ny←hND+f((⇔) untrail)> >hND+f((⇔) q)\n/u1D702(x++y)\n={- monad law - }\nfmap runStack◦fmap(fmap fst)◦hModify1 $do\nhND+f((⇔)( pushStack(Right())))\nx←hND+f((⇔) p)\nhND+f((⇔) untrail)\ny←hND+f((⇔) q)\n/u1D702(x++y)\n={- definition of hModify1 andLemma 17 -}\nfmap runStack◦fmap(fmap fst)$/u1D706s→do\n(,s1)←hModify1(hND+f((⇔)( pushStack(Right())))) s\n(x,s2)←hModify1(hND+f((⇔) p))s1\n(,s3)←hModify1(hND+f((⇔) untrail))s2\n(y,s4)←hModify1(hND+f((⇔) q))s3\n/u1D702(x++y,s4)\n={- definition of fmap (twice) -}\nfmap runStack $/u1D706s→do\n(,s1)←hModify1(hND+f((⇔)( pushStack(Right())))) s\n(x,s2)←hModify1(hND+f((⇔) p))s1\n(,s3)←hModify1(hND+f((⇔) untrail))s2\n(y,)←hModify1(hND+f((⇔) q))s3\n/u1D702(x++y)\n={- definition of fmap andrunStack -}\n/u1D706s→fmap fst◦flip h State1(Stack[])$do\n(,s1)←hModify1(hND+f((⇔)( pushStack(Right())))) s\n(x,s2)←hModify1(hND+f((⇔) p))s1\n(,s3)←hModify1(hND+f((⇔) untrail))s2\n(y,)←hModify1(hND+f((⇔) q))s3\n/u1D702(x++y)\n={- definition of hState1andLemma 4 -}\n/u1D706s→fmap fst $(/u1D706t→do\n((,s1),t1)←hState1(hModify1(hND+f((⇔)( pushStack(Right())))) s)t\n((x,s2),t2)←hState1(hModify1(hND+f((⇔) p))s1)t1\n((,s3),t3)←hState1(hModify1(hND+f((⇔) untrail))s2)t2\n((y,),t4)←hState1(hModify1(hND+f((⇔) q))s3)t3\n/u1D702(x++y,t4))(Stack[])\n={- function application - }\n/u1D706s→fmap fst $do\n((,s1),t1)←hState1(hModify1(hND+f((⇔)( pushStack(Right())))) s)(Stack[])\n((x,s2),t2)←hState1(hModify1(hND+f((⇔) p))s1)t1\n((,s3),t3)←hState1(hModify1(hND+f((⇔) untrail))s2)t2\n((y,),t4)←hState1(hModify1(hND+f((⇔) q))s3)t3\n/u1D702(x++y,t4)\n={- definition of fmap -}109\n/u1D706s→do\n((,s1),t1)←hState1(hModify1(hND+f((⇔)( pushStack(Right())))) s)(Stack[])\n((x,s2),t2)←hState1(hModify1(hND+f((⇔) p))s1)t1\n((,s3),t3)←hState1(hModify1(hND+f((⇔) untrail))s2)t2\n((y,),)←hState1(hModify1(hND+f((⇔) q))s3)t3\n/u1D702(x++y)\n={-Lemma 20 -}\n/u1D706s→do\n((x,),)←hState1(hModify1(hND+f((⇔) p))s)(Stack[])\n((y,),)←hState1(hModify1(hND+f((⇔) q))s)(Stack[])\n/u1D702(x++y)\n={- definition of runStack -}\n/u1D706s→do\n(x,)←runStack(hModify1(hND+f((⇔) p))s)\n(y,)←runStack(hModify1(hND+f((⇔) q))s)\n/u1D702(x++y)\n={- definition of hGlobalM -}\n/u1D706s→do\nx←runStack(hGlobalM p s)\ny←runStack(hGlobalM q s)\n/u1D702(x++y)\n={- definition of fmap -}\n/u1D706s→do\nx←(fmap runStack◦hGobalM)p s\ny←(fmap runStack◦hGobalM)q s\n/u1D702(x++y))\n={- definition of liftM2 -}\n/u1D706s→liftM2(++)(( fmap runStack◦hGobalM)p s)((fmap runStack◦hGobalM)q s)\n={- definition of algND\nLHS-}\nalgND\nLHS(Or((fmap runStack◦hGobalM)p)((fmap runStack◦hGobalM)q))\n={- definition of fmap -}\nalgND\nLHS(fmap(fmap runStack◦hGobalM)(Or p q))\nFor the last subcondition (5), we can define fwdLHSas follows.\nfwdLHS::Functor f⇒f(s→Free f[a])→( s→Free f[a])\nfwdLHSop=/u1D706s→Op(fmap($s)op)\nWe prove it by the following calculation for input op::f(Free(Modify Fs r:+:\nNondet F:+:f)a). In the corresponding case of Appendix 6.3 , we have calculated that\nhGlobalM(Op(Inr(Inr op)))=/u1D706s→Op(fmap($s)(fmap h GlobalM op))(★).\nfmap runStack $hGlobalM(fwd op)\n={- definition of fwd-}\nfmap runStack $hGlobalM(Op◦Inr◦Inr◦Inr$op)\n={- Equation (★)-}\nfmap runStack $/u1D706s→Op(fmap($s)(fmap h GlobalM(Inr op)))110\n={- fmap fusion - }\nfmap runStack $/u1D706s→Op(fmap(($s)◦hGlobalM)(Inr op))\n={- reformulation - }\nfmap runStack $/u1D706s→Op(fmap(/u1D706x→hGlobalM x s)(Inr op))\n={- definition of fmap -}\n/u1D706s→runStack $Op(fmap(/u1D706x→hGlobalM x s)(Inr op))\n={- definition of runStack -}\n/u1D706s→fmap fst◦flip h State1(Stack[])$\nOp(fmap(/u1D706x→hGlobalM x s)(Inr op))\n={- definition of hState1-}\n/u1D706s→fmap fst $(/u1D706t→\nOp(fmap($t)◦fmap(hState1)$fmap(/u1D706x→hGlobalM x s)op))(Stack[])\n={- fmap fusion and reformulation - }\n/u1D706s→fmap fst $(/u1D706t→\nOp(fmap(/u1D706x→hState1(hGlobalM x s)t)op))(Stack[])\n={- function application - }\n/u1D706s→fmap fst $\nOp(fmap(/u1D706x→hState1(hGlobalM x s)(Stack[]))op)\n={- definition of fmap -}\n/u1D706s→Op(fmap(/u1D706x→fmap fst(hState1(hGlobalM x s)(Stack[]))) op)\n={- reformulation - }\n/u1D706s→Op(fmap(/u1D706x→fmap(fmap fst◦flip h State1(Stack[]))◦ hGlobalM $x s)op)\n={- reformulation - }\n/u1D706s→Op(fmap(/u1D706x→(fmap runStack◦hGlobalM $x)s)op)\n={- fmap fission - }\n/u1D706s→Op(fmap($s)(fmap(fmap runStack◦hGlobalM)op))\n={- definition of fwdLHS-}\nfwdLHS(fmap(fmap runStack◦hGlobalM)op)\n7.4 Equating the Fused Sides\nWe observe that the following equations hold trivially.\ngenLHS=genRHS\nalgS\nLHS=algS\nRHS\nalgND\nLHS=algND\nRHS\nfwdLHS=fwdRHS\nTherefore, the main theorem ( Theorem 7 ) holds.\n7.5 Lemmas\nIn this section, we prove the lemmas used in Appendix 7.3 .111\nThe following lemma shows the relationship between the stat e and trail stack. Intuitively,\nthe trail stack contains all the deltas (updates) that have n ot been restored in the program.\nPrevious elements in the trail stack do not influence the resu lt and state of programs.\nLemma 18 (Trail stack tracks state) . For t ::Stack(Either r()), s::s, and p ::\nFree(Modify Fs r:+:Nondet F:+:f)a which does not use the restore operation, we have\nhState1((hModify1◦hND+f◦(⇔))( local2trail p)s)t\n=\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s)(Stack[])\n/u1D702((x,fplus s ys),extend t ys)\nfor some ys =[Left r n,...,Left r 1]. The functions extend and fplus are defined as follows:\nextend ::Stack s→[s]→Stack s\nextend(Stack xs)ys=Stack(ys++xs)\nfplus ::Undo s r⇒s→[Either r b]→s\nfplus s ys=foldr(/u1D706(Left r)s→s⊕r)s ys\nNote that an immediate corollary of Lemma 18 is that in addition to replacing the stack\ntwith the empty stack Stack[], we can also replace it with any other stack. The following\nequation holds.\nhState1((hModify1◦hND+f◦(⇔))( local2trail p)s)t\n=\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s)t′\n/u1D702((x,fplus s ys),extend t ys)\nWe will also use this corollary in the proofs.\nProof\nWe proceed by induction on p.\ncasep=Var y\nhState1((hModify1◦hND+f◦(⇔))( local2trail(Var y))s)t\n={- definition of local2trail -}\nhState1((hModify1◦hND+f◦(⇔))( Var y)s)t\n={- definition of(⇔)-}\nhState1((hModify1◦hND+f)(Var y)s)t\n={- definition of hND+f-}\nhState1(hModify1(Var[y])s)t\n={- definition of hModify1 and functiona application - }\nhState1(Var([y],s))t\n={- definition of hState1and functiona application - }\nVar(([y],s),t)\n={- similar derivation in reverse - }\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail(Var y))s)(Stack[])\nVar((x,s),t)112\ncaset=Op◦Inr◦Inl$Fail\nhState1((hModify1◦hND+f◦(⇔))( local2trail(Op◦Inr◦Inl$Fail))s)t\n={- definition of local2trail -}\nhState1((hModify1◦hND+f◦(⇔))( Op◦Inr◦Inl$Fail)s)t\n={- definition of(⇔)andhND+f-}\nhState1(hModify1(Var[])s)t\n={- definition of hModify1 and function application - }\nhState1(Var([],s))t\n={- definition of hState1and functiona application - }\nVar(([],s),t)\n={- similar derivation in reverse - }\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail(Op◦Inr◦Inl$Fail))s)(Stack[])\nVar((x,s),t)\ncaset=Op(Inl(MGet k))\nhState1((hModify1◦hND+f◦(⇔))( local2trail(Op(Inl(MGet k))))s)t\n={- definition of local2trail -}\nhState1((hModify1◦hND+f◦(⇔))( Op(Inl(MGet(local2trail◦k))))s)t\n={- definition of(⇔)andhND+f-}\nhState1(hModify1(Op(Inl(MGet(hND+f◦(⇔)◦ local2trail◦k))))s)t\n={- definition of hModify1 and function application - }\nhState1((hModify1◦hND+f◦(⇔)◦ local2trail◦k)s s)t\n={- reformulation - }\nhState1((hModify1◦hND+f◦(⇔))( local2trail(k s))s)t\n={- induction hypothesis on k s-}\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail(k s))s)(Stack[])\n/u1D702((x,fplus s ys),extend t ys)\n={- similar derivation in reverse - }\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail(Op(Inl(MGet k))))s)(Stack[])\n/u1D702((x,fplus s ys),extend t ys)\ncaset=Op(Inl(MUpdate r k))\nhState1((hModify1◦hND+f◦(⇔))( local2trail(Op(Inl(MUpdate r k))))s)t\n={- definition of local2trail -}\nhState1((hModify1◦hND+f◦(⇔))( do\npushStack(Left r)\nupdate r\nlocal2trail k\n)s)t\n={- definition of(⇔)andhND+f-}\nhState1(hModify1(do\nhND+f◦(⇔) $pushStack(Left r)113\nhND+f◦(⇔) $update r\nhND+f◦(⇔)◦ local2trail $k\n)s)t\n={- definition of hModify1 ,Lemma 17 and function application - }\nhState1(do\n(,s1)←( hModify1◦hND+f◦(⇔) $pushStack(Left r))s\n(,s2)←( hModify1◦hND+f◦(⇔) $update r)s1\n(hModify1◦hND+f◦(⇔)◦ local2trail $k)s2\n)t\n={- definition of hModify1 andupdate -}\nhState1(do\n(,s1)←( hModify1◦hND+f◦(⇔) $pushStack(Left r))s\n(,s2)←/u1D702([()],s1⊕r)\n(hModify1◦hND+f◦(⇔)◦ local2trail $k)s2\n)t\n={- monad law - }\nhState1(do\n(,s1)←( hModify1◦hND+f◦(⇔) $pushStack(Left r))s\n(hModify1◦hND+f◦(⇔)◦ local2trail $k)(s1⊕r)\n)t\n={- definition of hState1,Lemma 4 , and function application - }\ndo((,s1),t1)←hState1((hModify1◦hND+f◦(⇔) $pushStack(Left r))s)t\nhState1((hModify1◦hND+f◦(⇔)◦ local2trail $k)(s1⊕r))t1\n={- definition of pushStack -}\ndo((,s1),t1)←hState1((hModify1◦hND+f◦(⇔) $do\nStack xs←get\nput(Stack(Left r :xs)))s)t\nhState1((hModify1◦hND+f◦(⇔)◦ local2trail $k)(s1⊕r))t1\n={- definition of hState1,hModify1 ,hND+f,(⇔),get, and put; lett=Stack xs -}\ndo((,s1),t1)←/u1D702(([()],s),Stack(Left r :xs))\nhState1((hModify1◦hND+f◦(⇔))( local2trail k)(s1⊕r))t1\n={- monad law - }\nhState1((hModify1◦hND+f◦(⇔))( local2trail k)(s⊕r))(Stack(Left r :xs))\n={- by induction hypothesis on k, for some ysthe equation holds - }\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail k)(s⊕r))(Stack[Left r])\n/u1D702((x,fplus(s⊕r)ys),extend(Stack(Left r :xs))ys)\n={- definition of fplus andextend -}\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail k)(s⊕r))(Stack[Left r])\n/u1D702((x,fplus s(ys++[Left r])),extend(Stack xs)(ys++[Left r]))\n={- letys′=ys++[Left r]; Equation t=Stack xs -}\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail k)(s⊕r))(Stack[Left r])\n/u1D702((x,fplus s ys′),extend t ys′)114\n={- similar derivation in reverse - }\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail(Op(Inl(MUpdate r k))))s)(Stack[])\n/u1D702((x,fplus s ys′),extend t ys′)\ncaset=Op◦Inr◦Inl$Or p q\nhState1((hModify1◦hND+f◦(⇔))( local2trail(Op◦Inr◦Inl$Or p q))s)t\n={- definition of local2trail -}\nhState1((hModify1◦hND+f◦(⇔))(\n(pushStack(Right())> >local2trail p)/talloblong(untrail> >local2trail q))s)t\n={- definition of(/talloblong)-}\nhState1((hModify1◦hND+f◦(⇔))( Op◦Inr◦Inl$Or\n(pushStack(Right())> >local2trail p)\n(untrail> >local2trail q))s)t\n={- definition of hND+f,(⇔), and liftM2 -}\nhState1(hModify1(do\nx←hND+f((⇔)( pushStack(Right())))> >hND+f((⇔)( local2trail p))\ny←hND+f((⇔) untrail)> >hND+f((⇔)( local2trail q))\n/u1D702(x++y)\n)s)t\n={- monad law - }\nhState1(hModify1(do\nhND+f((⇔)( pushStack(Right())))\nx←hND+f((⇔)( local2trail p))\nhND+f((⇔) untrail)\ny←hND+f((⇔)( local2trail q))\n/u1D702(x++y)\n)s)t\n={- definition of hModify1 ,Lemma 17 , and function application - }\nhState1(do\n(,s1)←hModify1(hND+f((⇔)( pushStack(Right())))) s\n(x,s2)←hModify1(hND+f((⇔)( local2trail p)))s1\n(,s3)←hModify1(hND+f((⇔) untrail))s2\n(y,s4)←hModify1(hND+f((⇔)( local2trail q)))s3\n/u1D702(x++y,s4)\n)t\n={- definition of hState1,Lemma 4 , and function application - }\ndo((,s1),t1)←hState1(hModify1(hND+f((⇔)( pushStack(Right())))) s)t\n((x,s2),t2) ←hState1(hModify1(hND+f((⇔)( local2trail p)))s1)t1\n((,s3),t3)←hState1(hModify1(hND+f((⇔) untrail))s2)t2\n((y,s4),t4)←hState1(hModify1(hND+f((⇔)( local2trail q)))s3)t3\n/u1D702((x++y,s4),t4)\n={- definition of pushStack -}\ndo((,s1),t1)←hState1(hModify1(hND+f((⇔)( do\nStack xs←get115\nput(Stack(Right():xs))))) s)t\n((x,s2),t2) ←hState1(hModify1(hND+f((⇔)( local2trail p)))s1)t1\n((,s3),t3)←hState1(hModify1(hND+f((⇔) untrail))s2)t2\n((y,s4),t4)←hState1(hModify1(hND+f((⇔)( local2trail q)))s3)t3\n/u1D702((x++y,s4),t4)\n={- definition of hState1,hModify1,hND+f,(⇔),get,put; lett=Stack xs -}\ndo((,s1),t1)←/u1D702(([()],s),Stack(Right():xs))\n((x,s2),t2) ←hState1(hModify1(hND+f((⇔)( local2trail p)))s1)t1\n((,s3),t3)←hState1(hModify1(hND+f((⇔) untrail))s2)t2\n((y,s4),t4)←hState1(hModify1(hND+f((⇔)( local2trail q)))s3)t3\n/u1D702((x++y,s4),t4)\n={- monad law - }\ndo((x,s2),t2) ←hState1(hModify1(hND+f((⇔)\n(local2trail p)))s)(Stack(Right():xs))\n((,s3),t3)←hState1(hModify1(hND+f((⇔) untrail))s2)t2\n((y,s4),t4)←hState1(hModify1(hND+f((⇔)( local2trail q)))s3)t3\n/u1D702((x++y,s4),t4)\n={- reformulation - }\ndo((x,s2),t2) ←hState1((hModify1◦hND+f◦(⇔))\n(local2trail p)s)(Stack(Right():xs))\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔)) untrail s 2)t2\n((y,s4),t4)←hState1((hModify1◦hND+f◦(⇔))( local2trail q)s3)t3\n/u1D702((x++y,s4),t4)\n={- by induction hypothesis on p, for some ysthe equation holds - }\ndo((x,),) ← hState1((hModify1◦hND+f◦(⇔))\n(local2trail p)s)(Stack(Right()))\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔))\nuntrail(fplus s ys))(extend(Stack(Right():xs))ys)\n((y,s4),t4)←hState1((hModify1◦hND+f◦(⇔))( local2trail q)s3)t3\n/u1D702((x++y,s4),t4)\n={-Lemma 19 -}\ndo((x,),) ← hState1((hModify1◦hND+f◦(⇔))\n(local2trail p)s)(Stack(Right()))\n((y,s4),t4)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail q)(fminus(fplus s ys)ys))(Stack xs)\n/u1D702((x++y,s4),t4)\n={-Equation (7.1) gives fminus(fplus s ys)ys=s-}\ndo((x,),) ← hState1((hModify1◦hND+f◦(⇔))\n(local2trail p)s)(Stack(Right()))\n((y,s4),t4)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail q)s)(Stack xs)\n/u1D702((x++y,s4),t4)\n={- by induction hypothesis on p, for some ys′the equation holds - }\ndo((x,),) ← hState1((hModify1◦hND+f◦(⇔))\n(local2trail p)s)(Stack(Right()))116\n((y,),) ← hState1((hModify1◦hND+f◦(⇔))\n(local2trail q)s)(Stack[])\n/u1D702((x++y,fplus s ys′),extend(Stack xs)ys′)\n={- similar derivation in reverse - }\ndo((x,),) ← hState1((hModify1◦hND+f◦(⇔))\n(local2trail(Op◦Inr◦Inl$Or p q))s)(Stack[])\n/u1D702((x,fplus s ys′),extend t ys′)\ncaset=Op◦Inr◦Inr$y\nhState1((hModify1◦hND+f◦(⇔))( local2trail(Op◦Inr◦Inr$y))s)t\n={- definition of local2trail -}\nhState1((hModify1◦hND+f◦(⇔))( Op◦Inr◦Inr◦Inr◦fmap local2trail $y)s)t\n={- definition of(⇔)andhND+f-}\nhState1(hModify1(Op◦Inr◦Inr◦fmap(hND+f◦(⇔)◦ local2trail)$y)s)t\n={- definition of hModify1 and function application - }\nhState1(Op◦Inr◦fmap(($s)◦hModify1◦hND+f◦(⇔)◦ local2trail)$y)t\n={- definition of hState1and function application - }\nOp◦fmap(($t)◦hState1◦($s)◦hModify1◦hND+f◦(⇔)◦ local2trail)$y\n={- reformulation - }\nOp◦fmap(/u1D706k→hState1((hModify1◦hND+f◦(⇔))( local2trail k)s)t)$y\n={- by induction hypothesis on y, for some ysthe equation holds - }\nOp◦fmap(/u1D706k→do\n((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail k)s)t\n/u1D702((x,fplus s ys),extend t ys))$y\n={- definition of free monad - }\ndo((x,),)←Op◦fmap(/u1D706k→\nhState1((hModify1◦hND+f◦(⇔))( local2trail k)s)t)$y\n/u1D702((x,fplus s ys),extend t ys)\n={- similar derivation in reverse - }\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail(Op◦Inr◦Inr$y))s)t\n/u1D702((x,fplus s ys),extend t ys)\n/squaresolid\nThe following lemma shows that the untrail function restores all the updates in the trail\nstack until it reaches a time stamp Right().\nLemma 19 (UndoTrail undos) . For t=Stack(ys++(Right():xs))and ys=\n[Left r 1,...,Left r n], we have\nhState1((hModify1◦hND+f◦(⇔)) untrail s)t\n=\n/u1D702(([()],fminus s ys),Stack xs)\nThe function fminus is defined as follows:117\nfminus ::Undo s r⇒s→[Either r b]→s\nfminus s ys =foldl(/u1D706s(Left r)→s⊖r)s ys\nProof\nWe first calculate as follows:\nhState1((hModify1◦hND+f◦(⇔)) untrail s)t\n={- definition of untrail -}\nhState1((hModify1◦hND+f◦(⇔))( do\ntop←popStack\ncasetopof\nNothing→/u1D702()\nJust(Right())→/u1D702()\nJust(Left r)→dorestore r ;untrail\n)s)t\n={- definition of popStack -}\nhState1((hModify1◦hND+f◦(⇔))( do\ntop←doStack xs←get\ncasexsof\n[] → /u1D702Nothing\n(x:xs′)→doput(Stack xs′);/u1D702(Just x)\ncasetopof\nNothing→/u1D702()\nJust(Right())→/u1D702()\nJust(Left r)→dorestore r ;untrail\n)s)t\n={- monad law and case split - }\nhState1((hModify1◦hND+f◦(⇔))( do\nStack xs←get\ncasexsof\n[] → /u1D702()\n(Right():xs′)→doput(Stack xs′);/u1D702()\n(Left r :xs′) → doput(Stack xs′);restore r ;untrail\n)s)t\n={- definition of get-}\nhState1((hModify1◦hND+f◦(⇔))( Op◦Inr◦Inr◦Inl◦Get$/u1D706(Stack xs)→\ncasexsof\n[] → /u1D702()\n(Right():xs′)→doput(Stack xs′);/u1D702()\n(Left r :xs′) → doput(Stack xs′);restore r ;untrail\n)s)t\n={- definition of hND+fand(⇔)-}\nhState1(hModify1(Op◦Inr◦Inl◦Get$/u1D706(Stack xs)→\ncasexsof\n[] → /u1D702[()]118\n(Right():xs′)→hND+f◦(⇔) $doput(Stack xs′);/u1D702()\n(Left r :xs′) → hND+f◦(⇔) $doput(Stack xs′);restore r ;untrail\n)s)t\n={- definition of hModify1 and function application - }\nhState1(Op◦Inl◦Get$/u1D706(Stack xs)→\ncasexsof\n[] → /u1D702([()],s)\n(Right():xs′)→hModify1(hND+f◦(⇔) $doput(Stack xs′);/u1D702())s\n(Left r :xs′) → hModify1(hND+f◦(⇔) $doput(Stack xs′);restore r ;untrail)s\n)t\n={- definition of hState1and function application; let t=Stack(ys++(Right():xs))-}\ncase(ys++(Right():xs))of\n[] → /u1D702(([()],s),t)\n(Right():xs′)→/u1D702(([()],s),Stack xs′)\n(Left r :xs′) → hState1(hModify1(hND+f◦(⇔) $\ndoput(Stack xs′);restore r ;untrail)s)t\nThen, we proceed by an induction on the structure of ys.\ncaseys=[]\ncase(ys++(Right():xs))of\n[] → /u1D702(([()],s),t)\n(Right():xs′)→/u1D702(([()],s),Stack xs′)\n(Left r :xs′) → hState1(hModify1(hND+f◦(⇔) $\ndoput(Stack xs′);restore r ;untrail)s)t\n={- case split - }\n/u1D702(([()],s),Stack xs)\n={- definition of fminus -}\n/u1D702(([()],fminus s[]),Stack xs)\ncaseys=(Left r :ys′)\ncase(ys++(Right():xs))of\n[] → /u1D702(([()],s),t)\n(Right():xs′)→/u1D702(([()],s),Stack xs′)\n(Left r :xs′) → hState1(hModify1(hND+f◦(⇔) $\ndoput(Stack xs′);restore r ;untrail)s)t\n={- case split - }\nhState1(hModify1(hND+f◦(⇔) $\ndoput(Stack(ys′++(Right():xs)));restore r ;untrail)s)t\n={- definition of hState1,hModify1,hND+f,(⇔)and reformulation - }\nhState1((hModify1◦hND+f◦(⇔)) untrail(s⊖r))\n(Stack(ys′++(Right():xs)))\n={- induction hypothesis on ys′-}\n/u1D702(([()],fminus(s⊖r)ys′),Stack xs)\n={- definition of fminus -}\n/u1D702(([()],fminus s(Left r :ys′)),Stack xs)119\n={- definition of ys-}\n/u1D702(([()],fminus s ys),Stack xs)\n/squaresolid\nThe following lemma is obvious from Lemma 18 andLemma 19 . It shows that we can\nrestore the previous state and stack by pushing a time stamp o n the trail stack and use the\nfunction untrail afterwards.\nLemma 20 (State and stack are restored) .For t ::Stack(Either r()), s::s, and p ::\nFree(Modify Fs r:+:Nondet F:+:f)a which does not use the restore operation, we have\ndo((,s1),t1)←hState1((hModify1◦hND+f◦(⇔))( pushStack(Right()))s)t\n((x,s2),t2) ←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s1)t1\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔)) untrail s 2)t2\n/u1D702((x,s3),t3)\n=\ndo((x,),) ← hState1((hModify1◦hND+f◦(⇔))( local2trail p)s)(Stack[])\n/u1D702((x,s),t)\nProof Suppose t=Stack xs . We calculate as follows.\ndo((,s1),t1)←hState1((hModify1◦hND+f◦(⇔))( pushStack(Right()))s)(Stack xs)\n((x,s2),t2) ←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s1)t1\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔)) untrail s 2)t2\n/u1D702((x,s3),t3)\n={- definition of hState1,hModify1,hND+f,(⇔),pushStack -}\ndo((,s1),t1)←/u1D702(([()],s),Stack(Right():xs))\n((x,s2),t2) ←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s1)t1\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔)) untrail s 2)t2\n/u1D702((x,s3),t3)\n={- monad law - }\ndo((x,s2),t2) ←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s)\n(Stack(Right():xs))\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔)) untrail s 2)t2\n/u1D702((x,s3),t3)\n={- byLemma 18 , for some ys=[Left r 1,...,Left r n]the equation holds - }\ndo((x,s2),t2) ←do((x,),)←hState1((hModify1◦hND+f◦(⇔))\n(local2trail p)s)(Stack[])\n/u1D702((x,fplus s ys),extend(Stack(Right():xs))ys)\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔)) untrail s 2)t2\n/u1D702((x,s3),t3)\n={- monad law and definition of extend -}\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s)(Stack[])\n((,s3),t3)←hState1((hModify1◦hND+f◦(⇔))\nuntrail(fplus s ys))(Stack(ys++(Right():xs)))\n/u1D702((x,s3),t3)120\n={-Lemma 19 and monad law - }\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s)(Stack[])\n/u1D702((x,fminus(fplus s ys)ys),Stack xs)\n={-Equation (7.1) gives fminus(fplus s ys)ys=s-}\ndo((x,),)←hState1((hModify1◦hND+f◦(⇔))( local2trail p)s)(Stack[])\n/u1D702((x,s),Stack xs)\n/squaresolid"    },    {        "title": "1708.06874v2.Observation_of_Dirac_like_energy_band_and_ring_torus_Fermi_surface_associated_with_the_nodal_line_in_topological_insulator_CaAgAs.pdf",        "content": " 1 Observation of Dirac-like energy band and ring-torus  Fermi surface associated with the nodal line in topological insulator CaAgAs  Daichi Takane1, Kosuke Nakayama1, Seigo Souma2,3, Taichi Wada4, Yoshihiko Okamoto4,5, Koshi Takenaka4, Youichi Yamakawa5,6, Ai Yamakage5,6, Taichi Mitsuhashi1,7, Koji Horiba7, Hiroshi Kumigashira1,7, Takashi Takahashi1,2,3, and Takafumi Sato1,2* 1Department of Physics, Tohoku University, Sendai 980-8578, Japan 2Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 3WPI Research Center , Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan 5Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan 6Department of Physics, Nagoya University, Nagoya 464-8602, Japan 7Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan  Abstract One of key challenges in current material research is to search for new topological materials with inverted bulk-band structure. In topological insulators, the band inversion caused by strong spin-orbit coupling leads to opening of a band gap in the entire Brillouin zone, whereas an additional crystal symmetry such as point-group and nonsymmorphic symmetries sometimes prohibits the gap opening at/on specific points or line in momentum space, giving rise to topological semimetals. Despite many theoretical predictions of topological insulators/semimetals associated with such crystal symmetries, the experimental realization is still relatively scarce. Here, using angle-resolved  2 photoemission spectroscopy with bulk-sensitive soft x-ray photons, we experimentally demonstrate that hexagonal pnictide CaAgAs belongs to a new family of topological insulators characterized by the inverted band structure and the mirror reflection symmetry of crystal. We have established the bulk valence-band structure in three-dimensional Brillouin zone, and observed the Dirac-like energy band and ring-torus Fermi surface associated with the line node, where bulk valence and conducting bands cross on a line in the momentum space under negligible spin-orbit coupling. Intriguingly, we found that no other bands cross the Fermi level and therefore the low-energy excitations are solely characterized by the Dirac-like band. CaAgAs provides an excellent platform to study the interplay among low-energy electron dynamics, crystal symmetry, and exotic topological properties.  Introduction   Topological insulators (TIs) exhibit a novel quantum state with metallic edge or surface state (SS) within the bulk band gap generated by the strong spin-orbit coupling (SOC). The topological SS in three-dimensional (3D) TIs is characterized by a linearly dispersing Dirac-cone energy band,1-3 which hosts massless Dirac fermions protected by the time-reveal symmetry (TRS). The discovery of TIs triggered the search for new types of topological materials containing surface or bulk Dirac-cone bands protected by crystal symmetries, as represented by topological crystalline insulators (TCIs) with the Dirac-cone SSs protected by mirror symmetry,4-6 as well as 3D Dirac semimetals (DSMs) with bulk Dirac-cone bands protected by rotational symmetry (such as Cd3As2 and Na3Bi).7-11 While the Dirac  3 cone in DSMs is spin degenerate, breaking the TRS or space-inversion symmetry (SIS) leads to the Weyl-semimetal (WSM) phase with pairs of spin-split Dirac (Weyl) cones, as recently verified in transition-metal monopnictides.12-14 Such Dirac-cone states are known to provide a platform to realize outstanding physical properties such as extremely high mobility, gigantic linear magnetoresistance, and chiral anomaly.15-22 While the DSMs and WSMs are characterized by the crossing of bulk bands at the discrete points in k space (point nodes), there exists another type of topological semimetal characterized by the band crossing along a one-dimensional curve in k space (line node), called line-node semimetal (LNSM). The LNSMs are expected to show unique physical properties different from the DSMs and WSMs, such as a flat Landau level, the Kondo effect, long-range Coulomb interaction, and peculiar charge polarization and orbital magnetism.23-26 Despite many theoretical predictions of LNSMs in various material platforms,27-54 experimental studies on the LNSMs are relatively scarce.55-62 Recently, it was theoretically proposed by Yamakage et al. that noncentrosymmetric ternary pnictides CaAgX (X = P, As) are the candidate of LNSM and TI.41 These materials crystalize in the ZrNiAl-type structure with space group P6___2m (No. 189)63 (for crystal structure, see Fig. 1a). First-principles band-structure calculations have shown that, under negligible spin-orbit coupling (SOC), CaAgX displays a fairly simple band structure near the Fermi level (EF) with a ring-like line node (nodal ring) surrounding the G point of bulk hexagonal Brillouin zone (BZ) (bulk BZ is shown in Fig. 1b). The line node is associated with  4 the crossing of bulk conduction band (CB) and valence band (VB) with Ag s and P/As p character, respectively, and is protected by the mirror reflection symmetry of crystal. When the SOC is included in the calculation, CaAgP still keeps the line node due to the very small spin-orbit gap (~ 1 meV) while a relatively large spin-orbit gap (~ 75 meV) opens along the line node in CaAgAs to make this material a narrow-gap TI.41 The SOC thus plays a crucial role in switching the LNSM and TI phases in CaAgX. Transport measurements on the CaAgP and CaAgAs polycrystalline samples by Okamoto et al. have demonstrated the low-carrier-density nature of these samples, consistent with the existence of line node64 (note that this is further corroborated by the recent transport measurements on CaAgX single crystals62). They have further suggested that EF in hole-doped CaAgAs sample lies in the middle of linearly dispersive Dirac-like band, and concluded that CaAgAs is well suited to study the low-energy excitations related to the Dirac electrons. While the electronic states of CaAgX and its relationship to physical properties should be experimentally examined, there exist no experimental outputs on the band structure of CaAgX. It is thus urgently required to establish the fundamental electronic states. In this work, we report the ARPES study of CaAgAs. By utilizing bulk-sensitive soft-x-ray photons from synchrotron radiation, we established the bulk VB structure in the 3D bulk BZ. We suggest that CaAgAs is a narrow-gap TI with an ideal band structure suitable to study the low-energy excitations linked to the bulk Dirac-like band arising from the line node. This is demonstrated by observing the bulk Fermi surface which is solely derived from the VB and CB  5 associated with the single line node, consistent with our first-principles band structure calculations. We discuss the consequence of our observation in relation to the exotic physical properties.  Results and Discussion Samples and experimental High-quality single crystals of CaAgAs were grown on the sintered pellets of CaAgAs (for details, see Method). A typical photograph of our single crystal is shown in Fig. 1c. ARPES measurements were performed with synchrotron light at BL2 in Photon Factory, KEK. Samples were cleaved in situ along the (112___0) crystal plane (a shiny mirror plane in Fig. 1c) as confirmed by the Laue x-ray diffraction measurement on the cleaved surface (typical Laue pattern is shown in Fig. 1d) and the photon-energy (hn) dependence of the band dispersion. This indicates that the cleaved plane is the ky - kz plane in the hexagonal BZ (Fig. 1b). Figure 1e displays the energy distribution curve (EDC) in the wide energy region measured at hn = 580 eV. One can recognize several core-level peaks originating from the Ca (3s, 3p), Ag (4s, 4p, 4d), and As (3s, 3p, 3d) orbitals. No other core-level peaks were found in this energy range, confirming the clean sample surface.  Valence-band structure First, we present the overall VB structure of CaAgAs. We found that soft-x-ray photons are useful for revealing the bulk electronic states of CaAgAs as in the case of noncentrosymmetric Weyl semimetals such as TaAs,12-14 although we need to  6 sacrifice the energy/momentum resolution compared to the vacuum ultraviolet (VUV) photons. In fact, the obtained VUV data were found to suffer large broadening along wave vector perpendicular to the surface probably because of the final-state effect and rather rough nature of the cleaved surfaces, and therefore we concluded that the VUV photons are not best suited for resolving 3D electronic states of CaAgAs.\tFigure 2a displays the EDCs at the normal emission measured with various photon energies in the soft-x-ray region of 530 – 700 eV. One can identify several dispersive bands. For example, a band located at binding energy (EB) of ~ 1 eV at hn ~ 700 eV, which is attributed to the topmost bulk VB, disperses toward EF on decreasing hn to 650 eV, and it disperses back again toward ~ 1 eV at hn = 600 eV. On further decreasing hn down to ~ 550 eV, this band again approaches EF, similarly to the case of hn = 650 - 700 eV. Such periodic band dispersion was also observed for the bands located at EB ~1.5 and ~3 eV. To visualize the experimental band dispersion more clearly, we plot in Fig. 2b the band structure obtained from the second derivative of the ARPES intensity plotted as a function of wave vector perpendicular to the sample surface (kx), which corresponds to the GM cut in the bulk BZ (cut A in Fig. 2c). One can immediately recognize that the overall experimental band dispersion shows a reasonable agreement with the calculated bulk bands (red curves) regarding the periodicity and location of bands; this confirms that the cleaving plane is (112___0). The holelike dispersion approaching EF around the G point is well reproduced by the calculations, and therefore it is assigned as the topmost VB with the As 4p orbital character. Moreover, a good agreement of the band width between experiment and calculation signifies no apparent band-renormalization effect, suggesting the weak electron correlation. It is  7 noted that we observe a single holelike band within 1.5 eV of EF in the experiment, while the calculation predicts two holelike bands. Such difference may be due to the finite k/energy-broadening effect as well as the matrix-element effect of photoelectron intensity which turned out to be rather strong in this material. We comment here that our Hall conductivity measurement of CaAgAs single crystal suggests the existence of hole carriers with carrier concentration of ~ 1.6 × 1020 cm-3, which corresponds to the EF location of 0.27 eV below the VB top. On the other hand, the VB structure determined by ARPES shows a reasonable agreement with the calculated band structure without sizable EF shift. While we do not know the exact origin of such a difference, some possibilities like downward surface band-bending may be considered. To see the band dispersion along the in-plane wave vector, we used the photon energy of hn = 550 eV which traces the k cut crossing the G point of 13th BZ, and measured the EDC along the GKM (ky) cut (cut B in Fig. 2c), as shown in Fig. 2d. One can recognize highly dispersive holelike band centered at the G point, similarly to the case of GM cut in Figs. 2a and 2b. This band is better visualized in the ARPES-intensity plot in Fig. 2e in which a holelike band with linear dispersion shows up in the EB range of 0 ~ 1.5 eV (note that the intensity distribution around the M and G points is different between Figs. 2b and 2e and also among different BZs due to aforementioned matrix-element effect). We have confirmed by the band-structure mapping in 3D BZ that the Fermi surface exists only around the G point. This conclusion is supported by the experimental band dispersion along the  8 AHL cut in Fig. 2f (cut C in Fig. 2c) where the topmost VB always stays below EB ~ 1 eV without crossing EF.  Ring-torus Fermi surface Having established the overall VB structure, a next important issue is the electronic structure in the vicinity of EF responsible for the physical properties. Figure 3a displays the ARPES intensity at EF as a function of kx and ky (the GKM plane). One immediately finds a bright intensity pattern surrounding the G point, in particular in 13th BZ, confirming the absence of additional Fermi surface away from the G point. It is also obvious from Fig. 3b that no Fermi surface exists away from G in the ky-kz (GAKH) plane. As shown in Figs. 3a and 3b, when we overlaid the calculated Fermi surface (green curves) onto the ARPES intensity (note that we assumed the location of EF to be 0.05 eV below the VB top in the calculation to account for a small but finite hole-doping effect in experiment), the high-intensity region coincides with the k region where the calculated Fermi surface exists. To gain further insight into the Fermi-surface topology, we show in the top panels of Figs. 3c and 3d the ARPES intensity near EF and the intensity obtained by taking second derivative of the EDCs, respectively, along the k cut nearly crossing the G point (cut A in Fig. 3a). One finds a linearly dispersive holelike band originating from the As 4p states which is better visualized in the second-derivative plot in Fig. 3d (by a linear extrapolation of the band dispersion around EF, we have estimated the Fermi velocity to be vF = 2.1 ± 0.1 eVÅ). This band is reproduced by our calculation as shown by red curves in Fig. 3c, and is responsible for the outer  9 ring in Fig. 3a. As shown in cut A of Fig. 3c, there exists another electronlike band in the calculation which originates from the Ag 5s orbital; this band forms the inner ring in Fig. 3a. While the intensity of the electronlike band seems weak in the original intensity (Fig. 3c), the second-derivative image in Fig. 3d shows a finite spectral weight likely arising from the electronlike band. As shown in cut A of Fig. 3c, the calculated electronlike band intersects the holelike band at ~ 0.1 eV above EF, and forms the nodes at ky ~ ± 0.15 Å-1 under negligible SOC, since cut A is on the (0001) mirror plane (the kx-ky plane) and the nodes are protected by mirror reflection symmetry.41 This indicates that the electronic states within two opposite nodes across the G point have an inverted band character. Thus, our observation of electronlike feature can be regarded as a hallmark of the band inversion, which is a prerequisite for realizing LNSM or TI. It is remarked that with a finite SOC, an energy gap of 75 meV opens in the caluclation, as can be seen from a difference in the band dispersion with (solid curves) and without (dashed curves) SOC in Fig. 3c.41 The opening of a spin-orbit gap at the node is also seen in some other LNSM candidates such as Cu3(Pd,Zn)N,34,35 Ca3P2,37,49 ZrSiS,40 CaTe,50 and fcc alkaline-earth metal51. To clarify whether the node-like feature in CaAgAs is seen at a point or on a line in k space, it is necessary to measure the band dispersion along different k slices around the Fermi surface. For this sake, we show in the middle and bottom panels of Figs. 3c and 3d the intensity for cuts slightly away from the G point (cuts B and C in Fig. 3a) obtained with different hn’s. One can recognize that overall band structure along cut B is similar to that along cut A regarding the EF crossing of holelike band  10 and the presence of electronlike feature. This is reasonable since cut B is also on the mirror plane and still crosses the calculated nodal points. On the other hand, along cut C, the holelike band moves downward and shows no EF crossing. These behaviors are consistent with the presence of a ring-shaped nodal feature (nodal ring) on the mirror plane shown by a dashed curve in Fig. 3a. It should be stressed again that there exists a spin-orbit gap along the nodal line in the calculation. Since the gap is almost isotropic (75±1 meV) along the nodal ring (not shown), the low-energy excitations in CaAgAs are characterized by the excitations across the band gap in the k region involving the entire line node. Unfortunately, such a band gap (as well as the topological SSs) was not resolved in the ARPES experiment, likely due to the slightly hole-doped nature of crystal. Considering the fact that (i) the calculated spin-orbit gap is not so small compared to other TIs and (ii) the ARPES-derived band dispersion shows a reasonable agreement with the calculation near EF, it would be more reasonable to regard CaAgAs as a narrow-gap TI, rather than a LNSM. It is also emphasized here that the TI nature of CaAgAs should be distinguished from that of prototypical TIs such as Bi2Se3 since the node never shows up even without SOC in Bi2Se3 unlike CaAgAs. Figure 3e illustrates the calculated equi-energy contour maps in k space for selected energy slices (energy with respect to the VB top, EVB = 0.1, 0.27, and 0.6 eV). The energy contour around EF (EVB = 0.1 eV) shows a ring-torus shape. On increasing EVB, it gradually expands and fattens. Eventually it transforms into the spheroid as soon as the EVB passes the bottom of electonlike band. As shown in Fig.  11 3f, such evolution of the ring-torus contours can be seen from the ARPES-intensity mapping for representative EB slices compared with the calculated energy contours.  From all these experimental results, we concluded that CaAgAs is likely a narrow-gap TI characterized by the bulk Dirac-like band and ring-torus Fermi surface associated with the line node on the (0001) mirror plane, in line with the first-principles band-structure calculations. Intriguingly, we revealed that the bulk Fermi surface is solely derived from the Dirac-like bands associated with a single line node (which appears under negligible SOC), and no other bands cross EF. In this regard, the CaAgX family can be distinguished from some other LNSM candidates which contain additional normal bands crossing EF and multiple line nodes55-61. Thus, CaAgX is a promising platform to study the interplay among mirror symmetry, low-energy excitations, and transport properties in LNSMs. Also, it would provide an excellent platform to study the influence of SOC to the low-energy excitations involving the line node, because the SOC strength can be controlled by replacement of P and As in the crystal. Since the bulk electronic states of CaAgAs have been established in this study, a next important challenge is to clarify the nature of predicted topological SSs in the band inverted region.33 Such an experiment would require an access to the band dispersion above the line node via the fabrication of electron-rich CaAgAs.      12 Methods Sample preparation High-quality single crystals of CaAgAs were synthesized by the following procedure. An equimolar mixture of calcium chips, silver powder, and arsenic chunks were put in an alumina crucible and sealed in an evacuated quartz tube. The tubes were kept at 773 K for 12 h and then at 1273 K for 12 h, followed by furnace cooling to room temperature. The obtained samples were pulverized, pressed into pellets, and sealed in quartz tubes. The pellets were sintered at 1173 K for 2 h and cooled to room temperature at a rate of 30 K h-1, resulting in that shiny hexagonal-prismatic single crystals of CaAgAs were grown on the pellets. The quality of the crystal was checked by X-ray diffraction technique using a RIGAKU R-AXIS IP diffractometer.  ARPES experiments ARPES measurements were performed with a Scienta-Omicron SES2002 electron analyzer with energy-tunable synchrotron light at BL2 (Multiple Undulator beamline for Spectroscopic Analysis on Surface and HeteroInterface; MUSASHI) in Photon Factory, KEK. We used linearly polarized light (horizontal polarization) of 500-975 eV. The energy and angular resolutions were set at 150 meV and 0.2º, respectively. Samples were cleaved in situ in an ultrahigh vacuum better than 1×10-10 Torr along the (112___0) crystal plane, as confirmed by the Laue x-ray diffraction measurement on the cleaved surface and the photon-energy dependence of the band dispersion shown in Fig. 2b. Sample temperature was kept at T = 40 K  13 during the ARPES measurements. The Fermi level (EF) of samples was referenced to that of a gold film evaporated onto the sample holder.  Calculations Electronic band-structure calculations were carried out by means of first-principles band structure calculations by using WIEN2k code65 with the full-potential linearized augmented plane-wave method within the generalized gradient approximation. We used the experimental structural parameters for the calculations.63 24 × 24 × 36 k-points sampling was used for the self-consistent calculations.41  Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request.  Acknowledgements We thank H. Oinuma and T. Nakamura for their assistance in the ARPES measurements and T. Yajima for his assistance in the single-crystal X-ray diffraction experiments. This work was supported by Grant-in-Aid for Scientific Research on Innovative Areas \"Topological Materials Science\" (JSPS KAKENHI No: JP15H05853), Grant-in-Aid for Scientific Research (JSPS KAKENHI No: JP17H01139, JP15H02105, JP26287071, JP25287079, JP25220708, JP16K13664, and JP16K17725), KEK-PF (Proposal No: 2015S2-003 and 2016G555), and UVSOR (Proposal No: 28-542 and 28-828).  14 Author Contributions The work was planned and proceeded by discussion among D.T., A.Y., Y.O. and T.S., and T.W., Y.O., and K.T. carried out the samples’ growth and their characterization. D.T., K.N. S.S. T.M., K.H., H.K, T.T., and T.S. performed ARPES measurements. Y.Y . and A.Y. performed the first-principles band structure calculations. D.T. and T.S. finalized the manuscript with inputs from all the authors.  References 1. Ando, Y. Topological insulator materials. J. Phys. Soc. Jpn. 82, 102001 (2013). 2. Qi, X.-L., & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057-1110 (2011). 3. Hasan, M. Z., & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045-3067 (2010). 4. Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011). 5. Hsieh, T. H. et al. Topological crystalline insulators in the SnTe material class. Nat. Commun. 3, 982 (2012). 6. Tanaka, Y. et al. Experimental realization of a topological crystalline insulator in SnTe. Nat. Phys. 8, 800-803 (2012). 7. Borisenko, S. et al. Experimental realization of a three-dimensional Dirac semimetal. Phys. Rev. Lett. 113, 027603 (2014). 8. Neupane, M. et al. Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2. Nat. Commun. 5, 3786 (2014).  9. Liu, Z. K. et al. Discovery of a three-dimensional topological Dirac semimetal,  15 Na3Bi. Science 343, 864-867 (2014). 10. Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012). 11. Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Three-dimensional Dirac semimetal and quantum transport in Cd3As2. Phys. Rev. B 88, 125427 (2013).  12. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613-617 (2015). 13. Lv, B. Q. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015). 14. Yang, L. X. et al. Weyl semimetal phase in the non-centrosymmetric compound TaAs. Nat. Phys. 11, 728-732 (2015). 15. Zyuzin, A. A. & Burkov, A. A. Topological response in Weyl semimetals and the chiral anomaly. Phys. Rev. B 86, 115133 (2012). 16. Liu, C.-X., Ye, P. & Qi, X.-L. Chiral gauge field and axial anomaly in a Weyl semi-metal. Phys. Rev. B 87, 235306 (2013). 17. Wang, Z. & Zhang, S.-C. Chiral anomaly, charge density waves, and axion strings from Weyl semimetals. Phys. Rev. B 87, 161107(R) (2013). 18. Landsteiner, K. Anomaly related transport of Weyl fermions for Weyl semimetals. Phys. Rev. B 89, 075124 (2014). 19. Chernodub, M. N., Cortijo, A., Grushin, A. G., Landsteiner, K. & Vozmediano, M. A. H. Condensed matter realization of the axial magnetic effect. Phys. Rev. B 89, 081407(R) (2014). 20. Liang, T. et al. Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2. Nat. Mater . 14, 280-284 (2015).  16 21. Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413-416 (2015). 22. Weng, H., Dai, X. & Fang, Z. Topological semimetals predicted from first-principles calculations. J. Phys.: Condens. Matter 28, 303001 (2016). 23. Rhim, J.-W. & Kim, Y. B. Landau level quantization and almost flat modes in three-dimensional semimetals with nodal ring spectra. Phys. Rev. B 92, 045126 (2015). 24. Mitchell, A. K. & Fritz, L. Kondo effect in three-dimensional Dirac and Weyl systems. Phys. Rev. B 92, 121109 (2015). 25. Huh, Y., Moon, E.-G. & Kim, Y. B. Long-range Coulomb interaction in nodal-ring semimetals. Phys. Rev. B 93, 035138 (2016). 26. Ramamurthy, S. T. and Hughes, T. L. Quasitopological electromagnetic response of line-node semimetals. Phys. Rev. B 95, 075138 (2017). 27. Mikitik, G. P. & Sharlai, Y. V. Band-contact lines in the electron energy spectrum of graphite. Phys. Rev. B 73, 235112 (2006). 28. Phillips, M. & Aji, V. Tunable line node semimetals. Phys. Rev. B 90, 115111 (2014). 29. Heikkilä, T. T. & Volovik, G. E. Nexus and Dirac lines in topological materials. New J. Phys. 17, 093019 (2015). 30. Mullen, K., Uchoa, B. & Glatzhofer, D. T. Line of Dirac nodes in hyperhoneycomb lattices. Phys. Rev. Lett. 115, 026403 (2015). 31. Weng, H. et al. Topological node-line semimetal in three-dimensional graphene networks. Phys. Rev. B 92, 045108 (2015). 32. Chen, Y. et al. Nanostructured carbon allotropes with Weyl-like loops and points.  17 Nano Lett. 15, 6974-6978 (2015). 33. Zeng, M. et al. Topological semimetals and topological insulators in rare earth monopnictides. Preprint at https://arxiv.org/abs/1504.03492 (2015). 34. Yu, R., Weng, H., Fang, Z., Dai, X. & Hu, X. Topological node-line semimetal and Dirac semimetal state in antiperovskite Cu3PdN. Phys. Rev. Lett. 115, 036807 (2015). 35. Kim, Y., Wieder, B. J., Kane, C. L. & Rappe, A. M. Dirac line nodes in inversion-symmetric crystals. Phys. Rev. Lett. 115, 036806 (2015). 36. Fang, C., Chen, Y., Kee, H.-Y. & Fu, L. Topological nodal line semimetals with and without spin-orbital coupling. Phys. Rev. B 92, 081201 (2015). 37. Xie, L. et al. A new form of Ca3P2 with a ring of Dirac nodes. APL Mater. 3, 083602 (2015). 38. Chen, Y., Lu, Y.-M. & Kee, H.-Y. Topological crystalline metal in orthorhombic perovskite iridates. Nat. Commun. 6, 6593 (2015). 39. Kim, H.-S., Chen, Y. & Kee, H.-Y. Surface states of perovskite iridates AIrO3: Signatures of a topological crystalline metal with nontrivial Z2 index. Phys. Rev. B 91, 235103 (2015). 40. Xu, Q. et al. Two-dimensional oxide topological insulator with iron-pnictide superconductor LiFeAs structure. Phys. Rev. B 92, 205310 (2015). 41. Yamakage, A., Yamakawa, Y., Tanaka, Y. & Okamoto, Y. Line-node Dirac semimetal and topological insulating phase in noncentrosymmetric pnictides CaAgX (X = P, As). J. Phys. Soc. Jpn. 85, 013708 (2016). 42. Liu, J. et al. Strain-induced nonsymmorphic symmetry breaking and removal of Dirac semimetallic nodal line in an orthoperovskite iridate. Phys. Rev. B 93, 085118  18 (2016).  43. Liang, Q.-F., Zhou, J., Yu, R., Wang, Z. & Weng, H. Node-surface and node-line fermions from nonsymmorphic lattice symmetries. Phys. Rev. B 93, 085427 (2016). 44. Huang, H., Liu, J., Vanderbilt, D. & Duan, W. Topological nodal-line semimetals in alkaline-earth stannides, germanides, and silicides. Phys. Rev. B 93, 201114(R) (2016). 45. Zhu, Z., Li, M. & Li, J. Topological semimetal to insulator quantum phase transition in the Zintl compounds Ba2 X (X = Si, Ge). Phys. Rev. B 94, 155121 (2016). 46. Li, R. et al. Dirac node lines in pure alkali earth metals. Phys. Rev. Lett. 117, 096401 (2016). 47. Wang J.-T. et al. Body-centered orthorhombic C16: A novel topological node-line Semimetal. Phys. Rev. Lett. 116, 195501 (2016). 48. Zhao, J., Weng, H. & Fang, Z. Topological node-line semimetal in compressed black phosphorus. Phys. Rev. B 94, 195104 (2016). 49. Chan, Y.-H., Chiu, C.-K., Chou, M. Y. & Schnyder, A. P. Ca3P2 and other topological semimetals with line nodes and drumhead surface states. Phys. Rev. B 93, 205132 (2016). 50. Du, Y. et al. CaTe: a new topological node-line and Dirac semimetal. Npj Quantum. Matter. 2, 3 (2017). 51. Hirayama, M., Okugawa, R., Miyake, T. & Murakami, S. Topological Dirac nodal lines and surface charges in fcc alkaline earth metals. Nat. Commun. 8, 14022 (2017). 52. Xu, Q., Yu, R., Fang, Z., Dai, X. & Weng, H. Topological nodal line semimetals in  19 the CaP3 family of materials. Phys. Rev. B 95, 045136 (2017). 53. Takahashi, R., Hirayama, M. & Murakami, S. Topological nodal-line semimetals arising from crystal symmetry. Phys. Rev. B 96, 155206 (2017). 54. Li, J. et al. Topological nodal line states and a potential catalyst of hydrogen evolution in the TiSi family. Preprint at https://arxiv.org/abs/1704.07043 (2017). 55. Bian, G. et al. Topological nodal-line fermions in spin-orbit metal PbTaSe2. Nat. Commun. 7, 10556 (2016). 56. Schoop, L. M. et al. Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS. Nat. Commun. 7, 11696 (2016). 57. Neupane, M. et al. Observation of topological nodal fermion semimetal phase in ZrSiS. Phys. Rev. B 93, 201104(R) (2016). 58. Wu, Y. et al. Dirac node arcs in PtSn4. Nat. Phys. 12, 667-671 (2016). 59. Takane, D. et al. Dirac-node arc in the topological line-node semimetal HfSiS. Phys. Rev. B 94, 121108(R) (2016). 60. Hosen, M. M. et al. Tunability of the topological nodal-line semimetal phase in ZrSiX-type materials (X=S, Se, Te). Phys. Rev. B 95, 161101(R) (2017). 61. Chen, C. et al. Dirac line nodes and effect of spin-orbit coupling in the nonsymmorphic critical semimetals MSiS (M = Hf, Zr). Phys. Rev. B 95, 125126 (2017). 62. Emmanouilidou, E. et al. Magneto-transport properties of the \"hydrogen atom\" nodal-line semimetal candidates CaTX (T = Ag, Cd, X = As, Ge). Phys. Rev. B 95, 245113 (2017). 63. Mewis, A. & Naturforsch. Z. CaAgP und CaAgAs –Zwei Verbindungen mit Fe2P-Stuktur. (CaAgP and CaAgAs –Two Compounds with Fe2P-Structure.) Z.  20 Naturforsch. B 34, 14-17 (1979). 64. Okamoto, Y., Inohara, T., Yamakage, A., Yamakawa, Y. & Takenaka, K. Low carrier density metal realized in candidate line-node Dirac semimetals CaAgP and CaAgAs. J. Phys. Soc. Jpn. 85, 123701 (2016). 65. Blaha, P., Schwarz, K., Madsen, G., Kvasnicka, D. & Luiz, J. WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Tech. Univ. Wien, Vienna, 2001).    21                  Fig. 1  Characterization of CaAgAs single crystal. a Two different views of crystal structure of CaAgAs. b Bulk Brillouin zone of CaAgAs. c Photograph of CaAgAs single crystal. Scale bar: 200 µm. d Laue x-ray diffraction pattern for the cleaved surface. e The EDC in the wide energy region measured at T = 40 K with hn = 580 eV.   \n 22                  Fig. 2  Valence-band structure of CaAgAs. a Photon-energy dependence of normal-emission EDCs in the VB region. b ARPES intensity in the VB region along the GM line (cut A) plotted as a function of wave vector (kx) and EB, together with the calculated band dispersions (red curves).41 ARPES intensity was obtained by taking second derivative of the EDCs. c Bulk BZ and measured k cuts. d The EDCs along the GKM line (cut B), and e corresponding second-derivative ARPES intensity. f Same as e but measured along the AHL line (cut C).   \n 23            Fig. 3  Line-node semimetallic nature of CaAgAs. a ARPES-intensity mapping at EF as a function of kx and ky, together with the calculated Fermi surface (solid green curves). We set the EF position in the calculation to be 0.05 eV below the VB top, to take into account slight hole-doping effect. Calculated ring-like line node (nodal ring) is shown by a dashed circle in BZ. b Same as a but as a function of ky and kz (GAHK plane). c and d ARPES intensity and the intensity obtained by taking second derivative of the EDCs, respectively, measured along three representative cuts (cuts A-C in a). Calculated bands with SOC are shown by solid red curves in c, while those without SOC are indicated by dashed curves around the node. Dashed red curves in d are a guide to the eyes to trace the experimental band dispersions. e Calculated equi-energy contour maps in k space for selected energy slices (energy with respect to the VB top, EVB = 0.1, 0.27, and 0.6 eV). f ARPES-intensity mapping as a function of two-dimensional wave vector (kx and ky) for representative EB slices compared with the calculated energy contours (green curves). ARPES intensity in a, b, and f was obtained by taking second derivative of the EDCs. \n"    },    {        "title": "1312.6223v1.Electronic_structure_of_CaFe2As2.pdf",        "content": "arXiv:1312.6223v1  [cond-mat.supr-con]  21 Dec 2013Electronic structure of CaFe 2As2\nG. Adhikary,1D. Biswas,1N. Sahadev,1R. Bindu,1N. Kumar,1S. K. Dhar,1A. Thamizhavel,1and K. Maiti1,a)\nDepartment of Condensed Matter Physics and Materials’ Scie nce, Tata Institute of Fundamental Research,\nHomi Bhabha Road, Colaba, Mumbai - 400 005, INDIA.\n(Dated: 14 August 2021)\nWe investigate the electronic structure of CaFe 2As2using high resolution photoemission spectroscopy. Exper-\nimental results exhibit three energy bands crossing the Fermi leve l making hole pockets around the Γ-point.\nTemperature variation reveal a gradual shift of an energy band a way from the Fermi level with the decrease\nin temperature in addition to the spin density wave (SDW) transition in duced Fermi surface reconstruction\nof the second energy band across SDW transition temperature. T he hole pocket in the former case eventually\ndisappears at lower temperatures while the hole Fermi surface of t he third energy band possessing finite p\norbital character survives till the lowest temperature studied. T hese results reveal signature of a complex\ncharge redistribution among various energy bands as a function of temperature that might be associated to\nthe exotic properties of this system.\nPACS numbers: 74.70.Xa, 74.25.Jb, 71.20.-b, 79.60.-i\nI. INTRODUCTION\nStudy of superconductivity has seen an explosive\ngrowth since the discovery of high temperature super-\nconductivity in cuprates in 19861. Discovery of su-\nperconductivity in Fe-based systems2,3led to a resur-\ngence of interest in this direction, where charge car-\nrier doping in the parent compounds having spin den-\nsity wave (SDW) states to superconductivity via sup-\npression of long range magnetic order4. Interestingly,\nsome of these Fe-based compounds also exhibit pressure\ninduced superconductivity5enhancing the complexity of\nthe problems in these materials.\nAmong the Fe-based superconductors, AFe2As2(A =\nCa, Ba, SrandEu)classofmaterialsknownas‘122’com-\npounds can be grown easily with high quality and they\nhave been studied extensively in the recent past. These\nmaterials crystallize in the ThCr 2Si2type tetragonal\nstructure at room temperature, (space group I4/mmm).\nCaFe2As2is one such compound exhibiting spin den-\nsity wave (SDW) transition due to the long range mag-\nnetic ordering of the Fe moments at TSDW= 170 K\nalong with a concomitant structural transition to an or-\nthorhombic phase. High pressure5, substitution of Fe\nby Co, Ni6and other dopants induces superconductiv-\nity in CaFe 2As2(transition temperature, Tc∼15 K).\nThe SDW transition is found to accompany a nesting\nof the Fermi surface7,8along with a transition from two\ndimensional (2D) to three dimensional (3D) Fermi sur-\nface associated with the structural transition9–11. Fe 3d\nstates play a major role in the electronic properties of\nthese systems, while the doped holes in cuprates pos-\nsess dominant ligand 2 porbital character1. Clearly, the\nphysics of high temperature superconductors is complex\ndue to the significant differences among different classes\nof materials. In this article, we present our results on the\na)Correspondence to: kbmaiti@tifr.res.inelectronicstructureofCaFe 2As2obtained by high resolu-\ntion photoemission spectroscopy and show the signature\nof interesting gradual change in the electronic structure\nwith temperature that may be linked to the structural\nchanges of these materials. Such knowledge would be\nuseful to design new such materials with better specifi-\ncations for potential applications.\nII. EXPERIMENT\nHigh quality single crystals of CaFe 2As2were grown\nusing Sn flux. The grown crystals are flat platelet like,\nwhichcanbecleavedeasilyandthecleavedsurfacelooked\nmirror shiny. The sample quality was verified by var-\nious characterization methods such as x-ray diffraction\n(XRD) pattern and Laue pattern for determining the\ncrystal orientation, and energy dispersive analysis of x-\nrays (EDAX) for composition analysis. The XRD and\nLaue patterns were sharp and possess no spurious signal\nthat confirmed good crystallinity of the sample. EDAX\nresults showed the sample to be stoichiometric ensuring\ngood sample quality. A sharp transition to spin density\nwave state is observed at 170 K in both magnetic and\nspecific heat measurements.\nPhotoemission measurements were carried out us-\ning a Gammadata Scienta analyzer, R4000 WAL and\nmonochromaticphotonsources,Al Kα(hν=1486.6eV),\nHeI(hν= 21.2 eV) and He II(hν= 40.8 eV) sources.\nThe energy resolution and angle resolution were set to 2\nmeV and 0.3orespectively for ultraviolet photoemission\n(UP) studies and the energy resolution was fixed to 350\nmeV for x-ray photoemission (XP) measurements. The\ntemperature variation was carrier out using an open cy-\nclehelium cryostat,LT-3MfromAdvancedResearchSys-\ntems, USA.Thesamplewascleaved in situ(basepressure\n<3×10−11Torr) at each temperature several times to\nhave a clean well ordered surface for the photoemission\nstudies. Reproducibility of the data in both cooling and\nheating cycle was observed.2\n1 0 6 4 2 0710 708 706 44 42 40 \n(d) \n     10 K \n 100 K \n    300 K \n  \n(c)  \nBinding energy (eV) Int. (arb. units)  Fe d\n As p\n \n CA\nD B Expt  Int. (arb. units) \n Binding energy (eV) Fe 2 p(a) \n  \n   10 K \n300 K    10 K \n300 K \n As 3 d (b)    \nFIG. 1. (a) Fe 2 pand (b) As 3 dcore level spectra at 10 K\nand 300 K. (c) Valence band spectrum at 300 K (symbols).\nThe dashed line represents Fe 3 dand the solid line represents\nAs 4ppartial density of states. (d) Evolution of the valence\nband spectra near ǫFwith temperatures.\nThe energyband structure ofCaFe 2As2was calculated\nusing full potential linearized augmented plane wave\nmethod within the local density approximation (LDA)\nusing Wien2k software12. The energy convergence was\nachieved using 512 k-points within the first Brillouin\nzone. In order to calculate the band structure for the\nparamagnetic phase, we used the lattice parameters of\nthe tetragonal phase; a= 3.906˚A andc= 12.124 ˚A.\nIII. RESULTS\nAn overview of the electronic structure involving the\ncore levels and the valence band have been captured by\nx-ray photoemission spectroscopy with an energy resolu-\ntion of 350 meV. Fe 2 p3/2signal shown in Fig. 1(a) ex-\nhibits a highly asymmetric peak around 707 eV binding\nenergy typical for metallic Fe-like behavior - the asym-\nmetry arises due to the screening of the core holes in\nthe photoemission final states and associated low energy\nexcitations13,14. As 3dspectra shown in Fig. 1(b) are\nalso asymmetric exhibiting two distinct features around\n40.9 and 41.6 eV binding energies due to the spin-orbit\nsplitting. The feature at 44 eV binding energy is due\nto Ca 3score level photoemission. The origin of the\nweak shoulder at about 40.4 eV binding energy can be\nattributed to the surface effects15. Fe 2pand As 3 dspec-\ntra at 10 K and 300 K are almost identical indicating\nunchanged Madelung potential and/or effective valency\nof Fe and As in the whole temperature range despite the\nstructuraltransitionfromthe tetragonaltoorthorhombic\nphase at 170 K.\nThevalencebandspectrumcollectedusingAl Kαpho--1 0(b) px+ p y\n-1 0\nDJ (d) (d xz ,d yz ) O\u0015O\u0014\n-1 0\n* / 6 X PG(f) dz2\n-1 0\n* / 6 X P (g) dx2-y 2-6 -5 -4 -3 -2 -1 0\n* * / 6 ' H X P -1 0\n* / 6 X P D E\u000f\u0003J \nG(c) pz\n-1 0(e) dxy \nE(a) \nEnergy (eV) Energy (eV) Energy (eV) \nFIG.2. (a)Energybanddispersions calculatedusingFLAPW\nmethod. The orbital contributions of the bands near the\nFermi level are shown by the size of the symbols for (b) As\npx+py, (c) As pz, (d) Fe (3 dxz,3dyz) (e) Fe 3 dxy, (f) Fe 3 dz2\nand (g) Fe 3 dx2−y2states. In order to have clarity on As p\ncontributions, the corresponding symbol sizes are renorma l-\nized.\nton sourceat 300K is shownin Fig. 1(c). It exhibits four\ndistinct features denoted by A, B, C and D. A remark-\nably good representation of the experimental spectrum\nis given by the energy band structure of the tetrago-\nnal CaFe 2As2calculated by employing full potential lin-\nearized augmented plane wave method within the local\ndensity approximation (LDA) using Wien2k software12.\nThe orbitalpolarizationof the energybands areobtained\nby projecting the eigen states onto the constituting elec-\ntronic states, namely the Fe 3 dand As 4 pstates in the\npresent case. The calculated partial density of states\n(PDOS) shown by lines in the figure reproduce well the\nexperimental spectrum. The feature A exhibits domi-\nnant Fe 3 dcharacter and the As 4 pstates contribute at\nhigher binding energies. The hybridization between As\n4pand Fe 3 dstates is found to be strong. The tempera-\nture evolution of the valence band is shown in Fig. 1(d).\nA normalization by the spectral intensities in the energy\nrange beyond 1 eV binding energy exhibits a gradual en-\nhancement of the feature A intensity with the decrease\nin temperature.\nThe details of the calculated energy bands and their\norbital characters are shown in Fig. 2. The size of the\nsymbols represents the orbital contributions. The energy\nbands near the Fermi level, ǫFexhibit significant disper-3\nX X Γαβ, γ\nσHe II\nεF0.5 0.0He I\nαβ,γ\n0.5 0.0He IIα\nβ,γ\n800 400 0Γ\nXλσα\nβ, γ\nHe IIntensity (arb. units) \nΓ X Xαβ, γ\nσ\nHe IεF\n800 400 0Γ\nX300 K α\nβ, γ\nσ\nλ\nHe II\n Binding energy (meV) k||(1-10) (Å-1)Γ (0,0)X (π,π) M(0,π)\n k||(1-10) (Å-1)Intensity (arb. units) (a)\n(b)\n(c)(d) (e)\n(f) (g)\nFIG. 3. (a) A typical Brillouin zone of CaFe 2As2. MDCs at\n300 K employing (b) He IIand (c) He Iphoton energies. A\ntypical fit of the MDC at 140 meV to find peak positions in\nthe (d) He Iand (e) He IIphoton energies. EDCs at 300 K\nusing (f) He Iand (g) He IIphoton energies.\nsion (≥0.5 eV). Three bands denoted by α,βandγ\ncrossǫFaround the Γ point forming three hole-pockets.\nλ1andλ2bands form the electron pockets around the\nX-point. Fig. 2(b) and 2(c) show finite As 4 pcontri-\nbutions in these energy bands arising due to the Fe 3 d\n- As 4phybridizations. In Figs. 2(d) - 2(g), we show\nthe (dxz,dyz),dxy,dz2anddx2−y2contributions, respec-\ntively.αandγbands exhibit large ( dxz,dyz) symmetry,\nwhile the βband possesses primarily dxysymmetry. dz2\nstates appear as δbands shown in Fig. 2(f). The Fermi\nsurfaces correspondingto λ2andγbands possessing sim-\nilar symmetry are well nested.\nThe momentum distribution curves (MDCs) along Γ X\ndirection [see the Brillouin zone in Fig. 3(a)] obtained at\n300 K exhibit significant differences when probed with\nHeIIand He Iphoton energies as shown in Figs. 3(b)\nand 3(c), respectively. Crystal structure of CaFe 2As2is\ntetragonal at 300 K and it exhibits an essentially two-\ndimensional Fermi surface9. Thus, different photon ener-\ngies differing in the kzvalues (kz∼9.5π/cfor He Iand\n∼12.5π/cfor He II) will have a weak influence on the\nenergy position of the spectral features. In any case, the\nchange in the relative intensity of the spectral features\nwith the change in photon energy arises primarily due\nto the matrix element effect associated with the photo-\nexcitation process that reflects the orbital character of\nthe features16. The simulations of each of the MDCs re-\nquire at least two peaks representing α-band and ( β,γ)-\nbands as shown in Figs. 3(d) and 3(e) for the MDCs at\n140 meV binding energy. The large intensity of the α-\nband in the He II-spectra and its reduction with respect\nto the intensity of ( β,γ)-bands in the He Ispectra sug-0.4 0.8 \n150 100 50 00.4 0.8 0.0 0.4 0.8 \n0.8 0.4 0.0 0.0 0.4 0.8  \n  \nHe I\nHe II (b) \n 10 K \n 50 K \n150 K \n200 K \n300 K  Intensity (arb. units) \n Binding energy (meV) \n \n   Intensity (arb. units) He I\nHe II (a)  \n Binding energy (eV) \n  10 K \n300 K \nFIG. 4. (a) High resolution photoemission spectra at 10 K\nand 300 K using He Iand He IIphoton energies. (b) The\nsymmetrized spectral density of states (SDOS) of the He I\nand He IIspectra.\ngests larger As 4 p-contribution in ( β,γ)-bands relative\nto that in the α-band. This behavior is consistent with\nthe observation in other compounds in the same class of\nmaterials17. The influence of the matrix element is also\nevident in the energy distribution curves (EDCs) shown\nin Figs. 3(f) and 3(g).\nTheα,βandγbands form hole-pockets around the Γ\npoint at300K.The δbandspossessing dz2-symmetryap-\npear around 400 meV below ǫF. AtX-point, signature\nof the two bands denoted by σandλare observed in\nthe figure. The λband forms an electron pocket around\nX-point. The bandwidth of the experimentally observed\nenergy bands is significantly smaller than the calculated\nones manifesting the signature of electron correlation in-\nduced effects in the electronic structure.\nIn order to investigate the temperature evolution of\nthe electronic states at ǫFcritically, we employed high\nenergy resolution and the angle integrated mode for bet-\nter signal to noise ratio (acceptance angle ±15o) setting\nthek-vector along the Γ Xdirection. In such a case, the\nintensity at ǫFwill be contributed by the energy bands\ncrossing the Fermi level. The photoemission spectra ob-\ntained by He Iand He IIphoton energies at 10 K and 300\nK are shown in Fig. 4(a). All the spectra are normalized4Intensity (arb. units) \n160 80 0300 KE\u000f\u0003J \nD(d)\nE\u000f\u0003J He I\n*160 80 0300 K*X\nXD\nE\u000fJ (a) He II \n160 80 0*X\nX\n150 KDE\u000fJ (b)\nEVc He II \n160 80 010 KE\u000fJ \nD(c) \nEVc He II \n160 80 0D\nE\u000fJ \n10 K(f)\nEHe I\nVc \n160 80 0*\nXX\n150 KE\u000f\u0003J D(e) \nEHe I\nVc \n-0.6 0.0 0.610 K (h)\nDE\u000fJ He I100\n50 \n0\nIntensity (arb. units) \n-0.6 0.0 0.616080 010 K (g)D\nE\u000fJ Vc He II Intensity (arb. units) Binding energy (meV) Binding energy (meV) Binding energy (meV) \nX\n k || (1-10)  (Å -1 )\nFIG. 5. He IIEDCs along Γ Xat (a) 300 K, (b) 150 K and\n(c) 10 K. He IEDCs along Γ Xat (d) 300 K, (e) 150 K and\n(f) 10 K. Color plot of the same bands at 10 K using (g) He\nIIand (h) He Ienergies.\nby the intensity beyond 1 eV binding energy. The spec-\ntral changes are most profound in the He Ispectra while\nthe He IIspectra exhibit modification essentially at ǫF.\nThis is attributed to the higher degree of thermal sen-\nsitivity of the electronic states with porbital character\nrelative to that of the dcharacter.\nThe spectral intensity at ǫFcan be estimated quite ac-\ncurately by symmetrization of the experimental spectra\n(I(ǫ) =Iexpt(ǫ−ǫF)+Iexpt(ǫF−ǫ);Iexptis the experi-\nmental spectrum). The symmetrized spectra at different\ntemperatures, shown in Fig. 4(b), exhibit a jump in the\nintensity at ǫFacrossthe SDW transition temperature in\nboth He IandHe IIspectraasexpected. Furtherlowering\nof temperature leads to an unusually sharp dip below 100\nK although no phase/structural transitions have been re-\nported in this temperature range.\nThe details of the changes can be found in the angle\nresolved data. The energy bands obtained by He Iand\nHeIIexcitations are shown in Fig. 5 along Γ Xdirection\nat 300 K, 150 K and 10 K. The spectra exhibit an inter-\nesting evolutionacrossthe SDW transition at 170K. The\nsignature of the hole pocket corresponding to the αband\ndistinctly seen in the 300 K He IIspectra vanishes at 150\nK with the top of the band appearing below ǫF. In ad-\ndition, the intensity around 80 meV grows significantly\nin the 150 K data relative to the 300 K data suggest-ing SDW phase transition induced band-folding of the γ\nandλbands consistent with the literature18. Such folded\nband denoted by σ′in the figure appears due to the nest-\ning of the Fermi surfaces corresponding to the γandλ\nbands in the SDW phase. The signature of the σ′band\nis also observed in the He Ispectra shown in Fig. 5(e).\nHowever, the hole pocket corresponding to the α-band\nsurvives in the He Ispectra ( kz≈9.5π/c), which is dif-\nferentfromtheHe IIspectra( kz≈12.5π/c). Thisreflects\nthekz-dependence of the α-band Fermi surface undergo-\ning transition from the effective two dimensional nature\nof the Fermi surface to a three dimensional one with the\nα-band hole pocket centering around kz∼2(2n+1)π/c\nand its absence around 4 nπ/cin thek-plane containing\nkzaxis8,9.\nFurtherdecreasein temperature revealsanunusualbe-\nhavior; the αband shifts further below in energy. The\ngradual energy shift is most evident in the spectral den-\nsity of states (SDOS) obtained from He IIspectra shown\nin Fig. 5(c), where the Fe 3 dstates contribution is sig-\nnificantly enhanced in the photoemission signal due to\nthe matrix element effect. The top of the α-band moves\nbelow the Fermi level at 10 K even in the He Ispec-\ntra shown in Figs. 5(f) and 5(h) suggesting vanishing of\nthe corresponding Fermi surface even at about 9.5 π/c,\nwhich is close to the middle of the α-band hole pocket\nin reciprocal plane containing kz-axis. The β-band still\ncrossesǫF, which is observed most prominently in the He\nIspectra indicating its large pcharacter.\nIV. DISCUSSION\nTreating electron correlation induced effects in the\nstrongly correlated electron systems has been a major\nchallengein the contemporarycondensed matter physics.\nIt is observed that the spectral functions obtained by the\ndynamical mean field theory often provide a good de-\nscription of the experimental scenario19,20. Within this\ndescription, the spectral function of a correlated system\nin the intermediate coupling regime consists of three fea-\ntures; upper (unoccupied) and lower (occupied) Hub-\nbard bands representing the correlation induced local-\nized states and a central band at the Fermi level, termed\nas the coherent feature representing the itinerant carri-\ners often captured well by the ab initio band structure\ncalculations21. The decrease in temperature leads to an\nenhancement of the coherent feature intensity at the cost\nof the Hubbard band intensities19,22. The temperature\ninduced changes in the valence band shown in Fig. 1\nexhibit quite similar scenario, where the intensity at the\nFermi level, well reproduced in the band structure cal-\nculations, increases in intensity relative to the intensities\nat higher binding energies with the decrease in tempera-\nture. Effect of the electron correlation is also manifested\nby the narrowing of the energy bands shown in Fig. 3.\nWe calculated the effective mass, m∗/me(me= mass\nof an electron) corresponding to αandβ,γbands, and5\nfound it to be 1.4 and 3.2, respectively at 300 K suggest-\ning moderate mass enhancement due to electron corre-\nlation induced effects supporting the above view. The\nenergy bands close to the Fermi level are the antibond-\ning bands arising due to the hybridization between Fe 3 d\nand As 4 pstates. While these eigen states possess large\nFe 3dcharacter, As 4 pcontributions appear to be sig-\nnificant, which is observed in the band structure results\nshown in Fig. 2 and comparison of the experimental He\nI& He IIspectra shown in Fig. 3. Since the As lay-\ners are above and below the Fe-layers, the hybridization\nof Aspstates with the Fe dxz,dyzstates will be strong\nand plays an important role in the SDW phase transi-\ntion. This is manifested via the nesting of the Fermi\nsurface corresponding to the γandλbands possessing\n(dxz,dyz) symmetry. Signatureofthe Fermi surface nest-\ning observed in the present results is consistent with the\nearlier studies. The importance of the coupling of the\nelectronic states with the lattice degrees of freedom has\nbeen found in this class of compounds via inverse isotope\neffect23, spin-phonon coupling24,25, phonon softening26,\nsensitivity of the SDW phase to the anomalous phonon\ndispersion27etc. The photoemission and band structure\nresults provide a direct evidence of the importance of the\nhybridization and the electron correlation effects in the\nelectronic properties of these systems.\nAt 150 K, which is below the SDW transition temper-\nature, the top of the αband appears below ǫFin the\nHeIIspectra ( kz∼12.5π/c) while it is still crossing\nǫFin the He Ispectra ( kz∼9.5π/c). This suggests a\nkz-dependence of the α-band Fermi surface, which is a\nsignature of the three dimensionality of the Fermi sur-\nface consistent with the earlier findings9. Suchkzde-\npendence can be attributed to the structural transition\nfrom tetragonal to orthorhombic phase occurring at the\nsame SDW transition temperature. While energy bands\nat 300 K are narrower than their band structure results\ndue to electron correlation induced effects, the spectra at\n10 K exhibit further narrowing of the dispersions. We\ncalculated the effective mass, m∗/mecorresponding to α\nandγbands and found it to increase to 2.6 and 4.3 at 10\nK, respectively. This suggests that the electrons acquire\nmore local character at lower temperatures.\nInterestingly, the αband shifts below ǫFat 10 K even\nin the He Ispectra as shown in Fig. 5. This is unusual as\nthe He Ienergy corresponds to the kzvalue close to the\nmiddle of the α-band hole pocket. In order to exemplify\nthis point, we compare the EDCs at Γ point at different\ntemperatures obtained by He Iand He IIphoton energies\ninFig. 6exhibitingdistinct signatureofagradualshiftof\ntheαband with the decrease in temperature. Sharp dip\nobservedinthe highresolutionspectramaybe attributed\nto the shift of the αband completely below ǫFwhich\nmeans that the α-band Fermi surface disappears around\n10 K.\nTwo conclusions can be drawn from the above results.\n(i) Although the structural transition from tetragonal to\northorhombic phase occurs at the onset of SDW phase150 10050 015 K\n150 K\n300 KDD(a) \nVc He  II\nVc \n150 10050 0Vc (b) He I\n300 K150 K10 K\nD\nDIntensity (arb. units) \nBinding energy (meV) \nFIG. 6. Temperature evolution of the (a) He IIand (b) He I\nspectra at the Γ point.\ntransition, the transition is slow with the coexistence of\ntwo structural phases in the intermediate temperature\nrange as observed in other systems such as manganites28.\n(ii) The second observation is the signature of the van-\nishing of the Fermi surface corresponding to the αband\nand its consequences on the energy band, βthat crosses\nthe Fermi level in order to preserve charge count.\nIt is to note here that Lifshitz transition, a tran-\nsition from a hole-type Fermi surface to electron-type\nFermi surface or vice versa due to subtle change in\ncharge carrier concentration has been found to be im-\nportant in cuprates29as well as in electron doped\nBa(Fe1−xCox)2As218. Evidently, such Fermi surface re-\nconstruction seem to be important in CaFe 2As2too sug-\ngesting a generic nature of such an effect in the un-\nconventional superconductors. It is to be noted here\nthat a homologoussystem, SrFe 2As230exhibits largevol-\nume fraction of superconducting region even at ambient\npressure if the prepared sample possess significant struc-\ntural strain. In CaFe 2As2, it is observed5that a quasi-\nhydrostatic pressure leads to superconductivity suggest-\ning the importance of structural strain for such exotic\nground state.\nV. CONCLUSIONS\nIn summary, we have studied the evolution of the elec-\ntronic structure of CaFe 2As2, a parent compound of the\nFe-basedsuperconductorsemployinghighresolutionpho-\ntoemission spectroscopy. We discover a sharp dip at the\nFermi level appearing much below the SDW transition\ntemperature in addition to the dip associated with the\nSDW transition. Angle resolved photoemission results\nexhibit signature of a gradual shift of the αband with\nthedecreaseintemperature-the holepocketcorrespond-\ning to the αband vanishes near 10 K indicating renor-\nmalization of the βband hole pocket to preserve charge\ncount. These results, thus, reveal signature of interesting\nFermi surface reconstructions in these complex systems\nthat might be responsible for exotic properties of these\nmaterials.6\nVI. ACKNOWLEDGEMENTS\nThe authors, K. M. and N. S. acknowledge financial\nsupport from the Department of Science and Technology,\nGovernment of India under the ‘Swarnajayanti Fellow-\nship programme’. K. M. acknowledges the Department\nof Atomic Energy, Government of India for financial sup-\nport.\n1A. Damascelli, Z. Hussain, and Z. X-. Shen, Rev. Mod. Phys. 75,\n473 (2003).\n2Y. Kamihara et al. J. Am. Chem. Soc. 128, 10012 (2006).\n3Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.\nChem. Soc. 130, 3296-3297 (2008).\n4K.Ishida, Y.Nakai, andH.Hosono, J.Phys.Soc.Jpn. 78, 062001\n(2009).\n5T. Park et al., J. Phys.: Condens. Matter 20, 322204 (2008); H.\nLee et al., Phys. Rev. B 80, 024519 (2009); S.-H. Baek et al.,\nPhys. Rev. Lett. 102, 227601 (2009).\n6N. Kumar et al., Phys. Rev. B 79, 012504 (2009); N. Kumar, et\nal. Phys. Rev. B 80, 144524 (2009).\n7T. Kondo, et al. Phys. Rev. B 81, 060507(R), (2010).\n8Q. Wang, et al. arXiv:1009.0271v1.\n9C. Liu, et al. Phys. Rev. Lett. 102, 167004 (2009).\n10S. Thirupathaiah, et al. Phys. Rev. B 84, 014531 (2011).\n11S. de Jong, et al. Europhys. Letts. 89, 27007 (2010).\n12P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J.\nLuitz, [WIEN2k An Augmented Plane Wave + Local OrbitalsProgram for Calculating Crystal Properties ] [Schwarz, K. (ed.)]\n(Techn. Universit¨ at Wien, Austria, 2001).\n13J. F. van Acker, et al.Phys. Rev. B 37, 6827 (1988).\n14M. Takahashi and J.-I. Igarashi, Phys. Rev B 85, 085128 (2012).\n15S. Thakur, et al. J. Appl. Phys. 114, 053904 (2013); S. Patil, G.\nAdhikary, G. Balakrishnan, and K. Maiti, Solid State Commun .\n151, 326-328 (2011); K. Maiti, et al. Phys. Rev. B 73, 052508\n(2006);ibid.Phys. Rev. B 70, 195112 (2004).\n16J. J. Yeh and I. Lindau, At. Data and Nucl. Data Tables 32, 1\n(1985); as observed in varieties of oxides - for example, K. M aiti,\nP. Mahadevan, and D. D. Sarma, Phys. Rev. B 59, 12457 (1999);\nK. Maiti and D. D. Sarma, Phys. Rev. B 58, 9746 (1998).\n17G. Adhikary, et al. J. Phys.: Condens. Matter 25, 225701 (2013).\n18C. Liu,et al. Nat. Phys. 6, 419 (2010).\n19A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev.\nMod. Phys. 68, 13 (1996);\n20M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70,\n1039-1263 (1998).\n21K. Maiti, et al. Europhys. Letts. 55, 246-252 (2001).\n22K. Maiti, R. S. Singh, and V. R. R. Medicherla, Europhys. Lett .\n78, 17002 (2007)\n23P. M. Shirage, et al. Phys. Rev. Lett. 103, 257003 (2009).\n24P. Kumar, et al. J. Phys.: Condens. Mater 23, 255403 (2011).\n25P. Kumar, et al. Solid State Commun. 150, 557 (2010).\n26P. Kumar, et al. J. Phys.: Condens. Mater, 22, 255402 (2010).\n27R. Mittal, et al. Phys. Rev. Lett. 102, 217001 (2009).\n28R. Bindu, et al. Phys. Rev B 86, 140104(R) (2012); ibid., Phys.\nRev. B.84, 052407 (2011); ibid., New J. Phys. 12, 033026 (2010).\n29K. -S. Chen, Z. Y. Meng, T. Pruschke, J. Moreno, and M. Jarrell ,\nPhys. Rev. B 86, 165136 (2012).\n30S. R. Saha, N. P. Butch, K. Kirshenbaum, J. Paglione, and P. Y.\nZavalij, Phys. Rev. Lett. 103, 037005 (2009)."    },    {        "title": "1709.09854v1.Enhanced_ultrafast_relaxation_rate_in_the_Weyl_semimetal_phase_of___mathbf_MoTe_2___measured_by_time_and_angle_resolved_photoelectron_spectroscopy.pdf",        "content": "Enhanced ultrafast relaxation rate in the Weyl semimetal phase of MoTe 2measured by time-and\nangle-resolved photoelectron spectroscopy\nA. Crepaldi,1G. Aut `es,1, 2G. Gatti,1S. Roth,1A. Sterzi,3G. Manzoni,3M. Zacchigna,4C. Cacho,5R.\nT. Chapman,5E. Springate,5E. A. Seddon,6, 7Ph. Bugnon,1A. Magrez,1H. Berger,1I. V obornik,8\nM. Kall ¨ane,9A. Quer,9K. Rossnagel,9F. Parmigiani,3, 10, 11O. V . Yazyev,1, 2and M. Grioni1\n1Institute of Physics, Ecole Polytechnique F ´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\n2National Centre for Computational Design and Discovery of Novel Materials MARVEL,\nEcole Polytechnique F ´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland\n3Universit ´a degli Studi di Trieste - Via A. Valerio 2, Trieste 34127, Italy\n4C.N.R. - I.O.M., Strada Statale 14, km 163.5, Trieste 34149, Italy\n5Central Laser Facility, STFC Rutherford Appleton Laboratory, Harwell OX11 0QX, United Kingdom\n6The Photon Science Institute, The University of Manchester, Manchester, United Kingdom\n7The Cockcroft Institute, Sci-Tech Daresbury, Daresbury, Warrington, United Kingdom\n8CNR. - IOM., Strada Statale 14, km 163.5, Trieste 34149, Italy\n9Institut f ¨ur Experimentelle und Angewandte Physik, Christian-Albrechts-Universit ¨at zu Kiel, D-24098 Kiel, Germany\n10Elettra - Sincrotrone Trieste S.C.p.A., Strada Statale 14, km 163.5, Trieste 34149, Italy\n11International Faculty - University of K ¨oln, 50937 K ¨oln, Germany\n(Dated: September 29, 2017)\nMoTe 2has recently been shown to realize in its low-temperature phase the type-II Weyl semimetal (WSM).\nWe investigated by time- and angle- resolved photoelectron spectroscopy (tr-ARPES) the possible influence of\nthe Weyl points in the electron dynamics above the Fermi level EF, by comparing the ultrafast response of\nMoTe 2in the trivial and topological phases. In the low-temperature WSM phase, we report an enhanced relax-\nation rate of electrons optically excited to the conduction band, which we interpret as a fingerprint of the local\ngap closure when Weyl points form. By contrast, we find that the electron dynamics of the related compound\nWTe 2is slower and temperature-independent, consistent with a topologically trivial nature of this material. Our\nresults shows that tr-ARPES is sensitive to the small modifications of the unoccupied band structure accompa-\nnying the structural and topological phase transition of MoTe 2.\nThe recent discovery of Weyl fermions as low-energy\nquasiparticles in TaAs [1–3] and other related compounds\n[4, 5] has boosted the interest in topological semimetals\n(TSMs) [6]. Type-II Weyl semimetals (WSMs) are a novel\nclass of materials that host fermions violating Lorentz invari-\nance [7]. These quasiparticles are realized in the strongly\ntilted cones that form in momentum space around special\nWeyl points (WPs) where the valence and conduction bands\ntouch. WTe 2[7] and MoTe 2[8–10] have been proposed\nas possible type-II WSMs. While the topological phase of\nWTe 2is still under debate, the existence of the WSM phase in\nthe low temperature non-centrosymmetric structure of MoTe 2\nis supported by the observation of surface Fermi arcs [11–\n15]. The Weyl points, however, are located above the Fermi\nlevel EFand this hinders a direct observation by conventional\nangle-resolved photoelectron spectroscopy (ARPES). To cir-\ncumvent this difficulty, one would need a probe that is sensi-\ntive to the presence/absence of small energy band gaps in the\nunoccupied density of states.\nIn this Letter we show that time- and angle-resolved pho-\ntoelectron spectroscopy (tr-ARPES) can provide such infor-\nmation. We report a shortening of the relaxation time in the\ngapless type-II Weyl phase of MoTe 2, which reflects the en-\nhanced interband scattering from the conduction band (CB) to\nthe valence band (VB) mediated by electron – electron scat-\ntering along the Weyl cone. These scattering processes are\nactive only when the band gap is locally closed. This conclu-\nsion is supported by the observation of a slower, temperature-independent dynamics in WTe 2, indicative of a local direct\nband gap, which acts as an effective bottleneck for the elec-\ntron relaxation.\nMoTe 2is a layered material. It can be cleaved to expose\nlarge flat (001) terraces, ideal for ARPES studies. Figure 1\nillustrates the WSM phase, which is only realized in the low-\ntemperature orthorhombic (space group P mn21) structure –\nhereafter referred to as the 1T’ phase [17] – where inver-\nsion symmetry is broken. The crystal structure is sketched in\nFig. 1 (a), along with the 3D Brillouin zone (BZ) (Fig. 1 (b)).\nAbove 250K the 1T’ structure transforms to the centrosym-\nmetric monoclinic (space group P 12=m1) 1T” structure [16]\nRaman spectroscopy and resistivity measurements indicate a\nstructural phase transition at T* \u0018257:5K [18]. Details about\nthe crystals growth and their characterization can be found in\nthe Supplementary Information [19].\nThe ARPES experiments in the UV photon energy range\nhave been performed at the APE beamline at Elettra, and soft\nx-ray ARPES experiments at the end station ASPHERE III of\nbeamline P04 at PETRA III (DESY). The tr-ARPES experi-\nments have been carried out at the ARTEMIS facility, using\nultrafast pulses at 17:5eV photon energy, produced by laser-\ninduced high harmonics generation (HHG) in a gas. The op-\ntical excitation was driven by a 2eV pump pulse with fluence\n\u00180:3 mJ=cm2. The energy and temporal resolutions were\n150meV and 50fs, respectively [19].\nFigure 1 (c) illustrates the band structure of the 1T’ phase\nalong the \u0000Xdirection. The VB consists of several hole-likearXiv:1709.09854v1  [cond-mat.mtrl-sci]  28 Sep 20172\n(a)\n(b)\nkx ky kz \nT e\nMo\n2000 1000 01T'\n(d)E - E    (eV)F\n0 0.4 0.2 0 0.2 0.400.08\n-0.08k  (A  )y°-1\n00.08\n-0.081T ' phase, T= 50K 1T '' phase, T = 300K\nCB\nVB\n0 0.2 0.4 0.6 0.8\nk  (A  )x°-1(e) (f)\n(g) (h)\n(i) (j)\nCB\nVB+50 meV +50 meVCB VB CB VBWP\n+5 meV +5 meVCB VB CB VBWP \nWP\n6  BZth\nCBSS\nVB0\n-0.4-0.2\nk  (A  )x°-10 0.2 0.4 0.6(c)\nk  (A  )x°-1k  (A  )x°-10 0.4 0.2 0 0.2 0.4\nk  (A  )x°-1k  (A  )x°-1k  (A  )y°-1\n0 0.01 0.1 10 100CB\nVB\nFIG. 1: Crystal structure (a) and three-dimensional Brillouin zone (b) of 1T’ MoTe 2[16]. (c, d) Experimental and calculated band dispersion\nof 1T’ MoTe 2along the \u0000Xhigh-symmetry direction. (e - h) Calculated k-resolved bulk density of states, projected on the [001] surface\nBZ, for the low- (e, g) and high- (f, h) temperature phases of MoTe 2, at two different binding energies. The presence of two pairs of WPs\nis revealed in the 1T’ phase, at 50 meV (e) and 5 meV (g), respectively. In the centrosymmetric 1T” phase, the WPs vanish and the VB and\nthe CB are well separated in momentum and in energy, as shown in the the CEMs at 50 meV (f) and 5 meV (h). (i, j) The cartoon shows that\nelectron - electron interaction can mediate the scattering between CB and VB along the Weyl cone in the WSM phase (i). Interband scattering\nrequires momentum and energy exchange in the presence of the gap (j).\nstates, in good agreement with the literature [11–14, 20]. The\nbottom of the CB forms a shallow pocket around kx=\u00060:4\n˚A\u00001, Fig. 1 (c). A detailed mapping of the bulk band structure\nis shown in [19]. Our ab initio band structure calculations of\n1T’MoTe 2predict four pairs of WPs of two different types\nwhere the VB and the CB touch at kx\u00180:2˚A\u00001, slightly off\nthe\u0018\u0000Xdirection, respectively 5meV and 50meV above EF\n(Fig. 1 (d)). The WPs are located near the local minimum of\nthe CB, which suggests the possibility of electron accumula-\ntion. Their ( kx;ky) locations are illustrated in Fig. 1 (e, g). By\ncontrast, a similar calculation yields no WP in the trivial high-\ntemperature 1T” phase, where a local gap opens between the\nVB and the CB, Fig. 1 (f, h) [19].\nUnlike conventional ARPES, tr-ARPES can probe states\nabove EF. Namely, a recent measurement has provided indi-\nrect hints of the presence of Weyl nodes in Mo0:25W0:75Te2\nvia a line shape analysis [21]. Following a different approach,\nwe have measured by tr-ARPES the out-of-equilibrium dy-\nnamics of MoTe 2, which we expect to change across the topo-\nlogical phase transition. The rationale for our experiment is\nillustrated by Fig. 1 (i, j) which shows a schematic view of the\nbands near a type-II WP. When the VB and the CB touch and\na WP appears in the WSM phase, electrons can scatter from\nthe CB to the VB via electron – electron scattering along the\nWeyl cone, as schematized in Fig. 1 (i). In the trivial phase the\nlocal gap between the VB and the CB is a bottleneck for the\nrelaxation of electrons optically excited in the CB, Fig. 1 (j).\nElectrons can thermalize within the CB through electron –\nelectron scattering, but interband scattering towards the VB\nrequires the exchange of energy and momentum and is medi-\nated by phonons, with a corresponding longer relaxation time.\nFigure 2 (a) illustrates the band dispersion along theky= kWPdirection measured 200fs before optical excita-\ntion. The comparison of the data at low and high temper-\nature shows no significant differences between the occupied\nelectronic structure for the 1T’ and 1T” phases. This is con-\nfirmed by the high-resolution ARPES data [19] and by a de-\ntailed temperature dependent study [20]. The effect of the\noptical excitation in the low-temperature 1T’ phase is illus-\ntrated in Fig. 2 (b). It shows the difference between data taken\n80fs after and 200fs before the arrival of the pump pulse. The\n1T '  T = 50K\n1T''T = 300Kfit 1T '  T = 50K\nfit 1T '' T = 300K = 320   20 fs  = 200   10 fs \n = 355   20 fs  = 205   10 fs \n = 465  20 fs  = 325  10 fs \nt = 80 fs0\n-0.50.5\n-1.0\n-1.5\n-2.0E - E    (eV)F1T ', \nT=50K\nk  (A  )x°-10 -0.3 0.3 0 -0.3(a) (b)\n0.30Intensity (arb. unit)0.40.81.5\n00.51.0\n10 0\n10 0\n800 400 0\ndelay time (fs)800 400 0\ndelay time (fs)(e)\n+-+-+-\n = 540   25 fs  = 375   20 fs (f)\n+-+- +-+- +-1T '',\nT=300K\n1\n2\n3\n41T ', T=50K\nk  (A  )x°-1t = -200 fs(c) (d)\n1.5\n00.51.0\n00.40.81.2\nFIG. 2: (a) Band dispersion of MoTe 2along \u0000Xat low- (left) and\nhigh- (right) temperatures, measured with 17.5 eV HHG photons\n200n fs before optical excitation. (b) Effect of optical excitation on\n1T’MoTe 2. Difference between ARPES images acquired 80 fs af-\nter and 200 fs before the arrival of a 2eV optical excitation with\nfluence \u00180:3 mJ=cm2. Red and blue indicate the increase and de-\ncrease in the electron population, respectively. (c - f) Dynamics in\nthe regions 1 - 4 of panel (b) near the WPs for the 1T’ (blue) and 1T”\n(red) phases. A fit to the traces (dashed green and black lines) yields\nquantitative estimates of the characteristic relaxation times, \u001c.3\n T = 50K\n T = 300Kfit  T = 50K\nfit  T = 300K(a)\n0\n-0.50.5\n-1.0\n-1.5\n-2.0E - E    (eV)FT = 50K T = 300K T = 50K\nk  (A  )x°-10 -0.3 0.3\nk  (A  )x°-10 -0.3E - E    (eV)F0\n-0.5\n0\nk  (A  )x°-10.4 0.20 0.2 0.400.08\n-0.08k  (A  )y°-100.08\n-0.08+50 meV +50 meV\n+5 meV +5 meVCB VB\nCB VBt = -200 fs\n0 0.2 0.4 0.6 0.8\nk  (A  )x°-1\n(b) (c)\n(d) (e)(f)\n0.3t = 80 fs (g)\n = 900   50 fs  = 960   50 fs \n = 1070   50 fs  = 1060   50 fs \n0Intensity (arb. unit)0.5\n00.5\n1000 500 0\ndelay time (fs)+-+-+-+-\n1000 500 0\ndelay time (fs)\n = 1180   50 fs  = 1230   50 fs \n = 1320   50 fs  = 1350   50 fs (j) (k)\n+-+ -+- +-\n00.51.0\n00.51.0\n1000 500 0\ndelay time (fs)1000 500 0\ndelay time (fs)(h) (i) (l)\nCB VB\nCB VB\n10 0 10 0\nCB\nVB1\n2\n3\n4\n T = 50K T = 300K\n0 0.01 0.1 10 100phonon\nEgh  >E gphonon   he - e\ne - phMoT e 1T'2\nMoT e  1T''2\n  WT e2\nCBVB\nCB\nVB\nCB\nVB\nFIG. 3: (a) Band structure of WTe 2along \u0000Xcalculated for the low-temperature crystal lattice parameters. (b - e) Calculated k-resolved bulk\ndensity of states, projected on the [001] surface BZ, at two different binding energies for the crystal structure at low and high temperatures.\nUnlike the case of MoTe 2the CB and VB are always well separated in momentum and energy. (f) Band structure along \u0000XforWTe 2\nmeasured at 50K (left) and at 300K (right) with 17:5eV HHG photons. (g) The difference between ARPES images acquired 80 fs after and\n200 fs before the arrival of a 2eV optical excitation with fluence \u00180:3 mJ=cm2. (h - k) Comparisons of the dynamics in regions 1 - 4 at\nlow (blue) and high (red) temperature. A fit to the traces (dashed green and black lines) yields quantitative estimations of the characteristic\nrelaxation time, \u001c, which is three times larger than the gapped 1T” MoTe 2. (l) Close-up on the calculated band structures of 1T’ MoTe 2, 1T”\nMoTe 2andWTe 2in the region of the WPs. The possible electron – electron and electron – phonon scattering events are shown schematically.\ncolor scale encodes the transfer of electrons from the occupied\n(blue) to the unoccupied (red) density of states, i.e. the tran-\nsient electronic population in the optically excited state.\nFigure 2 (c-f) presents our main results. The blue and red\nlines show the temporal evolution of the electronic population\nextracted from the differential intensity map of Fig. 2 (b) and,\nrespectively, from analogous data measured at 300K in the\n1T” phase. The signal is integrated in the small rectangular\nregions labeled ‘1’ to ‘4’. The data show a clear shortening of\nthe relaxation time at the lower temperature, predicted by the\nscenario of Fig. 1(i) as consequence of the local gap closing\nupon the emergence of the Weyl nodes in the 1T’ phase.\nFor a quantitative analysis we fitted each curve by a step\nfunction multiplied by a single decaying exponential, and\nbroadened by a resolution-limited gaussian (green and black\ndashed lines). We find that the characteristic relaxation times\n\u001cextracted from the fits are \u001830\u000050% larger in the 1T”\ntopologically trivial phase (black lines) than in the 1T’ WSM\nphase (green lines), depending on the region of the band struc-\nture where the intensity is analyzed. Possible changes of\nthe electronic band structure and band velocity near the WPs\ncould affect the electron dynamics. However, our calculations\nshow that the only noticeable change is the emergence of WPs\n[19]. Therefore we conclude that any trivial changes of the\nband structure have only a minor effect, and that the observed\nenhanced scattering rate mainly reflects the local gap closure.\nThe data of Fig. 2 show that the dynamics of the excited\nelectronic population is sensitive to the change in the un-\noccupied band structure. The formation of the Weyl nodes\nshort-circuits the local bottleneck in the dynamics of the triv-\nial phase. There is a clear analogy between the fast dynam-ics at the WPs and the role played by conical intersections in\nthe non-adiabatic relaxation of chemical systems through non-\nradiative transitions [22]. The bottleneck behavior associated\nwith the formation of a local band gap has been reported in\nthe out-of-equilibrium properties of high-temperature super-\nconductors [23] and charge-density-wave systems [24]. It is\nalso similar to the slowdown of the dynamics in gapped bi-\nlayer graphene with respect to gapless single layer graphene\n[25].\nHaving identified the change in electron dynamics across\nthe topological phase transition in MoTe 2, we now turn to the\nrelated compound WTe 2, whose topological nature is still un-\nder debate [26, 27]. Even high-resolution laser-based ARPES\ncould not provide evidence of topological Fermi arcs [27] or\nof WPs, which, if present, would be located above EF. Fig-\nure 3 summarizes our results for WTe 2. The calculated elec-\ntronic structure shows no WPs. The band structure calculated\nfor both the experimental low-temperature ( 113K) [28] and\nhigh-temperature ( 300K) [29] lattice parameters predict the\nCB and VB to be always well separated in energy and momen-\ntum [19], Fig. 3 (a - e). Unlike MoTe 2,WTe 2has no struc-\ntural phase transition and remains in the non-centrosymmetric\n1T’ phase. The changes in the lattice constants between low\nand high temperature have only a small effect on the band dis-\npersion, which cannot be resolved in our experiment. By com-\nparing WTe 2andMoTe 2, we find two important differences\nin the dynamics of WTe 2: i) the relaxation time is tempera-\nture independent; ii) the dynamics is much slower than in the\ngapped 1T” MoTe 2phase.\nA quantitative analysis, summarized in Fig. 3 (h - k) yields\ncharacteristic times that are \u00183times larger than for 1T”4\nMoTe 2. This longer relaxation time is the signature of the\nlocal gap, which acts as a bottleneck for the electron dynam-\nics, as similarly observed for the case of 1T” MoTe 2. All\nthese results are consistent with the topologically trivial na-\nture of WTe 2. Notice that WTe 2andMoTe 2have very sim-\nilar phonon dispersions [30] and comparable Debye [31, 32]\nand superconducting [33, 34] temperatures. Hence, electron -\nphonon coupling alone cannot explain the different tempera-\nture dependence measured in the two materials.\nFigure 3 (1) further illustrates the scenario that emerges\nfrom the tr-ARPES results. It sketches the relevant scattering\nmechanisms that determine the ultrafast dynamics in the two\nphases of MoTe 2and in WTe 2. Blue and red indicate electron\n– electron and electron – phonon scattering processes, respec-\ntively. The former can efficiently scatter electrons from the\nCB to the VB only in the WSM 1T’ phase of MoTe 2. Inter-\nestingly, the calculated bands of 1T” MoTe 2andWTe 2show\nthat the local band gap is different in the two systems. The\ngap of 1T” MoTe 2is indirect and the material is, also at this\nlocal level, a semimetal, so that the scattering between the CB\nand the VB can be mediated by phonons with no energy con-\nstraint. By contrast, WTe 2is locally a semiconductor, with\na local direct band gap EG= 12(8) meV , for the high-(low)\ntemperature crystal parameters. As a consequence, the low-\nest energy optical phonons with h\u0017 < 4meV [30] cannot\ncontribute to the interband scattering. Hence, we attribute\nthe longer relaxation time observed in WTe 2to the reduced\nphase space available for the electron – electron and electron\n– phonon interband scattering processes between CB and VB.\nThis confirms the sensitivity of tr-ARPES to the small details\nof the unoccupied band structure.\nIn summary, tr-ARPES results bear a fingerprint of the\ntopological transition in the relaxation dynamics of MoTe 2.\nWe have measured an enhancement of the relaxation rate\nwhen the local gap closes and Weyl points emerge in the low-\ntemperature Weyl semimetal phase of MoTe 2. In comparison,\nthe dynamics of WTe 2exhibits a slower dynamics which is\ncompatible with a topologically trivial phase. Our work shows\nthat the measured ultrafast dynamics is sensitive to small de-\ntails of the unoccupied band structure, which are difficult to\ndetect by other methods. Our results show that tr-ARPES can\nprobe the consequences of topological order in novel materi-\nals. We expect ongoing developments of theoretical micro-\nscopic models of out-of-equilibrium electron dynamics to fur-\nther strengthen the impact of this emerging technique.\nWe acknowledge financial support by the SNF. The re-\nsearch leading to these results has received funding from\nLASERLAB-EUROPE (grant agreement no. 654148, Eu-\nropean Union’s Horizon 2020 research and innovation pro-\ngramme). This work was supported in part by the Italian Min-\nistry of University and Research under Grant Nos. FIRBR-\nBAP045JF2 and FIRB-RBAP06AWK3 and by the European\nCommunity Research Infrastructure Action under the FP6\n”Structuring the European Research Area” Program through\nthe Integrated Infrastructure Initiative ”Integrating Activity on\nSynchrotron and Free Electron Laser Science”, Contract No.RII3-CT-2004-506008. This work has been partly performed\nin the framework of the nanoscience foundry and fine analysis\n(NFFA-MIUR Italy) project. Parts of this research were car-\nried out at the light source PETRA III at DESY , a member of\nthe Helmholtz Association (HGF). We would like to thank G.\nHartmann, J. Buck, F. Scholz, J. Seltmann and J. Viefhaus for\nassistance in using beamline P04, and H. Bentmann, S. Rohlf,\nF. Diekmann, S. Jarausch and T. Riedel for support in using\nthe endstation ASPHERE III. G.A. and O.V .Y . acknowledge\nsupport by the NCCR Marvel and the ERC Starting Grant\n”TopoMat” (Grant No. 306504). First-principles calculations\nhave been performed at the Swiss National Supercomputing\nCentre (CSCS) under Project No. s675. We thank Phil Rice\nfor technical support.\n[1] S.-Y . Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian,\nC. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, et al., Sci-\nence 349, 613 (2015).\n[2] L. X. Yang, Z. K. Liu, Y . Sun, H. Peng, H. F. Yang, T. Zhang,\nB. Zhou, Y . Zhang, Y . F. Guo, M. Rahn, et al., Nat. Phys. 11,\n728 (2015).\n[3] B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang,\nL. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, et al., Nat. Phys.\n11, 724 (2015).\n[4] N. Xu, H. M. Weng, B. Q. Lv, C. E. Mat, J. Par, F. Bisti, V . N.\nStrocov, D. Gawryluk, E. Pomjakushina, K. Conder, et al., Nat.\nCommun. 7, 11006 (2016).\n[5] G. Aut `es, D. Gresch, M. Troyer, A. A. Soluyanov, and O. V .\nYazyev, Phys. Rev. Lett. 117, 066402 (2016).\n[6] A. A. Burkov, Nat. Mater. 15, 1145 (2016).\n[7] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer,\nX. Dai, and B. A. Bernevig, Nature 527, 495 (2015).\n[8] Y . Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, Phys. Rev.\nB92, 161107 (2015).\n[9] Z. Wang, D. Gresch, A. A. Soluyanov, W. Xie, S. Kshwaha,\nX. Dai, M. Troyer, R. J. Cava, and B. A. Bernevig, Phys. Rev.\nLett. 117(2016).\n[10] T.-R. Chang, S.-Y . Xu, G. Chang, C.-C. Lee, S.-M. Huang,\nB. Wang, G. Bian, H. Zheng, D. S. Sanchez, I. Belopolski, et al.,\nNat. Commun. 7(2016).\n[11] A. Tamai, Q. S. Wu, I. Cucchi, F. Y . Bruno, S. Ricco, T. Kim,\nM. Hoesch, C. Barreteau, E. Giannini, C. Besnard, et al., Phys.\nRev. X 6, 031021 (2016).\n[12] L. Huang, T. M. McCormick, M. Ochi, Z. Zhao, M.-T. Suzuki,\nR. Arita, Y . Wu, D. Mou, H. Cao, J. Yan, et al., Nat. Mater. 15,\n1155 (2016).\n[13] K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang,\nM. Yan, H. Huang, H. Zhang, Z. Xu, et al., Nat. Phys. 12, 1105\n(2016).\n[14] J. Jiang, Z. K. Liu, Y . Sun, H. F. Yang, R. Rajamathi, Y . P.\nQi, L. X. Yang, C. Chen, H. Peng, C.-C. Hwang, et al., Nat.\nCommun. 8, 13973 (2017).\n[15] M. Sakano, M. S. Bahramy, H. Tsuji, I. Araya, K. Ikeura,\nH. Sakai, S. Ishiwata, K. Yaji, K. Kuroda, A. Harasawa, et al.,\nPhys. Rev. B 95, 121101 (2017).\n[16] H. P. Hughes and R. H. Friend, J. Phys. C: Solid State Phys. 11,\nL103 (1978).\n[17] Notice that the orthorhombic phase is also referred to as \rphase5\n[11] or T dphase [12–14], and that the 1T” phase is also referred\nto as\fphase [11] and also as 1T’ phase [12–14].\n[18] K. Zhang, C. Bao, Q. Gu, X. Ren, H. Zhang, K. Deng, Y . Wu,\nY . Li, J. Feng, and S. Zhou, Nat. Commun. 7, 13552 (2016).\n[19] See the supplementary information for more details about the\nMoTe2 crystal characterization and the calculated and mea-\nsured band structure of MoTe2 in the 1T’ and1T” phases.\n[20] A. Crepaldi, G. Aut `es, A. Sterzi, G. Manzoni, M. Zacchigna,\nF. Cilento, I. V obornik, J. Fujii, P. Bugnon, A. Magrez, et al.,\nPhys. Rev. B 95, 041408 (2017).\n[21] I. Belopolski, D. S. Sanchez, Y . Ishida, X. Pan, P. Yu, S.-Y .\nXu, G. Chang, T.-R. Chang, H. Zheng, N. Alidoust, et al., Nat.\nCommun. 7, 13643 (2016).\n[22] D. Polli, P. Alto `e, O. Weingart, K. M. Spillane, C. Manzoni,\nD. Brida, G. Tomasello, G. Orlandi, P. Kukura, R. A. Mathies,\net al., Nature 467, 440 (2010).\n[23] C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, F. Parmigiani,\nand D. Mihailovic, Adv. Phys. 65(2016).\n[24] S. Mathias, S. Eich, J. Urbancic, S. Michael, A. Carr,\nS. Emmerich, A. Stange, T. Popmintchev, T. Rohwer,\nM.Wiesenmayer, et al., Nat. Commun. 7(2016).\n[25] S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi,\nM. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mam-madov, et al., Phys. Rev. Lett. 112, 257401 (2014).\n[26] Y . Wu, D. Mou, N. H. Jo, K. Sun, L. Huang, S. L. Bud’ko, P. C.\nCanfield, and A. Kaminski, Phys. Rev. B 94, 121113 (2016).\n[27] F. Y . Bruno, A. Tamai, Q. S. Wu, I. Cucchi, C. Barreteau,\nA. de la Torre, S. McKeown Walker, S. Ricc `o, Z. Wang, T. K.\nKim, et al., Phys. Rev. B 94, 121112 (2016).\n[28] A. Mar, S. Jobic, and J. Ibers, Journal of the American Chemi-\ncal Society 114(1992).\n[29] B. E. Brown, Acta Crystallographica 20, 268 (1966).\n[30] X. Ma, P. Guo, C. Yi, Q. Yu, A. Zhang, J. Ji, Y . Tian, F. Jin,\nY . Wang, K. Liu, et al., Phys. Rev. B 94, 214105 (2016).\n[31] J. E. Callanan, G. A. Hope, R. D. Weir, and E. F. W. Jr., J. Chem.\nThermodynamics 24, 627 (1992).\n[32] F. C. Chen, X. Luo, R. C. Xiao, W. J. Lu, B. Zhang, H. X. Yang,\nJ. Q. L. andQ. L. Pei, D. F. Shao, R. R. Zhang, L. S. Lin, et al.,\nAppl. Phys. Lett. 108, 162601 (2016).\n[33] D. Kang, Y . Zhou, W. Yi, C. Yang, J. Guo, Y . Shi, S. Zhang,\nZ. Wang, C. Zhang, S. Jiang, et al., Nat. Commun. 6, 7804\n(2015).\n[34] Y . Qi, P. G. Naumov, M. N. Ali, C. R. Rajamathi, W. Schnelle,\nO. Barkalov, M. Hanfland, S.-C. Wu, C. Shekhar, Y . Sun, et al.,\nNat. Commun. 7, 11038 (2016)."    },    {        "title": "2001.08348v2.Fermi_liquid_theory_for_nonlinear_transport_through_a_multilevel_Anderson_impurity.pdf",        "content": "arXiv:2001.08348v2  [cond-mat.mes-hall]  5 Nov 2020Fermi liquid theory for nonlinear transport through a multi level Anderson impurity\nYoshimichi Teratani\nDepartment of Physics, Osaka City University, Sumiyoshi-k u, Osaka 558-8585, Japan\nRui Sakano\nThe Institute for Solid State Physics, the University of Tok yo, Kashiwa, Chiba 277-8581, Japan\nAkira Oguri\nDepartment of Physics, Osaka City University, Sumiyoshi-k u, Osaka 558-8585, Japan and\nNambu Yoichiro Institute of Theoretical and Experimental P hysics, Osaka City University, Osaka 558-8585, Japan\n(Dated: November 6, 2020)\nWe present a microscopic Fermi-liquid view on the low-energ y transport through an Anderson im-\npurity with Ndiscrete levels, at arbitrary electron filling Nd. It is applied to nonequilibrium current\nfluctuations, for which the two-quasiparticle collision in tegral and the three-body correlations that\ndetermine the quasiparticle energy shift play important ro les. Using the numerical renormalization\ngroup up to N= 6, we find that for strong interactions the three-body fluctu ations are determined\nby a single parameter other than the Kondo energy scale in a wi de filling range 1 /lessorsimilarNd/lessorsimilarN−1. It\nsignificantly affects the current noise for N >2 and the behavior of noise in magnetic fields.\nIntroduction.— Highly correlated low-energy states of\nthe Kondo systems show fascinating universal behavior\n[1], which can be described by a Fermi liquid (FL) the-\nory in zero dimension [2–6]. FL behaviors have been ob-\nserved for the nonlinear current through quantum dots\n[7, 8] and also the current noise [9–12] which is now one\nof the most important probes to explore quantum states.\nFurthermore, in addition to the spin, internal degrees of\nfreedom such as orbital, flavor, etc., bring an interest-\ning variety in the Kondo effect, occurring in a carbon\nnanotube [12, 13] and novel quantum systems such as\nultracold atomic gases [14] and quark matters [15].\nTransport properties of the local FL have successfully\nbeen described by the renormalized quasiparticles and\ntheir collisions due to the residual interaction, especially\natthesymmetric point whereboth theparticle-hole(PH)\nand time-reversal (TR) symmetries are present [16–19].\nThese symmetries are broken in real systems by external\nfields, such as a gate voltageand a magnetic field. In this\ncase, a single quasiparticle captures the quadratic depen-\ndence on frequency ω, temperature T, and bias voltage\nVnot only through the well-investigated damping rate\nbut also through the energy shift . It has recently been\nclarified that the quadratic energy shift is determined by\nthe three-body correlations between the impurity elec-\ntrons [20–24]. It shows that the three-body correlations\nare essential parameters for describing the FL transport.\nDespite its importance, the current noise [25–31] has\nbeen still less elucidated away from the symmetric point .\nA major milestone was achieved by Mora et al[20],\nwho have extended Nozi` eres phenomenological FL the-\nory [3] to give the formula of the nonlinear noise for a PH\nasymmetric single-orbital Anderson model at zero mag-\nnetic field. Further investigation, however, is required to\nclarify the physics of nonequilibrium fluctuations in the\nKondo systems with various internal degrees of freedom.In this letter, we give a microscopic view on the low-\nenergytransportthrough amultilevel Andersonimpurity\nfor a wide range of electron fillings Nd. It is described in\nterms of fiveFL parameters, which can be calculated us-\ning the numerical renormalization group (NRG) [2] up\ntoN= 6. We find that for strong interactions the\nthree-body correlations for Ndegenerate levels are de-\ntermined by a single parameter over a wide filling range\n1/lessorsimilarNd/lessorsimilarN−1, which includes the intermediate valence\nregions. We also provide a current-noise formula for the\nFL, taking into account all the two-quasiparticle colli-\nsion processes [32, 33]. It satisfies a Ward identity [4–6]\nfor the Keldysh vertex function, and resolves an essen-\ntial problem of the current conservation of the correlated\nelectrons under a nonequilibrium condition [16, 25]. We\nalso calculate the nonlinear noise using the NRG, and\ndemonstrate that the internal degrees of freedom give a\nwide variety to the filling dependence. We also examine\nthe effect of a magnetic field that breaks the TR symme-\ntry, and show that the noise of a spin-1/2 quantum dot\nexhibits a universal Kondo scaling behavior.\nModel.— We consider an N-level Anderson impurity\ncoupled to two leads on the left ( L) and right ( R):\nH=N/summationdisplay\nσ=1ǫdσndσ+/summationdisplay\nλ=L,RN/summationdisplay\nσ=1vλ/parenleftBig\nψ†\nλσdσ+d†\nσψλσ/parenrightBig\n+/summationdisplay\nλ=L,RN/summationdisplay\nσ=1/integraldisplayD\n−Ddǫǫc†\nǫλσcǫλσ+U\n2/summationdisplay\nσ/negationslash=σ′ndσndσ′.(1)\nd†\nσcreates an impurity electron with energy ǫdσ,ndσ≡\nd†\nσdσ, andUthe Coulomb repulsion. Conduction elec-\ntronsare normalizedas {cǫλσ,c†\nǫ′λ′σ′}=δλλ′δσσ′δ(ǫ−ǫ′).\nThe coupling vλbetweenψλσ≡/integraltextD\n−Ddǫ√ρccǫλσandd†\nσ\nyields a resonance of the width ∆ ≡ΓL+ ΓR, with\nΓλ=πρcv2\nλ,ρc= 1/(2D), andDthe half band width.2\nTABLE I. Coefficients C’s introduced in Eq. (5). W’s and Θ’s represent the two- and three-body contributions, respectively.\nCS=π2\n192/bracketleftBig\nWS−cos2δ/braceleftBig\nΘI+3(N−1)ΘII/bracerightBig/bracketrightBig\nWS≡cos4δ+/bracketleftBig\n4+5cos4 δ+3\n2/parenleftbig\n1−cos4δ/parenrightbig\n(N−2)/bracketrightBig\n(N−1)(R−1)2\nCV=π2\n64/bracketleftbig\nWV+ ΘI+3(N−1)ΘII/bracketrightbig\nWV≡ −/bracketleftbig\n1+5(N−1)(R−1)2/bracketrightbig\ncos2δ\nCT=π2\n48/bracketleftbig\nWT+ ΘI+(N−1)ΘII/bracketrightbig\nWT≡ −/bracketleftbig\n1+2(N−1)(R−1)2/bracketrightbig\ncos2δ\nCQD\nκ=7π2\n80/bracketleftbig\nWQD\nκ+ ΘI+5\n21(N−1)ΘII/bracketrightbig\nWQD\nκ≡10−11cos2 δ\n21−6\n7(N−1)(R−1)2cos2δ\nIn this work, we study the nonlinear current noise [25]\nSQD\nnoise=/integraldisplay∞\n−∞dt/angbracketleftBig\nδ/hatwideJ(t)δ/hatwideJ(0)+δ/hatwideJ(0)δ/hatwideJ(t)/angbracketrightBig\nV.(2)\nHere,δ/hatwideJ(t)≡/hatwideJ(t)− ∝an}bracketle{t/hatwideJ(0)∝an}bracketri}htVis the current fluctuation\noperator through the quantum dot [34], and ∝an}bracketle{t···∝an}bracketri}htVis\nthe Keldysh steady-state average defined at finite bias\nvoltageseV≡µL−µRwithµλthe chemical potential\nforλ=L,R[35]. The average current J≡ ∝an}bracketle{t/hatwideJ(0)∝an}bracketri}htVis\ngiven by [16],\nJ=e\nh/summationdisplay\nσ/integraldisplay∞\n−∞dω/bracketleftbig\nfL(ω)−fR(ω)/bracketrightbig\nTσ(ω).(3)\nHere,fλ(ω)≡[e(ω−µλ)/T+ 1]−1the Fermi function,\nTσ(ω)≡ −4ΓLΓR\nΓL+ΓRImGr\nσ(ω) the transmission probabil-\nity, andGr\nσ(ω) = [ω−ǫdσ+i∆−Σr\nσ(ω)]−1the retarded\nGreen’s function with Σr\nσ(ω) the self-energy. From this\nTσ(ω), we can also deduce the thermal conductivity κQD\n[36] for the heat current JQ=−κQDδT, induced by the\ntemperature difference δTbetween the two leads [37].\nFermi-liquid parameters.— We investigate low-energy\ntransport up to next leading order. To this end, we ex-pand Σr\nσ(ω) up to terms of order ω2,T2, and (eV)2for\ngeneralN, extending the latest FL description for spin\n1/2 case [22, 23]. The expansion coefficients play an im-\nportant role as the FL parameters.\nThe phase shift δσ≡cot−1(ǫ∗\ndσ/∆) is a parameter of\nprimary importance, with ǫ∗\ndσ≡ǫdσ+ Σr\nσ(0)|T=eV=0\nthe effective impurity level. It determines the occupa-\ntion number ∝an}bracketle{tndσ∝an}bracketri}ht=δσ/π, and the density of states\nρdσ≡sin2δσ/(π∆). The renormalization factor is given\nbythe firstderivative zσ≡/bracketleftbig\n1−∂Σr\nσ(ω)\n∂ω|ω=0/bracketrightbig−1, definedat\nT=eV= 0. It is also related to the static susceptibility\nχσ1σ2≡/integraltext1/T\n0dτ∝an}bracketle{tδndσ1(τ)δndσ2∝an}bracketri}ht, asχσσT→0− −− →ρdσ/zσ,\nwithδndσ≡ndσ−∝an}bracketle{tndσ∝an}bracketri}ht[4–6]. The second derivative is\na complex number, the imaginary part of which corre-\nsponds to the single-quasiparticle damping rate of order\nω2,T2, and (eV)2[16, 17]. The real part corresponds\nto the quadratic energy shift that is determined by the\nnonlinear susceptibility defined at equilibrium [22, 23]:\nχ[3]\nσ1σ2σ3≡−/integraldisplay1/T\n0dτ3/integraldisplay1/T\n0dτ2∝an}bracketle{tTτδndσ3(τ3)δndσ2(τ2)δndσ1∝an}bracketri}ht,\nwithTτtheimaginary-timeorderingoperator. Itcanalso\nbe written as χ[3]\nσ1σ2σ3=∂χσ1σ2/∂ǫdσ3, andcontributesto\nthe transport when the PH or TR symmetry is broken.\nSU(N) symmetric case.— In the case at which the\nNimpurity levels are degenerate ǫdσ≡ǫd, the lin-\near susceptibility χσσ′has only two independent com-\nponents. The diagonal element determines the energy\nscaleT∗≡1/(4χσσ), by which the T-linear specific heat\nis scaled as Cimp=Nπ2\n12(T/T∗). It can also be identified\nas the Kondo temperature in the strong-coupling limit.\nThe other one is the off-diagonalelement χσσ′forσ∝ne}ationslash=σ′,\nwhich is related to the Wilson ratio R≡1−χσσ′/χσσ\n[38]. Similarly, the nonlinear susceptibility has three in-\ndependent components for N≥3: the diagonal element\nχ[3]\nσσσand two off-diagonal ones, which can also be ex-\npressed in the following form for σ∝ne}ationslash=σ′∝ne}ationslash=σ′′∝ne}ationslash=σ,\n−(N−1)χ[3]\nσσ′σ′=χ[3]\nσσσ−∂χσσ\n∂ǫd, (4)\n(N−1)(N−2)\n2χ[3]\nσσ′σ′′=χ[3]\nσσσ−∂χσσ\n∂ǫd+N−1\n2∂χσσ′\n∂ǫd.In this work, we obtain the low-energy expansion of\nSQD\nnoise,JandκQDuptonextleadingorder,specificallyfor\nsymmetric junctions Γ L= ΓRandµL=−µR=eV/2:\nSQD\nnoise=2Ne2|eV|\nh/bracketleftBigg\nsin22δ\n4+CS/parenleftbiggeV\nT∗/parenrightbigg2\n+···/bracketrightBigg\n,(5)\ndJ\ndV=Ne2\nh/bracketleftBigg\nsin2δ−CT/parenleftbiggπT\nT∗/parenrightbigg2\n−CV/parenleftbiggeV\nT∗/parenrightbigg2\n+···/bracketrightBigg\n,\nκQD=Nπ2T\n3h/bracketleftBigg\nsin2δ−CQD\nκ/parenleftbiggπT\nT∗/parenrightbigg2\n+···/bracketrightBigg\n.\nThe explicit expressions of the coefficients CS,CV,CT\nandCQD\nκarelistedintableI.Eachofthese C’sconsistsof\ntwo parts, denoted as Wand Θ. The W-part represents\ntwo-body contributions which can be described in terms\nofRandδ. The Θ-part represents dimensionless three-3\nbody contributions:\nΘI≡sin2δ\n2πχ[3]\nσσσ\nχ2σσ,ΘII≡sin2δ\n2πχ[3]\nσσ′σ′\nχ2σσ.(6)\nTherefore, the low-energy transport of the SU( N) Fermi\nliquid are determined completely by fiveparameters: δ,\nT∗,R, ΘI, and ΘII. These FL parameters can also be\ndeduced experimentally throughmeasurementsofthe co-\nefficientsC’s. We note that another parameter for three\ndifferent levels, ΘIII≡sin2δ\n2πχ[3]\nσσ′σ′′\nχ2σσ, doesnotaffectC’s\nfor symmetric junctions. Nevertheless, it contributes to\nthe transport for N≥3 when the tunneling couplings or\nthe chemical potentials are asymmetric.\nThe nonlinear noise of the Fermi liquid is determined\nnot only by a single-quasiparticle excitation but also by\ntwo-quasiparticlecollisionsdescribed by the Keldysh ver-\ntex corrections [25]. In this work, we calculate the vertex\nfunction up toorder eV[35], extendingthe diagrammatic\napproach of Yamada-Yosida [4–6]. Consequently, the\ncollision contributions Ccoll\nSand the single-quasiparticle\nonesCqp\nSyield the nonlinear noise CS=Cqp\nS+Ccoll\nS:\nCcoll\nS=/bracketleftbigg7+5cos4δ\n2+3\n2/parenleftbig\n1−cos4δ/parenrightbig\n(N−2)/bracketrightbigg/tildewideK2\nN−1\nwith/tildewideK≡(N−1)(R−1). The second term in the bracket\nemergesthroughthecollisionsspecifictomultilevelimpu-\nrities forN >2, and it vanishes in the SU(2) symmetric\ncase or the PH symmetric case at which δ=π/2.\nFilling dependence of the FL state.— How does the FL\nstate evolve as the number of levels Nand their position\nǫdvary? As the electron configuration Nd≡/summationtext\nσ∝an}bracketle{tndσ∝an}bracketri}ht\ncontinuouslyvaries with ǫd, a different classof the Kondo\nand valence-fluctuation states emerge for multilevel sys-\ntemsN >2. To our knowledge, however, the behavior\nof three-body correlations Θ’s that determine the non-\nlinear transport has not been explored so much, whereas\nthe two-body correlations have been well investigated for\nN= 4 [30, 39, 40]. In this work, we calculate the FL\nparameters for N= 4,6 with the NRG, using the in-\nterleaved algorithm particularly for N= 6 [41]. To be\nspecific, we choose the Coulomb interaction to be much\nlarger than the hybridization energy scale: U/(π∆) =\n5.0. The results are plotted vs ξd≡ǫd+(N−1)U/2 in\nFig. 1 for (left panels) N= 4 and (right panels) N= 6.\nThe top panels of Fig. 1 show the two-body correla-\ntions, relating to ∝an}bracketle{tndσ∝an}bracketri}ht,χσσ, andχσσ′. We see that\nsin2δ, which determines Tσ(0) atT=eV= 0, shows a\nflat Kondo ridge of the unitary limit δ≃π/2 near the\nPH symmetric point |ξd|/lessorsimilarU/2 where the occupation\nnumber is almost locked at Nd≃N/2. The other Kondo\nridges also emerge at ξdwhereNdapproaches an integer:\nξd≃ ±UforN= 4, and also ξd≃ ±U,±2UforN= 6.\nTherenormalizationfactor z, which determinesthe en-\nergy scaleT∗=zπ∆/(4sin2δ), is also shown in the top□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□\n□\n□\n□\n□\n□\n□\n□\n□□□□��□□\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆-2 \u000010120.00.20.40.60.81.0\nξd\n/U2body FLparameters U\u0001πΔ=5.0N\u00024K\nz\nsin2δ\n□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□ □□□□□□□□□□□□□□□ □□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□\n□\n□\n□\n□\n□\n□\n□\n□\n□\n□□□□□□□□□□□□□□□□\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n-3-2-101230.00.20.40.60.81.0\nξd/U2body FLparametersU/πΔ=5.0N=6 K˜z\nsin2δ\n◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼\n◼\n◼\n◼\n◼\n◼\n◼\n◼◼◼◼◼◼◼◼◼◼◼\n◼\n◼\n◼\n◼\n◼\n◼◼◼◼◼◼◼◼◼◼◼◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆●●●●●●●●\n●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●\n-2-1012-2.0-1.5-1.0-0.50.0\nξd/U3 bodycorrelations\nU/πΔ=5.0N=4\nΘI-Θ˜\nIIΘ˜\nIII(N-1)(N-2)\n2ΘIII=\u0003 \u0004 \u0005\n\u0006 \u0007 \b\n\t \n\u000b\n\f \r\n\u000e\n\u000f\n\u0010\n\u0011\n\u0012\n\u0013\n\u0014\n\u0015\n\u0016\n\u0017\n\u0018\n\u0019\n\u001a\n\u001b\n\u001c\n\u001d\n\u001e\n\u001f\n \n!\n\"\n#\n$\n%\n&\n'\n(\n)\n*\n+\n,\n.\n0\n1\n2\n3\n4\n5\n6\n7\n8\n9\n:\n;\n<\n>\n?\n@\nA\nB\nC\nD\nE\nF\nG\nH\nI J KLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~      ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n   ¡¢£¤¥¦§¨©ª«¬®¯\n°\n±\n²\n³\n´\nµ\n¶\n·\n¸\n¹\nº\n»\n¼\n½\n¾\n¿\nÀ\nÁ\nÂ\nÃ\nÄ\nÅ\nÆ\nÇ\nÈ\nÉ\nÊ\nË\nÌ\nÍ\nÎ\nÏ\nÐ\nÑ\nÒ Ó\nÔÕ Ö×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øù\nú\nû\nü\ný\nþ\nÿ\n\u0003\n\u0000\n\u0001\n\u0002\n\u0004\n\u0005\n\u0006\n\u0007\n\b\n\t\n\n\u000b\n\f\n\r\n\u000e\n\u000f\n\u0010\n\u0011\n\u0012\n\u0013 \u0014\n\u0015 \u0016 \u0017\n\u0018\n-3-2-10123-2.0-1.5-1.0-0.50.0\nξd/U3 bodycorrelations\nU/πΔ=5.0N=6\nΘI-Θ˜\nIIΘ˜\nIII(N-1)(N-2)\n2ΘIII=\n◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼\n◼\n◼\n◼\n◼◼◼◼◼◼◼◼◼◼◼◼◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n○\n○○○○○○○○○○○○○○\n○\n○\n○\n○○○○○○○○○○○○□□□□□□□\n□\n□\n□\n□\n□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□\n□\n□\n□\n□\n□\n□\n□□□□□□□□□□□□□□□□□□□□□□□□□□□\n-2-1012-3-2-101234\nξd/UCT,CV,Cκ Q\u0019&Cs\nU/πΔ=5.0N=44 \u001a\nπ2CT\n\u001b \u001c\nπ2CV8 \u001d7π2Cκ\n\u001e \u001f\n1  !\nπ2Cs\n◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○□□□□□□□□□□□□□□□□□\n□\n□\n□\n□\n□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□\n-3-2-10123-3-2-101234\nξd/UCT,CV,Cκ \"#&Cs\nU/πΔ=5.0N=$% &\nπ2CT\n' (\nπ2CV) *+π2Cκ\n, -\n. / 0\nπ2Cs\nFIG. 1. Fermi-liquid parameters for SU( N) Anderson model\nare plotted vs ξd≡ǫd+(N−1)U/2 forU/(π∆) = 5.0,N= 4\n(left panels) and N= 6 (right panels). Top panels: sin2δ,\nrenormalization factor z, and/tildewideK≡(N−1)(R−1). Middlepan-\nels: ΘI,−/tildewideΘII≡ −(N−1)ΘII, and/tildewideΘIII≡(N−1)(N−2)\n2ΘIII.\nBottom panels:48\nπ2CT,64\nπ2CV,80\n7π2CQD\nκ, and192\nπ2CS.\npanels. It is significantly suppressed over a wide range of\ngatevoltages |ξd|/lessorsimilarN−1\n2U,andappearsasabroadvalley.\nThis valley becomes shallow as Nincreases, and vanishes\nin the large Nlimit [40]. Inside the valley, zhas mini-\nmumsatξd≃N−2M\n2UforM= 1,2,...,N−1,wherethe\noccupation number approaches an integer Nd=M. At\nthese minimums, the low-energy states can be described\nby the SU(N) Kondo model in the strong-coupling limit\nU≫∆. We find that zis also suppressed at local max-\nimums corresponding to the intermediate valence states,\nfor bothN= 4 and 6. In the top panels, the rescaled\nWilson ratio /tildewideKis also shown. It is almost saturated to\ntheuniversalvalue /tildewideK= 1anditsderivativebecomesvery\nsmall∂/tildewideK/∂ǫd∼0 in the whole region of the broad valley\n1/lessorsimilarNd/lessorsimilarN−1. It reveals the fact that not only the\ncharge susceptibility χc≡ −∂∝an}bracketle{tndσ∝an}bracketri}ht/∂ǫd=χσσ(1−/tildewideK)\nbut its derivative ∂χc/∂ǫdis suppressed in this region.\nThe three-body correlation ΘIis plotted in the mid-\ndle panels of Fig. 1, together with the other two rescaled\nones:−/tildewideΘII≡ −(N−1)ΘII, and/tildewideΘIII≡(N−1)(N−2)\n2ΘIII.\nThese Θ’s also show plateau structures due to the Kondo\neffect at the values of ξdcorrespondingto integer Nd, and\nalmost vanish at |ξd|/lessorsimilarU\n2. We find that these three pa-\nrameters ΘI,−/tildewideΘII, and/tildewideΘIIIapproach each other very\nclosely over a wide gate-voltage range |ξd|/lessorsimilarN−2\n2U, at4\n◇◇◇◇◇◇◇◇◇◇◇◇◇\n◇ ◇◇ ◇\n◇ ◇◇\n◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇◇\n◇\n◇\n◇\n◇\n◇◇\n◇\n◇\n◇\n◇\n◇◇◇◇◇◇◇\n○○○○○○○○○○○○○○○○○○○○\n○○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n○○○○○○○○○○○○○○○○○○○○○◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆\n◆◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n-3-2-10123-2-1012\nξd/UFano\nF2359:; FK=\n<Cs/CV\nN=4\nN= =\nU/πΔ> ? @ A B\n5.0◇,\n○,◆C\n● D\nEGHIJKLMNOPRSTUV\nW\nX\nY\nZ\n[\n\\\n]\n^\n_ ` a\nb cd e f g△△\n△\n△\n△\n△\n△\n△\n△△△△△△△△△△△△△△△△△△△△△△△△△△△▽▽\n▽\n▽\n▽\n▽\n▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽⊕\n⊕\n⊕\n⊕\n⊕⊕⊕⊕⊕⊕⊕⊕⊕\n0.00.51.01.52.0-0.10.00.10.20.30.40.50.6\nb/TKCSbU/πΔ\nϵd=-U/2□:2.5\n△:3.0\n▽:3.5\n⊕:4.0N=2\nFIG. 2. Nonlinear current-current correlations. Left pane l:\nFK≡CS\nCV/3vsξd/Ufor SU(N) symmetric case for N= 4\n(•,/diamondsolid) andN= 6 (◦,♦), forU/(π∆) = 1/3 (diamonds) and\nU/(π∆) = 5 (circles). Right panel: Cb\nSvsb/TKforN= 2 at\nhalf filling ǫd=−U/2, forU/(π∆) = 2.5,3.0,3.5,4.0, withb\nthe magnetic field and TKthe Kondo temperature at b= 0.\nwhich 1 /lessorsimilarNd/lessorsimilarN−1. This indicates that contribu-\ntions of the diagonal element χ[3]\nσσσdominate the terms\nin the right-hand side of Eq. (4); i.e. χ[3]\nσσσbecomes much\ngreater than∂χσσ\n∂ǫdandN−1\n2∂χσσ′\n∂ǫd. It also reveals the fact\nthat not only ∂χc/∂ǫdbut also∂χs/∂ǫd, the derivative\nof the spin susceptibility χs∝χσσ−χσσ′, becomes much\nsmaller than ( T∗)−2. Thus, for large U, the FL proper-\nties are characterized by threeparameters δ,T∗and ΘI\nover the wide filling range 1 /lessorsimilarNd/lessorsimilarN−1. Outside\nthis region, the Θ’s approach the noninteracting values:\nΘI→ −2, and the other two vanish as |ξd| → ∞.\nNonequilibrium FL fluctuations.— We show in the\nfollowing how the transport coefficients evolve as Nd\nvaries continuously. The NRG results are also plotted\nin Fig. 1. The difference between the C’s near half filling\n|ξd|/lessorsimilarU\n2is caused by the two-body contributions W’s\nas the Θ’s almost vanish. In particular, the T2conduc-\ntanceCTis determined by WToverthe wide filling range\n1/lessorsimilarNd/lessorsimilarN−1 as the three-body contributions almost\ncancelout ΘI+(N−1)ΘII≈0, reflectingthe suppression\nof∂χc/∂ǫdand∂χs/∂ǫdmentioned above. For the ther-\nmal conductivity, the three-body contributions become\nnegative in this region, ΘI+5\n21(N−1)ΘII≈ −16\n21/tildewideΘII,\nbut otherwise CQD\nκshowsa similar ξddependence to that\nofCT. The three-body contributions on the ( eV)2con-\nductanceCVare given by ΘV≡ΘI+3(N−1)ΘII, which\ntakes a value ΘV≈2/tildewideΘIIin the same filling range. Thus,\nCVis significantly enhanced at Nd≃1 andN−1 where\n−/tildewideΘIIshows a deep valley. It pushes the tail of the CV\ncurve outside than that of the CTin the valence fluctu-\nation region towards the empty or fully-occupied limit.\nThe|eV|3current noise also exhibits the Kondo\nplateau structures as shown in Fig. 1. For CS, the\nthree-body contributions enter through ΘVwith a sinu-\nsoidal factor: −ΘVcos2δ≈sin4δ\n2πχ[3]\nσσσ\nχ2σσover the range of\n1/lessorsimilarNd/lessorsimilarN−1. In the valence fluctuation regions men-\ntioned above, CShas a minimum caused by the higher-\nharmonic sin4 δand cos4δcontributions. We also findthatCSapproaches zero almost simultaneously with CV\nat|ξd| ≃1.4UforN= 4, and at |ξd| ≃2.4UforN= 6.\nThis proximity of the zero points affects the behavior of\nan extended Fano factor FK, defined as the ratio of order\n(eV)3current noise to the nonlinear current [20, 29, 35]:\nFK≡lim\n|eV|→0SQD\nnoise−2Ne2|eV|\nhsin22δ\n4\n−2|e|/parenleftBig\nJ−Ne2|V|\nhsin2δ/parenrightBig=CS\nCV/3.(7)\nThis formulafor the SU( N) Andersonmodel includes the\nresult of Mora et al, obtained for N= 2 at zero magnetic\nfield [20], as a special case. In the strong coupling limit\nat integerNd, it also agrees with another noise formula\nof Mora et alfor the SU( N) Kondo model [29].\nThe Fano factor for N= 4,6 is plotted vs ξdfor two\ndifferent values of Uin the left panel of Fig. 2. It reaches\nthe local maximum FK→(N−1+9/tildewideK2)/(N−1+5/tildewideK2)\natξd= 0 [30], and has positive plateaus for large Uat\nintegerNd. In the limit of |ξd| → ∞, the ratio becomes\nnegative and takes the noninteracting value FK→ −1.\nBy definition, FKchanges sign at the zero points of CS.\nIt also diverges at the zero point of CV, where the non-\nlinear component of Jchanges direction from backward\nto forward. Such a singularityalready exists for U= 0 at\n|ξd|= ∆/√\n3. Forlarge U,FKdivergesnear |ξd| ≃N−1\n2U\nin the valence fluctuation region towards the empty or\nfully-occupied limit. We can see that sign of the coeffi-\ncientCSat the singular points becomes positive for large\nU, whereas it is negative for small U. Sign change oc-\ncurs, for both N= 4 and 6, at a finite Ubetween the\ntwoexamined cases U/(π∆) = 1/3and 5. It is associated\nwith the large enhancement of three-body contributions\nΘVoccurring in the Kondo regime at Nd≃1 andN−1\nforN >2. In contrast, the NRG calculations examined\nso far indicate that sign is always negative in the SU(2)\ncaseforany U≥0 [20, 35]. The main difference is that in\nthe SU(2) case the three-body correlations evolve in the\nvalence fluctuation region where electron correlationsbe-\ncome less important and the two-body contributions WS\ndominateCSnear the singular point.\nMagnetic-field dependence.— We next consider effects\nof a magnetic field bthat breaks the SU( N) and TR\nsymmetries: specifically at half filling for N= 2, where\nǫd↑=−U\n2−b,ǫd↓=−U\n2+b, and the electron filling is\nfixed at ∝an}bracketle{tnd↑∝an}bracketri}ht+∝an}bracketle{tnd↓∝an}bracketri}ht= 1. In this case, the transport\ncoefficients can be described also by five FL parameters:\nmagnetization md≡ ∝an}bracketle{tnd↑∝an}bracketri}ht − ∝an}bracketle{tnd↓∝an}bracketri}ht, susceptibilities χ↑↑\n=χ↓↓andχ↑↓=χ↓↑, andthree-bodycorrelations χ[3]\n↑↑↑=\n−χ[3]\n↓↓↓andχ[3]\n↑↓↓=−χ[3]\n↑↑↓. The nonlinear current for\nthis case has previously been studied [21–23]. However,\nbehavior of its fluctuations has not been clarified so far.\nHere, we examine the current noise at T= 0 [42]:\nSQD\nnoise=4e2|eV|\nh/bracketleftBigg\nsin2(πmd)\n4+Cb\nS/parenleftbiggeV\nTK/parenrightbigg2\n+···/bracketrightBigg\n.5\nNote that the second term is scaled by TK≡T∗|b=0, the\nKondo temperature defined at zero field. Thus, the co-\nefficientCb\nSincludes all effects of b, which enter through\nthe FL parameters. In the right panel of Fig. 2, NRG\nresults forCb\nSare plotted as a function of b/TKfor sev-\neral different values of U. We find that the nonlinear\nnoise exhibits a universal behavior for U/(π∆)/greaterorsimilar2.0 in a\nsimilar way that the nonlinear current shows [22, 23]. It\ndecreases rapidly as bincreases for small fields, changes\nsign atb≈0.36TK, takes a minimum at b≈0.5TK, and\nthen approaches zero at b/greaterorsimilarTK. We note that order T3\nthermal conductivity also exhibits the scaling behavior\n[35]. These observations reflect the fact that the three-\nbody fluctuations show the universal scaling behavior in\nthe Kondo regime without the TR symmetry.\nConclusion.— Nonlineartransportthroughthe SU( N)\nAnderson impurity has been described in a unified way\nwith five FL parameters. We have demonstrated how\nthe FL state evolves as electron filling Ndvaries, using\nthe NRG up to N= 6. For strong interactions U, not\nonlychargefluctuationsbut alsothe derivativesofcharge\nand spin susceptibilities aresuppressed overa wide filling\nrange 1/lessorsimilarNd/lessorsimilarN−1. It reduces the number of variable\nFL parameters from five to three, and causes the Kondo\nplateau structures emerging for all the coefficients C’s.\nIn particular, the three-body contributions on CVare\nsignificantly enhanced at Nd≃1 andN−1 forN >2.\nIt also affects the behavior of nonlinear Fano factor FK\nin the valence fluctuation region. We have also shown\nthat the nonlinear current noise exhibits the universal\nmagnetic-field scaling in the Kondo regime. The FL pa-\nrameters can also be deduced from experiments and can\nbe used to predict behaviors of unmeasured observables.\nWe would like to thank K. Kobayashi, T. Hata, M.\nFerrier, R. Deblock, and A. C. Hewson for valuable dis-\ncussions. This work was supported by JSPS KAKENHI\nGrand Numbers JP18J10205,JP18K03495,JP26220711,\nand JST CREST Grant No. JPMJCR1876.\n[1] A. C. Hewson, The Kondo Problem to Heavy Fermions\n(Cambridge University Press, 1993).\n[2] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n[3] P. Nozi` eres, J. Low Temp. Phys. 17, 31 (1974).\n[4] K. Yamada, Prog. Theor. Phys. 54, 316 (1975).\n[5] H. Shiba, Prog. Theor. Phys. 54, 967 (1975).\n[6] A. Yoshimori, Prog. Theor. Phys. 55, 67 (1976).\n[7] M. Grobis, I. G. Rau, R. M. Potok,\nH. Shtrikman, and D. Goldhaber-Gordon,\nPhys. Rev. Lett. 100, 246601 (2008).\n[8] G. D. Scott, Z. K. Keane, J. W. Ciszek, J. M. Tour, and\nD. Natelson, Phys. Rev. B 79, 165413 (2009).\n[9] O. Zarchin, M. Zaffalon, M. Heiblum, D. Mahalu, and\nV. Umansky, Phys. Rev. B 77, 241303(R) (2008).\n[10] T. Delattre, C. Feuillet-Palma, L. G. Herrmann,P. Morfin, J.-M. Berroir, G. FShu ve, B. PlaXun ais,\nD. C. Glattli, M.-S. Choi, C. Mora, and T. Kontos,\nNature Physics 5, 208 (2009).\n[11] Y. Yamauchi, K. Sekiguchi, K. Chida, T. Arakawa,\nS. Nakamura, K. Kobayashi, T. Ono, T. Fujii, and\nR. Sakano, Phys. Rev. Lett. 106, 176601 (2011).\n[12] M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. Dela-\ngrange, R. Weil, R. Deblock, R. Sakano, A. Oguri, and\nK. Kobayashi, Nat. Phys. 12, 230 (2016).\n[13] E. A. Laird, F. Kuemmeth, G. A. Steele, K. Grove-\nRasmussen, J. Nyg˚ ard, K. Flensberg, and L. P. Kouwen-\nhoven, Rev. Mod. Phys. 87, 703 (2015).\n[14] K. Ono, J. Kobayashi, Y. Amano, K. Sato, and Y. Taka-\nhashi, Phys. Rev. A 99, 032707 (2019).\n[15] S. Yasui and S. Ozaki, Phys. Rev. D 96, 114027 (2017).\n[16] S. Hershfield, J. H. Davies, and J. W. Wilkins,\nPhys. Rev. B 46, 7046 (1992).\n[17] A. Oguri, Phys. Rev. B 64, 153305 (2001).\n[18] A.A.Aligia, J. Phys.: Condens. Matter 24, 015306 (2012).\n[19] E. Mu˜ noz, C. J. Bolech, and S. Kirchner,\nPhys. Rev. Lett. 110, 016601 (2013).\n[20] C. Mora, C. P. Moca, J. von Delft, and G. Zar´ and,\nPhys. Rev. B 92, 075120 (2015).\n[21] M. Filippone, C.P.Moca, A.Weichselbaum, J. vonDelft,\nand C. Mora, Phys. Rev. B 98, 075404 (2018).\n[22] A. Oguri and A. C. Hewson,\nPhys. Rev. Lett. 120, 126802 (2018).\n[23] A. Oguri and A. C. Hewson,\nPhys. Rev. B 97, 035435 (2018).\n[24] A. Oguri and A. C. Hewson,\nPhys. Rev. B 98, 079905(E) (2018).\n[25] S. Hershfield, Phys. Rev. B 46, 7061 (1992).\n[26] A. O. Gogolin and A. Komnik,\nPhys. Rev. Lett. 97, 016602 (2006).\n[27] E. Sela, Y. Oreg, F. von Oppen, and J. Koch,\nPhys. Rev. Lett. 97, 086601 (2006).\n[28] E. Sela and J. Malecki, Phys. Rev. B 80, 233103 (2009).\n[29] C. Mora, P. Vitushinsky, X. Leyronas, A. A. Clerk, and\nK. Le Hur, Phys. Rev. B 80, 155322 (2009).\n[30] R. Sakano, T. Fujii, and A. Oguri,\nPhys. Rev. B 83, 075440 (2011).\n[31] D. B. Karki, C. Mora, J. von Delft, and M. N. Kiselev,\nPhys. Rev. B 97, 195403 (2018).\n[32] L. P. Pitaevskii and E. M. Lifshitz, Physical Kinetics\n(Butterworth-Heinemann, 1981).\n[33] H. Haug and A.-P. Jauho, Quantum Kinetics in Trans-\nport and Optics of Semiconductors (Springer, 2008).\n[34]/hatwideJ≡(ΓL/hatwideJR+ΓR/hatwideJL)/(ΓL+ΓR), with/hatwideJthecurrentflowing\nbetween the dot and lead on λ(=L,R) side.\n[35] See supplemental material for details.\n[36]κQD≡1\nhT/bracketleftbig/summationtext\nσLQD\n2,σ−/parenleftbig/summationtext\nσLQD\n1,σ/parenrightbig2//parenleftbig/summationtext\nσLQD\n0,σ/parenrightbig/bracketrightbig\n, and\nLQD\nn,σ=/integraltext∞\n−∞dωωn(−∂f\n∂ω)Tσ(ω) forn= 0,1,2.\n[37] T. A. Costi and V. Zlati´ c,\nPhys. Rev. B 81, 235127 (2010).\n[38] A.C.Hewson,J. Phys.: Condens. Matter 13, 10011 (2001).\n[39] Y. Nishikawa, D. J. G. Crow, and A. C. Hewson,\nPhys. Rev. B 82, 245109 (2010).\n[40] A. Oguri, Phys. Rev. B 85, 155404 (2012).\n[41] K. M. Stadler, A. K. Mitchell, J. von Delft, and A. We-\nichselbaum, Phys. Rev. B 93, 235101 (2016).\n[42]Cb\ns=π2\n192/bracketleftbig\ncos(2πmd)+{4+5cos(2 πmd)}(R−1)2\n+Θb\nVcos(πmd)/bracketrightbig\n(TK\nT∗)2, and Θb\nV≡ −sin(πmd)\n2πχ[3]\n↑↑↑+3χ[3]\n↑↓↓\nχ2\n↑↑.6\nFermi liquid theory for nonlinear transport through a multi level Anderson impurity\n(Supplemental Material)\nYoshimichi Teratani1, Rui Sakano2, and Akira Oguri1,3\n1Department of Physics, Osaka City University, Sumiyoshi-k u, Osaka 558-8585, Japan\n2The Institute for Solid State Physics, the University of Tok yo, Kashiwa, Chiba 277-8581, Japan\n3NITEP, Osaka City University, Sumiyoshi-ku, Osaka 558-858 5, Japan\nI. DERIVATION OF THE TRANSPORT COEFFICIENTS CT,CVANDCQD\nκ\nWe describe here outline of the derivation of the coefficients C’s, listed in table I in the main text. The steady\ncurrentJthrough the quantum dots has been calculated using the formula giv en in Eq. (3) with the transmission\nprobability, defined by\nTσ(ω) =−4ΓLΓR\nΓL+ΓRImGr\nσ(ω), Gr\nσ(ω) =1\nω−ǫdσ+i∆−Σrσ(ω). (8)\nWe have also calculated the thermal conductivity κQD, which can be expressed in the following form at eV= 0,\nκQD=1\n2π/planckover2pi1T\n/summationdisplay\nσLQD\n2,σ−/parenleftBig/summationtext\nσLQD\n1,σ/parenrightBig2\n/summationtext\nσLQD\n0,σ\n, LQD\nn,σ≡/integraldisplay∞\n−∞dωωnTσ(ω)/parenleftbigg\n−∂f(ω)\n∂ω/parenrightbigg\n. (9)\nThe coefficients for the charge and heat currents, CT,CV, andCQD\nκ, can be deduced form the low-energyexpansion of\nthe retarded self-energy Σr\nσ(ω) obtained up to terms of order ω2,T2, and (eV)2. We note that, in order to determine\nalso the thermopower of quantum dots SQD\nTPup to next leading order, additional terms of order ω3andωT2of the\nself-energy are necessary. This is because the leading term of SQD\nTPalready includes the derivative ρ′\ndσ≡∂ρdσ(ω)\n∂ω/vextendsingle/vextendsingle\nω=0which describes a variation from the ground state:\nSQD\nTP≡−1\n|e|T/summationtext\nσLQD\n1,σ/summationtext\nσLQD\n0,σ=−π2\n3|e|/summationtext\nσρ′\ndσ/summationtext\nσρdσT+O(T3), ρ′\ndσ=χσσsin2δσ\n∆. (10)\nThe expansion coefficients of Σr\nσ(ω) can be expressed in terms of the linear and nonlinear susceptibilities ,\nImΣr\nσ(ω) =−π(N−1)\n2χ2\nσσ′\nρdσ/bracketleftbigg\nω2+3\n4(eV)2+(πT)2/bracketrightbigg\n+···, (σ′∝ne}ationslash=σ), (11)\nǫd+ReΣr\nσ(ω) = ∆ cot δ+(1−/tildewideχσσ)ω+1\n2∂/tildewideχσσ\n∂ǫdσω2+N−1\n6χ[3]\nσσ′σ′\nρdσ/bracketleftbigg3\n4(eV)2+(πT)2/bracketrightbigg\n+···,(12)\nfor ΓL= ΓR= ∆/2 andµL=−µR=eV/2 in the SU( N) symmetric case. Note that /tildewideχσσ′′≡δσσ′′+∂Σr\nσ(0)\n∂ǫdσ′′/vextendsingle/vextendsingle/vextendsingle\nT=eV=0,\nχσσ′′=ρdσ/tildewideχσσ′′, andρdσ= sin2δ/(π∆) forσ= 1,2,3,...,N. These results are obtained by extending further the\nlatest version of Fermi-liquid description [A. Oguri and A. C. Hewson, Phys. Rev. B 97, 035435] to the multilevel\ncasesN >2. AtT=eV= 0, the self-energy satisfies the Ward identity of the following form , which yields the\nFermi-liquid relations between the expansion coefficients,\n/parenleftbigg\nδσσ′∂\n∂ω+∂\n∂ǫdσ′/parenrightbigg\nΣ−−\nσ(ω) =−Γ−−;−−\nσσ′;σ′σ(ω,0;0,ω)ρdσ′. (13)\nHere, Σ−−\nσ(ω) = ReΣr\nσ(ω)+isgn(ω)ImΣr\nσ(ω), and Γ−−;−−\nσσ′;σ′σ(ω,ω′;ω′,ω) is the causal vertex function at T=eV= 0.\nIt has been determined up to linear order terms with respect to ωandω′,\nΓ−−;−−\nσσ;σσ(ω,ω′;ω′,ω)ρ2\ndσ=iπ/summationdisplay\nσ′(/negationslash=σ)χ2\nσσ′/vextendsingle/vextendsingleω−ω′/vextendsingle/vextendsingle+···, (σ′∝ne}ationslash=σ), (14)\nΓ−−;−−\nσσ′;σ′σ(ω,ω′;ω′,ω)ρdσρdσ′=−χσσ′+ρdσ∂/tildewideχσσ′\n∂ǫdσω+ρdσ′∂/tildewideχσ′σ\n∂ǫdσ′ω′+iπχ2\nσσ′/parenleftBig/vextendsingle/vextendsingleω−ω′/vextendsingle/vextendsingle−/vextendsingle/vextendsingleω+ω′/vextendsingle/vextendsingle/parenrightBig\n+···.(15)7\nThe Ward identity itself follows from the current conservation betw een the dot and leads:\n∂\n∂t(endσ)+/hatwideJR,σ−/hatwideJL,σ= 0,/hatwideJL,σ≡ievL/parenleftBig\nψ†\nLσdσ−d†\nσψLσ/parenrightBig\n,/hatwideJR,σ≡ −ievR/parenleftBig\nψ†\nRσdσ−d†\nσψRσ/parenrightBig\n.(16)\nII. FERMI-LIQUID CORRECTIONS FOR NONLINEAR CURRENT NOISE SQD\nnoise\nIn contrast to the average current Jand thermal conductivity κQD, the current noise depends also on the two-\nquasiparticle collisions which correspond to the vertex corrections for the current-current correlation function Kν′ν\nσ′σ:\nSQD\nnoise=/integraldisplay∞\n−∞dt/summationdisplay\nσσ′i/bracketleftBig\nK+−\nσ′σ(t,0)+K−+\nσ′σ(t,0)/bracketrightBig\n, δ /hatwideJσ(t)≡/hatwideJσ(t)−∝an}bracketle{t/hatwideJσ(t)∝an}bracketri}htV, (17)\nK+−\nσ′σ(t,0)≡ −i/angbracketleftbig\nδ/hatwideJσ′(t)δ/hatwideJσ(0)/angbracketrightbig\nV, K−+\nσ′σ(t,0)≡ −i/angbracketleftbig\nδ/hatwideJσ(0)δ/hatwideJσ′(t)/angbracketrightbig\nV. (18)\nHere,/hatwideJσ≡(ΓL/hatwideJR,σ+ΓR/hatwideJL,σ)/(ΓL+ΓR) is a symmetrized current operator. In this work, we have expand edSQD\nnoiseup\nto terms of order ( eV)3, using the diagrammatic representation illustrated in Fig. 3. To this e nd, all the components\nof Keldysh Green’s function Gν′ν\nσ(ω) have been deduced up to order ω2and (eV)2, and the Keldysh vertex function\nΓν1ν2;ν3ν4\nσσ′;σ′σ(ω,ω′;ω′ω) have been calculated up to linear order in ω,ω′andeV. We have checked that the result satisfies\nthe nonequilibrium Ward identity [A. Oguri, Y. Tetatani, and S. Sakano , unpublished],\n/parenleftbigg\nδσσ′∂\n∂ω+∂\n∂ǫdσ′/parenrightbigg\nΣν4ν1\nσ(ω) =−/integraldisplay∞\n−∞dω′\n2π/summationdisplay\nν2ν3Γν1ν2;ν3ν4\nσσ′;σ′σ(ω,ω′;ω′,ω) 2∆Gr\nσ′(ω′)Ga\nσ′(ω′)/parenleftbigg\n−∂feff(ω′)\n∂ω′/parenrightbigg\n.(19)\nHere,feff(ω) =/braceleftbig\nΓLfL(ω)+ΓRfR(ω)/bracerightbig\n/(ΓL+ΓR), and Σν4ν1σ(ω) is the Keldysh self-energy. The expansion coefficient\nfor order (eV)3current noise is given in table I. It can be separated into two parts CS=Cqp\nS+Ccoll\nS, as mentioned\nin the main text. Here, Cqp\nSandCcoll\nSrepresent respectively the contributions of the bubble diagram an d that of the\nvertex corrections shown in Fig. 3.\nThe nonlinear Fano factor, FK=CS\nCV/3, can be expressed in the following form for the SU( N) Anderson impurity\nat arbitrary electron fillings,\nFK=cos4δ+/bracketleftBig\n4+5cos4δ+3\n2/parenleftbig\n1−cos4δ/parenrightbig\n(N−2)/bracketrightBig/tildewideK2\nN−1−cos2δ/bracketleftBig\nΘI+3(N−1)ΘII/bracketrightBig\n−/bracketleftBigg\n1+5/tildewideK2\nN−1/bracketrightBigg\ncos2δ+ ΘI+3(N−1)ΘII.(20)\nForN= 2, it reproduces the previous result, obtained by Mora et al[Eq. (11) of Phys. Rev. B 92, 075120 (2015)]:\ntheir notation and our one correspond to each other such that α(1)\nσ/π=χσσ,φ(1)\nσσ′/π=−χσσ′,α(2)\nσ/π=−1\n2χ[3]\nσσσ,\nandφ(2)\nσσ′/π= 2χ[3]\nσσ′σ′forσ′∝ne}ationslash=σ. In Fig. 4, we have also plotted FKand other Fermi-liquid parameters for the SU(2)\nsymmetric case as functions of ǫdfor comparisons with those for N= 4 and 6 shown in the main text.\nω,σ\nω,σν\nνν'\nν'ω,σ\nω,σω',σ'\nω',σ'ν1ν2\nν4ν3νν'\nνν\n'Γ\nFIG. 3. The Feynman diagrams for the correlation function/integraltext∞\n−∞dtKν′ν\nσ′σ(t,0). The solid lines denote the Keldysh Green’s\nfunctions Gν′ν\nσ(ω). The shaded region in the diagram on the right represents th e Keldysh vertex function Γν1ν2;ν3ν4\nσσ′;σ′σ(ω,ω′;ω′ω).\nThe superscripts ν,ν′andνi(i= 1,2,3,4) specify the branches of Kedysh time-loop contour. We are u sing the notation in\nwhichν=−and + represent the forward and return paths, respectively.8h\ni\nj\nk\nl\nm\nn\no\np\nq\nr\ns\nt\nu\nv\nw\nx\ny\nz\n{ | } ~      ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n-1.0-0.50.00.51.00.00.20.40.60.81.0\nξd/U2body FLparametersU/πΔ=5.0N=2\n(a)K˜=\n(N-1)(R-1)z\nsin2δ\n◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼\n◼\n◼\n◼\n◼\n◼\n◼◼◼◼◼◼◼◼◼◼◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆\n-1.0-0.50.00.51.0-2.0-1.5-1.0-0.50.0d/U3 body correlationsU/πΔ=5.0N=2(b)ΘI-Θ˜ =\n-(N-1)Θ \n◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼\n◼\n◼\n◼\n◼◼◼◼◼◼◼◼◼◼◼◼ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n○\n○\n○\n○○○○○○○○○○○○□□□□□□□□□□\n□\n□\n□□□□□□□□□□□□□□□\n□\n□\n□\n□\n□□□□□□□□□□□□□□\n-1.0-0.50.00.51.0--202\n\n\n10\nξd/UCT\nCV\n¡Cκ¢£\n¤ Cs\nU/πΔ=5.0N=2 (c)¥ ¦\nπ2CT\n§ ¨\nπ2CV\n© ª «\nπ2Cs¬ ®π2Cκ\n¯ °\n±\n²\n³\n´\nµ\n¶\n·\n¸\n¹\nº» ¼\n½¾\n¿\nÀ\nÁ\nÂ\nÃ\nÄ\nÅ\nÆ\nÇ\nÈ\nÉ\nÊ\nË\nÌ\nÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜ\nÝÞßàáâãäå\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●\n●\n●●\n●\n●\n●●●●●●●●●●●●●●●●●●\n-1.5-1.0-0.50.00.51.01.5-2-10123\nξd/U\næçèéêëìíîïð FK=3Cs/CV\n(d)N=2U\nπΔ=1\n3\nU\nπΔ=5\nFIG. 4. Fermi-liquid parameters for the SU(2) symmetric cas e are plotted vs ξd≡ǫd+U/2 forU/(π∆) = 5.0. (a): sin2δ,\nrenormalization factor z, and/tildewideK≡(N−1)(R−1). (b): thee-body correlatins ΘI, and−/tildewideΘII≡ −(N−1)ΘII. (c):48\nπ2CT,64\nπ2CV,\n192\nπ2CS, and80\n7π2CQD\nκ. (d): nonlinear Fano factor FK=CS\nCV/3is plotted also for U/(π∆) = 1/3,5.\nEquation (20) also reproduces the previous result in the particle-h ole symmetric case, at which δ=π/2 and the\nthree-body contributions vanish Θ I= ΘII= 0 [Sakano et alPhys. Rev. B 83, 075440 (2011)] :\nFKǫd→−(N−1)U/2− −−−−−−−−−− →1+9/tildewideK2\nN−1\n1+5/tildewideK2\nN−1U→∞− −−− →1+9\nN−1\n1+5\nN−1. (21)\nIn the strong coupling limit U→ ∞, the occupation number Nd≡/summationtext\nσ∝an}bracketle{tndσ∝an}bracketri}htbecomes integer M= 1,2,...,N−1\natǫd=−(M−1/2)Uand the phase shift is locked at δ=πM/N. The charge and spin susceptibilities satisfy the\nstationary conditions in this case, and thus Eq. (20) can be rewritt en as\nFK→1+sin2/parenleftbig2πM\nN/parenrightbig\n+9−13sin2(2πM\nN)\nN−1+ 2ΘIcos/parenleftbig2πM\nN/parenrightbig\n−/bracketleftBig\n1+5\nN−1/bracketrightBig\ncos/parenleftbig2πM\nN/parenrightbig\n−2ΘI. (22)\nThis expression is consistent with the corresponding noise formula f or the SU(N) Kondo model, obtained by Mora et\nal[Eq. (51) of Phys. Rev. B 80, 155322 (2009), after inserting some parenthesis for correctin g minor typos].\nIII. NRG CALCULATIONS\nNRGcalculationsfortheSU( N) Andersonmodelfor N= 2,4,6havebeencarriedout, dividing Nchannelsinto N/2\npairs and exploiting the SU(2) spin and U(1) charge symmetries for e ach of the pairs, i.e. using/producttextN\n2\nk=1{SU(2)⊗U(1)}k\nsymmetries. The discretization parameter Λ and the number of ret ained low-lying excited states Ntruncare chosen\nsuch that (Λ ,Ntrunc) = (2,4000) forN= 2, (6,10000) forN= 4, and (20 ,30000) forN= 6. We have also exploited\nmethods of Stadler’s et al[Phys. Rev. B 93, 235101 (2017)] for N= 6. The truncation is performed at each step after\nadding states from each pair of the channels, using Olivera’s Z-trick [Phys. Rev. B 49, 11986 (1994)] and choosing\ndifferent Zvalues for different pairs: Zi= 1/2+i/Nfor thei-th pair (i= 1,2,...,N/2).\nTABLE II. The coefficients C’s at finite magnetic fields bforN= 2 at half filling ǫd=−U\n2.\nSee also, A. Oguri and A. C. Hewson, Phys. Rev. B 98, 079905 (E).\nCb\nS=π2\n192/bracketleftbig\nWb\nS+/parenleftbig\nΘb\nI+3Θb\nII/parenrightbig\ncos(πmd)/bracketrightbig\nWb\nS≡cos(2πmd)+/bracketleftbig\n4+5cos(2 πmd)/bracketrightbig\n(R−1)2\nCb\nV=π2\n64/bracketleftbig\nWb\nV+ Θb\nI+3Θb\nII/bracketrightbig\nWb\nV≡/bracketleftbig\n1+5(R−1)2/bracketrightbig\ncos(πmd)\nCb\nT=π2\n48/bracketleftbig\nWb\nT+ Θb\nI+Θb\nII/bracketrightbig\nWb\nT≡/bracketleftbig\n1+2(R−1)2/bracketrightbig\ncos(πmd)\nCQD\nκ,b=7π2\n80/bracketleftbig\nWQD\nκ,b+ Θb\nI+5\n21Θb\nII/bracketrightbig\nWQD\nκ,b≡/bracketleftbig\n1+6\n7(R−1)2/bracketrightbig\ncos(πmd)9\nIV. MAGNETIC-FIELD DEPENDENCE OF CURRENT NOISE FOR N= 2\nWe describe here supplemental information about the\nnonlinear current noise at finite magnetic field b, specif-\nically forN= 2 at half filling, where the impurity level\nis given by ǫdσ≡ −U\n2−sgn(σ)bwith sgn( ↑) = +1 and\nsgn(↓) =−1. In this case, the phase shift takes the\nformδσ=π{1+sgn(σ)md}/2 withmd≡ ∝an}bracketle{tnd↑∝an}bracketri}ht−∝an}bracketle{tnd↓∝an}bracketri}ht,\nand the other correlation functions have symmetry prop-\nerties:χ↑↑=χ↓↓,χ↑↓=χ↓↑,χ[3]\n↑↓↓=−χ[3]\n↑↑↓, and\nχ[3]\n↓↓↓=−χ[3]\n↑↑↑. Thus, the transport coefficients up to\nthe next leading order can be described by five parame-\nters, for instance, md,T∗= 1/(4χ↑↑),R= 1−χ↑↓/χ↑↑\nand the following 3-body correlation functions,\nΘb\nI≡ −sin(πmd)\n2πχ[3]\n↑↑↑\nχ2\n↑↑,Θb\nII≡ −sin(πmd)\n2πχ[3]\n↑↓↓\nχ2\n↑↑.(23)\nThe low-energy expansion of the current noise SQD\nnoise,\nconductance dJ/dV, and thermal conductivity κQDfor\nthis case can be written in the following form, with the\ncoefficients C’s listed in table II,\nSQD\nnoise= 22e2\nh|eV|/bracketleftBigg\nsin2(πmd)\n4+Cb\nS/parenleftbiggeV\nT∗/parenrightbigg2\n+···/bracketrightBigg\n,(24)\ndJ\ndV=2e2\nh/bracketleftBigg\ncos2/parenleftBigπmd\n2/parenrightBig\n−Cb\nT/parenleftbiggπT\nT∗/parenrightbigg2\n−Cb\nV/parenleftbiggeV\nT∗/parenrightbigg2\n+···/bracketrightBigg\n,\nκQD=2π2T\n3h/bracketleftBigg\ncos2/parenleftBigπmd\n2/parenrightBig\n−CQD\nκ,b/parenleftbiggπT\nT∗/parenrightbigg2\n+···/bracketrightBigg\n.\nñòóôõö÷øùúûüý\nþÿ\u0001\u0000\u0002\u0003\u0004\u0005\u0006\u0007\b\t\n\u000b\f\r\u000e\u000f\u0010△△△△△△\n△\n△△\n△\n△△△△△△△△△△△△△△△△△△△△△△△△▽▽▽▽▽\n▽▽\n▽\n▽\n▽\n▽▽▽▽▽▽▽▽▽▽▽⊕⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕⊕⊕⊕\n0.00.51.01.52.0-1.0-0.50.00.51.01.52.0\nb/TKFKbU=0\nU=0U/πΔ\nϵd=-U/2 □ :2.5\n△\u00113.0\n▽ \u00123.5\n⊕\u0013 4 \u0014 \u0015N=2\n\u0016\u0017\u0018\u0019\u001a\u001b\u001c\u001d\u001e\u001f !\"#$%&'\n(\n)\n*\n+\n,\n-\n.\n/\n0\n1\n2\n3\n5\n6\n7△△△\n△\n△\n△\n△\n△\n△\n△△△△△△△△△△△△△△△△△△△△△△△△△△▽▽\n▽\n▽\n▽\n▽\n▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽⊕\n⊕\n⊕\n⊕\n⊕\n⊕⊕⊕⊕⊕⊕⊕⊕\n0.00.51.01.52.0-0.50.00.51.01.5\nb/TKCκ\n8bQ9U/πΔ\nϵd=-U/2; <2.5= >3.0? @3.5\n⊕A B C DN=2\nFIG. 5. Fb\nK=Cb\nS\nCb\nV/3andCQD\nκ,b≡(TK/T∗)2CQD\nκ,bfor\nN= 2 are plotted vs b/TKat half filling for U/(π∆) =\n2.5,3.0,3.5,4.0:TKvaries with U, andTKU→0− −− →π∆/4.\nIn order to see the magnetic field dependences in the\nKondo regime, it is preferable to rescale the next leading\n(eV)2andT2contributions by the Kondo temperature\ndefined at zero field TK= limb→0T∗. This is because\nall effects of bare absorbed into the coefficients redefined\nsuch thatCb\nV≡(TK/T∗)2Cb\nV,Cb\nS≡(TK/T∗)2Cb\nS, and\nCQD\nκ,b≡(TK/T∗)2CQD\nκ,b. We have presented the NRG\nresults for the nonlinear current noise Cb\nSin the main\ntext;Cb\nVwas examined previously [A. Oguri and A. C.\nHewson, Phys. Rev. Lett. 120, 126802 (2018)]. In Fig.\n5,Fb\nK≡Cb\nS\nCb\nV/3andCQD\nκ,bare plotted as functions of b/TK\nfor several different values of U. The nonlinear Fano\nfactorFb\nKshows the Kondo scaling behavior for strong\ninteractions U/(π∆)/greaterorsimilar2.0. The universal curve of Fb\nK\ndeviates significantly from the curve for U= 0 keeping\nits qualitative characteristics unchanged. We also find\nthat the thermal conductivity CQD\nκ,bexhibits the universal\nscaling behavior."    },    {        "title": "1503.02648v2.A_Quantum_Gas_Microscope_for_Fermionic_Atoms.pdf",        "content": "A Quantum Gas Microscope for Fermionic Atoms\nLawrence W. Cheuk,1Matthew A. Nichols,1Melih Okan,1Thomas Gersdorf,1Vinay\nV. Ramasesh,1Waseem S. Bakr,1Thomas Lompe,1and Martin W. Zwierlein1\n1Department of Physics, MIT-Harvard Center for Ultracold Atoms, and\nResearch Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA\nStrongly interacting fermions de\fne the properties of complex matter throughout nature, from\natomic nuclei and modern solid state materials to neutron stars. Ultracold atomic Fermi gases have\nemerged as a pristine platform for the study of many-fermion systems. Here we realize a quantum gas\nmicroscope for fermionic40K atoms trapped in an optical lattice, which allows one to probe strongly\ncorrelated fermions at the single atom level. We combine 3D Raman sideband cooling with high-\nresolution optics to simultaneously cool and image individual atoms with single lattice site resolution\nat a detection \fdelity above 95%. The imaging process leaves the atoms predominantly in the 3D\nmotional ground state of their respective lattice sites, inviting the implementation of a Maxwell's\ndemon to assemble low-entropy many-body states. Single-site resolved imaging of fermions enables\nthe direct observation of magnetic order, time resolved measurements of the spread of particle\ncorrelations, and the detection of many-fermion entanglement.\nThe collective behavior of fermionic particles gov-\nerns the structure of the elements, the workings of\nhigh-temperature superconductors and colossal magneto-\nresistance materials, and the properties of nuclear mat-\nter. Yet our understanding of strongly interacting Fermi\nsystems is limited, due in part to the antisymmetry re-\nquirement on the many-fermion wavefunction and the\nresulting \\fermion sign problem\". In recent years, ul-\ntracold atomic quantum gases have enabled quantita-\ntive experimental tests of theories of strongly interact-\ning fermions [1{4]. In particular, fermions trapped in\noptical lattices can directly simulate the physics of elec-\ntrons in a crystalline solid, shedding light on novel phys-\nical phenomena in materials with strong electron corre-\nlations. A major e\u000bort is devoted to the realization of\nthe Fermi-Hubbard model at low entropies, believed to\ncapture the essential aspects of high- Tcsuperconductiv-\nity [5{11]. For bosonic atoms, a new set of experimen-\ntal probes ideally suited for the observation of magnetic\norder and correlations has become available with the ad-\nvent of quantum gas microscopes [12{14], enabling high-\nresolution imaging of Hubbard-type lattice systems at the\nsingle atom level. They allowed the direct observation\nof spatial structures and ordering in the Bose-Hubbard\nmodel [13, 15] and of the intricate correlations and dy-\nnamics in these systems [16, 17]. A longstanding goal\nhas been to realize such a quantum gas microscope for\nfermionic atoms. This would enable the direct probing\nand control at the single lattice site level of strongly cor-\nrelated fermion systems, in particular the Fermi-Hubbard\nmodel, in regimes that cannot be described by current\ntheories. These prospects have sparked signi\fcant experi-\nmental e\u000bort to realize site-resolved, high-\fdelity imaging\nof ultracold fermions, but this goal has so far remained\nelusive.\nIn the present work, we realize a quantum gas micro-\nscope for fermionic40K atoms by combining 3D Raman\nsideband cooling with a high resolution imaging system.The imaging setup incorporates a hemispherical solid im-\nmersion lens optically contacted to the vacuum window\n(Fig. 1(a)). In combination with a microscope objec-\ntive with numerical aperture (NA) of 0.60, the system\nachieves an enhanced NA of 0.87 while eliminating aber-\nrations that would arise from a planar vacuum window.\nIn order to keep the atoms localized while performing \ru-\norescence imaging, one must simultaneously cool them in\norder to mitigate the heating from spontaneously emitted\nimaging photons. Previous microscope experiments in\nHubbard-type lattices [12{14] cool via optical molasses.\nIn contrast, we employ 3D Raman sideband cooling [18{\n26], in which Raman transitions on vibration-lowering\nsidebands are combined with optical pumping to provide\ncooling. Our method therefore not only achieves site-\nresolved imaging, but also leaves a large fraction of the\natoms (72(3)%) in the 3D motional ground state of each\nlattice site. This opens up prospects for the prepara-\ntion of low entropy many-body states, by measuring the\natoms' initial positions and rearranging them into the\ndesired con\fguration [27].\nRaman sideband cooling has previously been used to\ncool87Rb and133Cs atoms in lattices and in optical\ntweezers to large ground state populations [19{26]. Here,\nwe realize continuous Raman sideband cooling of40K\nusing two states from the ground hyper\fne manifolds,\njai=jF= 9=2;mF=\u00009=2iandjbi=j7=2;\u00007=2i,\nwhich form an approximate two-level system. To make\njaiandjbinon-degenerate with other hyper\fne states,\nand to provide a quantization axis for optical pump-\ning, we apply a bias \feld of 4 :2 G along the xdirection\n(Fig. 1(b)). A pair of Raman beams collinear with xand\nylattice beams, but not retro-re\rected, drives vibration-\nlowering Raman transitions from jaitojbi(Fig. 1(c)).\nThe Raman lasers are detuned \u000041 GHz from the D2\ntransition. The optical pumping is performed on the D1\ntransition, 3 nm away from the D2 line, allowing us to\n\flter out stray Raman light while transmitting atomicarXiv:1503.02648v2  [cond-mat.quant-gas]  10 Mar 20152\n(b)\nBxy (a)\nxz\nNA = 0.6\n(c)zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1\nzxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\n13% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n23% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n23% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n23% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n2\nFIG. 1. (a) High-resolution imaging setup. A solid immersion\nlens is formed by a spherical cap and a super-polished sub-\nstrate contacted on either side of the vacuum window. Using\nan objective with NA = 0.60, the system achieves an e\u000bective\nNA = 0.87. The substrate re\rects 1064 nm light while trans-\nmitting D1 and D2 light of40K. The lattice beams are shown\nin red; the optical pumping and x-Raman beams are shown in\nblue. (b) Top view of Raman beams (blue) and optical pump-\ning beam (green). (c) Raman cooling scheme. The Raman\nbeams detuned near the D2 line (solid blue) drive vibration-\nlowering transitions. The optical F-pumping beam (dashed\ngreen) is tuned to the D1 line. Not shown is the mF-pumping\nbeam.\n\ruorescence. By collecting the photons that are spon-\ntaneously scattered during this optical pumping process,\nwe can image the atoms without using additional reso-\nnant light.\nTo prepare a cold cloud of fermionic atoms under the\nmicroscope,40K is \frst sympathetically cooled with23Na\nin a plugged magnetic quadrupole trap [28], centered\n\u00189 mm below a super-polished substrate that forms the\nbottom of the solid immersion lens. After removal of\n23Na, the cloud of\u00181 million40K atoms is magnetically\ntransported to the substrate and trapped in a vertical lat-\ntice formed by a 1064 nm laser beam re\rected o\u000b the sub-\nstrate at an angle of 5 :9\u000e. A single layer 7 :8\u0016m from the\nsurface is selected using a radiofrequency sweep in a verti-\ncal magnetic gradient followed by a resonant light pulse\nthat removes atoms in the remaining layers. Next, we\nprepare a 50/50 mixture of j9=2;\u00009=2iandj9=2;\u00007=2i\nto allow thermalization, and transfer the atoms to a ver-\ntical (zdirection) 1064 nm beam, forming a lattice along\nzwith a spacing of 532 nm (Fig. 1(a)). After evaporat-\ning by lowering the power of the z-lattice, the z-depth is\n(a) (c) (b)zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n1zxy\nπσ + σ−\n|7/2,−7/2/angbracketright|7\n2,−7\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle7\n2,−7\n2/angbracketrightBig\n|9/2,−9/2/angbracketright|9\n2,−9\n2/angbracketright/vextendsingle/vextendsingle/vextendsingle9\n2,−9\n2/angbracketrightBig\nΩ=0 ,δ=0 Ω >0,δ=0 δ/greatermuchΩ\nδ/greatermuchΩ,Γ\n|g,N+1/angbracketright| e, N/angbracketright| g,N/angbracketright| e, N−1/angbracketright\n|D/angbracketright Γee Γeg Γge Γgg\nσ++σ−\nσ++σ− π\nσ−\n13% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n23% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n23% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n23% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n23% 97% 770 .1 nm\n4P1/2 4P3/2\nD1 D2\nD2 D1\n2FIG. 2. (a) Electronic ground ( jg;N i) and excited ( je;N i)\nstates dressed by Noptical pumping photons. (b) On res-\nonance, all dressed states are equally populated and expe-\nrience an anti-trapping potential. The four decay channels\nbetween the dressed states are equal. (c) The dressed states\nwith \n=\u000e= 0:175, shown with the dark state jDi. Only a\nsmall fraction \n4=\u000e4of the steady-state population resides in\nanti-trapping states. For the atoms in the trapping state, the\nbranching ratios between trapping transitions (solid arrows)\nand anti-trapping transitions (dashed arrow) are shown.\nincreased to 180 \u0016K. We simultaneously ramp up two ad-\nditional 1064 nm beams (Fig. 1(a)) re\rected o\u000b the sub-\nstrate at 10 :8\u000eand retro-re\rected. These form a lattice\nin the horizontal plane with a spacing of 541 nm.\nDuring imaging, the atoms are trapped in a deep\nlattice, where the potential at each lattice site can be\napproximated by a harmonic well with vibrational fre-\nquency!. At our imaging depth, the vibrational fre-\nquencies for the three axes are ( !x;!y;!z) = 2\u0019\u0002\n(280;300;260) kHz, corresponding to lattice depths of\n210\u0016K, 240\u0016K and 180\u0016K respectively. The Rabi cou-\npling for transitions that change the vibrational num-\nber by one is proportional to the Lamb-Dicke param-\neter,\u0011= \u0001ka, wherea=q\n~\n2m!is the harmonic\noscillator length and ~\u0001kis the momentum transfer\ndue to the Raman beams. Along our lattice directions,\n\u0001kx= 8:0\u0016m\u00001, \u0001ky= 8:0\u0016m\u00001and \u0001kz= 3:1\u0016m\u00001,\nyielding Lamb-Dicke parameters of 0.17 for xandy, and\n0.068 forz. The polarizations of both Raman beams are\nlinear and parallel to the substrate (Fig. 1(b)), in or-\nder to avoid di\u000berential e\u000bective magnetic \felds between\njaiandjbithat would arise for circularly polarized light.\nThe Raman beam along the y-axis contains a single fre-\nquency, whereas the Raman beam along the x-axis con-\ntains three frequencies, allowing us to address the cooling\nsidebands of the three directions simultaneously. The re-\nsulting two-photon detunings from the bare jai ! jbi\ntransition are 400 kHz, 450 kHz and 360 kHz for cooling\nalongx,yandzrespectively. These frequencies compen-\nsate for di\u000berential Stark shifts that arise in the presence\nof optical pumping light. The x-Raman beam intensi-3\n(b) (a)\nyx\n3.0 μm0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0Transfer probability\n-300 -200 -100 0 100 200 300\nTwo-photon Raman detuning (kHz)XY\nZ\nFIG. 3. (a) Site-resolved imaging of fermonic atoms on a densely-\flled 541 nm-period optical lattice, with an exposure time of\ntwo seconds; one can clearly discern the lattice structure and individual atoms. (b) Raman spectrum after cooling. The lattice\ndepths are chosen such that the vibrational sidebands are well-resolved. The heating sidebands for the three lattice axes are\nlabeled Z, X and Y. We observe a large sideband asymmetry, from which we extract a 3D ground state occupation of 72(3)%.\nties of the three frequency components are 0 :79 W=cm2,\n0:47 W=cm2and 0:49 W=cm2respectively; the intensity\nof they-Raman beam is 2 :0 W=cm2.\nIn addition to these Raman beams, optical pumping\nlight is present to complete the cooling cycle. During op-\ntical pumping, atoms enter electronically excited states,\nand preferentially decay into the desired state. Typi-\ncally, the excited states experience an anti-trapping po-\ntential when the ground state experiences a trapping po-\ntential. For our 1064 nm lattice, the anti-trapping poten-\ntial for the atoms in the 4 P 1=2states is 5:4 times stronger\nthan the trapping potential for atoms in the 4 S 1=2states,\ndue to the 4 P 1=2!3 D 3=2transition at 1169 nm. This\nstrong anti-trapping would lead to heating and di\u000busion\nof atoms through the lattice during imaging.\nA solution to this problem is to detune the optical\npumping light away from resonance. This reduces the\npopulation in the anti-trapping states and favors tran-\nsitions into trapping states. One can understand this\nby considering the dressed states of a driven two-level\nsystem. At zero intensity, the dressed states, jg;Niand\nje;Ni, correspond to the bare trapping and anti-trapping\nstates with Nphotons respectively (Fig. 2(a)). In the\npresence of resonant pumping light, neither of the dressed\nstates is trapping (Fig. 2(b)), which leads to heating.\nHowever, at large detunings \u000e\u001d\n, where \n is the\nRabi frequency, one dressed state becomes trapping. Fur-\nthermore, spontaneous decay among the dressed states\nfavors population in the trapping states. Speci\fcally, the\ndecay rates \u0000 eg, \u0000ee, \u0000ggand \u0000 ge, de\fned in Fig. 2(b),\nare proportional to 1, s,sands2respectively, where\ns= \n2=\u000e2. The ratio of the anti-trapped population to\nthe trapped population is suppressed, because in steady\nstate, it is given by \u0000 ge=\u0000egwhich scales as s2. An-other bene\ft of large detunings involves the state jDi\ninto which atoms are optically pumped (Fig. 2(c)). This\nstate is dark to the optical pumping light, and hence\nhas no excited state admixture; consequently, it expe-\nriences a trapping potential. Atoms in trapping states\njg;Nidecay preferentially into jDi, since the ratio of\nanti-trapping transitions to dark state transitions scales\nas \u0000 ge=\u0000gg=s, which is small at large \u000e.\nIn light of these considerations, we detune the\nhyper\fne-changing ( F) pumping light\u000080 MHz from the\nStark-shifted F= 7=2!F0= 9=2 transition and the\nZeeman-level ( mF) pumping light \u000080 MHz from the\nStark-shifted F= 9=2!F0= 7=2 transition. The\noptical pumping beam co-propagates with the x-Raman\nbeam, and has its polarization optimized for minimal \u001b+\nadmixture. The FandmFcomponents have intensities\nof 5:8 mW=cm2and 1:6 mW=cm2respectively; the Lamb-\nDicke parameters for optical pumping are 0 :17 forxand\ny, and 0.18 for z.\nThis Raman cooling scheme allows us to collect \ruo-\nrescence while keeping the atoms con\fned to their lattice\nsites (Fig. 3(a)). Furthermore, we \fnd that atoms are\ncooled predominantly into their motional ground state.\nIndeed, Raman spectroscopy reveals a 3D ground state\npopulation of 72(3)% after cooling (Fig. 3(b)). Note that\nparameters are optimized for imaging \fdelity rather than\nfor a large ground state population. We measure a \ruo-\nrescence rate of\u00185000 photons =atom\u0001s with a lifetime\nof\u001830 seconds. With a photon collection and detection\ne\u000eciency of 20%, about 1000 photons per atom can be\ncollected in an exposure time of one second, which is suf-\n\fcient to detect single atoms with high \fdelity. Using the\nnumber of scattered photons and the Lamb-Dicke param-\neters for spontaneous emission, we estimate the cooling4\n(e)\n (g) (f)\n(a) 600\n500\n400\n300\n200\n100\n0Intensity (a.u.)\n541nm(b)\n1200\n800\n400\n0Counts\n(d)(c)\n1.0g⁽/two.superior⁾(r)1.1\n0.9\n1.0 0.8 0.6 0.4 0.2 0.0\nDistance (μm)30 20 10 0\nIntensity (a.u.)15 10 5 0\nDistance (541nm)0.10\n0.05\n0.00Rate (s/hyphen.superior/one.superior)\n15 10 5 0\nTime (s)\nFIG. 4. (a) Radially averaged PSF extracted from isolated atoms; the FWHM is 640 nm. (b) Intensity histogram after binning\nthe deconvoluted images by lattice site. The threshold for reconstruction is shown by the dashed line. (c) Loss and hopping\nrates, shown in red squares and blue circles respectively, as extracted from 20 consecutive one-second exposures. (d) Correlation\nmeasurement g(2)(r) of a thermal cloud, showing the absence of distance-dependent loss. (e) Image of sparsely \flled lattice with\ngrid lines showing lattice spacing and orientation. The exposure time is one second. (f,g) The same image after deconvolution\nand with the \flled sites identi\fed.\nrate to be\u00184\u0016K=ms.\nTo verify that we can resolve individual lattice sites,\nwe measure the point spread function (PSF) of our imag-\ning system using isolated atoms from sparsely populated\nimages. The measured PSF has a full-width-half-max\n(FWHM) of 640 nm (Fig. 4(a)). Images are deconvolved\nwith the PSF to achieve sub-lattice-site resolution. From\nsuch images, we also extract the lattice axes and spacings\nnecessary to reconstruct the atomic distribution. Binning\nthe intensity of the deconvolved image by lattice site re-\nveals a clear bimodal distribution (Fig. 4(b)), which is\nused to determine whether a site is \flled (Fig. 4(e,f,g)).\nThis bimodality gives a reconstruction error of <%1.\nAn important aspect of quantum gas microscopy is the\n\fdelity of the imaging process, which can be character-\nized by hopping and loss rates. To this end, we take a\nseries of images of the same atomic cloud and observe\nchanges in the site occupations between images. Sites\nthat are empty but become occupied in a subsequent im-\nage are counted as hopping events; sites that become\nempty are counted as loss events. The Raman coolingparameters are optimized for low hopping and loss rates\nwhile maintaining a \fxed level of \ruorescence. For opti-\nmized parameters, we achieve loss rates of <4:8\u00060:2%\nand hopping rates of <1:2\u00060:2% for one-second ex-\nposures of clouds with \fllings between 0.10 and 0.20\n(Fig. 4(c)). These rates, which include reconstruction\nerrors, give an detection \fdelity of >95% for sparse\nclouds. At higher \fllings, hopping events lead to loss of\nadditional atoms in doubly-occupied sites due to light-\nassisted collisions. However, even for unity \flling, we\nestimate the imaging \fdelity to still be >94%, because\nof the low hopping rate.\nTo ensure that the imaging does not cause additional\nlosses for neighboring atoms, one can measure the 2-\npoint correlation function g(2)(r) =hn(x)n(x+r)i=hni2\nof thermal clouds, since distance-dependent loss will pro-\nduce anti-correlations at short distances. For a dilute\nthermal cloud with a \flling of 0.19, one 29 \u000232 site im-\nage givesg(2)(r) = 1:00(7) for distances from r= 1 to\n10 lattice spacings, indicating that the imaging does not\ncause signi\fcant distance-dependent loss (Fig. 4(d)).5\nIn conclusion, we have realized high-\fdelity site-\nresolved imaging of40K fermionic atoms in a Hubbard-\ntype optical lattice by combining 3D Raman sideband\ncooling with high resolution photon collection. In con-\ntrast to existing boson microscopes, the technique leaves\natoms predominantly in the absolute 3D ground state\nof a given lattice site. This opens up new ways to\nassemble low-entropy Fermi-Hubbard systems atom by\natom [27, 29, 30]. Combining site-resolved imaging with\non-site manipulation would allow one to deterministically\ncreate localized excitations and follow their time evolu-\ntion [17]. Finally, the presence of23Na in our system\ninvites the realization of a quantum gas microscope for\nultracold fermionic NaK molecules [31], which have been\nproposed as a new resource for quantum information pro-\ncessing and quantum simulation of lattice models with\nlong-range dipolar interactions.\nWe would like to thank Katherine Lawrence for experi-\nmental assistance and critical readings of the manuscript.\nThis work was supported by the NSF, AFOSR-PECASE,\nAFOSR-MURI on Exotic Phases of Matter, ARO-MURI\non Atomtronics, ONR, a grant from the Army Research\nO\u000ece with funding from the DARPA OLE program, and\nthe David and Lucile Packard Foundation.\n[1] M. Inguscio, W. Ketterle, and C. Salomon, eds., Ul-\ntracold Fermi Gases , Proceedings of the International\nSchool of Physics \"Enrico Fermi\", Course CLXIV,\nVarenna, 20 - 30 June 2006 (IOS Press, Amsterdam,\n2008).\n[2] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.\n80, 885 (2008).\n[3] W. Zwerger, ed., The BCS-BEC crossover and the uni-\ntary Fermi gas , Vol. 836 (Springer, 2011).\n[4] M. W. Zwierlein, in Novel Super\ruids, Vol. 2 , edited by\nK.-H. Bennemann and J. B. Ketterson (Oxford Univer-\nsity Press, Oxford, 2014).\n[5] T. Esslinger, Annual Review of Condensed Matter\nPhysics 1, 129 (2010).\n[6] J. Chin, D. Miller, Y. Liu, C. Stan, W. Setiawan, C. San-\nner, K. Xu, and W. Ketterle, Nature 443, 961 (2006).\n[7] R. J ordens, N. Strohmaier, K. G unter, H. Moritz, and\nT. Esslinger, Nature 455, 204 (2008).\n[8] U. Schneider, L. Hackerm uller, S. Will, T. Best, I. Bloch,\nT. A. Costi, R. W. Helmes, D. Rasch, and A. Rosch,\nScience 322, 1520 (2008).\n[9] D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and\nT. Esslinger, Science 340, 1307 (2013).\n[10] J. Imri\u0014 ska, M. Iazzi, L. Wang, E. Gull, D. Greif,T. Uehlinger, G. Jotzu, L. Tarruell, T. Esslinger, and\nM. Troyer, Phys. Rev. Lett. 112, 115301 (2014).\n[11] R. A. Hart, P. M. Duarte, T. L. Yang, X. X. Liu, T. Paiva,\nE. Khatami, R. Scalettar, N. Trivedi, D. A. Huse, and\nR. G. Hulet, Preprint arXiv:1407.5932 (2014).\n[12] W. S. Bakr, J. I. Gillen, A. Peng, S. F olling, and\nM. Greiner, Nature 462, 74 (2009).\n[13] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau,\nI. Bloch, and S. Kuhr, Nature 467, 68 (2010).\n[14] M. Miranda, R. Inoue, Y. Okuyama, A. Nakamoto, and\nM. Kozuma, arXiv:1410.5189 (2014).\n[15] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I.\nGillen, S. F olling, L. Pollet, and M. Greiner, Science\n329, 547 (2010).\n[16] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg,\nP. Schau\u0019, C. Gross, L. Mazza, M. C. Banuls, L. Pollet,\nI. Bloch, and S. Kuhr, Science 334, 200 (2011).\n[17] M. Cheneau, P. Barmettler, D. Poletti, M. Endres,\nP. Schau\u0019, T. Fukuhara, C. Gross, I. Bloch, C. Kollath,\nand S. Kuhr, Nature 481, 484 (2012).\n[18] C. Monroe, D. M. Meekhof, B. E. King, S. R. Je\u000berts,\nW. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev.\nLett. 75, 4011 (1995).\n[19] S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H.\nDeutsch, and P. S. Jessen, Phys. Rev. Lett. 80, 4149\n(1998).\n[20] V. Vuleti\u0013 c, C. Chin, A. J. Kerman, and S. Chu, Phys.\nRev. Lett. 81, 5768 (1998).\n[21] A. J. Kerman, V. Vuleti\u0013 c, C. Chin, and S. Chu, Phys.\nRev. Lett. 84, 439 (2000).\n[22] D.-J. Han, S. Wolf, S. Oliver, C. McCormick, M. T.\nDePue, and D. S. Weiss, Phys. Rev. Lett. 85, 724 (2000).\n[23] K. D. Nelson, X. Li, and D. S. Weiss, Nat. Phys. 3, 556\n(2007).\n[24] A. M. Kaufman, B. J. Lester, and C. A. Regal, Phys.\nRev. X 2, 041014 (2012).\n[25] Y. S. Patil, S. Chakram, L. M. Aycock, and M. Ven-\ngalattore, Phys. Rev. A 90, 033422 (2014).\n[26] J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuleti\u0013 c,\nand M. D. Lukin, Phys. Rev. Lett. 110, 133001 (2013).\n[27] D. S. Weiss, J. Vala, A. V. Thapliyal, S. Myrgren,\nU. Vazirani, and K. B. Whaley, Phys. Rev. A 70, 040302\n(2004).\n[28] J. W. Park, C.-H. Wu, I. Santiago, T. G. Tiecke, S. Will,\nP. Ahmadi, and M. W. Zwierlein, Phys. Rev. A 85,\n051602 (2012).\n[29] S. Murmann, A. Bergschneider, V. Klinkhamer, G. Z urn,\nT. Lompe, and S. Jochim, Phys. Rev. Lett. 114, 080402\n(2015).\n[30] A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L.\nWall, M. Foss-Feig, K. R. A. Hazzard, A. M. Rey, and\nC. A. Regal, Science 345, 306 (2014).\n[31] C.-H. Wu, J. W. Park, P. Ahmadi, S. Will, and M. W.\nZwierlein, Phys. Rev. Lett. 109, 085301 (2012)."    },    {        "title": "1905.03983v1.Diverse_quantization_phenomena_in_AA_bilayer_silicene.pdf",        "content": "Diverse quantization phenomena in AA bilayer silicene\nPo-Hsin Shih1, Thi-Nga Do1, Godfrey Gumbs2;3,\nDanhong Huang4, Hai Duong Pham1, and Ming-Fa Lin1\n1Department of Physics, National Cheng Kung University, Tainan, Taiwan 701\n2Department of Physics and Astronomy,\nHunter College of the City University of New York,\n695 Park Avenue, New York, New York 10065, USA\n3Donostia International Physics Center (DIPC),\nP de Manuel Lardizabal, 4, 20018 San Sebastian, Basque Country, Spain\n4US Air Force Research Laboratory, Space Vehicles Directorate,\nKirtland Air Force Base, New Mexico 87117, USA\n(Dated: May 13, 2019)\nAbstract\nThe rich magneto-electronic properties of AA-bottom-top (bt) bilayer silicene are investigated\nusing a generalized tight-binding model. The electronic structure exhibits two pairs of oscilla-\ntory energy bands in which the lowest conduction and highest valence states of the low-lying pair\nare away from the K point. The quantized Landau levels (LLs) are classi\fed into various sep-\narated groups by the localization behaviors of spatial distributions. The LLs in the vicinity of\nFermi energy do not present simple wave function modes which are quite di\u000berent from other two-\ndimensional systems. The geometry symmetry, intralayer and interlayer atomic interactions, and\ne\u000bect of a perpendicular magnetic \feld are responsible for the peculiar LL energy spectra in AA-bt\nbilayer silicene. This work provides a better understanding of the diverse magnetic quantization\nphenomena in 2D condensed-matter materials.\n1arXiv:1905.03983v1  [physics.comp-ph]  10 May 2019I. INTRODUCTION\nSilicene, an isostructure to graphene, is purely made of silicon atoms through both the\nsp2andsp3bondings. Up to this point of time, silicene systems have been successfully\nsynthesized by the epitaxial growth on the di\u000berent substrate surfaces. Monolayer silicene\nwith di\u000berent sizes of the unit cells was produced on various substrates, such as Ag(111)\n(4\u00024)1,2, Ir(111) (p\n3\u0002p\n3)3and ZrB 2(0001) (2\u00022)4. Such a buckled single-layer\nhoneycomb lattice was clearly identi\fed by the high-resolution measurements of STM and\nlow-energy electron di\u000braction. It should be noticed that the examined system might present\nan enlarged unit cell because of the signi\fcant e\u000bect of substrates, e.g., the non-negligible\norbital hybridization between silicon atoms and substrate ones. The energy band shows\na Dirac-cone structure with a graphene-like Fermi velocity but in the presence of a small\nspin-orbital coupling of \u0018;2 meV, as directly con\frmed by the ARPES experiments1,2.\nBilayer silicene is predicted to exhibit four distinct kinds of stacking con\fgurations, namely\nAB-bt, AB-bb, AA-bt, and AA-bb5{12. Experimental growth has been done for the two AB\nstable bucklings5, however, there is still lack of similar experimental study for AA-stacked\nbilayer silicene.\nTheoretical studies have focused on the essential properties of monolayer and bilayer with\nor without adatom chemisorptions or guest atom substitutions based on various approaches,\ncovering the \frst-principles calculations6{12, the generalized tight-binding model13,14, and\nthe e\u000bective-mass approximation15. While the \frst method is suitable in studying the\noptimal geometries, the later two are powerful tools for the exploration of the magnetic\nquantizations. Nevertheless, the e\u000bective-mass model reveals certain limitations in dealing\nwith complicated systems such as ABC trilayer graphene16, AB-bt bilayer silicene17and\nothers. A single-layer silicene is predicted to possesses a relatively narrow gap of \u00185 meV6,\nas a result of the weak spin-orbital coupling. The interplay between intrinsic interactions\nand electric or magnetic \felds might cause the destruction of z= 0-plane mirror symmetry\n(electric \feld) and the appearance of periodical Peierles phases (magnetic \feld), respec-\ntively. The former gives rise to the spin-dominated splitting energy band and the dramatic\nchange of band gap15; the latter yields highly degenerate Landau levels15. Recently,\nsilicene with adatoms (X) systems have been investigated for the chemical modi\fcations\n2by the remarkable multi-orbital hybridizations in Si-Si, Si-X and X-X bonds by VASP\ncalculations18,19. Such phenomena might induce the metallic or semiconducting behaviors in\nmaterials. However for bilayer silicene, the phenomenological models might not be suitable\nfor solving the magnetic-\feld-dominated fundamental properties due to the non-negligible\nbucklings, the largely enhanced spin-orbital interactions and the complex interlayer hopping\nintegrals. Speci\fcally, the optimization of reliable tight-binding parameters which are\nnecessary to reproduce the low-lying two pairs of valence and conduction bands are likely\nimpossible because of the complex in the \frst-principle results for energy dispersions6{12.\nBilayer silicene with AA and AB stackings are predicted to present the non-monotonous\nenergy dispersions and the irregular valleys at the non-high-symmetry points. Moreover,\nthe free carrier densities are expected to be quite sensitive to the buckling and stacking\ncon\fgurations.\nIn this paper, we explore the diverse quantization phenomena in AA-bt silicene by em-\nploying a generalized tight-binding model. Calculations and analysis will focus on the band\nproperties across the Fermi level, energy dispersions, critical points in energy-wave-vector\nspace, distinct valleys, special structures of van Hove singularities in density of states,\nsigni\fcant magnetic-\feld dependences, classi\fcation of valence and conduction LL groups\nand their main features. Especially, the valley-enriched magnetic quantization will be\ninvestigated by the detailed examinations on the spatial oscillation modes of magnetic wave\nfunctions. The interesting combined e\u000bects of distinct LL groups are clari\fed from di\u000berent\nstable or metastable valleys. Furthermore, we also discuss the signi\fcant di\u000berences in\nthe essential physical properties between AA-bt bilayer silicene and monolayer silicene and\ngraphene in the electronic valley structure point of view.\nII. THEORETICAL METHOD\nWe develop a generalized tight-binding model for AA-bt bilayer silicene in the presence\nof a uniform perpendicular magnetic \feld and explore the magnetoelectronic properties.\nThe crystal structure of a AA-bt bilayer silicene with hopping interaction terms is clearly\nillustrated in Fig. 1(a). The two layers possess opposite buckling order, in which the [A1,\n3A2] sublattices lie in the inner planes while the [B1, B2] sublattices are located at the outer\nplanes. The lattice constant and bond length are a= 3:83\u0017Aandb= 2:21\u0017A, respectively.\nA primitive unit cell contain four silicon atoms. Accordingly, the critial Hamiltonian is built\nfrom the four tight-binding functions of Si-3p zorbitals. It can be written as\nH=X\ni;lUl\nicyl\nicl\ni+X\nhi;ji;l;l0\rll0\nijcyl\nicl0\nj: (1)\nHere,cyl\ni(cl\ni) is the creation (annihilation) operator which could generate (destroy)\nan electronic state at the i-th site of the l-th layer,Ul\ni(Al;Bl) is the buckled-sublattice-\nheight-dependent Coulomb potential energy due to the applied gate voltage. \rll0\nijrepresents\nthe intra- and inter-layer atomic interactions, in which the former comes from the the\nnearest-neighbor interactions in the [ Al;Bl] sublattices ( \r0= 1:004 eV) and the latter stand\nfor interactions between sublattices from di\u000berent layers ( \r1=\u00002:110 eV,\r2=\u00001:041\neV, and\r3= 0:035 eV), as shown in Fig. 1(a). Interestingly, our optimization show that\n\r1is much higher than while \r2is comparable to \r0. This result, in addition to the high\nsymmetry of stacking con\fguration, might be responsible for the negligible spin-orbital\ncouplings. Apparently, the above Hamiltonian, in the absence of spin-orbital couplings, is\nsu\u000ecient for the investigation of certain essential physical properties.\nThe application of a uniform perpendicular magnetic \feld evidently changes the main\ncharacteristic of lattice. The original unit cell is remarkably enlarged to be come a long\nrectangular due to the \feld-induced extra Peierls phases17. The extended unit cell includes\n16\u001e0=\u001eSi atoms, in which \u001e0=hc=e is the magnetic \rux quantum and \u001e=Bzp\n3a2=2\nis the magnetic \rux. As a result, the magnetic Hamiltonian is a huge Hermitian matrix,\ne.g., the size is a \u001813000\u000213000 for a ( kx= 0;ky= 0) state at Bz= 20 T. For AA-bt\nbilayer silicene, the model calculation takes into account the buckled honeycomb structure\nand complicated intra- and inter-layer atomic interactions. The combined e\u000bect of those\ningredients and an external magnetic \feld is expected to generate the diverse physical\nphenomena, especially the magneto-electronic properties.\n4FIG. 1: (Color online) The (a) side view of geometric structure including the signi\fcant hopping\nintegrals for AA-bt bilayer silicene. The low-energy band structures are shown for (b) 3D and (c)\n2D along the high symmetry points. Energy band of monolayer silicene is also plotted in (c) for\ncomparison.\nIII. RESULTS AND DISCUSSION\nAA-bt bilayer silicene presents the unique and feature-rich energy dispersion, as demon-\nstrated in Figs. 1(b) and 1(c) for 3D and 2D views, respectively. There are two pairs of\nvalence and conduction bands due to the Si-3 pz-orbital\u0019bondings and they are slighly\nasymmetric about the Fermi level of EF= 0. It shows the semimetallic behavior with zero\ngap and a \fnite density of states (DOS) at EF, as clearly illustrated in Fig. 2(a). The outer\npair of conduction and valence energy bands is originated at higher and deeper energy\nranges (jEc;vj\u00152 eV). On the other hand, the pair of energy bands near EFis expected\n5to dominate the low-energy essential properties of the system. Interestingly, the stable and\nnon-stable electronic valleys are formed from the electronic states near the high-symmetry\npoints of the hexagonal \frst Brillouin zone. In particular, there exist the M, K and \u0000\nvalleys with di\u000berent conduction and valence band-edge state energies of about (0.5 eV,\n-0.51 eV), (1.19 eV, -1.21 eV), and (1.43 eV, -1.50 eV), respectively. These special valleys\nare expected to be closely related to the magnetic quantizations of the initial Landau levels,\nas discussed later.\nThe M valley belongs to the saddle point while the K and \u0000 valleys correspond to the\nlocal extreme points, therefore they lead to the di\u000berent van Hove singularities (Fig. 2(a))\nand the diverse LL energy spectra. Furthermore the Dirac-cone energy dispersion is absent\nbecause the low-lying electronic states closest to the Fermi level are not formed at K/K0\nvalleys. Instead of upward conduction and downward valence Dirac cone initiated from\nthe K/K0points as in monolayer graphene20, there exist the concave-downward conduction\nand concave-upward valence valley in AA-bt bilayer silicene. Such a signi\fcant di\u000berence\nbetween the two system might come from the more complicated and stronger atomic\ninteractions of the latter. It should be noticed that, the lowest conduction and highest\nvalence electronic states in AA-bt stacking are located at the midway of M and \u0000 points,\nmoreover, they are separated by a very small energy spacing of \u00188 meV. As a result, the\ndensity of states near EFpresent the special structure with a \fnite value. Furthermore,\nthe low-lying Landau levels do not correspond to the initial magneto-electronic states. In\ngeneral, the electronic properties in AA-bt bilayer silicene are in great contrast with those\nof monolayer silicene2and a AA-stacked bilayer graphene21.\nThe van Hove singularities in the density of states are closely associated with three kinds\nof band-edge states, as illustrated in Fig. 2(a). The DOS spectrum exhibit feature-rich\nstructures, including the asymmetric peaks in the square-root divergent form, shoulder\nstructures, and logarithmically divergent peaks. There exist a pair of temple-like cusp\nstructures crossing the Fermi level and they are separated by a quite small energy spacing\nof\u00188 meV. These structures come from the extraordinary conduction and valence band\nstructures near EFwhich could be considered as the one-dimensional parabolic dispersions.\nAway from the Fermi level, the two pairs of prominent symmetric peaks arising from the\n6FIG. 2: (Color online) The (a) van Hove singularities in density of states. The wave-vector-\ndependent wave functions are shown in (b). The solid and dashed blue curves of the A1and B1\nsublattices, respectively.\n2D saddle point (M point) are located at (0 :50 eV &\u00000:52 eV) and (2.60 eV & \u00002:65 eV),\nrespectively. Regarding the extreme points, there appear the special shoulder structures\nat (1.10 eV,\u00001:15 eV) and (2.16 eV, \u00002:20 eV) for the K valley and (1.43 eV, \u00001:5 eV)\nfor the \u0000 one. The special DOS spectrum in AA-bt bilayer silicene re\rects the unique\nenergy dispersion, it could be examined from the high-resolution STS experiments22{25.\nThis measurement method is useful for the investigation of the interplay between the\nbuckled structure and atomic interactions in bilayer silicene. For AA-bt stacking, the DOS\nis relatively high near the M point compared with other high-symmetry points, leading\n7to the very complicated LL energy spectrum, dissimilarly to those of monolayer graphene\nand silicene26,27. Roughly speaking, the M valley is regarded as the unstable one for the\nmagnetic quantization where LLs could not be initiated from.\nThe Bloch wave functions consisting of the Si-3 pz-orbital tight-binding functions on\nthe four sublattices strongly depend on the wave vectors, as clearly indicated in Fig.\n2(b). Those on the A and B sublattices are represented by the orange and green curves,\nrespectively; the result for monolayer silicene is also shown as solid and dashed blue curves\nfor comparison. The low-lying conduction band (and valence band; not shown) exhibit\nthe equal distribution probability on the B1& B2[A1& A2] sublattices because of the\nsame (x;y)-plane projections with similar chemical environment. Speci\fcally, electronic\nstates near the K point exhibit the vanishing A1and A2components and the identical B1\nand B2ones. Along the K !M!\u0000 directions, the Bl- and Al-sublattice probabilities\nvary as 0:5!0:375!0:265 and 0!0:125!0:2355, respectively. Such behavior is\nquite di\u000berent from those in monolayer silicene where the distribution of all sublatices\nbecome identical along M !\u0000. Generally, the distribution probability of Blsublattices are\nevidently dominated the low-lying energy spectrum. The opposite is true for the higher\nconduction and deeper valence bands. That is to say, the the equivalence of the intralayer\n[Al;Bl] sublattices is thoroughly broken by the very strong interlayer hopping integrals\n(\r1and\r2). On the other hand, monolayer silicene with signi\fcant spin-orbital coupling\npresent similar distribution probabilities of the sublattices of the same spin state during the\nvariation of wave vector due to the honeycomb lattice symmetry.\nAA-bt stacking possesses the distinctive and diverse magnetic quantization phenomena,\nmainly owing to the feature-rich valley structures of the low-lying pair of conduction and\nvalence bands. The LL energy spectra exhibit a number of interesting characteristics,\nincluding the LL state degeneracy, localization behavior of wave functions, sublattice\ndependence of localized oscillation modes, the well-behaved and non-well-behaved LLs,\nthe complicated magnetic-\feld dependence, and the LL crossing phenomenon. It is\nworth mentioning that the localization centers of LLs are continuously changed with the\nvariation of ( kx;ky), however, similar phenomena are observed for di\u000berent electronic states.\nOur numerical calculations in this work mainly focus on the magnetic quantization at\n8FIG. 3: (Color online) The Landau levels and wave functions at Bz= 40 T for the (a) conduction\nand (b) valence bands of the \frst group.\n9(kx= 0;ky= 0) state, which is su\u000ecient in understanding the essential magneto-electronic\nproperties. LLs could be classi\fed into di\u000berent groups based on the electronic valleys and\nthe main features of spatial distributions.\nAs for LL initiated from the \u0000-point top and bottom valley, the conduction and valence\nenergy spectra present the explicit asymmetry behavior. The originated conduction and\nvalence Landau levels are, respectively, located at 1 :43 eV and\u00001:53 eV forBz= 40 T,\nas shown in Figs. 3(a) and 3(b). LLs exhibits a nearly uniform energy spectrum, as a\nresult of the isotropically parabolic energy dispersion near the \u0000 point, similarly to that\nof a 2D electron gas28. The well-behaved magnetic subenvelope functions on the four\nsublattices are localized at 0 and 1/2 of the Bz-enlarged unit cell, and they are degenerate.\nThe oscillation modes of LLs on all four subenvelope functions are equivalent. Moreover,\nthe LL wave functions on (A1& A2) as well as (B1& B2) sublattices are observed to\nbe idential. These special characteristics of LLs might be related to the equivalence of\nthe intralayer [Al, Bl] sublattices and the same chemical environment for the interlayer\n[A1, A2] & [B1, B2] sublattices. The quantum number of each LL, nc;v\n1, is determined\nby the number of zero modes of its wave function on the dominated B isublattices. The\nfact that Blsublattices clearly dominate the energy spectra at \u0000 valley is consistent\nwith the zero-\feld wave-vector-dependent wave functions as shown in Fig. 2(b). In\ngeneral, each LL is four-fold degenerate by the spin and localization degrees of freedom.\nThis is dissimilar to the eight-fold degeneracy of LLs in monolayer graphene and silicene26,27.\nThe conduction and valence Landau levels magnetically quantized from electronic states\nnear the K/K0points display diverse characteristics. Here, there are LLs corresponding to\nboth low-lying and outer pairs of energy bands, however, the former is the center of interest,\nas illustrated in Figs. 4(a) and 4(b). Though the LL energy spectrum presents roughly\nuniform spacing, similarly to that near the \u0000 valley, LLs show quite di\u000berent behaviors.\nThe magnetic subenvelope functions are revealed at four distinct localization centers (1/6,\n2/6 4/6 and 5/6) of the extended unit cell. Especially, the LL spatial distributions are\nidentical for 1/6 and 4/6 as well as 2/6 and 5/6 areas, leading to the eight-fold degeneracy\nof LLs, as observed in monolayer graphene?. The \frst few LLs are well-behaved and their\nquantum numbers ( nc;v\n2) could be determined based on the wave functions of the evidently\n10FIG. 4: (Color online) The Landau levels and wave functions at Bz= 40 T for the (a) conduction\nand (b) valence bands of the second group.\n11FIG. 5: (Color online) The Landau levels and wave functions at Bz= 40 T of the \frst group near\nthe Fermi energy. The conduction and valence LLs are presented in green and pink, respectively.\ndominated Blsublattices. The spatial distributions on the Aland Blsublattices exhibit\none-mode di\u000berence, in which those on Blis higher (for 1/6 center) or less (for 2/6 center)\nthan the corresponding ones on Al, depending on the localization centers. Apparently,\nthe initialnc;v\n2= 0 LLs at 2/6 center are exceptionally four-fold degenerate because such\nmagneto-electronic states only originate from the Blsublattices.\nAway from the initiated level, LLs quantized from \u0000 and K valleys are slightly distorted\nbecause of the strong interlayer atomic interactions ( \r1and\r2), leading to the di\u000eculty\nin de\fning their quantum numbers. Regarding the nc;v\n1LLs, though their oscillation\ndistributions become much wider, which are \u001835% of a unit cell, they still remain two\ndegenerate localization centers of 0 and 1/2. This further illustrates the monotonous\n12variation of the stable \u0000 valley along the various directions. For the nc;v\n2LLs, the splitting of\nLL states at di\u000berent localization centers of (1/6, 2/6, 4/6, and 5/6) is no longer observable.\nTheir quantum numbers are expected to be much smaller than those of the nc;v\n1group\ndue to the higher DOS in the K-related valleys (Fig. 2(a)). In addition, the magnetic\nquantization demonstrates the fact that the M-point saddle structure belongs to the K/K0\nvalley.\nIn the vicinity of the Fermi level, there exist only conduction and valence LLs quantized\nfrom the \u0000 valley, as clearly shown in Fig. 5. Here, the magneto-electronic states are\nquite dense and complicated, therefore it is quite di\u000ecult to characterize the LLs. The\nLL spatial distributions present thousands of oscillation modes which are localized at 0\nand 1/2 centers. Such unique magnetic quantization phenomenon is never observed in the\nother condensed-matter systems according to the previous theoretical and experimental\nstudies13,26,27. Apparently, the high-resolution STS measurements22{25might not be able to\ndirectly identify the low-energy LLs in AA-bt bilayer silicene. Nevertheless, the complex\nof low-energy magneto-electronic states in the AA-bt bilayer silicene should be associated\nwith the other essential physical properties, such as, the delta-function-like van Hove\nsingularities, magneto-optical absorption spectra with speci\fc selection rules, quantum Hall\ntransports, and inter-Landau-level damping and magnetoplasmon modes. These should\nbe worthy of a systematic investigation despite of possibly high barriers in the numerical\ntechnique and detailed analysis.\nThe magnetic-\feld-dependent energy spectra are critical for the particularized compre-\nhension of the magnetic quantization phenomena. Though the conduction and valence LLs\nexhibit asymmetric spectra, their main features are equivalence, therefore the following\ndiscussion will focus on the conduction bands. The quantized LLs of the outer pair of energy\nbands show the linear Bz-dependence, as demonstrated in Fig. 6(a) for the conduction\nspectrum. The split energy of the neighboring LLs gradually increases with the growth of\n\feld strength. These quantization characteristics indicate the stable K/K0valleys for such\nelectronic structure. On the other hand, the dependence of LL groups corresponding to the\nlow-lying energy bands on the variation of magnetic \feld is much more complicated. The\nenergy spectra are the combination of both nc;v\n1andnc;v\n2LL groups with frequent intragroup\n13FIG. 6: (Color online) The magnetic-\feld-dependent conduction Landau level energy spectra for\n(a) the \frst (green curves) and (b) third (black curves) groups.\nand intergroup crossing behaviors, as illustrated in Figs. 6(b), 7 and 8.\nConcerning the nc\n1LLs originated from the \u0000 point, the initiated LLs are linearly\ndependent on Bzwithout any intragroup crossing (Fig. 6(b)), similarly to that of the\nouter energy bands but with smaller LL energy spacing. With decreasing energy, the\nBz-dependent spectrum contain of both the nc\n1(green curves) and nc\n2(red curves) LLs,\nas shown in Figs. 7(a) and 7(b). Interestingly, the two groups of LLs present crossing\nphenomenon without any hybridization of magneto-electronic states. This is because the\ntwo stable valleys of K and \u0000 are independent of each other. The evolution of nc\n2LLs with\nthe increase of magnetic \feld strength have simply linear behavior for su\u000eciently high\n14FIG. 7: (Color online) The magnetic-\feld-dependent conduction Landau level energy spectra for\nthe \frst (green curves) and second (red curves) groups at (a) higher and (b) lower energy ranges.\nenergy,Ec\u00150:5 eV, as clearly presented in Fig. 7(a). On the contrary, LLs at Ec\u00180:45\neV are formed in the shoulder-like structure, corresponding the energy dispersion around\nthe M point. The main reason for this is that the electronic states at very high DOS can\nnot be quantized into the well-behaved LLs.\nThe magneto-electronic properties of AA-bt bilayer system are in great contrast with\n15those in monolayer silicence and graphene26,27. While the former shows the initial LLs at\nhigher and lower energy ranges, those of the latter begin near EF= 0. Apparently, the\nmain di\u000berences lie on the main characteristics of LLs around the Fermi level which are very\ncomplicated for AA-bt stacking. Monolayer systems possess the conduction and valence\nDirac cones with the Dirac point or extremely narrow gap at the K/K0point. The low-lying\nLLs present the well-behaved spatial distributions and therefore their quantum numbers\nare easily determined based on the number of zero modes. Moreover, the Bzdependence\nof LL energies exhibit linear form in the absence of crossing phenomenon, dissimilarly to\nthe frequent crossing LLs in AA-bt stacking. The above-mentioned divergent points of\nAA-bt bilayer silicene compared with other monolayer systems mainly come from its unique\nintrinsic properties, such as the buckling structure with opposite ordering and signi\fcant\ninterlayer atomic interactions.\nIV. CONCLUDING REMARKS\nWe have presented a thorough investigation of the unique and diverse magnetic quanti-\nzation phenomena in AA-bt bilayer silicene using a generalized tight-binding model. This\nmaterial possesses the special electronic structure with two pairs of energy bands, in which\nthe low-lying pair shows interesting oscillation shape. Remarkably, the lowest conduction\nand highest valence states near the Fermi level are away from the K point, dissimilarly\nto that of graphene and other 2D systems. The complicated energy dispersion is closely\nrelated to the feature-rich magnetically quantized Landau levels. LLs originated from K\nand \u0000 valleys are quite di\u000berent in the main characteristics, covering the LL degeneracy,\nthe spatial distributions, the dominated sublattices, and the magnetic-\feld dependence.\nEspecially in the vicinity of Fermi energy, there are so many magneto-electronic states and\nthey do not present simple wave function modes. The geometry symmetry, intralayer and\ninterlayer atomic interactions, and e\u000bect of a perpendicular magnetic \feld are responsible\nfor the peculiar LL energy spectra in AA-bt bilayer silicene.\n16Acknowledgments\nWe would like to acknowledge the \fnancial support from the Ministry of Science and\nTechnology of Taiwan (R.O.C.) under Grant No. MOST 105-2112-M-017-002-MY2. D.H.\nwould like to thank the supports from the AFOSR and from the DoD Lab-University Col-\nlaborative Initiative (LUCI) Program. G.G. would like to acknowledge the support from the\nAir Force Research Laboratory (AFRL) through Grant #12530960.\n17References\n1P. Vogt, P. D. Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio et al., Phys.\nRev. Lett. 108, 155501 (2012).\n2B. Feng, Z. Ding, S. Meng, Y. Yao, X. He and P. Cheng et al., Nano Lett. 12 (7) , 35073511\n(2012).\n3L. Meng, Y. Wang, L. Zhang, S. Du, R. Wu and L. Li et al., Nano Lett. 13 (2) , 685 (2013).\n4A. Fleurence, R. Friedlein, T. Ozaki, H. Kawai, Y. Wang and Y. Yamada-Takamura, Phys. Rev.\nLett. 108, 245501 (2012).\n5R. Yaokawa, T. Ohsuna, T. Morishita, Y. Hayasaka, M. J. S. Spencer and H. Nakano, Nat.\nComm. 7, 10657 (2016).\n6B. Mohan, A. Kumar and P. K. Ahluwalia, Physica E 53, 233 (2013).\n7H. S. Liu, N. Han and J. J. Zhao, Phys.: Condens. Matter 26, 475303 (2014).\n8J. J. Liu and W. Q. Zhang, RSC Adv. 3, 21943 (2013).\n9F. Liuy, C. C. Liuy, K. H. Wu, F. Yang and Y. G. Yao, Phys. Rev. Lett. 111, 066804 (2013).\n10J. E. Padilha and R. B. Pontes, J. Phys. Chem. C 119, 3818 (2015).\n11H. X. Fu, J. Zhang, Z. J. Ding, H. Li and S. Menga, Appl. Phys. Lett. 104, 131904 (2014).\n12X. Q. Wang and Z. G. Wu, Phys. Chem. Chem. Phys. 19, 2148 (2017).\n13T. N. Do, P. H. Shih, G. Gumbs, D. Huang, C. W. Chiu and M. F. Lin, Phys. Rev. B 97,\n125416 (2018).\n14T. N. Do, G. Gumbs, P. H. Shih, D. Huang, C. W. Chiu and C. Y. Chen et al., Scienti\fc\nReports 9, 624 (2019).\n15C. J. Tabert and E. J. Nicol, Phys. Rev. B 88, 085434 (2013).\n16Y. P. Lin, C. Y. Lin, Y. H. Ho, T. N. Do, and M. F. Lin, Phys. Chem. Chem. Phys., 17,\n15921-15927 (2015).\n17T.N. Do, P. H. Shih, G. Gumbs, D. Huang, C. W. Chiu, M. F. Lin, Phys. Rev. B, 97 (12) ,\n125416 (2018).\n18X. Lin and J. Ni, Phys. Rev. B 86, 075440 (2012).\n19H. Sahin and F. M. Peeters, Phys. Rev. B 87, 085423 (2013).\n20M. Sprinkle, D. Siegel, Y. Hu, J. Hicks, A. Tejeda and A. Taleb-Ibrahimi et al., Phys. Rev.\n18Lett. 103, 226803 (2009).\n21A. L. Rakhmanov, A. V. Rozhkov, A. O. Sboychakov and F. Nori, Phys. Rev. Lett. 109, 206801\n(2012).\n22S. H. Jhang, M. F. Craciun, S. Schmidmeier, S. Tokumitsu, S. Russo and M. Yamamoto et al.,\nPhys. Rev. B 84 (16) , 161408 (2011).\n23D. L. Miller, K. D. Kubista, G. M. Rutter, M. Ruan and W. A. de Heer, Science 324 (5929) ,\n924 (2009).\n24G. M. Rutter, S. Jung, N. N. Klimov, D. B. Newell, N. B. Zhitenev and J. A. Stroscio, Nature\nPhysics 7 (8) , 649 (2011).\n25G. Li and E. Y. Andrei, Nature Physics 3 (9) , 623 (2007).\n26L. J. Yin, S. Y. Li, J. B. Qiao, J. C. Nie and L. He , Phys. Rev. B 91, 115405 (2015).\n27C. J. Tabert and E. J. Nicol , Phys. Rev. B 88, 085434 (2013).\n28C. Kallin and B. I. Halperin , Phys. Rev. B 30, 5655 (1984).\n19"    },    {        "title": "1610.08633v1._U_1___Fermi_liquid_theory___A_Fermi_liquid_state_that_supports_exclusion_statistics.pdf",        "content": "arXiv:1610.08633v1  [cond-mat.str-el]  27 Oct 2016U(1)Fermi liquid theory - A Fermi liquid state that supports excl usion statistics\nTai-Kai Ng\nDepartment of Physics, Hong Kong University of Science and T echnology,\nClear Water Bay Road, Kowloon, Hong Kong, China\n(Dated: June 12, 2018)\nWe propose in this paper an effective low-energy theory for in teracting fermion systems which\nsupports exclusion statistics. The theory can be viewed as a n extension of Landau Fermi liquid\ntheory where besides quasi-particle energy ξk, the kinetic momentum kof quasi-particles depends\nalso on quasi-particle occupation numbers as a result of mom entum (k)-dependent current-current\ninteraction. The dependence of kinetic momentum on quasi-p articles excitations leads to change in\ndensity of states and exclusion statistics. The properties of this new Fermi liquid state is studied\nwhere we show that the state (which we call U(1)-Fermi liquid state) has Fermi-liquid like properties\nexcept that the quasi-particles are notadiabatically connected to bare fermions in the system and\nthe state may not satisfy Luttinger theorem.\nPACS numbers: 05.30.Pr, 67.10.Fj, 71.10.Hf, 71.10.Ay\nI: Introduction\nThe notion of statistical interaction was first proposed\nby Haldane[1]. Let the number of available single par-\nticle states for spin σfermions from momentum range\nk tok+dkbe (dk\n2π)dzσ(k) wherezσ(k) = 1 for normal\nfermions. Then following Haldane[1, 2], the exclusion\nstatistics between particles is defined through the rela-\ntion\nzσ(k) = 1−/summationdisplay\nσ′/integraldisplayddk′\n(2π)dαkσk′σ′nk′σ′(1)\nwhereαkσk′σ′is a function determined by the micro-\nscopic theory giving rise to exclusion statistics. It is now\nunderstood that exclusion statistics is a general feature\nofinteractingfermion/bosonsystemsdescribedbyBethe-\nAnsatz solution in one dimension[3], and also exists in a\nnumber of Quantum-Hall type two-dimensional systems\nwith exact known ground state wave-functions[4]. De-\nspite the existence of these examples, a general under-\nstanding of the microscopic mechanism behind exclusion\nstatistics is still lacking[5].\nIn thispaperweshowhowexclusionstatistics canarise\nin amodified effective Fermi-liquid-type low-energy the-\nory. In usual Fermi liquid theory[6], the quasi-particle\nenergyξk[n] are shifted by quasi-particle interaction and\nexternal scalar potential with,\nξk[δn] =εk+φk[δn]+Φ, (2a)\nwhere Φ is an external scalar potential and\nφk=1\nV/summationdisplay\nk′σ′fkσ;k′σ′δnk′σ′, (2b)\nεk=δE\nδnkσ|n=n(0)andfkσ;k′σ′=δ2E\nδnkσδnk′σ′|n=n(0)are\nthe usually defined (single-particle) quasi-particle energy\nand Landau interaction, respectively[6]. E[n] is the totalenergy functional and nkσ=n(0)\nkσ+δnkσis the quasi-\nparticleoccupation number forthe state ( kσ), where n(0)\nis the ground state quasi-particleoccupation number and\nδnis the correctionto n(0). In this paperweproposethat\ninadditiontotheabove, theeffectiveinteractionbetween\nquasi-particles may also leads to shift in the kinetic mo-\nmentum kof the quasi-particles given by\nk[n] =k0[n(0)]+ak[δn]+A, (3a)\nwhereAis the external vector (EM) field and\nak[δn] =1\nV/summationdisplay\nk′σ′gkσ;k′σ′δnk′σ′ (3b)\nrepresents an interaction-induced effective k-dependent\nreal space gauge field at small δn.gkσ;k′σ′is a phe-\nnomenologicalvectorfunction. The microscopicmeaning\nofakandgkσ;k′σ′will be discussed in section III where\nthe Lagrangian and Hamiltonian of the system are in-\ntroduced. We note that in general a nonzero ak[n(0)]\nmay also exists in the ground state which is absorbed in\nthe definition of k0[n(0)]. We shall assume here a spin-\nisotropic system such that εk,φkandakare independent\nof spinσ. We shall call the resulting fermion liquid state\naU(1) Fermi liquid state since the interaction induced\nφkandakfields are the scalar and vector components of\nan effective k-dependent U(1) gauge field acting on the\nfermions, respectively.\nII: Dynamical properties of U(1)Fermi liquid state\nSpecifically we consider a fermion system in external\nEM fields Φ and Awith an effective low energy Hamil-\ntonian\nE[δnk,Ak] =/summationdisplay\nk0σεkδnkσ+Φ/summationdisplay\nk0σδnkσ (4)\n+1\n2V/summationdisplay\nk0σ,k′\n0σ′fkσ,k′σ′δnkσδnk′σ′.2\nwherek(r) =k0+Ak(r),Ak=ak+Aand the other\nterms have their usual meaning. E[δnk,Ak] describes\na Landau-Fermi liquid Hamiltonian[6] except that the\nquasi-particlekinetic momenta are shiftedin the presence\nof (k-dependent) real-space gauge field Ak. We shall\nassume that akis given by Eq. (3b) and examine its\nconsequences in this section.\nThe relation between the k-dependent effective gauge\nfieldAkand exclusion statistics can be seen most easily\nby noting that the shift in kinetic momentum k0→kre-\nsults in a coordinate transformation of the system when\nthe quasi-particle occupation numbers nkσ(r) are char-\nacterized by kandr, where\n1\nV/summationdisplay\nk0=/integraldisplayddk0\n(2π)d→/integraldisplay\nz(k)ddk\n(2π)d(5)\nwherez(k) = (Det|M(k)|) is the Jacobian for the trans-\nformation between the coordinates k0andk,Mij(k) =\n∂k0i\n∂kj=δij−∂kjAkiis ad×dmatrix. The Jacobian z(k)\ndescribesarenormalizationinthedensityofstates(DOS)\nof the system when the phase space is transformed from\ncanonical- to kinetic- momentum representation[7]. The\ndependence of z(k) onδnis precisely exclusion statistics\n(see Eq. (1) and section IV).\nlow energy dynamics\nTheeffectivelowenergytheorywedescribehereissim-\nilar to the Landau-Silin theory[8] for charged fermions in\nexternal gauge fields Φ and Aexcept that the effective\ngaugefield Akismomentum( k)-dependent. Thedynam-\nics of the quasi-particles are given by the usual Hamilton\nequations of motion\n˙r=∇k0ξk[δn]; (6)\n˙k0=−∇rξk[δn]\nwhereξk[n] is given by Eq. (2). We obtain after some\nstraightforward algebra\n˙rµ=/summationdisplay\nν(∂kν\n∂k0µ)∂kνξk[δn(r)] (7)\n=/summationdisplay\nν/bracketleftbig\nM(k)−1/bracketrightbig\nνµ/parenleftBigg\nvkν+/summationdisplay\nk′σ′∂kνfkσ;k′σ′δnk′σ′(r)/parenrightBigg\n,\nand\n˙k= =−∇rξk[δn]+˙Ak (8)\n=−/summationdisplay\nk′σ′fkσ;k′σ′∇rδnk′σ′(r)−∇rΦ\n+∂Ak\n∂t−˙r×Bk(r)+(˙k.∇k)Ak\ntoleadingorderinΦand δn, wherevk=∇kεkandBk=\n∇r×Ak. Eqs. (7) and (8) are the same as the Landau-\nSilin equations[8] for charged quasi-particles in externalEM field except the presence of[ M(k)]−1, (˙k.∇k)Akand\n∇k.Aktermscomingfromthe k-dependenceofthe gauge\nfieldAk. In particular,\n˙r→vk, (9)\n˙k→ −/summationdisplay\nk′σ′fkσ;k′σ′∇rδnk′σ′(r)−∇rΦ+∂Ak\n∂t\nin the linearized Boltzmann equation\ndnkσ\ndt=∂nkσ\n∂t+˙r.∇rnkσ+˙k.∇knkσ=/parenleftbigg∂nkσ\n∂t/parenrightbigg\ncol,\n(10)\nandAkintroduces only an effective k-dependent electric\nfield on the system.\nquasi-particles charge and current\nUsing Eq. (5)), the charge carried by quasi-particles is\ngiven in the linear response regime by\nQ(r,t) =/summationdisplay\nσ/integraldisplayddk0\n(2π)dnkσ(r,t) =/summationdisplay\nσ/integraldisplayddk\n(2π)dz(k)nkσ(r,t)\nIt is straightforward to show that\nz(k) = 1−∇k·Ak+O(Ak)2(11)\nand we obtain in the linear response regime\nQ(r,t) =/summationdisplay\nσ/integraldisplayddk\n(2π)d/parenleftBig\nδnkσ(r,t)−(∇k·Ak(r))n(0)\nk/parenrightBig\n(12)\n=/summationdisplay\nσ/integraldisplayddk\n(2π)de∗\nkσδnkσ(r,t)\nwhere\ne∗\nkσ= 1−/summationdisplay\nσ′/integraldisplayddk′\n(2π)dn(0)\nk′(∇k′·gk′σ′kσ) (13)\n= 1+/summationdisplay\nσ′/integraldisplayddk′\n(2π)d(∇k′n(0)\nk′)·gk′σ′kσ\nis the effective charge carried by the quasi-particles. We\nnote that in general e∗/negationslash= 1 as long as ∇k′·gk′σ′kσ/negationslash= 0,\nindicating that the quasi-particles in U(1)-Fermi liquids\nare not adiabatically connected to the ”bare” particles.\nIn particular, the Fermi sea volume in U(1) Fermi liquid\nmay not satisfy the Luttinger Theorem[9] and the U(1)\nFermi liquid state is not a Landau Fermi liquid state. We\nnote also that although exclusion statistics is in general\na bulk effect that affects the whole Fermi sea, it can be\nexpressed in terms of properties on the Fermi surface\nalone at zero temperature when ∇k′n(0)\nk′∼ −δ(ξk).\nWenextconsiderthecurrentcarriedbyquasi-particles.\nWe start by examining the equation of motion of the3\nchargeQ(r,t). Putting together Eq. (9), (10) and (12),\nwe obtain in the linear response regime,\n∂Q\n∂t=/summationdisplay\nσ/integraldisplayddk\n(2π)d∂\n∂t/parenleftBig\nδnkσ(r,t)+Ak(r)).∇kn(0)\nk/parenrightBig\n(14)\n=−/summationdisplay\nσ/integraldisplayddk\n(2π)d/parenleftbigg\nvk.∇rδnkσ+(˙k−∂Ak\n∂t).∇kn(0)\nkσ/parenrightbigg\n=−∇r./summationdisplay\nσ/integraldisplayddk\n(2π)djkδnkσ\nwhere\njk=vk−/summationdisplay\nk′σ′vk′∂n(0)(ε′)\n∂ε′|ε′=µfk′σ′kσ.(15)\nis the current carried by quasi-particles as in usual Lan-\ndau Fermi liquid theory. Notice that the charge and cur-\nrent of quasi-particles are renormalized separately by the\nspace- and time components of effective gauge field ak\nandφk, respectively in U(1) Fermi liquid.\nsolution of the transports equation\nWe nowconsiderthe generalsolutionofthe Boltzmann\ntransport equation in the collisionless regime ωτ >>1\nwhere the collision term can be neglected. We first con-\nsider the system in an external scalar potential Φ( r,t)∼\nΦ0ei(q.r−ωt). In this case δnkσ(r,t)∼δnkei(q.r−ωt)is\nspin-independent. Putting these into the transport equa-\ntions (9) and (10), we obtain\n(ω−q·vk)δnk+∂n0\nk\n∂εk/bracketleftBigg\n(q·vk)/parenleftBigg\nΦ0+/summationdisplay\nk′fs\nkk′δnk′/parenrightBigg\n(16)\n+ωvk./summationdisplay\nk′gs\nkk′δnk′/bracketrightBigg\n= 0,\nwheref(g)s\nkk′=f(g)k↑k′↑+f(g)k↑k′↓. Rearranging\nterms, the equation can be rewritten as\n(ω−q·vk)/parenleftBigg\nδnk+vk.∂n0\nk\n∂εk/summationdisplay\nk′g(s)\nkk′δnk′/parenrightBigg\n(17)\n=−(q·vk)∂n0\nk\n∂εk/bracketleftBigg\nΦ0+/summationdisplay\nk′(fs\nkk′+vk·gs\nkk′)δnk′/bracketrightBigg\n.\nIt is clear that we can write δnk=−∂n0\nk\n∂εkνk. For\nisotropic systems in three dimensions, we can expand[6,\n10]\nfs(a)\nkk′=∞/summationdisplay\nl=0fs(a)\nlPl(cosθkk′) (18)\nvk·g(s)\nkk′=∞/summationdisplay\nl=0gs(a)\nlPl(cosθkk′)\nνk=/summationdisplay\nlPl(cosθkq)νlwherePl’s areLegendre polynomials and θkpdenotes the\nangle between kandp. Eq. (17) can be simplified using\nthe properties of spherical harmonics. We obtain after\nsome lengthy but standard algebra[10],\nνl\n2l+1/parenleftbigg\n1−Gs\nl′\n2l+1/parenrightbigg\n+/summationdisplay\nl′˜Fs\nl′Ωll′(s)νl′\n2l′+1=−Ωl0(s)Φ0,\n(19)\nwheres=ω/qvF,G(F)s\nl=N(0)g(f)s\nl,˜Fs\nl=Fs\nl+Gs\nl,\nN(0) is the DOS on Fermi surface and\nΩll′(s) =1\n2/integraldisplay1\n−1dµPl(µ)µ\nµ−sPl′(µ).\nUsing eqs. (13) and (15) we alsoobtain for the physical\ncharge and current,\nQ= (1−Gs\n0)N(0)ν0 (20)\nJ= (1+Fs\n1\n3)N(0)ν1\n3.\nThe second equation (current renormalization) is the\nstandard Landau Fermi liquid result[6, 10]. The first\nequation (charge renormalization) is a new property of\ntheU(1) Fermi liquid state.\nThe density-density response of the system χdcan be\nobtained easily from Eq. (19) if we keep only l= 0,1\ncomponents. We obtain after some straightforward alge-\nbra,\nχd(s) =χ0(s)\n1−/parenleftBig˜fs\n0\n(1−Gs\n0)+(˜fs\n1\n1+Fs\n1/3)s2/parenrightBig\nχ0(s)(21)\nwhereχ0(s) is the Lindhard function.\nThe (transverse) conductivity σ(s) can be obtained\nsimilarly with −vk·qΦ→ωvk·EandE.q= 0, where\nE=iωAis the electricfield and θkEis the anglebetween\nkandE. Expanding νk=/summationtext\nlPl(cosθkE)νl, we obtain\nafter some similar algebra, J=σ(s)Ewhere\nσ(s) =σ0(s)\n1−(˜fs\n1\n1+Fs\n1/3)(iω)σ0(s), (22)\nσ0(s) is the conductivity for non-interacting fermions.\nWe note that the (extended)-Landau interaction terms\nfsandgstogether produceeffective interactionsbetween\ncharges and currents. However they play different roles\nin the renormalization of the quasi-particle charge and\ncurrent.\nIII: Lagrangian and Hamiltonian formulation\nThe transports equation we study in section II is sup-\nported by a proper Lagrangian and Hamiltonian formu-\nlation we derive here. For simplicity we shall consider4\na spinless system in this section. The extension to the\nspin-ful case is straightforward.\nWe start with the following phenomenological\nLagrangian[7, 11]\nL[n,A,Φ] =LG+L1[δn;a,φ]+L2[a,φ] (23)\nL1=−/summationdisplay\nkδnk/parenleftBig\n˙k.rk+ ˙rk.ak+εk+φk/parenrightBig\nL2=1\n2V/summationdisplay\nk;k′\n/summationdisplay\nµν=1,..dakµ[G−1]kµk′νak′ν\n+φk[f−1]kk′φk′/parenrightbig\nwhereLGis the Lagrangian describing the ground state\nandL1describes the low energy dynamics of fermions\nin effective gauge fields φkandak.rkis the position of\nthe particle with momentum kand kinetic energy εk.L2\nprovides the dynamics for the effective gauge fields.\nThe Euler-Lagrange equationsδL\nδk−d\ndtδL\nδ˙k= 0,δL\nδr−\nd\ndtδL\nδ˙r= 0 andδL\nδak= 0,δL\nδφk= 0 give rise to the equation\nof motions (7) and (8) in the main text with\nφk=1\nV/summationdisplay\nk′fkk′δnk′ (24a)\nand\nak=1\nV/summationdisplay\nk′Gkk′·˙rk′δnk′∼1\nV/summationdisplay\nk′Gkk′·vk′δnk′(24b)\nto leading order in δn, which are just Eqs. (2b) and (3b)\nif we identify fkk′as the Landau parameters and Gkk′·\nvk′=gkk′(with spin neglected), confirming that Lis\nthe Lagrangian for the U(1) Fermi liquid state.\nEliminating φkandakdirectly from Eq, (23), we also\nobtain\nL[n,A,Φ]→LG+L0[δn]+LI[δn] (25a)\nwhere\nL0=/summationdisplay\nkσδnk/parenleftBig\n−˙k.rk−εk/parenrightBig\n(25b)\nLI=−1\n2V/summationdisplay\nk;k′δnk[fkk′+˙rk·Gkk′·˙rk′]δnk′.\nIt is obvious that L0describes the single particle dynam-\nics of quasi-particles in U(1) Fermi liquid theory and L2\ndescribes interaction between quasi-particles. In particu-\nlar,thek-dependent vector potential akis originated from\ncurrent-current interaction between quasi-particles . The\ncorresponding Hamiltonian of the system is\nH[δn,ak] =EG+E[δnk,ak]−1\n2V/summationdisplay\nkk′µ,νakµ[G−1]kµk′νak′ν.\n(26)\nwhereEGis the ground state energy for the U(1) Fermi\nliquid. Notice that δH/δak= 0 leads to Eq. (24b) for\nak.Thermodynamics\nThe low temperature thermodynamics of the system\ncan be obtained by expanding H[δn,ak] to second order\ninδnk. Using\nεkδnk∼(εk0+ak·∇k0εk0)δnk0\nto second order in δn, we obtain\nH→EG+/summationdisplay\nk0εk0δnk0 (27)\n+1\n2V/summationdisplay\nk0;k′\n0δnk0/bracketleftbig\nfk0k′\n0+vk0·Gk0k′\n0·vk′\n0/bracketrightbig\nδnk′\n0\n=/summationdisplay\nk0εk0δnk0+1\n2V/summationdisplay\nk0;k′\n0δnk0˜fk0k′\n0δnk′\n0,\nwhich is a standard Fermi liquid energy functional with\neffective Landau interaction ˜fkk′. This result suggests\nthat the low energy thermodynamical behaviors of U(1)\nFermi liquid states are the same usual Fermi liquid\nstates[6, 10]. In particular, a linear specific heat CV=\nγT(T= temperature) is expected when the system is\nstable, i.e. when\n1+˜Fl\n2l+1>0,\nfor alll. The only difference between U(1) Fermi liquid\nandusualFermiliquidisthattheLandauinteraction fkk′\nis replaced by the extended Landau interaction ˜fkk′.\nIV: Summary and Discussion\nIn this paper we study a class of fermion liquids de-\nscribed by a modified Landau Fermi liquid type low\nenergy theory where besides quasi-particle energy, the\nquasi-particle kinetic momentum is also renormalized by\ninteractionbetween quasi-particles. We showthat the re-\nsultingU(1) Fermi liquid state exhibits exclusion statis-\ntics and the quasi-particles are notadiabatically con-\nnected to the non-interacting fermion states as a result.\nThe renormalization of kinetic momentum is a conse-\nquence of momentum-dependent current-current interac-\ntion between quasi-particles.\nThe dynamics of the U(1) Fermi liquid state is stud-\nied where we show that the resulting density-density and\ncurrent-current response functions are Fermi-liquid like,\nexcept the appearance of ‘extended’ Landau parameters\nwhich renormalizenot only current but also chargein the\nFermi liquid state. This result is rather non-trivial since\nexclusion statistics is in general a bulk effect that affects\nthe whole Fermi sea whereas only quasi-particle states\non the Fermi surface are important in Fermi liquid the-\nory. We find that the effects of exclusion statistics can5\nbe expressed in terms of Fermi surface properties at zero\ntemperature, when ∇kn(0)\nk∼ −δ(ξk) (Eq. (13)).\nThermodynamically the U(1) Fermi liquid is also\nFermi-liquid like except that the Fermi sea volume may\nnot satisfy the Luttinger Theorem and the Landau in-\nteraction fkk′is replaced by the extended Landau inter-\naction˜fkk′. TheU(1) Fermi liquid state is an example\nof ‘marginal’ Fermi liquid state where the low tempera-\nture properties of the system are Fermi liquid like except\nthat the quasi-particles are notadiabatically connected\nto non-interacting fermions. This is perhaps not too sur-\nprising since we know that in one dimension, exclusion\nstatistics is consistent with effective Luttinger-liquid-like\nlow energy theories[2].\nmicroscopic origin of exclusion statistics\nThe factor z(k) describes a modification of the DOS\nas a result of transformation from canonical coordinates\n(r,k0) to kinetic coordinates ( r,k). Comparing Eq. (11)\nwith Eq. (1) we obtain an excitation-driven exclusion\nstatistics effect with nkσ→δnkσand\nαkσk′σ′→ ∇k.gkσk′σ′.\nThe modified DOS can also be understood quan-\ntum mechanically. For simplicity we consider a one-\ndimensional system where the usual local Berry curva-\ntures are absent. In this case, the gauge field Ak(x) gives\nrise to a k- andx- dependent phase of the quasi-particle\nwave-function\nuk(x)→uk(x)eiθ(k;x), (28)\nwhereAk(x) =dθ(k;x)/dx. We note that since Ak(x)\nis a pure gauge, it does not have any effect on the local\ndynamics of the system. However, it may have a physical\neffect through modifying the boundary condition, i.e. a\nlarge gauge transformation.\nFor ordinary periodic systems, the allowed momentum\nvaluek’sarequantizedbecauseoftheboundarycondition\neikL=ei0= 1, whichgivesrisetothequantizationcondi-\ntionkL= 2mπ,L=length of system and the spacing be-\ntweenallowedvalueofmomentum isgivenby δk= 2π/L.\nInthepresenceofnonzero θ(k;x)field, theboundarycon-\ndition is modified to kL+θ(k;L)−θ(k;0) = 2mπand\nthe spacing between allowed value of momentum is given\nby\nδk(1+1\nLd(θ(k;L)−θ(k;0))\ndk) =2π\nL.(29)\nWe see that the DOS g(k) inkspace is modified from\ng(k) =L\n(2π)tog(k) =L\n(2π)+z(k), where\nz(k) =1\n2πd(θ(k;L)−θ(k;0))\ndk(30)\n=1\n2πd\ndk/integraldisplayL\n0dxdθ(k;x)\ndx=1\n2π/integraldisplayL\n0dxd\ndkAk(x),showingthatwemayidentify z(k,x) = (2π)−1dAk(x)/dk\nas the correction to localDOS of the system, which\nis exactly what we obtained from the U(1) Fermi liq-\nuid state. The analysis can be extended to higher di-\nmension systems easily in linear-response regime where\n∂(Ak(r))i/∂kj(i/negationslash=j) do not contribute[7, 11].\nIt is interesting to note that similar renormalization\nof DOS has been observed by Niu et al.[7] in study-\ning the semi-classical dynamics of electrons in periodic\nlattices when effective real- andk- space gauge fields\n(Berry curvatures) are present. let Ak(r) andBk(r) be\nthe Berry connection in real- and k- space, respectively.\nThe crossed r−kspace Berry curvatures are defined by\nΩrk\nµν=∂rµBν−∂kνAµ[7, 11], etc. Renormalization in\nDOS appears when Ωrk\nµν/negationslash= 0. The non-vanishing k-space\nderivative ∇k.Ak/negationslash= 0 we discuss in this paper corre-\nsponds to a special case where B= 0 but Ωkr\nµµ/negationslash= 0. The\ncrossed Berry-curvature is not a band structure effect as\nstudied by Niu et al., but an interaction effect driven by\nquasi-particle excitations. Our result in this paper sug-\ngests that interaction-induced crossed Berry curvature in\nk−rphase space may provide an effective mechanism of\ngenerating non-Fermi liquid states. The U(1) Fermi liq-\nuid state we propose in this paper is probably just one\nexample in this class of states.\n[1] F.D.M. Haldane, Phys. Rev. Lett. 67, 937 (1991).\n[2] Y.S. Wu, Phys. Rev. Lett. 73, 922 (1994).\n[3] see Y.S. Wu and Y. Yu, Phys. Rev. Lett. 75890 (1995).\n[4] W-P. Su, Y.S. Wuand J. Yang, Phys. Rev.Lett. 77, 3423\n(1996);\n[5] see for example, C. Nayak and F. Wilczek, Phys. Rev.\nLett.73, 2740 (1994); S.B. Isakov, Phys. Rev. Lett. 73,\n2150 (1994); M. V. N. Murthy and R. Shankar, Phys.\nRev. Lett. 73, 3331 (1994).\n[6] L.D. Landau, Zh. Eksp. Teor. Fiz. 30, 1058 (1956) [Sov.\nPhys, JETP 3, 920 (1956)]; ibid.8, 70 (1959).\n[7] D. Xiao, M-C. Chang and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010); D. Xiao, J. Shi, and Q. Niu, Phys. Rev.\nLett.95, 137204 (2005).\n[8] V. P. Silin, Zh. Eksp. Teor. Fiz. 33, 495 (1957)[Soviet\nPhys.-JETP 6, 387 (1958)].\n[9] J.M. Luttinger and J. C. Ward, Physical Review 118,\n1417 (1960); J.M. Luttinger, Physical Review 119, 1153\n(1960).\n[10] see for example G, Baym and C. Pethick, Landau Fermi-\nLiquid Theory: Concepts and Applications , WILEY-\nVCH Verlag GmbH and Co. KGaA, Weinheim (2004).\n[11] G. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999)."    },    {        "title": "2209.00785v1.Linear_in_Frequency_Optical_Conductivity_over_a_broad_range_in_the_three_dimensional_Dirac_semimetal_candidate_Ir__2_In__8_Se.pdf",        "content": "Linear-in-Frequency Optical Conductivity over a broad range in the three-dimensional Dirac\nsemimetal candidate Ir 2In8Se\nS. X. Xu,1,\u0003H. Q. Pi,2, 3,\u0003R. S. Li,1T. C. Hu,1Q. Wu,1D. Wu,4H. M. Weng,2, 3,yand N. L. Wang1, 4, 5,z\n1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China\n2Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China\n4Beijing Academy of Quantum Information Sciences, Beijing 100913, China\n5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China\nThe optical conductivity of the new Dirac semimetal candidate Ir 2In8Se is measured in a frequency range\nfrom 40 to 30000 cm\u00001at temperatures from 300 K down to 10 K. The measurement reveals that the compound\nis a low carrier density metal. We find that the real part of the conductivity \u001b1(!) is linear in frequency over a\nbroad range from 500 to 4000 cm\u00001at 300 K and varies slightly with cooling. This linearity strongly suggests the\npresence of three-dimensional linear electronic bands with band crossings near the Fermi level. Band structure\ncalculations indicate the presence of type-II Dirac points. By comparing our data with the optical conductivity\ncomputed from the band structure, we conclude that the observed linear dependence mainly originates from the\nDirac cones and the transition between the Dirac cones and the next lower bands. In addition, a weak energy\ngap feature is resolved below the charge density wave phase transition temperature in reflectivity spectra. An\nenhanced structure arising from the imperfect Fermi surface nesting is identified in the electronic susceptibility\nfunction, suggesting a Fermi surface nesting driven instability.\nI. INTRODUCTION\nThree-dimensitional Dirac semimetals (3DDSMs) have at-\ntracted tremendous interests in recent years due to their exotic\nphysical properties such as large linear magnetoresistivity1,2,\nhigh carrier mobilities3and strong SdH oscillations4–7. The\n3DDSMs are characterized by a fourfold degenerate Dirac\npoint where conduction and valence bands touch each other\nin momentum space. The Dirac point is usually formed by\ntwo spin-degenerated and linearly dispersed bands along all\nmomentum directions resulting in the three dimensionality.\nThe 3DDSM state is protected by the time-reversal symme-\ntry and inversion symmetry. When the inversion or time-\nreversal symmetry is broken, the 3DDSMs are driven into\nWeyl semimetals with di \u000berent transport behaviour and Fermi\nsurface states compared to 3DDSMs8–10.\nSo far, Only several materials have been identified\nas 3DDSMs. The typical examples are the hexagonal\nphase of Cd 3As2, tetragonal structure of Na 3Bi and or-\nthorhombic structure of ZrTe 5which were confirmed by\nangle-resolved photoemission spectroscopy11–16and optical\nspectrocopy17–19, respectively. The Dirac point close to the\nFermi level in 3DDSMs are easier to be tuned by outer param-\neters such as pressure and laser pulse, which usually induces\ninteresting structure and electronic transitions20–25. Therefore,\nit is significant to explore more unique 3DDSMs candiate ma-\nterials and identify their topological properties for advancinng\nthe knowledge of how Dirac and Weyl fermions behave.\nRecently, exotic subchalcogenides Ir 2In8Q (Q =S, Se, Te)\nwere synthesized successfully and suggested to be 3DDSMs\ncandidate based on transport experiments and density func-\ntional theory (DFT) calculations26,27. Subchalcogenides pos-\nsesses not only metal-metal bonds to each other but covalent\nbonds to the chalcogen atoms which easily induce abundant\nquantum states including charge density wave (CDW), su-\nperconductivity and topological state. Some of these quan-tum phenomena have been observed in subchalcogenides\nsuch as superconductor Bi 2Rh3Se228, toppological insulator\nBi14Rh3I929and magnetic Weyl semimetal Co 3Sn2S230. Pre-\nvious studies showed that Ir 2In8Q crystallizes in P4 2/mnm\nspace group and has two Dirac crossing along the \u0000-Z direc-\ntion of the Brillouin zone. In addition, an phase transition as-\nsociated with the commensuratedly 6 \u00026 (10\u000210) modulated\nstructure was observed in Ir 2In8Se (Ir 2In8Te) near 203 K (150\nK) along ab plane26,27. Incomprehensibly, the single-crystal\ndi\u000buse X-ray scattering measurements show that the supercell\nordering of Ir 2In8Se and Ir 2In8Te become very weak below\n100 K.\nTo confirm the Dirac cone structure and understand the ex-\notic behaviour of supercell ordering in Ir 2In8Q, various spec-\ntroscopy techniques are needed. On the one hand, there are\nonly SdH oscillations and theoretical works suggested that\nDirac crossings may exist in this novel material system. On\nthe other hand, the driving force of CDW phase transition is\nstill unknown. As we known that optical spectroscopy are\nsensitive to probe the electronic properties of bulk materials.\nTherefore, it is significant to reveal the nature of electronic\nstructure and phase transition utilizing optical spectroscopy\nmeasurements.\nIn this present work, we synthesized single-crystal Ir 2In8Se\nand performed temperature-dependent optical spectroscopy\nmeasurements. The optical reflectivity spectra reveal a rela-\ntively sharp plasma edge a low energy scale, suggesting that\nthe compound is a low carrier density metal. Furthermore, the\nspectral weight of the Drude response decreases with temper-\nature cooling. By subtracting a single sharp Drude peak and\ntwo narrow phonons peaks from the real part of the optical\nconductivity at 300 K, we reveal approximatedly linear con-\nductivity over a large frequency range (from 500 cm\u00001to 4000\ncm\u00001). And the slope of linear conductivity varies slightly\nwhen the temperature decreases. However, a weak suppres-\nsion feature in reflectance related to CDW formation is ob-\nserved below 750 cm\u00001. We further calculated the band struc-arXiv:2209.00785v1  [cond-mat.str-el]  2 Sep 20222\nture and optical conductivity of Ir 2In8Se. By comparing our\nexperimetal and calculated optical conductivity, we find the\nobserved nearly linear-in-frequency optical conductivity over\na broad range mainly originates from the Dirac cones and the\ntransition between the Dirac cones and the next lower bands.\nIn addition, the imaginary (real) part of electronic suscepti-\nbility of hole (electron) pocket peaks at CDW wave vector,\nindicating the Fermi surface nesting contributes to the forma-\ntion of CDW. These experimental and theoretical results show\nthat Ir 2In8Se indeed has the linear dispersion feature of 3D\nDirac semimetal but the energy gap associated to charge den-\nsity wave is very weak. Our work provides new insights for\nexploring the new 3DDSMs and understanding the phase tran-\nsition of the charge density wave states.\nII. EXPERIMENTAL METHODS\nThe single crystals of Ir 2In8Se were synthesized by self-\nflux method with indium as flux26. High-purity Ir powder\n(99.99%), In balls (99.999%) and Se powder (99.999%) were\nput into the crucible and sealed into a quartz tube with the ratio\nof Ir:In:Se =1:20:4. Then the quartz tube was heated to 1100\n\u000eC at 10 h and held for 24 h, then cooled to 650\u000eC at 1\u000eC/h.\nThe flux was removed by centrifugation. The atomic composi-\ntion of the single crystal was checked to be Ir:In:Se =2:8:1 by\nenergy dispersive x-ray spectroscopy. The large shiny poly-\nhedra single crystals with maximal length \u00187 mm (inset of\nFig.1(b)) were obtained. The temperature-dependent resistiv-\nity measurement was performed in a Quantum Design physi-\ncal property measurement system (PPMS) by a standard four-\nprobe method with current parellel to ab plane. The optical\nreflectance measurements of Ir 2In8Se (001) crystal were per-\nformed on the Fourier transform infrared spectrometer Bruker\n80V in the frequency range from 40 to 30000 cm\u00001. The value\nof reflectance R(!) was obtained by an in-situ gold and alu-\nminum evaporation technique.\nThe calculation of the electronic structure of Ir 2In8Se\nwas performed using the Vienna ab initio simulation pack-\nage (V ASP)31with the generalized gradient approximation\nof Perdew-Burke-Ernzerhof exchange-correlation potential32.\nThe self-consistent calculation was carried out on an 8 Ö8Ö8\nk-mesh with the energy cuto \u000bof 500 eV . The maximally-\nlocalized Wannier functions33–35were generated using Ir d\norbital and In p orbital. With the tight-binding Hamiltonian\nconstructed by the WANNIER90 package36, we calculated the\noptical conductivity based on the Kubo-Greenwood formula\nand employed the adaptive scheme for the broadened delta\nfunction with the dimensionless factor of 0.8. The calculation\nwas performed on the 200 Ö200Ö200 k-mesh, the convergence\nof which had been tested. The Fermi surface was calculated\nusing the WANNIERTOOLS package37. A 100 Ö100Ö100 k-\nmesh was used to obtain the electron susceptibility. We used\nthe Gaussian function to approximate the delta function and\nthe smearing factor is 10\u00003.III. RESULTS AND DISCUSSION\nFigure 1(a) displays the crystal structure of Ir 2In8Se which\nis featured by the IrIn 8polyhedra and Se atoms chains along\ncaxis. The IrIn 8polyhedra are corner sharing along the aand\nbaxes but alternate between corner and edge sharing along\nthecaxis forming a tetragonal structure. Figure 1(b) shows\nthe temperature-dependent resistivity curve of Ir 2In8Se single\ncrystal which demenstrates metallic behavior with a small re-\nsistivity value of about 2.7 Ö10\u00004\n\u0001cm at 300 K and the resid-\nual resistance ratio (RRR =\u001a300 K=\u001a2 K)is about 70 which is\n14 times larger than that from previous report26indicating the\nhigh quality of the single crystal sample. The resistivity curve\nshows a significant kink near 203 K without thermal hystere-\nsis which suggests the phase transition is of the second order.\nAll these sample fundamental characterizations are consistent\nwith the previous report26.\n01 002 003 000100200300\nSe\nIn\nIr\n/s114 (µΩ cm)T\nemperature (K)(a)(\nb)R\nRR/s32/s8776/s32/s55/s48\nFIG. 1. (a) Crystal structure of tetragonal Ir 2In8Se characterized\nby distorted corner-sharing or edge-sharing IrIn 8polyhedra. (b)\nTemperature-dependent resistivity with current parallel to abplane,\na charge density wave transition is evident near 210 K. Inset shows\nthe picture of the grown Ir 2In8Q crystal.\nFigure 2(a) shows the reflectivity up to 2000 cm\u00001at six dif-\nferent temperatures. In the low-frequency region ( !\u00141000\ncm\u00001),R(!) exhibits clearly a metallic response which is con-\nsistent with the results of transport26. Besides, R(!) shows a\nrelatively sharp plasma edge, and approaches unity at the low\nfrequency limit. The screened plasma edge frequency !scr\np\nwhich is related to the density nand e \u000bective mass m* of3\n05 001 000150020000.40.60.81.01\n011021031040.00.51.00\n5 001 00015002000-3000-2000-100004\n00500600700800-80-400400\n5 001 000150020000.000.040.080\n10020030050060070080001 002 003 000.40.50.60.7ReflectivityW\nave number (cm-1) 10 \n100 \n150 \n200 \n250 \n300T (K)(a)( b)(\nc)R\nω\n (cm-1)T = 300 Kε1 (ω)W\nave number (cm-1)ε1 (ω)ω\n (cm-1)I\nm[-1/ε(ω)]W\nave number (cm-1)ωpscr (cm-1)T\n (K) ε1 = 0 \nloss 05 0100150200Photon energy (meV)0\n5 0100150200Photon energy (meV)0\n5 0100150200Photon energy (meV)ReflectivityT\nemperature (K)/s119 = 600 cm-1(\nd)\nFIG. 2. (a) Temperature-dependent optical reflectivity spectra of a (001)-oriented crystal below 2000 cm\u00001. Inset shows the reflectivity during\nlarge energy scale range of 40-30000 cm\u00001at 300 K. (b) Temperature-dependent reflectivity at about 600 cm\u00001(c) Temperature dependence of\nthe real part of dielectric function \"1(!). Inset shows the enlargeed view of \"1(!) of which the zero crossings correspond to screened plasma\nfrequency at various temperatures. (d) The loss function Im[-1 /\"(!)] at di \u000berent tamperatures. The inset shows the screened plasma frequency\nof the free carriers obtained from zero crossings of \"1(!) (black square) and the peak of the loss function (red circle).\nfree carriers ( !scr2\np/n=m\u0003) shows a significant temperature\ndependence. As the temperature decreases, the edge contin-\nuously shifts to lower frequency and becomes steeper, indi-\ncating the reduction of both the free carrier density and the\nscattering rate, which are consistent with the experimental re-\nsults of Hall transport26. The rather lower plasma frequency\nindicates that the compound is a metal with small carrier den-\nsity or Fermi surfaces. In addition, two sharp features at 100\nand 250 cm\u00001are associated with IR-active phonon modes.\nThe temperature-dependent reflectivity at about 600 cm\u00001isplotted in Fig. 2(b). We found that the value of reflectivity de-\ncreases with temperature cooling and a kink is observed near\n200 K close to the CDW transition temperature. The suppres-\nsion of reflectivity coincides with the CDW transition from the\ntransport measurements which indicates the suppression fea-\nture of reflectivity below 200 K is related to the CDW energy\nformation.\nFigure 2(c) shows the real part of dielectric function\n\"1(!) which is derived from R(!) through Kramers-Kronig\ntransformation. The Hagen-Rubens relation was used for4\nthe low energy extrapolation of R(!) and the x-ray atomic\nscattering functions were employed for the high frequency\nextrapolation38. At low frequency, \"1(!) is negative which\nsatisfies the defining property of a metal. Thus the dielec-\ntric function \"1(!) can be described by the Drude model\n\"(!)=\"1\u0000!2\np=(!2+i!\r), where\"1,!pand\ris the high-\nfrequency dielectric constant, the Drude plasma frequency,\nand the electronic scattering rate respectively. The inset of\nFig. 2(c) clearly displays the zero-crossings of \"1(!) at dif-\nferent temperatures. The zero crossing of \"1(!) corresponds\nto the screened plasma frequency !scr\npwhich has a relation-\nship with the Drude plasma frequency !pthrough!scr\np=\n!p=p\"1. Moreover, the temperature dependent loss function\nIm[-1 /\"(!)] is plotted in the Fig. 2(d). The peak position\nof loss function gives rise to the value of plasma frequency\n!scr\np\u0019!p=p\"1indicating the peak position of loss function\ncan also be used to estimate the screened plasma frequency.\nAs shown in the inset of Fig. 2(d), the screened plasma fren-\nquency!scr\npobtained from zero crossing of \"1(!) is basically\nclose to the peak position of Im[-1 /\"(!)] which consistently\nshow that!scr\npdecreases from about 700 cm\u00001at 300 K to\n550 cm\u00001at 10 K.\nThe real part of optical conductivity \u001b1(!) is displayed in\nFig. 3. Fig. 3(a) displays \u001b1(!) below 300 cm\u00001at dif-\nferent temperatures. The Drude-type conductivity was ob-\nserved in all spectra and appeared mainly at the extrapolated\nregion. Obviously, the Drude spectral weight decreases sig-\nnificantly as temperature reduces, which are consistent with\nthe significant shift of plasma edge towards low frequency in\nR(!). Furthermore, two phonon peaks are identified at 100,\n225 cm\u00001, respectively ( T=10 K). The position of phonon\nmodes at T=10 K are marked by vertical black lines. The\nphonon modes at di \u000berent temperatures can be well depicted\nby Lorentzians instead of asymmetric (Fano-like) models.\nUsually, an asymmetric Fano-like phonon lineshape indicates\nstrong coupling between the phonon mode and electronic con-\ntinuum. Here, the symmetric Lorentzian lineshape suggests\nthat the phonon in Ir 2In8Se is not strongly coupled to other\nexcitations. The phonon modes display a usual broadening as\ntemperature increases. Additionally, the central frequency of\nhigh-frequency mode softens explicitly with warming, which\nusually originates from the thermal expansion. However, the\npeak position of sharp low-frequency mode varies slightly\nwith temperature compared to high-frequency mode. Ac-\ncording to previous report, the phase transition appeared near\n203 K in Ir 2In8Se is related to the formation of charge den-\nsity wave26. As we known, phase transitions associated with\nCDW order usually result in energy gap formation, leading to\nthe spectral weight suppression below the energy gap. The\nsuppressed spectral weight piles up above the energy gap fre-\nquency. These features have been observed in lots of conven-\ntional CDW systems driven by fermi surface nesting such as\nRTe 3(R=rare-earth elements)39, LaAgSb 240, and CuTe41.\nFor Ir 2In8Se, from measured reflectance spectra, we find a\ncontinuous suppression of reflectance values below 750 cm\u00001\nnear the plasma edge below 200 K. It is very likely that the\nsuppression feature is related to the CDW energy formation.\nThere exists 4 2screw axis which is responsible for protecting\n01 002 003 0001230\n2000400060008000100000246σ1 (103 Ω-1 cm-1)W\nave number (cm-1)(a)(\nb)σ1 (103 Ω-1 cm-1)W\nave number (cm-1) 10 K \n100 K \n150 K \n200 K \n250 K \n300 K01 02 03 0Photon energy (meV)0\n20040060080010001200Photon energy (meV)FIG. 3. The optical conductivity spectra at di \u000berent temperatures\nup to (a) 300 cm\u00001and (b) 10000 cm\u00001. The short chromatic dotted\nline are low frequency extrapolations according to the Hagen-Rubens\nrelation. The vertical black dotted lines display the peak position of\ntwo phonons at 10 K.\nthe Dirac points. When the symmetry of structure is broken,\nthe 4 2screw axis would be perturbed. Therefore, the band\nwould open an energy gap near the Dirac point in the charge\ndensity wave state. However, because of the small carrier\ndensity, the energy scale is very close to the screened plasma\nedge, the e \u000bect in the optical conductivity spectra is too small\nto be di \u000berentiated. The additional reduction of the carrier\ndensity below the CDW order makes it further di \u000ecult to be\ndetected in conductivity spectra. The optical conductivity in\nthe energy scale up to 10000 cm\u00001was presented in Fig. 3(b).\nAlthough there is no clear features about the CDW phase tran-\nsition in optical conductivity, we observed another interesting\nfeature, that is, the optical conductivity contributed from in-\nterband transitions shows the nearly perfect linearity over a\nwide frequency range from 500 to 4000 cm\u00001.\nIn order to obtain more explicit information about the linear\nfeature of interband transitions, the interband transition contri-\nbution from the experimental data was obtained by subtracting\nthe sharp Drude and phonon responses at low frequency. For\na noninteracting systems, the optical conductivity has power-\nlaw frequency dependence with \u001b1(!)/(~!\n2)d\u00002\nz, where dis5\n01 002 003 004.44.64.85.00\n10020030001002003000100020003000400050000120\n100020003000400050000120\n1 0002000300040000.00.51.01.5T\nemperature (K)ωP (103 cm-1)(a)( b)(\nc)( d)0\n50100150 \n1/τD (cm-1)ρ (µΩ cm)T\n (K)σ1 (103 Ω-1 cm-1)W\nave number (cm-1) Exp. data \nDrude \nPhonon1 \nPhonon2 \nSub. D&Ph \nT = 300 Kσ\n1 (103 Ω-1 cm-1)W\nave number (cm-1) Exp. data \nDrude \nPhonon1 \nPhonon2 \nSub. D&Ph \nT = 10 K/s68σ1 (103 Ω-1 cm-1)W\nave number (cm-1) 10 K \n100 K \n150 K \n200 K \n250 K \n300 K0100200300400500600Photon energy (meV)0\n100200300400500600Photon energy (meV)0\n100200300400Photon energy (meV)\nFIG. 4. Optical conductivity of Ir 2In8Se at (a) 300 K and (b) 10 K. The solid lines represnt the experimental data at 10 K (red) and 300 K (blue).\nThe thin dashed lines represent the fit from Drude term (sapphire) and phonon modes (pink) and interband contribution after substracting the\nDrude and phonon terms. The black solid lines refer to the linear conductivity. (c) Temperature-dependent interband contranbutions obtaind\nby substracting the Drude and phonon terms. (d) Temperature dependence of the unscreened plasma (left axis) and scattering rate (right axis).\nInset: comparision of the dc resistivity (blue line) and the conductivity value at zero frequency (black circle) from the Drude fits.\nthe dimension of the system and zrefers to the power-law\nterm of the band dispersion42,43. The above empirical for-\nmula has been confirmed in several experiments18,44–48. For\nexample, a frequency independent conductivity has been ob-\nserved in two-dimensional graphene ( d=2 and z=1) while\nthree-dimensional ZrTe 519and CoSi48(d=3 and z=1) con-\nsistently display a linear rising conductivity up to 1000 cm\u00001.\nFor Ir 2In8Se, a possible 3D Dirac semimetal ( d=3 and z=1),\nthe conductivity from interband contributions ought to satisfy\nlinear frequency dependence. To identify this, we fit the intra-\nband and phonon responses with Drude and Lorentz function\nrespectively. Fig. 4(a) and (b) shows the conductivity spectra\nand fitting curves at two typical temperatures 300 K and 10 K,\nrespectively. For simplicity, we use only one Drude compo-\nnent to fit the low frequency free carrier spectral weight. Ap-\nparently, this is not a strict approach for a multiple band sys-\ntem, but it is already good enough to see the spectral features.\nThe spectral weight of Drude term (the area between sapphireline and horizontal axis) at 300 K decreases obviously when\ntemperature reaches 10 K but the phonon modes strenghen.\nThe frequency regime of linear conductivity reach about 4000\ncm\u00001(496 meV) at 300 K and still keep up to 3000 cm\u00001(372\nmeV) at 10 K which indicates the presence of 3D linear bands\non a large energy scale. The linear extrapolation of the in-\nterband conductivity to zero can be seen as the onset of the\ninterband transitions. According to previous theory42,43,49, for\nDirac /Weyl semimetal, when the node point is located right\nat the Fermi level, the real part of the interband optical con-\nductivity can be written as \u001b1(!)=e2NW\n12h!\nvF, where NWis the\nnumber of Weyl points (For a single Dirac node, NW=2),his\nthe Planck constant, vFis the Fermi velocity. If the node po-\nsition is not at the Fermi level, \u001b1(!)=e2NW\n12h!\nvF\u0012(!\u00002jEFj)\nwhere\u0012(x) is the Heaviside step function. Although the in-\nterband conductivity can still be extrapolated to zero, the in-\nterband transitions will be terminated below !=2jEFjdue6\nkz\nkxkyZ R\nX\nMA\nГ\n53 meV\n(a) (b) (c) (d)\n(e) (f) (g)Г─\n9\nГ─\n8\nГ─\n6LD7─\nLD6─\n─LD6\nFIG. 5. (a) Brillouin zone (BZ) and (b) band structure with spin-orbit coupling (SOC). The red and green bands form the Dirac bands and\nbelong to LD 6andLD 7irreducible representations on \u0000-Zpath, respectively. They are the highest occupied bands (VBs) and the lowest two\nunoccupied bands (CBs). The blue bands are the second-highest VBs. The green box denotes the region where only the Dirac bands contribute\nto the optical conductivity. (c-d) Band structure along the \u0000-Z(c) and \u0000-R(d) path. The green arrows in (d) indicate the transitions excited by\nphotons of ~!=300 meV . (e) The real part of optical conductivity \u001bxx(!) including both the interband and intraband contribution. The red\ndashed line is the extrapolation of the linear region of \u001bxx(!) that intercepts the horizontal axis at 53 meV . (f) The interband part of \u001bxx(!)\ncalculated with all the bands (black lines), bands forming the Dirac cone (red lines) corresponding to the red and green bands in (b), and the\nhighest two doubly-degenerate VBs plus the lowest two doubly-degenerate CBs (blue lines) corresponding to the red, green and blue bands in\n(b). (g) Distribution of \u001b\u000b\f;k(!) in the BZ on the photon energy of ~!=300 meV .\nto Pauli blockade. Here, for Ir 2In8Se, the interband conduc-\ntivity is extrapolated to a finite intercept. This intercept is\nlocated around 270 cm\u00001(33 meV) at 300 K and decreases to\n230 cm\u00001(29 meV) at 10 K. Fig. 4(c) displays the interband\nconductivity spectra at di \u000berent temperatures. As we can see,\nboth intercept and slope of linearly interband conductivity de-\ncreases slightly with cooling. To understand the origin of this\nfinite intercept, theory calculation has been performed and we\nwill discuss in detail later. Considering Ir 2In8Se has two pos-\nsible Dirac nodes and there are four molecular formula per\nunit cell, we can obtain the Fermi velocity vF=2:03\u0002106\ncm/s (2:22\u0002106cm/s) at T=300 K (10 K) in terms of the\nslope of interband optical conductivity according to the above\ntheoretical formula. It is worth mentioning that the Fermi ve-\nlocity of Ir 2In8Se enhances slightly with cooling and lower\nthan the value of Dirac semimetal ZrTe 5(5:17\u0002106cm/s)19.\nFurthermore, the plasma frequency !p=p4\u0019n=m\u0003and\ncarrier scattering rate \r=1=\u001cDare obtained from intergrating\nthe Drude spectral weight by !2\np=8R!\n0\u001b1d!and identifying\nthe peak width of the Drude profile at half maximum respec-\ntively. Fig. 4(d) displays the temperature dependent plasma\nfrequency and scattering rate. The plasma frequency drops\nfrom 4900 cm\u00001(0.61 eV) at 300 K to 4500 cm\u00001(0.56 eV) at\n10 K which indicates that free carriers are reduced assumingthe e\u000bective mass doesn’t change with cooling. Meanwhile,\nas evidenced by the narrowing of the Drude peak, the scat-\ntering rate drops monotonically from 120 cm\u00001(15 meV) at\n300 K to 10 cm\u00001(1 meV) at 10 K. The reduction of free car-\nriers may be related to CDW formation. The inset of shows\nthe dc resistivity \u001a=1=\u001b1(!=0), derived from the fitted zero-\nfrequency value (black circles), which accords well with the\ntransport result (blue curve) indicating our fitting model is re-\nliable.\nIr2In8Se belongs to P42=mnm (No.136) space group. The\ndistribution of Brillouin zone are depicted in Fig. 5(a). To\ngain insight into the electronic properties of Ir 2In8Se, we ob-\ntain the band structure with SOC as shown in Fig. 5(b). Due to\nthe coexistence of the time-reversal and inversion symmetry,\nevery energy band is doubly-degenerate throughout the BZ.\nThere exist two linear band crossings on the \u0000-Zpath shown\nin Fig. 5(c). By analyzing the little group representation of\nthe high symmetry path, the crossings are both formed by two\nbands belonging to the LD 6andLD 7irreducible represen-\ntations respectively50,51. Thus Ir 2In8Se is a Dirac semimetal\nwith Dirac points protected by the fourfold screw symmetry,\nwhich is in agreement with previous report26.\nThe real part of calculated optical conductivity \u001b1(!) is il-\nlustrated in Fig. 5(e), including the interband contribution7\nkxkz\nky\n(a) (b) (c)\n(d) (e)\n(f) (g)\nFIG. 6. Fermi surfaces consisting of hole pockets (a) and electron pockets (b-c). (d-e) The imaginary part of the susceptibility contributed by\nthe hole pocket (d) and electron pockets (e). (f-g) The real part of the susceptibility contributed by the hole pocket (f) and electron pockets (g).\nThe red arrows in (d-g) indicate the location of CDW wave vector q=1\n6a\u0003+1\n6b\u0003. Data in (d-g) has been rescaled.\ncomputed with the Kubo-Greenwood formula and the intra-\nband part simulated with the Drude model.\nRe\u001bDrude (!)=!\n4\u0019Im\u000fDrude (!)=!\n4\u0019!2\np\r\n!3+!\r2(1)\n\u001binter\u000b\f=ie2~\nNk\ncX\nk\u001b\u000b\f;k(!) (2)\u001b\u000b\f;k(!)=X\nn;mfmk\u0000fnk\n\"mk\u0000\"nkh nkjv\u000bj mkiD\n mk\f\f\fv\f\f\f\f nkE\n\"mk\u0000\"nk\u0000(~!+i\u0011)(3)\nwhere\u000b,\fdenote Cartesian directions, \ncis the cell vol-\nume, Nkdenotes the number of k-points sampling the BZ,\n\"mkis the band energy and fmkis the Fermi-Dirac distribution\nfunction. Constrained by symmetry operations in P42=mnm\ngroup,\u001b1(!) has one nonvanishing independent component\n\u001bxx=\u001byywhen the incident light propagates in ˆ cdirection.\nObtaining the plasma frequency !p=0.63 eV from the first\nprinciple calculation which is close to the experimental data8\n(0.56 eV), we parameterized the scattering rate as \r=7 meV\nto fit the measured spectra at 10 K. However, this scattering\nrate (7 meV) is larger than the experimental data (1 meV).\nThe main reason is that the experimental data is analyzied by\none Drude model inevitably introducing some fitting errors.\nIf we fit the data using two Drude model, the value of scat-\ntering rate become about 5 meV which is basically consistent\nwith the therotical value. In addition, the optical conductivity\nshows a linear dependence on the incident frequency ranging\nfrom 100 meV to 500 meV . Apart from this, the extrapolation\nof the linear part has an interception on the horizontal axis.\nThose features are well consistent with the experimental data\nin Fig. 3(b) and Fig. 4.\nGenerally, the interband optical response of the Dirac\nfermions is supposed to follow a power-law dependence on\nthe frequency18,43,\u001b1(!)/!(d\u00002)=z. Besides, the extrap-\nolation of optical conductivity should pass the origin. It is\nregardless of the position of the Dirac points relative to the\nFermi level, the deviation of which only brings a cut o \u000bon\nthe conductivity42,49,52,53. Although the linearity in conductiv-\nity of Ir 2In8Se seems to originate from the linear Dirac cones\nin the screw axis, the extrapolation does not go through the\norigin but intercepts the horizontal axis.\nTo explore the origin of linearity and the interception, we\ncalculate the interband optical conductivity with only selected\nbands involved. As shown in Fig. 5(f), when three doubly-\ndegenerate bands forming the Dirac points are included, the\nconductivity spectrum in the low-energy range below 100\nmeV is dominantly contributed by those bands, which we de-\nnote with a green box in Fig. 5(b). Due to presence of multiple\ninterband transitions, the conductivity spectrum displays weak\npeak structures. The blue band in Fig. 5(b) are involved when\nconsidering a larger energy range. If we take the blue band\ninto consideration, the linear region starting from 100 meV to\n380 meV can be recovered as the blue curve in Fig. 5(f), indi-\ncating that the contribution in this energy interval comes from\nnot only the Dirac cones but also the transition between the\nDirac cones and the next lower bands. In addition, the slope\nof optical conductivity attributed from Dirac band (blue line)\nbelow 100 meV is close to that slope from all bands (black\nline) which means the slope of linear optical conductivity be-\nlow 500 meV could be used to estimate the value of Fermi\nvelocity, which is about 8 :2\u0002106cm/s for the calculated op-\ntical conductivity, which is basically consistent with the value\nobtained from experiment (2 :2\u0002106cm/s). To further ana-\nlyze the contribution to the conductivity in the linear region,\nwe depict the distribution of \u001b\u000b\f;k(!) in the BZ on the photon\nenergy of ~!=300 meV as presented in Fig. 5(g). It illus-\ntrates that the electron transitions concentrate in kx=0 plane,\nespecially along \u0000\u0000Rpath where large joint density of states\n(JDOS) exists as shown in Fig. 5(d). Therefore, a larger por-\ntion of interband conductivity stems from transitions involv-\ning non-Dirac bands and the linear dispersion of those bands\nover a broad energy range leads to the linearity in \u001b1(!). It\nis also the additional contribution that results in the horizontal\ninterception.\nThe Fermi surface is presented in shown in Fig. 6(a-c).\nThe flower-like hole pockets in Fig. 6(a) consist of four petalsonZ\u0000Xpath and a spherical pistil on \u0000\u0000Zpath while the\nelectron pockets in Fig. 6(b-c) form two ellipsoids centered\nat\u0000point, which is similar with the reported Fermi surface\nstructure of Ir 2In8S26. Ir 2In8Se was discovered to undergo\none structural phase transition upon cooling from room tem-\nperature. It has a commensurately modulated structure with q\nvector q=1\n6a\u0003+1\n6b\u0003at 110-203 K26. To understand whether\nthe modulated structure, i.e., the commensurate charge den-\nsity wave phase, is driven by the Fermi surface nesting (FSN),\nwe calculate the imaginary part of generated electronic sus-\nceptibility,\u001f00\n0(q), as a quantitative measure of the FSN and the\nreal part of generated electronic susceptibility, \u001f0\n0(q), which\ndefines the stability of the electronic subsystem54,55.\nlim\n!!0\u001f00\n0(q;!)=!=X\nk\u000e(\"k\u0000\"F)\u000e\u0010\n\"k+q\u0000\"F\u0011\n(4)\nlim\n!!0\u001f0\n0(q)=X\nkf(\"k)\u0000f\u0010\n\"k+q\u0011\n\"k\u0000\"k+q(5)\nFig. 6(d-g) show the contribution to \u001f00\n0(q) and\u001f0\n0(q) from\nhole pockets and electron pockets in the qz=0 plane. We\nfind that\u001f00\n0(q) of hole pockets [see Fig. 6(d)] and \u001f0\n0(q) of\nelectron pockets [see Fig. 6(g)] both peak at the CDW wave\nvector q=1\n6a\u0003+1\n6b\u0003, indicating that FSN possibly accounts\nfor the formation of CDW. Furthermore, if we translate the\nflower-like hole pockects in Fig. 6(a) by j1\n6a\u0003jorj1\n6b\u0003jalong\n[110] direction, these translated petals partially overlaps with\noriginal hole pockets. However, the translated electron pock-\nets with the same operating condition as hole pocket don’t\noverlap with its original pockets at all as shown in Fig. 6(b-c).\nIt seems that only partial hole pockets satisfy the 6 \u00026 Fermi\nnesting structure. This imperfect FSN causes that CDW gap\ndoesn’t form completely below CDW phase transition which\nare consistent with the experiment.\nIV . SUMMARY\nIn summary, we have performed temperature-dependent\noptical spectroscopy measurements and electronic structure\ncalculation on single-crystal Ir 2In8Se. We observed nearly\nlinear-in-frequency optical conductivity over a extreme broad\nrange (500\u00184000 cm\u00001, 60\u0018500 meV) and varies slightly\nwith cooling. By comparing our experimetal and calculated\noptical conductivity, we find the observed linear \u001b1(!) mainly\noriginates from the Dirac cones and the transition between the\nDirac cones and the next lower bands. By linearly fitting the\ninterband optical conductivity with the empirical formula, we\nobtain the Fermi velocity: vF\u0018106cm/s. In addition, both our\nexperimental and therotical results demonstrate there aren’t\nobvious CDW gap opening below the CDW phase transition\ndue to the imperfect Fermi surface nesting.\nACKNOWLEDGMENTS\nThis work was supported by the National Natural Sci-\nence Foundation of China (No. 11888101), the National9\nKey Research and Development Program of China (No. 2017YFA0302904).\n\u0003These authors contributed equally to this work.\nyhmweng@iphy.ac.cn\nznlwang@pku.edu.cn\n1J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan,\nM. Hirschberger, W. D. Wang, R. J. Cava, and N. P. Ong, Sci-\nence 350, 413 (2015).\n2H. Li, H. T. He, H. Z. Lu, H. C. Zhang, H. C. Liu, R. Ma, Z. Y .\nFan, S. Q. Shen, and J. N. Wang, Nat. Commun. 7, 10301 (2016).\n3T. Liang, Q. Gibson, M. N. Ali, M. H. Liu, R. J. Cava, and N. P.\nOng, Nat. Mater. 14, 280 (2015).\n4L. P. He, X. C. Hong, J. K. Dong, J. Pan, Z. Zhang, J. Zhang, and\nS. Y . Li, Phys. Rev. Lett. 113, 246402 (2014).\n5J. Z. Cao, S. H. Liang, C. Zhang, Y . W. Liu, J. W. Huang, Z. Jin,\nZ. J. Chen, Z. G. Wang, Q. S. Wang, J. Zhao, S. Y . Li, X. Dai,\nJ. Zou, Z. C. Xia, L. Li, and F. X. Xiu, Nat. Commun. 6, 7779\n(2015).\n6Z. J. Xiang, D. Zhao, Z. Jin, C. Shang, L. K. Ma, G. J. Ye, B. Lei,\nT. Wu, Z. C. Xia, and X. H. Chen, Phys. Rev. Lett. 115, 226401\n(2015).\n7Y . Zhao, H. Liu, C. Zhang, H. Wang, J. Wang, Z. Lin, Y . Xing,\nH. Lu, J. Liu, Y . Wang, S. M. Brombosz, Z. Xiao, S. Jia, X. C.\nXie, and J. Wang, Phys. Rev. X 5, 031037 (2015).\n8K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D\n78, 074033 (2008).\n9A. A. Burkov, Phys. Rev. Lett. 113, 247203 (2014).\n10P. Hosur and X.-L. Qi, Phys. Rev. B 91, 081106 (2015).\n11Z. Wang, Y . Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng,\nX. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012).\n12Z. K. Liu, B. Zhou, Y . Zhang, Z. J. Wang, H. M. Weng, D. Prab-\nhakaran, S. K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and\nY . L. Chen, Science 343, 864 (2014).\n13S. Y . Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Be-\nlopolski, M. Neupane, G. Bian, N. Alidoust, T. R. Chang, H. T.\nJeng, C. Y . Huang, W. F. Tsai, H. Lin, P. P. Shibayev, F. C. Chou,\nR. J. Cava, and M. Z. Hasan, Science 347, 294 (2015).\n14S. Borisenko, Q. Gibson, D. Evtushinsky, V . Zabolotnyy,\nB. B ¨uchner, and R. J. Cava, Phys. Rev. Lett. 113, 027603 (2014).\n15Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y . Zhang, H. M. Weng,\nD. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch,\nZ. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain, and Y . L.\nChen, Nat. Mater. 13, 677 (2014).\n16M. Neupane, S. Y . Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu,\nI. Belopolski, T. R. Chang, H. T. Jeng, H. Lin, A. Bansil, F. Chou,\nand M. Z. Hasan, Nat. Commun. 5, 3786 (2014).\n17G. S. Jenkins, C. Lane, B. Barbiellini, A. B. Sushkov, R. L. Carey,\nF. Liu, J. W. Krizan, S. K. Kushwaha, Q. Gibson, T.-R. Chang,\nH.-T. Jeng, H. Lin, R. J. Cava, A. Bansil, and H. D. Drew, Phys.\nRev. B 94, 085121 (2016).\n18D. Neubauer, J. P. Carbotte, A. A. Nateprov, A. L ¨ohle, M. Dressel,\nand A. V . Pronin, Phys. Rev. B 93, 121202 (2016).\n19R. Y . Chen, S. J. Zhang, J. A. Schneeloch, C. Zhang, Q. Li, G. D.\nGu, and N. L. Wang, Phys. Rev. B 92, 075107 (2015).\n20M. Gam ˙za, P. Abrami, L. V . D. Gammond, J. Ayres, I. Osmond,\nT. Muramatsu, R. Armstrong, H. Perryman, D. Daisenberger,\nS. Das, and S. Friedemann, Phys. Rev. Materials 5, 024209\n(2021).\n21G. Zhai, C. Ma, J. Xiang, J. Ye, T. Li, Y . Li, P. Sun, G. Chen,\nX. Wu, and X. Zhang, Phys. Rev. B 101, 174310 (2020).22H. T. Chorsi, S. Y . Yue, M. Iyer, P. P. Goyal, T. Schumann,\nS. Stemmer, B. L. Liao, and J. A. Schuller, Adv. Opt. Mater. 8,\n1901192 (2020).\n23D. Shao, J. Ruan, J. Wu, T. Chen, Z. Guo, H. Zhang, J. Sun,\nL. Sheng, and D. Xing, Phys. Rev. B 96, 075112 (2017).\n24Y . Qi, W. Shi, P. G. Naumov, N. Kumar, W. Schnelle, O. Barkalov,\nC. Shekhar, H. Borrmann, C. Felser, B. Yan, and S. A. Medvedev,\nPhys. Rev. B 94, 054517 (2016).\n25C. Vaswani, L.-L. Wang, D. H. Mudiyanselage, Q. Li, P. M.\nLozano, G. D. Gu, D. Cheng, B. Song, L. Luo, R. H. J. Kim,\nC. Huang, Z. Liu, M. Mootz, I. E. Perakis, Y . Yao, K. M. Ho, and\nJ. Wang, Phys. Rev. X 10, 021013 (2020).\n26J. F. Khoury, A. J. E. Rettie, M. A. Khan, N. J. Ghimire, , I. Ro-\nbredo, J. E. Pfluger, K. Pal, C. Wolverton, A. Bergara, J. S. Jiang,\nL. M. Schoop, M. G. Vergniory, J. F. Mitchell, D. Y . Chung, , and\nM. G. Kanatzidis, J. Am. Chem. Soc. 141, 19130 (2019).\n27J. F. Khoury, A. J. E. Rettie, I. Robredo, M. J. Krogstad,\nC. D. Malliakas, A. Bergara, M. G. Vergniory, R. Osborn,\nS. Rosenkranz, D. Y . Chung, and M. G. Kanatzidis, J. Am. Chem.\nSoc.142, 6312 (2020).\n28T. Sakamoto, M. Wakeshima, Y . Hinatsu, and K. Matsuhira, Phys.\nRev. B 75, 060503 (2007).\n29C. Pauly, B. Rasche, K. Koepernik, M. Richter, S. Borisenko,\nM. Liebmann, M. Ruck, J. van den Brink, and M. Morgenstern,\nACS Nano 10, 3995 (2016).\n30Q. Wang, Y . F. Xu, R. Lou, Z. H. Liu, M. Li, Y . B. Huang, D. W.\nShen, H. M. Weng, S. C. Wang, and H. C. Lei, Nat. Commun. 9,\n3681 (2018).\n31G. Kresse and J. Furthm ¨uller, Phys. Rev. B 54, 11169 (1996).\n32J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,\n3865 (1996).\n33N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).\n34I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109\n(2001).\n35N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vander-\nbilt, Rev. Mod. Phys. 84, 1419 (2012).\n36A. A. Mostofi, J. R. Yates, G. Pizzi, Y .-S. Lee, I. Souza, D. Van-\nderbilt, and N. Marzari, Computer Physics Communications 185,\n2309 (2014).\n37Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov,\nComputer Physics Communications 224, 405 (2018).\n38D. B. Tanner, Phys. Rev. B 91, 035123 (2015).\n39B. F. Hu, B. Cheng, R. H. Yuan, T. Dong, and N. L. Wang, Phys.\nRev. B 90, 085105 (2014).\n40R. Y . Chen, S. J. Zhang, M. Y . Zhang, T. Dong, and N. L. Wang,\nPhys. Rev. Lett. 118, 107402 (2017).\n41R. S. Li, L. Yue, Q. Wu, S. X. Xu, Q. M. Liu, Z. X. Wang, T. C.\nHu, X. Y . Zhou, L. Y . Shi, S. J. Zhang, D. Wu, T. Dong, and N. L.\nWang, Phys. Rev. B 105, 115102 (2022).\n42P. Hosur, S. A. Parameswaran, and A. Vishwanath, Phys. Rev.\nLett. 108, 046602 (2012).\n43A. B ´acsi and A. Virosztek, Phys. Rev. B 87, 125425 (2013).\n44B. Xu, Y . M. Dai, L. X. Zhao, K. Wang, R. Yang, W. Zhang,\nJ. Y . Liu, H. Xiao, G. F. Chen, A. J. Taylor, D. A. Yarotski, R. P.\nPrasankumar, and X. G. Qiu, Phys. Rev. B 93, 121110 (2016).\n45S.-i. Kimura, H. Yokoyama, H. Watanabe, J. Sichelschmidt,\nV . S ¨uß, M. Schmidt, and C. Felser, Phys. Rev. B 96, 075119\n(2017).10\n46I. Crassee, E. Martino, C. C. Homes, O. Caha, J. Nov ´ak,\nP. T¨uckmantel, M. Hakl, A. Nateprov, E. Arushanov, Q. D. Gib-\nson, R. J. Cava, S. M. Koohpayeh, K. E. Arpino, T. M. McQueen,\nM. Orlita, and A. Akrap, Phys. Rev. B 97, 125204 (2018).\n47F. H ¨utt, A. Yaresko, M. B. Schilling, C. Shekhar, C. Felser,\nM. Dressel, and A. V . Pronin, Phys. Rev. Lett. 121, 176601\n(2018).\n48B. Xu, Z. Y . Fang, M. A. Sanchez-Martinez, J. W. F. Venderbos,\nZ. L. Ni, T. Qiu, K. Manna, K. F. Wang, J. Paglione, C. Bernhard,\nC. Felser, E. J. Mele, A. G. Grushin, A. M. Rappe, and L. Wu,\nProc. Natl. Acad. Sci. U. S. A. 117, 27104 (2020).\n49P. E. C. Ashby and J. P. Carbotte, Phys. Rev. B 89, 245121 (2014).\n50J. Gao, Q. Wu, C. Persson, and Z. Wang, Computer Physics Com-munications 261, 107760 (2021).\n51L. Elcoro, B. Bradlyn, Z. Wang, M. G. Vergniory, J. Cano,\nC. Felser, B. A. Bernevig, D. Orobengoa, G. de la Flor, and M. I.\nAroyo, Journal of Applied Crystallography 50, 1457 (2017).\n52T. Timusk, J. P. Carbotte, C. C. Homes, D. N. Basov, and S. G.\nSharapov, Phys. Rev. B 87, 235121 (2013).\n53C. J. Tabert, J. P. Carbotte, and E. J. Nicol, Phys. Rev. B 93,\n085426 (2016).\n54M. D. Johannes and I. I. Mazin, Phys. Rev. B 77, 165135 (2008).\n55Y . Liu, D. F. Shao, L. J. Li, W. J. Lu, X. D. Zhu, P. Tong, R. C.\nXiao, L. S. Ling, C. Y . Xi, L. Pi, H. F. Tian, H. X. Yang, J. Q. Li,\nW. H. Song, X. B. Zhu, and Y . P. Sun, Phys. Rev. B 94, 045131\n(2016)."    },    {        "title": "0904.2164v2.Chiral_asymmetry_of_the_Fermi_surface_in_dense_relativistic_matter_in_a_magnetic_field.pdf",        "content": "arXiv:0904.2164v2  [hep-ph]  15 Sep 2009Chiral asymmetry of the Fermi surface in dense relativistic matter in a magnetic field\nE. V. Gorbar,1,∗V. A. Miransky,2,†and I. A. Shovkovy3,‡\n1Bogolyubov Institute for Theoretical Physics, 03680, Kiev , Ukraine\n2Department of Applied Mathematics, University of Western O ntario, London, Ontario N6A 5B7, Canada\n3Department of Applied Sciences and Mathematics,\nArizona State University, Mesa, Arizona 85212, USA\n(Dated: September 18, 2018)\nIt is revealed that in the normal phase of dense relativistic matter in a magnetic field, there exists\na contribution to the axial current associated with a relati ve shift of the longitudinal momenta in the\ndispersion relations of opposite chirality fermions. Unli ke the topological contribution in the axial\ncurrent at the lowest Landau level, recently discussed in th e literature, the dynamical one appears\nonly in interacting matter and affects the fermions in allLandau levels, including those around the\nFermi surface. The induced axial current and the shift of the Fermi surfaces of the left-handed and\nright-handed fermions are expected to play an important rol e in transport and emission properties\nof matter in various types of compact stars as well as in heavy ion collisions.\nPACS numbers: 12.39.Ki, 12.38.Mh, 21.65.Qr\nIntroduction.— At zero density and temperature, the\nstructure of the ground state in relativistic chiral in-\nvariant theories in a magnetic field is dictated by the\nmagnetic catalysis phenomenon: the magnetic field is a\nstrong catalyst of spontaneous chiral symmetry breaking\n[1, 2]. The situation becomes much more complicated in\nthe case of dense relativistic matter in a magnetic field\n[3]. The main goal of this Rapid Communication is to\nreveal and describe some universal properties of such dy-\nnamics.\nThe recent studies [4] of similar dynamics in graphene\nin 2 + 1 dimensions have revealed several types of or-\nder parameters whose analogs have not been discussed\nin the context of relativistic models in 3+1 dimensions.\n(For earlier applications of the magnetic catalysis phe-\nnomenon in graphene, see Ref. [5].) This motivated us\nto reexamine the properties of dense relativistic matter\nin an external magnetic field in 3+1 dimensions. As\nwe show, an external magnetic field induces two quali-\ntatively different contributions to the net axial current\nin the normal phase of dense relativistic matter. The\nfirst one is a topological contribution due to the lowest\nLandau level (LLL) that was previously discussed in the\nliterature [6, 7, 8, 9]. It exists even in free theories. (For\nrelated studies in hot quark-gluon plasma, see Ref. [10].)\nIn this Rapid Communication we find that there is also\nan additional dynamical contribution to the axialcurrent\nthat appears only in interacting matter. The crucial new\npoint is that allLandau levels, including those around\nthe Fermi surface, contribute to this interaction-driven\ncontribution. Its origin is related to a relative shift of\nthe longitudinal momenta in the dispersion relations of\n∗Electronic address: gorbar@bitp.kiev.ua\n†Electronic address: vmiransk@uwo.ca; On leave from Bogoly ubov\nInstitute for Theoretical Physics, 03680, Kiev, Ukraine.\n‡Electronic address: igor.shovkovy@asu.eduopposite chirality fermions. The amount of the shift is\nproportional to a coupling constant, the magnetic field,\nand the fermion chemical potential. Notably, it is almost\nindependent of temperature. As we discuss below, this\neffect may have profound implications for the physics of\ncompact stars as well as heavy ion collisions.\nModel.— In this paper, in order to illustrate the effect\nin the clearest way, a simple model will be utilized. We\nperform a study of a Nambu-Jona-Lasinio model with\nthe Lagrangian density\nL=¯ψ/parenleftbig\niDν+µ0δ0\nν/parenrightbig\nγνψ+Gint\n2/bracketleftBig/parenleftbig¯ψψ/parenrightbig2+/parenleftbig¯ψiγ5ψ/parenrightbig2/bracketrightBig\n,\n(1)\nwhereγ5≡iγ0γ1γ2γ3andµ0is the chemical potential.\nThismodelpossessesthe chiral U(1)L×U(1)Rsymmetry.\nThe covariant derivative Dν=∂ν−ieAνis taken in the\nLandau gauge, i.e., Aν=xBδ2\nνwhereBis the strength\nof the external magnetic field pointing in the z-direction.\nThe structure of the full fermion propagator is given\nby a (3+1)-dimensional generalization of the ansatz used\nfor the description of the quantum Hall effect dynamics\nin graphene [4], namely\niG−1(u,u′) =/bracketleftBig\n(i∂t+µ)γ0−(π·γ)−π3γ3\n+i˜µγ1γ2+∆γ3γ5−m/bracketrightBig\nδ4(u−u′),(2)\nwhereπk=i(∂k−ieAk) is the canonical momentum\nandu= (t,r). The above propagator contains several\ndynamical parameters that are absent in the tree level\npropagator,\niS−1(u,u′) =/bracketleftbig\n(i∂t+µ0)γ0−(π·γ)−π3γ3/bracketrightbig\nδ4(u−u′).\n(3)\nThe physical meaning of the parameters mandµis\nstraightforward: mis the Dirac mass and µis the full\nchemical potential in interacting theory. From the struc-\nture of the inverse propagator in Eq. (2), it is clear that\n˜µplays the role of the anomalous magnetic moment [in2\ngraphene, it also can be interpreted as a chemical poten-\ntial related to a conserved (pseudospin) current [4]].\nThe meaning of the last parameter, ∆, is more subtle.\nIn the context of graphene, ∆ is a mass parameter that is\noddbothundertime-reversalandparitytransformations.\nIn 2+1 dimensional models, this mass is responsible for\ninducing the Chern-Simons term in the effective action\nfor gauge fields [11]. In 3+1 dimensions, as suggested by\nEq. (2), it is related to an induced axial current along\nthe direction of the magnetic field, ¯ψγ3γ5ψ. As will be\nshown below, ∆ determines a shift of the longitudinal\nmomenta in the dispersion relations of opposite chirality\nfermions in the chiral limit. We will call it a chiral shift\nparameter. Let us emphasize that, in the presence of\nan external magnetic field, the time-reversal and parity\nsymmetries are broken and a state with the vanishing ∆\nis not protected by any symmetry.\nThe parameters m,µ, ∆ and ˜µare self-consistently\ndetermined from the gap equation, which takes the fol-\nlowing form in the mean field approximation:\nG−1(u,u′) =S−1(u,u′)−iGint/braceleftbig\nG(u,u)−γ5G(u,u)γ5\n−tr[G(u,u)]+γ5tr[γ5G(u,u)]/bracerightbig\nδ4(u−u′).\n(4)\nWhile the parameters ∆ and µdo not break the chiral\nU(1)L×U(1)Rsymmetry, nonzero values of mand ˜µ\nbreak it down to U(1)L+R. In the mean field approxi-\nmation used here, we find that ˜ µ= 0 in a self-consistent\nsolution to the gap equation. While this fact simplifies\nthe analysis, we note that ˜ µmay be non-vanishing in\nmore refined approximations and in models with other\ntypes of interactions [4, 12]. On the other hand, as sug-\ngested by the analysis in graphene, a nonzero ˜ µshould\nnot change the main qualitative features of the phase\nwith an induced ∆ [4].\nThe spectrum offermionicquasiparticlesisdetermined\nby the poles in the full propagator (2) with ˜ µ= 0. As in\nRefs. [2, 4], expanding the propagator over the Landau\nlevels, we arrive at the following dispersion relation:\nωn,σ=−µ±/radicalBigg/bracketleftbigg/radicalBig\nm2+k2\n3+σ∆sign(eB)/bracketrightbigg2\n+2n|eB|,\n(5)\nwherenlabels the Landau levels, σ=±1, andk3is the\nmomentum along the direction of the magnetic field. In\nthe chiral limit with m= 0, the states with the quan-\ntum number σ=±1 have a shifted longitudinal momen-\ntum,k3→k3±∆sign(eB). As will be shown below,\nin this limit, the two different values of σcorrespond to\nfermions with opposite chiralities. As a result, the Fermi\nsurfaces for left-handed and right-handed fermions be-\ncome shifted.\nResults.— In accordance with the magnetic catalysis\nscenario[1, 2], the ground state in the NJL model at van-\nishingµ0is characterized by a nonzero Dirac mass m0\nthat breaks the chiral U(1)L×U(1)Rsymmetry. Such\na vacuum state can withstand a finite stress due to anonzero chemical potential. However, when µ0exceeds\na certain critical value µc, the chiral symmetry restora-\ntion and a new ground state are expected. As we show\nhere, the new state is characterized by a non-vanishing\nchiral shift parameter ∆ and a nonzero axial current in\nthe direction of the magnetic field. Since no symmetry\nof the theory is broken, this state describes the normal\nphase of matter that happens to have a rather rich chiral\nstructure.\nThe value of the Dirac mass m0in the vacuum state\nwas calculated in Ref. [2]. In the weakly coupled regime,\ng≡GintΛ2/(4π2)≪1, the solution reads\nm2\n0=1\nπl2exp/parenleftbigg\n−Λ2l2\ng/parenrightbigg\n, (6)\nwherel= 1//radicalbig\n|eB|is the magnetic length and Λ is the\nultraviolet cutoff in the model at hand. This zero tem-\nperature solution exists for µ0< m0. In this solution,\nthe full chemical potential µ=µ0.\nOur analysis shows that, in addition to the solution\nwith a nonzero Dirac mass m, the gap equation also al-\nlows a solution with m= 0 and a nonzero chiral shift\nparameter ∆,\n∆ =gµeB\nΛ2[1+2ag]+g|eB|, (7)\nwhereais a dimensionless constant of order 1, which is\ndeterminedbytheregularizationschemeusedintheanal-\nysis (see below). Interestingly, the temperature depen-\ndence of ∆ comes only through the chemical potential,\nwhich has a weak temperature dependence when T≪µ\n[13]. AtT= 0, the chemical potential satisfies the fol-\nlowing equation:\nµ=µ0−g\n(Λl)2[µ−∆sign(eB)]\n−2gsign(µ)\n(Λl)2∞/summationdisplay\nn=1/radicalbig\nµ2−2n|eB|θ/parenleftbig\nµ2−2n|eB|/parenrightbig\n.(8)\nThe approximate solution to this equation is µ≃µ0up\nto power corrections in small g.\nThe free energies of the two states are given by [13]\nΩm≃ −m2\n0\n2(2πl)2/parenleftbig\n1+(m0l)2ln|Λl|/parenrightbig\n(9)\nand\nΩ∆≃ −µ2\n0\n(2πl)2/parenleftbigg\n1−g|eB|\nΛ2/parenrightbigg\n, (10)\nrespectively. In deriving the last expression, we used\nthe approximate relations µ≃µ0and ∆≃gµ0eB/Λ2.\nBy comparing the free energies in Eqs. (9) and (10), we\nsee that the ground state with a nonzero ∆ becomes fa-\nvorable when µ0>∼m0/√\n2. This is analogous to the\nClogston relation in superconductivity [14].3\nThe properties of the two types of quasiparticles corre-\nsponding to σ=±1 in the dispersion relation in Eq. (5)\nwithm= 0 are further clarified by the structure of the\nfull propagator:\nG(u,u) =G−\n0P−+∞/summationdisplay\nn=1/parenleftbig\nG−\nnP−+G+\nnP+/parenrightbig\n,(11)\nwhereP±≡1\n2/bracketleftbig\n1±iγ1γ2sign(eB)/bracketrightbig\nare the spin projec-\ntors. (Foru′/negationslash=u, the propagator will be presented else-\nwhere [13].) The functions G±\nnwithn≥0 are given by\nG±\nn=i|eB|γ0\n2π/integraldisplaydωdk3\n(2π)2\n×/bracketleftBigω+µ±[k3−∆sign(eB)]\n(ω+µ)2−2n|eB|−[k3−∆sign(eB)]2P−\n5\n+ω+µ∓[k3+∆sign(eB)]\n(ω+µ)2−2n|eB|−[k3+∆sign(eB)]2P+\n5/bracketrightBig\n,\n(12)\nwhereP±\n5≡1\n2/bracketleftbig\n1±γ5sign(eB)/bracketrightbig\nare the chirality projec-\ntors for a fixed sign of eB. As follows from this equa-\ntion, the quasiparticles of opposite chiralities have dis-\npersion relations that differ from those in the free the-\nory by the shift of their longitudinal momentum k3→\nk3±∆sign(eB). This has profound implications for the\nphysical properties of matter.\nThegroundstatewith∆ /negationslash= 0ischaracterizedbyanon-\nvanishing expectation value of the axial current density,\n/angbracketleftj3\n5(u)/angbracketright=−tr/bracketleftbig\nγ3γ5G(u,u)/bracketrightbig\n=eB\n2π2µ−|eB|\n2π2∆\n−|eB|\nπ2∆∞/summationdisplay\nn=1κ(/radicalbig\n2n|eB|,Λ), (13)\nwhereκ(x,Λ) is a smooth cutoff function defined by the\nvalue of the cut-off energy Λ and a certain width of the\nregion in which the value of the function drops from 1\ndown to 0 (i.e., κ(x,Λ)≃1 forx≪Λ andκ(x,Λ)≃\n0 forx≫Λ). Taking this into account, we find that/summationtext∞\nn=1κ(/radicalbig\n2n|eB|,Λ) =aΛ2/|eB|anda=O(1).\nWhile the first two terms in the current density (13)\ncome from the LLL, the last term is due to the higher\nLandau levels, n≥1. Notably, all higher Landau levels\nbelow the (smeared) cut-offenergygive (nearly) identical\ncontributions to the induced axial current. The contri-\nbution proportional to µis topological in nature and ap-\npears even in the free theory [7]. All other terms, which\nare proportional to ∆, are the result of interactions and\nhave not been revealed in the literature before.\nWe expect that the interaction-driven contributions\ncan strongly modify transport and emission properties\nof dense relativistic matter in a magnetic field. Indeed,\nthe corresponding induced axial currents are the conse-\nquence of a relative longitudinal flow of opposite chiral-\nityquasiparticles, including thoseinhigherLandaulevels\naround the Fermi energy. This is in contrast to the roleof the topological contribution that is exclusively due to\nthe LLL, which is to a large degree quenched from the\nlow-energy dynamics by the Pauli exclusion principle in\nmany realistic cases.\nDiscussion and Summary.— In this Rapid Communi-\ncation we found that, in accordance with the magnetic\ncatalysis scenario, the vacuum state of relativistic matter\nin a magnetic field is characterized by a non-zero Dirac\nmass given by Eq. (6) at weak coupling. However, when\nthe chemical potential exceeds a certain critical value,\nµc≃m0/√\n2, such a state is replaced by the normal\nground state that is characterized by the following two\nproperties: (i) the presence of an induced axial current\nalong the magnetic field and (ii) the presence of the dy-\nnamically generated chiral shift parameter ∆, which is a\n3+1 dimensional analog of the parity odd mass term in\n2+1 dimensions leading to the Chern-Simons term. We\nfind that, in addition to the previouslyknown topological\nterm in the induced current, there are also interaction-\ndriven contributions from the lowest as well as from the\nhigher Landau levels. In fact, the newly found contri-\nbutions are directly related to a dynamically generated\nvalue of the chiral shift parameter ∆ ≃gµ0eB/Λ2. This\nparameter quantifies the relative shift of the longitudinal\nmomenta in the dispersion relations of opposite chirality\nquasiparticles.\nIt might be appropriate to mention that, for instruc-\ntional purposes, our study here is simplified: we used an\nNJL-typelocalinteractionandutilizedthemean-fieldap-\nproximation. These limitations may lead to the results\nthat are not always quantitatively reliable, e.g., in the\ncontext of stellar matter. Nevertheless, it is expected\nthat our results should remain qualitatively the same\neven when more realistic models are used. Indeed, the\nfact that the expression for ∆ in Eq. (7) is linear in g\nin the lowest order indicates that the corresponding dy-\nnamics is essentially perturbative. Apparently this is a\ngeneral feature that should not depend on whether the\ninteractionsare shortrange, asin the NJL model, or long\nrange, as in QCD or QED. In either case, a vanishing ∆\nis not protected by any symmetry.\nAnother limiting assumption of this study is the exact\nchiral symmetry. However, most of the results are not\nmodified much when the symmetry is at least approxi-\nmate, i.e., when the fermions have non-zero bare Dirac\nmasses, but such masses are small compared to the value\nof the chemical potential [13]. For the applications in\nprotoneutron stars suggested below, this approximation\nis justified, but the general study will be of interest, e.g.,\nin relation to the electron plasma in white dwarfs.\nIn the future, it will be also of interest to address a\npossible interference of the dynamics responsible for the\ngeneration of the chiral shift parameter with color super-\nconductivity in quark matter [15]. Here we just mention\nthat the normal ground state with a nonzero ∆ seems to\nbe the only possibility at higher temperatures, which are\nof main interest for us here.\nWe expect that the generation of the chiral shift pa-4\nrameter may affect physical properties of the quark mat-\nter in quark and/or hybrid stars, the electron gas in neu-\ntron stars, and possibly even the electron gas in white\ndwarfs. The corresponding fermionic systems are degen-\nerate (T≪µ) and the results of this Rapid Communica-\ntionapplydirectly. Itisappropriatetomention,however,\nthat the chiral shift parameter does not vanish even in\nthe non-degenerate limit ( T≫µ), although the analysis\nof the dynamics becomes more involved [13]. Therefore,\nthe generation of a non-zero ∆ can also affect the chiral\nmagnetic effect in heavy ion collisions [10].\nOne of the consequences of the phenomenon discussed\nin this Rapid Communication is the possibility of a qual-\nitatively new mechanism for the pulsar kicks [16]. In the\npresence of a magnetic field, almost any type of relativis-\ntic matter inside a protoneutron star should develop ax-\nial currents as in Eq. (13). The main carriers of such\ncurrents are the electrons in the nuclear matter, and\nthe quarks together with the electrons in the deconfined\nquark matter. Since the induced currents and the chiral\nshift parameter have only a weak temperature depen-\ndence (assuming T≪µ), this phenomenon may provide\na robust anisotropic medium even at the earliest stages\nof the protoneutron star. This is important because\nmoderately hot matter with 10 MeV <∼T<∼50 MeV,\npresent during the first few tens of seconds of the pro-\ntoneutron star evolution [17], may have a large enough\namount of the thermal energy to power the strongest\n(withv>∼1000 km/s) pulsar kicks observed [16]. In con-\ntrast, the constraints of the energy conservation makeit hard, if not impossible, to explain such kicks if the\ninterior matter is cold ( T<∼1 MeV). The common diffi-\nculty of using a hot matter, however, is the very efficient\nthermal isotropization that washes out a non-isotropic\ndistribution of neutrinos produced by almost any mech-\nanism [18, 19]. In the mechanism proposed in this Rapid\nCommunication, however,the asymmetricdistributionof\nthe neutrinos develops as a result of their multiple elastic\nscattering on the left-handed electrons or quarks, flowing\nin the whole bulk of the stellar matter in one direction\nalong the magnetic field.\nIn passing, let us mention that the robustness of the\naxialcurrents in hot magnetized matter may alsoprovide\nan additional neutrino push to facilitate successful super-\nnova explosions as suggested in Ref. [20]. Further details\nof this mechanism will be discussed elsewhere [13].\nAcknowledgments.— The authors would like to thank\nValery Gusynin for fruitful discussions. They also thank\nJ. Noronha, A. Schmitt and D. T. Son for useful com-\nments on the manuscript. V.A.M. acknowledges fruitful\ndiscussionswith DimaKharzeevandEricZhitnitskycon-\ncerning the results in Refs. [7, 10]. The work of E.V.G.\nwas supported by the State Foundation for Fundamen-\ntal Research under the grant F/16-457-2007, and by the\nProgram of Fundamental Research of the Physics and\nAstronomy Division of the National Academy of Sciences\nof Ukraine. The work of V.A.M. was supported by the\nNatural Sciences and Engineering Research Council of\nCanada. The work of I.A.S. was supported in part by\nthe start-up funds from the Arizona State University.\n[1] V.P.Gusynin,V.A.Miransky, andI.A.Shovkovy,Phys.\nRev.Lett. 73, 3499(1994); Phys.Rev.D 52, 4718(1995).\n[2] V.P.Gusynin,V.A.Miransky, andI.A.Shovkovy,Phys.\nLett. B349, 477 (1995); Nucl. Phys. B 462, 249 (1996).\n[3] D. Ebert, K. G. Klimenko, M. A. Vdovichenko, and\nA. S. Vshivtsev, Phys. Rev. D 61, 025005 (1999);\nE. J. Ferrer, V. de la Incera, and C. Manuel, Phys. Rev.\nLett.95, 152002 (2005); E. J. Ferrer and V. de la In-\ncera, Phys. Rev. D 76, 045011 (2007); K. Fukushima and\nH. J. Warringa, Phys. Rev. Lett. 100, 032007 (2008);\nJ. L. Noronha and I. A. Shovkovy, Phys. Rev. D 76,\n105030 (2007); D. T. Son and M. A. Stephanov, Phys.\nRev. D77, 014021 (2008).\n[4] E. V. Gorbar, V.P. Gusynin, and V. A. Miransky,\nLow Temp. Phys. 34, 790 (2008); E. V. Gorbar,\nV.P.Gusynin, V.A.Miransky, andI.A.Shovkovy,Phys.\nRev. B78, 085437 (2008).\n[5] D. V. Khveshchenko, Phys. Rev. Lett. 87, 206401\n(2001); E. V. Gorbar, V. P. Gusynin, V. A. Miran-\nsky, and I. A. Shovkovy, Phys. Rev. B 66, 045108\n(2002); V. P. Gusynin, V. A. Miransky, S. G. Shara-\npov, and I. A. Shovkovy, Phys. Rev. B 74, 195429\n(2006); I. F. Herbut, Phys. Rev. Lett. 97, 146401 (2006);\nPhys. Rev. B 75, 165411 (2007); ibid.76, 085432 (2007);\nM. Ezawa, J. Phys. Soc. Jpn. 76, 094701 (2007).\n[6] D. T. Son and A. R. Zhitnitsky, Phys. Rev. D 70, 074018\n(2004).[7] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D 72,\n045011 (2005).\n[8] J. Charbonneau and A. Zhitnitsky, arXiv:0903.4450.\n[9] A. Rebhan, A.SchmittandS.A.Stricker, J. HighEnergy\nPhys. 05 (2009) 084.\n[10] D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A 797, 67\n(2007); D. E. Kharzeev, L. D. McLerran, and H. J. War-\nringa, Nucl. Phys. A 803, 227 (2008); K. Fukushima,\nD. E. Kharzeev, and H. J. Warringa, Phys. Rev. D 78,\n074033 (2008).\n[11] A. J. Niemi and G. W. Semenoff, Phys. Rev. Lett. 51,\n2077 (1983); A. N. Redlich, Phys. Rev. Lett. 52, 18\n(1984); Phys. Rev. D 29, 2366 (1984).\n[12] E. J. Ferrer and V. de la Incera, Phys. Rev. Lett. 102,\n050402 (2009).\n[13] E. V. Gorbar, V. A. Miransky, and I. A. Shovkovy, in\npreparation.\n[14] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962).\n[15] M. G. Alford, A. Schmitt, K. Rajagopal and T. Sch¨ afer,\nRev. Mod. Phys. 80, 1455 (2008).\n[16] A. G. Lyne and D. R. Lorimer, Nature 369(1994) 127;\nJ. M. Cordes and D. F. Chernoff, Astrophys. J. 505, 315\n(1998); B. M. S. Hansen and E. S. Phinney, Mon. Not.\nRoy. Astron. Soc. 291, 569 (1997); C. Fryer, A. Burrows,\nand W. Benz, Astrophys. J. 496, 333 (1998); Z. Arzou-\nmanian, D. F. Chernoffs, and J. M. Cordes, Astrophys.\nJ.568, 289 (2002); S. Chatterjee et al., Astrophys. J.5\n630, L61 (2005).\n[17] D. Page and S. Reddy, Ann. Rev. Nucl. Part. Sci. 56,\n327 (2006).\n[18] A. Kusenko, G. Segre, and A. Vilenkin, Phys. Lett. B\n437, 359 (1998).[19] I. Sagert and J. Schaffner-Bielich, J. Phys. G 35, 014062\n(2008); arXiv:astro-ph/0612776; arXiv:0708.2352.\n[20] C. L. Fryer and A. Kusenko, Astrophys. J. Suppl. 163,\n335 (2006)."    },    {        "title": "1509.07865v1.Composite_fermions_and_the_field_tuned_superconductor_insulator_transition.pdf",        "content": "Composite fermions and the \feld-tuned superconductor-insulator transition\nMichael Mulligan1and S. Raghu1;2\n1Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA and\n2SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA\n(Dated: October 31, 2021)\nIn several two-dimensional \flms that exhibit a magnetic \feld-tuned superconductor to insulator\ntransition (SIT), stable metallic phases have been observed. Building on the `dirty boson' description\nof the SIT, we suggest that the metallic region is analogous to the composite Fermi liquid observed\nabout half-\flled Landau levels of the two-dimensional electron gas. The composite fermions here\nare mobile vortices attached to one \rux quantum of an emergent gauge \feld. The composite vortex\nliquid is a 2D non-Fermi liquid metal, which we argue is stable to weak quenched disorder. We\ndescribe several experimental consequences of the emergent composite vortex liquid.\nI. INTRODUCTION\nHow is superconductivity destroyed by a perpendicu-\nlar magnetic \feld at T= 0 in a disordered thin \flm1{13?\nThere are two approaches to addressing this question.\nThe \frst is a fermionic description where the destruction\nof superconductivity is identi\fed as the loss of super-\nconducting pairing amplitude14. The second, so called\n`dirty boson' approach, assumes the magnetic \feld in-\nduces a superconductor to insulator transition (SIT),\nwhere Cooper-pair localization occurs, while the ampli-\ntude remains \fnite across the transition15,16.\nIn the presence of both disorder and a strong mag-\nnetic \feld (H >Hc1), superconductors and insulators are\nsharply de\fned only at T= 0 in 2D17. Thus, the \feld-\ntuned SIT is a rather unconventional quantum critical\npoint. In contrast, the zero-\feld disorder-tuned SIT1,18\nhas a line of \fnite-temperature superconducting tran-\nsitions that terminate at the zero-\feld disorder-tuned\nSIT19,20. This, along with the preservation of time-\nreversal symmetry, implies that the zero-\feld disorder-\ntuned SIT must be in a distinct universality class. Our\nfocus here will be on \feld-tuned transitions.\nIn strongly disordered \flms, where the normal\nstate resistance is well in excess of the Cooper-pair\nquantum of resistance, RQ=h=4e2'6:45k\n=\u0003,\nmany predictions of the dirty boson theory have been\ncon\frmed2,3,10,12,13,21. However, in somewhat cleaner\n\flms, a direct transition from superconducting to insu-\nlating behavior is lost. Instead, an intervening metal-\nlic phase with substantial superconducting \ructuations22\nhas been observed4,6,7,10{12: the resistance approaches a\nconstant, \feld-dependent value R(T!0;H)\u001cRQas\nT!0. To the extent that such metallic behavior ob-\nserved at \fnite temperature uncovers the properties of a\ntrue zero temperature metal, it lies outside the scope of\ndirty bosons.\nHere, we address - at a phenomenological level - how\na metallic phase can emerge starting from the strongly\ndisordered dirty boson limit. Our analysis builds on the\nanalogy between direct SITs and quantum Hall plateau\ntransitions, which has been discussed in the literature\nusing a composite boson mapping23,26,27. In contrast,\nBoseInsulatorElectron InsulatorComposite Vortex LiquidSuperconductor\u0000\nHextFIG. 1. Schematic T= 0 phase diagram in the vicinity of the\nSIT of disordered thin \flms as a function of external mag-\nnetic \feldHextand disorder strength \u000e. The solid lines denote\nphase transitions, while the dashed line denotes the bound-\nary (either transition or crossover) between a Bose insulator\nof Cooper pairs and an electron insulator. A similar phase\ndiagram obtains for a 2DEG with the relabeling: supercon-\nductor$integer quantum Hall e\u000bect, Bose insulator $Hall\ninsulator23, and composite vortex liquid $composite Fermi\nliquid24,25. Appendix B has a second possible phase diagram.\nwe employ the composite fermion description28, which\nmore naturally accounts for the metallic behavior. As\nsummarized in Fig. 1, we suggest the metallic point ob-\nserved at the SIT in dirtier samples is closely related to\nthe metallic phase in somewhat cleaner systems; in both\ncases, the metal is analogous to the composite Fermi\nliquid24,25observed in half-\flled Landau levels of a 2D\nelectron gas (2DEG)29,30. The composite fermions here\nare \feld-induced vortices attached to a unit \rux quantum\nof an emergent gauge \feld. The hypothesis that compos-\nite fermions determine the low-energy behavior in both\nsystems straightforwardly allows for a number of testable\npredictions as the two systems complement and inform\none another.arXiv:1509.07865v1  [cond-mat.str-el]  25 Sep 2015II. EFFECTIVE FIELD THEORY FOR THE SIT\nWe begin with an e\u000bective description of a disordered\n2D superconductor close to a \feld-tuned SIT at T=\n0. The e\u000bective Hamiltonian for the delocalized Cooper\npairs,Hs=H1+H2, takes the following form:\nH1=1\n2X\nr;r0\u0010\nns(r)\u0000ns\u0011\nVr;r0\u0010\nns(r0)\u0000ns\u0011\n;\nH2=\u0000X\ni;rJi;rcos\u0010\n\u0001i\u0012(r) +e\u0003Ai(r)\u0011\n: (1)\nHere, the unit lattice spacing plays the role of a short-\ndistance cuto\u000b, which is comparable to zero \feld, zero\ntemperature superconducting correlation length. The\namplitude of the Cooper-pair \feld,  \u0018ei\u0012, is assumed\nfrozen, while the Cooper-pair density ns(r) and phase\nvariable\u0012(r) at site r= (x;y) satisfy the equal-time com-\nmutation relations, [ ns(r);\u0012(r0)] =i\u000er;r0. In Eq. (1),\nAi(r) = (Ax(r);Ay(r)) is the vector potential for the\nbackground magnetic \feld, \u0001 i\u0012(r)\u0011\u0012(r+^i)\u0000(~ r) is the\nlattice derivative, Vr;r0parameterizes both the density-\ndensity interaction and the coupling to disorder, Ji;ris\nthe superconducting phase sti\u000bness, which can also vary\nspatially due to strong disorder, and the Cooper pairs\ncarry electrical charge e\u0003= 2e.\nFor a clean metal with Galilean invariance at T= 0,\nthe average Cooper-pair density hn(r)i \u0011ns=ne=2\nwhereneis the electronic carrier density of the metal.\nIf the same relation were to hold for the disordered \flms\nconsidered here, the \feld scales required to tune to the\nSIT would be incompatible with experiment. Instead,\nfor a disordered metal at T= 0, the average Cooper-pair\ndensity can be substantially less than ne, since many of\nthe Cooper pairs will be strongly localized, and do not\nparticipate in the low-energy e\u000bective \feld theory. An\nestimate of the relevant degrees of freedom can be made\nvia the formula: ns\u0018ne\n3`2\n~D=(kBTc)where`is the elec-\ntronic mean-free path, Dis the normal-state di\u000busion\nconstant, and Tcis the zero-\feld superconducting criti-\ncal temperature31. For disordered metals,`2\n~D=(kBTc)\u001c1\nand so the Cooper-pair density may be much reduced\nfrom its value in the clean limit.\nAn important test of the reasoning above is a crude\nestimation of SIT \feld scales. For a 40 \u0017Athick MoGe\n\flm of the type studied by Mason and Kapitulnik, the\n3D carrier density ne\u001810\u000026m\u00003,`\u00186\u000210\u000010m,\nD\u00185\u000210\u00005m2=s, andTc\u00181K32. We estimate\na 2D Cooper-pair density ns\u00182\u00021014m\u00002, which\nis three orders of magnitude less than the clean esti-\nmate of 2\u00021017m\u00002. The estimated Cooper pair den-\nsity de\fnes a magnetic \feld scale bv=ns\b0\u00180:5T\nat which the Cooper pairs are at unit \flling fraction\n\u0017=ns=(Hext=\b0), where the magnetic \rux quantum\n\b0=hc=e\u0003\u00112\u0019=e\u0003. The metallic behavior observed\nin MoGe \flms occurs at \feld scales, which are roughly\ncomparable to bv33.Having estimated the average density of mobile\nCooper-pairs in a disordered system, we implement the\nfollowing duality transformation,\nns(r) =e\u0003\n2\u0019b(r);\u0001i\u0012(r) +e\u0003Ai(r) =e\u0003\n2\u0019^z\u0002e(r);(2)\nonly to these mobile, low energy degrees of freedom. The\nduality transformation relates the density and current\noperators of the mobile Cooper pairs to the magnetic\nb(r) and electric e(r) \feld of an emergent gauge \feld\nthat is minimally coupled to the dual vortex degrees of\nfreedom. The emergent electric \feld is constrained by\nGauss' Law:\n\u0001\u0001e(r) = 2\u0019\u0010\nnv(r)\u0000nv\u0011\n; (3)\nwhere the average vortex density nv\u0011 hnv(r)i=\nHext=\b0is sourced by the external magnetic \feld. The\nresulting dual vortex Hamiltonian34,~Hv=~H1+~H2+~H3:\n~H1=1\n2X\nr;r0\u0010\nnv(r)\u0000nv\u0011\n~Vr;r0\u0010\nnv(r0)\u0000nv\u0011\n;\n~H2=\u0000X\ni;r~Ji;rcos\u0010\n\u0001i\u001e(r)\u0000e\u0003ai(r)\u0011\n;\n~H3=X\nre(r)\u0001e(r) +X\nr;r0b(r)Vr;r0b(r0): (4)\nIn Eq. (4), \u001e(r) is the number operator conjugate to\nnv(r),ai(r) = (ax(r);ay(r)) are the spatial components\nof the emergent gauge \feld introduced in Eq. (2) with av-\nerage emergent magnetic \rux hbi=ns\b0. The emergent\ngauge charge carried by the vortices is equal and oppo-\nsite to the electromagnetic charge carried by the Cooper\npairs. The vortices are electrically neutral. ~Vr;r0and ~Ji;r\nparameterize the density and phase interactions. Dual-\nity implies that the dimensionless electrical resistivity \u0016 \u001a\nis equal to the dimensionless vortex conductivity (\u0016 \u001av)\u00001\nin units ofh=4e2and vice-versa . As the external \feld is\nincreased, nucleated vortices, pinned at \frst by disorder,\nbecome mobile at the SIT due to quantum \ructuations,\nand lead to non-zero resistance.\nIII. SELF-DUALITY AND COMPOSITE\nVORTICES\nA useful theoretical anchorpoint is the notion of self-\nduality which implies nv=ns35. Since the supercurrent\nis de\fned as I=e\u0003_ns, whereas the Josephson relation\nimplies that the voltage, V=h\ne\u0003_nvin the limit of van-\nishingly small Hall resistivity, a spectacular prediction of\nself-duality is that the resistance at the SIT must be uni-\nversal and equal to RQ. A triumph of the dirty boson\ntheory is the observation of roughly this value for the\ncritical resistance at the SIT in a variety of \flms3,12,13.\nIn our phenomenological treatment here, we take the\nobservation of self-duality as an empirical fact and con-\nsider its consequences. This in turn will enable us to\n2speculate on the nature of the ground state when self-\nduality is broken by reducing \flm disorder.\nSincens=nvat a SIT with self-duality, the transition\nrepresents the point where both the Cooper pairs and\nvortices are on the verge of condensation. To extend this\ndescription to the metallic phase, we must consider the\nstability of the above picture. Since Cooper pairs `see'\na vortex as a unit of \rux and vice-versa , and because\nboth particles are mutual bosons, it follows that in a\nmean-\feld description, the ground state involves bosons\nat\u0017= 1. However, in the presence of disorder andinter-\nactions, such a mean-\feld treatment favors the mutually\ncontradictory ground states consisting either of localized\nCooper pairs or vortices.\nA much more stable mean-\feld solution is obtained\nvia \rux attachment28. Bosons at \u0017= 1 map onto a com-\nposite fermion metal in zero net background magnetic\n\rux. Importantly, the Pauli principle is operative in the\nfermionic description, which substantially stabilizes the\nmean-\feld ground state. Thus, we propose that a com-\nposite fermion metal provides an e\u000bective description for\na self-dual SIT with a \fnite T= 0 resistance. How this\npicture quantitatively results in the observed experimen-\ntal behavior requires additional work; nevertheless, it is\npossible to explore a few of the immediate implications\nof this hypothesis.\nA. Emergent Metallic Phases\nThe most important consequence is the possibility that\nthe metallic point broadens into an emergent metallic\nphase . To be speci\fc, we consider a composite Fermi\nliquid of composite vortices, which are bosonic vortices\nbound to a unit \rux of the emergent gauge \feld a, whose\ncurl equals the Cooper pair density. We propose the fol-\nlowing scenario. At strong disorder, the \fnite density\nof composite vortices is driven to a strong disorder \fxed\npoint that describes the SIT. As the disorder is weakened\nand the e\u000bects of a Fermi surface of composite vortices\nbecomes better de\fned, the \fxed point broadens into a\nmetallic phase.\nTo motivate this hypothesis, we draw upon a well-\nknown analogy26between the SIT and quantum Hall\nplateau transitions. Consider an integer quantum Hall\nplateau transition from \u0017!\u0017\u00001. Upon increasing the\nexternal \feld, the electron chemical potential is lowered,\nand holes are nucleated in the \flled Landau level. These\nholes are the precise analogs of the vortices nucleated\nby a \feld in a superconductor. The holes are localized\nuntil the \feld is tuned to its critical value at which the\nplateau transition occurs. A consequence of Landau level\nparticle-hole symmetry { the analog of particle-vortex\nsymmetry { is that \u001bxy=\u0000\u0017+ 1=2. In addition, the di-\nagonal conductivity takes a universal value of e2=h36,37.\nThe metallic point that obtains at a plateau transition is\nclosely related to the composite Fermi liquid of half-\flled\nLandau levels. In samples with somewhat less disorder,there is a metallic phase intervening between the two\nplateaus38. The fact that the metallic point in dirtier\n2DEGs broadens into a metallic phase in cleaner 2DEGs\nre\rects the stability of the composite fermion metal rela-\ntive to either quantum Hall state with Hall conductance\nof\u0000\u0017or\u0000\u0017+ 1 (in units of e2=h). In the same manner,\nwe suggest here that the metallic phase related to the\ncomposite vortex liquid can be a more stable phase than\neither the superconductor or the insulator.\nB. Composite Vortex Lagrangian\nTo test these ideas more quantitatively, we provide an\ne\u000bective Hamiltonian for the composite vortex degrees\nof freedom, generalizing ~Hv. At unit \flling fraction, the\ncomposite vortices see zero e\u000bective magnetic \feld, form\na Fermi surface39,40, and interact via an emergent gauge\n\feld ~ain addition to the already present \feld aintro-\nduced in ~Hv. We consider the following \\working\" e\u000bec-\ntive Lagrangian, written in the continuum, for the com-\nposite vortices  ,Lcv=L0+Lgauge +Lint:\nL0= y\u0010\ni@t+ (~at\u0000at) +1\n2mv(@j\u0000i(~aj\u0000aj))2\u0011\n ;\nLgauge =\u00001\n4\u0019\u000f\u0016\u0017\u001a~a\u0016@\u0017~a\u001a+e\u0003\n2\u0019\u000f\u0016\u0017\u001aA\u0016@\u0017a\u001a;\nLint=\u00001\n2Z\nd2r0\u0010\n y (r)\u0000nv\u0011\n~Vr;r0\u0010\n y (r0)\u0000nv\u0011\n:\n(5)\nWe take the composite vortex kinetic term L0to be\nthat of a non-relativistic fermion, applicable to a metal-\nlic phase. The \frst term in Lgauge implements the unit\n\rux attachment, while the second term describes the cou-\npling of the Cooper-pair current J\u0016=e\u0003\n2\u0019\u000f\u0016\u0017\u001a@\u0017a\u001ato the\nexternal electromagnetic \feld A\u0016. The third term Lint\ndescribes the composite vortex density-density interac-\ntions inherited from ~Hv. Because the composite fermion\n only couples to the linear combination ~A= ~a\u0000a, we\nmay simplifyLcvand integrate out ato obtain41:\nLgauge =\u0000e\u0003\n2\u0019\u000f\u0016\u0017\u001a~A\u0016@\u0017A\u001a+e2\n\u0003\n4\u0019\u000f\u0016\u0017\u001aA\u0016@\u0017A\u001a: (6)\nThe \frst term in Lgauge describes the induced coupling\nbetween the external gauge \feld Aand the composite\nvortices through ~A. The presence of the second term is\nreminiscent of a composite fermion treatment of the holes\n{ the analog of the vortices of a superconductor { in a\n\flled Landau level42(See also43).\nC. Composite Vortex Response\nConsider the consistency of the above scenario with the\ngeneral phase diagram depicted in Fig. 1. As explained\nin Appendix A, we can relate the electrical conductivity\n3tensor\u001bijto the composite vortex conductivity \u001bcv\nij:\n\u001bij=e2\n\u0003\n2\u0019\u0010\n\u0000\u000fjk+ (\u001bcv)\u00001\njk\u0011\n: (7)\nWe concentrate on the DC response as T!0.\n1. Superconductor\nTo obtain an electrical superconductor, we must take\nthe composite vortices to be insulating, \u001bcv\nxx(T!0) = 0\nand\u001bcv\nxy(T!0) = 0. An Anderson insulator of compos-\nite vortices is expected for low composite fermion density\nwhen the external magnetic \feld is small. The Hall con-\nductivity is sensitive to the rate at which the various\ncomponents of the composite vortex conductivity tensor\nvanish: (1) when lim T!0(\u001bcv\nxx)2\n\u001bcvxy!0,\u001bxy(T!0) di-\nverges, while (2) when lim T!0\u001bcv\nxy\n(\u001bcvxx)2!0,\u001bxy(T!0) is\n\fnite. (Note that we always assume lim T!0\u001bcv\nxx\n(\u001bcvxy)2!0\nin order to ensure superconducting behavior.) The sec-\nond case may be referred to as a Hall superconductor.\nThe Hall resistivity, however, vanishes in both cases.\n2. Insulator\nAn electrical insulator obtains when the composite vor-\ntices exhibit the integer quantum Hall e\u000bect, \u001bcv\nxx(T!\n0) = 0 and \u001bcv\nxy(T!0) =\u00001. Analogous to the situa-\ntion with the superconductor, the precise value of the\nHall resistivity depends upon the rates by which the\ncomponents of the composite vortex conductivity ten-\nsor approach their zero temperature values: (1) when\nlimT!0(\u001bcv\nxx)2\n(1+\u001bcvxy)!0,\u001axy(T!0) diverges, while (2)\nwhen lim T!0(1+\u001bcv\nxy)\n(\u001bcvxx)2!0,\u001axy(T!0) is \fnite. (We as-\nsume lim T!0\u001bcv\nxx\n(1+\u001bcvxy)2!0 in order to ensure insulating\nbehavior.) The \frst case represents a trivial insulator,\nwhile the second case may be called a Hall insulator23,44.\nAs parameters in the laboratory are varied, we might\nexpect a crossover from a Hall insulator to a trivial insu-\nlator (along with an analogous crossover in the supercon-\nducting phase), consistent with the various phases fur-\nnished by the composite vortex Lagrangian.\nD. Composite Vortices in the Weak Disorder Limit\nWe now address the stability of the composite vortex\nliquid to disorder. The scaling theory of localization only\napplies to Fermi liquids coupled to quenched chemical po-\ntential disorder. By contrast, chemical potential disorder\nin~Hvtranslates into random \rux disorder of zero mean\nand random chemical potential disorder. To see this, con-\nsider the composite vortex Lagrangian in Eq. (5) wherethe coupling of the composite vortex density to random\nchemical potential disorder is implicitly included in Lint.\nThe equation of motion for the time-component ~ atof the\nemergent gauge \feld,\n y (r) =1\n2\u0019~b(r); (8)\nrelates the local composite vortex density to the emergent\ngauge \rux ~b=@x~ay\u0000@y~ax, thereby tying \ructuations\nin the local chemical potential to those of the emergent\ngauge \feld45.\nHappily, the problem of an electron hopping on a lat-\ntice in the midst of random chemical potential and ran-\ndom \rux of zero mean was studied in24,46with the con-\nclusion that localization is avoided. Random \rux of zero\nmean is of crucial importance. Random \rux with non-\nzero average e\u000bectively acts as an additional contribution\nto the random chemical potential and results in localiza-\ntion of non-interacting electrons. Thus, one of the most\nimportant simplifying features of the composite fermion\ntransformation wherein vortices at unit \flling fraction\nmap to composite vortices in zero background \rux is also\none of the most important in guaranteeing stability of the\nresulting metal to weak disorder.\nIV. EXPERIMENTAL CONSEQUENCES\nWe now brie\ry discuss a few experimental conse-\nquences, many of which are readily adapted from the\ncorresponding implications of a composite fermion metal\nin the 2DEG.\n1. Heat Capacity\nA simple, but remarkable thermodynamic signature of\nthe metallic phase should, in principle, arise in the heat\ncapacity. A linear in temperature heat capacity is ex-\npected for a Fermi liquid-like metal; additional correc-\ntions due to the interactions of the composite vortices\nwith the Chern-Simons gauge \feld may be expected to\ntake the form \u000ecV\u0018T2=3or\u000ecV\u0018Tlog(T) depend-\ning upon the e\u000bective composite vortex density-density\ninteraction. Distilling such behavior in 2DEGs has thus\nfar proven di\u000ecult47,48.\n2. Quantum Oscillations\nQuantum oscillations indicative of the composite vor-\ntex Fermi surface with wave vector determined by the\napplied external magnetic \feld would provide a striking\ncon\frmation of the picture presented in this paper. The\nfrequency of oscillation will monotonically depend on the\ndeviation of the external \feld from the value at which the\ncomposite vortices experience zero \rux. Precisely this\nbehavior is observed in the 2DEG about half-\flling28.\n43. Tunneling Density of States\nSuppression of the bulk electron tunneling density of\nstates at half-\flling in a 2DEG is a direct result of the\ncomposite fermion picture49,50. However, for the com-\nposite vortex metal, a suppression might be expected\nsimply because of the \fnite superconducting amplitude.\nTo probe the composite vortex liquid more directly, one\nshould consider instead the tunneling between a super-\nconducting probe and the composite vortex metal, which\nwe expect to be exponentially suppressed. Within the\ndirty boson framework, there would be un-suppressed\ntunneling at the SIT for a superconducting tip.\n4. Thermopower and Nernst E\u000bect\nThermopower and the Nernst e\u000bect may be utilized to\ndetermine whether self-duality is realized in the thin \flm.\nSelf-duality, obtained at the SIT or about unit \flling frac-\ntion of the vortices, requires a vanishing thermopower\nand Nernst signal51. Likewise, particle-hole symmetry\nwithin a single Landau level requires the thermopower to\nvanish52. We note that the thermopower is observed to\nbe non-zero for T > 0 in 2D hole systems at half-\flling\nin the lowest and \frst Landau levels53,54. Taken at face\nvalue, this indicates a breakdown of particle-hole sym-\nmetry. An obvious guess for the cause of the breakdown\nis Landau level mixing which is not typically small in\nthe 2DEG; a second possibility is the spontaneous break-\ndown of particle-hole symmetry in the limit of vanish-\ningly small Landau level mixing. Thus, if the relation\nbetween the 2DEG and thin \flm \flm is taken seriously,\nwe may abstract an important lesson: the breakdown\nof particle-hole symmetry observed in experimentally-\nrealized 2DEGs may imply and provide a mechanism for\na similar breakdown of particle-vortex duality in the thin\n\flm.\nV. DISCUSSION\nWe have suggested that the 2D metallic phase observed\nin the vicinity of the magnetic \feld-tuned SIT is a com-\nposite vortex liquid, analogous to the composite Fermi\nliquid found in 2DEGs near half-\flling. This hypoth-\nesis entails the statistical transmutation of interacting\nbosons in a magnetic \feld into a Fermi sea of composite\nvortices and provides a natural explanation for the emer-\ngent metal with a variety of experimental consequences.\nThere have been several previous studies formulating\na theory of the metallic phase55{59. The work by Gal-\nitski et al.59whose theoretical formulation most closely\nresembles ours, identi\fed the metal with a gas of neutral\nspinons coupled to vortices by an emergent U(1) statis-\ntical gauge \feld60. By contrast, our approach and moti-\nvation rely on the observation of self-duality at the \feld-\ntuned SIT, the analogy between the SIT and quantumHall plateau transitions, and the substantial supercon-\nducting \ructuations observed within the metal22.\nThe superconductor-metal transition has been well\nstudied in thin MoGe and InO \flms. It would be of great\ninterest to better understand the nature of the weak in-\nsulator in systems where a metal is observed, similar to\nthat in Ta \flms. In disordered \flms where the metallic\nphase has shrunk to a point, the proximate insulator is\na Bose insulator of localized Cooper pairs with non-zero\nsuperconducting correlations. Upon increasing the mag-\nnetic \feld, the Cooper pairs are broken and an electron\ninsulator results. By continuity, we expect the Bose insu-\nlator to continue to border the emergent metal for as long\nas the Bose insulator persists as the disorder is decreased\nas depicted in Fig. 1. A second possible phase diagram is\npresented in Fig. 2 of Appendix B. Here, the supercon-\nducting, Bose insulating, electron insulating, and metal-\nlic phases meet at a multicritical point that is of higher\ndegree than occurs in Fig. 1. Distinguishing the two\npossibilities merits further study.\nInterestingly, there are indications13that the Bose in-\nsulator observed in InO \flms is a Hall insulator with\nvanishing conductivity tensor, diverging linear resistivity,\nbut \fnite Hall resistivity as T!0. Hall insulators were,\nnot surprisingly, \frst predicted23,44and observed61{63in\nthe 2DEG. Indications of a similar state in the thin \flm\nfurther unites the two systems and gives us further hope\nfor a relation between the metals.\nSeveral open theoretical questions remain, in light of\nthe observations made here. Building upon previous\nwork64,65, further investigation of self-duality in numeri-\ncal studies of Josephson junction arrays in the presence\nof strong disorder and magnetic \felds is desired. Sec-\nond, the emergence of a metallic phase as the disorder\nstrength is reduced would be a clear validation of the\nideas presented here. Furthermore, a deeper understand-\ning of the role of disorder in composite Fermi liquids,\ntaking into account both the random \rux disorder due\nto the Chern-Simons \feld, as well as non-Fermi liquid\ne\u000bects is needed66. Lastly, the application, if any, of re-\ncent developments considering particle-hole symmetry {\nthe analog of particle-vortex duality { in half-\flled Lan-\ndau levels67{70, in the context of the emergent metal near\nthe SIT, is an intriguing possibility.\nACKNOWLEDGMENTS\nWe thank P. Armitage, E. Berg, S. Chakravarty, D.\nFisher, M. P. A. Fisher, S. Kivelson, R. Laughlin, O.\nVafek, A. Vishwanath, J. Yoon, and especially A. Ka-\npitulnik for helpful discussions. We thank S. Kachru, R.\nLaughlin, and C. Nayak for comments on an early draft\nof this manuscript. This work was supported in part\nby DOE O\u000ece of Basic Energy Sciences, contract DE-\nAC02-76SF00515 (SR) and the John Templeton Founda-\ntion (MM).\n5Appendix A: Response Derivation\nTo relate the composite vortex conductivity to the\nelectrical conductivity, we formally integrate out the\ncomposite vortices of Lcv, while working in the gauge\n~At=At= 0:\nLcv=1\n2~Aj\u0005cv\njk~Ak+e\u0003i!\n2\u0019\u000fjk~AjAk\u0000e2\n\u0003i!\n2\u0019\u000fjkAjAk;\n(A1)\nwhere the composite vortex response \u0005cvand conductiv-\nity\u001bcvtensors:\n\u0005cv=i!\n2\u0019\u0012\u001bcv\nxx\u001bcv\nxy\n\u0000\u001bcv\nxy\u001bcv\nxx\u0013\n: (A2)\nFinally, we integrate out ~Ato obtain the e\u000bective elec-\ntronic response Lagrangian,\nLcv=i!\n2e2\n\u0003\n2\u0019Aj\u0010\n\u0000\u000fjk+ (\u001bcv)\u00001\njk\u0011\nAk; (A3)\nfrom which we may read o\u000b the electrical conductivity\nand resistivity tensors:\n\u001bxx=e2\n\u0003\nh\u001bcv\nxx\n(\u001bcvxx)2+ (\u001bcvxy)2;\n\u001bxy=\u0000e2\n\u0003\nh\u0010\n1 +\u001bcv\nxy\n(\u001bcvxx)2+ (\u001bcvxy)2\u0011\n;\n\u001axx=h\ne2\u0003\u001bcv\nxx\n(\u001bcvxx)2+ (1 +\u001bcvxy)2;\n\u001axy=h\ne2\u0003\u0010\n1\u00001 +\u001bcv\nxy\n(\u001bcvxx)2+ (1 +\u001bcvxy)2\u0011\n: (A4)Appendix B: Alternative Phase Diagram\n\u0000\nHextBoseInsulatorElectron InsulatorComposite        Vortex LiquidSuperconductor\nFIG. 2.\nIn Fig. 1, the superconducting, Bose insulating, and\nmetallic phases meet at a point, while in Fig. 2 the same\nthree phases along with the electron insulator meet at\na point. Fig. 2 displays a transition between the Bose\ninsulator and electron insulator. If instead, there hap-\npened to be a crossover, the degree of multicriticality at\nwhich the various phases meet would be the same in both\n\fgures.\n1A. M. Goldman, International Journal of Modern Physics\nB24, 4081 (2010).\n2A. F. Hebard and M. A. Paalanen, Phys. Rev. Lett. 65,\n927 (1990).\n3M. A. Paalanen, A. F. Hebard, and R. R. Ruel, Phys. Rev.\nLett. 69, 1604 (1992).\n4A. Yazdani and A. Kapitulnik, Phys. Rev. Lett. 74, 3037\n(1995).\n5V. Gantmakher, M. Golubkov, V. Dolgopolov, G. Tsy-\ndynzhapov, and A. Shashkin, Journal of Experimental\nand Theoretical Physics Letters 68, 363 (1998).\n6N. Mason and A. Kapitulnik, Phys. Rev. Lett. 82, 5341\n(1999).\n7N. Mason and A. Kapitulnik, Phys. Rev. B 64, 060504\n(2001).\n8E. Bielejec and W. Wu, Phys. Rev. Lett. 88, 206802 (2002).\n9G. Sambandamurthy, L. W. Engel, A. Johansson, and\nD. Shahar, Phys. Rev. Lett. 92, 107005 (2004).\n10M. Steiner and A. Kapitulnik, Physica C: Superconductiv-\nity422, 16 (2005).\n11Y. Qin, C. L. Vicente, and J. Yoon, Phys. Rev. B 73,100505 (2006).\n12M. A. Steiner, N. P. Breznay, and A. Kapitulnik, Phys.\nRev. B 77, 212501 (2008).\n13N. P. Breznay, M. A. Steiner, S. A. Kivelson, and A. Ka-\npitulnik, ArXiv e-prints (2015), arXiv:1504.08115 [cond-\nmat.supr-con].\n14M. V. Feigel'man, A. I. Larkin, and M. A. Skvortsov,\nPhys. Rev. Lett. 86, 1869 (2001).\n15M. P. A. Fisher, G. Grinstein, and S. M. Girvin, Phys.\nRev. Lett. 64, 587 (1990).\n16M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990).\n17D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev.\nB43, 130 (1991).\n18N. Markovi\u0013 c, C. Christiansen, and A. M. Goldman, Phys.\nRev. Lett. 81, 5217 (1998).\n19A. A. Abrikosov and L. Gor'Kov, Zhur. Eksptl'. i Teoret.\nFiz.35(1958).\n20P. Anderson, Journal of Physics and Chemistry of Solids\n11, 26 (1959).\n21R. Crane, N. P. Armitage, A. Johansson, G. Samban-\ndamurthy, D. Shahar, and G. Gr uner, Phys. Rev. B 75,\n6184530 (2007).\n22W. Liu, L. Pan, J. Wen, M. Kim, G. Sambandamurthy,\nand N. P. Armitage, Phys. Rev. Lett. 111, 067003 (2013).\n23S. Kivelson, D.-H. Lee, and S.-C. Zhang, Phys. Rev. B 46,\n2223 (1992).\n24V. Kalmeyer and S.-C. Zhang, Phys. Rev. B 46, 9889\n(1992).\n25B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47,\n7312 (1993).\n26D.-H. Lee, S. Kivelson, and S.-C. Zhang, Phys. Rev. Lett.\n67, 3302 (1991).\n27S. C. Zhang, Int. J. Mod. Phys. B6, 25 (1992).\n28J. K. Jain, Composite Fermions (Cambridge University\nPress, 2007).\n29R. Willett, J. P. Eisenstein, H. L. St ormer, D. C. Tsui,\nA. C. Gossard, and J. H. English, Phys. Rev. Lett. 59,\n1776 (1987).\n30H. W. Jiang, H. L. Stormer, D. C. Isui, L. N. Pfei\u000ber, and\nK. W. West, Phys. Rev. B 40, 12013 (1989).\n31A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski,\nMethods of Quantum Field Theory in Statistical Physics\n(Dover Publications, 1975).\n32N. Mason, Superconductor-Metal-Insulator Transitions in\nTwo Dimensions , Ph.D. thesis, Stanford University (2001).\n33We have checked that a similar estimate for the Cooper-\npair \flling fraction where metallic behavior is observed can\nbe made for InO and Ta \flms via the methods described\nabove, along with an independent estimate using more di-\nrect super\ruid sti\u000bness measurements in InO \flms22, as-\nsuming a reduction of the electronic carrier mass from its\nfree value in the range 10\u00001\u000010\u00002.\n34M. P. A. Fisher and D. H. Lee, Phys. Rev. B 39, 2756\n(1989).\n35A more precise consequence of self-duality is the relation\n\u0016\u001av\nij= \u0016\u001ajiwhich implies \u001a2\nxx+\u001a2\nxy=R2\nQ.\n36D. Shahar, D. C. Tsui, M. Shayegan, R. N. Bhatt, and\nJ. E. Cunningham, Phys. Rev. Lett. 74, 4511 (1995).\n37D. Shahar, D. C. Tsui, M. Shayegan, E. Shimshoni, and\nS. L. Sondhi, Science 274, 589 (1996).\n38L. W. Wong, H. W. Jiang, and W. J. Scha\u000b, Phys. Rev.\nB54, R17323 (1996).\n39V. Pasquier and F. D. M. Haldane, Nucl. Phys. B 516, 719\n(1998).\n40N. Read, Phys. Rev. B 58, 16262 (1998).\n41We safely ignore the Maxwell terms contained in ~H3be-\ncause they are irrelevant at low-energies compared to the\nChern-Simons term in Lgauge.\n42M. Barkeshli, M. Mulligan, and M. P. A. Fisher,\n\\Particle-hole symmetry and the composite fermi liquid,\"\nArXiv:1502.05404 [cond-mat.str-el].\n43J. Alicea, O. Motrunich, M. Hermele, and M. Fisher, Phys.\nRev. B 72, 064407 (2005).\n44S.-C. Zhang, S. Kivelson, and D.-H. Lee, Phys. Rev. Lett.\n69, 1252 (1992).\n45Recall that the mean-\feld ansatz for Lcvis one where ~b=\nbwithb=nv\b0so the constant o\u000bset cancels and the\n\ructuations in both the chemical potential and \rux ~bare\ncorrelated.\n46V. Kalmeyer, D. Wei, D. P. Arovas, and S. Zhang, Phys.\nRev. B 48, 11095 (1993).\n47E. Gornik, R. Lassnig, G. Strasser, H. L. St ormer, A. C.\nGossard, and W. Wiegmann, Phys. Rev. Lett. 54, 1820(1985).\n48J. K. Wang, D. C. Tsui, M. Santos, and M. Shayegan,\nPhys. Rev. B 45, 4384 (1992).\n49S. He, P. M. Platzman, and B. I. Halperin, Phys. Rev.\nLett. 71, 777 (1993).\n50Y. B. Kim and X.-G. Wen, Phys. Rev. B 50, 8078 (1994).\n51This conclusion follows from the transformation of the elec-\ntric \feldEi7!\u0000Eiand thermal gradient ( rT)i7!(rT)i\nunder the particle-vortex transformation. Alternatively, we\ncan see this by mapping the superconductor in zero \feld to\nthe integer quantum Hall e\u000bect where the particle-vortex\ntransformation is traded for the particle-hole transforma-\ntion. In the quantum Hall problem, the thermopower and\nNernst signal are odd under the anti-unitary particle-hole\ntransformation de\fned in the next footnote.\n52This follows from the transformation of the Ei7!\u0000Eiand\n(rT)i7!(rT)iunder the anti-unitary particle-hole trans-\nformation generated by CTwhereCis charge-conjugation\nandTis time-reversal. The thermopower remains odd if\ninsteadCPwithPthe generator spatial re\rection along\nthey-axis is identi\fed as the generator of particle-hole\ntransformations. The Nernst signal, however, is even under\nCP. If a measurement of the thermopower indicated un-\nbroken particle-hole symmetry, a further measurement of\nthe Nernst signal could determine whether the physically-\nrealized particle-hole symmetry were anti-unitary or uni-\ntary in the 2DEG.\n53X. Ying, V. Bayot, M. B. Santos, and M. Shayegan, Phys.\nRev. B 50, 4968 (1994).\n54V. Bayot, E. Grivei, H. C. Manoharan, X. Ying, and\nM. Shayegan, Phys. Rev. B 52, R8621 (1995).\n55D. Das and S. Doniach, Phys. Rev. B 60, 1261 (1999).\n56A. Kapitulnik, N. Mason, S. A. Kivelson, and\nS. Chakravarty, Phys. Rev. B 63, 125322 (2001).\n57D. Dalidovich and P. Phillips, Phys. Rev. B 64, 052507\n(2001).\n58B. Spivak, P. Oreto, and S. A. Kivelson, Phys. Rev. B 77,\n214523 (2008).\n59V. M. Galitski, G. Refael, M. P. A. Fisher, and T. Senthil,\nPhys. Rev. Lett. 95, 077002 (2005).\n60M. V. Feigelman, V. B. Geshkenbein, L. B. Io\u000be, and A. I.\nLarkin, Phys. Rev. B 48, 16641 (1993).\n61V. J. Goldman, B. Su, and J. K. Wang, Phys. Rev. B 47,\n10548 (1993).\n62S. V. Kravchenko, J. E. Furneaux, and V. M. Pudalov,\nPhys. Rev. B 49, 2250 (1994).\n63L. W. Wong, H. W. Jiang, N. Trivedi, and E. Palm, Phys.\nRev. B 51, 18033 (1995).\n64S. Chakravarty, G.-L. Ingold, S. Kivelson, and A. Luther,\nPhys. Rev. Lett. 56, 2303 (1986).\n65S. Chakravarty, G.-L. Ingold, S. Kivelson, and G. Zimanyi,\nPhys. Rev. B 37, 3283 (1988).\n66M. Mulligan and S. Raghu, Work in progress.\n67D. T. Son, Phys. Rev. X 5, 031027 (2015).\n68C. Wang and T. Senthil, \\Dual dirac liquid on the surface\nof the electron topological insulator,\" ArXiv: 1505.05141.\n69M. A. Metlitski and A. Vishwanath, \\Particle-vortex du-\nality of 2d dirac fermion from electric-magnetic duality of\n3d topological insulators,\" ArXiv: 1505.05142.\n70S. Kachru, M. Mulligan, G. Torroba, and H. Wang, \\Mir-\nror symmetry and the half-\flled Landau level,\" ArXiv:\n1506.01376 cond-mat.str-el.\n7"    },    {        "title": "1007.2058v1.Modeling_of_complex_oxide_materials_from_the_first_principles__systematic_applications_to_vanadates_RVO3_with_distorted_perovskite_structure.pdf",        "content": "arXiv:1007.2058v1  [cond-mat.str-el]  13 Jul 2010Noname manuscript No.\n(will be inserted by the editor)\nModeling of complex oxide materials from the first\nprinciples: systematic applications to vanadates RVO3\nwith distorted perovskite structure\nIgor Solovyev\nReceived: date / Accepted: date\nAbstract “Realistic modeling” is a new direction of electronic struc ture calculations,\nwhere the main emphasis is made on the construction of some eff ective low-energy\nmodel entirely within a first-principle framework. Ideally , it is a model in form, but\nwith all the parameters derived rigorously, on the basis of fi rst-principles electronic\nstructure calculations. The method is especially suit for t ransition-metal oxides and\nother strongly correlated systems, whose electronic and ma gnetic properties are pre-\ndetermined by the behavior of some limited number of states l ocated near the Fermi\nlevel. After reviewing general ideas of realistic modeling , we will illustrate abilities of\nthis approach on the wide series of vanadates RVO3(R= La, Ce, Pr, Nd, Sm, Gd, Tb,\nYb, and Y) with distorted perovskite structure. Particular attention will be paid to\ncomputational tools, which can be used for microscopic anal ysis of different spin and\norbital states in the partially filled t2g-band. We will explicitly show how the lifting\nof the orbital degeneracy by the monoclinic distortion stab ilizes C-type antiferromag-\nnetic (AFM) state, which can be further transformed to the G- type AFM state by\nchanging the crystal distortion from monoclinic to orthorh ombic one. Two microscopic\nmechanisms of such a stabilization, associated with the one -electron crystal field and\nelectron correlation interactions, are discussed. The flex ibility of the orbital degrees of\nfreedom is analyzed in terms of the magnetic-state dependen ce of interatomic magnetic\ninteractions.\nKeywords First-principle calculations ·Effective models ·Perovskite vanadates ·\nSpin-orbital order\nPACS71.15.-m ·71.10.-w ·75.25.-j·75.25.Dk\nThis work is partly supported by Grant-in-Aid for Scientific Research (C) No. 20540337 from\nthe Ministry of Education, Culture, Sport, Science and Tech nology of Japan.\nI. Solovyev\nNational Institute for Materials Science, 1-2-1 Sengen, Ts ukuba, Ibaraki 305-0047, Japan\nTel.: +81-29-859-2619\nFax: +81-29-859-2601\nE-mail: SOLOVYEV.Igor@nims.go.jp2\n1 Introduction\nThe transition-metal oxides are currently regarded as some promising materials for the\nfuture generation of electronic devises. The interest to th ese systems was spurred by\nthe discoveries in them of such key phenomena as\n1. high-temperature superconductivity (Cu- and Fe-based o xides) [1],\n2. colossal magnetoresistance (doped manganites) [2],\n3. multiferroelectricity (BiMnO 3, TbMnO 3) [3],\nand many others, which can be potentially used in applicatio ns. One important aspect\nof the transition-metal oxides is realization of the so-cal led “switching phenomena”,\nwhere electronic properties can be controlled or easily swi tched between different states\nby various external factors, such as the hydrostatic pressu re, electric or magnetic field,\netc. For instance, the colossal magnetoresistance is the gi gantic suppression of resis-\ntivity by the magnetic field, the multiferroelectricity pro vides a unique possibility for\ncontrolling the electric polarization by the magnetic field and vice versa, etc.\nBesides the applications, there exists a long-standing fun damental interest to the\ntransition-metal oxides, related to understanding of the a bove-mentioned phenomena\non the microscopic level.\nHistorically, theoretical developments around the transi tion-metal oxides went al-\nmost parallel in two directions:\n1. model approaches, based on the solution and analysis of al l possible model Hamil-\ntonians (the typical example is the Hubbard model, which is t he basic and widely\nused model in the physics of strongly correlated systems) [4 ];\n2. first-principles calculations, many of which are based on the density-functional\ntheory (DFT) [5].\nEach of these directions has merits and demerits. For exampl e, the main advantage of\nthe model analysis is the simplicity and transparency, whil e the main disadvantage is\nthe use of adjustable parameters, which are typically chose n to fit the experimental\ndata. On the contrary, the undisputable advantage of first-p rinciples calculations is\nthe lack of adjustable parameters. However, as calculation s become more and more\ncomplex, we inevitably face the question about the interpre tation of the obtained re-\nsults. The problem is complicated by the fact that many first- principles methods are\nsupplemented with some additional approximation for the ex change-correlation inter-\nactions. The most typical one is the local-density approxim ation (LDA), which is based\non the model of homogeneous electron gas and undermines the p hysics of short-range\nCoulomb correlations. Thus, in the first-principles calcul ations for the transition-metal\noxides we frequently face the question: are the obtained res ults physical or artifacts of\nadditional approximations, which were employed in the proc ess of calculations?\nThe primary goal of the new project, which is called “realist ic modeling”, is the\nconstruction of “intelligent” models, which would combine the principles of simplicity\nand transparency with the rigorous first-principle basis fo r determination of the model\nparameters.\nWhat is the main idea of realistic modeling?\nOf course, the crystal and electronic structure of many oxid e materials can be very\ncomplex. Nevertheless, as longas we are mainlyconcerned wi thelectronic andmagnetic\nproperties of these systems, in many cases we can limit our co nsideration to a small\ngroup of states (the so-called low-energy states), which ar e located near the Fermi level3\nFig. 1 Electronic structure for the orthorhombic (top) and monocl inic (bottom) phases of\nSmVO 3in the local-density approximation. The shaded area shows t he contributions of the\nV-3dstates. Right part of the figure shows the enlarged behavior o f thet2g-bands computed\nfrom the original LMTO basis functions [6] (solid curves) an d downfolded bands obtained in\nthe process of construction of the low-energy model (dot-da shed curves). The corresponding\nbands in the left part of the figure are shown by arrows. The Fer mi level is at zero energy.\nand which are mainly responsible for the considered propert ies. The typical example\nof the LDA band structure of SmVO 3is shown in Fig. 1. In this case, the low-energy\nstates are the t2g-bands. Thus, the main idea is to\n1. construct an effective model for the low-energy states (ty pically, the multiorbital\nHubbard model);\n2. include the effect of all other states in some approximate f orm, through the defini-\ntion of the model parameters for the low-energy states;\n3. determine all model parameters rigorously, on the basis o f first-principles calcula-\ntions of the electronic structure.\nWhat are the main goals of the realistic modeling?\n1. It provides a possibility to go beyond the local-density a pproximation and to sys-\ntematically study the effects of electron-electron correla tions in the narrow bands.\nThe solution of such a many-body problem is an extremely diffic ult task, even for\nthe present computational facilities. Therefore, if the is a chance to formulated this\nproblem rigorously in some restricted Hilbert space of states, near the Fermi level,\nthere is a chance to solve it, at least numerically.4\n2. It is certainly true, that the first-principles calculati ons of the electronic structure\nare currently on the rise. However, as the complexity of such calculations also rises,\nwe will inevitably need some tool for the analysis and interp retation of the obtained\nresults. We would like to emphasize that the final goal of comp utational physics is\nnot to reproduce some complex experimental trend. The final g oal is to understand\nit and to come up with some transparent explanation. From thi s point of view, the\nrealistic modeling will continue to play an important role, as a convenient tool for\nthe analysis and interpretation of results of complex calcu lations.\nFinally, the construction of the low-energy model is always conjugated with some\napproximations (and, as a matter of fact, the form of the mode l itself is the main\napproximation). However, we would like to emphasize from th e very beginning that\napart from these approximations, we do not use any adjustabl e parameters. Instead,\nthe realistic modeling brings the state of the discussion to a qualitatively new level:\nif the model does not work, we need to reconsider the approxim ations underlying the\ndefinition of the model parameters or maybe the model itself.\n2 Construction of multiorbital Hubbard model for the low-en ergy states\nThe first step of realistic modeling is the construction of an effective multiorbital Hub-\nbard model in the low-energy part of spectrum:\nˆH=/summationdisplay\nij/summationdisplay\nαβtαβ\nijˆc†\niαˆcjβ+1\n2/summationdisplay\ni/summationdisplay\nαβγδUαβγδˆc†\niαˆc†\niγˆciβˆciδ, (1)\nwhich is regarded as the basic model in the physics of strongl y correlated systems.\nOther models can be derived by considering some limiting cas es of (1) (an example of\nthe Heisenberg model will be considered in the next section) . The Hubbard model is\nspecified in the basis of Wannier orbitals, which are denoted by Greek symbols, each of\nwhich is a combination of spin ( s=↑or↓) and orbital ( m) variables. The site-diagonal\npart ofˆtijdescribes the crystal-field splittingof the atomic levels, while theoff-diagonal\n(i/ne}ationslash=j) elements stand for transfer integrals. If the relativisti c spin-orbit interaction is\nnot included, ˆtijis diagonal with respect to the spin indices. Uαβγδare the matrix\nelements of screened Coulomb interactions for the low-ener gy states.\nThe most straightforward way to derive /bardblˆtij/bardbland/bardblUαβγδ/bardblis to use a complete\nbasis of localized orbitals in the low-energy part of spectr um. By the definition, this is\nthe basis of Wannier functions [7]. There is a number of moder n techniques, which can\nbe used for the construction of Wannier functions, starting from one-electron Kohn-\nSham orbitals in LDA. In the basis of localized pseudoatomic orbitals, which are used\nfor example in the linear muffin-tin orbital (LMTO) method [6] , it is convenient to use\nthe projector-operator technique [8]. In the plane-wave ba sis, the Wannier functions\ncan be constructed by minimizing the matrix elements of some local operators, such\nas the square of the position operator [9].\nThen, the one-electron part /bardblˆtij/bardblof the Hubbard model (1) is identified with the\nmatrix elements of the Kohn-Sham Hamiltonian in the basis of Wannier functions.1\n1Indeed, the Kohn-Sham Hamiltonian is also one-electron one . Moreover, since the exchange\ncorrelation potential in LDA is local, it does not contribut e to the matrix elements between\ndifferent atomic sites, so that the latter can be identified wi th the kinetic transfer integrals.\nMeanwhile, the site-diagonal contributions of the LDA pote ntial to/bardblˆtij/bardblshould be subtracted,5\nSince the Wannier basis is complete in the low-energy part of spectrum, the con-\nstruction is exact, and the one-electron energies derived f rom/bardblˆtij/bardblcoincide with the\neigenenergies of the original LDA Hamiltonian (Fig. 1). His torically, the derivation of\n/bardblˆtij/bardblwas based on the downfolding technique [10]. However, after some modifications,\naiming to get rid of the frequency-dependence of ˆtij, this approach becomes equivalent\nto the projector-operator technique [8].\nThe effective Coulomb interaction in solids is defined as the e nergy cost for trans-\nferring an electron from one atomic site to another:\n2(dn)⇀↽dn+1+dn−1. (2)\nEven if such redistribution of electrons occurs in the low-e nergy part of spectrum (be-\ntween appropriate Wannier orbitals), the corresponding ch ange of the electron density\ncan interact with the other electrons and change the electro nic structure in the entire\nrange of both low-energy and high-energy states. The change of the electronic structure\nin the high-energy part contributes to the screening of Coul omb interactions /bardblUαβγδ/bardbl\nin the low-energy part. Some of these effects can be described in the framework of the\nrandom-phase approximation (RPA) [11], which takes into ac count the screening of\nUαβγδdue to the deformation of the Kohn-Sham orbitals in the cours e of the reaction\n(2).\nSomewhat heuristically, in the pseudoatomic basis, the scr eening of Coulomb in-\nteractions, associated with the reaction (2), can be divide d in two parts [10,12]:\n1. the screening, caused by relaxation of the pseudoatomic ( basis) orbitals;\n2. the self-screening of the low-energy states by the atomic states of the same origin,\nwhich contribute to other parts of spectrum due to the hybrid ization (the typical\nexample is the screening of the t2g-bands of the 3 d-origin by the oxygen- and eg-\nbands, which have a large weigh of the atomic 3 d-states due to the hybridization\neffects – see Fig. 1).\nThese two channel of screening can be easily incorporated in the framework of the con-\nstrained density-functional theory(DFT) and RPA,respect ively [10]. Sucha separation\nconsiderably facilitates the calculations and make them mo re transparent.\n3 Solution of the model Hamiltonian\nThe simplest way to solve the many-body problem (1) is to use t he mean-field Hartree-\nFock (HF) approach, where the ground-state wavefunction is approximated by the\nsingle Slater determinant, constructed from the one-elect ron orbitals {φk}. The latter\nare obtained from the solution of the one-electron equation s (in the reciprocal space):\n/parenleftbigˆtk+ˆV/parenrightbig\n|ϕk/an}bracketri}ht=εk|ϕk/an}bracketri}ht, (3)\nwhereˆtkis the Fourier image of ˆtij,kis a collective index combining the momentum\nkof the first Brillouin zone, the band number, and the spin ( s=↑or↓) of the particle,\nandˆVis the HF potential\nVαβ=/summationdisplay\nγδ/parenleftbig\nUαβγδ−Uαδγβ/parenrightbig\nnγδ, (4)\nbecause the splitting of the atomic levels caused by electro n-electron interactions is explicitly\nincluded in the second part of the Hubbard model [12].6\nexpressed through the density matrix\nˆn≡ /bardblnγδ/bardbl=occ/summationdisplay\nk|ϕk/an}bracketri}ht/an}bracketle{tϕk|.\nThe latter is obtained self-consistently. By knowing {φk}and{εk}, it is easy to con-\nstruct the one-electron (retarded) Green function ˆG↑,↓\nij(ω) for the spin ↑and↓, which\ncan be used in many applications. Particularly, one useful a pplication is related to the\nanalysis of interatomic magnetic interactions, which desc ribe infinitesimal rotations of\nthe spins near different magnetic equilibriums [13]. In this case, the total energy change\ncan be mapped onto the Heisenberg model\nEHeis=−1\n2/summationdisplay\nijJijei·ej (5)\n(eiandejbeing the directions of the spin moments), and the parameters are given by\nJij=1\n2πIm/integraldisplayεF\n−∞dωTrL/braceleftBig\nˆG↑\nij(ω)∆ˆVˆG↓\nji(ω)∆ˆV/bracerightBig\n. (6)\nIn these notations, ∆ˆV= TrS{ˆσzˆV}is the spin part of the Hartree-Fock potential, Tr S\n(TrL) denotes the trace over the spin (orbital) indices, and ˆ σzis the Pauli matrix.2\nThe simplest way to go beyond the HF approximation is to consi der the regular\nperturbation theory for correlation interactions. The lat ter are defined as the difference\nbetweentrueoperator ofelectron-electroninteractions i ntheHubbardmodel(1)andits\ncounterpart in the HF approximation (i.e., the combination of Coulomb and exchange\ninteractions):\nˆHC=/summationdisplay\ni\n1\n2/summationdisplay\nαβγδUαβγδˆc†\niαˆc†\niγˆciβˆciδ−/summationdisplay\nαβVαβˆc†\niαˆciβ\n. (7)\nFor instance, in the second order, the correlation energy is given by:\nE(2)\nC=−/summationdisplay\nS/an}bracketle{tG|ˆHC|S/an}bracketri}ht/an}bracketle{tS|ˆHC|G/an}bracketri}ht\nEHF(S)−EHF(G), (8)\nwhere|G/an}bracketri}htis the ground-state wavefunction in the HF approximation an d|S/an}bracketri}htis the\nexcited state, which is obtained from |G/an}bracketri}htby replacing two orbitals in the occupied part\nof spectrum by two unoccupied orbitals.\nOf course, this strategy is not universe. Nevertheless, it c an justified when the de-\ngeneracy of the ground state is already lifted (for instance , by the lattice distortions),\nso that the ground state can be described reasonably well by t he single Slater determi-\nnant in the HF approximation and the correlation interactio ns can be included after\nthat by considering the regular (non-degenerate) perturba tion theory expansion.\nIn principle, one can go beyond the second order and consider higher-order effects\nfor correlations interactions in the framework of the T-mat rix theory [14,15].\n2More precisely, the parameters Jijin (6) define the localstability of the given magnetic\nconfiguration, which is stable if Jij>0 and unstable if Jij<0. In order to be consistent with\ntheglobaldefinition of the parameters of the Heisenberg model (5), the y should be additionally\nmultiplied by ei·ej= 1 and −1, correspondingly for the ferromagnetic and antiferromag netic\nbonds.7\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99\n/s67/s45/s65/s70/s77\n/s97/s97 /s98/s98/s99\n/s99\n/s71/s45/s65/s70/s77/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99\n/s49/s50/s51/s52\n/s49/s50/s51/s52\nFig. 2G- and C-type antiferromagnetic ordering realized in the or thorhombic and monoclinic\nstructure of YVO 3.\n4 Applications to vanadates RVO3(R= La, Ce, Pr, Nd, Sm, Gd, Tb, Yb,\nand Y)\nThe vanadates RVO3have attracted a considerable experimental and theoretica l at-\ntention. All these compounds crystallize in the highly dist orted perovskite structure.\nHowever, even small change of the crystal structure in RVO3can lead to dramatic\nchange of the magnetic structure. Thus, this behavior can be regarded as a prototype\nof the “switching phenomena”. Below the magnetic transitio n temperature, the crystal\nstructureof RVO3canbeeitherorthorhombic(thespacegroup D16\n2h)ormonoclinic(the\nspace group C5\n2h). The orthorhombic structure typically coexists with the G -type anti-\nferromagnetic (AFM) ordering (Fig. 2), where all nearest-n eighbor V-sites are coupled\nantiferromagnetically, while the monoclinic structure co exists with the C-type AFM\nordering (ferromagnetic chains propagating along the c-axis and antiferromagnetically\ncoupled in the ab-plane).\nDepending on the size of the R3+ions, which control the magnitude of the lattice\ndistortion (particularly, the angles V-O-V, etc.), one can distinguish three types of\nbehavior of RVO3:\n1. The materials containing large ions, such as La3+, Ce3+, Pr3+, and Nd3+, below\nthemagnetictransition pointcrystallize inthemonoclini c structureanddevelopthe\nC-type AFM ordering. The transition to the orthorhombic pha se typically occurs\nright above the N´ eel temperature ( TN) [16,17];\n2. The low-temperature properties of RVO3withR= Sm, Gd, and Tb are marked\nby a coexistence of orthorhombic and monoclinic phases. The monoclinic phase is\nformed above TN. The orthorhombic phase starts to develop near magnetic tra n-\nsition and its proportion gradually increases with decreas ing temperature [18,19,8\n20]. For example, at T= 5 K, the proportion of the orthorhombic fraction is 25 %\nin SmVO 3[18] and about 70 % in GdVO 3[20].\n3. In YVO 3, the monoclinic structure is also formed well above TN= 116 K. With\ndecreasingtemperature,firstC-typeAFMorderingdevelops belowTN,withingiven\nmonoclinic symmetry. Then, at Ts= 77 K the crystal structure suddenly changes\nto the orthorhombic one. The corresponding magnetic struct ure also changes from\nC- to G-type AFM [21]. Similar behavior is observed in YbVO 3.\nDetails of the experimental crystal structure and the magne tic properties of RVO3\ncan be found in [16,17,18,19,21]. In the present work, we wil l use distorted vanadates\nin order to illustrate the main ideas of realistic modeling. There are several reasons\nwhy we have selected RVO3for these purposes:\n1. The microscopic origin of two magnetic structures, which are realized in RVO3, is\nstill under debates. Although main details of the magnetic s tructures can be as-\ncribed to the lattice distortions, which restrict the varia tion of the orbital degrees\nof freedom near some particular configurations [22,23], the role of orbital fluctua-\ntions away from these configurations is also actively discus sed in the literature [24].\nParticularly, the magnetic interactions in these systems a re organized in such a way\nthat the spin degrees of freedom are tightly coupled to the or bital ones, where any\nchange of the orbital configuration affects the magnetic stru cture and vice versa.\nIn the present work, we will illustrate how the computationa l tools, introduced in\nSec. 3, can be used for the analysis of these properties.\n2. The vanadates RVO3are currently regarded as some test compounds for various\ntheories aiming to describe the close interplay among spin, orbital, and lattice\ndegrees of freedom. Therefore, it is very important to test t he ideas of realistic\nmodeling on these systems and to apply them to the whole serie s of compounds\nRVO3with different R. All previous studies were mainly limited by LaVO 3and\nYVO3[12,22,23,25].\n3. We will also use RVO3in order to discuss some prototypical examples of the\n“switching phenomena”, which could be important in applica tions. Particularly,\nwe will show how the magnetic structure of RVO3can be switched by chang-\ning the lattice distortions. An interesting example of swit ching between different\nspin and orbital states in RVO3by optical pulses was considered recently in [20,\n26,27]. Another interesting example is the “magnetic field s witching”, realized in\nDyVO3[28].\n4.1 Parameters of the model Hamiltonian\nIn all the applications, the model was constructed for the t2g-bands, located near the\nFermi level (see Fig. 1).3In vanadates RVO3, each of V-site donates two electrons to\nthese bands.\nThe first important set of parameters is related to the crysta l field (CF), which is\nspecified by the site-diagonal elements ˆtiiof the one-electron Hamiltonian. It breaks\nthe degeneracy of the atomic levels and define the type of occu piedt2g-orbitals in\nthe atomic limit. The interactions, which may act against th e crystal field and deform\n3We did not consider the magnetic effects associated with the r are-earth 4 f-states, which\ncan be also interesting and important [33]. It could be a good subject for a separate study, but\nin the present work, the 4 f-states were treated as non-polarized quasiatomic core sta tes.9\n/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48\n/s32/s67/s114/s121/s115/s116/s97/s108/s45/s70/s105/s101/s108/s100/s32/s83/s112/s108/s105/s116/s116/s105/s110/s103/s32/s40/s109/s101/s86/s41\n/s76/s97/s32/s32/s67/s101/s32/s32/s32/s80/s114/s32/s32/s78/s100/s32/s83/s109/s32/s71/s100/s32/s32/s84/s98/s32/s32/s89/s98/s32/s32/s89/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99\n/s32\n/s83/s109/s32/s71/s100/s32/s32/s84/s98/s32/s32/s89/s98/s32/s32/s89/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99\nFig. 3 The crystal-field splitting of three t2g-levels in monoclinic (left) and orthorhombic\n(right) phases of RVO3. The energies are measured relative to the middle level, cor responding\nto the highest occupied orbital in the atomic limit. The spli tting for two vanadium sublattices\nin the monoclinic phase is shown by different colors, where pa rameters for the less distorted\nsublattice (according to the energy difference between midd le and highest orbitals) are denoted\nby the light blue color, while the ones for the more distorted sublattice – by the dark brown\ncolor.\nthe occupied orbitals are controlled by the intraatomic Cou lomb repulsion U, Hund’s\nrule coupling Jand transfer integrals ˆtijbetween different atomic sites. They will be\ndiscussed below.\nThe scheme of the t2g-level splitting, obtained from the diagonalization of ˆtii, is\nshown in Fig. 3. The physically relevant parameter is the spl itting between middle\nand highest orbitals (corresponding to the highest occupie d and lowest unoccupied\norbitals in the atomic limit). In the most cases this splitti ng is at least 100 meV.\nSomewhat exceptional behavior is observed in the monoclini c phase of LaVO 3and\nin the orthorhombic phases of SmVO 3and GdVO 3, were the characteristic splitting\ncan be as small as 60-80 meV. Some consequences of this behavi or will be discussed\nbelow. In the monoclinic phase, the splitting can be very diff erent in two inequivalent\nV-sublattices, which are denoted as (1,2) and (3,4) in Fig. 2 . For example, in LaVO 3\nthis splitting is 78 meV and 152 meV, respectively. An exampl e of the CF-orbitals,\nobtained after the diagonalization of ˆtiifor SmVO 3, is presented in Table 1.\nAn example of the transfer integrals in the local coordinate frame, corresponding to\nthe diagonal representation of the crystal field at each atom ic site, is shown in Table 2\nfor SmVO 3. The form of these integrals is very different from the one exp ected for the\nundistorted cubic structure.4Thus, the crystal distortions have a profound effect not\nonly on the t2g-level splitting, but also on the form of the transfer integr als. All these\ndetails are very important for the analysis of the low-energ y properties of vanadates.\n4In the cubic structure, the hoppings are allowed only betwee n alike orbitals, lying in the\nplane of the bond. For example, in the z-direction, the hoppings can take place only between\nt2g-orbitals of either yzorzxsymmetry [29].10\nTable 1 Eigenenergies (measured in meV from the middle level) and ei genvectors obtained\nfrom the diagonalization of the crystal-field Hamiltonian ˆtiifor orthorhombic (‘o’) and mon-\noclinic (‘m’) phases of SmVO 3. The eigenvectors are expanded over the basis of xy,yz,z2,\nzx, andx2-y2Wannier-orbitals, in the orthorhombic coordinate frame. P ositions of atomic\nsites are explained in Fig. 2. The eigenvectors for other sit es can be obtained using symmetry\noperations.\nphase site energies orbitals\n−95 0 ,0.22,−0.39,0.84,0.30\n‘o’ 1 0 0 .06,0.78,0.22,−0.28,0.51\n63 0 .32,0.52,0.06,0.17,−0.77\n−65−0.07,0.70,0.30,0.63,−0.14\n‘m’ 1 0 −0.37,−0.05,0.03,0.20,0.91\n142 0 .12,0.62,0.18,−0.72,0.23\n−188 −0.39,0.02,0.04,0.05,0.92\n‘m’ 3 0 0 .12,−0.87,0.30,0.37,0.04\n132 0 .20,0.34,−0.21,0.89,0.04\nTable 2 Transfer integrals (measured in meV) between nearest neigh bors in the orthorhombic\n(‘o’) and monoclinic (‘m’) phases of SmVO 3in the local coordinate frame corresponding to the\ndiagonal representation of the crystal field. Positions of a tomic sites are explained in Fig. 2.\nphase ˆt12ˆt34ˆt13\n‘o’/parenleftBigg\n69 157 4\n2−86−94\n25 25−123/parenrightBigg /parenleftBigg\n69 157 4\n2−86−94\n25 25−123/parenrightBigg /parenleftBigg\n86−53−4\n−53 56−64\n−4−64 12/parenrightBigg\n‘m’/parenleftBigg\n42−29−142\n−36−118−40\n−5 23 −43/parenrightBigg /parenleftBigg\n−121−43−24\n41 22 −12\n−19−124−10/parenrightBigg /parenleftBigg\n54 23 −155\n15−25 74\n−21−139−102/parenrightBigg\nWithout help of the first-principles electronic structure c alculations, it is practically\nimpossible to fix all the parameters in an unambiguous manner .\nThe behavior of screened Coulomb and exchange interactions in thet2g-band is\nillustrated in Fig. 4. For these (purely explanatory) purpo ses, the whole matrices of\nscreened Coulomb interactions were fitted in terms of two Kan amori parameters [14]:\nthe intraorbital Coulomb interaction Uand the exchange interaction J.5The Coulomb\ninteraction Uis slightly larger than 3 eV and somewhat sensitive to the loc al environ-\nment of the V-sites in solids. In orthorhombic systems, Uincreases with the lattice\ndistortion in the direction Sm →Y. In monoclinic systems, this dependence is not so\nobvious. Also in the monoclinic systems, the values of Uare slightly different for the in-\nequivalent V-sublattices. On the other hand, the screening of the exchange interaction\nJis practically insensitive to the local environment, and fo r all considered compounds\nJis close to 0 .62 eV.\nAnother important parameter, which controls the orbital st ate and can compete\nwith the CF splitting, is the energy gain caused by the supere xchange (SE) processes:\nTrL(ˆtijˆtji)/(U−3J) and Tr L(ˆtijˆtji)/U, correspondingly for the ferromagnetically and\nantiferromagnetically coupled bonds [30]. Using the value s of transfer integrals between\nnearest neighbors (Table 2), Coulomb Uand exchange Jinteractions (Fig. 4), this\n5For third Kanamori parameter – the interorbital Coulomb int eraction U′– we used the\nrelation U′=U −2J, which holds for the t2g-states in the cubic environment.11\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s74\n/s32/s75/s97/s110/s97/s109/s111/s114/s105/s32/s80/s97/s114/s97/s109/s101/s116/s101/s114/s115/s32/s40/s101/s86/s41\n/s76/s97/s32/s32/s67/s101/s32/s32/s32/s80/s114/s32/s32/s78/s100/s32/s83/s109/s32/s71/s100/s32/s32/s84/s98/s32/s32/s89/s98/s32/s32/s89/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99\n/s85\n/s74/s85\n/s32\n/s83/s109/s32/s71/s100/s32/s32/s84/s98/s32/s32/s89/s98/s32/s32/s89/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99\nFig. 4Parameters of screened intraorbital Coulomb interaction Uand the exchange interac-\ntionJderived for the t2g-bands of RVO3in monoclinic (left) and orthorhombic (right) phase.\nThe parameters for two different V-sublattices in the monocl inic phase are shown by different\nsymbols: the values corresponding to the more distorted env ironment (according to the scheme\nof the crystal-field splitting in Fig. 3) are shown by circles , while the values corresponding to\nthe less distorted environment are shown by crosses.\nenergy gain can be estimated (very roughly) as 7-40 meV per on e V-V bond. Thus, the\nenergy gain caused by the SE processes is expected to be small er or at least comparable\nwith the CF splitting (Fig. 3).6Thus, there is no predominant mechanism and both\ncrystal field and SE processes can play some role in the format ion of the orbital (and\nrelated to it magnetic) ground state of vanadates RVO3. More detailed analysis will\nbe given in the next sections.\n4.2 Relative stability of magnetic structures: total energ ies\nIn this section we will discuss abilities of theories, which implies that the orbital degrees\nof freedom are frozen in some peculiar configuration by the la ttice distortions, and\nthe magnetic structure simply follow them, basically accor ding to the conventional\nGoodenough-Kanamori [31,32].7\nA typical example of the orbital ordering in shown in Fig. 5 fo r TbVO 3. We have\nselected this material because it has the largest CF splitti ng (Fig. 3) and therefore it\n6Note that since each V-site participate in six bonds, the ene rgy gain recalculated “per\nsite” will be larger. On the other hand, the simple estimates in this section do not take into\naccount the Pauli principle, which forbids the hoppings ont o already occupied orbitals and\nmakes energy gain smaller. Thus, this is indeed only an “orde r or magnitude estimate”.\n7In the present contents, the “Goodenough-Kanamori rules” m eans that the maximal\noverlap between occupied orbitals in some particular bond f avors AFM interactions, while the\nminimal overlap between occupied orbitals (or the maximal o verlap between occupied and\nunoccupied orbitals) favors FM interactions.12\nFig. 5Distribution of charge densities associated with occupied t2g-orbitals (the orbital or-\ndering) realized in the monoclinic (left) and orthorhombic (right) phases of TbVO 3in the\nHartree-Fock approximation. Different magnetic sublattic es associated with opposite direc-\ntions of spins in the C- (left) and G- (right) type antiferrom agnetic ground state are shown by\ndifferent colors.\nis the most appropriate for the analysis in the present conte nts. In the orthorhombic\nphase of TbVO 3, the occupied orbitals have a large overlap in all three dire ctions.\nTherefore, it is reasonable to expect the AFM character of in teractions and the G-type\nAFM ground state. In the monoclinic phase, the overlap betwe en occupied orbitals\nalong the c-axis is considerably weaker. Thus, this orbital configurat ion will favor FM\ncharacter of interactions in the c-direction and the C-type AFM ground state.\nAnother important point of the CF theory is that as long as the orbitals degen-\neracy is lifted by the lattice distortions, there is a hope th at the correct magnetic\nground state can be reproduced already at the level of the HF a pproximation. In some\nsense, the situation is similar to the closed atomic shell, w here the ground state is\nalso non-degenerate and can be described by the single Slate r determinant. Then, if\nnecessary, the correlation interactions can be taken into a ccount on the top of the HF\napproximation, using regular perturbation theory expansi on.\nAll these trends are clearly seen in the total energy calcula tions for the RVO3(R=\nLa-Nd) compounds, which below the magnetic transition temp erature stabilize in only\none – monoclinic structure. Fig. 6 shows the behavior of stab ilization energy (minus\ntotal energy measured relative to the FM state) for the AFM co nfigurations of the A-,\nC-, and G-type.8We note the following:\n1. The HF approximation predicts the correct C-type AFM grou nd state for all four\ncompounds;\n2. The correlation interactions, which were taken into acco unt using regular pertur-\nbation theory expansion near the HF solution for each magnet ic state, additionally\nstabilize the experimentally observed C-typeAFM ordering . Moreover, both second\norder perturbation and the T-matrix theory provide very con sistent explanation for\nthe behavior of correlation energies, although the energie s obtained in the T-matrix\n8The C- and G-type AFM structures are explained in Fig. 2. The A -type AFM structure\nconsists of the FM ab-planes, which are coupled antiferromagnetically along th ec-axis.13\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65/s40/s97/s41\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s40/s98/s41\n/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s40/s99/s41\n/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s40/s100/s41\n/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\nFig. 6Stabilization energies of the main antiferromagnetic stat es in LaVO 3(a), CeVO 3(b),\nPrVO3(c), and NdVO 3(d) measured relative to the ferromagnetic state as obtaine d in the\nHartree-Fock approximation (dark blue area) and after taki ng into account the correlation\ninteractions in the second order of perturbation theory (li ght blue area) and in the framework\nof the T-matrix theory (hatched area).\ntheory for the AFM states are systematically smaller due to t he higher-order cor-\nrelation effects, which are included to the T-matrix, but not to the second-order\nperturbation theory.\nThus, not only the one-electron crystal field, which break th e degeneracy of the HF\nstates in some peculiar way, but also the correlation intera ctions play an important role\nin stabilizing the experimentally observed C-type AFM orde ring in compounds RVO3\n(R= La-Nd). The stabilization energies of the correlation ori gin are systematically\nlarger for the C-type AFM state, irrespectively on the appro ximation employed for\ntreating the correlation interactions.\nHow general are these results?\nLet us consider the next group of compounds RVO3(R= Sm, Gd, and Tb), which\nbelow the magnetic transition temperature exist simultaneously in the monoclinic and\northorhombic phases. The experimental crystal structure p arameters, which have have\nbeen refined for both phases, can be found in [18,19]. In the pr esent study we use the\ndata for T= 5 K. The stabilization energies for several AFM states, cal culated for two\ncrystallographic modifications, are shown in Fig. 7. Their b ehavior is quite consistent\nwith the expectations based on the simplified CF theory: the m onoclinic structure\ntends to stabilize the C-type AFM ordering, while the orthor hombic structure – G-\ntype AFM ordering, although with some exceptions. For examp le, in the orthorhombic\nphase of SmVO 3, the total energy is lower for the C-type AFM state, although the\nnext G-type AFM state is very close in energy (the energy diffe rence between G- and\nC-type AFM states is only 0.8 meV per one V-site). It seems tha t the quasidegeneracy\nof the G- and C-type AFM states is the generic feature of SmVO 3and GdVO 3(but not\nof TbVO 3). In the monoclinic phase of GdVO 3, the C-type AFM state has the lowest\nenergy. However, the next G-type AFM state is only 1.4 meV hig her than the C-type14\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s53/s75/s41/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s53/s75/s41/s40/s97/s41\n/s71/s67\n/s65\n/s32\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s40/s98/s41\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s53/s75/s41/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s53/s75/s41/s71\n/s67\n/s65\n/s32\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s40/s99/s41\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s53/s75/s41/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s53/s75/s41/s71\n/s67\n/s65\n/s32\nFig. 7 Stabilization energies of the main antiferromagnetic stat es in SmVO 3(a), GdVO 3\n(b), and TbVO 3(d) relative to the ferromagnetic state as obtained in the Ha rtree-Fock ap-\nproximation (dark blue area) and after taking into account t he correlation interactions in the\nsecond order of perturbation theory (light blue area) and in the framework of the T-matrix\ntheory (hatched area). Results for the orthorhombic and mon oclinic phases (obtained using\nthe experimental crystal structure at T= 5 K) are shown correspondingly in the left and right\nparts of the figure.\nAFM state. This picture seems to be consistent with the exper imentally observed phase\ncoexistence, which means that there should be several phase s competing in a narrow\nenergy range. For example, the proximity of G- and C-type AFM states in one of the\ncrystallographic phases may indicate at the importance of m agnetic forces in driving\nthe transition to another phase. More specifically, in SmVO 3atT= 5 K, the smaller\northorhombic fraction ( ∼25% of the volume [18]) should be nearly unstable, because\nthe magnetic transition to the C-type AFM state can cause the structural transition.\nOn the contrary, in GdVO 3the smaller volume fraction is monoclinic ( ∼30% atT= 5\nK [20]), which is also nearly unstable, because the change of the crystal structure in\nthis case can be induced by the magnetic transition to the G-t ype AFM state. Such a15\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s40/s97/s41\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s55/s53/s75/s41/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s49/s53/s75/s41/s71\n/s67\n/s65\n/s32\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s40/s98/s41\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s49/s48/s48/s75/s41/s71/s67\n/s32/s83/s116/s97/s98/s105/s108/s105/s122/s97/s116/s105/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s86/s41/s65\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s32/s112/s104/s97/s115/s101/s32/s40 /s84 /s61/s54/s53/s75/s41/s71\n/s67\n/s65\n/s32\nFig. 8Stabilization energies of the main antiferromagnetic stat es in YbVO 3(a) and YVO 3\n(b) relative to the ferromagnetic state as obtained in the Ha rtree-Fock approximation (dark\nblue area) and after taking into account the correlation int eractions in the second order of\nperturbation theory (light blue area) and in the framework o f the T-matrix theory (hatched\narea). Results for the orthorhombic and monoclinic phases a re shown correspondingly in the\nleft and right part of the figure.\ntransition can be caused either by the exchange striction (b asically, the change of the\nlattice parameters associated with the change of the magnet ic state) or bythe change of\nthe orbital structure, which would further minimize the ene rgy of SE interactions [30].\nThe relative role of these two mechanisms will be clarified in the next section. The\nquasidegeneracy of magnetic states in the monoclinic phase of GdVO 3may be also\nresolved through the additional symmetry-breaking transi tion and emergence of a new\nordered phase, as it was suggested in some experimental repo rts [20].\nSimilar to the previous group of compounds, the correlation interactions in RVO3\n(R= Sm, Gd, and Tb) play an important role and additionally stab ilize the “correct”\nmagnetic ground state: C-type AFM for monoclinic systems an d G-type AFM for\northorhombic ones.\nTypical representatives of the last group of compounds are Y bVO3and YVO 3.\nSimilar to the previous case, below the N´ eel temperature, Y bVO3and YVO 3can be\nfound both in orthorhombic and monoclinic modifications. Ho wever, contrary to RVO3\n(R= Sm, Gd, and Tb), these two phases emerges consequently , in two different temper-\nature intervals. For examples, below Ts= 77 K, YVO 3crystallizes in the orthorhombic\nmodification, and above Ts– in the monoclinic one. The construction and solution of\nthe model Hamiltonian for YVO 3have been discussed in details in [12,15,25]. Here we\nsummarize the main results for the total energies. They are s hown in Fig. 8 (together\nwith YbVO 3, which exhibits quite a similar behavior). In this case, the type of the\nmagnetic ground state realized in each temperature range is controlled by the crystals16\nstructure: the low-temperature orthorhombic phase coexis ts with the G-type AFM or-\ndering, while the high-temperature monoclinic phase coexi sts with the C-type AFM\nordering. Each magnetic ground state can be reproduced at th e level of the HF ap-\nproximation and is additionally stabilized by correlation interactions, considered in the\nframework of the second-order perturbation theory as well a s in the T-matrix theory.\n4.3 Interatomic magnetic interactions as a probe of the orbi tal state\nInthis sectionwe will discuss some limitations oftheCF the ories. SincetheCFsplitting\nis finite, the orbital degrees of freedom are not fully quench ed and can be affected by\nother interactions, which generally compete with the cryst al field. Particularly, another\nmajor factor “reshaping” the orbitals is related to the supe rexchange processes [30].\nIn this case, the occupied orbitals will tend to additionall y adjust their form for each\nmagnetic state by minimizing the energy of SE interactions. The magnitude of this\neffect is controlled by the ratio of Tr L(ˆtijˆtji)/Uto the CF splitting.\nThus, the main question, which will be addressed in this sect ion ishow flexible\nare the orbital degrees of freedom in RVO3? A very useful tool for the analysis of\nthis kind of problem is the interatomic magnetic interactio ns (6). Since interatomic\nmagnetic interactions are defined locally, via infinitesimal rotations of spins, they carry\nan information about the details of the orbital state in each magnetic state. Thus, the\nparameters of the Heisenberg model (5) depend on the magneti c state in which they\nare calculated. However, it is not an artifact of calculatio ns or the Heisenberg model\nitself. This dependence has a clear physical meaning and refl ects the changes of the\norbital configuration in each magnetic state.\nThis effect is clearly seen in the behavior of nearest-neighb or magnetic interactions\ninRVO3(R= La, Ce, Pr, and Nd), crystallizing in the monoclinic struct ure (Fig. 9).\nLet us consider first the least distorted compound CeVO 3(similar situation occurs\nin LaVO 3, which was considered in [25]). Distribution of the charge d ensities associ-\nated with the occupied t2g-orbitals (the so-called orbital ordering), which was obta ined\nby minimizing the HF energy for different magnetic states, is shown in Fig. 10. This\ncompound has two V-sublattices: “less distorted” (accordi ng to the values of the CF-\nsplitting – Fig. 3) sublattice (1,2) and “more distorted” on e (3,4). Then, one can see\neven visually that the shape of the orbitals changes dependi ng on the magnetic state.\nAs expected, the orbitals in the “less distorted” sublattic e (1,2) are more flexible. The\nlargest change of the occupied orbitals occurs in the G-type AFM state. All these\ntendencies are reflected in the behavior of interatomic magn etic interactions (Fig. 9),\nwhich tend to additionally stabilize the magnetic state in w hich they are calculated.\nParticularly, the FM interaction J13decreases drastically in the G-type AFM state.\nOn the other hand, the strength of the AFM interaction J12increases. Similar situ-\nation occurs in LaVO 3[25]: due to the rearrangement of the occupied orbitals, not\nonly C-, but also G-type AFM state becomes locally stable (al l nearest-neighbor inter-\nactions are antiferromagnetic in the G-type AFM state, whil eJ12is ferromagnetic in\nthe C-type AFM state). Nevertheless, the C-type AFM state ha s lower energy and is\nrealized as the true magnetic ground state of LaVO 3(Fig. 6). Furthermore, the orbital\nordering in the A-type AFM state of LaVO 3further breaks the monoclinic symmetry\nand makes two inequivalent sublattices in the plane (1,2) (w hich is reflected in the\nsplitting of each of the magnetic interaction J12andJ13in two types – ferromagnetic\nand antiferromagnetic). When the crystal distortion incre ases in the direction LaVO 317\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s74\n/s51/s52/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s76/s97/s86/s79\n/s51\n/s74\n/s49/s51\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s74\n/s51/s52\n/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s67/s101/s86/s79\n/s51\n/s74\n/s49/s51\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s74\n/s51/s52/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s80/s114/s86/s79\n/s51\n/s74\n/s49/s51\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s74\n/s51/s52/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s78/s100/s86/s79\n/s51\n/s74\n/s49/s51\nFig. 9Nearest-neighbor magnetic interactions in LaVO 3, CeVO 3, PrVO 3, and NdVO 3cal-\nculated in the ferromagnetic (F) and in the A-, C, and G-type a ntiferromagnetic states using\nthe expression (6) for the infinitesimal rotations of spins.\n→CeVO3→PrVO3→NdVO3(see Fig. 3), the orbitals become less flexible, and the\nmagnetic interactions in PrVO 3and NdVO 3only weakly depend on the magnetic state\nin which they are calculated. Thus, in the last two compounds , we have more or less\nconventional CF scenario of the orbital ordering and relate d to it interatomic magnetic\ninteractions.\nThe behavior of interatomic magnetic interactions in RVO3(R= Sm, Gd, and\nTb), crystallizing simultaneously in the monoclinic and or thorhombic structures, is ex-\nplained in Fig. 11. As was pointed out in the previous section , one of the interesting\nfeatures in this regime is the quasidegeneracy of the C- and G -type AFM states, re-\nalized in the orthorhombic phase of SmVO 3and in the monoclinic phase of GdVO 3.\nTherefore, we have to address the question whether this quas idegeneracy is related\nto the reconstruction of the orbitals ordering, which will f urther affect the interatomic\nmagnetic interactions and make them specific for each magnet ic state, or to some other\neffects, such as the exchange striction. The typical example of the orbital ordering re-\nalized in the orthorhombic phase of SmVO 3is shown in Fig. 12. The orbital degrees of\nfreedom in these compounds are indeed rather flexible and to s ome extent are able to\nadjust the change of the magnetic state. Nevertheless, one i mportant aspect is that the\norbital ordering does not significantly change between the C - and G-type AFM states.\nIn fact, the orbital ordering pattern in the orthorhombic ph ase of SmVO 3can be of\ntwo types: one type is realized for the C- and G-type AFM state s and another type –18\nFig. 10 Orbital ordering realized in the Hartree-Fock approximati on for the ferromagnetic\n(F) and A-, C-, and G-type antiferromagnetic states of CeVO 3. Different magnetic sublattices\nare shown by different colors.\nfor the F-state and the A-type AFM state. The orbital orderin g within each group of\nstates is practically identical. This behavior ofthe orbit al degrees of freedom is reflected\nin the behavior of interatomic magnetic interactions (Fig. 11): similar to the orbital\nordering, there are two sets of the parameters {Jij}, one of which acts in the F- and\nA-states and another one – in the C- and G-states. The differen ce is mainly reflected\nin the behavior of in-plane interaction J12. Similar situation occurs in the monoclinic\nphase of GdVO 3, where two sets of the in-plane interactions J34are associated with\ndifferent orbital states, realized correspondingly for the FM and A-type AFM ordering\nand for the C- and G-type AFM ordering. Thus, the quasidegene racy of the C- and\nG-type AFM states in SmVO 3and GdVO 3cannot be related to the change of the\norbital state. A more likely scenario is the exchange strict ion, where the change of the\nlattice parameters can easily change the sign of the weak int er-plane interaction J13.\nThe interatomic magnetic interactions for the last two comp ounds, YbVO 3and\nYVO3, which exhibits the consecutive monoclinic-to-orthorhom bic transition with the\ndecrease of temperature are shown in Fig. 13. Details of the o rbital ordering and inter-\natomic magnetic interactions for YVO 3can be found in [25]. In this case, the orbitals\ndegrees of freedom can remain flexible and the interatomic ma gnetic interactions de-\npend on the magnetic states (especially for the orthorhombi c phase of YVO 3). Never-\ntheless, the effect is not so strong as in SmVO 3and GdVO 3. Moreover, the form of the19\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s74\n/s51/s52/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s83/s109/s86/s79\n/s51\n/s74\n/s49/s51\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s74\n/s49/s50/s74\n/s49/s51\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s74\n/s51/s52/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s71/s100/s86/s79\n/s51\n/s74\n/s49/s51\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s74\n/s49/s50/s74\n/s49/s51\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s74\n/s51/s52\n/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s84/s98/s86/s79\n/s51\n/s74\n/s49/s51\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s74\n/s49/s50/s74\n/s49/s51\nFig. 11 Nearest-neighbor magnetic interactions in the monoclinic (left) and orthorhombic\n(right) phases of SmVO 3, GdVO 3, and TbVO 3, calculated in the ferromagnetic (F) and in\nthe A-, C, and G-type antiferromagnetic states using the exp ression (6) for the infinitesimal\nrotations of spins.20\nFig. 12 Orbital ordering realized in the Hartree-Fock approximati on for the ferromagnetic\n(F) and A-, C-, and G-type antiferromagnetic states of ortho rhombic SmVO 3(atT= 5 K).\nDifferent magnetic sublattices are shown by different colors .\norbital ordering and interatomic magnetic interactions, o btained for the C- and G-type\nAFM states in each of thecrystallographic modification, is p ractically the same (similar\nto SmVO 3and GdVO 3). Thus, the orbital fluctuations donot seem tobe thetrigger for\nthe C-G-type AFM transition and the concomitant monoclinic -to-orthorhombic phase\ntransition. The present results are more consistent with th e scenario where the struc-\ntural transition goes first, while the spin and orbital degre es of freedom mainly follow\nthe change of the crystal structure.\n5 Summary and Conclusions\nWe have reviewed the main ideas of realistic modeling of the s trongly correlated sys-\ntems. This is a new direction of the electronic structure cal culations, which is espe-\ncially suit for complex oxides materials. In the present con text, the term “complex”\nmeans the computational complexity (the necessity to deal w ith highly distorted crys-\ntal structures having very low symmetry and many atoms in the unit cell) as well as\nthe methodological complexity (the necessity to go beyond c onventional approxima-\ntions in the electronic structure calculations, such as the local-density approximation).\nThe basic idea of realistic modeling is to construct an effect ive low-energy model for21\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s74\n/s51/s52\n/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s89/s98/s86/s79\n/s51\n/s74\n/s49/s51\n/s74\n/s49/s50/s74\n/s49/s51\n/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s111/s114/s116/s104/s111/s114/s104/s111/s109/s98/s105/s99\n/s109/s111/s110/s111/s99/s108/s105/s110/s105/s99/s74\n/s51/s52/s74\n/s49/s50/s32/s70\n/s32/s65\n/s32/s67\n/s32/s71/s69/s120/s99/s104/s97/s110/s103/s101/s32/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s115/s32/s40/s109/s101/s86/s41/s89/s86/s79\n/s51\n/s74\n/s49/s51\n/s74\n/s49/s50\n/s74\n/s49/s51\nFig. 13 Nearest-neighbor magnetic interactions in the monoclinic (left) and orthorhombic\n(right) phases of YbVO 3and YVO 3, calculated in the ferromagnetic (F) and in the A-, C,\nand G-type antiferromagnetic states using the expression ( 6) for the infinitesimal rotations of\nspins.\nthe states close to the Fermi level, derive all the parameter s rigorously, on the basis of\nfirst-principles electronic structure calculations, and t o solve this model using modern\nmany-bodytechniques. Thus, realistic modeling combines t he accuracy and predictable\npoweroffirst-principleselectronic structurecalculatio ns withtheflexibilityandinsights\nof the model analysis.\nIt is certainly true that approximations are inevitable in s uch an approach (and the\nform of the low-energy model itself is the main approximatio n!). However, we would\nlike to emphasize again that apart from these approximation s, the entire procedure is\nparameter-free. Namely, we do not need anymore to deal with n umerous adjustable\nparameters. Instead, the state of the discussion is brought to a qualitatively new level:\nhow to improve the approximations used for the definition and calculation of the model\nparameters.\nThe abilities of this method were demonstrated on the wide se ries of vanadates\nRVO3(R= La, Ce, Pr, Nd, Sm, Gd, Tb, Yb, and Y) with distorted perovski te struc-\nture. Particular attention was paid to computational tools , which can be used for the\nmicroscopic analysis of different spin and orbital states re alized in the partially filled\nt2g-band.Meanwhile, wewere abletosolveanumberoffundament al andmaterialogical22\nproblems related to the origin of the C- and G-type AFM states in these compounds.\nThe first applications of realistic modeling are very encour aging. We hope that these\nideas will continue to develop in the future to become a power ful tool for theoretical\nanalysis of complex oxide materials and other strongly corr elated systems.\nAcknowledgements I am grateful to G.R. Blake for providing details of the exper imental\ncrystal structure for the series of compounds RVO3. I also wish to thank the support from the\nFederal Agency for Science and Innovations, grant No. 02.74 0.11.0217, during my stay at the\nUral State Technical University (Ekaterinburg, Russia).\nReferences\n1. Norman, M.R.: High-temperature superconductivity in th e iron pnictides. Physics 1, 21\n(2008)\n2. Tokura, Y.: Critical features of colossal magnetoresist ive manganites. Rep. Prog. Phys. 69,\n797 (2006)\n3. Khomskii, D.: Classifying multiferroics: Mechanisms an d effects. Physics 2, 20 (2009)\n4. Imada, M., Fujimori, A., Tokura, Y.: Metal-insulator tra nsitions. Rev. Mod. Phys. 70, 1039\n(1998)\n5. Kohn, W.: Nobel Lecture: Electronic structure of matter: wave functions and density func-\ntionals. Rev. Mod. Phys. 71, 1253 (1999)\n6. Andersen, O.K., Jepsen, O.: Explicit, First-Principles Tight-Binding Theory. Phys. Rev.\nLett.53, 2571 (1984)\n7. Wannier, G.H.: The Structure of Electronic Excitation Le vels in Insulating Crystals. Phys.\nRev.52, 191 (1937)\n8. Solovyev, I.V., Pchelkina, Z.V., Anisimov, V.I.: Constr uction of Wannier functions from\nlocalized atomiclike orbitals. Phys. Rev. B. 75, 045110 (2007)\n9. Marzari, N., Vanderbilt, D.: Maximally localized genera lized Wannier functions for com-\nposite energy bands. Phys. Rev. B. 56, 12847 (1997)\n10. Solovyev, I.V.: First-principles Wannier functions an d effective lattice fermion models for\nnarrow-band compounds. Phys. Rev. B. 73, 155117 (2006)\n11. Aryasetiawan, F., Imada, M., Georges, A., Kotliar, G., B iermann, S., Lichtenstein, A.I.,\nFrequency-dependent localinteractions and low-energyeff ective models fromelectronic struc-\nture calculations. Phys. Rev. B. 70, 195104 (2004)\n12. Solovyev, I.V.: Combining DFT and Many-Body Methods to U nderstand Correlated Ma-\nterials. J. Phys.: Condens. Matter 20, 293201 (2008)\n13. Liechtenstein, A.I., Katsnelson, M.I., Antropov, V.P. , Gubanov, V.A.: Local spin density\nfunctional approach to the theory ofexchange interactions inferromagnetic metalsand alloys.\nJ. Magn. Magn. Matter. 67, 65 (1987)\n14. Kanamori,J.:Electron Correlationand Ferromagnetism ofTransitionMetals. Prog.Theor.\nPhys.30, 275 (1963)\n15. Solovyev, I.V.: Correlation Energies in Distorted 3 d-t2gPerovskite Oxides. J. Exp. Theor.\nPhys.105, 46 (2007)\n16. Zubkov,V.G,Bazuev,G.V.,Perelyaev, V.A.,Shveikin,G .P.:Magnetic structure ofLaVO 3.\nSov. Phys. Solid State 15, 1079 (1973)\n17. Bordet, P., Chaillout, P., Marezio, M., Huang, Q., Santo ro, A., Cheong, S.-W., Takagi, H.,\nOglesby, C.S., Batlogg, B.: Structural Aspects of the Cryst allographic-Magnetic Transition\nin LaVO 3around 140 K. J. Solid State Chem. 106, 253 (1993)\n18. Sage, M.H., Blake, G.R., Nieuwenhuys, G.J., Palstra, T. T.M.: Evidence for Electronic\nPhase Separation between Orbital Orderings in SmVO 3. Phys. Rev. Lett. 96, 036401 (2006)\n19. Sage, M.H., PhD Thesis “Orbital, charge and magnetic ord er ofRVO3perovskites”, 155.\nPrint Partners Ipskamp B.V., Enschede, the Netherlands (20 06)\n20. Yusupov, R.V.,Mihailovic,D.,Colin,C.V.,Blake, G.R. , Palstra,T.T.M.:Criticalphenom-\nena and femtosecond ordering dynamics associated with elec tronic and spin-ordered phases\nin YVO 3and GdVO 3Phys. Rev. B. 81, 075103 (2010)\n21. Blake, G.R., Palstra, T.T.M., Ren, Y., Nugroho, A.A., Me novsky, A.A.: Neutron diffrac-\ntion, x-ray diffraction, and specific heat studies of orbital ordering in YVO 3. Phys. Rev. B\n65, 174112 (2002)23\n22. Sawada, H., Terakura, K.: Orbital and magnetic ordering s in localized t2gsystems, YTiO 3\nand YVO 3: Comparison with a more itinerant egsystem LaMnO 3. Phys. Rev. B. 58, 6831\n(1998)\n23. Fang, Z., Nagaosa, N.: Quantum Versus Jahn-Teller Orbit al Physics in YVO 3and LaVO 3.\nPhys. Rev. Lett. 93, 176404 (2004)\n24. Khaliullin, G.: Orbital Order and Fluctuations in Mott I nsulators. Prog. Theor. Phys.\nSuppl.160, 155 (2005)\n25. Solovyev, I.V.: Lattice distortion and magnetism of 3 d-t2gperovskite oxides. Phys. Rev.\nB.74, 054412 (2006)\n26. Tomimoto, S., Miyazaka, S., Ogasawara, T., Okamoto, H., Tokura, Y.: Ultrafast photoin-\nduced melting of orbital order in LaVO 3. Phys. Rev. B. 68, 035106 (2003)\n27. Mazurenko, D.A., Nugroho, A.A., Palstra, T.T.M., van Lo osdrecht, P.H.M.: Dynamics of\nSpin and Orbital Phase Transitions in YVO 3. Phys. Rev. Lett. 101, 245702 (2008)\n28. Miyasaka, S., Yasue, T., Fujioka, J., Yamasaki, Y., Okim oto, Y., Kumai, R., Arima, T.,\nTokura, Y.: Magnetic Field Switching between the Two Orbita l-Ordered States in DyVO 3.\nPhys. Rev. Lett. 99, 217201 (2007)\n29. Slater, J.C., Koster, G.F.: Simplified LCAO Method for th e Periodic Potential Problem.\nPhys. Rev. 94, 1498 (1954)\n30. Kugel, K.I., Khomskii, D.I.: The Jahn-Teller effect and m agnetism: transition metal com-\npounds. Sov. Phys. Usp. 25, 231 (1982)\n31. Goodenough, J.B., Magnetism and the Chemical Bond, 394. Interscience, New York (1963)\n32. Kanamori, J.: Superexchange interaction and symmetry p roperties of electron orbitals. J.\nPhys. Chem. Solids 10, 87 (1959)\n33. Reehuis,M.,Ulrich,C.,Pattison, P.,Ouladdiaf,B.,Rh einst¨ adter, M.C.,Ohl,M.,Regnault,\nL.P., Miyasaka, M., Tokura, Y., Keimer, B.: Neutron diffract ion study of YVO 3, NdVO 3,\nand TbVO 3. Phys. Rev. B. 73, 094440 (2006)"    },    {        "title": "0804.0001v1.2D_skew_scattering_in_the_vicinity_and_away_from_resonant_scattering_condition.pdf",        "content": "arXiv:0804.0001v1  [cond-mat.mes-hall]  31 Mar 20082D skew scattering in the vicinity and away from resonant sca ttering condition\nV. V. Mkhitaryan and M. E. Raikh\nDepartment of Physics, University of Utah, Salt Lake City, U T 84112\nWe studied the energy dependence of the 2D skew scattering fr om strong potential, for which\nthe Born approximation is not applicable. Since the skew sca ttering cross section is zero both\nat low and at high energies, it exhibits a maximum as a functio n of energy of incident electron.\nWe found analytically the shape of the maximum for an exactly solvable model of circular-barrier\npotential. Within a rescaling factor, this shape is univers al for strong potentials. If the repulsive\npotential has an attractive core, the discrete levels of the core become quasilocal due to degeneracy\nwith continuum. For energy of incident electron close to the quasilocal state with zero angular\nmomentum, the enhancement of the net cross section is accomp anied by resonant enhancement of\nthe skew scattering. By contrast, near the resonance with qu asilocal states having momenta ±1, the\nskew scattering cross section is an oddfunction of energy deviation from the resonance, and passes\nthrough zero, i.e., it exhibits a sign reversal . In the latter case, in the presence of the Fermi sea, the\nKondo resonance manifests itself in strong temperature dep endence of the skew scattering.\nPACS numbers: 72.25.Dc, 73.43.-f, 72.25.Rb, 72.20.Dp\nI. INTRODUCTION\nThe phenomenon of skew scattering has its origin in\na spin-orbit part ∝ˆσ·[k×∇V(ρ)] of the scattering po-\ntential,V(ρ). Due to this part, the scattering ampli-\ntude between the states with momenta kandk′acquires\na contribution, fa∝ˆσ·[k×k′], which is asymmetric\nwith respect to the scattering angle, θ, between kand\nk′. Therefore, in two dimensions, differential cross sec-\ntion acquires an asymmetric contribution\ndσa\ndθ= 2ℜe[(fs)∗fa]∝σsinθ (1)\nwherefsis the symmetric part of the scattering ampli-\ntude, i.e., the scattering amplitude in absence of spin-\norbit potential, and σ=±1 is the spin projection on the\nnormal to the 2D plane.\nSkew scattering, introduced almost 80 years ago1,2,\nhad recently attracted a lot of interest, since it is a key\ningredient of the anomalous Hall effect3,4,5,6,7,8as well\nas of the spin-Hall effect9,10,11,12,13,14,15,16,17. Detailed\ntheoretical calculations of σawere carried out for atomic\nsystems18. Complexity of these calculations stems from\nthe fact that σa≡0 in the lowest Born approximation\nforfs. The results of the second Born approximation for\nfs, yielding a finite σa, are presented in Ref. 18 for the\nmodel of the Thomas-Fermi screening of the charge of a\nnucleus.\nCalculationsofRef. 18wererecentlyutilizedinRef. 12\nto estimate the magnitude of skew scattering by a donor\nimpurity in 2D electron gas. Concerning the energy de-\npendence of the skew-scattering, σa(E), in calculation of\nthe anomalous Hall and spin-Hall effects, it should be\ntaken at E=EF, whereEFis the Fermi energy. Thus,\nthe skew scattering is sensitive to the electron density\nviaEF. Then the question about the explicit form of\nσa(E) in two dimensions arises. Nontriviality of this de-\npendence stems from the fact that at low energy fs(E),\nis dominated by the angular momentum, l= 0. As aEV0\n0b a0ΓEρ\nρV( )\nFIG. 1: Schematic plot of circular-barrier potential with\nheight,V0, and radius, b. Attractive core at ρ < acontains a\nlocalized state with energy, E0, which has a width, Γ, due to\ndegeneracy with continuum.\nresult, the skew-scattering cross section, σa(E), turns to\nzero1atE→0. Since σa(E) turns to zero at large Eas\nwell, we conclude that it should pass through a maximum\nat a certain finite energy.\nStudy of the energy dependence of the skew scattering\nis the focus of the present paper. We demonstrate that\nthis dependence exhibits especially rich behavior for a\n“strong” scattering potential with characteristic magni-\ntude,V0, and characteristic radius, b, satisfying the con-\nditionV0≫/planckover2pi12/mb2, wheremis the electron mass. For\nsuch potentials, the Born approximation does not apply\nat low energies. Then the calculation of the skew scat-\ntering requires the knowledge of scattering phases in the\nabsence of spin-orbit coupling , while the spin-orbit cou-\npling should still be treated perturbatively .\nEven more interesting energy dependence of the skew\nscattering emerges in the case when the scattering po-2\ntential possesses a core, as shown in Fig. 1. Then the\nquasilocal state, E0, due to this core, has a small width,\nΓ. It is known, that spin-independent cross section ex-\nhibits a resonant behavior in the vicinity of E0. We will\ndemonstrate that skew scattering is also resonantly en-\nhanced near E=E0. A particular interesting situation\nrealizes when the quasilocal state at E=E1corresponds\nto nonzero momenta, l=±1. The peculiarity of this\nsituation is that weak spin-orbit-induced splitting of the\nstates with l= 1 and l=−1 can be comparable to Γ.\nThen the dependence, σa(E), is nonperurbative with re-\nspect to spin-orbit coupling. Moreover, we demonstrate\nthat the skew scattering can change sign within a narrow\nenergy interval around E=E1. Finally we show, in the\npresence of the Fermi sea, the Kondo physics, that mod-\nifies the elastic scattering with l= 0 also modifies the\nenergy dependence of the skew scattering near the Fermi\nlevel within the narrow interval ( E−EF) of the order of\nthe Kondo temperature.\nTo the best of our knowledge, the “exact” (without\nutilizing the Born approximation) energy dependence of\nthe 2D skew scattering has been previously studied by\nHankiewicz and Vignale13and by Huang et al.19In both\npapers the study of σa(E) was carried out without ex-\npansion with respect to a weak spin-orbit coupling; the\nresults for the skew scattering are presented in the form\nof numerical plots for certain parameters of the scatter-\ning potential and the magnitude of the spin-orbit cou-\npling. This complicates the inference of general features\nofσa(E). We, on the other hand, made use of the small-\nness of the spin-orbit coupling at the first step . This\nallowed us to obtain closed analytical results for the skew\nscattering as a function of energy. For strong scatter-\ning potential, V(ρ), we show that the energy dependence\nofσais universal within two parametrically wide do-\nmains of low energies before the Born approximation be-\ncomes valid. Characteristics of V(ρ) determine the en-\nergy scales, but not the form of σa(E) in these domains.\nConcerning the resonances, around energies E=E0and\nE=E1of the quasilocal states, σa(E) exhibits a uni-\nversal behavior as well, in the same way as the spin-\nindependent scattering. Since quasilocal states manifest\nthemselves only in the narrow energy intervals around\nE=E0andE=E1, we consider resonant and non-\nresonant scattering separately.\nThe paper is organized as follows. In Sec. II we derive\na general expression for skew-scattering part of the 2D\ntransport scattering cross section. This expression is lin-\near in spin-orbit coupling and contains all the scattering\nphases in the absence of spin-orbit coupling. In Sec. III\nwe extract the energy dependence of the skew scattering\nin different domains for a model circular-barrier poten-\ntial. Resonant skew scattering is studied in detail in Sec.\nIV. In Sec. V. we consider the skew scattering in the\nKondo regime. Concluding remarks are presented in Sec.\nVI.II. GENERAL FORMALISM\nAs a result of the spin-orbit term\nˆHso=λσ1\nρdV(ρ)\ndρˆLz (2)\nin the Hamiltonian, where λis the spin-orbit constant\nandˆLz=−i/planckover2pi1∂/∂θis thez-component of the orbital an-\ngularmomentum, we have δl,σ∝negationslash=δl,−σ, whereδl,σare the\nscattering phases in the channel with orbital momentum\nl.\nCharacteristics of the skew scattering, relevant for\ntransport13is\nIa(E) =σ\nπ/integraldisplay2π\n0dθ/parenleftbiggdσa\ndθ/parenrightbigg\nsinθ, (3)\nwheredσa/dθis the asymmetric part of the differential\nscattering cross section\ndσc(E)\ndθ=dσs(E)\ndθ+dσa(E)\ndθ. (4)\nWhileσis explicitly present in Eq. (3), Ia(E) is, in fact,\nσ-independent. Expression for dσa/dθin terms of scat-\ntering phases reads\ndσa\ndθ=i\n2πk/summationdisplay\nl,l′(e2iδlσ−1)(e−2iδl′σ−1)sin[(l−l′)θ].(5)\nEq. (5) yields for Ia(E) the following general relation:\nIa(E) =2σ\nπk/summationdisplay\nl=l′±1isign(l−l′)\n(cotδl,σ−i)(cotδl′,σ+i)(6)\n=σ\nπk/summationdisplay\nl/braceleftBig\nsin[2(δl,σ−δl+1,σ)]−sin[2δl,σ]+sin[2δl+1,σ]/bracerightBig\n.\nObviously, the two last terms ofthe second line ofEq. (6)\nhave vanishing contribution, and we will omit them later\non.\nIt is convenient to separate the spin-dependent and\nspin-independent parts of scattering phases\nδl,σ(E) =δ0\nl(E)+σδ1\nl(E). (7)\nSpin-orbit correction, δ1\nl(E), can be expressed in terms\nof radial eigenfunctions of the continuous spectrum,\nχl(ρ,E), in the absence ˆHsoas follows\nδ1\nl(E) =2mλl\n/planckover2pi12/integraldisplay∞\n0dρ1\nρdV(ρ)\ndρχ2\nl(ρ,E).(8)\nLarge-ρbehavior of properly normalized χl(ρ,E) is\nχl(ρ,E) =/radicalbiggπρ\n2/bracketleftBig\ncosδ0\nl(k)J|l|(kρ)��sinδ0\nl(k)N|l|(kρ)/bracketrightBig\n.\n(9)3\nSubstitution of Eq. (7) into Eq. (6) and expansion over\nδ1\nlyields\nIa(E) =2\nπk/summationdisplay\nl/parenleftbig\nδ1\nl−δ1\nl+1/parenrightbig\ncos2/parenleftbig\nδ0\nl−δ0\nl+1/parenrightbig\n.(10)\nVia simple transformation of the sum in Eq. (10) and\ninsertion of the expression for δ1\nl, Eq. (8), we find\nIa(E) =4mλ\nπk/planckover2pi12∞/summationdisplay\nl=−∞/integraldisplay∞\n0dρ\nρdV(ρ)\ndρ/bracketleftBig\nlχ2\nl(ρ,E)\n−(l+1)χ2\nl+1(ρ,E)/bracketrightBig\ncos[2(δ0\nl−δ0\nl+1)].(11)\nIn Eq. (11), it is also convenient to make use of the iden-\ntity\ncos2(δ0\nl−δ0\nl−1) = 1−2(tanδ0\nl−tanδ0\nl−1)2\n(1+tan2δ0\nl)(1+tan2δ0\nl−1).\n(12)\nThe advantage of using this identity is that the first term\ndoes not contribute to the sum in Eq. (11), and the ex-\npression for Ia(E) acquires the form\nIa(E)=16mλ\nπk/planckover2pi12∞/summationdisplay\nl=1/integraldisplay∞\n0dρ\nρdV(ρ)\ndρ/bracketleftBig\nlχ2\nl(ρ,E) (13)\n−(l−1)χ2\nl−1(ρ,E)/bracketrightBig(tanδ0\nl−tanδ0\nl−1)2\n(1+tan2δ0\nl)(1+tan2δ0\nl−1).\nThe form Eq. (13) clearly illustrates that, for a weak\nscattering potential, skew scattering is ∝V3. Indeed,\nwhen the scattering phases are small, the last fraction\nin Eq. (13) can be replaced by ( δl−δl−1)2∝V2,\nwhile for χl(ρ) one can use the free radial wave func-\ntionsχ0\nl(ρ,E) = (πρ/2)1/2Jl(kρ), Another power of V\ncomes fromdV(ρ)\ndρ. Eq. (13) is also convenient for the\nanalysis of the energy dependence of the skew scattering\nfrom a strong potential. We perform this study in the\nnext Section.\nIII. CIRCULAR-BARRIER POTENTIAL\nIt is a textbook knowledge20that spin-independent\nscattering cross section from a strong potential, V≫\n/planckover2pi12/mb2, has different forms in three energy intervals:\n(i) low-energy scattering, E≪/planckover2pi12\nmb2, in which only zero\norbital momentum, l= 0, contributes to the cross sec-\ntion;\n(ii) intermediate energies (semiclassical regime),/planckover2pi12\nmb2≪\nE≪V/bracketleftBig\nV//parenleftBig\n/planckover2pi12\nmb2/parenrightBig/bracketrightBig\n, where the scattering cross section is\ndetermined by high l≫1, and finally,\n(iii) high-energy scattering, E≫V/bracketleftBig\nV//parenleftBig\n/planckover2pi12\nmb2/parenrightBig/bracketrightBig\n, which\ncorresponds to the Born approximation.As we will see below, the skew scattering exhibits dif-\nferent universal behaviors in the above three domains.\nWe will see that, within the interval (ii), in contrast to\nthe spin-independent scattering, the skew scattering has\nan additional scale at E∼V0, where it passes through\nthe maximum value. Note that, for strong potential,\nthe energy E∼V0is intermediate between /planckover2pi12/mb2and\nmV2\n0b2//planckover2pi12, which are the boundaries of the interval (ii).\nWe will perform calculations for a model of circular-\nbarrier potential V(ρ) =V0θ(b−ρ). In principle,\nthis model allows to incorporate spin-orbit interaction\nnonperturbatively13. However, we will use the fact that\nthis interaction is weak and evaluate Eq. (13) expanded\nwith respect to the coupling parameter, λ.\nWithin the model V(ρ) =V0θ(b−ρ), the phases δ0\nl\nare found from matching at ρ=bthe expressions for\nχ′(ρ)/χ(ρ) inside and outside the well. This yields\ntanδ0\nl(k) =J′\n|l|(kb)I|l|(νb)−ν\nkI′\n|l|(νb)J|l|(kb)\nN′\n|l|(kb)I|l|(νb)−ν\nkI′\n|l|(νb)N|l|(kb),(14)\nwhere\nν=/radicalbig\nv0−k2, v0= 2mV0//planckover2pi12. (15)\nIt is convenient to rewrite Eq. (14) in a different form\nusing the recurrence relations for Jl(z) andNl(z)\ntanδ0\nl(k) =/parenleftBig\nl\nkb−ν\nkI′\n|l|(νb)\nI|l|(νb)/parenrightBig\nJ|l|(kb)−J|l|+1(kb)\n/parenleftBig\nl\nkb−ν\nkI′\n|l|(νb)\nI|l|(νb)/parenrightBig\nN|l|(kb)−N|l|+1(kb).(16)\nAlso, within a model of a circular well, we havedV\ndρ∝\nδ(ρ−b), so that the general expression Eq. (11) assumes\nthe form\nIa(E) =4λv0\nk(17)\n×∞/summationdisplay\nl=1(tanδ0\nl−tanδ0\nl−1)2\n(1+tan2δ0\nl)(1+tan2δ0\nl−1)/bracketleftbiggl\nZl−l−1\nZl−1/bracketrightbigg\n.\nHere we have introduced a notation\nZ|l|=1+tan2δ0\nl(k)\n/bracketleftbig\nJ|l|(kb)−tanδ0\nl(k)N|l|(kb)/bracketrightbig2,(18)\nso that the spin-orbit correctionsto the scattering phases\nare given by\nδ1\nl=πλlv0\n2Z|l|. (19)\nFrom Eq. (16), we can express Z|l|via the Bessel func-\ntions4\nZ|l|=/parenleftbiggπkb\n2/parenrightbigg2\n\n/bracketleftBigg/parenleftBigg\nl\nkb−ν\nkI′\n|l|(νb)\nI|l|(νb)/parenrightBigg\nJ|l|(kb)−J|l|+1(kb)/bracketrightBigg2\n+/bracketleftBigg/parenleftBigg\nl\nkb−ν\nkI′\n|l|(νb)\nI|l|(νb)/parenrightBigg\nN|l|(kb)−N|l|+1(kb)/bracketrightBigg2\n\n.(20)\nWe start the analysis of Eq. (17) for the strong poten-\ntialV0≫/planckover2pi12\nmb2from the low-energy domain (i), kb≪1.\nIn this domain, spin-independent scattering is dominated\nby a single phase\nδ0\n0≈arctan/bracketleftbiggπ\n2ln(kb)/bracketrightbigg\n, (21)\nwhich follows from z≪1 behavior of N0(z). The phases\nwith higher momenta are much smaller, namely δ0\nl(E)∼\n(kb)2l. On the other hand, parameters Zlgrow rapidly\nwithl,\nZl=v0b2Γ2(l)\nπ2/parenleftbigg2\nkb/parenrightbigg2l\n, (22)\nas follows from Eq. (20) (here Γ( l) is the Gamma-\nfunction). For this reason we can replace the sum in\nEq. (17) by a single term,/parenleftbig\nδ0\n0/parenrightbig2/Z1, so that\nIa(E)/vextendsingle/vextendsingle/vextendsingle\nkb≪1=4λv0\nk/parenleftbig\nδ0\n0/parenrightbig2\nZ1=/parenleftbiggπ2λ\nb/parenrightbiggkb\nln2(kb).(23)\nFirst we emphasize that, unlike σs, the low-energy skew\nscatteringisgovernedby twophases,δ0\n0andδ0\n1; the latter\nphase enters into Z1, as can be seen from Eq. (18). This\nis a natural consequence of the form Eq. (2) of the spin-\norbit Hamiltonian. We also note that, compared to σs,\nwhich behaves as σs(E)∝1/ln2(kb), Eq. (23) contains\nanadditionalfactor( kb)∝E1/2, growingrapidlywith E.\nAlthough derived for particular model of a circular-well\npotential, this low- Eresult is universal within a factor\nfor a general “strong” potential, v0b2≫1.\nNext we consider the energy domain 1 ≪kb≪v1/2\n0b,\nwhich belongs to (ii). In this domain, the following sim-\nplifications are possible. Firstly, since νb≈v1/2\n0b≫kb,\nwe can replace I′\nl(νb)/Il(νb) in Eq. (20) by 1. Such a re-\nplacement is valid only for l≪νb. However, the relevant\nrange of momenta in this domain is narrower, namely\nl < kb≪νb. This is because, for l < kb, the Bessel\nfunctions, Jl(kb) andNl(kb) oscillate, while for l > kb\nthey behave as Jl(kb)∼(kb/l)landNl(kb)∼(kb/l)−l.\nThis behavior suggests that, tan δ0\nlfalls off very rapidly\noncelexceedskb, and the sum over lin Eq. (17) should\nbe terminated21atl=kb. Note now, that in the interval\n1< l < kb, the ratio l/(kb) in Eq. (16) can be neglected\ncompared to ν/k. This yields the following simplified\nexpression for the phases, δ0\nl(E),\ntanδ0\nl=−tan/braceleftbigg\nkb+arctan/parenleftBigν\nk/parenrightBig\n−πl\n2−π\n4/bracerightbigg\n.(24)On the other hand, neglecting l/(kb) in Eq. (20) leads us\nto the conclusion that in the interval 1 < l < kb allZl\nare equal to each other and are equal to\nZl=πkb\n2/parenleftbigg\n1+ν2\nk2/parenrightbigg\n=πv0b\n2k. (25)\nThen the combination [ l/Zl−(l−1)/Zl−1] simplifies to\n2k/(πv0b). Lastly, we notice that δ0\nlandδ0\nl−1are related\nas\nδ0\nl−δ0\nl−1=π/2. (26)\nThis follows from Eq. (24). Using the latter relation, it\nis easy to see that all combinations containing phases in\nfront of the square brackets in the sum Eq. (17) are equal\nto 1. Thus, the sum over lin Eq. (17) is simply equal to\n2k2/(πv0), and the result for Ia(E) reads\nIa(E)/vextendsingle/vextendsingle/vextendsingle\n1≪kb≪√v0b=8λ\nπbkb. (27)\nWe see that the two asymptotes Eq. (23) and Eq. (27)\nmatch at kb∼1. In both limits Ia(E) increases essen-\ntially as ∝E1/2withE. Curiously, the magnitude of\npotential, v0,drops out from the asymmetric part of the\nscattering rate in both limits.\nThe growth of Ia(E) with energy is terminated at\nE∼V0, i.e., at kb∼νb, which also belongs to do-\nmain (ii). This can be seen from Eqs. (16), (20) in the\nfollowing way. For l∼kb∼νb≫1 we can use the\nasymptotic expression/bracketleftbig\nI′\nl(z)/Il(z)/bracketrightbig\n≈/parenleftbig\n1+l2/z2/parenrightbig1/2for\nthe ratio I′\nl(νb)/Il(νb). Then Eq. (20) yields\nZl=πkb\n2\n1+/parenleftBigg\nl\nkb−ν\nk/radicalbigg\n1+l2\nν2b2/parenrightBigg2\n.(28)\nUsing the same asymptotic form in the expression\nEq. (16) for phases, δ0\nlamounts to replacement of ν/k\nby/parenleftbig√\nν2b2+l2−l/parenrightbig\n/(kb) in the argument of the arctan-\ngent in Eq. (24). Still, the difference ( δ0\nl−δ0\nl−1) remains\nπ/2 with accuracy of a small parameter 1 /(kb). This\nis sufficient to replace by 1 all combinations containing\nphasesinfrontofthesquarebracketsinthe sumEq.(17).\nUpon substituting Eq. (28) into Eq. (17), the summation\noverlcan be easily performed, and we obtain\nIa(E) =4λv0\nkkb/summationdisplay\nl=1/bracketleftbig\nl/Zl−(l−1)/Zl−1/bracketrightbig\n(29)\n=4λv0\nkkb\nZkb=8λ√v0\nπ√v0/k\n1+/parenleftbig\n1−√v0/k/parenrightbig2.5\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50\n/s120/s73/s32/s97/s32\nFIG. 2: (Color online) Intermediate energy domain:\n/planckover2pi12/(mb2)≪E∼V0≪mV2\n0b2//planckover2pi12. The skew-scattering\ncontribution, Ia(E), to the scattering rate, in the units\n8λ√v0/π, is plotted from Eq. (29) versus dimensionless en-\nergyx=E/V0.\nNow we see that Ia(E) reaches a maximum at k=\n(v0/2)1/2, which corresponds to E=V0/2. The maxi-\nmal value is equal to\nIa(V0/2) =4(√\n2+1)\nπλ√v0. (30)\nFor smaller energies, Eq. (29) reproduces the result\nEq. (27). Although Eq. (29) was derived for E < V 0,\nit remains applicable also for above-barrier scattering,\nE > V 0. It is seen that, after passing the maximum,\nIa(E) fallsoffas E−1/2. Theenergydependence of Ia(E)\nis illustrated in Fig. 2.\nWe note that the position of maximum in Ia(E) is\nmodel-dependent, in the sense, that for a general poten-\ntial with radius, ∼b, and magnitude, ∼V0, the position\nof maximum can differ from V0/2 by a numerical fac-\ntor∼1. However, the existence of maximum in Ia(E),\nfollowed by ∝E−1/2decrease, is model-independent.\nWithin ascalingfactor, Eq.(29) applies up tothe bound-\nary,E∼mV2\n0b2//planckover2pi12, of the domain (iii), where the Born\napproximation applies. From E∼V0to this boundary\nIa(E) drops by a large factor ∼√v0b2. In the high-\nenergy tail (iii), the behavior of Ia(E) depends strongly\non the shape of the potential.\nIV.Ia(E)IN THE PRESENCE OF QUASILOCAL\nSTATES\nThe simplest model in which the quasilocal states\nemerge, is the repulsive scattering potential with attrac-\ntive core, as shown in Fig. 1. Quasilocal state does not\naffect the scatteringprocesswhen the deviationof energy\nof the incident electron from the resonance exceeds thewidth, Γ, of the quasilocal state. For the model poten-\ntial Fig. 1 the calculation of the width (with prefactor)\nis presented in the Appendix. Below we consider skew\nscattering near the resonance for two particular cases:\n1. Resonance for zero angular momentum, l= 0. In\nthis case, the phase δ0(E) changes by πasEis swept\nacross the resonance. As we have seen above, in the\nlow-energy limit, kb≪1, the phases δ0\nlfall off rapidly\nwithl. In calculating Ia, the phases, δ±1(E), should be\nretained, since there is no skew scattering without them.\nAll phases δl(E) with|l| ≥2 can be neglected.\n2. Resonance for angular momenta, l=±1. Now the\nphasesδ±1(E) exhibit resonant behavior. One has to\nretain the phase δ0(E); all phases δl(E) with|l| ≥2 can\nbe neglected.\nCases 1 and 2 are dramatically different because ˆHso\ndoes not split the level l= 0, but doessplit levels l=±1.\nFor this reason, in the case 1, δ1\n±1(E) can be calculated\nperturbatively from Eq. (8), so that for δ0we have\nδ0=−arctan/bracketleftbiggΓ0\n2(E−E0)/bracketrightbigg\n. (31)\nIn the case 2, δ0is non-resonant and is still not affected\nbyˆHso, whilebothscattering phases δ1andδ−1exhibit\na resonant behavior\nδ1,σ=δ−1,−σ=−arctan/bracketleftbiggΓ1\n2(E−E1−δE1,σ)/bracketrightbigg\n.(32)\nNowwehavetoexpressthesplitting δE1,±σthrough ˆHso.\nδE1,σ≡σδE1=λσl/integraltext∞\n0dρdV(ρ)\ndρ/braceleftbig\nR0\n1(ρ,E1)/bracerightbig2\n/integraltext∞\n0dρρ/braceleftbig\nR0\n1(ρ,E1)/bracerightbig2,(33)\nwhereR0\n1(ρ,E) is the radial wave function of the discrete\nstate, i.e., the coupling to continuum is neglected. Im-\nportantis that the width, Γ1, ofthe state E1is unaffected\nby the spin-orbit interaction.\nNow we evaluate Eq. (3) for the cases 1 and 2. Upon\nneglecting δlwith|l| ≥2, we obtain from Eq.(6) a sim-\nplified form of Ia(E)\nIa(E) =σ\nπk/bracketleftbig\nsin(2δ1,σ)+sin2( δ0−δ1,σ) (34)\n+sin2(δ−1,σ−δ0)−sin(2δ−1,σ)/bracketrightbig\n=4\nπksin(δ1−δ−1)sin(δ0−δ1−δ−1)sinδ0.\nSinceIais independent of σ, in the last identity we had\ndropped the subindex, σ, in the scattering phases. In\nthe case 1, the phases δ1andδ−1are small, and Eq. (34)\nassumes the form\nIa(E) =4\nπk(δ1−δ−1)sin2δ0 (35)\n≈1\nπk(δ1−δ−1)Γ2\n0\n(E−E0)2+Γ2\n0/4.6\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s50 /s52 /s54 /s56\n/s45/s48/s44/s50/s45/s48/s44/s49/s48/s44/s49/s48/s44/s50/s32\n/s32/s120/s73/s32/s97\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s48/s46/s49\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s49\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s50/s46/s50/s53\nFIG. 3: (Color online) The shape of Iafor small E,E1≪\n/planckover2pi12/(mb2), is plotted from Eq. (37) versus dimensionless devi-\nationx= 2(E−E1)/Γ1from the resonance, for three different\nvalues of dimensionless spin-orbit splitting, δE1/Γ1.\nThe difference ( δ1−δ−1) is proportional to spin-orbit\nconstant, λ, and can be evaluated from Eqs. (19) and\n(22), namely,\nδ1−δ−1=π3λ\n4b2(kb)2. (36)\nEq. (35) indicates that in the case 1 the energy depen-\ndence of the skew scattering is the same as the energy\ndependence of σs(E), i.e., it exhibits a resonant enhance-\nment near E=E0.In the case 2, δ0is small and nonresonant, while δ1\nandδ−1have resonances at E=E1±δE1. Upon using\nEq. (32), we find\nIa(E)≈ −4δ0\nπk(37)\n×Γ2\n1(E−E1)δE1\n[(E−E1−δE1)2+Γ2\n1/4][(E−E1+δE1)2+Γ2\n1/4].\nOurprimeobservationisthat,inthecase2,thebehaviors\nofσs(E) andIa(E) are vastly different. While σs(E) is\n∝Γ2\n1//bracketleftbig\n(E−E1)2+Γ2\n1/4/bracketrightbig\n, the skew scattering is an odd\nfunctionof( E−E1). Actualshapeof Ia(E)isdetermined\nby the relation of two small energies δE1and Γ 1. For\nδE1≪Γ1, position of a maximum (and minimum) and\nthe width of Ia(E) are all ∼Γ1. In the opposite limit,\nδE1≫Γ1, the extrema are located at ( E−E1) =±δE1,\nwhile the width ∼Γ1is much smaller. Behavior of the\nskew scattering near the resonance is illustrated in Fig. 3\nfor both limits.\nAt this point we emphasize a significant difference be-\ntween the two cases. In the case 1 we have Ia∝dV/dρ\natρ=bwhile in the case 2, the characteristic scale δE1\nis∝dV/dρatρ=a.\nIn the end of this Section we consider the case when\nthe resonance E=E1falls into the domain of “semi-\nclassical” scattering, 1 ≪kb/lessorsimilarv1/2\n0b. Nonresonant skew\nscattering is described by Eq. (27) in this domain and\nrequires taking into account all the angular momenta up\ntol=kb. For the same reason, in calculating resonant\nskew scattering, one has to keep all four terms in the sum\nEq. (6) which contain either δ1orδ−1. We have\nδIar(E)=1\nπk/bracketleftbig\nsin2(δ1−δ2)+sin2( δ0−δ1)+sin2( δ−1−δ0)+sin2( δ−2−δ−1)/bracketrightbig\n(38)\n=2\nπk/bracketleftbig\nsin(δ1−δ−1−δ2+δ−2)−sin(δ1−δ−1)/bracketrightbig\ncos(δ1+δ−1−2δ0) = (δ1\n−2−δ1\n2)/bracketleftbig\ncos2(δ1−δ0)+cos2( δ−1−δ0)/bracketrightbig\n.\nIn the second identity we used the fact that δ2+δ−2≈\n2δ0\n0, which follows from Eq. (24); in the third identity\nwe replaced sin( δ2−δ−2), which is nonzero only due to\nspin-orbit-induced corrections, by/parenleftbig\nδ1\n2−δ1\n−2/parenrightbig\n≈4λk/b.\nWe note now, that the nonresonant parts of differences\n(δ1−δ0) and (δ−1−δ0) areπ/2 and−π/2, respectively\n[see Eq. (26)], while the resonant parts of δ1andδ−1are\ngiven by Eq. (32). This leads us to the final result\nδIar(E)≈4λ\nπb/braceleftbigg(E−E1−δE1)2−Γ2\n1/4\n(E−E1−δE1)2+Γ2\n1/4(39)\n+(E−E1+δE1)2−Γ2\n1/4\n(E−E1+δE1)2+Γ2\n1/4/bracerightbigg\n.It is seen from Eq. (39) that the characteristic mag-\nnitude of the resonant skew scattering contribution is\nδIar(E)∼λ/b. On the other hand, the backgroundvalue,\nIa(E)∼λk[seeEq.(27)], ismuch larger. Thisis because\nthe background value is the sum of kb≫1 contribu-\ntions. Although δIar(E) constitutes a small correction, it\nhas a lively energy dependence. This dependence is illus-\ntrated in Fig. 4 for different values of dimensionless ratio\nδE1/Γ1. We see that, as δE1/Γ1increases, the structure\ninδIar(E) crossesoverfromone minimum to twominima.7\n/s45/s49/s48 /s45/s53 /s53 /s49/s48\n/s45/s49/s44/s53/s45/s49/s44/s48/s45/s48/s44/s53/s48/s44/s53/s49/s44/s48/s49/s44/s53/s50/s44/s48\n/s120/s73\n/s114/s97\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s48/s46/s49\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s48/s46/s52/s51\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s48/s46/s56\nFIG.4: (Color online)Theshape oftheresonantcontributio n\nto the skew-scattering part of the scattering rate, δIar, in the\ndomain, /planckover2pi12/(mb2)≪E/lessorsimilarV0, is plotted from Eq. (39) versus\ndimensionless deviation, x= 2(E−E1)/Γ1, for three different\nvalues of dimensionless ratio δE1/Γ1.\nV. MANY-BODY EFFECTS IN THE\nRESONANT SKEW SCATTERING\nAs it was mentioned in the Introduction, the quantity\nrelevant for transport is Ia(E) atE=EF. Then the\nresonance condition, EF=E0, can be satisfied either\nif electron concentration or the potential of the attrac-\ntive core can be controlled. On the other hand, even\naway from resonance, at EF> E0, the symmetric part\nof scattering cross section, σs(EF), experiences a strong\nenhancement when the temperature, T, is lower than the\nKondo temperature, TK. This prominent many-body ef-fect stems from the Hubbard repulsion of two electrons\nin the quasilocal level.\nSpin-orbit interaction does not lift the degeneracy of\nthe ground state, and thus, does not affect σs(EF). On\nthe other hand, since the skew scattering is essentially\ndue to interference of l= 0 and l=±1 scattering chan-\nnels, one should expect the Kondo enhancement of σa\nalongside with enhancement of σs.\nQuantitatively, this enhancement can be found in the\nfollowing way. Kondo effect modifies Eq. (35). Namely,\nthe scattering phase, δ0, should be replaced by the\ntemperature-dependent Kondo phase, δ0(T). The low-\nTand high- Tasymptotes of δ0(T) are the following\nsin2δ0=\n\n1−π2/parenleftBig\nT\nTK/parenrightBig2\n, T≪TK,\n3π2\n161\nln2(T/TK), T≫TK,(40)\nEq. (40) describes how Iadrops with increasing tem-\nperature from its “unitary” value Ia=π2λkF, wherekF\nis the Fermi momentum, to the background value, given\nby Eq. (23) with k→kF.\nA very lively behavior of the skew scattering emerges\nwhen the quasilocal level, E1, is close to the Fermi level,\nwhileE0is well below the Fermi level. Then the Kondo\nresonancedevelops in l= 0 channel at T < T K, while the\nphasesδ±1(EF) are strongly sensitive to the deviation\n(E1−EF).\nA particularly interesting issue is what happens to the\nsign reversal of Ia, found in the previous Section. In\nthis case, both δ0andδ±1are resonant; sin δ0is defined\nby Kondo resonance Eq. (40), whereas δ±1are given by\nEq. (32). Upon setting E=EFin Eq. (32) and substi-\ntuting it into Eq. (34), we obtain\nIa(E1,EF,T)≈4δE1Γ1\nπksin2δ0(T)Γ1(EF−E1)cotδ0(T)+/bracketleftbig\n(EF−E1−δE1)(EF−E1+δE1)−Γ2\n1/4/bracketrightbig\n/bracketleftbig\n(EF−E1−δE1)2+Γ2\n1/4/bracketrightbig/bracketleftbig\n(EF−E1+δE1)2+Γ2\n1/4/bracketrightbig.(41)\nAt “high” temperature, δ0is small and we return to\nEq. (37). However, deep in the Kondo regime, the be-\nhavior of Ia(EF) changes drastically. As it is seen from\nEq.(41), the dominantcontributionto Iacomesfromthe\nsecond term in the numerator, which is evenfunction of\n(EF−E1), unlike the first term, which is odd. Therefore,\nupon decreasing temperature, the curve, Ia(EF), evolves\nfrom asymmetric to symmetric.\nForδ0=π/2, the shape of IaversusEF, which is pro-\nportional to the electron density, is shown in Fig. 5. We\nsee that symmetric shape undergoes a strong transfor-\nmation as the splitting δE1/Γ1increases. For each valueof splitting there are twopoints of the sign change posi-\ntioned symmetrically with respect to the point EF=E1.\nVI. CONCLUDING REMARKS\nThe Lorentzian shape of Ia(E), described by Eq. (35),\nsuggests that it should be accompanied by the Fano\nfeature22, sinceIa(E) must assume its nonresonantvalue\naway from E=E0. The issue of Fano resonance near\nE=E1is more delicate. As can be seen from Eq. (37)\nat the Fano-resonance condition δ1≈δ−1≈δ0/2 the8\n/s45/s49/s53 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s53 /s49/s48 /s49/s53\n/s45/s48/s44/s56/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50/s48/s44/s50\n/s32/s32/s120/s73/s32/s97\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s48/s46/s50\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s49/s46/s53\n/s32 /s69\n/s49/s47\n/s49/s32/s61/s32/s52\nFIG. 5: (Color online) The shape of Iain the Kondo regime,\nT≪TK, and resonance with quasilocal level, E=E1, is\nplotted from Eq. (41) versus dimensionless deviation x=\n2(E1−EF)/Γ1ofE1from the Fermi level for three values\nof the dimensionless spin-orbit splitting, δE1/Γ1, of the level\nE=E1.\nfactor sin( δ1−δ−1) turns to zero. This observation can\nbe interpreted as “cancellation” of Fano resonances due\nto interference in l= 1 and l=−1 channels. Simi-\nlar “cancellation” was pointed out in Refs. 23,24. In\nthese papers the photo-current, caused by infrared ex-\ncitation of electron either from impurity into the con-\nduction band23or between two Zeeman subbands24of\nInSbin a strong magnetic field, was studied experimen-\ntally. In fact, there is a general similarity between the\nsine reversal of the skew scattering with energy and the\nsign reversal of photocurrent as a function of magnetic\nfield, observedin Refs.23,24, theunderlyingreasonbeing\nthe interference of a non-resonant and twosplit resonant\nchannels.\nVII. ACKNOWLEDGEMENT\nThis work was supported by NSF under Grant No.\nDMR-0503172.\nAPPENDIX A\nIn this Appendix we derivean analyticalexpressionfor\nthewidthofa2Dquasilocalstate. TheSchr¨ odingerequa-\ntion for the radial part of the wave function, Rl(ρ,E), in\na azimuthally-symmetric potential, V(ρ), reads\nR′′\nl+1\nρR′\nl+/parenleftbigg2m\n/planckover2pi12/bracketleftBig\nEl−V(ρ)/bracketrightBig\n−l2\nρ2/parenrightbigg\nRl= 0.(A1)Upon substitution\nRl(ρ,E) =χl(ρ,E)√ρ, (A2)\nEq. (A1) acquires a Hermitian form\nχ′′\nl+/parenleftbigg2m\n/planckover2pi12/bracketleftBig\nEl−V(ρ)/bracketrightBig\n−l2−1/4\nρ2/parenrightbigg\nχl= 0.(A3)\nConsider now the potential, depicted in Fig. 1 with\nV(ρ) = 0 for ρ > b. To calculate the width, Γ, we con-\nsider an auxiliary potential, ˜V(ρ), which coincides with\nV(ρ) forρ < band is a constant ˜V(ρ) =V0forρ > b.\nLocalized state, E=˜Elin this potential is stationary.\nDenote with ˜ χl(ρ) the corresponding radial wave func-\ntion, so that\n˜χ′′\nl+/parenleftbigg2m\n/planckover2pi12/bracketleftBig\n˜El−˜V(ρ)/bracketrightBig\n−l2−1/4\nρ2/parenrightbigg\n˜χl= 0.(A4)\nUpon multiplying Eq. (A3) by ˜ χland Eq. (A4) by χl,\nsubtracting and integrating from atobwe obtain the\nrelation\n/bracketleftbig\n˜χlχ′\nl−χl˜χ′\nl/bracketrightbig/vextendsingle/vextendsingle/vextendsingleb\na=2m\n/planckover2pi12(˜El−El)b/integraldisplay\nadρχl˜χl.(A5)\nThe fact that the difference ( ˜El−El) is much smaller\nthanElallows us to keep in the left-hand side only the\ncontribution from ρ=b. Then it is convenient to rewrite\nEq. (A5) as\n/bracketleftbiggχ′\nl\nχl(b)−˜χ′\nl\n˜χl(b)/bracketrightbigg\n=2m(˜El−El)\n/planckover2pi12χl(b)˜χl(b)b/integraldisplay\nadρχl(ρ)˜χl(ρ).(A6)\nTo find the imaginary part of Elwe use the continuity of\nthe logarithmic derivatives of χlatρ=b\nχ′\nl\nχl(b) =1\n2b+kH+′\nl(kb)\nH+\nl(kb), (A7)\nwhereH+\nl(kρ) is the Hankel function corresponding to\noutgoing wave at ρ→ ∞. Since ˜χl(ρ) is real, the second\nterm in the left-hand side ofEq. (A6) does not contribute\ntoℑmEl. The imaginary part of Eloriginates from log-\narithmic derivative of the Hankel function in Eq. (A7).\nFinal expression for Γ l=ℑmElemerges upon setting\nχl(ρ) = ˜χl(ρ) =√ρKl(νρ), where Klis the Macdon-\nald function, which is the solution of Eq. (A4) in the\nbarrier region, and extending the upper limit of integra-\ntion in Eq. (A6) to infinity. Both steps are justified if\nℑmEl≪El. Then we obtain\nΓl(E) =/planckover2pi12\nπm·1\nJ2\nl(kb)+N2\nl(kb)·K2\nl(νb)/integraltext∞\nadρρK2\nl(νρ).(A8)9\nFurther simplification is achieved when the core is wide\nenough, so that νa=/bracketleftbig\n2m(V0−El)a2/bracketrightbig1/2//planckover2pi1is large.\nThen we can use the large- ρasymptote of the Macdonald\nfunction in the integrand in Eq. (A8), which yields\nΓl(E) =2/planckover2pi12\nπmb·νe−2ν(b−a)\nJ2\nl(kb)+N2\nl(kb). (A9)In the opposite limit, νa≪1, the exponent in the width,\nΓl, is exp(−2νb), while the dependence on aenters into\nthe prefactor.\n1N. F. Mott, Proc. R. Soc. A 124, 425 (1929); N. F. Mott\nand H. S. W. Massey, The Theory of Atomic Collisions ,\n(Oxford University Press, 1964).\n2J. Smit, Physica (Amsterdam) 21, 877 (1955); ibid.24, 39\n(1958).\n3R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154\n(1954).\n4P. W¨ olfle and K. A. Muttalib, Ann. Phys. 15, 508 (2006).\n5K. A. Muttalib and P. W¨ olfle, Phys. Rev. B 76, 214415\n(2007).\n6J. Cumings, L. S. Moore, H. T. Chou, K. C. Ku, G. Xiang,\nS. A. Crooker, N. Samarth, and D. Goldhaber-Gordon,\nPhys. Rev. Lett. 96, 196404 (2006).\n7A.G. AronovandS.Hikami, Phys.Rev.B 41, 9548 (1990).\n8for a comprehensive review, see N. A. Sinitsyn, J. Phys.;\nCond. Mat. 20, 023201 (2008), and references therein.\n9M. I. Dyakonovand V. I. Perel, Phys. Lett. 35, 459 (1971).\n10M. I. Dyakonov and V. I. Perel, Zh. Eksp. Teor. Fiz. 13,\n657 (1971).\n11J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n12H.-A. Engel, B. I. Halperin, and E. Rashba, Phys. Rev.\nLett.95, 166605 (2005).\n13E. M. Hankiewicz andG. Vignale, Phys. Rev.B 73, 115339\n(2006).\n14E. M. Chudnovsky, Phys. Rev. Lett. 99, 206601 (2007).15E. M. Hankiewicz and G. Vignale, Phys. Rev. Lett. 100,\n026602 (2008).\n16E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys.\nRev. Lett. 93, 226602 (2004).\n17for review, see H.-A. Engel, E. I. Rashba, and B. I.\nHalperin, in Handbook of Magnetism and Advanced Mag-\nnetic Materials , ed. by H. Kronm¨ uller and S. Parkin, J.\nWiley and Sons Ltd, Chichester, UK, 2858 (2007).\n18J. W. Motz, H. Olsen, and H. W. Koch, Rev. Mod. Phys.\n36, 881 (1964).\n19H. C. Huang, O. Voskoboynikov, and C. P. Lee, Phys. Rev.\nB67, 195337 (2003).\n20L. D. Landau and E. M. Lifshitz, Course of Theoretical\nPhysics, Vol. III (Butterworth-Heinemann, Oxford, 1964).\n21In fact, the cosine in Eq. (10) crosses over from −1 to 1\nwithin a narrow interval of momenta |l−(kb)| ∼(kb)1/3≪\nkb. This justifies cutting the summation over lin Eq. (17)\natl=kb.\n22U. Fano, Phys. Rev. 124, 1866 (1961).\n23A. P. Dmitriev, S. A. Emel’yanov, Ya. V. Terent’ev, and\nI. D. Yaroshetskii, JETP Lett. 49, 584 (1989).\n24A. P. Dmitriev, S. A. Emel’yanov, Ya. V. Terent’ev, and\nI. D. Yaroshetskii, JETP Lett. 51, 445 (1990)."    },    {        "title": "2401.15302v1.On_the_origin_of_metal_insulator_transitions_in_the_parent_compounds_of_ruthenium_pnictide_superconductors.pdf",        "content": "On the origin of metal-insulator transitions in the parent compounds of\nruthenium-pnictide superconductors\nNiraj Aryal,∗Emil S. Bozin, and Weiguo Yin†\nCondensed Matter Physics and Materials Science Division,\nBrookhaven National Laboratory, Upton, New York 11973, USA\n(Dated: January 30, 2024)\nWe study the interplay of the structural phase transition, flat electronic band dispersion, and\nmetal-to-insulator transition (MIT) in the parent compounds of the Ru-pnictide superconductors\nby using first-principles calculations. Our electron and phonon calculations reveal that Ru(P,As)\nundergo MIT accompanied by orthorhombic to monoclinic distortion at low temperature, but RuSb\nstays orthorhombic and metallic in agreement with the experimental findings. We find that although\nsmall monoclinic distortion can remove the van Hove singularity at the Fermi level, it does not\nimmediately gap out the Fermi surface and a large value of monoclinic distortion is necessary for\na clear MIT suggesting the possibility of an intermediate pseudogapped monoclinic metallic phase.\nFurthermore, we predict a light-induced two-step insulator-to-metal and structural transitions in the\nmonoclinic phases of RuP and RuAs, which can be tested in future ultrafast pump-probe experiments\nas an alternative ideal play ground to VO 2.\nIntroduction. —Understanding the origin of supercon-\nductivity and its interplay with charge-density-wave\n(CDW) instability, band topology, and metal-to-insulator\ntransition (MIT) continues to be an important problem\nin condensed matter physics [1–3]. The discoveries of\nhigh-temperature superconductivity and many compet-\ning phases in layered copper oxides [4, 5] and iron pnic-\ntides (Fe Pn) [6–9] have triggered numerous investigations\nto find superconductivity in similar materials such as lay-\nered nickelates [10–12] and pnictides [13–15] for possible\nhints on the mechanism of high-temperature supercon-\nductivity and its relationship with the interplay of the\ncharge, spin, orbital, and lattice degrees of freedom. To\ndiscern this complex interplay via light-induced transient\nphase transition is of particular interest due to recent ad-\nvancements in ultrafast measurements [16–18]; however,\nit turns out to be highly challenging to even understand\nthe light-induced MIT and melting of CDW in VO 2, es-\npecially concerning the emergence of the instantaneous\nmonoclinic metallic phase [19–24]. Here, we show that\nthe recently discovered ruthenium-pnictide (Ru Pn) fam-\nily of superconductors Ru 1−xRhxAs, Ru 1−xRhxP, and\nRuSb near a CDW-driven MIT [25–30] provides an al-\nternative ideal play ground.\nRuPn(Pn=P, As, Sb) materials crystallize in the\nMnP-type orthorhombic structure with space group of\nPnma (no. 62) at high temperature. Upon cooling, two\nsuccessive MITs have been reported for Ru(P,As). At\nthe first transition temperature T1, these two materials\nundergo metal to pseudo-gap (PG) phase transition and\natT2, there is a transition to the non-magnetic mono-\nclinic insulating phase. The presence of the PG feature\nand dome-shaped superconductivity upon doping is rem-\niniscent of the behavior of Fe Pn[31–33] and cuprate su-\nperconductors [34–36]. In addition, recently Koch et al.\nreported the observation of fluctuating local Ru-trimers\nwell above T1in synchrotron x-ray total scattering mea-surements [37]. This finding, in contrast to VO 2where\nno such evidence of high-temperature fluctuating local\nV-dimers are present [38], might facilitate the emergence\nof the much debated instantaneous monoclinic metallic\nphase in pump-probe experiments as well as could shed\nlight on the PG behaviour. RuSb, on the other hand,\nstays metallic with no structural phase transition and\nis superconducting without doping. Unexpectedly, ex-\nperiments have indicated higher density of states at the\nFermi level in RuSb; hence the finding of monoclincity in\nRuP and RuAs appears to contradict the common notion\nthat higher DOS drives instability [25]. Therefore, it is\nimportant to thoroughly investigate the electronic struc-\nture, the driving mechanism for the CDW transition and\nits relation to the MIT in these family of materials. As\nRu lies one row below Fe in the periodic table, Ru Pnand\nFePncould share similar properties; one the other hand,\nthe 4dvalence electronic states in Ru Pnare expected to\nbe less localized and less magnetic than the 3 dstates in\nFePn, making Ru Pnmore tractable theoretically.\nIn this letter, we perform a systematic study of the\nelectronic bands, phonon bands, and electron-lattice cou-\npling in Ru(P,As,Sb) materials using first-principles cal-\nculations in order to understand the origin of metal-to-\ninsulator transition. We find that the high temperature\northorhombic phase of these systems hosts symmetry-\nenforced multiple Dirac band crossings and multi-fold\nband degeneracies in the vicinity of the Fermi level. For\nRuP and RuAs, the symmetry-enforced degenerate flat\nbands are present right at the Fermi level giving rise to\nthe van Hove singularity whereas the bands are less flat\nand shifted away from the Fermi level for RuSb. The\nmonoclinic distorted phases in Ru(P,As) are energetically\nfavorable compared to the orthorhombic phase whereas\nRuSb prefers orthorhombicity which is consistent with\nexperiments. Furthermore, we predict a light-induced\ntwo-step insulator-to-metal and structural transitions inarXiv:2401.15302v1  [cond-mat.mtrl-sci]  27 Jan 20242\n 0 2 4\n-8-6-4-2 0 2 4DOS (States/eV/f.u.)Energy (eV)orthorhombicmonoclinic 0 1 2-0.5 0 0.5 0 2 4\n-8-6-4-2 0 2 4DOS (States/eV/f.u.)Energy (eV)orthorhombicmonoclinic 0 1 2-0.5 0 0.5(a)(b)\n(c)(d)\nFIG. 1. (a) and (b) Crystal structure of the Ru-pnictides\nfor orthorhombic and monoclinic phases, respectively, shown\nfrom different perspectives (see text for details). (c) and (d)\nDensity of states plot for RuP and RuAs showing comparison\nbetween the orthorhombic and monoclinic phases. The inset\nplots show DOS in ±0.5 eV window around the Fermi level.\nthe monoclinic phases of RuP and RuAs—with an in-\ntermediate monoclinic metallic phase—which can be at-\ntested in future ultrafast pump-probe experiments as an\nalternative ideal play ground to VO 2.\nCrystal structure and lattice relaxation. —Fig. 1(a) and\n(b) shows the crystal structure of Ru-pnictides in both\northorhombic and monoclinic distorted phase. Above T1,\nthe system crystallizes in an orthorhombic Pnma space\ngroup with 4 formula units in an unit cell. The Ru-atoms\nare surrounded by 6 Pn-atoms in a distorted octahe-\ndral environment with face-sharing octahedra along the\naortho-direction and edge-sharing ones along the bortho-\ndirection and bc-plane. A zig-zag chain of nearest neigh-\nbour (nn) Ru-atoms runs along the a-direction. In addi-\ntion, the next nn and third nn Ru-atoms form a triangu-\nlar lattice in the bc-plane. Below T2, P and As members\nof the family undergo orthorhombic to monoclinic distor-\ntion. The unit cell triples both along the orthorhombic c-\ndirection and in the ab-plane thereby increasing the num-\nber of atoms in the monoclinic unit cell by 9 times. The\ntransformation matrix that maps from orthorhombic to\nmonoclinic unit cell is given by V mono =MVortho where,\nV’s are the lattice vectors for the monoclinic and or-\nthorhombic unit cells nciteCommentCrystalConventionandM=\n1 2 0\n−1 1 0\n0 0 3\n.The monoclinic distorted phase is\ncharacterized by trimerization along the bortho-direction\nconsisting of two short (S 1, S2) and one long (L) bonds in\nthe order —LS 1S2LS1S2L—. Such trimerization results\nin tightly bound Ru 6clusters or hexamers. In addition to\nthe trimerization and hexamerization, the zig-zag chain\nalong aorthodirection has a weak modulation of the bond\ndistances with a period of 6.\nFig. 1 (c) and (d) shows the density of states compar-\nison between the orthorhombic and monoclinic distorted\nphase for both RuP and RuAs obtained from Density\nFunctional Theory (DFT) calculation. The orthorhom-\nbic phases show peak DOS at the Fermi level thereby\nsignalling the presence of lattice instability. As we will\nsee below, such peak DOS arises from the presence of\nRu-dxyderived flat bands at the Fermi level. The fully\nrelaxed monoclinic distorted phases correctly reproduces\nthe insulating phase for both systems with gap-size com-\nparable to the experimental findings which is ∼100 meV\nfor RuP [28, 37, 39], in agreement with a recent DFT\nstudy of RuP [30], and ∼30 meV for RuAs [40]. In con-\ntrast, the experimentally refined lattice parameters fail to\nshow the gap opening in both systems within DFT cal-\nculations despite using different flavors of exchange cor-\nrelation functional [37, 40]. It is indeed incredible that\nthe relaxed structures can reproduce the gap-sizes cor-\nrectly despite the well-known issue of underestimation of\ngap-size in plain LDA/GGA calculations, in sharp con-\ntrast to the DFT studies of VO 2, which remains metallic\nin the LDA/GGA and requires additional cares of elec-\ntronic correlation to become an insulator [41–44]. We\nattribute the difference to the general expectation that\nthe 4 dvalence electronic states in Ru Pnare less local-\nized and less magnetic than the 3 dstates in VO 2, making\nRuPnmore tractable in DFT.\nThe lattice parameters and energy differences for var-\nious orthorhombic and monoclinic phases obtained from\nthe DFT calculations are shown in Table S1 of Supple-\nmentary Material [45]. For RuP and RuAs, the DFT\nrelaxed monoclinic phases are lower in energy compared\nto the orthorhombic phases which is consistent with the\nfact that the systems prefer monoclinic distortion at low\ntemperature. For RuSb, our calculation finds that the\nsystem prefers the orthorhombic phase even when the re-\nlaxation with large monoclinic distortion was attempted.\nOur results from phonon calculation which is discussed\nbelow are consistent with the above findings thereby uni-\nfying our discussion. The DFT relaxed lattice parame-\nters and unit-cell volume obtained using GGA exchange-\ncorrelation functional are consistently larger than the ex-\nperimental values due to the well-known delocalization\nproblem resulting in the overestimation of the lattice\nparamaters in GGA. The LDA calculations, on the other\nhand, show decrease in the unit-cell volume due to the3\nEnergy (eV)\nEnergy (eV)Phonon Frequency (THz)\nPhonon Frequency (THz)-4        -3       -2        -1          0         1         2 \n-4        -3       -2        -1          0         1         2 \nFIG. 2. Comparison of the electronic band structure, Fermi surface and phonon band dispersion between the orthorhombic\nphases of RuP and RuSb. Red (blue) lines in panels (a) and (e) are obtained with (without) the inclusion of SOC. Panels (b) and\n(f) show Ru-d band character plots in a small window around the flat bands with the following color convention: black ( dxy),\nred (dz2), green ( dx2−y2), blue ( dxz), and yellow ( dyz). (c) and (f) show Fermi surface plots. The red-green (blue-magenta)\ncolors show the hole (electron) Fermi surface pieces. (d) and (h) show the phonon band dispersion. The colors denote the\nweight of the normal mode atomic displacements along the crystallographic aortho(red), bortho(green) and cortho(blue) direction\nfor Ru-atoms. Black color shows the weight of pnictogen atoms.\nwell-known overestimation of the binding energy prob-\nlem [46]. Despite the opposite trends in the structural\noptimization, both functionals correctly reproduces the\nMIT.\nElectronic bands, Fermi surface and phonon disper-\nsion.—Fig. 2 presents the electronic bands and Fermi sur-\nface plots for RuP and RuSb in the orthorhombic phase.\nSimilar plots for RuAs are presented in the SM [45]. RuP\nand RuAs share similar electronic structure with degen-\nerate flat dispersion close to the Fermi level, however the\nbands are less flat and slightly downshifted for RuSb.\nThe presence of such flat dispersion gives rise to the van\nHove singularity right at the Fermi level. The flat bands\nare derived mainly from Ru- dxyand Ru- dz2orbitals [47].\nThe effect of spin-orbit coupling (SOC) is small; its main\nrole being lifting of the degeneracy of the band cross-\nings throughout the spectrum. However, the flat band\nand peak DOS remains intact [see DOS comparison with\nand without SOC in SM]. Hence, in the following, we\nwill present the results in the absence of the SOC. Note\nthat the presence of non-symmorphic symmetry gives rise\nto band degeneracies and band crossing along different\nhigh symmetry directions. Some of the band crossings\nare topological in the sense that they are unavoidable\nand protected by symmetries. Such phenomena of “band\nsticking” has been studied in detail in other systems be-\nlonging to the same space group [48, 49].\nFigs. 2(c) and (f) show the Fermi surface plots for RuP\nand RuSb, respectively. The FS of RuP consists of a large\ncorrugated cylinder-like electron pocket at the BZ center\nand small dispersionless hole FS sheets at the BZ bound-\nary. RuAs shows similar FS topology. However, the FS\nof RuSb has more structure; the electron FS at the BZcenter is smaller and more elliptical, whereas the hole FS\ncovers a much larger volume with a complicated topol-\nogy. The nesting property of the FS will be investigated\nin the subsequent sections.\nFigs. 2(d) and (h) present the phonon dispersion for\nRuP and RuSb along high symmetry direction. For RuP,\nthere is a clear separation of the phonon modes; the low\nlying modes correspond to the Ru-atom displacement and\nhigh-energy modes are from the lighter P atoms. How-\never, such distinction is less clear in RuAs and becomes\neven more fuzzy for RuSb owing to the larger mass of\npnictogens. Consistent with our conclusions from the\nlattice relaxation calculations and experimental findings,\nwe find that orthorhombic phases of RuP and RuAs give\nimaginary phonon frequencies (soft modes) signalling dy-\nnamical lattice instability. We find four soft phonon\nmodes with Ru-displacements mainly along the zig-zag\ndirection ( bortho). The soft mode at T high symmetry\npoint introduces dimerization between Ru-atoms in the\nzig-zag direction and is more likely to be the driver of\nthe lattice instability. RuSb, on the contrary, does not\nshow such imaginary phonon modes confirming that it’s\northorhombic phase is stable at low temperature.\nIn order to understand more about the potential driv-\ning factors for the CDW instability, we first assess the\nrole of the Fermi surface nesting (FSN) by calculating\nthe real and imaginary part of the bare charge suscepti-\nbility within constant matrix approximation [50, 51] [see\nSM for definition]. The CDW vectors in the orthorhom-\nbic basis are q= (1\n3,1\n3,0) and (0 ,0,1\n3). Hence, if FSN\nis the driving mechanism for the CDW transition, peaks\nare expected in both the real and imaginary parts of the\nsusceptibility at these q-values [51, 52] which correspond4\nFIG. 3. Evolution of the electronic structure as a function of the monoclinic distortion ( x). (a-d) Unfolded band dispersion\nof the monoclinic phases of RuP. (e-h) & (i-l) Unfolded RuP Fermi surface on the kx-kzplane for ky= 0.0 and 0 .5 planes (in\nunits of 2 π/b), respectively. The intensity shows the spectral weights. (m) Total energy cost per formula unit for RuP and\nRuAs. The dashed lines are quadratic fits to the energy. (n) Band gap and DOS at the Fermi level. All phases, except x=0,\nhave monoclinic distortion. See text for details.\nto the Γ-S, Γ-Z and Γ-R directions. However, we find\nthat the peak position of the real and imaginary parts of\ncharge susceptibility has no correspondence to the CDW\nvector which excludes FSN as the driving force for the\nCDW transition (see Fig. S2 in SM [45]). The other\npossible mechanism for CDW transition is the electron-\nphonon coupling (EPC) effect. For this, we compared\nthe relative strength of the electron-phonon coupling con-\nstant for RuP and RuSb. As orthorhombic phase of\nRuP gives imaginary phonon modes, this creates prob-\nlem while calculating coupling constant. However, we\nwere able to stabilize the orthorhombic phase by means\nof electronic smearing [see Fig.S4 in SM [45]]; such elec-\ntronically smeared orthorhombic phases can give an es-\ntimate of the strength of the EPC parameter λ[53, 54].\nFor RuP, we estimate λto be∼1.2, whereas for RuSb,\nwe find λto be less than 1. This hints that EPC could be\nresponsible for the CDW instability in RuP and RuAs.Monoclinic distortion. — To explore the relationship\nbetween the monoclinic distortion and MIT, we study\nthe evolution of the electronic structure and total energy\ncost as a function of the strength of the monoclinic dis-\ntortion by linearly combination of the orthorhombic and\nmonoclinic phases in the ratio of (1 −x) and x, respec-\ntively. Figs. 3(a-d) and (e-l) show the unfolded band\nstructure and Fermi surface plots, respectively, in the\nBZ of the orthorhombic unit cell, for various values of x.\nThe folded band structure in the BZ of the monoclinic\nunit cell along with the orbital character is presented in\nthe SM [45]. We find that the flat bands do not imme-\ndiately gap-out; instead there is rather a large critical\nvalue of monoclinic distortion after which the flat bands\nopen gap. The initial reaction of the system to the mon-\noclinic distortion is to gap-out the Γ-centered electron\nFermi surface [Figs. 3(e-h)] rather than gapping out the\nflat bands located on the ky= 0.5 plane and the flat5\nFIG. 4. Variation of the Ru-Ru nearest neighbour bond dis-\ntance (upper panels) and bond angle (lower panels) along the\ntrimerization direction for (a, b) RuP and (c, d) RuAs as a\npngction of the electronic temperature ( Te). The gray dashed\nlines are guide to the eye. The black dashed line shows the\nMIT.\nbands undergo massive reconstruction. At x≈0.5, the\nFS gaps out completely at ky= 0 plane but does not\ngap out at ky= 0.5 plane showing pseudo-gap like fea-\nture. The DOS at the Fermi surface [Fig. 3(f)] keeps\ndecreasing when monoclinic distortion increases with a\nnoticeable sharp drop at the value of distortion when the\nflat bands gap-out. The total energy gain as a function\nofx[Fig. 3(f)] confirms that the system favors mono-\nclinic distortion. In addition, we find that closer to the\ninsulating phase ( x > 0.4), quadratic equation of form\nE(x)∼ −1\n2gx+1\n2kx2, where gandkdenote the electron-\nlattice ( e-l) and lattice-lattice ( l-l) coupling, respectively,\nyields excellent fit to the data points, suggesting that e-l\ncoupling stabilizes the monoclinic phase [55]. From the\nfits, we find the value of gandkto be of similar order\nfor RuP and RuAs.\nUltrafast laser induced MIT. —Owing to the small\nband-gap in the monoclinic phases of RuP and RuAs, it\nis plausible to induce insulator-to-metal transition (IMT)\nin these systems under ultra-fast laser irradiation simi-\nlar to VO 2[19–24]. In order to mimic the lattice re-\nsponse of the system to the ultrafast high energy high\nfluence pump, we performed DFT lattice relaxation cal-\nculations at several values of electronic temperatures ( Te)\nand studied the variation of the atomic coordinates as\na function of Te[20, 21, 56]. We found that a nomi-\nnalTeof∼0.1 eV is sufficient to induce insulator to\nmetal transition in RuAs whereas for RuP, the critical\nTeis∼0.15 eV owing to its slightly larger band-gap.\nIn Fig. 4, we show the variation of the bond distances\nand bond angles along the trimerization direction as a\nfunction of Tefor RuP and RuAs. The trimer bond dis-tances and bond angles decrease monotonically with Te\nand finally reach the non-trimerized phase for larger val-\nues of Te. However, the IMT happens well before the\nsystem reaches the non-trimerized phase, i.e., the mono-\nclinic insulating phase can enter the monoclinic metallic\nphase—similar to what has been reported in Ref. [22] on\nVO2—with less than 10% decrease in the trimerization\nbond distance and bond angle. This is consistent with\nthe results from linear interpolation which shows simi-\nlar behavior [Fig. 3(n)]. It will be interesting to verify\nthis prediction of two-step light-induced IMT in future\nultrafast pump-probe experiments.\nIn summary, we perform a systematic study of the elec-\ntron, phonon, and electron-lattice interplay in the par-\nent compounds of Ru Pn superconductors using first-\nprinciples calculations. Our calculations reveal a direct\ncorrelation between the coexistence of the flat-bands,\nCDW instability and MIT in this family. The electron\nand phonon calculations find that RuP and RuAs mem-\nbers of the family host flat bands at the Fermi surface and\nundergo monoclinic distortion at lower temperature ac-\ncompanied by MIT, whereas RuSb remains orthorhombic\nand metallic at low temperature which is in agreement\nwith the experimental observation. We predict a two-\nstep insulator to metal and structural phase transition\nin the monoclinic phases of RuP and RuAs which can\nbe attested in future ultrafast experiments.In addition,\nit will be interesting in the future to study the stability\nof the monoclinic phases of RuP and RuAs and the pos-\nsible emergence of monoclinicility in RuSb by means of\nsmall external perturbations like pressure, chemical sub-\nstitution, etc.\nThis work was supported by the U.S. Department of\nEnergy (DOE) the Office of Basic Energy Sciences, Ma-\nterials Sciences and Engineering Division under Contract\nNo. DE-SC0012704.\n∗naryal@bnl.gov\n†wyin@bnl.gov\n[1] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev.\nMod. Phys. 87, 457 (2015).\n[2] A. F. Kusmartseva, B. Sipos, H. Berger, L. Forr´ o, and\nE. Tutiˇ s, Phys. Rev. Lett. 103, 236401 (2009).\n[3] F. H. Yu, D. H. Ma, W. Z. Zhuo, S. Q. Liu, X. K. Wen,\nB. Lei, J. J. Ying, and X. H. Chen, Nature Communi-\ncations 12, 3645 (2021).\n[4] J. G. Bednorz and K. A. M¨ uller, Zeitschrift f¨ ur Physik B\nCondensed Matter 64, 189 (1986).\n[5] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L.\nMeng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W.\nChu, Phys. Rev. Lett. 58, 908 (1987).\n[6] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,\nJournal of the American Chemical Society 130, 3296\n(2008), pMID: 18293989.\n[7] C.-H. Lee, A. Iyo, H. Eisaki, H. Kito, M. Teresa6\nFernandez-Diaz, T. Ito, K. Kihou, H. Matsuhata,\nM. Braden, and K. Yamada, Journal of the Physical\nSociety of Japan 77, 083704 (2008).\n[8] X. H. Chen, T. Wu, G. Wu, R. H. Liu, H. Chen, and\nD. F. Fang, Nature 453, 761 (2008).\n[9] G. F. Chen, Z. Li, D. Wu, G. Li, W. Z. Hu, J. Dong,\nP. Zheng, J. L. Luo, and N. L. Wang, Phys. Rev. Lett.\n100, 247002 (2008).\n[10] D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R.\nLee, Y. Cui, Y. Hikita, and H. Y. Hwang, Nature 572,\n624 (2019).\n[11] G. A. Pan, D. Ferenc Segedin, H. LaBollita, Q. Song,\nE. M. Nica, B. H. Goodge, A. T. Pierce, S. Doyle, S. No-\nvakov, D. C´ ordova Carrizales, A. T. N’Diaye, P. Shafer,\nH. Paik, J. T. Heron, J. A. Mason, A. Yacoby, L. F.\nKourkoutis, O. Erten, C. M. Brooks, A. S. Botana, and\nJ. A. Mundy, Nature Materials 21, 160 (2022).\n[12] H. Sun, M. Huo, X. Hu, J. Li, Z. Liu, Y. Han, L. Tang,\nZ. Mao, P. Yang, B. Wang, J. Cheng, D.-X. Yao, G.-M.\nZhang, and M. Wang, Nature 621, 493 (2023).\n[13] W. Wu, J. Cheng, K. Matsubayashi, P. Kong, F. Lin,\nC. Jin, N. Wang, Y. Uwatoko, and J. Luo, Nature Com-\nmunications 5, 5508 (2014).\n[14] H. Kotegawa, S. Nakahara, H. Tou, and H. Sugawara,\nJournal of the Physical Society of Japan 83, 093702\n(2014).\n[15] J.-G. Cheng, K. Matsubayashi, W. Wu, J. P. Sun, F. K.\nLin, J. L. Luo, and Y. Uwatoko, Phys. Rev. Lett. 114,\n117001 (2015).\n[16] D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst,\nM. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and\nA. Cavalleri, Science 331, 189 (2011).\n[17] R. Mankowsky, A. Subedi, M. F¨ orst, S. O. Mariager,\nM. Chollet, H. T. Lemke, J. S. Robinson, J. M. Glow-\nnia, M. P. Minitti, A. Frano, M. Fechner, N. A. Spaldin,\nT. Loew, B. Keimer, A. Georges, and A. Cavalleri, Na-\nture516, 71 (2014).\n[18] D. Nevola, N. Zaki, J. M. Tranquada, W.-G. Yin, G. D.\nGu, Q. Li, and P. D. Johnson, Phys. Rev. X 13, 011001\n(2023).\n[19] C. Xu, C. Jin, Z. Chen, Q. Lu, Y. Cheng, B. Zhang,\nF. Qi, J. Chen, X. Yin, G. Wang, D. Xiang, and D. Qian,\nNature Communications 14, 1265 (2023).\n[20] J. Li, L. Wu, S. Yang, X. Jin, W. Wang, J. Tao, L. Boat-\nner, M. Babzien, M. Fedurin, M. Palmer, W. Yin, O. De-\nlaire, and Y. Zhu, Phys. Rev. X 12, 021032 (2022).\n[21] S. Wall, S. Yang, L. Vidas, M. Chollet, J. M. Glow-\nnia, M. Kozina, T. Katayama, T. Henighan, M. Jiang,\nT. A. Miller, D. A. Reis, L. A. Boatner, O. Delaire, and\nM. Trigo, Science 362, 572 (2018).\n[22] V. R. Morrison, R. P. Chatelain, K. L. Tiwari, A. Hen-\ndaoui, A. Bruh´ acs, M. Chaker, and B. J. Siwick, Science\n346, 445 (2014).\n[23] D. Wegkamp, M. Herzog, L. Xian, M. Gatti, P. Cudazzo,\nC. L. McGahan, R. E. Marvel, R. F. Haglund, A. Rubio,\nM. Wolf, and J. St¨ ahler, Phys. Rev. Lett. 113, 216401\n(2014).\n[24] P. Baum, D.-S. Yang, and A. H. Zewail, Science 318,\n788 (2007).\n[25] D. Hirai, T. Takayama, D. Hashizume, and H. Takagi,\nPhys. Rev. B 85, 140509 (2012).\n[26] R. Y. Chen, Y. G. Shi, P. Zheng, L. Wang, T. Dong, and\nN. L. Wang, Phys. Rev. B 91, 125101 (2015).\n[27] G.-Z. Fan, R.-Y. Chen, N.-L. Wang, and J.-L. Luo, Chi-nese Physics Letters 32, 077203 (2015).\n[28] S. Li, Y. Kobayashi, M. Itoh, D. Hirai, and H. Takagi,\nPhys. Rev. B 95, 155137 (2017).\n[29] Y. Nakajima, Z. Mita, H. Watanabe, Y. Ohtsubo, T. Ito,\nH. Kotegawa, H. Sugawara, H. Tou, and S.-i. Kimura,\nPhys. Rev. B 100, 125151 (2019).\n[30] D. Hirai, K. Kojima, N. Katayama, M. Kawamura,\nD. Nishio-Hamane, and Z. Hiroi, Journal of the Ameri-\ncan Chemical Society , Journal of the American Chemical\nSociety 144, 17857 (2022).\n[31] Y. Ishida, T. Shimojima, K. Ishizaka, T. Kiss, M. Okawa,\nT. Togashi, S. Watanabe, X. Wang, C. Chen, Y. Kami-\nhara, M. Hirano, H. Hosono, and S. Shin, Journal of the\nPhysical Society of Japan 77, 61 (2008).\n[32] T. Sato, K. Nakayama, Y. Sekiba, T. Arakane,\nK. Terashima, S. Souma, T. Takahashi, Y. Kamihara,\nM. Hirano, and H. Hosono, Journal of the Physical So-\nciety of Japan 77, 65 (2008).\n[33] T. Shimojima, T. Sonobe, W. Malaeb, K. Shinada,\nA. Chainani, S. Shin, T. Yoshida, S. Ideta, A. Fuji-\nmori, H. Kumigashira, K. Ono, Y. Nakashima, H. An-\nzai, M. Arita, A. Ino, H. Namatame, M. Taniguchi,\nM. Nakajima, S. Uchida, Y. Tomioka, T. Ito, K. Kihou,\nC. H. Lee, A. Iyo, H. Eisaki, K. Ohgushi, S. Kasahara,\nT. Terashima, H. Ikeda, T. Shibauchi, Y. Matsuda, and\nK. Ishizaka, Phys. Rev. B 89, 045101 (2014).\n[34] W. W. Warren, R. E. Walstedt, G. F. Brennert, R. J.\nCava, R. Tycko, R. F. Bell, and G. Dabbagh, Phys.\nRev. Lett. 62, 1193 (1989).\n[35] H. Alloul, T. Ohno, and P. Mendels, Phys. Rev. Lett.\n63, 1700 (1989).\n[36] M. R. Norman, D. Pines, and C. Kallin, Advances in\nPhysics 54, 715 (2005).\n[37] R. J. Koch, N. Aryal, O. Ivashko, Y. Liu, M. Abeykoon,\nE. D. Bauer, M. v. Zimmermann, W. Yin, C. Petrovic,\nand E. S. Bozin, Phys. Rev. B 106, 214516 (2022).\n[38] S. A. Corr, D. P. Shoemaker, B. C. Melot, and R. Se-\nshadri, Phys. Rev. Lett. 105, 056404 (2010).\n[39] D. Ootsuki, K. Sawada, H. Goto, D. Hirai, D. Shi-\nbata, M. Kawamoto, A. Yasui, E. Ikenaga, M. Arita,\nH. Namatame, M. Taniguchi, T. Toriyama, T. Konishi,\nY. Ohta, N. L. Saini, T. Yoshida, T. Mizokawa, and\nH. Takagi, Phys. Rev. B 101, 165113 (2020).\n[40] H. Kotegawa, K. Takeda, Y. Kuwata, J. Hayashi, H. Tou,\nH. Sugawara, T. Sakurai, H. Ohta, and H. Harima, Phys.\nRev. Materials 2, 055001 (2018).\n[41] Z. Zhu and U. Schwingenschl¨ ogl, Phys. Rev. B 86, 075149\n(2012).\n[42] X. Yuan, Y. Zhang, T. A. Abtew, P. Zhang, and\nW. Zhang, Phys. Rev. B 86, 235103 (2012).\n[43] R. Grau-Crespo, H. Wang, and U. Schwingenschl¨ ogl,\nPhys. Rev. B 86, 081101 (2012).\n[44] V. Eyert, Phys. Rev. Lett. 107, 016401 (2011).\n[45] See Supplemental Material for calculation details, re-\nlaxed lattice parameters, susceptibility plots, minimal\nmodel Hamiltonian and other information which includes\nRefs. [37, 40, 50, 51, 57–65].\n[46] A. J. Cohen, P. Mori-S´ anchez, and W. Yang, Chemical\nReviews 112, 289 (2012), pMID: 22191548.\n[47] The local coordinate system is chosen such that one of\nthe lobes of the dxyorbitals is oriented along the zig-\nzag chain in the bortho direction and dz2is along the\naortho direction.\n[48] S. Mozaffari, N. Aryal, R. Sch¨ onemann, K.-W.7\nChen, W. Zheng, G. T. McCandless, J. Y. Chan,\nE. Manousakis, and L. Balicas, Phys. Rev. B 102, 115131\n(2020).\n[49] G. Cuono, F. Forte, M. Cuoco, R. Islam, J. Luo, C. Noce,\nand C. Autieri, Phys. Rev. Materials 3, 095004 (2019).\n[50] W. E. Pickett and P. B. Allen, Phys. Rev. B 16, 3127\n(1977).\n[51] M. D. Johannes and I. I. Mazin, Phys. Rev. B 77, 165135\n(2008).\n[52] M. D. Johannes, I. I. Mazin, and C. A. Howells, Phys.\nRev. B 73, 205102 (2006).\n[53] W. Wang, B. Wang, Z. Gao, G. Tang, W. Lei, X. Zheng,\nH. Li, X. Ming, and C. Autieri, Phys. Rev. B 102, 155115\n(2020).\n[54] J. G. Si, W. J. Lu, H. Y. Wu, H. Y. Lv, X. Liang, Q. J.\nLi, and Y. P. Sun, Phys. Rev. B 101, 235405 (2020).\n[55] W.-G. Yin, D. Volja, and W. Ku, Phys. Rev. Lett. 96,\n116405 (2006).\n[56] W. Wang, L. Wu, J. Li, N. Aryal, X. Jin, Y. Liu, M. Fe-durin, M. Babzien, R. Kupfer, M. Palmer, C. Petrovic,\nW. Yin, M. P. M. Dean, I. K. Robinson, J. Tao, and\nY. Zhu, npj Quantum Materials 6, 97 (2021).\n[57] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\n[58] P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n[59] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n[60] A. Togo and I. Tanaka, Scripta Materialia 108, 1 (2015).\n[61] V. Popescu and A. Zunger, Phys. Rev. B 85, 085201\n(2012).\n[62] Https://github.com/QijingZheng/VaspBandUnfolding.\n[63] P. Giannozzi, S. Baroni, and N. B. et al, Journal of\nPhysics: Condensed Matter 21, 395502 (2009).\n[64] P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H.\nMadsen, and L. D. Marks, The Journal of Chemical\nPhysics 152, 074101 (2020).\n[65] D. R. Hamann, Phys. Rev. B 88, 085117 (2013)."    }]
\ No newline at end of file